Great books of the Western world, vol. 34 [34]

Citation preview

iSiaiimMSMUiUmMmmMmtimMMKamk

'^

GREAT BOOKS OF THE WESTERN WORLD 5 »«i

n >ii nm t>

m mm «i

»

n

«

mnm m »«

'»

Thompson

MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY

BIOGRAPHICAL NOTE Sir Isaac

Newton, 642-1 727 i

Newton was

problems from Vieta and Van Schooten, an-

shire,

notations out of Wallis' Arithmetic of Infinities, together with observations on refraction, on the grinding of spherical optic glasses, on the

born at Woolsthorpe, Lincolnon Christmas Day, 1642. His father, a small farmer, died a few months before his birth, and when in 1645 his mother married the rector of North Witham, Newton was left with his maternal grandmother at Woolsthorpe. After having acquired the rudiments of education at small schools close by,

errors of lenses, and on the extraction of all kinds of roots. It was around the time of his taking the Bachelor's degree, in 1665, that Newton discovered the binomial theorem and made the first notes on his discovery of the

Newton

was sent at the age of twelve to the grammar school at Grantham, where he lived in the house of an apothecary. By his own account, Newton was at first an indifferent scholar until a successful fight with another boy aroused a spirit of emulation and led to his becoming first in

the school.

He

"method

of fluxions."

When the Great Plague spread from London Cambridge in 1665, college was dismissed, and Newton retired to the farm in Lincolnshire, where he conducted experiments in optics and chemistry and continued his matheto

displayed very early a

matical speculations. From this forced retirement in 1666 he dated his discovery of the gravitational theory: "In the same year I began to think of gravity extending to the orb of the Moon, compared the force requisite to

taste and aptitude for mechanical contrivances

he made windmills, water-clocks, kites, and sun-dials, and he is said to have invented a four-wheel carriage which was to be moved by the rider.

.

.

.

Moon

After the death of her second husband in 1656, Newton's mother returned to Woolsthorpe and removed her eldest son from school so that he might prepare himself to manage the farm. But it was soon evident that his in-

keep the

were not in farming, and upon the advice of his uncle, the rector of Burton Goggles,

work he accomplished

in her orb with the force of gravity at the surface of the earth and found them to answer pretty nearly." At about the same time his work on optics led to his explanation of the composition of white light. Of the

terests

he was sent to Trinity GoUege, Cambridge, where he matriculated in 1661 as one of the boys who performed menial services in return for their expenses. Although there is no record of his formal progress as a student, Newton is known to have read widely in mathematics and mechanics. His first reading at Cambridge was in the optical works of Kepler. He turned to Euclid because he was bothered by his inability to comprehend certain diagrams in a book on astrology he had bought at a fair; finding its propositions self-evident, he put it aside as "a trifling book," until his teacher, Isaac Barrow, induced him to take up the book again. It appears to have been the study of Descartes' Geometry which inspired him to do original mathematical work. In a small commonplace book kept by Newton as an undergraduate, there are several articles on angular

and the squaring calculations about musical

sections

in these years

Newton

remarked: "All this was in the two years of 1665 and 1666, for in those years I was in the prime of my age for invention and minded Mathematics and Philosophy more than at any time since." later

On the re-opening of Trinity College in Newton was

elected a fellow,

1667,

and two years

later, a little before his twenty-seventh birthday, he was appointed Lucasian professor of mathematics, succeeding his friend and teacher. Dr. Barrow. Newton had already built a reflecting telescope in 1668; the second tele-

scope of his making he presented to the Royal Society in December, 1671. Two months later, as a fellow of the Society, he communicated his discovery on light and thereby started a controversy which was to run for many years and to involve Hooke, Lucas, Linus, and others. Newton, who always found controversy distasteful,

"blamed

my own

imprudence

for

parting with so substantial a blessing as my quiet to run after a shadow." His papers on

of curves, several

notes, geometrical

IX

Biographical Note most important of which were communicated to the Royal Society between 1672 and 1676, were collected in the Optics (1704). It was not until 1684 that Newton began to think of making known his work on gravity. Hooke, Halley, and Sir Christopher Wren had independently come to some notion of the law of gravity but were not having any success in optics, the

explaining the orbits of the planets. In that

year Halley consulted Newton on the problem and was astonished to find that he had already solved it. Newton submitted to him four theorems and seven problems, which proved to be the nucleus of his major work. In some seventeen or eighteen months during 1685 and 1686 he wrote in Latin the Mathematical Principles of Natural Philosophy. Newton thought for some time of suppressing the third book, and it was only Halley's insistence that preserved it. Halley also took upon himself the cost of publishing the work in 1687 after the Royal Society proved unable to meet its cost. The book caused great excitement throughout Europe, and in 1689 Huygens, at that time the

most famous

scientist,

came

to

England to

make

the personal acquaintance of Newton. While working upon the Principles, Newton had begun to take a more prominent part in university affairs. For his opposition to the attempt of James II to repudiate the oath of allegiance and supremacy at the university, Newton was elected parliamentary member for Cambridge. On his return to the university, he suffered a serious illness which incapacitated him for most of 1692 and 1693 and caused considerable concern to his friends and fellowworkers. After his recovery, he left the university to

work

for the

friends Locke,

government. Through

his

Wren, and Lord Halifax, New-

ton was made Warden of the Mint in 1695 and four years later. Master of the Mint, a position he held until his death. For the last thirty years of his life Newton produced little original mathematical work. He kept his interest and his skill in the subject; in

1696 he solved overnight a problem offered by Bernoulli in a competition for which six months

had been allowed, and again in 1716 he worked in a few hours a problem which Leibnitz had proposed in order to "feel the pulse of the English analysts." He was much occupied, to his own distress, with two mathematical controversies, one regarding the astronomical observations of the astronomer royal, and the other with Leibnitz regarding the invention of cal-

culus.

He also worked on

revisions for a second

edition of the Principles, which appeared in 1713.

Newton's scientific work brought him great He was a popular visitor at the Court and was knighted in 1705. Many honors came to him from the continent; he was in correspondence with all the leading men of science, and visitors became so frequent as to prove a fame.

serious discomfort. Despite his fame,

Newton

maintained his modesty. Shortly before his death, he remarked: "I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

From an early period of been much interested in

his life

Newton had

theological studies

and before 1690 had begun to study the prophecies. In that year he wrote, in the form of a letter to Locke, an Historical Account of Two Notable Corruptions of the Scriptures, regarding two passages on the Trinity. He left in manuscript Observations on the Prophecies of Daniel the Apocalypse and other works of exegesis. After 1725 Newton's health was much impaired, and his duties at the Mint were discharged by a deputy. In February, 1727, he presided for the last time at the Royal Society, of which he had been president since 1703, and

and

died on year.

March

He was

20,

1727, in his eighty-fifth

buried in Westminster

after lying in state in the Jerusalem

Abbey

Chamber.

CONTENTS Biographical Note Prefaces to the First, Second and Third Editions Definitions Axioms, or Laws of Motion

BOOK

SECTION I.

II.

Method

of first

and

I.

ix 1

5 14

The Motion of Bodies

last ratios

Determination of centripetal forces

Motion of bodies in eccentric conic sections IV. Finding of elliptic, parabolic, and hyperbolic orbits from the focus given V. How the orbits are to be found when neither focus is given

III.

VI.

How

the motions are to be

found in given

orbits

VII. Rectilinear ascent and descent of bodies

25 32 42 50 55 76 81

VIII. Determination of orbits in which bodies will revolve, being acted upon

by any sort of centripetal force IX. Motion of bodies in movable orbits; and the motion of the apsides X. Motion of bodies in given surfaces; and the oscillating pendulous

motion of bodies

XI. Motions of bodies tending

101 to

each other with centripetal forces

XII. Attractive forces of spherical bodies XIII. Attractive forces of bodies which are not spherical XIV. Motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body

BOOK I.

II.

III.

II.

The Motion of Bodies

(In Resisting

131

144

Mediums)

IV. Circular motion of bodies in resisting mediums V. Density and compression of fluids; hydrostatics

VI. Motion and resistance of pendulous bodies VII. Motion of fluids, and the resistance made to projected bodies VIII. Motion propagated through fluids

159 165 183

189 194

203

219 247

IX. Circular motion of fluids III.

111

152

Motion of bodies that are resisted in the ratio of the velocity Motion of bodies that are resisted as the square of their velocities Motion of bodies that are resisted partly in the ratio of the velocities, and partly as the square of the same ratio

BOOK

88 92

259

The System of the World

Rules of Reasoning in Philosophy

(In Mathematical Treatment)

270 272

Phenomena Propositions Motion of the Moon's Nodes

276

315 369

General Scholium

XI

PREFACE TO THE FIRST EDITION Since the ancients

(as

we

are told

by Pappus) esteemed the

science of

meand

chanics of greatest importance in the investigation of natural things, the moderns, rejecting substantial forms and occult qualities, have endeavored to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics as far as it relates to philosophy. The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry that what is perfectly accurate is called geometrical what is less so, is called mechanical. However, the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic; and if any could work with perfect accuracy, he would be the most perfect mechanic of all, for the description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn, for it requires that the learner should first be taught to describe these accurately before he enters upon geometry, then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics, and by geometry the use of them, when so solved, is shown; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is ;

founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly employed in the moving of bodies, it happens that geometry is commonly referred to their magnitude, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. This part of mechanics, as far as it extended to the five powers which relate to manual arts, was cultivated by the ancients, who considered gravity (it not being a manual power) no otherwise than in moving weights by those powers. But I consider philosophy rather than arts and write not concerning manual but natural powers, and consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to consist in this from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena; and to this end the general propositions in the first and second books are directed. In the third book I give an example of this in the explication of the System of the World; for by the propositions mathematically demonstrated in the former books in the third I derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and

—

1

Mathematical Principles

2

Then from

by other propositions which are also mathematical, I deduce the motions of the planets, the comets, the moon, and the sea. I wish we could derive the rest of the phenomena of Nature by the same kind of reasoning from mechanical principles, for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede from one another. These forces being unknown, philosophers have hitherto attempted the search of Nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me in correcting the errors of the press and preparing the geometrical figures, but it was through his solicitations that it came to be published; for when he had obtained of me my demonstrations of the figure of the celestial orbits, he continually pressed me to communicate the same to the Royal Society, who afterwards, by their kind encouragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures of gravity and other forces; and the figures that would be described by bodies attracted according to given laws; and the motion of several bodies moving among themselves; the motion of bodies in resisting mediums; the forces, densities, and motions, of mediums; the orbits of the comets, and such like, I deferred that publication till I had made a search into those matters, and could put forth the whole together. What relates to the lunar motions (being imperfect), I have put all together in the corollaries of Prop. 66, to avoid being obliged to propose and distinctly demonstrate the several things there contained in a method more prolix than the subject deserved and interrupt the series of the other propositions. Some the several planets.

these forces,

found out after the rest, I chose to insert in places less suitable, rather than change the number of the propositions and the citations. I heartily beg that what I have here done may be read with forbearance; and that my labors in a subject so difficult may be examined, not so much with the view to censure, things,

as to

remedy

their defects. Is.

Cambridge, Trinity College,

May

8,

Newton

1686

PREFACE TO THE SECOND EDITION In this second edition of the Principia there are many emendations and some additions. In the second section of the first book, the determination of forces, by which bodies may be made to revolve in given orbits, is illustrated and enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the lunar theory and the precession of the equinoxes were more fully deduced from their principles; and the theory of the comets was

Preface

3

confirmed by more examples of the calculation of their orbits, done also with greater accuracy. Is.

London, March

Newton

28, 1713

PREFACE TO THE THIRD EDITION mth much

by Henry Pemberton, M.D., a man of the greatest skill in these matters, some things in the second book on the resistance of mediums are somewhat more comprehensively handled than before, and new experiments on the resistance of heavy bodies falling in air are In this third edition, prepared

care

added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is more fully stated and there are added new observations made by Mr. Pound, concerning the ratio of the diameters of Jupiter to one another. Some observations are also added on the comet which appeared in the year 1680, made in Germany in the month of November by Mr. Kirk; which have lately come to my hands. By the help of these it becomes apparent how nearly parabolic orbits represent the motions of comets. The orbit of that comet is determined somewhat more accurately than before, by the computation of Dr. Halley, in an ellipse. And it is sho^vn that, in this elliptic orbit, the comet took its course through the nine signs of the heavens, with as much accuracy as the planets move in the elliptic orbits given in astronomy. The orbit of the comet which appeared in the year 1723 is also added, computed by Mr. Bradley, Professor of Astronomy at Oxford. ;

Is.

London, Jan.

12,

1725-6

Newton

DEFINITIONS DEFINITION

I

The quantity of matter is the measure of the same, arising from its density and bulk conjointly. Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or lique-

and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter. faction,

DEFINITION

II

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. The motion of the whole is the sum of the motions of all the parts and therefore in a body double in quantity, with equal velocity, the motion is double; ;

with tmce the velocity,

it is

quadruple.

DEFINITION The

III

power of resisting, hy which every present state, whether it be of rest, or of

vis insita, or innate force of matter, is a

body, as

much

as in

it lies,

continues in

its

moving uniformly forwards in a right line. This force is always proportional to the body whose force it is and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inert nature of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita may, by a most significant name, be called inertia {vis inertioe) or force of inactivity. But a body only exerts this force when another force, impressed upon it, endeavors to change its condition and the exercise of this force may be considered as both resistance and impulse it is resistance so far as the body, for maintaining its present state, opposes the force impressed; it is impulse so far as the body, by not easily giving way to the impressed force of another, endeavors to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion but motion and rest, as commonly conceived, are only relatively distinguished nor are those bodies always truly at rest, which commonly are taken to be so. ;

;

;

;

5

Mathematical Principles

6

DEFINITION IV An

impressed force

is

an

action exerted

upon a

body, in order

to

change

its state,

uniform motion in a right line. This force consists in the action only, and remains no longer in the body when the action is over. For a body maintains every new state it acquires, by its inertia only. But impressed forces are of different origins, as from percus either of rest, or of

sion,

from pressure, from centripetal

force.

DEFINITION V

A centripetal force is that by which bodies are drawn or impelled, towards a point as

Of

to

a

or

any way

tend,

centre.

this sort is gravity,

by which bodies tend

to the centre of the earth

;

mag-

by which iron tends to the loadstone; and that force, whatever it is, by which the planets are continually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavors to recede from the hand that turns it and by that endeavor, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as it is let go, flies away. That force which opposes itself to this endeavor, and by which the sling continually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavor to recede from the centres of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right hnes, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, but would go off from it in a right line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside continually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velocity of its motion. The less its gravity is, or the quannetism,

;

tity of its matter, or the greater the velocity

with which

it is

projected, the less

from a rectilinear course, and the farther it will go. If a leaden ball, projected from the top of a mountain by the force of gunpowder, mth a given velocity, and in a direction parallel to the horizon, is carried in a curved line to the distance of two miles before it falls to the ground the same, if the resistance of the air were taken away, with a double or decuple velocity, would will it deviate

;

twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the Une which it might describe, till at last it should fall at the distance fly

even might go quite round the whole earth before it so that it might never fall to the earth, but go forwards into the celestial spaces, and proceed in its motion in infinitum. And after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued \vith gravity, or by any other force, that impels it towards the earth, may be continually drawn aside towards the earth, out of the rectilinear way which by its innate force it would pursue and would be made of 10, 30, or 90 degrees, or

falls; or lastly,

;

Definitions to revolve in the orbit

some such

which

it

now

7

describes; nor could the

force be retained in its orbit. If this force

was too

moon without

small,

it

would not

moon out of a rectilinear course; if it was too great, it too much, and draw down the moon from its orbit towards the earth. It is necessary that the force be of a just quantity, and it belongs to the mathematicians to find the force that may serve exactly to retain a body in a given orbit mth a given velocity; and vice versa, to determine the curvilinear way into which a body projected from a given place, mth a given velocity, may be made to deviate from its natural rectilinear way, by means of a given sufficiently turn the

would turn

it

force.

The quantity

of

any centripetal force may be considered as and motive.

of three kinds:

absolute, accelerative,

DEFINITION VI The absolute quantity of a centripetal force to the efficacy

is the

of the cause that propagates

it

measure of the same, proportional

from

the centre,

through the spaces

round about.

Thus the magnetic

force

is

greater in one loadstone

and

less in another, ac-

cording to their sizes and strength of intensity.

DEFINITION The

VII

a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time. Thus the force of the same loadstone is greater at a less distance, and less at a greater also the force of gravity is greater in valleys, less on tops of exceeding high mountains; and yet less (as shall hereafter be shown), at greater distances from the body of the earth; but at equal distances, it is the same everywhere; because (taking away, or allowing for, the resistance of the air), it equally acaccelerative quantity of

:

celerates all falling bodies,

whether heavy or

DEFINITION

light, great or small.

VIII

The motive quantity of a centripetal force is the measure of the same, proportional motion which it generates in a given time. Thus the weight is greater in a greater body, less in a less body and, in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight and it is always known by the quantity of an equal and contrary force just sufficient to hinder the descent of the body. These quantities of forces, we may, for the sake of brevity, call by the names of motive, accelerative, and absolute forces; and, for the sake of distinction, consider them with respect to the bodies that tend to the centre, to the places of those bodies, and to the centre of force towards which they tend that is to say, I refer the motive force to the body as an endeavor and propensity of the whole towards a centre, arising from the propensities of the several parts taken together the accelerative force to the place of the body, as a certain power diffused from the centre to all places around to move the bodies that are in them; and the absolute force to the centre, as endued with some cause, Avithout which those motive forces would not be propagated through the spaces round about whether that cause be some central body (such as is the magnet in the centre to the

;

;

;

;

Mathematical Principles

8

magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and of the

seats.

Wherefore the accelerative force will stand as celerity does to motion. For the quantity

in the

same

relation to the motive,

motion arises from the celerity multiplied by the quantity of matter; and the motive force arises from the accelerative force multiplied by the same quantity of matter. For the sum of the actions of the accelerative force,

Hence

upon the

of

several particles of the body,

is

the

that near the surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body; but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those regions, where the accelerative gravity is di-

motive force

of the whole.

it is,

minished into one-half, the weight of a body two or three times or six times less. I likewise call

attractions

and impulses,

in the

same

less, will

be four

sense, accelerative,

and

motive; and use the words attraction, impulse, or propensity of any sort towards a centre, promiscuously, and indifferently, one for another; considering those forces not physically, but mathematically: wherefore the reader is not to imagine that by those words I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points) when at any time I happen to speak of centres as attracting, or as endued with attractive powers. ;

Scholium Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place, and motion, as being well known to all. Only I must observe, that the common people conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which it will be convenient to distinguish them into absolute and relative, true and apparent,

mathematical and common. I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and externa' (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year. II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is commonly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. Absolute and relative space are the same in figure and magnitude; but they do not remain always numerically the same.

Definitions For

if

the earth, for instance, moves, a space of our

9

which relatively and in

air,

respect of the earth remains always the same, mil at one time be one part of the absolute space into which the air passes at another time it mil be another ;

part of the same, and so, absolutely understood, it will be continually changed. III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space; not the situation, nor the external surface of the body. For the places of equal solids are always equal but their surfaces, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing vnth the sum of the translations of the pa.rts out of their places; and therefore the place of the whole is the same as the sum of the places of the parts, and for that reason, it is internal, and in the whole body. IV. Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses; or that part of the cavity which the body fills, and which therefore moves together with the ship and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, \xi\\ really and absolutely move A^dth the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body Anil arise, partly from the true motion of the earth, in immovable space, partly from the relative motion of the ship on the earth; and if the body moves also relatively in the ship, its true motion Avill arise, partly from the true motion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth, as of the body in the ship and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved towards the east, mth a velocity of 10,010 parts; while the ship itself, Avith a fresh gale, and full sails, is carried towards the west, Avith a velocity expressed by 10 of those parts; but a sailor walks in the ship towards the east, A\-ith 1 part of the said velocity; then the sailor \nll be moved truly in immovable space towards the east, with a velocity of 10,001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts. Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the apparent time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality that they may measure the celestial motions by a more accurate time. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the floAnng of absolute time is not Uable to any change. The duration of perseverance of the existence of things remains the same, whether the motions are s^dft or slow, or none at all and therefore this duration ought to be distinguished from what are only sensible measures thereof; and from which we deduce it, by means of the astronomical equation. The :

;

:

Mathematical Principles

10

necessity of this equation, for determining

as well from the experiments of the

tiie

times of a phenomenon,

pendulum

clock, as

by

is

evinced

eclipses of the satel-

lites of Jupiter.

As the order

of the parts of time

immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they ^^^ll be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places; and that the primary places of things should be movable, is absurd. These are therefore the absolute places; and translations out of those places, are the only absolute is

motions. But because the parts of space cannot be seen, or distinguished from on another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that mthout any inconvenience in common affairs; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.

But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes, and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know, from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote body, it follows that absolute rest cannot be determined from the position of bodies in our regions. It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavor to recede from the axis of motion and the impetus of bodies moving forwards arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them will partake of their motion. Upon which account, the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, besides their translation from near the surrounding ones, partake likewise of their true motions; and though that translation were not made, they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the Hke relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel; but if the shell moves, the kernel will also move, as being part of the whole, mthout any removal from near the shell. A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon ;

Definitions

1

which account, all motions, from places in motion, are no other than parts of entire and absolute motions and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Wherefore, entire and absolute motions can be no otherwise determined than by immovable places; and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another; and upon this account must ever remain unmoved; and do thereby constitute immovable ;

space.

The

causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved; but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the relative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body but relative motion does not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be preserved, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Thus, true motion by no means consists in ;

such relations.

The

which distinguish absolute from relative motion are, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; thereupon, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untmsting itself, the vessel continues for some time in this motion; the surface of the water will at first be plain, as before the vessel began to move; but after that, the vessel, by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experienced), and the s^\ifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times mth the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavor to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, becomes known, and may be measured by this endeavor. At first, when the relative motion of the water in the vessel was greatest, it produced no endeavor to recede from the axis; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But aftereffects

Mathematical Principles

12

when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavor to recede from the axis; and this endeavor showed the real circular motion of the water continually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavor does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavoring to recede from its axis of motion, as its proper and adequate effect; but relative motions, in one and the same body, are innumerable, according to the various relations it wards,

bears to external bodies, and, like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps partake of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavor to recede from the axis of their motions. Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to be determined by their use, then by the names time, space, place, and motion, their [sensible] measures are properly to be understood; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. On this account, those violate the accuracy of language, which ought to be kept precise, who interpret these words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quantities with their relations and sensible measures. It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavor of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented; that is, we might discover their hindmost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise ;

Definitions

13

the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the translation of the globes among the bodies, we should find the determination of their motions. But how we are to obtain the true motions from their causes, effects, and apparent differences, and the converse, shall be explained more at large in the following treatise. For to this end it was that I composed it.

know

AXIOMS, OR LAWS OF MOTION LAW

I

Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it. Projectiles continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are continually drawn aside from rectilinear motions, does not cease its rotation, othermse than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time.

LAW The change of motion

is

proportional

II

to the

motive force impressed;

and

is

made in

which that force is impressed. force generates a motion, a double force will generate double the

the direction of the right line in If

any

mo-

a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire mth or are directly contrary to each other; or obliquely joined, when they are obhque, so as to produce a new motion compounded from the determination of both. tion,

LAW

III

To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. Whatever draws or presses another is as much drawn or pressed by that

you press a stone with your finger, the finger is also pressed by the draws a stone tied to a rope, the horse (if I may so say) mil be equally drawn back towards the stone; for the distended rope, by the same endeavor to relax or unbend itself, Avill draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equahty of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions other. If

stone. If a horse

are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered b.y any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are inversely proportional to the bodies. This law takes place also in attractions,

as will be proved in the next SchoHum. 14

Axioms, or Laws of Motion

COROLLARY

15

I

A

body, acted on hy two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately.

M

impressed apart in the place A, a body in a given time, by the force should Avith an uniform motion be carried from A to B, and by the force N impressed apart in the same place, should be carried from A to C, let the parallelogram ABCD be completed, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force If

N

acts in the direction of the line

BD,

AC,

parallel to

second Law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the hne BD in the same time, whether the force N be impressed or not and therefore at the end of that time it will be found somewhere in the Hne BD. By the same argument, at the end of the same time it will be found somewhere in the hne CD. Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law i. this force (by the

;

COROLLARY

II

explained the composition of any one direct force AD, out of any two oblique forces AC and CD; and, on the contrary, the resolution of any one direct into two oblique forces AC and CD: which composition and resolution are force abundantly confirmed from mechanics. drawn from the centre O of any wheel, and As if the unequal radh and the forces of and should sustain the weights A and P by the cords

And hence

is

AD

OM

ON

MA

NP

;

draw those weights to move the wheel were required. Through the centre and L; and from the right hne KOL, meeting the cords perpendicularly in and OL, describe a cirthe centre 0, mth OL the greater of the distances cle, meeting the cord in D; and drawing OD, make AC parallel and perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be fixed to the plane of the

K

OK

MA

DC

wheel or not, the weights will have the same effect whether they are suspended from the points and L, or from D and L. Let the whole force of the weight A be represented by the line AD, and let it be resolved into the forces AC and CD, of which the force AC, dramng the radius OD directly from the centre, will have no effect to move the wheel; but the other force DC, drawing the radius DO perpendicularly, will have the same effect

K

drew perpendicularly the radius OL equal to OD; that is, it will have the same as

if it

effect as the

weight P,

if

P A = DC :DA, :

ADC and DOK are similar, DC DA = OK OD = OK: OL.

but because the triangles

:

:

Mathematical Principles

16 Therefore,

OK

P A = radius :

As these

radii lie in the

in equilibrium

the wheel. If

;

which

same

:

radius

OL. and so remain the balance, the lever, and

right line they will be equipollent,

the well-known property of is greater than in this ratio,

is

either weight

its force

to

move

the

be so much greater. If the weight p = P, is partly suspended by the cord Np, partly sustained by the oblique plane pG; draw pH, NH, the former perpendicular to the horizon, the latter to the plane pG and if the force of the weight p tending downwards is represented by the line pH, it may be resolved into the forces pN, HN. If there was any plane pQ, perpendicular to the cord pN, cutting the other plane pG in a line parallel to the horizon, and the weight p was supported only by those planes pQ, pG, it would press those planes perpendicularly with the forces pN, HN; to wit, the plane pQ with the force pN, and the plane pG with the force HN. And therefore if the plane pQ was taken away, so that the weight might stretch the cord, because the cord, now sustaining the weight, supplied the place of the plane that was removed, it would be strained by the same force pN which pressed upon the plane before. Therefore, the

wheel

will

;

tension of

pN

:

PN = line pN

tension of

Therefore, if p is to A in a ratio which least distances of their cords pN and

is

:

line

pH.

the product of the inverse ratio of the

AM from the centre of the wheel, and of

to pN, then the weights p and A will have the same effect towards moving the wheel, and will, therefore, sustain each other; as anyone

the ratio

may

pH

by experiment. But the weight p pressing upon those two oblique planes, may be considered as a wedge between the two internal surfaces of a body split by it and hence the forces of the wedge and the mallet may be determined: because the force with which the weight p presses the plane pQ is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the find

;

direction of the line

pH

towards both the planes, as

pN :pH; and to the

force with

which

it

presses the other plane pG, as

pN NH. :

thus the force of the screw may be deduced from a like resolution of forces it being no other than a wedge impelled with the force of a lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is further confirmed. For on what has been said depends the whole doctrine of mechanics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded of wheels, puUies, levers, cords, and weights, ascending directly or obliquely, and other mechanical powers; as also the force of the tendons to move the bones of animals.

And

;

COROLLARY

III

The quantity of motion, which is obtained by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves. For action and its opposite reaction are equal, by Law iii, and therefore, by Law II, they produce in the motions equal changes towards opposite parts.

Axioms, or Laws of Motion

17

Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subtracted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both and therefore the difference of the motions directed towards opposite parts will remain the same. Thus, if a spherical body A is 3 times greater than the spherical body B, and has a velocity = 2, and B follows in the same direction with a velocity = 10, then the ;

motion of A motion of B = 6 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will lose as many; and therefore after reflection A will proceed with 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11, or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, will either proceed with 1 part, having lost 9, or stop and remain at rest, as having lost its whole progressive motion of 10 parts; or it will go back with 1 part, having not only lost its whole motion, but (if I may so say) one part more or it will go back with 2 parts, because a progressive motion of 12 parts is taken off. And so the sums of the conspiring :

:

;

motions, 15-t-l

and the

16+0,

or

differences of the contrary motions,

17-1

and

18-2,

always be equal to 16 parts, as they were before the meeting and reflection of the bodies. But the motions being known with which the bodies proceed after reflection, the velocity of either will be also known, by taking the velocity aftertothe velocity before reflection, as the motion after is to the motion before. As in the last case, where the motion of A before reflection (6) motion of A after (18) will

:

= velocity that

of

A

before (2)

:

velocity of

A

after {x)

is,

6 18 = 2 :x, x = Q. But if the bodies are either not spherical, or, moving in different right lines, impinge obliquely one upon the other, and their motions after reflection are required, in those cases we are first to determine the position of the plane that touches the bodies in the point of impact, then the motion of each body (by Cor. ii) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, because the bodies act upon each other in the direction of a line perpendicular to this plane, the parallel motions are to be retained the same after reflection as before and to the perpendicular motions we are to assign equal changes towards the contrary parts; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflections sometimes arise also the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows and it would be too tedious to demonstrate every particular case that relates to this subject. :

;

;

Mathematical Principles

18

COROLLARY The common

centre of gravity of two or

or rest by the actions of the bodies

IV

more bodies does not

among

themselves;

and

alter its state of

therefore the

motion

common

centre of gravity of all bodies acting upon each other {excluding external actions impediments) is either at rest, or moves uniformly in a right line.

For

if

two points proceed with an uniform motion

in right lines,

and

and

their

distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated hereafter in Lem. 23

and Corollary, when the points are moved in the same plane and by a like way of arguing, it may be demonstrated when the points are not moved in the same plane. Therefore if any number of bodies move uniformly in right lines, the common centre of gravity of any two of them is either at rest, or proceeds uniformly in a right line because the line which connects the centres of those two bodies so moving is divided at that common centre in a given ratio. In like manner the common centre of those two and that of a third body will be either at rest or moving uniformly in a right line because at that centre the distance between the common centre of the two bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a right line; because the distance between the common centre of the three bodies, and the centre of the fourth, is there also divided in a given ratio, and so on in infinitum. Therefore, in a system of bodies where there is neither any mutual action among themselves, nor any foreign force impressed upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is either at rest or moves uniformly forwards in a right line. Moreover, in a system of two bodies acting upon each other, since the distances between their centres and the common centre of gravity of both are reciprocally as the bodies, the relative motions of those bodies, whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither accelerated nor retarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting upon each other suffers no change in its state by that action; and much less the common centre of gravity of the others with which that action does not intervene; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts inversely proportional to the total sums of those bodies whose centres they are; and therefore while those two centres retain their state of motion or ;

;

;

rest,

the

common

centre of

all

does also retain

its state

:

it is

manifest that the

common

centre of all never suffers any change in the state of its motion or rest the actions of any two bodies between themselves. But in such a system from all the actions of the bodies among themselves either happen between two bodies, or are composed of actions interchanged between some two bodies; and therefore they do never produce

any alteration

in the

common

centre of

all

as

Wherefore since that centre, when the bodies do not act one upon another, either is at rest or moves uniformly forwards in some right line, it will, notwithstanding the mutual actions of the bodies among to its state of

motion or

rest.

Axioms, or Laws of Motion

19

themselves, always continue in its state, either of rest, or of proceeding uniformly in a right line, unless it is forced out of this state by the action of some power impressed from without upon the whole system. And therefore the same law takes place in a system consisting of many bodies as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion, whether of one single body, or of a whole system of bodies, is always to be estimated from the motion of the centre of gravity.

COROLLARY V The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion. For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same; and it is from those sums and differences that the collisions and impulses do arise with which the bodies impinge one upon another. Wherefore (by Law 2), the effects of those collisions will be equal in both cases; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the motions of the bodies among themselves in the other. A clear proof of this we have from the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line.

COROLLARY

VI

If bodies, moved in any manner among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by those forces. For these forces acting equally (with respect to the quantities of the bodies to be moved) and in the direction of parallel lines, will (by Law 2) move all the bodies equally (as to velocity) and therefore will never produce any change in the positions or motions of the bodies among themselves. ,

,

Scholium Hitherto

I

have

laid

down such

maticians, and are confirmed

principles as

by abundance

have been received by matheBy the first two

of experiments.

Laws and the first two Corollaries,

Galileo discovered that the descent of bodies varied as the square of the time (in duplicata ratione temporis) and that the

motion

of projectiles

was

in the curve of a parabola; experience agreeing with

both, unless so far as these motions are a little retarded by the resistance of the air. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal intervals of time, equal forces upon that body, and therefore generates equal velocities;

and

in the

whole time impresses a whole

force,

and generates a whole velocity proportional to the time. And the spaces described in proportional times are as the product of the velocities and the times; that is, as the squares of the times. And when a body is thrown upwards, its uniform gravity impresses forces and reduces velocities proportional to the times; and the times of ascending to the greatest heights are as the velocities to be taken away, and those heights are as the product of the velocities and the times, or as the squares of the velocities. And if a body be projected in any

Mathematical Principles

20

motion arising from its projection is compounded with the motion arising from its gravity. Thus, if the body A by its motion of projection alone could describe in a given time the right Hne AB, and with its motion of falling alone could describe in the same time the altitude AC; complete the parallelogram ABCD, and the body by that compounded B motion \\ill at the end of the time be found in the place D; and the curved line AED, which that body describes, uill be a parabola, to which the right line AB will be a tangent at A; and whose ordinate BD vn\\ be as the square of the line AB. On the same Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experic ments of pendulum clocks. By the same, together A\ith Law 3, Sir Christopher Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the impact and reflection of hard bodies, and about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was somewhat earlier in the pubhcation; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiments on pendulums, which M. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have due regard as well to the resistance of the air as to the elastic force of the concurring bodies. Let the spherical bodies A, B be suspended by the parallel and equal strings AC, BD, from the centres C, D. About these cendirection, the

mth those lengths as radii, describe the semicircles EAF, GBH, bisected respectively

tres,

radii CA, DB. Bring the body A to any point R of the arc EAF, and (mthdrawing the body B) let it go from thence, and

by the

the point air.

Of

V

this

:

then

after one oscillation suppose it to return to be the retardation arising from the resistance of the be a fourth part, situated in the middle, namely, so that

RV mil

RV let ST

RS = TV, and

RS :ST = 3

:2,

then will ST represent very nearly the retardation during the descent from S to A. Restore the body B to its place and, supposing the body A to be let fall from :

the point S, the velocity thereof in the place of reflection A, \\dthout sensible be the same as if it had descended in vacuo from the point T. Upon

error, vn\\

which account this velocity may be represented by the chord of the arc TA. For it is a proposition well kno^\^l to geometers, that the velocity of a pendulous body in the lowest point is as the chord of the arc which it has described in its descent. After reflection, suppose the body A comes to the place s, and the body B to the place k. Withdraw^ the body B, and find the place v, from which the

body A, being

should after one oscillation return to the place r, st may be a fourth part of rv, so placed in the middle thereof as to leave rs equal to tv, and let the chord of the arc ^A represent the velocity which the body A if

let go,

Axioms, or Laws of Motion

21

had in the place A immediately after reflection. For t ^vill be the true and correct place to which the body A should have ascended, if the resistance of the air had been taken off. In the same way we are to correct the place k to which the body B ascends, by finding the place / to which it should have ascended in vacuo. And thus everything may be subjected to experiment, in the same manner as if we were really placed in vacuo. These things being done, we are to take the product (if I may so say) of the body A, by the chord of the arc TA (which represents its velocity), that we may have its motion in the place A immediately before reflection; and then by the chord of the arc tK, that we may have its motion in the place A immediately after reflection. And so we are to take the product of the body B by the chord of the arc BZ, that we may have the motion of the same immediately after reflection. And in like manner, when two bodies are let go together from different places, we are to find the motion of each, as well before as after reflection; and then we may compare the motions between themselves, and collect the effects of the reflection. Thus trying the thing with pendulums of 10 feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of

8, 12, or 16 feet, I found always, without an error of 3 inches, that when the bodies concurred together directly, equal changes towards the contrary parts were produced in their motions, and, of consequence, that the action and reaction were alwaj^s equal. As if the body A impinged upon the body B at rest with 9 parts of motion, and losing 7, proceeded after reflection with 2, the body B was carried backwards with those 7 parts. If the bodies concurred with contrary motions, A with 12 parts of motion, and B with 6, then if A receded with 2, B receded with 8; namely, with a deduction of 14 parts of motion on each side. For from the motion of A subtracting 12 parts, nothing will remain; but subtracting 2 parts more, a motion will be generated of 2 parts towards the contrary way; and so, from the motion of the body B of 6 parts, subtracting 14 parts, a motion is generated of 8 parts towards the contrary way. But if the bodies were made both to move towards the same way, A, the swifter, with 14 parts of motion, B, the slower, Avith 5, and after reflection A went on with 5, B likemse went on with 14 parts; 9 parts being transferred from A to B. And so in other cases. By the meeting and collision of bodies, the quantity of motion, obtained from the sum of the motions directed towards the same way, or from the difference of those that were directed towards contrary ways, was never changed. For the error of an inch or two in measures may be easily ascribed to the difficulty of executing everything \vith accuracy. It was not easy to let go the two pendulums so exactly together that the bodies should impinge one upon the other in the lowermost place AB nor to mark the places s, and k, to which the bodies ascended after impact. Nay, and some errors, too, might have happened from the unequal density of the parts of the pendulous bodies themselves, and from the irregularity of the texture proceeding from other causes. But to prevent an objection that may perhaps be alleged against the rule, for the proof of which this experiment was made, as if this rule did suppose that the bodies were either absolutely hard, or at least perfectly elastic (whereas no such bodies are to be found in Nature), I must add, that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to diminish the reflection in such a certain ;

Mathematical Principles

22

proportion as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies absolutely hard return one from another with the same velocity with which they meet. But this may be affirmed with more certainty of bodies perfectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminished together with the elastic force; because that force (except when the parts of bodies are bruised by their impact, or suffer some such extension as happens under the strokes of a hammer) is (as far as I can perceive) certain and determined, and makes the bodies to return one from the other with a relative velocity, which is in a given ratio to that relative velocity with which they met. This I tried in balls of wool, made up tightly, and strongly compressed. For, first, by letting go the pendulous bodies, and measuring their reflection, I determined the quantity of their elastic force; and then, according to this force, estimated the reflections that ought to happen in other cases of impact. And with this computation other experiments made afterwards did accordingly agree the balls always receding one from the other with a relative velocity, which was to the relative velocity with which they met as about 5 to 9. Balls of steel returned with almost the same velocity; those of cork with a a velocity something less; but in balls of glass the proportion was as about 15 to 16. And thus the third Law, so far as it regards percussions and reflections, is proved by a theory exactly agreeing with experience. In attractions, I briefly demonstrate the thing after this manner. Suppose an obstacle is interposed to hinder the meeting of any two bodies A, B, attracting one the other: then if either body, as A, is more attracted towards the other body B, than that other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of the body A than by the pressure of the body B, and therefore will not remain in equilibrium: but the stronger pressure will prevail, and will make the system of the two bodies, together with the obstacle, to move directly towards the parts on which B lies; and in free spaces, to go forwards in infinitum with a motion continually accelerated; which is absurd and contrary to the first Law. For, by the first Law, the system ought to continue in its state of rest, or of moving uniformly forwards in a right line; and therefore the bodies must equally press the obstacle, and be equally attracted one by the other. I made the experiment on the loadstone and iron. If these, placed apart in proper vessels, are made to float by one another in standing water, neither of them will propel the other; but, by being equally attracted, they ^vill sustain each other's pressure, and rest at last in an equilibrium. So the gravitation between the earth and its parts is mutual. Let the earth FI by cut by any plane EG into two parts EGF and EGI, and their weights one towards the other will be mutually equal. For if by another plane HK, parallel to the former EG, the greater part EGI is cut into two parts and HKI, whereof HKI is equal to the part EFG, first cut off, it is evident that the middle part will have no propension by its proper weight towards either side, but will hang as it were, and rest in an equilibrium between both. But the one extreme part HKI will with its whole weight bear upon and press the middle part towards the other extreme part EGF; and therefore the force with which EGI, the sum of the parts HKI and EGKH, tends towards the ;

EGKH

EGKH

Axioms, or Laws of Motion

23

equal to the weight of the part HKI, that is, to the weight of the third part EGF. And therefore the weights of the two parts EGI and EGF, one towards the other, are equal, as I was to prove. And indeed if those weights were not equal, the whole earth floating in the nonresisting ether would give way to the greater weight, and, retiring from it, would be carried off in third part

EGF,

is

infinitum.

And

and reflection, whose veso in the use of mechanic instruments

as those bodies are equipollent in the impact

innate forces, those agents are equipollent, and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are inversely as the forces. So those weights are of equal force to move the arms of a balance, which during the play of the balance are inversely as their velocities upwards and downwards; that is, if the ascent or descent is direct, those weights are of equal force, which are inversely as the distances of the points at which they are suspended from the axis of the balance but if they are turned aside by the interposition of oblique planes, or other obstacles, and made to ascend or descend obliquely, those bodies will be equipollent, v/hich are inversely as the heights of their ascent and descent taken according to the perpendicular; and that on account of the determination of gravity downwards. And in like manner in the pulley, or in a combination of pulleys, the force of a hand drawing the rope directly, which is to the weight, whether ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight. In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are inversely as the velocities of the parts of the wheel on which they are impressed, will mutually sustain each other. The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body. The forces by which the w^edge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the progress of the wedge in the direction of the force impressed upon it by the mallet is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines pei:'pendicular to the sides of the wedge. And the like account is to be given of all machines. locities are inversely as their

;

The power and use of machines consist only in this, that by diminishing the velocity we may augment the force, and the contrary; from whence, in all sorts of proper machines, we have the solution of this problem: To move a given weight with a given power, or with a given force to overcome resistance.

For

if

machines are so contrived that the velocities

any other given of the agent and

resistant are inversely as their forces, the agent will just sustain the resistant,

but with a greater disparity of velocity

overcome

So that if the disparity of velocities is so great as to overcome all that resistance which commonly arises either from the friction of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those will

it.

24

Mathematical Principles

an acceleration of motion proportional thereto, as well in the parts of the machine as in the resisting body. But to treat of mechanics is not my present business. I was aiming only to show by those examples the great extent and certainty of the third Law of Motion. For if we estimate the action of the agent from the product of its force and velocity, and likewise the reaction of the impediment from the product of the velocities of its several parts, and the forces of resistance arising from the friction, cohesion, weight, and acceleration of those parts, the action and reaction in the use of all sorts of machines will be found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon the resisting body, the ultimate action will be always contrary to the resistances are overcome, will produce

reaction.

BOOK ONE

THE MOTION OF BODIES SECTION

I

The method of first and last ratios of

we

quantities, by the help of which demonstrate the propositions that follow

Lemma Quantities,

and

1

which in any finite time converge continend of that time approach nearer to each other than

the ratios of quantities,

ually to equality,

and

before the

by any given difference, become ultimately equal. If you deny it, suppose them to be ultimately unequal, and let be their ultimate difference. Therefore they cannot approach nearer to equality than which is contrary to the supposition. by that difference

D

D

;

Lemma

2

If in any figure AacE, terminated by the right lines Aa, AE, and the curve acE, there be inscribed any number of parallelograms Ah, Be, Cd, &c., comprehended under equal

AB, BC, CD, &c., and the sides, Bb, Cc, Dd, &c., Aao/ the figure; and the parallelograms aKbl, bLcm, cMdn, &c., are completed: then if the breadth bases

parallel to one side

of those parallelograms be supposed to be diminished, and their number to be augmented in infinitum, / say, that the

ultimate ratios which the inscribed figure AKbLcMdD, the circumscribed figure AalbmcndoE, and curvilinear figure

AabcdE,

will have to one another, are ratios of equality.

and circumscribed figures is the sum of Lm, Mn, Do, that is (from the equality of all their bases), the rectangle under one of their bases K6 and the sum of their altitudes Aa, that is, the rectangle ABZa. But this rectangle, because its breadth AB is supposed diminished in infinitum, becomes less than any given space. And therefore (by Lem. 1) the figures inscribed and circumscribed become ultimately equal one to the other; and much more will the intermediate curvilinear For the difference

of the inscribed

the parallelograms KZ,

figure be ultimately equal to either.

q.e.d.

Lemma The same ultimate

BC, DC,

3

ratios are also ratios of equality,

&c., of the parallelograms are unequal,

when

and are

all

the

breadths

AB,

diminished in in-

finitum.

For suppose AF equal to the greatest breadth, and complete the parallelogram FAa/. This parallelogram will be greater than the difference of the in25

Mathematical Principles

26

and circumscribed figures; but, because its breadth in infinitum, it will become less than any given rectangle. Cor. I. Hence the ultimate sum of those evanescent

scribed

parallelograms

^\ill

in all parts coincide with the curvi-

AF

is

diminished q.e.d.

a

Unear figure. Cor. II. Much more will the rectihnear figure comprehended under the chords of the evanescent arcs ah, he, cd, &c., ultimately coincide with the curvilinear figure.

Cor. III. And also the circumscribed rectilinear figure comprehended under the tangents of the same arcs. Cor. IV. And therefore these ultimate figures (as to their perimeters acE) are not rectilinear, but curvihnear Umits of rectilinear figures.

Lemma

A

BF

C

D

E

4

If in two figures AacE, PprT, there are inscribed {as hefore) two series of parallelograms, an equal numher in each series, and, their breadths being diminished in infinitum, if the ultimate ratios of the parallelograms in one figure to those in the other, each to each respectively, are the same: I say, that those two figures, AacE, PprT, are to each other in that same ratio.

Book

I

:

The Motion of Bodies

Lemma

27

5

All homologous sides of similar figures, whether curvilinear or rectilinear, are proportional; and the areas are as the squares of the homologous sides.

Lemma

^

6

If any arc ACB, given in position, is subtended by its chord AB, and in any point A, in the middle of the continued curvature, is

AD, produced both ways; then if the points A and B approach one another and meet, I say, the angle BAD, touched by a right line

contained between the chord and the tangent, will be diminished in infinitum, and ultimately will vanish.

For if that angle does not vanish, the arc ACB will contain with, the tangent AD an angle equal to a rectilinear angle and therefore the curvature at the point A ^\^ll not be continued, which is against the supposition.

Lemma

7

The same things being supposed, I say that the ultimate ratio of the arc, chord, and tangent, any one to any other, is the ratio of equality. For while the point B approaches towards the point A, consider always AB and AD as produced to the remote points b and d; and parallel to the secant BD draw bd; and let the arc k.cb be always similar to the arc ACB. Then, supposing the points A and B to coincide, the angle dkb will vanish, by the preceding Lemma; and therefore the right lines AJb, Kd (which are always finite), and the intermediate arc Ac6, will coincide, and become equal among themselves. Wherefore, the right lines AB, AD, and the intermediate arc ACB (which are always proportional to the former), ^vill vanish, and ultimately acquire the ratio of equality.

q.e.d.

Cor. I. Whence if through B we draw BF parallel to the tangent, always cutting any right line AF passing through A in F, this line BF \^ill be ultimately in the ratio of equality with the evanescent arc ACB; because, completing the parallelogram AFBD, it is always in a ratio of equality with AD. Cor. II. And if through B and A more right Unes are drawn, as BE, BD, AF, AG, cutting the tangent

AD and its

parallel

BF

;

__

"^^^^

„

-^1

/^

the ulti''

mate ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the ratio of equaUtJ^ CoR.

III.

freely use

And

therefore in

any one

all

f/

our reasoning about ultimate ratios, we maj^

of those lines for

any

other.

Lemma 8 If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB, RACB, RAD, and the points A and B approach and meet: I say, that the ultimate form of these evanescent itude, and their ultimate ratio that of equality.

triangles is that of simil-

Mathematical Principles

28

For while the point B approaches towards the point A, consider always AB, AD, AR, as produced to the remote points h, d, and r, and rhd as drawn parallel to RD, and let the arc Ach be always similar to the arc ACB. Then supposing the points A and B to coincide, the angle bAd will vanish; and therefore the three triangles rAb, rAcb, rAd (which are always finite), mil coincide, and on that account become both similar and equal. And therefore the triangles

RAB, RACB, RAD,

which are always similar and proportional to these, vAW ultimately become both similar and equal among themselves. Cor. And hence in all reasonings about ultimate one of those triangles for any other.

Lemma 9 // a right line AE, and a curved line ABC, both a given angle, A; and

to that right line,

Q.E.D. ratios,

we may

use any

given by position, cut each other in

in another given angle,

BD,

CE

are ordi-

and the points B and C together approach towards and meet in the point A: I say, that the areas of the triangles ABD, ACE, will ultimately be to each other as the squares of homologous sides. For while the points B, C, approach towards the point A, suppose always to be produced to the remote points d and e, so as Ad, Ae may be proporc tional to AD, AE; and the ordinates db, ec, to e g and EC, be drawTi parallel to the ordinates

nately applied, meeting the curve in B, C;

AD

DB

and AC produced in b and c. Let the curve Abe be similar to the curve ABC, and draw the right line Ag so as to touch both curves in A, and cut the ordinates DB, EC, db, ec, in F, G, /, g. Then, supposing the length Ae to remain the same, let the points B and C meet in the point A; and the angle cAg vanishing, and meeting

AB

the curvilinear areas Abd, Ace will coincide with the rectilinear areas Afd, Age; and therefore (by Lem. 5) mil be one to the other in the duplicate ratio of the sides Ad, Ae. But the areas ABD, ACE are always proportional to these areas; and so the sides AD, AE are to these sides. And therefore the areas ABD, ACE are ultimately to each other as the squares of the sides

AD, AE.

q.e.d.

Lemma

10

The spaces which a body describes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminished, are in the very beginning of the motion to each other as the squares of the times.

Let the times be represented by the lines AD, AE, and the velocities generated in those times by the ordinates DB, EC. The spaces described with

Book

The Motion of Bodies areas ABD, ACE, described by

I

:

29

these velocities vdW be as the those ordinates, that is, at the very beginning of the motion (bj^ Lem. 9), in the duphcate ratio of the times AD, AE. q.e.d. Cor. I. And hence one may easilj^ infer, that the errors of bodies describing similar parts of similar figures in proportional times, the errors being generated by any equal forces similarly applied to the bodies, and measured by the distances of the bodies from those places of the similar figures, at which, ^\4thout the action of those forces, the bodies would have arrived in those proportional times are nearly as the squares of the times in which they are generated. CoR. II. But the errors that are generated by proportional forces, similarly applied to the bodies at similar parts of the similar figures, are as the product of the forces and the squares of the times. CoR. III. The same thing is to be understood of any spaces whatsoever described by bodies urged with different forces all which, in the very beginning of the motion, are as the product of the forces and the squares of the times. CoR. IV. And therefore the forces are directly as the spaces described in the very beginning of the motion, and inversely as the squares of the times. CoR. V. And the squares of the times are directly as the spaces described, and inversely as the forces.

—

;

Scholium comparing with each other indeterminate quantities of different sorts, any one is said to be directly or inversely as any other, the meaning is, that the former is augmented or diminished in the same ratio as the latter, or as its reciprocal. And if any one is said to be as any other two or more, directly or inversely, the meaning is, that the first is augmented or diminished in the ratio compounded of the ratios in which the others, or the reciprocals of the others, are augmented or diminished. Thus, if A is said to be as B directly, and C directly, and D inversely, the meaning is, that A is augmented or diminished If in

in the

same

ratio as

B C •

1

•

•

^, that

to say, that

is

A

RO and -^ are to each other

in a given ratio.

Lemma The evanescent subtense of

11

the angle of contact, in all curves

which

at the point of contact have a finite curvature, is ultimately as the square of the subtense of the con-

terminous arc.

be that arc, AD its tangent, BD the subtense of the angle of contact perpendicular on the tangent, AB the subtense of the arc. Draw BG perpendicular to the subtense AB, and AG perpendicular to the tangent AD, meeting in G then let the points D, B, and G approach to the points d, b, and g, and suppose J to be the ultimate intersection of the lines BG, AG, when the points D, B have come to A. It is evident that the distance GJ may be less than any assignable distance. But (from the nature of the circles passing through the points A, B, G, and through A, b, g),

Case

1.

Let

AB

;

AB2 = AGBD, and .W- = Kg-bd.

Mathematical Principles

30

But because GJ may be assumed

than any assignable, the ratio of AG to A^ may be such as to differ from unity by less than any assignable difference and therefore the ratio of AB^ to Ab'^ may be such as to differ from the ratio of BD to 6c? by less than any assignable difference. Therefore, by of less length

;

Lem.

ultimately,

1,

^^, ^^, ^ ^^ ,

.

^^

^^^

Now let BD be incHned to AD

in any given angle, and the ultimate same as before, and therefore the same with always the will be hd ratio of to q.e.d. the ratio of AB^ to Ab^. not to be given, but that the right Case 3. And if we suppose the angle

Case

2.

BD

D

BD

converges to a given point, or is determined by any other condition whatever; nevertheless the angles D, d, being determined by the same law, will always draw nearer to equality, and approach nearer to each other than by an}'- assigned difference, and therefore, by Lem. 1, will at last be equal; and therefore the lines BD, bd are in the same ratio to each other as before, q.e.d. Cor. I. Therefore since the tangents AD, Ad, the arcs AB, Ab, and their sines, BC, be, become ultimately equal to the chords AB, Ab, their squares will ultimately become as the subtenses BD, bd. Cor. II. Their squares are also ultimately as the versed sines of the arcs, bisecting the chords, and converging to a given point. For those versed sines are as the subtenses BD, bd. Cor. III. And therefore the versed sine is as the square of the time in which a body will describe the arc with a given line

velocity.

Cor.

IV.

The ultimate

AADB is

:

proportion,

AAdb = AD^:Ad^ = 'DB'i^:d¥i%

derived from

AADB AAd6 = AD-DB :

:

Ad

db

and from the ultimate proportion

AD2 Ad2 = DB :

So also

is

:

db.

obtained ultimately

A ABC

:AAbc = BC^:bc\ Cor. v. And because DB, db are ultimately parallel and as the squares of the lines AD, Ad, the ultimate curvilinear areas ADB, Adb mil be (by the nature of the parabola) two-thirds of the rectilinear triangles ADB, Adb, and the segments, AB, Ab will be one-third of the same triangles. And thence those areas and those segments will be as the cubes of the tangents AD, Ad, and also of the chords

and

arcs

AB, A6.

Scholium But we have

along supposed the angle of contact to be neither infinitely greater nor infinitely less than the angles of contact made by circles and their tangents; that is, that the curvature at the point A is neither infinitely small nor infinitely great, and that the interval A J is of a finite magnitude. For may be taken as AD^ in which case no circle can be drawn through the point A, between the tangent and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. And by a like reasoning, if DB be made successfully as AD^, AD^, AD^ AD^ &c., we shall have a series of angles of contact, proceeding in infinitum, wherein every succeeding term is inall

DB

:

AD

The Motion of Bodies preceding. And if DB be made

Book

I

31

:

successively as AD^, than the AD^/^ AD^/^ AD^/*, AD«/^ AD^/^ &c., we shall have another infinite series of angles of contact, the first of which is of the same sort with those of circles, the second infinitely greater, and every succeeding one infinitely greater than the preceding. But between any two of these angles another series of intermediate angles of contact may be interposed, proceeding both ways in infinitum, wherein every succeeding angle shall be infinitely greater or infinitely less than the preceding. As if between the terms AD^ and AD^ there were interposed the series kD^^i\ KW^i'', kT)^i\ kT>"\ KT>'>i^, AD^/^ AD^^i\ KD^^i\ KWi\ &c. And again, between any two angles of this series, a new series of intermediate angles may be interposed, differing from one another by infinite intervals. Nor is Nature confined to any bounds. Those things which have been demonstrated of curved fines, and the surfaces which they comprehend, may be easily appfied to the curved surfaces and contents of solids. These Lemmas are premised to avoid the tediousness of deducing involved demonstrations ad absurdum, according to the method of the ancient geometers. For demonstrations are shorter by the method of indivisibles; but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following Propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the Hmits of those sums and ratios, and so to premise, as short as I could, the demonstrations of those Hmits. For hereby the same thing is performed as by the method of indivisibles and now those principles being demonstrated, we may use them with greater finitely less

safety. Therefore

up

of particles, or

if

happen to consider quantities as made curved lines for right ones, I would not be

hereafter I should

should use

mean

little

but evanescent divisible quantities; not the sums and ratios of determinate parts, but always the limits of sums and ratios; and that the force of such demonstrations always depends on the method laid understood to

down

indivisibles,

in the foregoing

Lemmas.

may

be objected, that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument it may be alleged that a body arriving at a certain place, and there stopping, has no ultimate velocity; because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, there is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in fike manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished) There is a fimit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like Umit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geo-

Perhaps

it

;

.

32

Mathematical Principles

metrical we may use in determining and demonstrating any other thing that is also geometrical. It may also be objected, that if the ultimate ratios of evanescent quantities are given, their ultimate magnitudes mil be also given and so all quantities :

which is contrary to what Euclid has demonstrated concerning incommensurables, in the tenth book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing wthout limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum. This thing will appear more evident in quantities infinitely great. If two quantities, whose difference is given, be augmented in infinitum., the ultimate ratio of these quantities will be given, namely, the ratio of equality; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, Avill be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end. will consist of indivisibles,

SECTION

II

The determination of centripetal forces Proposition The areas which

1.

Theorem

revolving bodies describe by radii

of force do lie in the same immovable planes, which they are described.

1

drawn

to

an immovable centre

and are proportional

to the

times in

For suppose the time to be divided into equal parts, and in the first part of that time let the body by its innate force describe the right line AB. In the second part of that time, the same would (by Law i), if not hindered, proceed directly to c, along the line Be equal to AB; so that by the radii AS, BS, cS, drawn to the centre, the equal areas ASB, BSc, would be described. But when the body is arrived at B, suppose that a centripetal force acts at once with a great impulse, and, turning aside the body from the right line Be, compels it afterwards to continue its motion along the right line BC. Draw eC parallel to BS, meeting BC in C; and at the end of the second part of the time, the body (by Cor. i of the Laws) will be found in C, in the same plane with the triangle ASB. Join SC, and, because SB and Ce are parallel, the triangle SBC will be equal to the triangle SBe, and therefore also to the triangle SAB. By the like argument, if the centripetal force acts successively in C, D, E, &c., and makes the body, in each single particle of time, to describe the right lines CD, DE, EF, &c., they will all lie in the same plane; and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore, in equal times, equal areas are described in one immovable plane: and, by composition, any sums SADS, SAFS, of those areas, are to each other as the times in which they are described. Now let the number of those triangles be aug-

Book

I

:

The Motion of Bodies

33

and (by Cor. iv, Lem. 3) their ultimate perimeter ADF ^\^ll be a curved hne: and therefore the centripetal force, by which the body is continually drawn back from the tangent of merited,

and

their breadth diminished in infinitum;

this curve, will act continually;

and any described areas SADS, SAFS, which are always pro-

.If

/....

portional to the times of description, mil, in this case also,

be proportional to those times. Q.E.D.

Cor.

I.

The

velocity of a

body attracted towards an immovable centre, in spaces void of resistance, is inversely as the

perpendicular let fall from that centre on the right line that touches the orbit. For the velocities in those places A, B, C, D, E, are as the bases AB, BC, CD, DE, EF, of equal triangles; and these bases are inversely as the perpendiculars let fall

upon them.

AB,

BC

two

arcs, successively described in equal void of resistance, are completed into a times by the same body, in spaces parallelogram ABCV, and the diagonal BV of this parallelogram, in the position which it ultimately acquires when those arcs are diminished in infinitum, is produced both ways, it will pass through the centre of force. CoR. III. If the chords AB, BC, and DE, EF, of arcs described in equal times, in spaces void of resistance, are completed into the parallelograms ABCV, DEFZ, the forces in B and E are one to the other in the ultimate ratio of the diagonals BV, EZ, when those arcs are diminished in infinitum. For the motions BC and EF of the body (by Cor. 1 of the Laws) are compounded of the motions Be, BV, and E/, EZ; but BV and EZ, which are equal to Cc and F/, in the demonstration of this Proposition, were generated by the impulses of the centripetal force in B and E, and are therefore proportional to those impulses. CoR. IV. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are to each other as the versed sines of arcs described in equal times; which versed sines tend to the centre of force, and bisect the chords when those arcs are diminished to infinity. For such versed sines are the halves of the diagonals mentioned in

CoR.

Cor.

II.

If

the chords

of

III.

And

therefore those forces are to the force of gravity as the said versed sines to the versed sines perpendicular to the horizon of those parabolic arcs which projectiles describe in the same time. CoR. VI. And the same things do all hold good (by Cor. v of the Laws) when

CoR.

V.

the planes in which the bodies are moved, together with the centres of force which are placed in those planes, are not at rest, but move uniformly forwards in right lines.

Mathematical Principles

34

Proposition

Theorem

2.

2

Every body that moves in any curved line described in a plane, and by a radius drawn to a point either immovable, or moving forwards with an uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.

(by Law i) turned aside from its rectilinear course by the action of some force that impels it. And that force by which the body is turned off from its rectilinear course, and is made to describe, in equal g times, the equal least triangles

Case

1.

For every body that moves in a curved

line is

,

SAB, SBC, SCD, &c., about the immovable point S (by Prop. 40, Book 1, Elements of Euclid, and

Law ii),

acts in the

place B, according to the direction of a line parallel tocC, that is,

;

;

/

/

..

,

\

,_

_..,c

in the direction of the line

BS; and in the place C, according to the direction of a line parallel to dD, that is, in the direction of the line CS, &c.; and therefore acts always in the direction of lines tending to the immovable point S. q.e.d.

Case

2.

And

(by Cor. v of

the Laws) it is indifferent whether the surface in which a

body

describes a curvilinear figure be at rest, or

moves together with the body,

the figure described, and

its point S, uniformly forwards in a right line. In nonresisting spaces or mediums, if the areas are not proportional to the times, the forces are not directed to the point in which the radii meet, but deviate therefrom towards the part to which the motion is directed, if the description of the areas is accelerated, and away from that part, if retarded. CoR. II. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the part to which the motion tends.

Cor.

I.

Scholium

A body may be urged by a centripetal force compounded of several forces; in which case the meaning of the Proposition is, that the force which results out of all tends to the point S. But if any force acts continually in the direction of lines perpendicular to the described surface, this force will make the body to deviate from the plane of its motion but wll neither augment nor diminish the area of the described surface, and is therefore to be neglected in the composition ;

of forces.

Book

I

:

The Motion of Bodies

Proposition Every body, that by a radius drawn

3.

to the

Theorem

35

3

centre of another body, howsoever moved,

describes areas about that centre proportional to the times, is urged by a force

compounded

of the centripetal force tending to that other body, celerative force by which that other body is impelled.

and of

all the ac-

Let L represent the one, and T the other body; and (by Cor. vi of the Laws) both bodies are urged in the direction of parallel lines, by a new force equal if and contrary to that by which the second body T is urged, the first body L will go on to describe about the other body T the same areas as before: but the force by which that other body T was urged will be now destroyed by an equal

and contrary force; and therefore (by Law i) that other body T, now left to itself, mil either rest, or move uniformly forwards in a right line and the first body L, impelled by the difference of the forces, that is, by the force remaining, will go on to describe about the other body T areas proportional to the times. :

And

therefore (by Theor. 2) the difference of the forces

body

T

Cor.

is

directed to the other q.e.d.

as its centre. I.

Hence

if

the one

body

L,

by a

radius

drawn

to the other

body T,

describes areas proportional to the times; and from the whole force, by which the first body L is urged (whether that force is simple, or, according to Cor. ii

compounded out of several forces) we subtract (by the same Cor.) that whole accelerative force by which the other body is urged; the whole remaining force by which the first body is urged will tend to the other body T, of the Laws,

,

as its centre.

And, if these areas are proportional to the times nearly, the remaining force will tend to the other body T nearly. CoR. III. And vice versa, if the remaining force tends nearly to the other body Cor.

II.

T, those areas

^vill

be nearly proportional to the times.

IV. If the body L, by a radius drawn to the other body T, describes which, compared with the times, are very unequal; and that other body areas, T be either at rest, or moves uniformly forwards in a right line: the action of the centripetal force tending to that other body T is either none at all, or it is mixed and compounded mth very powerful actions of other forces: and the whole force compounded of them all, if they are many, is directed to another (immovable or movable) centre. The same thing obtains, when the other body is moved by any motion whatsoever; provided that centripetal force is taken, which remains after subtracting that whole force acting upon that other body T.

Cor.

Scholium Since the equable description of areas indicates that there is a centre to which tends that force by which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit, why may we not be allowed, in the following discourse, to use the equable description of areas as an indication of a centre, about which all circular motion is performed in free spaces? _, ^

Proposition

The

centripetal forces of bodies,

same

4.

Theorem

4

which by equable motions describe different

and are

circles,

each other as the squares of the arcs described in equal times divided respectively by the radii of the circles. tend

to the

centres of the

circles;

to

Mathematical Principles

36

These forces tend to the centres of the circles (by Prop. 2, and Cor. ii, Prop. in equal 1), and are to one another as the versed sines of the least arcs described times (by Cor. iv, Prop. 1) that is, as the squares of the same arcs divided by the diameters of the circles (by Lem. 7) and therefore since those arcs are as arcs described in any equal times, and the diameters are as the radii, the forces will be as the squares of any arcs described in the same time divided by the q.e.d. radii of the circles. Cor. I. Therefore, since those arcs are as the velocities of the bodies, the centripetal forces are as the squares of the velocities divided by the radii. Cor. II. And since the periodic times are as the radii divided by the velocities, the centripetal forces are as the radii divided by the square of the periodic ;

;

times.

the periodic times are equal, and the velocities therefore as the radii, the centripetal forces will be also as the radii; and conversely. Cor. IV. If the periodic times and the velocities are both as the square roots of the radii, the centripetal forces will be equal among themselves; and con-

Cor.

III.

Whence

if

versely.

the periodic times are as the radii, and therefore the velocities equal, the centripetal forces \vi\\ be inversely as the radii; and conversely. Cor. VI. If the periodic times are as the /^th powers of the radii, and therefore the velocities inversely as the square roots of the radii, the centripetal

Cor.

v. If

forces will be inversely as the squares of the radii; and conversely. CoR. VII. And universally, if the periodic time is as any power

R"

of the

radius R, and therefore the velocity inversely as the power R''-^ of the radius, the centripetal force will be inversely as the power R^"-^ of the radius; and conversely.

things hold concerning the times, the velocities, and the forces by which bodies describe the similar parts of any similar figures that have their centres in a similar position with those figures; as appears by applying the demonstration of the preceding cases to those. And the application is easy, by only substituting the equable description of areas in the place of

Cor.

VIII.

The same

equable motion, and using the distances of the bodies from the centres instead of the radii. CoR. IX. From the same demonstration it likewise follows, that the arc which a body, uniformly revolving in a circle with a given centripetal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would describe in the same given time.

Scholium

case of the sixth Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed) and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing as the squares of the distances from the centres.

The

;

Moreover, by means of the preceding Proposition and

its Corollaries,

we

discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by Cor. ix of this Prop.). And by

may

Book

I

:

The Motion of Bodies

37

such propositions, Mr. Huygens, in his excellent book De horologio oscillatorio, has compared the force of gravity mth the centrifugal forces of revolving bodies.

The preceding Proposition may be

likewise demonstrated after this manner.

In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points, the force, with which at every reflection it strikes the circle, will be as its velocity: and therefore the sum of the forces, in a given time, will be as the product of that velocity and the number of reflections; that is (if the species of the polygon be given), as the length described in that given time, and increased or diminished in the ratio of the same length to the radius of the circle; that is, as the square of that length divided by the radius; and therefore the polygon, by having its sides diminished in infinitum, coincides Avith the circle, as the square of the arc described in a given time divided by the radius. This is the centrifugal force, with which the body impels the circle; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal.

Proposition

5.

Problem

1

There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre. Let the three right lines PT, TQV, touch the figure described in as many points, P, Q, R, and meet in T and V. On the tangents erect the perpendiculars PA, QB, RC, inversely proportional to the velocities of the body in the points P, Q, R, from which the perpendiculars were raised; that is, so that PA may be to QB as the velocity in Q to the velocity in P, and QB to RC as the velocity in R to the velocity in Q. Through the ends A, B, C of the perpendiculars draw AD,

VR

DBE, EC,

at right angles, meeting in

D

and

E: and the right hnes TD, VE produced, will meet in S, the centre required. For the perpendiculars let fall from the centre S on the tangents PT, QT, are inversely as the velocities of the bodies in the points P and Q (by Cor. i. Prop. 1), and therefore, by construction, directly as the perpendiculars AP, BQ; that is, as the perpendiculars let fall from the point D on the tangents. Whence it is easy to infer that the points S, D, T are in one right line. And by the like argument the points S, E, V are also in one right line; and therefore q.e.d. the centre S is in the point where the right lines TD, VE meet.

Proposition

6.

Theorem

5

In a space void of resistance, if a body revolves in any orbit about an immovable centre, and in the least time describes any arc just then nascent; and the versed sine of that arc is supposed to be drawn bisecting the chord, and produced passing through the centre of force: the centripetal force in the middle of the arc will be and inversely as the square of the time. For the versed sine in a given time is as the force (by Cor. iv, Prop. 1) and augmenting the time in any ratio, because the arc will be augmented in the

directly as the versed sine

;

Mathematical Principles

38

the versed sine mil be augmented in the square of that ratio (by Cor. II and iii, Lem. 11), and therefore is as the force and the square of the time. Divide both sides by the square of the time, and the force will be directly q.e.d. as the versed sine, and inversely as the square of the time. easily demonstrated by Cor. iv, Lem. 10. also be may thing the same And Cor. I. If a body P revolving about the centre S describes a curved line APQ, which a right hne ZPR touches in any point P is and from any other point Q of the curve, drawn parallel to the distance SP, meeting the tangent in R and QT is drawn perpendicular to

same

ratio,

QR

;

the distance SP; the centripetal force will be inversely asthe solid

—QR — ^^.^^

,

if

the soHd be taken

'

magnitude which it ultimately acquires V when the points P and Q coincide. For QR is equal to the versed sine of double the arc QP, whose middle is P: and double the triangle SQP, or SP-QT is proportional to the time in which that double arc is described; and therefore of that

may

be used to represent the time. Cor. II. By a like reasoning, the centripetal force

gY2 OP^ — — QR .

^^-^

;

SY is

if

of the orbit.

is

inversely as the solid

a perpendicular from the centre of force on PR, the tangent

For the rectangles

SY-QP and SP-QT

are equal.

the orbit is either a circle, or touches or cuts a circle concentriCor. cally, that is, contains with a circle the least angle of contact or section, having the same curvature and the same radius of curvature at the point P and if PV be a chord of this circle, drawn from the body through the centre of force; the III.

If

;

centripetal force will be inversely as the solid

SY^ PV. For •

PV

Qp2 is

^^-.

QR

The same things being supposed, the centripetal force is as the square of the velocity directly, and the chord inversely. For the velocity is reciprocally as the perpendicular SY, by Cor. i, Prop. 1. Cor. v. Hence if any curvilinear figure APQ is given, and therein a point S is Cor.

IV.

tripetal force

which a centripetal force is continually directed, that law of cenmay be found, by which the body P will be continually drawn

back from a

rectilinear course, and, being detained in the perimeter of that

also given, to

figure, will describe

by computation,

the same by a continual revolution. That

either the solid

portional to this force.

Examples

—QK — y=r^

of this

Proposition

or the solid

we

are to find,

SY^-PV, inversely pro-

we shall give in the following Problems. 7.

If a body revolves in the circumference of a

Problem

2

circle, it is

centripetal force directed to

VQPA be the

is,

proposed

any given

to

find the law of

point.

circumference of the circle S the given point to which as to a centre the force tends; P the body moving in the circumference; Q the next place into which it is to move and PRZ the tangent of the circle at the preceding place. Through the point S draw the chord PV, and the diameter VA of the circle; join AP, and draw QT perpendicular to SP, which produced, may meet the tangent PR in Z; and lastly, through the point Q, draw LR parallel to SP,

Let

;

;

The Motion of Bodies and the tangent PZ in R. And, because

Book meeting the triangles

circle in L,

I

39

:

ZQR, ZTP, VPA, we

shall

of the similar

have

RP2:QT2 = AV2:PV2. RLQR-PV2 Since RP2 = RLQR,QT2: AV2

Multiply those equals by

P and PV; then we points

Q

;;-

VK

because and by composition, as B.V,

KS+A-S; and by subtraction, as VA--VK to A-S-KS, that is, as 2KX, and 2KX to 2SX, and therefore as VX to HX and HX to SX,

2VX to

to

the triangles

XH; and

VXH, HXS

therefore as

be similar; therefore VH will be to SH as VX to to KS. Wherefore VH, the principal axis of the de-

will

VK

same ratio to SH, the distance of the foci, as the prinwhich was to be described has to the distance of its foci; the same kind. And seeing VH, vH are equal to the principal

scribed conic, has the cipal axis of the conic

and

therefore of

is

axis, and VS, vS are perpendicularly bisected by the right lines TR, tr, it is evident (by Lem. 15) that those right lines touch the described conic. q.e.f. Case 3. About the focus S it is required to describe a conic which shall touch

a right

fine

TR in a given point R. On the right line TR let fall the lar

perpendicuST, which produce to V, so that TV may

be equal to ST; join VR, and cut the right VS indefinitely produced in and A-, so that VK SK, and VAmay be to to SA-, as the // -../ principal axis of the ellipse to be described to \ the distance of its foci; and on the diameter * KA- describing a circle, cut the right line VR produced in H; then with the foci S, H, and principal axis equal to VH, describe a conic: I say, the thing is done. For VH SH = SK, and therefore as the principal axis of the conic which was to be described to the distance of its foci (as appears from what we have demonstrated in Case 2) and therefore the described conic is of the same kind with that which was to be described; but that the right fine TR, by which the angle VRS is bisected, touches the conic in the point R, is certain from the jj^..--

\

^.:;;'--'"

...i.....".V.:.-.

VK

:

line

K

\

).

:

;

properties of the conic sections.

q.e.f.

About the focus S it is required to describe a conic APB that shall touch a right line TR, and pass through any given point P without the tangent, and shall be similar to the figure apb, described with the principal axis ah, and

Case

foci

s,

4.

On the tangent TR let fall the perpendicular ST, which produce to V, TV may be equal to ST; and making the angles hsq, shq, equal to the

h.

so that

VSP, SVP, about g as a centre, and with a radius which shall be to ab as to VS, describe a circle cutting the figure apb in p. Join sp, and draw SH

angles

SP

it may be to sh as SP is to sp, and may make the angle PSH equal to the angle psh, and the angle VSH equal to the angle psq. Then with the foci S, H, and principal axis AB, equal to the distance VH, describe a conic section: I say, the thing is done; for if sv is drawn so that it shall be to sp as sh is to sq,

such that

Book

I

:

The Motion of Bodies

53

and shall make the angle vsp equal to the angle hsq, and the angle vsh equal to the angle psq, the triangles svh, spq, will be similar, and therefore vh will be to pq as sh is to sq; that is (because of the similar triangles VSP, hsq), as VS is to SP, or as ab to pq. Wherefore vh and ab are equal. But, because of the similar

B

a

VH

is to SH as vh to sh; that is, the axis of the conic section VSH, vsh, described is to the distance of its foci as the axis ab to the distance of the foci sh; and therefore the figure now described is similar to the figure aph. But, because the triangle PSH is similar to the triangle psh, this figure passes through is equal to its axis, and VS is perpendicularlythe point P; and because bisected by the right line TR, the said figure touches the right line TR. q.e.f.

triangles

now

VH

Lemma

16

From three given points to draw to a fourth point that is not given three right lines whose differences either shall be given or are zero. Case 1. Let the given points be A, B, C, and Z the fourth point which we are to find; because of the given difference of the lines AZ, BZ, the locus of the point Z will be an hyperbola whose foci are A and B, and whose principal axis is the given difference. Let that axis be MN. Taking

PM

to

MA as MN to AB, erect PR per-

pendicular to AB, and let fall ZR perpendicular to PR; then from the nature of the hyperbola, AB. And by the like argument, ZR AZ = the locus of the point Z will be another hyperbola, whose foci are A, C, and whose principal axis is the difference between AZ and CZ; and QS a perpendicular on AC may be drawn, to which (QS) if from any point Z of this hyperbola a perpendicular ZS is let fall, (this ZS) shall be to AZ as the difference between AZ and CZ is to AC. Wherefore the ratios of ZR and ZS to AZ are given, and consequently the ratio of ZR to ZS one to the other; and therefore if the right lines RP, SQ, meet in T, and TZ and TA are drawn, the figure TRZS will be given in kind, and the right line TZ, in which the point Z is somewhere placed, will be given in position. There will be given also the right line TA, and the angle ATZ; and :

MN

:

Mathematical Principles

54

because the ratios of AZ and TZ to ZS are given, their ratio to each other is given also; and thence will be given likewdse the triangle ATZ, whose vertex is the point Z. q.e.i. Case 2. If two of the three lines, for example AZ and BZ, are equal, draw the right line TZ so as to bisect the right Hne AB; then find the triangle ATZ as above. q.e.i. Case 3. If all the three are equal, the point Z will be placed in the centre of a circle that passes through the points A, B, C. q.e.i. This problematic Lemma is likemse solved in the Book of Tactions of Apollonius [of Perga], restored

by

Vieta.

Proposition About a given

21.

Problem

13

focus, to describe a conic that shall pass through given points

and

tou^h right lines given by position.

TR

Let the focus S, the point P, and the tangent be given, and suppose that the other focus is to be found. On the tangent let fall the perpendicular ST, which produce to Y, so that may be equal to ST, and will be equal to the principal axis. Join SP, HP, and SP \\\\\ be the y. difference between and the principal axis. After this manner, if more tangents are given, or more points P, we shall always determine as many lines YH, or PH, drawn from the said points or P, to the focus H, which either shall be equal H to the axes, or differ from the axes by given lengths SP; and therefore which shall either be equal among themselves, or shall have given differences; from w^hence (by the preceding Lemma), that other focus is given. But ha\'ing the foci and the length of the axis (which is either YH, or, if the conic be an ellipse, PH+SP; or PH — SP, if it be an hyperbola), the conic is given. q.e.i.

H

YH

TY

HP

TR

Y

H

Scholium

When

the conic is an hyperbola, I do not include its conjugate hyperbola under the name of this conic. For a body going on with a continued motion can never pass out of one hyperbola into its conjugate hyperbola. The case when three points are given is more readily solved thus. Let B, C, be the given points. Join BC, CD, and produce them to E, F, so as EB may be to EC as as SC to SB to SC; and FC to

D

FD

SD. On let fall

and

EF drawn

and produced

the perpendiculars SO,

BH,

OS

produced indefinitely take GA to AS, and Ga to aS, as HB is to BS: then A will be the in

vertex,

and Aa the

principal axis

GA is greater than, equal to, or less than AS, be either an ellipse, a parabola, or an hyperbola; the point a in the first case falling on the same side of the line GF as the point A; in the second, going of the conic; which, according as

wall

Book

I

:

The Motion or Bodies

55

an infinite distance; in the third, falhng on the other side of the line GF. are let fall, IC ^^ill be to HB as EC to For if on GF the perpendiculars CI, EB; that is, as SC to SB; and by permutation, IC to SC as HB to SB, or as GA is to SD in the same to SA. And, b}^ the like argument, we may prove that ratio. "WTierefore the points B, C, D he in a conic section described about the focus S, in such manner that all the right lines draAATi from the focus S to the several points of the section, and the perpendiculars let fall from the same points on the right line GF, are in that given ratio. That excellent geometer M. de la Hire has solved this Problem much after the same way, in his Conies, Prop. 25, Book viii. off to

DK

KD

SECTION V how

the orbits aee to be found when neither focus

Lemma

is

given

17

If from any point P of a given conic section, to the four 'produced sides AB, CD, inscribed in that section, as many right lines AC, of any trapezium given angles, each line to each side; the rectangle PQ, PR, PS, PT are drawn in PQ-PR of those on the opposite sides AB, CD, will be to the rectangle PS-PT of those on the other two opposite sides AC, BD, in a given ratio.

ABDC

DB

Case

Let us suppose, first, that the lines dra\Mi to one pair of opposite sides are parallel 1.

PR

to either of the other sides; as PQ and the side AC, and PS and PT to the side AB

.

to

And

further, that one pair of the opposite sides, as

AC and BD,

are parallel between themselves; then the right line which bisects those parallel sides will be one of the diameters of the conic section, and will like\\ise bisect RQ. Let be the point in which RQ is bisected, and PO ^^dll be an ordinate to that diameter. Produce PO to K, so that OK may be equal to PO, and OK ^\-ill be an ordinate on the other side of that diameter. Since, therefore, the points A, B, P,

and

K are placed in

the conic section, and PK cuts AB in a given angle, the rectangle PQ (by Props. 17, 19, 21, and 23, Book iii. Conies of Apollonius) ^^i.]l be to the rectangle AQ-QB in a given ratio. But and Cr PR are equal, as being the differences of the equal fines OK, OP, and OQ, OR; whence the rectangles PQ and PQ PR are equal and therefore the rectangle PQ-PR is to the rectangle AQ QB, that is, to the rectangle PS PT, in a given ratio. q.e.d. Case 2. Let us next suppose that the opposite sides AC and of the trapezium are not par•

QK

QK

•

QK

•

•

;

•

BD

allel.

Draw Bd

parallel to

well the right line

and draw

ST

in

t,

AC, and meeting

as

cutting PQ in r, in N. Then (because of

as the conic section in d. Join

DM parallel to PQ, cutting Cd in M, and AB

Cd

Mathematical Principles

56

DBN

DN

the similar triangles BT^, T« = ) B^ or PQ NB. And so Rr AQ or PS = AN. Wherefore, by multiplying the antecedents by the antecedents, and the consequents by the consequents, as the rectangle PQ Rr is to the rectangle PS-T/., so will the rectangle be to the rectangle NA-NB; and (by Case 1) so is the rectangle PQ-Pr to the rectangle PS-P^, and, by division, so is the rectangle PQ-PR to the, rectangle PS-PT. q.e.d. Case 3. Let us suppose, lastly, the four lines PQ, PR, PS, PT not to be parallel to the sides AC, AB, but any way inclined to them. In their place draw Pg, Pr, parallel to AC; and Ps, P^ parallel to AB; and because the angles of the triangles PQg, PRr, PSs, PT^ are given, the ratios of PQ to Fq, PR to Pr, PS to Ps, PT to P^ will be also given; and therefore the compounded ratios PQ-PR to Pg-Pr,

DM

:

:

:

:

•

DN-DM

and PS-PT to Ps-P^ are given. But from what ratio of Fq Pr to Ps- P^ is given; and therefore also the ratio of PQ-PR to PS-PT. Q.E.D.

we have demonstrated before, the

-

Lemma The same things supposed,

if the rectangle

18

PQ PR -

of the lines

drawn to the two drawn to the other

opposite sides of the trapezium is to the rectangle PS PT of those two sides in a given ratio, the point P, from whence those lines are drawn, will be placed in a conic section described about the trapezium. -

Conceive a conic section to be described passing through the points A, B, C, D, and any one of the infinite number of points P, as for example p: I say, the point P will be always placed in this section. If you deny the thing, join AP cutting this conic section somewhere

else, if

than in P, as in b. Therefore if from those points p and b, in the given angles to possible,

the sides of the trapezium, lines pq, pr, ps, pt,

and

we draw the right we shall

bk, bn, bf, bd,

have, as bk-bn to bf-bd, so (by Lem. 17) pq-pr to ps-pt; and so (by supposition) PQ PR to PS PT. And because of the similar trapezia bkAf, PQAS, as bk to bf, so PQ to PS. Wherefore by dividing the terms of the preceding proportion by the correspondent terms of this, we shall have bn to bd as PR to PT. And therefore the equiangular trapezia Dnbd, DRPT, are similar, and consequently their diagonals D6, DP do coincide. Wherefore b falls in the intersection of the right Unes AP, DP, and consequently coincides with the point P. And therefore the point P, wherever it is taken, falls within the assigned conic section. q.e.d. Cor. Hence if three right lines PQ, PR, PS are drawn from a common point P, to as many other right lines given in position, AB, CD, AC, each to each, in as many angles respectively given, and the rectangle PQ PR under any two of the lines drawn be to the square of the third PS in a given ratio; the point P, from which the right lines are drawn, will be placed in a conic section that -

•

•

Book

I

:

The Motion of Bodies

57

CD

in A and C; and the contrary. For the position of the touches the hnes AB, three right hnes AB, CD, AC remaining the same, let the hne approach to and coincide with the hne AC; then let the line PT come likewise to coincide with the line PS and the rectangle PS PT will become PS^ and the right lines AB, CD, which before did cut the curve in the points A and B, C and D, can no longer cut, but only touch, the curve in those coinciding points.

BD

•

;

Scholium In this Lemma, the name of conic section

to be understood in a large sense, comprehending as well the rectilinear section through the vertex of the cone, is

as the circular one parallel to the base. For if the point p happens to be in a and D, or C and B are joined, the conic section right line, by which the points

A

be changed into two right lines, one of which is that right line upon which the point p falls, and the other is a right line will

that joins the other two of the four points. If the two opposite angles of the trapezium taken together are equal to two right angles, and if the four hnes PQ, PR, PS, PT are drawn to the sides thereof at right angles, or any other equal angles, and the rectangle PQ-PR under two of the lines drawn PQ and PR, is equal to the rectangle PS-PT under the other two PS and PT, the conic section will become a circle. And the same if the lines are drawn in any angles, and the rectangle thing will happen four PQ PR, under one pair of the lines drawn, is to the rectangle PS PT under the other pair as the rectangle under the sines of the angles S, T, in which the two last lines PS, PT are drawn, to the rectangle under the sines of the angles Q, R, in which the first two PQ, PR are drawn. In all other cases the locus of the point P mil be one of the three figures which pass commonly by the name of the conic sections. But in place of the trapezium ABCD, we may substitute a quadrilateral figure whose two opposite sides cross one another like diagonals. And one or two of the four points A, B, C, may be supposed to be removed to an infinite distance, by which means the sides of the figure which converge to those points, will become parallel and in this case the conic section will pass through the other points, and will go the same way as the parallels in infinitum. •

•

D

;

Lemma To find a point P from which

many

other right lines

19

if four right lines

AB, CD, AC, BD,

PQ, PR, PS,

PT

given by position, each

to

drawn

as each, at given

are

to

PQ PR,

under any two of the lines drawn, shall he to the rectangle PS PT, under the other two, in a given ratio. Suppose the lines AB, CD, to which the two right lines PQ, PR, containing one of the rectangles, are drawn to meet two other lines, given by position, in the points A, B, C, D. From one of those, as A, draw any right line AH, in which you would find the point P. Let this cut the opposite lines BD, CD, in and I; and, because all the angles of the figure are given, the ratio of PQ to PA, angles, the rectangle

•

•

H

Mathematical Principles

58

PA to PS, and therefore of PQ to PS, will be also given. This ratio taken as a divisor of the given ratio of PQ-PR to PS-PT, gives the ratio of PR to PT; and

and multiplying the given ratios of PI to PR, and PT to PH, the ratio of PI to PH, and therefore the point P, will be given. Q.E.I.

Hence also a tangent may be to any point D of the locus of all the P. For the chord PD, where the P and D meet, that is, where AH is through the point D, becomes a

Cor.

drawn points

points

drawn

I.

tangent. In which case the ultimate ratio of the evanescent hnes IP and PH will be

found as above. Therefore draw

meeting BD in F, and cut be the tangent; because CF

CF parallel to AD,

in E in the same ultimate ratio, then DE will and the evanescent IH are parallel, and similarly cut in E and P. Cor. II. Hence also the locus of all the points P may be determined. Through any of the points A, B, C, D, as A, draw AE touching the locus, and through any other point B, parallel to the tangent, draw BF meeting the locus in F; and find the point F by this Lemma. Bisect BF in G, and, it

drawing the indefinite line AG, this will be the position of the diameter to which BG and FG are ordinates. Let this AG meet the locus in H, and AH will be its diameter or latus transversum, to which the latus rectum will be as BG^ to AG-GH. If AG nowhere meets the locus, the line AH being infinite, the locus will be a parabola; and its latus rectum corresponding to the diameter AG will

BG2

be ^T^- But

if it

does meet

it

anywhere, the locus will be an hyperbola,

H

the points A and are placed on the same side of the point G; and an eUipse, if the point G falls between the points A and H; unless, perhaps, the angle is a right angle, and at the same time BG^ equal to the rectangle GA-GH, in which case the locus will be a circle. And so we have given in this Corollary a solution of that famous Problem of the ancients concerning four lines, begun by Euclid, and carried on by Apollonius; and this not an analytical calculus but a geometrical composition,

when

AGB

such as the ancients required.

Lemma 20 If the two opposite angular points A and P of any parallelogram ASPQ touch any conic section in the points A and P; and the sides AQ, AS of one of those angles, indefinitely produced, meet the same conic section in B and C and from the points of meeting B and C to any fifth point D of the conic section, two right lines BD, ;

CD are drawn meeting the two other sides PS, PQ of the parallelogram, indefinitely produced.in T and R; the parts PR and PT, cut off from the sides, will always be one to the other in a given ratio. And conversely, if those parts cut off are one to the

Book

I

:

The Motion of Bodies

59

D will be a conic section passing through

other in a given ratio, the locus of the point the four points A, B, C, P.

Case 1. Join BP, CP, and from the point D draw the two right Unes DG, DE, of which the first DG shall be parallel to AB, and meet PB, PQ, CA, in H, G; and the other DE shall be parallel to AC, and meet PC, PS, AB, in F, K, E; and (by Lem. 17) the rectangle DE DF will be I,

•

to the rectangle

DG DH in a given ratio. •

PQ is to DE (or IQ) as PB to HB, and PT to DH and by permutation PQ is to PT as DE to DH. Likewise

But

consequently as

;

PR is to DF

as RC to DC, and therefore PS to DG and by permutation as (IG or) PR is to PS as DF to DG and, by com;

;

pounding

those

PQ PR will •

the rectangle ratio.

PQ

But

and PS are

given,

the

ratios,

rectangle

be to the rectangle PS

DE DF •

is

PT as

•

to the rectangle

DG-DH, and consequently and therefore the ratio of PR to

a given

in

PT

is

given. Q.E.D.

Case

2.

But if PR and PT are supposed by going back again, by a like

to be in a given ratio one to the

the in a given ratio; and so the point rectangle DE DF is to the rectangle D (by Lem. 18) will lie in a conic section passing through the points A, B, C, P, other, then

reasoning,

it

will follow that

DG DH

•

•

Q.E.D.

as its locus.

CoR.

I.

Hence

if

BC cutting PQ in r and in PT take P^ to Pr in the

we draw

which PT has to PR; then B^ will touch the conic section in the point B. For suppose the point D to coalesce with the point B, so that the chord BD vanishing, BT shall become a tangent; and CD and BT will coincide with CB and B^. CoR. II. And, vice versa, if B^ is a tangent, and the lines BD, CD meet in any point D of a conic section, PR will be to PT as Pr to P^. And, on the contrary, if PR is to PT as Pr to P^, then BD and CD will meet in some point D of a

same

ratio

conic section.

CoR.

III.

One

four points. For,

conic section cannot cut another conic section in more than if it is possible, let two conic sections pass through the five

BD

cut them in the points D, d, points A, B, C, P, O; and let the right line the right line Cd cut the right line PQ in 5. Therefore PR is to PT as Pg to whence PR and Fq are equal one to the other, against the supposition.

Lemma 21 lines BM, CM drawn

and PT:

through given points B, // two movable and indefinite right given by describe a third right line C, as poles, do by their point of meeting making with the are drawn, lines BD, position; and other two indefinite right / say, that those former two at those given points B, C, given angles, MBD, describe a conic section will by their point of meeting two right lines BD,

MN

M

CD

MCD:

D

CD

CD

BD, do by passing through the points B, C. describe a conic section passing through the given points their point of meeting is always equal to the given angle ABC, as well as the B, C, A, and the angle

And

D

DBM

conversely, if the right lines

Mathematical Principles

60 angle

DCM

always equal

to the

given angle

ACB,

the point

M will

lie

in a right

line given by position, as its locus.

For

M

in the right line

MN let a point N be given, and when the movable point

on the immovable point N, able point P. Join CN, BN, CP, BP, and from the point P draw the right lines PT, PR meeting BD, CD in T and R, and falls

let

the movable point

D fall on an

immov-

BPT equal to BNM, and the

making the angle

the given angle angle CPR equal to the given angle CNM. Wherefore since (by supposition) the angles MBD, NBP are equal, as also the angles MCD, NCP, take and away the angles that are common, and there will remain the angles

NBD

NCD

NBM and PBT, NCM and PCR equal; and therefore the triangles NBM, PBT

NM

NCM, PCR.

Wherefore PT is to as PB N, are immovthe points B, C, P PR able: wherefore PT and PR have a given ratio to NM, and consequently a given ratio between themselves; and therefore, (by Lem. 20) the point continually concur, will be placed wherein the movable right lines BT and are similar, as also the triangles to as PC to to NB; and

NM

NC. But

D

CR

in a conic section passing

through the points B, C, P.

q.e.d.

D

And conversely, if the movable point lies in a conic section passing through is always equal to the given the given points B, C, A; and the angle angle ABC, and the angle always equal to the given angle ACB, and when the point D falls successively on any two immovable points p, P, of the conic secfalls tion, the movable point

DBM

DCM

M

successively on

two immovable

points n, N. Through these points n, N, draw the right line nN: this hne nN will be the continual locus of that movable point M. For, if

M

be placed Therefore the will be placed in a conic point section passing through the five continually placed in a curved is points B, C, A, p, P, when the point line. But from what was demonstrated before, the point D mil be also placed in a conic section passing through the same five points B, C, A, p, P, when placed in a right hne. Wherefore the two is continually the point possible, let the point

in

any curved

line.

D

M

M

conic sections will both pass through the

Lem.

20. It is

curved

fine.

same

five points, against Cor.

therefore absurd to suppose that the point

M

is

iii,

placed in a q.e.d.

Book

I

:

The Motion of Bodies

Proposition

To

22.

Problem

61

14

describe a conic that shall pass through five given points.

given points be A, B, C, P, D. From any one of them, as A, to any other two as B, C, which may be called the poles, draw the right lines AB, AC, and parallel to those the lines TPS, PRQ through the fourth point P. Then from the two poles B, C, draw two indefithrough the fifth point nite lines BDT, CRD, meeting with the last drawn fines TPS, PRQ (the former with the former, and the latter with the latter) in T and R. And then draw the right line tr parallel to TR, cutting off from the right lines PT, PR, any segments Ft,Fr, proportional B to PT, PR; and if through their ex-

Let the

five

D

tremities

t,

r,

and the poles B, C, the

right lines B^,

Cr

are drawn, meeting in

that point d \vin be placed in the conic required. For (by Lem. 20) that point d is placed in a coruc section passing through the four points A, B, C, P; and the lines Rr, Tt vanishing, the point d comes to coincide with the point D. Wherefore the conic section passes through the five points A, B, C, P, D. d,

Q.E.D.

The same otherwise. Of the given points join any three, as A, B, C; and about two of them B, C, as poles, making the angles ABC, ACB of a given magnitude to revolve, apply the legs BA, CA, first to the point D, then to the point P, and mark the points M, N, in which the other legs BL, CL intersect each other in both cases. Draw the indefinite right line MN, and let those movable angles revolve about their poles B, C, in such manner that the intersection, which is now supposed to be m, of the legs BL, CL, or BM, CM, may always fall in that indefinite right line and the intersection, which is now supposed to be d, of the legs BA, CA, or BD, CD, will describe the conic required, PADdB. For (by Lem. 21) the point d will be placed in a conic section passing through the points comes to coincide with the points L, M, N, the B, C; and when the point point d will (by construction) come to coincide mth the points A, D, P. Wherefore a conic section will be described that shall pass through the five

MN

;

m

points A, B, C, P, D. Cor. I. Hence a right line

Q.e.f.

may

be readily

drawn which

shall

be a tangent to

Mathematical Principles

62

the conic in any given point B. Let the point d come to coincide w-ith the point B, and the right line Brf will become the tangent required. Cor. II. Hence also may be found the centres, diameters, and latera recta of the conies, as in Cor. ii, Lem. 19.

Scholium

The former joining B, P,

of these constructions will

and

in that line, produced,

if

become something more simple by need be, taking

Bp to BP

as

PR is to

PT and through p draw the indefinite right line pe parallel to SPT, and in that line pe taking always pe equal to Pr; and draw the right lines ;

Cr to meet in d. For since Pr to P^, PR to PT, PB, pe to P^, are all in the same ratio, pe and Pr will be always equal. After this manner the points of the conic are most readily found, unless you would rather describe the curve meBe,

pB

to

chanically, as in the second construction.

Proposition

To

23.

Problem

15

describe a conic that shall pass through four given points,

and touch a given

right line.

Case

1.

Suppose that

HB is the given tangent, B the point

of contact,

and

BC, and draw PS parallel to BH, and PQ parallel to BC; complete the parallelogram BSPQ. Draw BD cutting SP in T, and CD cutting PQ in R. Lastly, draw any line tr parallel to TR, cutting off from PQ,

C, D, P, the three other given points. Join

PS, the segments Pr, Ft proportional

PR,

to

and draw

PT

(by Lem. 20) on the conic to be des-

intersection d will

always

fall

cribed.

The same otherwise. a given magnitude revolve

Let the angle CBH of about the pole B, as also the rectilinear radius DC, both ways produced, about the pole C. Mark the points M, N, on which the leg BC of the angle cuts that radius when BH, the other leg thereof, meets the same radius in the points P and D. Then drawing

CD

the indefinite Hne MN, let that radius CP or and the leg BC of the angle continually meet in this line; and the point of meeting of the other leg with the radius Avill delineate the conic required. For if in the constructions of the preceding Problem the point A comes to a coincidence with the point B, the lines CA and CB will coincide, and the line AB, in its last situation, will become the tan-

respectively,

Cr, B^; their point of

BH m /

N

Book

BH

I

:

The Motion of Bodies

63

and therefore the constructions there set down will become the same gent here described. Wherefore the intersection of the leg constructions with the BH with the radius will describe a conic section passing through the points C, q.e.f. D, P, and touching the hne BH in the point B. Case 2. Suppose the four points B, C, D, P, given, being situated without the tangent HI. Join each two by the lines BD, CP meeting in G, and cutting and I. Cut the tangent in A in such manner that may be the tangent in to lA as the product of the mean proportional between CG and GP, and the mean proportional between and is to the product of the mean proportional between and GB, and the mean proportional between PI and IC, and A will be the point of contact. For if HX, a parallel to the right line PI, cuts the conic in any points and Y, the point A (by the properties of the conic sections) will come to be so placed, that HA^ will become to AI^ in a ratio that to the rectangle is compounded out of the ratio of the rectangle GB and the ratio of HD, or of the rectangle CG GP to the rectangle to the rectangle PI IC. But after the point of contact A the rectangle BH is found, the conic will be described as in the first Case, q.e.f. But the point A may be taken either between or without the points and I, upon which account a two-fold conic may be described. ;

HA

H

HD

BH

GD

X

BH

HX-HY DG

•

•

•

•

HD

;

•

H

Proposition

To

24.

Problem

16

describe a conic that shall pass through three given points,

and touch two given

right lines.

Suppose HI, KL to be the given tangents and B, C, D the given points. Through any two of those points, as B, D, draw the indefinite right line BD meeting the tangents in the points H, K. Then likewise through any other two of these points, as C, D, draw the indefinite right line CD meeting the tangents in the points I, L. Cut the lines drawn in R and S,

HR

KR

may be to as the mean proso that and is to the portional between and KD, mean proportional between and IS to LS as the mean proportional be-

BH

HD BK

s

•

tween CI and ID is to the mean proportional between CL and LD. But you may cut, at pleasure, either within or between the points K and H, I and L, or without them. Then draw RS cutting the tangents in A and P, and A and P will be the points of contact. For if A and P are supposed to be the points of contact, situated anywhere else in the tangents, and through any of the points H, I, K, L, as I, situated in either tangent HI, a right line lY is drawn parallel to the other tangent KL, and meeting the curve in X and Y, and in that right fine there be taken IZ equal to a mean proportional between IX and lY, the rectangle XI I Y or IZ^ Avill (by the properties of the •

Mathematical Principles

64

conic sections) be to LP^ as the rectangle CI -ID is to the rectangle CL-LD; that is (by the construction), as SI is to SL^, and therefore IZ LP = SI SL. Wherefore the points S, P, Z are in one right line. Moreover, since the tangents meet in G, the rectangle XI lY or IZ^ will (by the properties of the :

:

•

conic sections) be to I A^ as GP^ is to GA^, and consequently IZ lA = GP GA. Wherefore the points P, Z, A lie in one right line, and therefore the points S, P, and A are in one right line. And the same argument will prove that the :

:

points R, P, and A are in one right line. Wherefore the points of contact A and P lie in the right line RS. But after these points are found, the conic may be q.e.f. described, as in the first Case of the preceding Problem. In this Proposition, and Case 2 of the foregoing, the constructions are the and Y, or not; neither do cuts the conic in same, whether the right line

XY

X

they depend upon that section. But the constructions being demonstrated where that right line does cut the conic, the constructions where it does not are also known; and therefore, for brevity's sake, I omit any further demonstration _ of them. -,Lemma 22

To transform figures into other figures of the same kind. Suppose that any figure HGI is to be transformed. Draw, at pleasure, two parallel Hnes AO, BL, cutting any given third line AB in A and B, and from any point G of the figure, draw out any right line GD, parallel to OA, till it meets the right line AB. Then from any given point O in the Une OA, draw to

D

the right line OD, meeting BL in d; and from the point of intersection raise the right line dg containing any given angle with the right line BL, and has to having such ratio to Od as OD; and g will be the point in the new figure hgi, corresponding to the point G. And in like manner the several points of the point

g

DG

the

first figure will

spondent points

give as

of the

new

many

corre-

figure. If

therefore conceive the point through all the points of the

G

B

we

to be carried along

D

by a continual motion

the point g will be hkewise carried along by a continual motion through all the points of the new figure, and dethe first ordinate, dg the scribe the same. For distinction's sake, let us call the the first abscissa, ad the new abscissa, O the pole, new ordinate, abscinding radius, OA the first ordinate radius, and Oa (by which the paralfirst figure,

DG

OD

AD

OABa

completed) the new ordinate radius. I say, then, that if the point G is placed in a given right Une, the point g will be also placed in a given right Une. If the point G is placed in a conic section, the point g will be likewise placed in a conic section. And here I understand the circle as one of the conic sections. But further, if the point G is placed in a line of the third analytical order, the point g will also be placed in a Une of the third order, and so on in curved lines of higher orders. The two fines in which the points G, g are placed, will be always of the same analytical order. For as lelogram

a

between k and z, i and y, h and x. Therefore / / / because of the analogy there is between the propa/ / gation of the rays of light and the motion of bodies, I thought it not amiss to add the follo^^^ng Propositions for optical uses; not at all considering the nature of the rays of light, or inquiring whether they are bodies or not; but only determining the curves of bodies which are extremely like the curves of the rays. fleeted

.••

Mathematical Principles

156

Proposition Supposing

the sine of incidence

97.

Problem 47

upon any surface

of emergence; and performed in a very short space, which

in a given ratio

to be

to the

sine

that the inflection of the paths of those bodies near that surface is

may

be considered as a point;

it

is

required

determine such a surface as may cause all the corpuscles issuing from any one given place to converge to another given place. Let A be the place from whence the corpuscles diverge B the place to which the curved line which by its revolution round the they should converge; axis AB describes the surface sought; D, E any two points of that curve; and to

;

CDE

EG

perpendiculars let fall oh the paths of the bodies AD, DB. Let the approach to and coalesce with point the point E; and the ultimate ratio of

EF,

D

AD DB

^

is increased, the line DF by which q is diminby which to the line ished, will be the same as that of the sine of incidence to the sine of emergence. Therefore the ratio of the increment of the line to the decrement of the is given; and therefore if in the axis AB there be taken anywhere the line must pass, and the increment of point C through which the curve the decrement of BC, and from the AC be taken in that given ratio to centres A, B, with the radii AM, BN, there be described two circles cutting will touch the curve sought CDE, and, by toucheach other in D; that point q.e.i. ing it anywhere at pleasure, will determine that curve. Cor. i. By causing the point A or B to go off sometimes in infinitum, and sometimes to move towards other parts of the point C, will be obtained all those figures which Descartes has exhibited in his Optics and Geometry relating to refractions. The invention of which Descartes having thought fit to conceal is here laid open in this Proposition. CoR. II. If a body lighting on any surface in the direction of a right line AD, drawn "\^ according to any law, should emerge in the

DG

^^

AD

DB

CM

CDE CN

D

CD

direction of another right line

CQ,

DK; and from

C

there be drawn curved lines CP, always perpendicular to AD, DK; the

the point

^

'^

increments of the lines PD, QD, and therefore the lines themselves PD, QD, generated by those increments, ^vill be as the sines of incidence and emergence to each other, and conversely.

Proposition The same things supposed; as

CD,

if

round

98.

the axis

Problem 48

AB any attractive surface be

described,

from the given place surface EF, which may make

regular or irregular, through which the bodies issuing

A must pass; it is required to find a second attractive those bodies converge to a given place B.

AB cut the first surface in C and

the second in E, the point being taken in any manner at pleasure. And supposing the sine of incidence on the first surface to the sine of emergence from the same, and the sine of emergence from the second surface to the sine of incidence on the same, to be then produce AB to G, so to another given quantity as any given quantity

Let a

line joining

D

M

N

;

Book

I

:

The Motion of Bodies

157

BG may be to CE as M — N to N and AD to H, so that AH may be equal to AG; and DF to K, so that DK may be to DH as N to M. Join KB, and about the centre D vnih the radius DH describe a circle meeting KB produced in L,

that

;

.\

H

and draw BF parallel to DL; and the point F will touch the Une EF, which, being turned round the axis AB, will describe the surface sought. q.e.f. For conceive the lines CP, CQ to be everywhere perpendicular to AD, DF, and the lines ER, ES to FB, FD respectively, and therefore QS to be always equal to CE; and (by Cor. ii. Prop. 97) PD will be to as to N, and there-

QD M fore as DL to DK, or FB to FK; and by subtraction, as DL - FB or PH - PD FB to FD or FQ-QD; and by addition as PH-FB to FQ, that is (because PH and CG, QS and CE, are equal), as CE+BG-FR to CE-FS. But (because BG is to CE as M — N to N) it comes to pass also that CE+BG is to CE as

M to N; and therefore, by subtraction, FR

Prop. 97) the surface direction DF, to go on in the Hne

(by Cor.

ii.

is

to

FS

as

EF

compels a body,

FR

to the place B.

M to N; and therefore falling

upon

it

in the q.e.d.

Scholium

may go on to three or more surfaces. But of all most proper for optical uses. If the object glasses of two glasses of a spherical figure, containing water be-

In the same manner one figures the spherical is the

telescopes were

made

of

tween them, it is not unlikely that the errors of the refractions made in the extreme parts of the surfaces of the glasses may be accurately enough corrected by the refractions of the water. Such object glasses are to be preferred before elhptic and hyperbolic glasses, not only because they may be formed mth more ease and accuracy, but because the pencils of rays situated without the axis of the glass would be more accurately refracted by them. But the different refrangibility of different rays is the real obstacle that hinders optics from being made perfect by spherical or any other figures. Unless the errors thence arising can be corrected, all the labor spent in correcting the others is quite thrown away.

BOOK TWO

THE MOTION OF BODIES MEDIUMS

IN RESISTING

SECTION The motion of bodies that are

I

resisted in the ratio of the velocity

Proposition

1.

Theorem

1

// a body is resisted in the ratio of its velocity, the motion lost hy resistance is as the space gone over in its motion. For since the motion lost in each equal interval of time is as the velocity,

that

is,

as the small increment of space gone over, then,

by composition, the

motion lost in the whole time will be as the whole space gone over. q.e.d. Cor. Therefore if the body, destitute of all gravity, move by its innate force only in free spaces, and there be given both its whole motion at the beginning, and also the motion remaining after some part of the way is gone over, there will be given also the whole space which the body can describe in an infinite time. For that space will be to the space now described as the whole motion at the beginning

is

to the part lost of that motion.

Lemma Quantities proportional

are continually proportional.

A: A-B = B:B-C = C:C-D = &c.;

Let then,

to their differences

1

by

subtraction,

A B = B :C = C D = &c. :

:

Proposition

2.

Theorem

q.e.d.

2

// a body is resisted in the ratio of its velocity, and moves, by its inertia only, through an homogeneous medium, and the times be taken equal, the velocities in the

beginning of each of the times are in a geometrical progression, and the spaces described in each of the times are as the velocities. Case 1. Let the time be divided into equal intervals; and if at the very beginning of each interval we suppose the resistance to act with one singlfe impulse which is as the velocity, the decrement of the velocity in each of the intervals of time will be as the same velocity. Therefore the velocities are proportional to their differences, and therefore (by Lem. 1, Book ii) continually proportional. Therefore if out of an equal number of intervals there be compounded any equal portions of time, the velocities at the beginning of those times will be as terms in a continued progression, which are taken by jumps, omitting everywhere an equal number of intermediate terms. But the ratios of these terms are 159

Mathematical Principles

160

compounded of the equal ratios of the intermediate terms equally repeated, and therefore are equal. Therefore the velocities, being proportional to those terms, are in geometrical progression. Let those equal intervals of time be diminished, and their number increased in infinitum, so that the impulse of resistance may become continual; and the velocities at the beginnings of equal times, always continually proportional, will be also in this case continually q.e.d.

proportional.

Case

2.

And, by

division, the differences of the velocities, that

is,

the parts

the times, are as the wholes; but the spaces

of the velocities lost in each of described in each of the times are as the lost parts of the velocities (by Prop. 1, Book i), and therefore are also as q.e.d. the wholes. Cor. Hence if to the rectangular asymptotes AC, CH,

DG

BG

be drawn is described, and AB, the hyperbola perpendicular to the asymptote AC, and both the velocity of the body, and the resistance of the medium, at the very beginning of the motion, be expressed by any given line AC, and, after some time is elapsed, by the indefinite line DC; the time may be expressed by the area ABGD, and the space described in that time by the line

AD. For

that area, by the motion of the point D, be uniformly increased in will decrease in a geometrical the same manner as the time, the right line ratio in the same manner as the velocity; and the parts of the right line AC, described in equal times, will decrease in the same ratio. if

DC

Proposition

To

3.

Problem

1

an homogeneous medium, ascends or and is resisted in the ratio of its velocity, and acted upon by

define the motion of a body which, in

descends in a right

line,

an uniform force of gravity. The body ascending, let the gravity be represented by any given rectangle BACH; and the resistance of the medium, at the beginning of the ascent, by the rectangle BADE, taken on the contrary side of the right line AB. Through the point B, with the rectangular asymptotes AC, CH, describe an hyperbola, cutting the perpendiculars DE, de in G, g; and the body ascending \vill in the time DGgd describe the space EG^e; in the time DGBA, the space of the whole ascent EGB in the time ABKI, the space of descent BFK; and in the time IKH the space of descent KFfk; and the velocities of the bodies (proportional to the resistance of the medium) in these periods of time will be ABED, ABed, o, ABFI, AB/i respectively; and the greatest velocity which the body can acquire by descending will be BACH. For let the rectangle BACH be resolved into innumerable rectangles AA-, K^, Lm, Mn, &c., which shall be as the increments of the velocities produced in so many equal times; then will o, Ak, Al, Am, An, &c., be as the whole velocities, and therefore (by supposition) as the resistances of the medium in the beginning of each of the equal times. Make AC to AK, or ABHC to ABA-K, as the ;

Book

II:

The Motion of Bodies

161

force of gravity to the resistance in the beginning of the second time then the force of gravity subtract the resistances, and ABHC, KA'HC, L/HC, ;

from

Mm-

HC, &c., ^^'ill be as the absolute forces ^\dth which the body is acted upon in the beginning of each of the times, and therefore (by Law i) as the increments of the velocities, that is, as the rectangles Air, K/, Lm, Mw, &c., and therefore (by Lem. 1, Book ii) in a geometrical progression. Therefore, if the right lines K/r, LZ, Mm, Nn, &c., are produced so as to meet the hyperbola in g, r, s, t, &c., the areas ABgK, KgrL, LrsM, Ms^N, &c., will be equal, and therefore analogous to the equal times and equal gravitating forces. But the area ABgK (by Cor. iii, Lems. 7 and 8, Book i) is to the area B/r? as Kg to 3^A-g', or AC to /^AK, that is, as the force of gravity to the resistance in the middle of the first time. And by the like reasoning, the areas H gKLr, rLMs, sMNi, &c., are to the areas qklr, m rims, smnt, &c., as the gravitating forces to the resistances in the middle of the second, A K LMN third, fourth time, and so on. Therefore since the equal areas BAKg, qKLr, rLMs, sMNf, &c., are analogous to the gravitating forces, the areas Bkq, qklr, rims, smnt, &c., vdW be analogous to the resistances in the middle of each of the times, that is (by supposition), to the velocities, and so to the spaces described. Take the sums of the analogous quantities, and the areas Bkq, Blr, Bms, Bnt, &c., will be analogous to the whole spaces described; and also the areas ABgK, ABrL, ABsM, ABv will be as -

pD2

is

AD2+Ap2,

that

is,

—

^-r:^

,

that

is

pD^ AD^+AD-AA",

Therefore tDv, the interval of the sector,

is

(because ^D

is

,

or

AD-CA-; and qDp

as

^; that

is,

But pD^ 3^AD-pg.

given) as ^-^^. is

directly as the least

decrement pq of the velocity, and inversely as the force Ck which diminishes the velocity; and therefore as the interval of time answering to the decrement

Mathematical Principles

172

And, by composition, the sum of all the intervals (Dv in the sector AD^ will be as the sum of the intervals of time answering to each of the lost intervals pq of the decreasing velocity Ap, till that velocity, being diminished into nothing, vanishes; that is, the whole sector AD^ is as the whole time of the velocity.

of ascent to the highest place.

q.e.d.

Draw DQV cutting off the least intervals TDV and PDQ of the sector DAV, and of the triangle DAQ; and these intervals will be to each other as DT2 to DP2, that is (if TX and AP are parallel), as DX^ to DA^ or TX^ to Case

2.

AP2; and, by subtraction, as

DX^-TX^

to

DA^-AP^. But, from

the nature

and, by the supposition, AP^ is AD- AK. Therefore the intervals are to each other as AD^ to AD^ — AD-AK; that is, of the sector as — or AC to CK; and therefore the interval to of the hyperbola,

AD

is

DX^-TX^ is AD^;

TDV

AD AK

—^^—

;

and therefore (because

AC

directly as the increment of the velocity,

and

AD

are given) as

and inversely as the

^^;

that

is,

force generating

the increment; and therefore as the interval of the time answering to the increment. And, by composition, the sum of the intervals of time, in which all the intervals PQ of the velocity AP are generated, ^vill be as the sum of the intervals of the sector ATD; that is, the whole time will be as the whole sector. Q.E.D.

Hence if AB be equal to a fourth part of AC, the space which a body will describe by faUing in any time will be to the space which the body could describe, by moving uniformly on in the same time with its greatest velocity AC, as the area ABNK, which expresses the space described in falling to the area ATD, which expresses the time. For since CoR.

I.

AC AP = AP: AK, :

and by Cor.

i,

Lem.

2, of this

:

:

therefore

and

since

Book,

LK PQ = 2AK AP = 2AP AC, LK >^PQ = AP MAC or AB, KN AC or AD = AD CK, :

:

:

:

:

multiplying together corresponding terms,

LKNO: DPQ = AP: CK.

Book As

II:

The Motion of Bodies

173

sho^^^l above,

DPQ:DTV = CK: AC. LKXO BT\ = AP AC

Hence, that

is,

:

:

as the velocity of the falUng

body by

falling

of the areas

body to the

greatest velocity which the

can acquire. Since, therefore, the moments

ABXK and ATD are as the velocities,

LKXO

and

DTV

the parts of those areas generated in the same time -snll be as the spaces described in the same time; and ADT, generated from the beginning, and therefore the Avhole areas will be as the whole spaces described from the beginning of the descent, q.e.d. Cor. II. The same is true also of the space described in the ascent. That is to say, that all that space is to the space described in the same time, ^\•ith the uniform velocity AC, as the area AB/iA' is to the sector AD^. CoR. III. The velocity of the body, falling in the time ATD, is to the velocity all

ABXK

would acquire

same time in a nonresisting space, as the triangle APD to the hyperbolic sector ATD. For the velocity in a nonresisting medium would be as the time ATD, and in a resisting medium is as AP, that is, as the which

it

in the

APD. And

those velocities, at the beginning of the descent, are equal among themselves, as well as those areas ATD, APD. CoR. IV. By the same argument, the velocity in the ascent is to the velocity with which the body in the same time, in a nonresisting space, would lose all its motion of ascent, as the triangle ApD to the circular sector A^D; or as the right line Ap to the arc A^. CoR. V. Therefore the time in which a body, by falling in a resisting medium, would acquire the velocity AP, is to the time in which it would acquire its greatest velocity AC, by falling in a nonresisting space, as the sector to the triangle ADC; and the time in which it would lose its velocity Ap, by ascending in a resisting medium, is to the time in which it would lose the same velocity by ascending in a nonresisting space, as the arc At to its tangent Ap. CoR. VI. Hence from the given time there is given the space described in the ascent or descent. For the greatest velocity of a body descending in infinitum is given (by Cor. ii and iii, Theor. 6, of this book) and thence the time is given in which a body would acquire that velocity by falling in nonresisting space. Taking the sector or ADt to the triangle in the ratio of the given time to the time just found, there ^nll be given both the velocity AP or Ap. and the area ABX^K or ABn/r, which is to the sector ADT, or AD^ as the space sought to that which would, in the given time, be uniformly described A\'ith that greatest velocity found just before. CoR. VII. And by going backwards, from the given space of ascent or descent ABnA- or ABXK, there will be given the time AD^ or ADT. triangle

ADT

;

ADT

ADC

Proposition Suppose

the

uniform force of gravity

to

10.

Problem

tend directly

3

to the

plane of the horizon, and

the resistance to be as the product of the density of the medium and the square of the velocity: it is proposed to find the density of the medium in each place, which

make

body move in any given curved line, the velocity of the body, and the medium in each place. Let PQ be a plane perpendicular to the plane of the scheme itself; PFHQ a curved line meeting that plane in the points P and Q; G, H, I ,K four places of the body going on in this curve from F to Q; and GB, HC, ID, four shall

the

resistance of the

KE

Mathematical Principles

174

from these points to the horizon, and standing on the at the points B, C, D, E; and let the distances BC, CD, DE

parallel ordinates let fall

horizontal line

PQ

of the ordinates be equal

among

themselves.

From

the points

G and H

let

the

GL, HN be drawn touching the curve in G and H, and meeting the ordinates CH, DI, produced upwards, in L and N and complete the parallelogram HCDM And the times in which the body describes right lines

;

the arcs GH, HI, will be as the square root of the altitudes LH, NI, which the bodies would describe in those times, by falling from the tangents and the velocities will be directly as the lengths described GH, HI, and inversely as the times. Let the times be represented by ;

and the in the

velocities

time

t

will

by

GH -7=-

and

HI

—

;

and the decrement

be represented by

—

-^^

.

of the velocity

T and

t,

produced

This decrement arises from the

resistance which retards the body, and from the gravity which accelerates it. Gravity, in a falling body, which in its fall describes the space NI, produces a velocity with which it would be able to describe tmce that space in the same

time, as Galileo hath demonstrated; that

describes the arc HI,

— —

it

is,

the velocity

——-: but

the body

augments that arc only by the length HI — HN or

and therefore generates only the velocity

;

^5^P

if

.

Let this velocity

be added to the before-mentioned decrement, and we shall have the decrement of the velocity arising

from the resistance alone, that

-=^

is,

——

H

—

:fj^

Therefore since, in the same time, the action of gravity generates, in a falling

—— the resistance will be to the gravity as 2MINI 2NI f-GH „^ 2MINI ^^„ ^j^to2NL ^oras^--HI-f—

body, the velocity

GH

HI

,

,

-r-T- + ^ll^*° Now,

,

,

,

CB, CD, CE, put —0, 0, 2o. For the ordinate CH series Qo+Ro2-|-So^+, &c. And all the terms of the that is, Ro^+So^-I-, &c., will be NI; and the ordinates DI,

for the abscissas

put P; and for

MI

put any

series after the first,

EK, and BG will be P-Qo-Ro^-So^-, &c., P-2Qo-4Ro2-8So3-, &c., and P + Qo — Ro^+So^ — &c., respectively. And by squaring the differences of ,

CH

and CH — DI, and to the squares thence produced and CD themselves, you will have 00+QQoo — and oo+QQoo-|-2QRo^+, &c., the squares of the arcs GH, HI;

the ordinates BG — adding the squares of

2QRo^ +

,

&c.,

BC

whose roots oa/(1+QQ)

QRoo

y(i+QQ)

QRoo

and oV(H-QQ)'^ ^^'

GH

;"

' '

CH

v(i+QQ)

are the

arcs and HI. Moreover, if from the ordinate there be subtracted half the sum of the ordinates and DI, and from the ordinate DI there be subtracted half the sum of the ordinates and EK, there will remain Roo and Roo-|-3So', the versed sines of the arcs GI and HK. And these are proportional to the short lines LH and NI, and therefore are as the squares of the infinitely

BG

CH

Book small times

— ^ R

or

T and ;

HI, MI, and

the resistance

:

:

The Motiox of Bodies

and thence the

— — — HIH

and

NI

/

II

^^ill

ratio =, varies as the square root of

Fp

=j^^

i

xli

just found,

be

now

175

becomes

by substitutmg the values

3Soo

2R

V(1+QQ). And 3Soo

+ QQ)

to the gra\dty as -?^^-V(l

2R

since

R+3So

R GH,

of ^, i

2NI

is

2Roo,

to 2Roo, that

is,

as

3SV(1+QQ) to4RR. And the velocity \nll be such,

that a body going off the^e^^'ith from any place H, in the direction of the tangent HX, would describe, in a vacuum, a parabola,

whose diameter

is

HC, and

its latus

rectum -^r^ or js i

And

the resistance

is

—R

.

medium and the square of the velocity; and therefore the density of the medium is directly as the resistance, and inversely as the square of the

as the product of the density of the

velocity; that

and

is,

inversely

directly as

—^^; R

as

3SV(l+QQ)

4RR that

IS,

Q.E.I.

RV(i+QQ)" Cor.

If

I.

the tangent

TTT both ways, so as to meet any ordinate xVF in T,

QQ), and therefore in what has gone before this

means the

may

TTT

ACyR'

;,

and the density '

of the

HN

be produced

^'^ be equal to \/(l +

^p

V(1+QQ)- By to 4RR-AC; the S- \C

be put for

resistance will be to the gravity as

velocity will be as

as

3S-HT

medium

be as

\\ill

R HT •

PFHQ

be defined by the relation beCoR. II. And hence, if the curved tween the base or abscissa AC and the ordinate CH, as is usual, and the value of the ordinate be resolved into a converging series, the Problem ^^-ill be expeditiously solved b}' the first terms of the series; as in the following Examples. Exam. 1. Let the fine PFHQ be a semicircle described upon the diameter PQ; to find the density of the medium that shall make a projectile move in line

that hne. Bisect the diameter

DP

or

PQ

in

A; and

call

AQ2 — AD2 = nn — oa — 2ao — 00,

extracted by our method, Anil give ao 00

°I = ^-7-2.-

Here put nn

for ee-\-aa,

and DI

\\\\\

AQ, or

n\

AC,

a;

CH,

ee — 2ao — oo;

aaoo

ad^

a^o^

2e^

2e^

2e^'

become

=ee

and CD,

then and the root being e;

o;

&c. ao

nnoo

anno''

e

2e^

2e

— ,&c.

In such a series I distinguish the successive terms after this manner: I call that the first term in which the infinitely small quantity o is not found; the second, in which that quantity is of one dimension only; the third, in which it arises to two dimensions; the fourth, in which it is of three; and so ad infinitum. And the first term, which here is e, \\\\\ always denote the length of

Mathematical Principles

176 the ordinate

CH,

erected at the starting point of the indefinite quantity

o.

The

— will denote the difference between CH and DN the short line MN which cut by completing the parallelogram

second term, which here

is

,

e

that

is,

HCDM;

and therefore always determines the position

in this case,

nnoo ,

off

is

by taking

Anil represent

MN HM = — :

o

:

=a

the short line IN, which

:

e.

lies

The

of the tangent

HN;

as,

third term, which here

is

between the tangent and the

curve; and therefore determines the angle of contact IHN, or the curvature which the curved line has in H. If that short line IN is of a finite magnitude, it will be expressed by the third term, together \viih those that follow in infinitum. T But if that short line be diminished in in•

finitum, the terms follo^^^ng become infinitely less than the third term, and there-

may

be neglected. The fourth term determines the variation of the curvature

fore

the variation of the variation; and so on. From this, by the way, appears the use, not to be disdained, which may be

the

fifth,

made

problems that depend upon tangents,

of these series in the solution of

and the curvature of curves. Now compare the series ao

nnoo

anno^

2e^

2e'

e

— &c.

with the series

P-Qo and

for P, Q,

R and S,

put

e, -,-?r-,

e

n

or -

.

;

le* .

and the density

of the

Roo — So^ — &c ann and and for V(1 + QQ) put 5 '

2e'

medium

wall

come out

as

—

;

that

is

ne

e

given); as - or

a

.

AC ^^,

that

is,

as that length of the tangent

V(^+^0

(because n

HT, which

is

is

term-

inated at the semidiameter AF standing perpendicularly on PQ: and the resistance A\dll be to the gravity as 3a to 2n, that is, as 3AC to the diameter PQ of the circle; and the velocity will be as VCH. Therefore if the body goes from the place F, with a due velocity, in the direction of a line parallel to PQ, and the density of the medium in each of the places is as the length of the tangent HT, and the resistance also in any place is to the force of gravity as SAC to PQ, that body vnll describe the quadrant of a circle. q.e.i. But if the same body should go from the place P, in the direction of a line perpendicular to PQ, and should begin to move in an arc of the semicircle PFQ, we must take AC or a on the contrary side of the centre A; and therefore its sign must be changed, and we must put — a for+a. Then the density of the

H

H

FHQ

medium would come out

—

.

But Nature does not admit

of a negative

is, a density which accelerates the motion of bodies; and therefore cannot naturally come to pass that a body by ascending from P should

density, that it

as

Book

II

:

The Motion of Bodies circle. To produce such an

describe the quadrant PF of a to be accelerated by an impelling

177 effect,

medium, and not impeded by a

a body ought resisting one.

Let the line PFQ be a parabola, having its axis AF perpendicular to the horizon PQ to find the density of the medium, which \\-ill make a pro-

Exam.

2.

;

move in that From the nature

jectile

line.

of the parabola, the rectangle

— PD DQ

equal to the if that right

is

•

rectangle under the ordinate DI and some given right line; that is, line be called h; PC, a; PQ, c; CH, e; and CD, o; the rectangle {a-\-o)(c — a — o)=ac — aa — 2ao-\-co — oo = h-'Dl; 2a 00 = ac — aa

0—

therefore

DI

Now the

second term

Q

^

and the third term ^

for Qo,

are

2a —— o of this

-

no more terms, the

series is to

for Roo.

But

be put

since there

S of the fourth term g vanish; and therefore the quantity :^ jjtttt;, coefficient

—

^\ill

—

Kv(l+QQ)

medium

to which the density of the

be nothing Therefore, where the medium is of no density, the projectile \d\\ move in a parabola; as Galileo hath heretofore demonstrated. q.e.i. Exam. 3. Let the line be an hyperbola, having its asymptote perpendicular to the horizontal plane AK to find the density of the medium that will make a projectile move in that line. Let be the other asymptote, meeting the ordinate produced in V; and from the nature of the hyperbola, the rectangle of into VG will be given. There is also given the ratio of to VX, and therefore the rectangle of into VG is given. Let that be hh; and, completing the parallelogram DNXZ, let BN be called a; BD, o; NX, c; and let the given ratio of VZ to ZX is

proportional,

^\^ll

NX

AGK

:

MX

DG XV

DN

DN

or

DN be n—

m -•{a-o) n

Then

.

GD

and

DN or

A\dll

be equal to a — o,

NX-VZ-VG equal to m

,

—m

hb

—a-\ 0n n a—o bb_

Let the term

be resolved into the con-

a—o verging senes hh hh ,66 ,

i

a

and

aa

66 0^—00^—0%

&G.,

a*

a^

GD will become equal to m a 66 w 66 -_«2_ hh hh 0'

—

„

,

c

„

,

,

n

\

a

n

The second term — o n

aa

aa

0^,

&c.

a"

a"

o of this series is to

hh

be used for Qo; the third

—gO^,

with

changed for Ro^; and the fourth

its

—jo^, a

sign ^\ith

M ,-

VG

equal to

a—o

,

VZ

equal to

Mathematical Principles

178 its

sign changed also for So*, bb bb bb

—

m

ficients

—

,

, ,

n

—^, ,

and

and

their coef-

are to be put

a aa a" for Q, R, and S in the former being done, the density of the come out as

Which medium vn.\\ rule.

a*

mm

+ a^\\ 1

2mbb naa

nn

'A + a'/

or

mm

,

aa-\

that

is, if

in

2mbb n

aa

nn

VZ you

take

b*

,

1

)

aa

VY equal to VG, as...^ XY

For aa and

—1^r a^ n''

n

aa

XZ and ZY. But the ratio of the resistance to gravity is found to be that of 3XY to 2YG and the velocity is that with which the body are the squares of

;

would describe a parabola, whose vertex

XY^ -^TfT-

is

DG,

G, diameter

Suppose, therefore, that the densities of the

VG G are inversely

medium in each

latus rectum of the places

G

as the distances XY, and that the resistance in any place is to the gravity as 3XY to 2YG and a body let go from the place A, with a due velocity, \vill describe that hyperbola AGK. q.e.i. ;

Exam.

AGK

Suppose, indefinitely, the line to be an hyperbola described with the centre X, and the asymptotes MX, NX, so that, having constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G and its asymptote in V, VG may be inversely as any power DN" of the line ZX or DN, whose index is the number n: to find the density of the medium in which a projected

body

4.

will describe this curve.

VZ be to XZ or DN as d to e, and VG be equal to fy^r-^] then DN will be equal to A — O, VG = -rr p^, For BN, BD,

NX,

put A, O, C, respectively, and

let

—

VZ = -(A-0)

GD

and

or

NX-VZ-VG equal to

e

bb e

Let the term

bb

(A-0)'

b_b^ nbb

^

be resolved into an infinite series nn-\-n

6A"+3

GD will be equal

^_d^_^

d

A"

e

e

_

,

660^^ for

n66 A"+i

4^m+n

,

&c.,

+n*+3nn+ 2n

2A«+2

Roo, the fourth '

bbO\

to

The second term -O — -r-r,0 2A«+2

n*+3nn+2n

6602

A" and

(A-0)"

e

6A»+3

of this series is to

^

,

^.

6A"+3

660

bbO\ &c.

be used for Qo, the third

for So'.

And

thence the density -^

Book of the

medium

II

S

RV(l+QQ)

,

:

in

The Motion of Bodies any place G,

179

be

will

n+2

—

2dnhb

"A>/

X

eA"

and therefore as

if

in

VZ you take VY equal to w VG, •

XY. For A^ and

are the squares of ^A2-?^A+^' eA" ee

the resistance in the same place

that

is,

as

XY to n+2 ^^^r-

the projected body would r^T. GD,

^ and

that density

1

. . rectumlatus 1

move or

XZ

and ZY. But

XY

G is to the force of gravity as 3S -^ to 4RR, •

VG. And

+ QQ ^ R

reciprocally

is

the velocity there

in a parabola,

is

the same wherewith

whose vertex

is

G, diameter

2XY2 {nn-\-n) -VG

Q.E.I.

Scholium In the same manner that the density of

medium comes

the Cor.

V"

I,

if

the resistance will

come out

/HT\"-i ,

in

put as any power

or of

S^ ^^to

to be as

W)

R-F And therefore, S

AC

/Acy-i

s

-T^tolj^]

is

S-

^

of the velocity V, the density of the

medium

that the ratio of

out to be as

if

a curve can be found, such

(l+QQ)"-i may be given; the

body, in an uniform medium, whose resistance is as the power V" of the velocity V, mil move in this curve. But let us return to more simple curves. Since there can be no motion in a parabola except in a nonresisting medium, but in the hyperbolas here described it is produced by a continual resistance it is evident that the Hne which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That Hne is certainX ly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hyperbolas here described. The difference, however, is not so great between the one and the other but that these latter may be commodiously enough used in practice instead of the former. And perhaps these may prove more useful than an hyperbola that is more accurate, and at the same time more comT plex. They may be made use of, then, in N this manner.

Mathematical Principles

180

Complete the parallelogram XYGT, and the right line GT hyperbola in G, and therefore the density of the medium in G the tangent

GT, and the

force of gravity as

^_ GT

velocity there as a/ttft)

—

2nn+2n to

.

»

will is

touch the

inversely as

and the resistance

is

to the

^,^

uV.

a body projected from the place A, in the direction of the right line AH, describes the hyperbola AGK, and AH produced meets the asymptote in H, and AI drawn parallel to it meets the other asymptote in I the density of the medium in A will be inversely as AH and the velocity of the Therefore

if

MX

NX

body as \-rT-, ^^^ the

resistance there to the force of gravity as

;

AH

to

^^-— -AI. Hence the following Rules are deduced.

n+2

Rule

the density of the medium at A, and the velocity with which the body is projected, remain be the same, and the angle changed; the lengths AH, AI, remain. Therefore if those will lengths, in any one case, are found, the hyperbola may afterwards be easily determined from any given 1.

If

NAH

HX

angle

NAH.

the angle NAH, and the density of the medium at A, remain the same, and the velocity with which the body is projected be will continue changed, the length

Rule

2.

If

AH

the same and AI will be changed inversely as the square of the velocity. Rule 3. If the angle NAH, the velocity of the body at A, and the accelerative gravity remain the same, and the proportion of the resistance at A to the to AI will be motive gravity be augmented in any ratio; the proportion of augmented in the same ratio, the latus rectum of the above-mentioned parab;

AH

ola remaining the same,

and

also the length

AH^ -r-j.-

proportional to

it;

and there-

AH will be diminished in the same ratio, and AI will be diminished as the square of that ratio. But the proportion of the resistance to the weight is augmented, when either the specific gravity is made less, the magnitude remaining equal, or when the density of the medium is made greater, or when, by diminishing the magnitude, the resistance becomes diminished in a less ratio than the weight. Rule 4. Because the density of the medium is greater near the vertex of the hyperbola than it is in the place A, that a mean density may be preserved, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT. is to be deRule 5. If the lengths AH, AI are given, and the figure fore

AGK

The Motion of Bodies 181 scribed, produce HN to X, so that HX may be to AI as n+1 to 1; and with the centre X, and the asymptotes MX, NX, describe an hyperbola through the point A, such that AI may be to any of the hues VG as XV" to XI". Book

M

II:

A

.••••£

number n is, so much the more accurate are these hyperbolas in the ascent of the body from A, and less accurate in its descent to K; and conversely. The conic hyperbola keeps a mean ratio between these, and is more simple than the rest. Therefore if the hyperbola be of this kind, and you are to find the point K, where the projected body falls upon any right line AN passing through the point A, let AN produced meet the

Rule

6.

By how much

the greater the

NX in M

and N, and take NK equal to AM. Rule 7. And hence appears an expeditious method of determining this hyperbola from the phenomena. Let two similar and equal bodies be projected with the same velocity, in different angles HAK, hAk, and let them fall upon the plane of the horizon in K and k; and note the proportion of AK to AA*. Let it be as d to e. Then erecting a perpendicular AI of any length, assume any length AH or Ah, and thence graphically, or by scale and compass, collect the lengths AK, Ak (by Rule asymptotes

MX,

AK

to Ak be the the ratio of same with that of d to e, the length 6). If

AH

was rightly assumed. If not, take on the indefinite right fine SM, the length SM equal to the assumed AH; and erect a perpendicular of

MN

equal to the difference ^ ratios multiplied

of the

-r-^

Ak

e

by any given

right

By the like method, from several assumed lengths AH, you may find

line.

several points

them

N; and draw through

all a regular curve cutting the right line

NNXN,

SMMM in X.

Mathematical Principles

182

AH

equal to the abscissa SX, and thence find again the length the lengths, which are to the assumed length AI, and this last AH, known by experiment, to the length last found, vdW be as the length the true lengths AI and AH, which were to be found. But these being given, there ^^'ill be given also the resisting force of the medium in the place A, it to /^AI. Let the density of the medium be being to the force of gravity as Lastly,

assume

AK; and

AK

AK

AH

increased

same

by Rule

ratio, it will

and if the resisting force become still more accurate. 4,

AH,

just found be increased in the

HX

being found; let there be now required the position of the line AH, according to which a projectile thrown with that given velocity shall fall upon any point K. At the points A and K, erect the lines AC, KF perpendicular to the horizon; whereof let AC be drawn downwards, and be equal to AI or 3^HX. With the asymptotes AK, KF, describe an hyperbola, whose conjugate shall pass through the point C; and from the centre A, mth the interval AH, describe a circle cutting that hyperbola in the point H; then will fall upon the the projectile thrown in the direction of the right line point K. Q.E.I. For the point H, because of the given length AH, must be someand where in the circumference of the described circle. Draw CH meeting AC, AI equal, AE A\ill are parallel, and KF in E and F; and because CH, be equal to AM, and therefore also equal to KN. But CE is to AE as FH to KN, and therefore CE and FH are equal. Therefore the point falls upon the hyperbolic curve described with the asymptotes AK, KF whose conjugate passes through the point C; and is therefore found in the common intersection of this hyperbolic curve and the circumference of the described circle, q.e.d. It is to be observed that this operation is the same, whether the right line be parallel to the horizon, or inclined thereto in any angle; and that from two intersections H, h, there arise two angles NAH, NA/i; and that in mechanical practice it is sufficient once to describe a circle, then to apply a ruler CH, of an indeterminate length, so to the point C, that its part FH, intercepted between the circle and the right line FK, may be equal to its part CE placed between the point C and the right line AK. What has been said of hyperbolas may be easily applied to parabolas. For if a parabola be represented in the vertex X, by XAGK, touched by a right fine and the ordinates lA, VG be as any powers XI", XV", of the abscissas XI, XV; draw XT, GT, AH, whereof touch the let be parallel to VG, and let GT, parabola in G and A: and a body projected from any place A, in the direction of the right line AH, with a due velocity, \vill describe this parabola, if the density of the medium in each of the places G be inversely as the tangent GT. In that case the velocity in G wll be the same as would cause a body, moving in a nonresisting space, to describe a conic parabola, having G for its vertex, VG produced downwards for its

Rule

8.

The

lengths

AH

AK

MX

H

AKN

XV

AH

XT

diameter, and will

2GT^ -,

{nn — n) ,

^^^ for

VG

its latus

be to the force of gravity as

sent an horizontal

line,

GT to

rectum. ^r-

and both the density

And

the resisting force in

VG. Therefore

of the

medium

if

G

NAK repre-

at A,

and the

Book

body

velocity with which the

be anyhow

II:

The Motion of Bodies

183

N AH

projected, remaining the same, the angle AH, AI, will remain; and thence mil be of the parabola, and the position of the right line XI and is

HX

altered, the lengths

X

given the vertex by taking VG to I A as XV" to XI", there will be given parabola, through which the projectile will pass.

SECTION

;

all

G

the points

of the

III

The motion of bodies that are resisted partly

in the ratio of

the

and partly as the square of the same ratio

velocities,

Proposition

11.

Theorem

8

// a body he resisted partly in the ratio and partly as the square of the ratio of its velocity, and moves in a similar medium by its innate force only; and the times be taken in arithmetical progression: then quantities inversely proportional to the increased by a certain given quantity, will be in geometrical progression.

velocities,

With the centre C, and the rectangular asymptotes CADd and CH, describe an hyperbola BEe, and let AB, DE, de be parallel to the asymptote CH. In the asymptote CD let A, G be given points; and if the time be represented by the hyperbolic area I say, that the velocity may be expressed by the length DF, whose reciprocal GD, together with the given line CG, compose the length increasing in a geometrical progression. For let the small area DEed be the least given increment of the time, and Dd will be inversely as DE, and therefore directly as CD. Therefore the decre-

ABED uniformly increasing,

— —

'-

CD

}

ment

—

of j::^,

which (by Lem.

CC-I-CD

—

j^TF^

creasing

that

is,

as

+ 7^^2-

^^a

by the addition same

Book

is t^tft^'

ii)

^^^ ^^

C^O

1 >

2,

ABED

Therefore, the time

of the given intervals

EDde,

^^^^ ^^

it

prj2

^^

uniformly in-

follows that

^^^pr

GD

de-

For the decrement of the velocity (by the supposition), as the sum of two quantities, whereof one is as the velocity, and the other as the square of the velocity; and 1 1 C^O the decrement of y::^^^ is as the sum of the quantities p^-p- and 777;^, whereof the creases in the

is

ratio with the velocity.

as the resistance, that

is

GD

first is 7^-pr itself,

GD

and the

last 7^pp-„ is as p;pp-„

the decrements of both being analogous. proportional to the time

therefore 7=7^

And

t:^Y\'

as the velocity,

CG;

GD

inversely

the

sum CD,

uniformly increasing, will increase in a geometrical progresQ.E.D.

sion.

Cor.

is

the quantity

if

^^ augmented by the given quantity

GD

ABED

:

(jrU

I.

Therefore,

if,

having the points

sented by the hyperbolic area the reciprocal of

GD.

ABED,

A

and

G

the velocity

given, the time be repre-

may

be represented by

GD

Mathematical Principles

184

Cor.

And by

taking

GA

GD

as the reciprocal of the velocity at the beginning to the reciprocal of the velocity at the end of any time ABED, the

point

G

II.

\nll

be found.

And

to

that point being found, the velocity

may

be found

from any other time given.

Proposition The same things being supposed, I

12.

Theorem

9

say, that if the spaces described are taken in

arithmetical progression, the velocities augmented by a certain given quantity will be in geometrical progression.

CD

In the asymptote let there be given the point R, and, erecting the perpendicular RS meeting the hyperbola in S, h let the space described be represented by the hyperbolic area RSED; and the velocity will be as the length GD, which, together with the decreasing given line CG, composes a length in a geometrical progression, while the space RSED increases in an arithmetical progression. D &c. With the centre

bA bB bU

S,

and the asymptotes SA,

hyperbola, cutting the perpendiculars

F

E

D C B

AH,

BI,

CK,

&c., in a,

Sx, describe h, c,

&c.,

any

and the

Book

II:

The Motion of Bodies

FN will be found at any height SF, thiu as the difference

Aa — F/

by taking the area

to the difference

201

thnz to that given area

Aa — Bb.

Scholium reasoning it may be proved, that if the gravity of the particles of a fluid diminishes as the cube of the distances from the centre, and the reciproS A^ g A 3 g A 3 cals of the squares of the distances SA, SB, SC, &c., (namely, ^-r^, ^)

By a like

^— — ,

be taken in an arithmetical progression, the densities AH, BI, CK, &c., will be in a geometrical progression. And if the gravity be diminished as the fourth power g A4 g A 4 of the distances, and the reciprocals of the cubes of the distances (as ^ tz

— — ,

SA^' SB^'

SA*

^^,

&c.) be taken in arithmetical progression, the densities

AH,

CK,

BI,

&c.,

be in geometrical progression. And so in infinitum. Again; if the gravity of the particles of the fluid be the same at all distances, and the distances be in arithmetical progression, the densities will be in a geometrical progression, as Dr. Halley hath found. If the gravity be as the distance, and the squares of the distances be in arithmetical progression, the densities will be in geometrical progression. And so in infinitum. These things will be so, when the density of will

the fluid condensed by compression is as the force of compression; or, which is the same thing, when the space possessed by the fluid is inversely as this force. Other laws of condensation may be supposed, as that the cube of the compressing force may be as the fourth power of the density, or the cube of the ratio of the force the same with the fourth power of the ratio of the density in which case, if the gravity be inversely as the square of the distance from the centre, the density will be inversely as the cube of the distance. Suppose that the cube of the compressing force be as the fifth power of the density; and if the gravity be inversely as the square of the distance, the density will be inversely as the 3^th power of the distance. Suppose the compressing force to be as the square of the density, and the gravity inversely as the square of the distance, then the density will be inversely as the distance. To run over all the cases that might :

be offered would be tedious. But as to our own air, this is certain from experiment, that its density is either accurately, or very nearly at least, as the compressing force; and therefore the density of the air in the atmosphere of the earth is as the weight of the whole incumbent air, that is, as the height of the mercury in the barometer.

Proposition

23.

Theorem

18

7/ a fluid he composed of particles fleeing from each other, and the density he as the compression, the centrifugal forces of the particles will he inversely proportional to the distances of their centres.

And,

conversely, particles fleeing

with forces that are inversely proportional

from each

other,

to the distances of their centres, compose whose density is as the compression. Let the fluid be supposed to be included in a cubic space ACE, and then to be reduced by compression into a lesser cubic space ace; and the distances of the particles retaining a like situation with respect to each other in both the spaces, mil be as the sides AB, ah of the cubes; and the densities of the mediums will be inversely as the containing spaces AB^, ah^. In the plane side of the

an

elastic fluid,

^^ F

Book plate. If in this

manner

II:

The Motion of Bodies

particles repel others of their

203

own kind

that lie next them, but do not exert their force on the more remote, particles of this kind ^^•ill compose such fluids as are treated of in this Proposition. If the force of any particle diffuse itself every way in infinitum, there ^nll be required a greater force to produce an equal condensation of a greater quantity of the fluid. But whether elastic fluids do really consist of particles so repelling each other, is a physical question. We have here demonstrated mathematically the property of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question.

SECTION VI The motion and resistance of pendulous bodies Proposition

24.

Theorem

19

The quantities of matter in 'pendulous bodies, whose centres of oscillation are equally distant from the centre of suspension, are in a ratio compounded of the ratio of the weights and the squared ratio of the times of the oscillations in a vacuum. For the velocity which a given force can generate in a given matter in a given time is directly as the force and the time, and inversely as the matter.

The

greater the force or the time

is,

or the less the matter, the greater the

is manifest from the second Law of Motion. Now if pendulums are of the same length, the motive forces in places equally distant from the perpendicular are as the weights: and therefore if two bodies by oscillating describe equal arcs, and those arcs are divided into equal parts; since the times in which the bodies describe each of the correspondent parts of

velocity generated. This

the arcs are as the times of the whole oscillations, the velocities in the correspondent parts of the oscillations \\dll be to each other directly as the motive forces and the whole times of the oscillations, and inversely as the quantities of matter and therefore the quantities of matter are directly as the forces and the times of the oscillations, and inversely as the velocities. But the velocities are inversely as the times, and therefore the times are directly and the velocities inversely as the squares of the times; and therefore the quantities of matter are as the motive forces and the squares of the times, that is, as the weights and the squares of the times. q.e.d. Cor. I. Therefore if the times are equal, the quantities of matter in each of the bodies are as the weights. Cor. II. If the weights are equal, the quantities of matter will be as the squares of the times. Cor. III. If the quantities of matter are equal, the weights mil be inversely as the squares of the times. Cor. IV. Since the squares of the times, other things being equal, are as the lengths of the pendulums, therefore if both the times and the quantities of matter are equal, the weights ^\ill be as the lengths of the pendulums. Cor. V. And, in general, the quantity of matter in the pendulous body is directly as the weight and the square of the time, and inversely as the length of the pendulum. :

Mathematical Principles

204

Cor.

VI.

But

dulous body

is

medium, the quantity of matter in the penand the square of the time, the pendulum. For the comparative weight is the any heavy medium, as was shown above; and

in a nonresisting

directly as the comparative weight

and inversely as the length of motive force of the body in therefore does the same thing in such a nonresisting medium as the absolute weight does in a vacuum. Cor. VII. And hence appears a method both of comparing bodies one with another, as to the quantity of matter in each; and of comparing the weights of the same body in different places, to know the variation of its gravity. And by experiments made with the greatest accuracy, I have always found the quantity of

matter in bodies to be proportional to their weight.

Proposition

25.

Theorem

20

Pendulous bodies that are, in any medium, resisted in the ratio of the moments of and pendulous bodies that move in a nonresisting medium of the same specific gravity, perform their oscillations in a cycloid in the same time, and describe time,

proportional parts of arcs together. be an arc of a cycloid, which a Let

AB

body D, by vibrating in a nonresisting

shall describe in any time. Bisect that arc in C, so that C may be the thereof; and the accelerative force with which the body is urged in point lowest any place D, or d, or E, Avill be as the length of the arc CD, or Cd, or CE. Let that force be expressed by that same arc; and since the resistance is as the moment of the time, and therefore given, let it be expressed by the given part CO of the cycloidal arc, and take the arc Od in the same ratio to the arc that the arc OB has to the arc CB and the force with which the body in d is urged in a resisting medium, being the excess of the force Cd above the resistance CO, will be expressed by the arc Od, and will therefore be to the force with

medium,

CD

:

which the body

D is urged in a non-

medium arc Od to

resisting

in the place

D,

the arc CD; and as the the therefore also in place B, as the arc OB to the arc CB. Therefore if two bodies D, d go from the place B, and are urged by these forces; since the forces at the beginning are as the arcs CB and OB, the first velocities and arcs first described will be in the same ratio. Let those arcs be and Bd, and the remaining arcs CD, Od will be in the same ratio. Therefore the forces, being proportional to those arcs CD, Od, Avill remain in the same ratio as at the beginning, and therefore the bodies mil continue describing together arcs in the same ratio. Therefore the forces and velocities and the remaining arcs CD, Od, will be always as the whole arcs CB, OB, and therefore those remaining arcs will and d will arrive together be described together. Therefore the two bodies at the places C and O; that which moves in the nonresisting medium, at the place C, and the other, in the resisting medium, at the place O. Now since the velocities in C and are as the arcs CB, OB, the arcs which the bodies describe

BD

D

when they go

farther mil be in the

same

ratio.

Let those arcs be

CE

and Oe.

Book

II:

The Motiox of Bodies

205

which the body D in a nonresisting medium is retarded in E is as CE. and the force A\-ith which the body d in the resisting medium is retarded in e. is as the sum of the force Ce and the resistance CO, that is, as Oe; and therefore the forces \nth which the bodies are retarded are as the arcs CB, OB, proportional to the arcs CE. Oe; and therefore the velocities, retarded in that given ratio, remain in the same given ratio. Therefore the velocities and the arcs described A\-ith those velocities are always to each other in that given ratio of the arcs CB and OB; and therefore if the entire arcs AB, aB are taken in the same ratio, the bodies D and d A\'ill describe those arcs together, and in the places A and a ^^•ill lose all their motion together. Therefore the whole oscillations are isochronal, or are performed in equal times; and any parts of the

The

force

^^•ith

BD,

Bd. or BE, Be, that are described together, are proportional to the whole arcs BA, Ba. q.e.d. Cor. Therefore the s"\\iftest motion in a resisting medium does not fall upon the lowest point C, but is found in that point 0, in which the whole arc described Ba is bisected. And the body, proceeding from thence to a, is retarded at the same rate with which it was accelerated before in its descent from B to O. arcs, as

Proposition Pendulous

26.

Theorem

21

bodies, that are resisted in the ratio of the velocity, have their oscillations

in a cycloid isochronal.

For

if

two

bodies, equally distant

from

their centres of suspension, describe,

unequal arcs, and the velocities in the correspondent parts of the arcs be to each other as the whole arcs; the resistances, proportional to the velocities, ^^ill be also to each other as the same arcs. Therefore if these resistances be subtracted from or added to the motive forces arising from gravity which are as the same arcs, the differences or sums mil be to each other in the same ratio of the arcs; and since the increments and decrements of the velocities are as these differences or sums, the velocities ^rill be always as the whole arcs; therefore if the velocities are in any one case as the whole arcs, they Arill remain always in the same ratio. But at the beginning of the motion, when the bodies begin to descend and describe those arcs, the forces, which at that time in oscillating,

are proportional to the arcs, will generate velocities proportional to the arcs.

Therefore the velocities ^^dll be always as the whole arcs to be described, and therefore those arcs ^\ill be described in the same time. q.e.d.

Proposition

27.

Theorem

22

If pendulous bodies are resisted as the squxire of their velocities, the differences between the times of the oscillations in a resisting medium, and the times of the oscillations in a nonresisting

medium

of the same specific gravity, will be proportional to the arcs described in oscillating, nearly.

equal pendulums in a resisting medium describe the unequal arcs A, B and the resistance of the body in the arc A will be tc the resistance of the body in the

For

let

;

Mathematical Principles

206

correspondent part of the arc

BB,

B

as the square of the velocities, that

is,

as

AA to

B were to the resistance in the arc A as and B would be equal (by the last Proposi-

nearly. If the resistance in the arc

A tion). Therefore the resistance AA in the arc A, or AB in the arc B, causes the excess of the time in the arc A above the time in a nonresisting medium and the resistance BB causes the excess of the time in the arc B above the time in a nonresisting medium. But those excesses are as the efficient forces AB and BB q.e.d. nearly, that is, as the arcs A and B.

AB

to

AA, the times

in the arcs

;

Cor.

I.

Hence from the times

of the oscillations in

unequal arcs in a resisting

medium, may be known

the times of the oscillations in a nonresisting medium of the same specific gravity. For the difference of the times will be to the excess of the time in the shorter arc above the time in a nonresisting medium as the difference of the arcs is to the shorter arc.

Cor. II. The shorter oscillations are more isochronal, and very short ones are performed nearly in the same times as in a nonresisting medium. But the times of those which are performed in greater arcs are a little greater, because the resistance in the descent of the body, by which the time is prolonged, is greater, in proportion to the length described in the descent than the resistance in the subsequent ascent, by which the time is contracted. But the time of the oscillations, both short and long, seems to be prolonged in some measure by the motion of the medium. For retarded bodies are resisted somewhat less in proportion to the velocity, and accelerated bodies somewhat more than those that proceed uniformly forwards; because the medium, by the motion it has received from the bodies, going forwards the same way with them, is more agitated in the former case, and less in the latter; and so conspires more or less with the bodies moved. Therefore it resists the pendulums in their descent more, and in their ascent less, than in proportion to the velocity; and these two causes concurring prolong the time.

Proposition

28.

Theorem

23

If a pendulous body, oscillating in a cycloid, he resisted in the ratio of the moments of the time, its resistance will he to the force of gravity, as the excess of the arc described in the whole descent ahove the arc described in the subsequent ascent is to twice the length of the

pendulum.

BC

represent the arc described in the descent, the ascent, and Aa the difference of the arcs: and things remaining as they were constructed and de-

Let

Qa

the arc described in

monstrated in Prop. 25, the force with which the oscillating body is urged in any place D will be to the force of resistance as the arc to the arc CO, which is half of that difference Aa. Therefore the force with which the oscillating body is urged at the beginning or the highest point of the cycloid, that is, the force of gravity, will be to the resistance as the arc of the cycloid, between that highest point and the lowest point C, is to the arc CO; that is

CD

Book

II

The Motion of Bodies

:

207

(doubling those arcs), as the whole cycloidal arc, or twice the length of the is to the arc Aa. q.e.d.

pendulum,

Proposition Supposing Let

that

29.

Problem

a body oscillating in a cycloid

6

is resisted

as the square of the

velocity; to find the resistance in each place. arc described in one entire oscillation, C the lowest point of the CZ half the whole cycloidal arc, equal to the length of the pen-

Ba be an

and dulum; and let it be required to find the resistance of the body in any place D. Cut the indefinite right line OQ in the points 0, S, P, Q, so that (erecting the perpendiculars OK, ST, PI, QE, and with the centre 0, and the asymptotes cycloid,

OK, OQ,

describing the hyperbola

TIGE

cutting the perpendiculars ST, PI, QE in T, I, and E, and through the point I drawing KF, parallel to the asympin K, and the perpendiculars ST and QE tote, OQ, meeting the asymptote in L and F) the hyperbolic area PIEQ may be to the hyperbolic area PITS as the arc BC, described in the descent of the body, is to the arc Ca described in

OK

the ascent; and that the area

lEF may be

to the area

O

S

P

ILT

as

OQ to OS. Then

M

rR Q

MN

with the perpendicular cut off the hyperbolic area PINM, and let that area be to the hyperbolic area PIEQ as the arc CZ to the arc BC described in the descent. And if the perpendicular RG cuts off the hyperbolic area PIGR, which shall be to the area PIEQ as any arc CD is to the arc BC described in the whole descent, the resistance in any place D will be to the force of gravity as the area

^ lEF-IGH

is

to the area

PINM.

For since the forces arising from gravity places Z, B, D, a are as the arcs CZ,

wth which the body is urged in the

CB, CD, Ca, and those

arcs are as the areas

PINM, PIEQ, PIGR, PITS; let those areas represent both the arcs and the forces respectively. Let Dd be a very small space described by the body in its descent; and let it be expressed by the very small area KGgr, comprehended between the parallels RG, rg; and produce rg to h, so that GKhg and RGgr may

be the contemporaneous decrements of the areas IGH, PIGR.

ment GHhg

~

RrHG-^ lEF, of the area ^ lEF

lEF, or

be to the decrement RGgr, or Rr-RG, ;

IGH

will

of the area

PIGR,

as

HG- lEF OQ

is

to

OR HG OR lEF is to OR-GR or OP -PI, that is (beOQ quantities ORHG, OR-HR-ORGR, ORHK-OPIK,

and therefore as

cause of the equal

the incre-

vyv^

vyv^

\J\il

RG

And

•

-

Mathematical Principles

208

PIHR and PIGR+IGH), the area

OR lEF — IGH

-^-^

as

PIGR+IGH OR lEF is to OPIK. OQ

be called Y, and

RGgr

Therefore

if

the decrement of the area

be given, the increment of the area Y mil be as PIGR — Y. Then if V represent the force arising from the gravity, proportional to the arc CD to be described, by which the body is acted upon in D, and R be put for the resistance, V — R will be the whole force with which the body is urged in D. Therefore the increment of the velocity is as V — R and the interval of time in which it is generated conjointly. But the velocity itself is directly as the contemporaneous increment of the space described and inversely as the same

PIGR

interval of time. Therefore, since the resistance is, by the supposition, as the square of the velocity, the increment of the resistance will (by Lem. 2) be as

the velocity and the increment of the velocity conjointly, that is, as the moment of the space and V — conjointly; and, therefore, if the moment of the space be given, as V — R; that is, if for the force V we put its expression PIGR, be expressed by any other area Z, as PIGR — Z. and the resistance Therefore the area PIGR uniformly decreasing by the subtraction of given increases in proportion of PIGR — Y, and the area Z in moments, the area

R

R

Y

Y

and Z begin together, and proportion of PIGR — Z. And therefore if the areas at the beginning are equal, these, by the addition of equal moments, will continue to be equal; and in like manner decreasing by equal moments, will vanish together. And, conversely, if they together begin and vanish, they will have equal moments and be always equal. For, if the resistance Z be augmented, then the velocity together with the arc Co, described in the ascent of the body, will be diminished; and, the point in which all the motion together with the resistance ceases, coming nearer to the point C, then the resistance vanishes sooner than the area Y. And the contrary will happen when the resistance is diminished. Now the area Z begins and ends where the resistance is nothing, that is, at is equal to the arc CB, and the the beginning of the motion where the arc falls upon the right hne QE; and at the end of the motion where right line falls upon the right line ST. And the equal to the arc Ca, and is the arc

CD

RG RG CD OR area Y or j^ lEF — IGH begins and ends also where the resistance is nothing, and therefore where

OR -^-^

lEFand IGH

are equal; that

is

(by the construction),

upon the right lines QE and ST. Therefore those areas begin and vanish together, and are therefore always where the right

line

RG

falls

successively

O

S

P

rR^

M

Book

resistance

is

expressed,

:

I.

gravity as the area ^^^ II.

is

to the gravity.

But

it

lEF

is

to the area

q.e.d.

C

is

to the force of

PINM.

PIHR is to the area lEF as PIGR — Y) becomes nothing.

becomes greatest where the area

OR is to OQ. For in that case its moment Cor.

equal to the area Z, by which the

Therefore the resistance in the lowest place

OP

CoR.

is

209

and therefore is to the area PINM, by which the gravity

expressed, as the resistance

Cor.

The Motion of Bodies

OR lEF — IGH -^

equal. Hence, the area

is

II

Hence

(that

is,

may be known

the velocity in each place, as varying as the square root of the resistance, and at the beginning of the motion being equal to the velocity of the body oscillating in the same cycloid wthout any III.

also

resistance.

However, by reason of the difficulty of the calculation by which the resistance and the velocity are found by this Proposition, we have thought fit to subjoin the Proposition follo^^^ng.

Proposition If a right line aB he equal and at each of its points

to the

pendulum as

Theorem

arc of a cycloid which

24

an

oscillating

body describes,

perpendiculars

DK be erected, which shall be to the

the resistance of the

body in the corresponding points of

D

length of the

30.

the

the arc is to the force of gravity: I say, that the difference between the arc described

in the whole descent and the arc described in the whole subsequent ascent multiplied by half the sum of the same arcs will be equal to the area BKa which all those perpendiculars take up.

Let the arc of the cycloid, described in one entire oscillation, be expressed by the right line aB, equal to it, and the arc which would have been described in a the length AB. Bisect AB in C, and the point C will represent the lowest point of the cycloid, and will be as the force arising from gravity, with which the body in is urged in the direction of the tangent of the cycloid, and will have the same ratio to the length of the pendulum as the force in has

vacuum by

CD

D

D

to

the force

of gravity.

Let that

be expressed by that length CD, and the force of gravity by the length of the pendulum; and if in in the same ratio you take to the length of the pendulum as the resistance is to the gravity, will the exponent of the resistance. be AM From the centre C with the interval CA or CB describe a semicircle BEeA. Let the body describe, in the least time, the space Dd; and, erecting the perpendiculars DE, de, meeting the circumference in E and e, they will be as the velocities which the body descending in a vacuum from the point B would acquire in the places and d. This appears by Prop. 52, Book i. Let, therefore, these velocities be expressed by those perpendiculars DE, de; and let DF be the velocity which it acquires in by falling from B in the resisting medium. And if from the centre C with the interval CF we describe the circle F/M meeting the right lines de and AB in / and M, then will be the place to which it Avould thenceforward, without further force, therefore,

DE

DK

DK

D

D

M

Mathematical Principles

210

and dj the velocity

would acquire in d. Hence, also, if which the body D, in describing the Yg least space Dd, loses by the resistance of the medium; and CN be taken equal to Cg; then will N be the place to which the body, if it met no further resiswill be the decrement of the tance, would thenceforward ascend, and ascent arising from the loss of that velocity. Draw Fm perpendicular to dj^ and the decrement F^ of the velocity DF generated by the resistance DK will be to the increment /m of the same velocity, generated by the force CD, as the genresistance, ascend,

moment

represent the

it

of the velocity

MN

DK

erating force angles Fm/, Yhg,

to the generating force

FDC, jm is to Fm

or

CD. But because

of the similar tri-

Dd as CD to DF; and, by multipHcation

DK to DF. Also Yh is to Yg as DF to CF; again by multiplication of corresponding terms, F/i or MN to T)d as DK

of corresponding terms,

and, to CF or

F^ to

Dd as

CM;

MN

and therefore the sum of all the -CM will be equal to the sum of all the T>d DK. At the movable point suppose always a rectangular erected equal the indeterminate ordinate to CM, which by a continual motion is multiplied by the whole length Aa; and the trapezium described by that motion, or its equal, the rectangle Aa-j^dB, will be equal to the sum of all the MN-CM, and therefore to the sum of all the Dd-DK, that is, to the area

M

BKVTa.

Q.E.D.

Cor. Hence from the law of resistance, and the difference Aa of the arcs may be derived the proportion of the resistance to the gravity,

Ca, CB, nearly.

For if the resistance DK be uniform, the figure BKTa will be a rectangle under Ba and DK; and hence the rectangle under J^Ba and Aa will be equal to the rectangle under Ba and DK, and DK will be equal to 3^Aa. Therefore since DK represents the resistance, and the length of the pendulum represents the gravity, the resistance will be to the gravity as 3^Aa is to the length of the pendulum; altogether as in Prop. 28 is demonstrated. If the resistance be as the velocity, the figure BKTa will be nearly an ellipse. For if a body, in a nonresisting medium, by one entire oscillation, should describe the length BA, the velocity in any place D would be as the ordinate DE of the circle described on the diameter AB. Therefore since Ba in the resisting medium, and BA in the nonresisting one, are described nearly in the same times; and therefore the velocities in each of the points of Ba are to the velcorresponding points of the length BA nearly as Ba is to BA, the velocity in the point in the resisting medium will be as the ordinate of the circle or ellipse described upon the diameter Ba; and therefore the figure BKVTa will be nearly an ellipse. Since the resistance is supposed proportional to the velocity, let OV represent the resistance in the middle point 0; and an ellipse BRVSa described with the centre O, and the semiaxes OB, OV, will be nearly equal to the figure BKVTa, and to its equal the rectangle Aa-BO. Therefore Aa BO is to OV BO as the area of this ellipse to OV BO; that is, Aa is to OV as the area of the semicircle is to the square of the radius, or as 11 to 7 nearly; and, therefore, ^/ii Aa is to the length of the pendulum as the resistance of the oscillating body in is to its gravity. Now if the resistance varies as the square of the velocity, the figure BKVTa will be almost a parabola having V for its vertex and OV for its axis, and therefore will be nearly equal to the rectangle under %Ba and OV. Therefore the rectangle under ^Ba and Aa is equal to the rectangle %Ba OV, and

ocities in the

D

•

•

DK

•

Book

OV

II:

The Motion of Bodies

211

equal to /^Aa; and therefore the resistance in O made to the is to its gravity as ^Aa is to the length of the pendulum. And I take these conclusions to be accurate enough for practical uses. For since an ellipse or parabola BRVSa falls in with the figure BKVTa in the middle point V, that figure, if greater towards the part BRV or VSa, is less towards the contrary part, and is therefore nearly equal to it. therefore

oscillating

is

body

Proposition

31.

Theorem 25

// the resistance made to an oscillating body in each of the proportional parts of the arcs described be augmented or diminished in a given ratio, the difference between the arc described in the descent

and

the arc described in the subsequent ascent will be

augmented or diminished in the same ratio. For that difference arises from the retardation of the pendulum by the resistance of the medium, and therefore is as the whole retardation and the retarding resistance proportional thereto. In the foregoing Proposition the rectangle under the right line 3^aB and the difference Aa of the arcs CB, Ca, was equal to the area BKTa. And that area, if the length aB remains, is aug-

mented

or diminished in the ratio of the ordinates DK; that is, in the ratio B AM of the resistance, and is therefore as the length aB and the resistance conjointly. And therefore the rectangle under Aa and 3^aB is as aB and the resistance conjointly, and therefore Aa is as the resistance. q.e.d. Cor. i. Hence if the resistance be as the velocity, the difference of the arcs in the same medium will be as the whole arc described; and conversely. Cor. II. If the resistance varies as the square of the velocity, that difference will vary as the square of the whole arc; and conversely. CoR. III. And generally, if the resistance varies as the third or any other power of the velocity, the difference will vary as the same power of the whole

and conversely. CoR. IV. If the resistance varies partly as the

arc;

power of the velocity and partly as the square of the same, the difference will vary partly as the first power and partly as the square of the whole arc; and conversely. So that the law and ratio of the resistance will be the same for the velocity as the law and first

ratio of that difference for the length of the arc.

CoR. V. And therefore if a pendulum describe successively unequal arcs, and we can find the ratio of the increment or decrement of this difference for the length of the arc described, there will be had also the ratio of the increment or decrement of the resistance for a greater or less velocity.

General Scholium From

these Propositions

we may

mediums by penduthe air by the following

find the resistance of

lums oscillating therein. I found the resistance of experiments. I suspended a wooden globe or ball weighing 57/^2 ounces troy, its diameter Qj/g London inches, by a fine thread on a firm hook, so that the distance between the hook and the centre of oscillation of the globe was 103^

Mathematical Principles

212

marked on the thread a point 10 feet and 1 inch distant from the centre of suspension; and even with that point I placed a ruler divided into inches, by the help of which I observed the lengths of the arcs described by the pendulum. Then I numbered the oscillations in which the globe would lose 3^ part of its motion. If the pendulum was drawn aside from the perpendicular to the distance of 2 inches, and then let go, so that in its whole descent it described an arc of 2 inches, and in the first whole oscillation, compounded of the descent and subsequent ascent, an arc of almost 4 inches, the pendulum in 164 oscillafeet. I

tions lost

}/s pa^rt

of its motion, so as in its last ascent to describe

an arc

of

1^

described an arc of 4 inches, it lost 3^ part of its motion in 121 oscillations, so as in its last ascent to describe an arc of 33^ inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches, it lost oscillations, respectively. Therefore y^ part of its motion in 69, 353^, 183/2» the difference between the arcs described in the first descent and the last ascent inches. If in the first descent

it

^%

in the 1st, 2d, 3d, 4th, 5th, 6th cases, }4, ^2, 1, 2, 4, 8 inches, respectively. Divide those differences by the number of oscillations in each case, and in one

was

120 inches was described, the difference of the arcs described in the descent and subsequent ascent will be ^656, ^242, ^69, /^7, ^/^9 parts of an inch, respectively. But these are as the square of the arcs described, greater oscillations in the differences nearly, but in lesser oscillations somewhat greater than in that ratio; and therefore (by Cor. ii. Prop. 31 of this Book) the resistance of the globe, when it moves very swiftly, varies as the square of the velocity, nearly; and when it moves slowly, in a somewhat greater ratio. Now let V represent the greatest velocity in any oscillation, and let A, B, and C be given quantities, and let us suppose the difference of the arcs to be AV+BV^'^-I-CV^. Since the greatest velocities are in the cycloid as 3^ the arcs described in oscillating, and in the circle as 3^ the chords of those arcs; and therefore in equal arcs are greater in the cycloid than in the circle in the ratio of 3^ the arcs to their chords; but the times in the circle are greater than in the cycloid, in a ratio inversely as the velocity; it is plain that the differences of the arcs (which are as the resistance and the square of the time conjointly) are nearly the same in both curves for in the cycloid those differences must be on the one hand augmented, with the resistance, in about the squared ratio of the arc to the chord, because of the velocity augmented in the simple ratio of the same; and on the other hand diminished, with the square of the time, in the same squared ratio. Therefore to reduce these observations to the cycloid, we must take the same differences of the arcs as were observed in the circle, and suppose the greatest velocities analogous to the half, or the whole arcs, that is, to the numbers 3^, 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and 6th cases put 1, 4, and 16 for V; and the difference of the arcs in the 2d case will become t^t = A C; in the 4th case, ^iy, =4A+8B 16C; in the 6th case, 9% = 16A+64B +256C. These equations reduced give A = 0.0000916, B = 0.0010847, and C = 0.0029558. Therefore the difference of the arcs is as 0.0000916V +0.0010847V»/2 -hO. 0029558 V^; and therefore since (by Cor., Prop. 30, apphed to this case) the resistance of the globe in the middle of the arc described in oscillating, where the velocity is V, is to its weight as MiAV+MoBV^/^^MCV^ is to the length of the pendulum, if for A, B, and C you put the numbers found, the resistance of the globe will be to its weight as 0.0000583V+0.0007593V»/2_|_o.o022169V2

mean

oscillation, in

which an arc of 3^,

Mb

:

+B +

+

73/^, 15, 30, 60,

Book

II:

The Motion of Bodies

213

to the length of the pendulum between the centre of suspension and the ruler, that is, to 121 inches. Therefore since V in the second case represents 1, in the 4th case 4 and in the 6th case 16, the resistance will be to the weight of is

the globe, in the 2d case, as 0.0030345 is to 121; in the 4th, as 0.041748 is to 121; in the 6th, as 0.61705 is to 121. The arc, which the point marked in the thread described in the 6th case, was 120—9^, or 119^ inches. And therefore since the radius was 121 inches, and the length of the pendulum between the point of suspension and the centre of the globe was 126 inches, the arc which the centre of the globe described was 124^ inches. Because the greatest velocity of the oscillating body, by reason of the resistance of the air, does not fall

on the lowest point

of the arc described,

but near the middle place of the Avhole arc, this velocity will be nearly the same as if the globe in its whole descent in a nonresisting medium should describe 62^ inches, the half of that arc, and that in a cycloid, to which we have above reduced the motion of the pendulum; and therefore that velocity will be equal to that which the globe would acquire by falling perpendicularly from a height equal to the versed sine of that arc. But that versed sine in the cycloid is to that arc 62^ as the same arc to twice the length of the pendulum 252, and therefore equal to 15.278 inches. Therefore the velocity of the pendulum is the same which a body would acquire by falling, and in its fall describing a space of 15.278 inches. Therefore with such a velocity the globe meets with a resistance which is to its weight as 0.61705 is to 121, or (if we take that part only of the resistance which is in the squared ratio of the velocity) as 0.56752 to 121. I found, by an hydrostatical experiment, that the weight of this wooden globe was to the weight of a globe of water of the same magnitude as 55 to 97; and therefore since 121 is to 213.4 in the same ratio, the resistance made to this globe of water, moving forwards with the above-mentioned velocity, will be to its weight as 0.56752 to 213.4, that is, as 1 to 376^0- Since the weight of a globe of water, in the time in which the globe with a velocity uniformly continued describes a length of 30.556 inches, will generate all that velocity in the falling globe, it is manifest that the force of resistance uniformly continued in the same time will take away a velocity, which will be less than the other in the ratio of 1 to 3765V, that is, the -gTeT';^ part of the whole velocity. And therefore in the time that the globe, with the same velocity uniformly continued, would describe the length of its semidiameter, or 3tV inches, it would lose the 33^2 part of its motion. I also counted the oscillations in which the pendulum lost }/i part of its motion. In the following table the upper numbers denote the length of the arc described in the first descent, expressed in inches and parts of an inch; the middle numbers denote the length of the arc described in the last ascent; and in the lowest place are the numbers of the oscillations. I give an account of this experiment, as being more accurate than that in which only Y^ part of the motion was lost. I leave the calculation to such as are disposed to make it. First descent

2

4

8

16

32

64

Last ascent

13^

3

6

12

24

48

83^

41%

22%

No. of

oscillations

374

272

1623^

I afterwards suspended a leaden globe of 2 inches in diameter, weighing 263^ ounces troy by the same thread, so that between the centre of the globe and

Mathematical Principles

214

the point of suspension there was an interval of IOJ/2 feet, and I counted the oscillations in which a given part of the motion was lost. The first of the following tables exhibits the number of oscillations in which }/s part of the whole motion was lost; the second the number of oscillations in which there was lost }4: part of the same. irst

descent

Book

II:

The Motion of Bodies

lation to the whole arc of 673/8 inches described

%

215

by the centre

of the globe in

one

oscillation; and so is the difference to a new difference 0.4475. If the length of the arc described were to remain, and the length of the pendulum should be augmented in the ratio of 126 to 1223^, the time of the oscillation would be augmented, and the velocity of the pendulum would be diminished as the square root of that ratio; so that the difference 0.4475 of the arcs described

mean

in the descent and subsequent ascent would remain. Then if the arc described be augmented in the ratio of 1243/31 to Q7}/8, that difference 0.4475 would be augmented as the square of that ratio, and so would become 1.5295. These things would be so upon the supposition that the resistance of the pendulum were as the square of the velocity. Therefore if the pendulum describe the whole arc of 1245/31 inches, and its length between the point of suspension and the centre of oscillation be 126 inches, the difference of the arcs described in the descent and subsequent ascent would be 1.5295 inches. And this difference multiplied by the weight of the pendulous globe, which was 208 ounces, produces 318.136. Again, in the pendulum above mentioned, made of a wooden globe, when its centre of oscillation, being 126 inches from the point of suspension, described the whole arc of 124/^1 inches, the difference of the arcs described in the descent and ascent was ^^/^2i into -§% This multiplied by the weight of the globe, which was 57/^2 ounces, produces 49.396. But I multiply these differences by the weights of the globes, in order to find their resistances. For the differences arise from the resistances, and are as the resistances directly and the weights inversely. Therefore the resistances are as the numbers 318.136 and 49.396. But that part of the resistance of the lesser globe, which is as the square of the velocity, was to the whole resistance as 0.56752 to 0.61675, that is, as 45.453 to 49.396, whereas that part of the resistance of the greater globe is almost equal to its whole resistance, and so those parts are nearly as 318.136 and 45.453, that is, as 7 and 1. But the diameters of the globes are 18^ and 6j^; and their squares 351/^6 and 47^/^4 are as 7.438 and 1, that is, nearly as .

the resistances of the globes 7 and 1. The difference of these ratios is barely greater than may arise from the resistance of the thread. Therefore those parts of the resistances which are, when the globes are equal, as the squares of the velocities, are also, when the velocities are equal, as the squares of the diameters of the globes. But the greatest of the globes I used in these experiments was not perfectly spherical, and therefore in this calculation I have, for brevity's sake, neglected some little niceties; being not very solicitous for an accurate calculus in an experiment that was not very accurate. So that I could msh that these experiments were tried again mth other globes, of a larger size, more in number, and more accurately formed; since the demonstration of a vacuum depends thereon. If the globes be taken in a geometrical proportion, whose diameters, let us suppose, are 4, 8, 16, 32 inches; one may infer from the progression observed in the experiments what would happen if the globes were still larger. In order to compare the resistances of different fluids with each other, I made the foUomng trials. I procured a wooden vessel 4 feet long, 1 foot broad, and 1

foot high. This vessel, being uncovered, I filled with spring water, and, having

immersed pendulums

therein, I

made them

oscillate in the water.

3^

And

I

found

that a leaden globe weighing 166^6 ounces, and in diameter inches, moved therein as it is set down in the following table the length of the pendulum from ;

64

16 12

Mathematical Principles

218 resolved into drops,

I

doubt not that the

rate enough, especially

if

rule already laid

the experiments be

made with

down may be

accularger pendulous bodies

and more swiftly moved. Lastly, since it is the opinion of some that there is a certain ethereal medium extremely rare and subtile, which freely pervades the pores of all bodies; and from such a medium, so pervading the pores of bodies, some resistance must needs arise; in order to try whether the resistance, Avhich we experience in bodies in motion, be made upon their outward surfaces only, or whether their internal parts meet with any considerable resistance upon their surfaces, I thought of the following experiment. I suspended a round deal box by a thread 11 feet long, on a steel hook, by means of a ring of the same metal, so as to make a pendulum of the aforesaid length. The hook had a sharp hollow edge on its upper part, so that the upper arc of the ring pressing on the edge might move the more freely; and the thread was fastened to the lower arc of the ring. The pendulum being thus prepared, I drew it aside from the perpendicular to the distance of about 6 feet, and that in a plane perpendicular to the edge of the hook, lest the ring, while the pendulum oscillated, should slide to and fro on the edge of the hook; for the point of suspension, in which the ring touches the hook, ought to remain immovable. I therefore accurately noted the place to which the pendulum was brought, and letting it go, I marked three other places, to which it returned at the end of the 1st, 2d, and 3d oscillation. This I

often repeated, that I might find those places as accurately as possible. Then the box A\'ith lead and other heavy metals that were near at hand. But

I filled

weighed the box when empty, and that part of the thread that went round it, and half the remaining part, extended between the hook and the suspended box; for the thread so extended always acts upon the pendulum, when drawn aside from the perpendicular, with half its weight. To this weight I added the weight of the air contained in the box. And this whole weight was about y^8 of the weight of the box when filled ^\ith the metals. Then because the box when full of the metals, by extending the thread with its weight, increased the length of the pendulum, I shortened the thread so as to make the length of the pendulum, when oscillating, the same as before. Then draAnng aside the pendulum to the place first marked, and letting it go, I reckoned about 77 oscillations before the box returned to the second mark, and as many afterwards before it came to the third mark, and as many after that before it came to the fourth mark. From this I conclude that the whole resistance of the box, when full, had not a greater proportion to the resistance of the box, when empty, than 78 to 77. For if their resistances were equal, the box, when full, by reason of its inertia, which was 78 times greater than the inertia of the same when empty, ought to have continued its oscillating motion so much the longer, and therefore to have returned to those marks at the end of 78 oscillations. But it returned to them at the end of 77 oscillations. Let, therefore, A represent the resistance of the box upon its external surface, and B the resistance of the empty box on its internal surface, and if the resistfirst, I

ances to the internal parts of bodies equally swift be as the matter, or the number of particles that are resisted, then 78B will be the resistance made to the internal parts of the box, when full; and therefore the whole resistance of the empty box mil be to the whole resistance A4-78B of the full box as 77 to 78, and, by subtraction, to 77B as 77 to 1; and thence to

A+B

A+B

A+B

The Motion of Bodies by subtraction, again, A to B as 5928

Book

B

219

II:

Therefore the resistance of the empty box in its internal parts will be above 5000 times less than the resistance on its external surface. This reasoning depends upon the supposition that the greater resistance of the full box arises not from any other latent cause, but only from the action of some subtile fluid upon the included as 77-77 to

1,

and,

to

1.

metal.

by memory, the paper being lost in which I had described it; so that I have been obhged to omit some fractional parts, which are shpped out of my memory; and I have no leisure to try it again. The first time I made it, the hook being weak, the full box was retarded sooner. The cause I found to be, that the hook was not strong enough to bear the weight of the box; so that, as it oscillated to and fro, the hook was bent sometimes this and sometimes that way. I therefore procured a hook of sufficient strength, so that the point of suspension might remain unmoved, and then all things hapThis experiment

pened as

is

is

related

above described.

SECTION VII The motion of

fluids,

and the resistance made to projected bodies

Proposition

32.

Theorem

26

Suppose two similar systems of bodies consisting of an equal number of particles, and let the correspondent particles be similar and proportional, each in one system to each in the other, and have a like situation among themselves, and the same given ratio of density to each other; and let them begin to move among themselves in proportional times, and with like motions {that is, those in one system among one another, and those in the other among one another). And if the particles that are in the same system do not touch one another, except in the moments of reflection; nor nor repel eo^h other, except with accelerative forces that are inversely as the diameters of the correspondent particles, and directly as the squares of the velocities: I say, that the particles of those systems will continue to move among themselves attract,

with like motions and in proportional times. Like bodies in like situations are said to be

moved among themselves ^ith

motions and in proportional times, when their situations at the end of those times are always found alike in respect of each other; as suppose we compare the particles in one system wdth the correspondent particles in the other. Hence the times ^\'ill be proportional, in which similar and proportional parts of similar figures \\\\\ be described by correspondent particles. Therefore if we suppose two systems of this kind, the correspondent particles, by reason of the simihtude of the motions at their beginning, %nll continue to be moved A\dth Uke motions, so long as they move A\dthout meeting one another; for if they are acted on by no forces, they \nll go on uniformly in right lines, by the first Law. But if they agitate one another \nth some certain forces, and those forces are inversely as the diameters of the correspondent particles and directly as the squares of the velocities, then, because the particles are in like situations, and their forces are proportional, the whole forces with which correspondent particles are agitated, and which are compounded of each of the agitating forces (by Cor. ii of the Laws), will have like directions, and have the same effect as like

Mathematical Principles

220

they respected centres places alike among the particles; and those whole forces will be to each other as the several forces which compose them, that is, inversely as the diameters of the correspondent particles and directly as the squares of the velocities: and therefore will cause correspondent particles to continue to describe like figures. These things mil be so (by Cor. i and viii, Prop. 4, Book i), if those centres are at rest; but if they are moved, yet, by reason of the similitude of the translations, their situations among the particles of the system \\i\\ remain similar, so that the changes introduced into the figures described by the particles will still be similar. So that the motions of correspondent and similar particles will continue similar till their first meeting \vith each other; and thence will arise similar coUisions, and similar reflections; which will again beget similar motions of the particles among themselves (by what was just now shown), till they mutually fall upon one another again, and q.e.d. so on ad infinitum. similar in like situations to which are and Cor. i. Hence if any two bodies, the correspondent particles of the systems, begin to move amongst them in like manner and in proportional times, and their magnitudes and densities be to each other as the magnitudes and densities of the corresponding particles, these bodies will continue to be moved in like manner and in proportional times; for the case of the greater parts of both systems and of the particles is the very same. Cor. II. And if all the similar and similarly situated parts of both systems be at rest among themselves; and two of them, which are greater than the rest, and mutually correspondent in both systems, begin to move in lines alike posited, \vith any similar motion whatsoever, they will excite similar motions in the rest of the parts of the systems, and will continue to move among those if

parts in like manner and in proportional times; and will therefore describe spaces proportional to their diameters.

Proposition

33.

Theorem

27

The same things being supposed, I say, that the greater parts of the systems are resisted in a ratio compounded of the squared ratio of their velocities, and the squared ratio of their diameters, and the simple ratio of the density of the parts of the systems.

For the resistance arises partly from the centripetal or centrifugal forces with which the particles of the system act on each other, partly from the collisions and reflections of the particles and the greater parts. The resistances of the first kind are to each other as the whole motive forces from which they arise, that is, as the whole accelerative forces and the quantities of matter in corresponding parts; that is (by the supposition), directly as the squares of the velocities and inversely as the distances of the corresponding particles, and directly as the quantities of matter in the correspondent parts: and therefore since the distances of the particles in one system are to the correspondent distances of the particles in the other, as the diameter of one particle or part in the former

system to the diameter of the correspondent particle or part in the other, and since the quantities of matter are as the densities of the parts and the cubes of the diameters, the resistances are to each other as the squares of the velocities and the squares of the diameters and the densities of the parts of the systems. Q.E.D.

The

resistances of the latter sort are as the

number

of correspondent

Book reflections

and the

The Motion of Bodies

II:

221

forces of those reflections conjointly; but the

number

of the

reflections are to each other directly as the velocities of the corresponding parts

and inversely as the spaces between their reflections. And the forces of the reflections are as the velocities and the magnitudes and the densities of the corresponding parts conjointly; that is, as the velocities and the cubes of the diameters and the densities of the parts. And, joining all these ratios, the resistances of the corresponding parts are to each other as the squares of the velocities and the squares of the diameters and the densities of the parts conjointly. Q.E.D.

Therefore if those systems are two elastic fluids, like our air, and their parts are at rest among themselves; and two similar bodies proportional in magnitude and density to the parts of the fluids, and similarly situated among those parts, be in any manner projected in the direction of lines similarly posited; and the accelerative forces ^\^th which the particles of the fluids act upon each other are inversely as the diameters of the bodies projected and directly as the squares of their velocities those bodies will excite similar motions in the fluids in proportional times, and ^vill describe similar spaces and proportional to their diameters. CoR. II. Therefore in the same fluid a projected body that moves swiftly meets with a resistance that is as the square of its velocity, nearly. For if the forces %\ith which distant particles act upon one another should be augmented as the square of the velocity, the projected body would be resisted in the same squared ratio accurately; and therefore in a medium, whose parts when at a distance do not act with any force on one another, the resistance is as the square of the velocity, accurately. Let there be, therefore, three mediums A, B, C, consisting of similar and equal parts regularly disposed at equal distances. Let the parts of the mediums A and B recede from each other ^\ith forces that are among themselves as T and V; and let the parts of the medium C be entirely destitute of any such forces. And if four equal bodies D, E, F, G move in these mediums, the two first and E in the two first A and B, and the other two F and G in the third C and if the velocity of the body be to the velocity of the body E, and the velocity of the body F to the velocity of the body G, as the square root of the ratio of the force T to the force V; then the resistance of the body to the resistance of the body E, and the resistance of the body F to the resistance of the body G, ^^ill be as the square of the velocities; and therefore the resistance of the body mil be to the resistance of the body F as the and resistance of the body E to the resistance of the body G. Let the bodies F be equally swift, as also the bodies E and G; and, augmenting the velocities of the bodies and F in any ratio, and diminishing the forces of the particles of the medium B as the square of the same ratio, the medium B will approach to the form and condition of the medium C at pleasure; and therefore the resistances of the equal and equally s^\ift bodies E and G in these mediums mil continually approach to equality, so that their difference mil at last become and F are less than any given. Therefore since the resistances of the bodies to each other as the resistances of the bodies E and G, those will also in like and F, when manner approach to the ratio of equality. Therefore the bodies they move with very great smftness, meet mth resistances very nearly equal; and therefore since the resistance of the body F is in a squared ratio of the velocity, the resistance of the body mil be nearly in the same ratio.

Cor.

I.

;

D

D

;

D

D

D

D

D

D

D

Mathematical Principles

222

Hence the resistance of a body moving very s\\dftly in an elastic fluid is almost the same as if the parts of the fluid were destitute of their centrifugal forces, and did not fly from each other; provided only that the elasticity of the fluid arise from the centrifugal forces of the particles, and the velocity be so great as not to allow the particles time enough to act. Cor. IV. Since the resistances of similar and equally swift bodies, in a medium whose distant parts do not fly from each other, are as the squares of the diameters, therefore the resistances made to bodies moving with very great and Cor.

III.

equal velocities in an elastic fluid

be as the squares of the diameters, nearly. Cor. v. And since similar, equal, and equally swift bodies, moving through mediums of the same density, whose particles do not fly from each other, will strike against an equal quantity of matter in equal times, whether the particles of which the medium consists be more and smaller, or fewer and greater, and therefore impress on that matter an equal quantity of motion, and in return (by the third Law of Motion) suffer an equal reaction from the same, that is, are equally resisted; it is manifest, also, that in elastic fluids of the same density, when the bodies move mth extreme swiftness, their resistances are nearly equal, whether the fluids consist of gross parts, or of parts ever so subtile. For the resistance of projectiles moving with exceedingly great celerities is not much diminished by the subtilty of the medium. Cor. VI. All these things are so in fluids whose elastic force takes its rise from the centrifugal forces of the particles. But if that force arise from some other cause, as from the expansion of the particles after the manner of wool, or the boughs of trees, or any other cause, by which the particles are hindered from moving freely among themselves, the resistance, by reason of the lesser fluidity of the medium, will be greater than in the Corollaries above. \vill

Proposition

34.

Theorem 28

If in a rare medium, consisting of equal particles freely disposed at equal distances from each other, a globe and a cylinder described on equal diameters move with equal velocities in the direction of the axis of the cylinder, the resistance of the globe will be but half as great as that of the cylinder. For since the action of the medium upon the body is the same (by Cor. v of the Laws) whether the body move in a quiescent medium, or whether the particles of the medium impinge with the same velocity upon the quiescent body, let us consider the body as if it were quiescent, and see with what force it would be impelled by the moving medium. G Let, therefore, ABKI represent a spherical body described from the centre C with the semidiameter CA, and let the particles of the medium impinge with a given velocity upon that spherical body in the directions of right lines parallel to

AC; and

those right

In

lines.

FB

FB be one of LB equal to the

let

take

BD touching the KC and BD let fall the

semidiameter CB, and draw

sphere in B. Upon perpendiculars BE, LD; and the force with which a particle of the medium, impinging on the globe obliquely in the direction FB, would strike the globe in B, \dll be to the force with which the same particle, meeting the cylinder

ONGQ

Book

II

:

The Motion of Bodies

223

described about the globe uith the axis ACI, would strike it perpendicularly in b, as LD is to LB, or BE to BC. Again; the efficacy of this force to move the globe, according to the direction of its incidence FB or AC, is to the efficacy of the same to move the globe, according to the direction of its determination, that is, in the direction of the right line BC in which it impels the globe directly, as BE to BC. And, joining these ratios, the efficacy of a particle, falling upon the globe obliquely in the direction of the right Hne FB, to move the globe in the direction of its incidence, is to the efficacy of the same particle falling in the same line perpendicularly on the cylinder, to move it in the same direction, as BE^ to BC^ Therefore if in 6E, which is perpendicular to the circular base of the cylinder

6H

NAO, and

equal to the radius AC,

we take 6H equal

BE^ CB'

to -p;—

be to 6E as the effect of the particle upon the globe to the effect upon the cylinder. And therefore the soUd which is formed by the right lines 6H \vill be to the solid formed by all the right lines 6E as the

then

\vill

of the particle all

upon the globe to the effect of all the particles upon But the former of these soHds is a paraboloid whose vertex is C, its axis CA, and latus rectum CA, and the latter solid is a cylinder circumscribing the paraboloid; and it is kno\vn that a paraboloid is half its circumscribed cylinder. Therefore the whole force of the medium upon the globe is half the entire force of the same upon the cyhnder. And therefore if the particles of the medium are at rest, and the cylinder and globe move with equal velocities, effect of all the particles

the cyhnder.

the resistance of the globe will be half the resistance of the cyhnder.

q.e.d.

Scholium the same method other figures may be compared together as to their resistance; and those may be found which are most apt to continue their motions in resisting mediums. As if upon the circular Q base from the centre O, with the radius OC, and the altitude OD, one would construct a frustum CBGF of a cone, which should meet with less resistance than any other frustum constructed with the o same base and altitude, and going forwards towards in the direction of its axis bisect the altitude in Q, and produce OQ to S, so that QS may be equal to QC, and S \vill be the vertex of the cone whose frustum is sought.

By

CEBH

D

:

D

OD

Mathematical Principles

224

provided that both move forwards in the direction of their axis AB, and that the extremity B of each go foremost. This Proposition I conceive may be of use in the building of ships. If the figure be such a curve, that if, from any point thereof, as N, be let fall on the axis AB, and from the given point G the perpendicular parallel to a right hne touching the figure in there be drawn the right Une N, and cutting the axis produced in R, becomes to as OR' to 4BR GB^, the solid described by the revolution of this figure about its axis AB, moving in the before-mentioned rare medium from A towards B, will be less resisted than any other circular solid whatsoever, described of the same length and breadth.

former

solid,

DNFG

NM

OR

Proposition

MN

35.

OR

Problem

•

7

// a rare medium consist of very small quiescent particles of equal magnitudes, and freely disposed at equal distances from one another: to find the resistance of a globe moving uniformly forwards in this medium. Case 1. Let a cylinder described with the same diameter and altitude be conceived to go forwards with the same velocity in the direction of its axis through the same medium; and let us suppose that the particles of the medium, on which the globe or cylinder falls, fly back with as great a force of reflection as possible. Then since the resistance of the globe (by the last Proposition) is but half the resistance of the cyhnder, and since the globe is to the cylinder as 2 to 3, and since the cylinder by falling perpendicularly on the particles, and reflecting them with the utmost force, communicates to them a velocity double to its own it follows that the cylinder in moving forwards uniformly half the length of its axis, will communicate a motion to the particles which is to the whole motion of the cylinder as the density of the medium to the density of the cylinder; and that the globe, in the time it describes one length of its diameter in moving uniformly forwards, will communicate the same motion to the particles; and, in the time that it describes two-thirds of its diameter, will communicate a motion to the particles which is to the whole motion of the globe as the density of the medium to the density of the globe. And therefore the globe meets with a resistance, which is to the force by which its whole motion may be either taken away or generated in the time in which it describes two-thirds of its diameter moving uniformly forwards, as the density of the medium is to the density of the globe. Case 2. Let us suppose that the particles of the medium incident on the globe or cylinder are not reflected; and then the cylinder falling perpendicularly on the particles will communicate its own simple velocity to them, and therefore meets a resistance but half so great as in the former case, and the globe also meets with a resistance but half so great. Case 3. Let us suppose the particles of the medium to fly back from the globe with a force which is neither the greatest, nor yet none at all, but with a certain mean force; then the resistance of the globe will be in the same mean ratio between the resistance in the first case and the resistance in the second. :

Q.E.I.

Hence

the globe and the particles are infinitely hard, and destitute and therefore of all force of reflection, the resistance of the globe will be to the force by which its whole motion may be destroyed or

Cor.

I.

if

of all elastic force,

Book

II

:

The Motion of Bodies

225

generated, in the time that the globe describes four third parts of its diameter, as the density of the medium is to the density of the globe. Cor. II. The resistance of the globe, other things being equal, varies as the

square of the velocity.

Cor.

III.

The

resistance of the globe, other things being equal, varies as the

square of the diameter.

The resistance of the globe, other things being equal, varies as the density of the medium. CoR. V. The resistance of the globe varies jointly as the square of the velocity, as the square of the diameter, and as the density of the medium. CoR. VI. The motion of the globe and its resistance may be thus represented. Let AB be the time in which the globe may, by its resistance uniformly continued, lose its whole motion. Erect AD, BC perpendicular to AB. Let BC be that whole motion, and through the point C, the asymptotes being AD, AB, CoR.

IV.

CF. Produce

AB

to any point E. Erect the perpendicular EF meeting the hyperbola in F. Complete the parallelogram CBEG, and draw AF meeting BC in H. Then if the globe in describe the hyperbola

any time BE, with

motion

BC

uniformly continued, describes in a nonresisting medium the space CBEG represented by the area of the parallelogram, the same in a resisting medium mil describe the space CBEF, represented by the area of the hyperbola; and its motion at the end of that time will be represented by EF, the ordinate of the hyperbola, there being lost of its motion the part FG. And its resistance at the end of the same time will be represented by the length BH, there being lost of its resistance the part CH. All these things appear by Cor. i and iii. Prop. 5, Book ii. CoR. VII. Hence if the globe in the time T by the resistance R uniformly continued to lose its whole motion M, the same globe in the time t in a resisting medium, wherein the resistance R decreases as the square of the velocity, will lose out of its

motion

describe a space which

uniform motion

M,

M is

the part

^M Tfrj-^)

its first

the part

TM

^=^-—-

to the space described in the

as the logarithm of the

number

remaining; and will

same time

t,

with the

multiplied

by the

number 2.302585092994 is to the number — because the hyperbolic area BCFE ,

is

to the rectangle

BCGE in that

proportion.

Scholium and retardation of spherical projectiles in mediums that are not continued, and shown that this resistance is to the force by which the whole motion of the globe may be destroyed or produced in the time in which the globe can describe two-thirds of its diameter, I

have exhibited in

this Proposition the resistance

with a velocity uniformly continued, as the density of the medium is to the density of the globe, provided the globe and the particles of the medium be perfectly elastic, and are endued with the utmost force of reflection; and that this force, where the globe and particles of the medium are infinitely hard and void of any reflecting force, is diminished one-half. But in continued mediums.

Mathematical Principles

226

quicksilver, the globe as it passes through them does not against all the particles of the fluid that generate the resistimmediately strike ance made to it, but presses only the particles that lie next to it, which press the particles beyond, which press other particles, and so on; and in these

as water, hot

oil,

and

mediums the resistance is diminished one other half. A globe in these extremely fluid mediums meets ^^^th a resistance that is to the force by which its whole motion may be destroyed or generated in the time wherein it can describe, with that motion uniformly continued, eight third parts of its diameter, as the density of the medium is to the density of the globe. This I shall endeavor to show in what follows. Proposition 36. Problem 8

To find

the

motion of water running out oj a cylindrical

vessel

through a hole

made

at the bottom.

ACDB

CD

vessel, AB the mouth of it, the bottom a circular hole in the middle of the bottom, G the the axis of the cylinder perpendicular to the horizon. centre of the hole, and And suppose a cylinder of ice APQB to be of the same breadth mth the cavity of the vessel, and to have the same axis, and to descend continually with an uniform motion, and that its parts, as soon as they touch the surface AB, dissolve into water, and flow down by their weight into the vessel, and in their fall compose the cataract or column of water ABNFEM, passing through the hole EF, and filHng up the same exactly. Let the uniform velocity of the descending ice and of the contiguous water in the circle AB be that which the water would acquire by falling through the space IH and let IH lie in the same right line; and through the and point I let there be drawn the right line KL parallel to the horizon, and meeting the ice on both the sides thereof in and L. Then the velocity of the water running out at the hole EF will be the same that it would acquire by falling from I through the space IG. Therefore, by Galileo's Theorems, IG mil be to IH as the square of the velocity of the water that runs out at the hole to the velocity of the water in the circle AB, that is, as the square of the ratio of the circle AB to the circle EF; those circles being inversely as the velocities of the water which in the same time and in equal quantities passes through each of them, and completely fills them both. We are now considering the velocity \vith which the water tends to the plane of the horizon. But the motion parallel to the same, by which the parts of the falling water approach to each other, is not here taken notice of; since it is neither produced by gravity, nor at all changes the motion perpendicular to the horizon which the gravity produces. We suppose, indeed, that the parts of the water cohere a little, that by their cohesion they may in falling approach to each other with motions parallel to the horizon in order to form one single cataract, and to prevent their being divided into several; but the motion parallel to the horizon arising from this cohesion does not come under our present

Let

be a cylindrical

parallel to the horizon,

EF

GH

;

HG

K

consideration.

Case

1.

Conceive

falhng water

now

ABNFEM,

the whole cavity in the vessel, which surrounds the may pass through

to be full of ice, so that the water

Book

II

:

The Motion of Bodies

227

the ice as through a funnel. Then if the water pass very near to the ice only, without touching it; or, which is the same thing, if by reason of the perfect smoothness of the surface of the ice, the water, though touching it, glides over it \vith the utmost freedom, and Avithout the least resistance; the water Anil run through the hole EF Avith the same velocity as before, and the whole weight of will be taken up as before in forcing out the the column of water water, and the bottom of the vessel will sustain the weight of the ice surrounding that column. Let now the ice in the vessel dissolve into water; but the efflux of the water will remain, as to its velocity, the same as before. It will not be less, because the ice now dissolved mil endeavor to descend; it will not be greater, because the ice, now become water, cannot descend without hindering the descent of other water equal to its own descent. The same force ought always to generate the same velocity in the effluent water. But the hole at the bottom of the vessel, by reason of the obhque motions of the particles of the effluent water, must be a little greater than before. For now the particles of the water do not all of them pass through the hole perpendicularly, but, flowing down on all parts from the sides of the vessel, and converging towards the hole, pass through it with obhque motions; and in tending downwards they meet in a stream whose diameter is a little smaller below the hole than at the hole itself; its diameter being to the diameter of the hole as 5 to 6, or as 53^ to Q}^, very nearly, if I measured those diameters rightly. I procured a thin flat plate, having a hole pierced in the middle, the diameter of the circular hole being five eighth parts of an inch. And that the stream of running water might not be accelerated in falling, and by that acceleration become narrower, I fixed this plate not to the bottom, but to the side of the vessel, so as to make the water go out in the direction of a line parallel to the horizon. Then, when the vessel was full of water, I opened the hole to let it run out; and the diameter of the stream, measured with great accuracy at the distance of about half an inch from the hole, was ^/^o of an inch. Therefore the diameter of this circular hole was to the diameter of the stream very nearly as 25 to 21. So that the water in passing through the hole converges on all sides, and, after it has run out of the vessel, becomes smaller by converging in that manner, and by becoming smaller is accelerated till it comes to the distance of half an inch from the hole, and at that distance flows in a smaller stream and with greater celerity than in the hole itself, and this in the ratio of 25 25 to 21-21, or 17 to 12, very nearly; that is, in about the ratio of \/2 to 1. Now it is certain from experiments, that the quantity of water running out in a given time through a circular hole made in the bottom of a vessel is equal to the quantity, which, flowing freely with the aforesaid velocity, would run out in the same time through another circular hole, whose diameter is to the diameter of the former as 21 to 25. And therefore this running water in passing through the hole itself has a velocity doAvnwards nearly equal to that which a heavy

ABNFEM

•

body would acquire in falling through half the height of the stagnant water in the vessel. But then, after it has run out, it is still accelerated by converging, till it arrives at a distance from the hole that is nearly equal to its diameter, and acquires a velocity greater than the other in about the ratio of \/2 to 1 this velocity a heavy body would nearly acquire by faUing freely through the whole height

of the stagnant

water in the

vessel.

Mathematical Principles

228

Therefore in what follows let the diameter of the stream be represented by above that lesser hole which we shall call EF. And imagine another plane placed a distance equal at the hole EF, and parallel to the plane thereof, to be to the diameter of the same hole, and to be pierced through with a greater hole ST, of such a magnitude that a stream which will exactly fill the lower hole EF may pass through it; the diameter of this hole will therefore be to the diameter of the lower hole nearly as 25 to 21. By this means the water will run perpendicularly out at the lower hole; and the quantity of the water running

VW

out will be, according to the magnitude of this last hole, very nearly the same as that which the solution of the Problem requires. The space included between the two planes and the falling stream may be con-

make

the solution more simple and alone for the bottom of the plane mathematical, it is better to take the lower vessel, and to suppose that the water which flowed through the ice as through a funnel, and ran out of the vessel through the hole EF made in the lower plane, preserves its motion continually, and that the ice continues at rest. Therefore in what follows let ST be the diameter of a circular hole described from the centre Z, and let the stream run out of the vessel through that hole, when the water in the vessel is all fluid. And let EF be the diameter of the hole, which the stream, in falUng through, exactly fills up, whether the water runs out of the vessel by that upper hole ST, or flows through the middle of the ice in the vessel, as through a funnel. And let the diameter of the upper hole ST be to the diameter of the lower EF as about 25 to 21, and let the perpendicular distance between the planes of the holes be equal to the diameter of the lesser hole EF. Then the velocity of the water downwards, in running out of the sidered as the

bottom

of the vessel.

But

to

vessel through the hole ST, will be in that hole the

same that a body may

ac-

from half the height IZ; and the velocity of both the falling streams will be in the hole EF, the same which a body would acquire by falling freely from the whole height IG. Case 2. If the hole EF be not in the middle of the bottom of the vessel, but in some other part thereof, the water will still run out with the same velocity as before, if the magnitude of the hole be the same. For though a heavy body takes a longer time in descending to the same depth, by an obUque fine, than by a perpendicular fine, yet in both cases it acquires in its descent the same velocity; as Galileo hath demonstrated. Case 3. The velocity of the water is the same when it runs out through a hole in the side of the vessel. For if the hole be small, so that the interval between the surfaces AB and KL may vanish as to sense, and the stream of water horizontally issuing out may form a parabolic figure; from the latus rectum of this parabola one may see, that the velocity of the effluent water is that which a body may acquire by falling the height IG or HG of the stagnant water in the vessel. For, by making an experiment, I found that if the height of the stagnant water above the hole were 20 inches, and the height of the hole above a plane parallel to the horizon were also 20 inches, a stream of water springing out from thence would fall upon the plane, at the distance of very nearly 37 inches, from a perpendicular let fall upon that plane from the hole. For Avithout resistance quire

by

falling freely

Book

II

:

The Motion of Bodies

229

the stream would have fallen upon the plane at the distance of 40 inches, the latus rectum of the parabolic stream being 80 inches. Case 4. If the effluent water tend upwards, it will still issue forth with the same velocity. For the small stream of water springing upwards, ascends with or GI, the height of the stagnant water in the a perpendicular motion to vessel; except so far as its ascent is hindered a little by the resistance of the air; and therefore it springs out with the same velocity that it would acquire in falling from that height. Every particle of the stagnant water is equally pressed on all sides (by Prop. 19, Book ii), and, yielding to the pressure, tends always with an equal force, whether it descends through the hole in the bottom of the vessel, or gushes out in an horizontal direction through a hole in the side, or passes into a canal, and springs up from thence through a little hole made in the upper part of the canal. And it may not only be inferred from reasoning, but is manifest also from the well-known experiments just mentioned, that the velocity with which the water runs out is the very same that is assigned in this Proposition. Case 5. The velocity of the effluent water is the same, whether the figure of the hole be circular, or square, or triangular, or of any other figure whatever equal to the circular; for the velocity of the effluent water does not depend upon the figure of the hole, but arises from such depth of the hole as it may have below the plane KL. Case 6. If the lower part of the vessel be immersed into stagnant water, and the height of the stagnant water above the bottom of the vessel be GR, the velocity with which the water that is in the

GH

ABDC

EF into the stagnant be the same which the water would acquire by falling from the height IR; for the weight of all the water in the vessel that is below the surface of the stagnant water will be sustained in equilibrium by the weight of the stagnant water, and therefore does not at all accelerate the motion of the descending water in the vessel. This case will also become evident from experiments, measuring the times in which the water will run out. Cor. i. Hence if CA, the depth of the water, be produced to K, so that AK may be to CK as the square of the ratio of the area of a hole made in any part of the bottom to the area of the circle AB, the velocity of the effluent water will be equal to the velocity which the water would acquire by falHng freely from vessel will run out at the hole

water

the height

Cor.

II.

will

KC.

And the force with which the whole motion of the effluent water may

be generated

is equal to the weight of a cylindric column of water, whose base the hole EF, and its altitude 2GI or 2CK. For the effluent water, in the time it becomes equal to this column, may acquire, by falling by its own weight from the height GI, a velocity equal to that with which it runs out. is

Cor. III. The weight of all the water in the vessel ABDC is to that part of the weight which is employed in forcing out the water as the sum of the circles AB and EF is to twice the circle EF. For let 10 be a mean proportional between IH and IG, and the water running out at the hole EF will, in the time that a drop falling from I would describe the altitude IG, become equal to a

Mathematical Principles

230

cylinder whose base is the circle EF and its altitude 2IG, that is, to a cylinder whose base is the circle AB, and whose altitude is 210. For the circle EF is to the circle AB as the square root of the ratio of the altitude IH to the altitude IG; that is, in the simple ratio of the mean proportional 10 to the altitude IG.

Moreover, in the time that a drop falling from I can describe the altitude IH, the water that runs out will have become equal to a cylinder whose base is the circle AB, and its altitude 21 H; and in the time that a drop falling from I through to G describes HG, the difference of the altitudes, the effluent water, that is, the water contained within the solid ABNFEM, will be equal to the difference of the cylinders, that is, to a cylinder whose base is AB, and its altitude 2H0. And therefore all the water contained in the vessel ABDC is to is to 2H0, as the whole falling water contained in the said solid 2IH. the weight of all the water 2H0, or IH IO to But that is, as HO + OG to is employed in forcing out the water; and therefore the in the solid weight of all the water in the vessel is to that part of the weight that is employed in forcing out the water as IH IO is to 2IH, and therefore as the sum of the circles EF and AB is to twdce the circle EF. Cor. IV. And hence the weight of all the water in the vessel ABDC is to the other part of the weight which is sustained by the bottom of the vessel as the sum of the circles AB and EF is to the difference of the same circles. Cor. v. And that part of the weight which the bottom of the vessel sustains is to the other part of the weight employed in forcing out the water as the difference of the circles AB and EF is to twice the lesser circle EF, or as the area of the bottom to twice the hole. Cor. VI. That part of the weight which presses upon the bottom is to the whole weight of the water perpendicularly incumbent thereon as the circle AB is to the sum of the circles AB and EF, or as the circle AB is to the excess of twdce the circle AB above the area of the bottom. For that part of the weight which presses upon the bottom is to the weight of the whole water in the vessel as the difference of the circles AB and EF is to the sum of the same circles (by Cor. iv) and the weight of the whole water in the vessel is to the weight of the whole water perpendicularly incumbent on the bottom as the circle AB is to the difference of the circles AB and EF. Therefore, multiplying together corresponding terms of the two proportions, that part of the weight which presses upon the bottom is to the weight of the whole water perpendicularly incumbent thereon as the circle AB to the sum of the circles AB and EF, or the excess of twice the circle AB above the bottom. Cor. vii. If in the middle of the hole EF there be placed the little circle PQ described about the centre G, and parallel to the , horizon, the weight of water which that little circle iH B sustains is greater than the weight of a third part of a cylinder of water whose base is that little circle be the catand its height GH. For let aract or column of falling water whose axis is GH, M N as above, and let all the water, whose fluidity is not requisite for the ready and quick descent of the water, be supposed to be congealed, as well round about the cataract, as above the little circle. And let PHQ be the column of water congealed above E pgq f d c

H

ABNFEM

HG

+

ABNFEM

+

;

'

;

ABNFEM

Book

II:

The Motion of Bodies H, and its altitude GH. And

231

the little circle, whose vertex is suppose this cataract to fall with its whole weight downwards, and not in the least to lie against or to press PHQ, but to glide freely by it without any friction, unless, perhaps, just at the very vertex of the ice, where the cataract at the beginning of its fall may tend to a concave figure. And as the congealed water AMEC, BNFD, lying round the cataract, is convex in its internal surfaces AME, towards the falHng cataract, so this column will be convex towards the cataract

BNF

PHQ

also, and be greater than a cone whose base is that little circle PQ and its altitude GH; that is, greater than a third part of a cylinder described with the same base and altitude. Now that Uttle circle sustains the

will therefore

weight of this column, that is, a weight greater than the weight of the cone, or a third part of the cyhnder. CoR. VIII. The weight of water which the circle PQ, when very small, sustains, seems to be less than the weight of two-thirds of a cylinder of water whose base is that little circle, and its altitude HG. For, things standing as above supposed, imagine the half of a spheroid described whose base is that little circle, and its semiaxis or altitude HG. This figure will be equal to twothirds of that cylinder, and will comprehend within it the column of congealed water PHQ, the weight of which is sustained by that httle circle. For though the motion of the water tends directly downwards, the external surfaces of that column must yet meet the base PQ in an angle somewhat acute, because the water in its fall is continually accelerated, and by reason of that acceleration becomes narrower. Therefore, since that angle is less than a right one, this column in the lower parts thereof mil lie within the hemispheroid. In the upper parts also it will be acute or pointed; because to make it otherwise, the horizontal motion of the water must be at the vertex infinitely more swdft than its motion towards the horizon. And the less this circle PQ is, the more acute will the vertex of this column be; and the circle being diminished in infinitum, the angle PHQ -will be diminished in infinitum, and therefore the column will lie within the hemispheroid. Therefore that column is less than that hemispheroid, or than two third parts of the cylinder whose base is that little circle, and its altitude GH. Now the little circle sustains a force of water equal to the weight of this column, the weight of the ambient water being employed in causing its efflux out at the hole. CoR. IX. The weight of water which the little circle PQ sustains, when it is very small, is very nearly equal to the weight of a cylinder of water whose base is that little circle, and its altitude 3^GH; for this weight is an arithmetical mean between the weights of the cone and the hemispheroid above mentioned. But if that little circle be not very small, but on the contrary increased till it be equal to the hole EF, it will sustain the weight of all the water lying perpendicularly above it, that is, the weight of a cyhnder of water whose base is that httle circle,

and

its altitude

GH.

CoR. X. And (as far as I can judge) the weight which this little circle sustains always to the weight of a cylinder of water whose base is that little circle, and its altitude 3^GH, as EF^ is to EF2-3/^PQ2, or as the circle EF is to the excess of this circle above half the little circle PQ, very nearly. is

Mathematical Principles

232

Lemma

4

If a cylinder moves uniformly forwards in the direction of its length, the resistance made thereto is not at all changed by augmenting or diminishing that length; and is therefore the same with the resistance of a circle, described with the same diameter, and moving forwards with the same velocity in the direction of a right line perpendicular

to its

plane.

opposed to the motion; and a cylinder becomes a diminished in infinitum.

For the sides are not at circle

when

its

length

is

all

Proposition

37.

Theorem

29

// a cylinder moves uniformly forwards in a compressed, fluid, in the direction of its length, the resistance arising

infinite,

from

transverse section is to the force by which its whole motion

the

and nonelastic

magnitude of

may

its

be destroyed or

it moves four times its length, as the density of the medensity of the cylinder, nearly. touch the surface of stagnant water with its bottom let the vessel

generated, in the time that

dium

is to the

For CD, and

ABDC

let the water run out of this vessel into the stagnant water through the cylindric canal EFTS perpendicular to the hori. I k l zon and let the little circle PQ be placed parallel to A the horizon anywhere in the middle of the canal; may be to and produce CA to K, so that as the square of the ratio, which the excess of the orifice C of the canal EF above the little circle PQ bears to the circle AB. Then it is manifest (by Case 5, Case 6, and Cor. i. Prop. 36) that the velocity of the water passing through the annular space between the little circle and the sides of the vessel will be the very same as that which the water would acquire by fallor IG. ing, and in its fall describing the altitude And (by Cor. x, Prop. 36) if the breadth of the vessel be infinite, so that the become equal; the force of short line HI may vanish, and the altitudes IG, the water that flows down and presses upon the circle will be to the weight of a cylinder whose base is that little circle, and the altitude 3^IG, as EF^ is to EF^ — 3^PQ^, very nearly. For the force of the water flowing downwards uniformly through the whole canal will be the same upon the little circle PQ in whatsoever part of the canal it be placed. Let now the orifices of the canal EF, ST be closed, and let the little circle ascend in the fluid compressed on every side, and by its ascent let it oblige the water that lies above it to descend through the annular space between the little circle and the sides of the canal. Then will the velocity of the ascending little circle be to the velocity of the descending water as the difference of the circles EF and PQ is to the circle PQ; and the velocity of the ascending little circle will be to the sum of the velocities, that is, to the relative velocity of the descending water with which it passes by the little circle in its ascent, as the difference of the circles EF and PQ is to the circle EF, or as EF2-PQ2 to EF^. Let that relative velocity be equal to the velocity with which it was shown above that the water would pass through the annular space, if the circle were to remain unmoved, that is, to the velocity which the water would acquire by faUing, and .

;

AK

CK

,

KC

HG

.

Book

II:

The Motion of Bodies

233

IG; and the force of the water upon the ascending circle will be the same as before (by Cor. v of the Laws of Motion) that is, the resistance of the ascending little circle will be to the weight of a cylinder of water whose base is that little circle, and its altitude 3^IG, as EFis to EF^ — 3^PQ^ nearly. But the velocity of the little circle will be to the velocity which the water acquires by falling, and in its fall describing the altitude IG, as EF^-PQ^ is to EFl Let the breadth of the canal be increased in infinitum; and the ratios between EF2-PQ2 and EF^, and between EF^ and EF^-i^PQ^ will become at last ratios of equality. And therefore the velocity of the little circle will now be the same as that which the water would acquire in falling, and in its fall describing the altitude IG; and the resistance will become equal to the weight of a cylinder whose base is that little circle, and its altitude half the altitude IG, from which the cylinder must fall to acquire the velocity of the ascending circle; and with this velocity the cylinder in the time of its fall will describe four times its length. But the resistance of the cylinder moving forwards with this velocity in the direction of its length is the same with the resistance of the little circle (by Lem. 4), and is therefore nearly equal to the force by which its motion may be generated while it describes four times its length. If the length of the cylinder be augmented or diminished, its motion, and the time in which it describes four times its length, will be augmented or diminished in the same ratio, and therefore the force by which the motion, so increased or diminished, may be destroyed or generated, will continue the same; because the time is increased or diminished in the same proportion; and therefore that force remains still equal to the resistance of the cylinder, because (by Lem. 4) that resistance will also remain the same. If the density of the cyHnder be augmented or diminished, its motion, and the force by which its motion may be generated or destroyed in the same time, will be augmented or diminished in the same ratio. Therefore the resistance of any cylinder whatsoever mil be to the force by which its whole motion may be generated or destroyed, in the time during which it moves four times its length, as the density of the medium is to the density of the cylinder, nearly, q.e.d. A fluid must be compressed to become continued; it must be continued and nonelastic, that all the pressure arising from its compression may be propagated in an instant; and so, acting equally upon all parts of the body moved, may produce no change of the resistance. The pressure arising from the motion of the body is spent in generating a motion in the parts of the fluid, and this creates the resistance. But the pressure arising from the compression of the fluid, be it ever so forcible, if it be propagated in an instant, generates no motion in the parts of a continued fluid, produces no change at all of motion therein; and therefore neither augments nor lessens the resistance. This is certain, that the action of the fluid arising from the compression cannot be stronger on the hinder parts of the body moved than on its fore parts, and therefore cannot lessen the resistance described in this Proposition. And if its propagation be infinitely swifter than the motion of the body pressed, it will not be stronger on the fore parts than on the hinder parts. But that action will be infinitely s^^'ifter, and propagated in an instant, if the fluid be continued and in its fall describing the altitude

nonelastic.

CoR.

I.

The

resistances,

made

to cylinders going uniformly forwards in the

234

K

Mathematical Principles

Book

II:

The Motion of Bodies

235

HG

DF

to acquire the velocity \nth which it moves, as to 3^AB. Let CF and be two other paraboUc arcs described \\ath the axis CD, and a latus rectum four times the former; and by the revolution of the figure about the axis EF let

there be generated a solid, whose middle part

speaking H'

C

'G

and

ABDC is the cylinder we are here and whose extreme parts ABE

of,

CDF contain the parts of the fluid at rest

A

among

B

hard bodies, adhering to the cylinder at each end like a head and tail. Then if this solid

F'.

•/.E

themselves, and concreted into two

EACFDB move in the direction of the length

FE

towards the parts beyond E, the resistance will be nearly the same as that which we have here determined in this Proposition; that is, it will have the same ratio to the force with which the whole motion of the cylinder may be destroyed or generated, in the time that it is describing the length 4AC with that motion uniformly continued, as the density of the fluid has to the of its axis

density of the cylinder, nearly. And (by Cor. vii, Prop. 36) the resistance must be to this force in the ratio of 2 to 3, at the least.

Lemma

5

// a cylinder, a sphere, and a spheroid, of equal breadths be placed successively in middle of a cylindric canal, so that their axes may coincide with the axis of the

the

mil equally hinder the passage of the water through the canal. For the spaces lying between the sides of the canal, and the cylinder, sphere, and spheroid, through which the water passes, are equal; and the water will pass equally through equal spaces. This is true, upon the supposition that all the water above the cylinder, sphere, or spheroid, whose fluidity is not necessary to make the passage of the water the quickest possible, is congealed, as was explained above in Cor. vii, canal, these bodies

Prop. 36.

Lemma The same supposition remaining,

6

the fore-mentioned bodies are equally acted

on by

the water flowing through the canal.

This appears by Lem. 5 and the third Law. For the water and the bodies upon each other mutually and equally.

act

Lemm.\ 7 If the water be at rest in the canal, and these bodies move with equal velocity and in opposite directions through the canal, their resistances will be equal among themselves.

This appears from the last

among

Lemma,

for the relative

motions remain the same

themselves.

Scholium

The

the same for all convex and round bodies, whose axes coincide with the axis of the canal. Some difference may arise from a greater or less friction; but in these Lemmas we suppose the bodies to be perfectly smooth, and the medium to be void of all tenacity and friction; and that those parts of the fluid which by their oblique and superfluous motions may disturb, hinder, case

is

236

Mathematical Principles

and retard the flux of the water through the canal, are at rest among themselves; being fixed like water by frost, and adhering to the force and hinder parts of the bodies in the manner explained in the Scholium of the last Proposition for in what follows we consider the very least resistance that round bodies described with the greatest given transverse sections can possibly meet with. Bodies swimming upon fluids, when they move straight forwards, cause the fluid to ascend at their fore parts and subside at their hinder parts, especially if they are of an obtuse figure and hence they meet with a little more resistance than if they were acute at the head and tail. And bodies moving in elastic fluids, if they are obtuse behind and before, condense the fluid a little more at their fore parts, and relax the same at their hinder parts; and therefore meet also with a little more resistance than if they were acute at the head and tail. But in these Lemmas and Propositions we are not treating of elastic but nonelastic fluids; not of bodies floating on the surface of the fluid, but deeply immersed therein. And when the resistance of bodies in nonelastic fluids is once known, we may then augment this resistance a little in elastic fluids, as our air; and in the surfaces of stagnating fluids, as lakes and seas. ;

;

Proposition

38.

Theorem

30

If a globe move uniformly forwards in a compressed, infinite, and nonelastic fluid, its resistance is to the force by which its whole motion may be destroyed or generated, in the time that it describes eight third parts of its diameter, as the density of the fluid is to the density of the globe, very nearly.

For the globe is to its circumscribed cylinder as 2 to 3; and therefore the force which can destroy all the motion of the cylinder, while the same cylinder is describing the length of four of its diameters, will destroy all the motion of the globe, while the globe is describing two-thirds of this length, that is, eight third parts of its own diameter. Now the resistance of the cylinder is to this force very nearly as the density of the fluid is to the density of the cylinder or globe (by Prop. 37), and the resistance of the globe is equal to the resistance of the cylinder (by Lems. 5, 6, 7). q.e.d. Cor. I. The resistances of globes in infinite compressed mediums are in a ratio compounded of the squared ratio of the velocity, and the squared ratio of the diameter, and the ratio of the density of the mediums. Cor. II. The greatest velocity, with which a globe can descend by its comparative weight through a resisting fluid, is the same as that which it may acquire by falhng with the same weight, and without any resistance, and in its fall describing a space that is to four third parts of its diameter as the density of the globe is to the density of the fluid. For the globe in the time of its fall, moving with the velocity acquired in falling, will describe a space that will be to eight third parts of its diameter as the density of the globe is to the density of the fluid; and the force of its weight which generates this motion will be to the force that can generate the same motion, in the time that the globe describes eight third parts of its diameter, with the same velocity as the density of the fluid is to the density of the globe; and therefore (by this Proposition) the force of weight will be equal to the force of resistance, and therefore cannot accelerate the globe. Cor. III. If there be given both the density of the globe and its velocity at the beginning of the motion, and the density of the compressed quiescent fluid

Book

II:

The Motion of Bodies

237

which the globe moves, there is given at any time both the velocity of the globe and its resistance, and the space described by it (by Cor. vii, Prop. 35). Cor. IV. A globe moving in a compressed quiescent fluid of the same density with itself will lose half its motion before it can describe the length of two of its in

diameters (by the same Cor.

vii).

Proposition

39.

Theorem

31

// a globe move uniformly forwards through a fluid inclosed and compressed in a

may

cylindric canal, its resistance is to the force by which its whole motion

be

generated or destroyed, in the time in which

it describes eight third parts of its in ratio compounded ratio diameter, a of the of the orifice of the canal to the excess above the greatest circle that orifice half of the globe; and the squared ratio of the of orifice of the canal to the excess of that orifice above the greatest circle of the globe;

and

the ratio of the density of the fluid to the density of the globe, nearly.

This appears by Cor.

Prop. 37, and the demonstration proceeds in the same manner as in the foregoing Proposition. ii,

Scholium In the last two Propositions we suppose (as was done before in Lem. 5) that all the water which precedes the globe, and whose fluidity increases the resistance of the same, is congealed. Now if that water becomes fluid, it will somewhat increase the resistance. But in these Propositions that increase is so small, that it may be neglected, because the convex surface of the globe produces the very same effect almost as the congelation of the water.

Proposition

40.

Problem

9

To find by experiment the resistance of a globe moving through a perfectly fluid compressed medium. Let A be the weight of the globe in a vacuum, B its weight in the resisting medium, the diameter of the globe, F a space which is to VsD as the density of the globe is to the density of the medium, that is, as A is to A — B, G the time in which the globe falling with the weight B without resistance describes the space F, and the velocity which the body acquires by that fall. Then will be the greatest velocity \vith which the globe can possibly descend with the weight B in the resisting medium, by Cor. ii, Prop. 38; and the resistance which the globe meets mth, when descending mth that velocity, mil be equal to its weight B and the resistance it meets udth in any other velocity will be to the weight B as the square of the ratio of that velocity to the greatest velocity H, by Cor. i, Prop. 38. This is the resistance that arises from the inactivity of the matter of the fluid. That resistance which arises from the elasticity, tenacity, and friction of its parts, may be thus investigated. Let the globe be let fall so that it may descend in the fluid by the weight B; and let P be the time of falling, and let that time be expressed in seconds, if the time G be given in seconds. Find the absolute number agreeing to the log-

D

H

H

;

N

arithm 0.4342944819

2P -^-^

and

let

L be the

logarithm of the number

N+1 '

;

and

Mathematical Principles

238

the velocity acquired in falling will be ,,

N+1

2PF -

G

may

1.3862943611F+4.605170186LF.

neglect the term 4.605170186LF;

altitude described, nearly.

If

H, and the height described

will

the fluid be of a sufficient depth,

and

2PF =^ 1.3862943611F

These things appear by Prop.

9,

Book

be

we

vdW be the ii,

and

its

and are true upon this supposition, that the globe meets with no other resistance but that which arises from the inactivity of matter. Now if it really meet with any resistance of another kind, the descent will be slower, and from the amount of that retardation will be known the amount of this new Corollaries,

resistance.

That the velocity and descent of a body falling in a I have composed the following table, the

be known, The Times

P

fluid first

might more easily column of which

Book

II

:

The Motion of Bodies

239

Scholium In order to investigate the resistances of fluids from experiments, I procured a square wooden vessel, whose length and breadth on the inside was 9 inches English measure, and its depth 93^ feet; this I filled with rain water; and having provided globes made up of wax, and lead included therein, I noted the times of the descents of these globes, the height through which they descended being 112 inches. A solid cubic foot of English measure contains 76 pounds troy weight of rain water; and a solid inch contains ^%6 ounces troy weight, or 253^ grains; and a globe of water of one inch in diameter contains 132.645 grains in air, or 132.8 grains in a vacuum; and any other globe will be as the excess of its weight in a vacuum above its weight in water. ExPER. 1. A globe whose weight was 1563^ grains in air, and 77 grains in water, described the whole height of 112 inches in 4 seconds. And, upon repeating the experiment, the globe spent again the very same time of 4 seconds in falling.

The weight of this globe in a vacuum is 156^3^8 grains; and the excess of this weight above the weight of the globe in water is 79^/^8 grains. Hence the diameter of the globe appears to be 0.84224 parts of an inch. Then it will be, as that excess to the weight of the globe in a vacuum, so is the density of the water to the density of the globe; and so is /^ parts of the diameter of the globe (viz., 2.24597 inches) to the space 2F, which will be therefore 4.4256 inches. Now a globe falUng in a vacuum with its whole weight of 156^/^8 grains in one second of time will describe 1933^ inches; and falling in water in the same time with the weight of 77 grains without resistance, will describe 95.219 inches; and in the time G, which is to one second of time as the square root of the ratio of the space F, or of 2.2128 inches to 95.219 inches, will describe 2.2128 inches, and will acquire the greatest velocity with which it is capable of descending in water. Therefore the time G is 0.15244 seconds. And in this time G, with that greatest velocity H, the globe will describe the space 2F, which is 4.4256 inches; and therefore in 4 seconds will describe a space of 116.1245 inches. Subtract the space 1.3862944 -F, or 3.0676 inches, and there will remain a space of 113.0569 inches, which the globe falling through water in a very wide vessel will describe in 4 seconds. But this space, by reason of the narrowness of the wooden vessel before mentioned, ought to be diminished in a ratio compounded of the square root of the ratio of the orifice of the vessel to the excess of this orifice above half a great circle of the globe, and of the simple ratio of the same orifice to its excess above a great circle of the globe, that is, in a ratio of 1 to 0.9914. This done, we have a space of 112.08 inches, which a globe falling through the water in this wooden vessel in 4 seconds of time ought nearly to describe by this theory; but it described 112 inches by the experiment. ExpER. 2. Three equal globes, whose weights were severally 763^ grains in air, and 53/i6 grains in water, were let fall successively; and every one fell through the water in 15 seconds of time, describing in its fall a height of 112

H

inches.

By

computation, the weight of each globe in a vacuum is 76^2 grains; the excess of this weight above the weight in water is 71 ^^8 grains; the diameter of the globe 0.81296 of an inch; parts of this diameter 2.16789 inches; the space 2F is 2.3217 inches; the space which a globe of 5/^6 grains in weight would describe in one second without resistance, 12.808 inches, and the time G

%

240

Mathematical Principles

0.301056 seconds. Therefore the globe, ^^^th the greatest velocity it is capable of receiving from a weight of 5/^6 grains in its descent through water, will describe in the time 0.301056 seconds the space 2.3217 inches; and in 15 seconds the space 115.678 inches. Subtract the space 1.3862944F, or 1.609 inches, and there remains the space 114.069 inches; which therefore the faUing globe ought to describe in the same time, if the vessel were very wide. But because our vessel was narrow, the space ought to be diminished by about 0.895 of an inch. And so the space will remain 113.174 inches, which a globe falling in this vessel ought nearly to describe in 15 seconds. But by the experiment it described 112 inches. The difference is not sensible. ExPER. 3. Three equal globes, whose weights were severally 121 grains in air, and 1 grain in water, were successively let fall; and they fell through the water in the times 46 seconds, 47 seconds, and 50 seconds, describing a height of 112 inches. By the theory, these globes ought to have fallen in about 40 seconds. Now whether their falling more slowly were occasioned from the consideration that in slow motions the resistance arising from the force of inactivity does really bear a less proportion to the resistance arising from other causes; or whether it is to be attributed to little bubbles that might chance to stick to the globes, or to the rarefaction of the wax by the warmth of the weather, or of the hand that let them fall; or, lastly, whether it proceeded from some insensible errors in weighing the globes in the water, I am not certain. Therefore the weight of the globe in water should be of several grains, that the experiment may be certain, and to be depended on. ExPER. 4. I began the foregoing Experiments to investigate the resistances of fluids, before I was acquainted with the theory laid down in the Propositions immediately preceding. Afterwards, in order to examine the theory after it was discovered, I procured a wooden vessel, whose breadth on the inside was S34 inches, and its depth 153^ feet. Then I made four globes of wax, mth lead included, each of which weighed 139^ grains in air, and 73^8 grains in water. These I let fall, measuring the times of their falling in the water with a pendulum oscillating to half-seconds. The globes were cold, and had remained so some time, both when they were weighed and when they were let fall; because warmth rarefies the wax, and by rarefying it diminishes the weight of the globe in the water; and wax, when rarefied, is not instantly reduced by cold to its former density. Before they were let fall, they were totally immersed under water, lest, by the weight of any part of them that might chance to be above the water, their descent should be accelerated in its beginning. Then, when after their immersion they were perfectly at rest, they were let go with the greatest care, that they might not receive any impulse from the hand that let them down. And they fell successively in the times of 473^, 483^, 50, and 51 oscillations, describing a height of 15 feet and 2 inches. But the weather was now a little colder than when the globes were weighed, and therefore I repeated the experiment another day; and then the globes fell in the times of 49, 493^2, 50, and 53; and at a third trial in the times of 493/2, 50, 51, and 53 oscillations. And by making the experiment several times over, I found that the globes fell mostly in the times of 493^ and 50 oscillations. When they fell slower, I suspect them to have been retarded by striking against the sides of the vessel.

Book

II:

The Motion of Bodies

241

theory, the weight of the globe in a vacuum is ISQ'/'o grains; the excess of this weight above the weight of the globe in water 132^/^0 grains; the diameter of the globe 0.99868 of an inch; /^ parts of the

Now, computing from the

diameter 2.66315 inches; the space 2F 2.8066 inches; the space which a globe weighing 73^ grains falling \Wthout resistance describes in a second of time 9.88164 inches; and the time G 0.376843 seconds. Therefore the globe with the greatest velocit}^ Anth which it is capable of descending through the water by the force of a weight of T^s grains, Anil in the time 0.376843 seconds describe a space of 2.8066 inches, and in one second of time a space of 7.44766 inches, and in the time 25 seconds, or in 50 oscillations, the space 186.1915 inches. Subtract the space 1.386294F, or 1.9454 inches, and there A^ill remain the space 184.2461 inches which the globe A\ill describe in that time in a very wide vessel. Because our vessel Avas narrow, let this space be diminished in a ratio compounded of the square root of the ratio of the orifice of the vessel to the excess of this orifice above half a great circle of the globe, and of the simple ratio of the same orifice to its excess above a great circle of the globe and we shall have the space of 181.86 inches, which the globe ought by the theory to describe in this vessel in the time of 50 oscillations, nearly. But it described the space of 182 inches, by experiment, in 493/^2 or 50 oscillations. ExpER. 5. Four globes weighing 154^ grains in air, and 21^^ grains in water, being let fall several times, fell in the times of 283^, 29, 29}4> and 30, and sometimes of 31, 32, and 33 oscillations, describing a height of 15 feet and 2 inches. They ought by the theory to have fallen in the time of 29 oscillations, nearly. ExPER. 6. Five globes, weighing 212^ grains in air, and 793^ in water, being several times let fall, fell in the times of 15, 153^, 16, 17, and 18 oscillations, describing a height of 15 feet and 2 inches. By the theory they ought to have fallen in the time of 15 oscillations, ;

nearly.

ExPER.

7.

water, being

and 33

Four

globes, weighing

let fall several

times,

2933^ grains in air, and 35J^ grains in in the times of 29^, 30, 303^, 31, 32,

fell

oscillations, describing a height of 15 feet

and 13^

inches.

By the theory they ought to have fallen in the time

of 28 oscillations, nearly. In searching for the cause that occasioned these globes of the same weight

and magnitude to

some

and some slower, I hit upon this that the first let go and began to fall, oscillated about their centres; that side which chanced to be the heavier descending first, and producing an oscillating motion. Xow by oscillating thus, the globe communicates a greater motion to the water than if it descended AA-ithout any oscillations; and by this communication loses part of its oaatl motion AA-ith AA'hich it should descend; and therefore as this oscillation is greater or less, it aaiII be more or less retarded. Besides, the globe alAA'ays recedes from that side of itself AA-hich is descending in the oscillation, and by so receding comes nearer to the sides globes,

fall,

swifter

:

when they were

of the vessel, so as even to strike against them sometimes. And the heavier the globes are, the stronger this oscillation is; and the greater they are, the more is the AA'ater agitated by it. Therefore to diminish this oscillation of the globes, I made neAv ones of lead and Avax, sticking the lead in one side of the globe very

near

and

the globe in such a manner, that, as near as possible, the heavier side might be loAA^est at the beginning of the descent. By this its

surface;

I let fall

Mathematical Principles

242

became much

than before, and the times in which the globes fell were not so unequal as in the following Experiments. ExPER. 8. Four globes weighing 139 grains in air, and 63^ in water, were let fall several times, and fell mostly in the time of 51 oscillations, never in more than 52, or in fewer than 50, describing a height of 182 inches. By the theory they ought to fall in about the time of 52 oscillations. ExPER. 9. Four globes weighing 2733^ grains in air, and 140^ in water, being several times let fall, fell in never fewer than 12, and never more than

means the

oscillations

less

:

13 oscillations, describing a height of 182 inches. These globes by the theory ought to have fallen in the time of 113^^ oscillations, nearly.

ExPER.

10.

Four

globes, weighing

384 grains in

and 1193^ in water, 183>^, and 19 oscilla-

air,

the times of 17^, 18, of inches. And when they fell in the time of 19 height tions, describing a 1813^ oscillations, I sometimes heard them hit against the sides of the vessel before

being

let fall several times, fell in

they reached the bottom. By the theory they ought to have fallen in the time of 15/^

oscillations,

nearly.

ExPER.

11.

and 3^/^2 in water, 443^, 45, and 46 oscilla-

Three equal globes, weighing 48 grains in

air,

being several times let fall, fell in the times of 433^, 44, tions, and mostly in 44 and 45, describing a height of 1823^2 inches, nearly. By the theory they ought to have fallen in the time of 46/^ oscillations, nearly.

ExPER. being

12.

air, and 4^ in water, the times of 61, 62, 63, 64, and 65 oscillations,

Three equal globes, weighing 141 grains in

let fall several times, fell in

describing a space of 182 inches.

And by the theory they ought to have From these Experiments it is manifest,

fallen in 643/^ oscillations, nearly.

that

when the

globes

fell

slowly, as

and twelfth Experiments, the by the theory; but when the globes fell and tenth Experiments, the resistance was

in the second, fourth, fifth, eighth, eleventh,

times of falling are rightly exhibited

more swiftly, as in the sixth, ninth, somewhat greater than the square of the velocity. For the globes in falhng oscillate a little; and this oscillation, in those globes that are light and fall slowly, soon ceases by the weakness of the motion; but in greater and heavier globes, the motion being strong, it continues longer, and is not to be checked by the ambient water till after several oscillations. Besides, the more swiftly the globes move, the less are they pressed by the fluid at their hinder parts; and if the velocity be continually increased, they will at last leave an empty space behind them, unless the compression of the fluid be increased at the same time. For the compression of the fluid ought to be increased (by Props. 32 and 33) as the square of the velocity, in order to maintain the resistance in the same squared ratio. But because this is not done, the globes that move swiftly are not so much pressed at their hinder parts as the others; and by the defect of this pressure it

comes to pass that

their resistance

is

a

little

greater than the

square of their velocity. So that the theory agrees with the experiments on bodies falling in water. It remains that we examine the observations of bodies falling in air. ExPER. 13. From the top of St. Paul's Church in London, in June, 1710, there were let fall together two glass globes, one full of quicksilver, the other

Book

II

:

The Motion of Bodies

243

and in their fall they described a height of 220 English feet. A wooden table was suspended upon iron hinges on one side, and the other side of the table was supported by a wooden pin. The two globes lying upon this table were let fall together by pulling out the pin by means of an iron wire reaching thence down to the ground; so that, the pin being removed, the table, which had then no support but the iron hinges, fell downwards, and turning round upon the hinges, gave leave to the globes to drop off from it. At the same instant, with the same pull of the iron wire that took out the pin, a pendulum oscillating to seconds was let go, and began to oscillate. The diameters and weights of the globes, and their times of falling, are exhibited in the accompanying table. of air;

The globes

filled

with mercury

244

Mathematical Principles

8 seconds 12 thirds of time nill describe 245 feet and 5}4 inches. Subtract 1.3863 -F, or 20 feet and }4 an inch, and there remain 225 feet 5 inches. This space, therefore, the falling globe ought by the theory to describe in 8 seconds 12 thirds. But by the experiment it described a space of 220 feet. The difference is inappreciable. By like calculations appHed to the other globes full of air, I composed the follo\\ing table.

The weights of the globes

Book

II:

The Motion of Bodies

245

19 seconds, 18^ seconds, 18^ seconds, 24 seconds, and 213^ seconds. But the bladders did not always fall directly down, but sometimes fluttered a little in the air, and waved to and fro, as they were descending. And by these motions the times of their falling were prolonged, and increased by half a second sometimes, and sometimes by a whole second. The second and fourth bladders fell most directly the first time, and the first and third the second time. The fifth bladder was wrinkled, and by its wrinkles was a little retarded. I found their diameters by their circumferences measured Avith a very fine thread wound about them tmce. In the follo^\^ng table I have compared the experiments \vith the theory; making the density of air to be to the density of rain water as 1 to 860, and computing the spaces which by the theory the globes ought to describe in faUing.

The weights of the bladders

Mathematical Principles

246 ever,

though

of the

most extreme

fluidity, are,

other things being equal, as the

densities of the fluids.

These things being thus estabUshed, we may now determine what part of motion any globe projected in any fluid whatsoever would nearly lose in a given time. Let D be the diameter of the globe, and V its velocity at the beginning of its motion, and T the time in which a globe with the velocity V can its

describe in a

vacuum a space

that

is

to the space

/^D

as the density of the the globe projected in that fluid will, in

globe to the density of the fluid; and ^V TV any other time t lose the part ijrr^,, the part 7yj—. remaining; and

a space, which

will

\w\\\

describe

be to that described in the same time in a vacuum with the

T+^

uniform velocity V, as the logarithm of the number —=r- multiplied by the

number 2.302585093

is

to the

number = by

Cor.

vii.

Prop. 35. In slow motions

the resistance may be a little less, because the figure of a globe is more adapted to motion than the figure of a cyUnder described with the same diameter. In swift motions the resistance may be a little greater, because the elasticity and compression of the fluid do not increase as the square of the velocity. But these little niceties I take no notice of. And though air, water, quicksilver, and the like fluids, by the division of their parts in infinitum, should be subtilized, and become mediums infinitely fluid, nevertheless, the resistance they would make to projected globes would be the same. For the resistance considered in the preceding Propositions arises from the inactivity of the matter; and the inactivity of matter is essential to bodies, and always proportional to the quantity of matter. By the division of the parts of the fluid the resistance arising from the tenacity and friction of the parts may be indeed diminished; but the quantity of matter will not be at all diminished by this division; and if the quantity of matter be the same, its force of inactivity will be the same; and therefore the resistance here spoken of will be the same, as being always proportional to that force. To diminish this resistance, the quantity of matter in the spaces through which the bodies move must be diminished; and therefore the celestial spaces, through which the globes of the planets and comets are continually passing towards all parts, with the utmost freedom, and without the least sensible diminution of their motion, must be utterly void of any corporeal fluid, excepting, perhaps, some extremely rare vapors and the rays of light. Projectiles excite a motion in fluids as they pass through them, and this motion arises from the excess of the pressure of the fluid at the fore parts of the projectile above the pressure of the same at the hinder parts; and cannot be less in mediums infinitely fluid than it is in air, water, and quicksilver, in proportion to the density of matter in each. Now this excess of pressure does, in proportion to its quantity, not only excite a motion in the fluid, but also acts upon the projectile so as to retard its motion; and therefore the resistance in every fluid is as the motion excited by the projectile in the fluid; and cannot be less in the most subtile ether in proportion to the density of that ether, than it is in air, water, and quicksilver, in proportion to the densities of those fluids.

Book

II:

The Motion of Bodies

SECTION

VIII

The motion propagated through Proposition

41.

247

Theorem

fluids

32

A

pressure is not propagated through a fluid in rectilinear directions except where the particles of the fluid lie in a right line.

a right line, the pressure may be indeed directly propagated from a to e; but then the particle e mil urge the obliquely posited particles / and g obliquely, and those particles / and g will not sustain this pressure, unless they be supported by the particles h and k lying beyond them; but the particles that support them are also pressed by them; and those particles cannot sustain that pressure, without being supported by, and pressing upon, those particles that lie still farther, as I and m, and so on in infinitum. Therefore the pressure, as soon as it is propagated to particles that he out of right lines, begins to deflect towards one hand and the other, will propagated obliquely in infinitum; and after it has begun to be proand be pagated obliquely, if it reaches more distant particles lying out of the right line, it will deflect again on each hand; and this it will do as often as it lights on particles that do not lie exactly in a right line. q.e.d. Cor. If any part of a pressure, propagated through a fluid from a given point, be intercepted by any obstacle, the remaining part, which is not intercepted, will deflect into the spaces behind the obstacle. This may be demonstrated also after the following manner. Let a pressure be propagated from the point A towards any part, and, if it be possible, in rectilinear directions; and the obstacle being perforated in BC, let all the pressure be intercepted If

the particles

a, h, c, d, e lie in

NBCK

Mathematical Principles

248

but the coniform part APQ passing through the circular hole BC. Let the cone APQ be divided into frustums by the transverse planes, de, fg, hi. Then while the cone ABC, propagating the pressure, urges the conic frustum degf beyond it on the surface de, and this frustum urges the next frustum fgih on the surface fg, and that frustum urges a third frustum, and so in infinitum; it is manifest (by the third Law) that the first frustum defg is, by the reaction of the second frustum fghi, as much urged and pressed on the surface fg, as it urges and presses that second frustum. Therefore the frustum degf is compressed on both sides, that is, between the cone Ade and the frustum fhig; and therefore (by Case 6, Prop. 19) cannot preserve its figure, unless it be compressed with the same force on all sides. Therefore with the same force with which it is pressed on the surfaces de, fg, it will endeavor to break forth at the sides df, eg; and there (being not in the least tenacious or hard, but perfectly fluid) it will run out, expanding itself, unless there be an ambient fluid opposing that endeavor. Therefore, by the effort it makes to run out, it will press the ambient fluid, at its sides df, eg, with the same force that it does the frustum fghi; and therefore, the pressure will be propagated as much from the sides df, eg, into the spaces NO, KL this way and that way, as it is propagated from the surface fg towards PQ. q.e.d.

Proposition

42.

Theorem

33

All motion propagated through a fluid diverges from a rectilinear progress into the

unmoved

spaces.

A through the hole BC, BCQP according to right

Case L Let a motion be propagated from the point and,

if it

be possible, let it proceed in the conic space from the point A. And let us first suppose this motion to be

lines diverging

"'"""//,,

that of weaves in the surface of standing water; and let de, fg, hi, kl, &c., be the tops of the several waves, divided from each other by as many intermediate valleys or hollows. Then, because the water in the ridges of the waves is higher

Book

II

:

The Motion of Bodies KL, NO,

249

run down from off the tops of those ridges, e, g, i, I, &c., d, f, h, k, &c., this way and that way towards KL and NO; and because the water is more depressed in the hollows of the waves than in the unmoved parts of the fluid KL, NO, it will run down into those hollows out of those unmoved parts. By the first deflux the ridges of the waves will dilate themselves this way and that way, and be propagated towards KL and NO. And because the motion of the waves from A towards PQ is carried on by a continual deflux from the ridges of the waves into the hollows next to them, and therefore cannot be swifter than in proportion to the celerity of the descent; and the descent of the water on each side towards KL and NO must be performed with the same velocity it follows that the dilatation of the waves on each side towards KL and NO mil be propagated with the same velocity as the waves themselves go forwards directly from A to PQ. And therefore the whole space this way and that way towards KL and NO will be filled by the dilated waves rfgr, shis, tklt, vmnv, &c. q.e.d. That these things are so, anyone may find by making the experiment in still water. Case 2. Let us suppose that de, fg, hi, kl, mn represent pulses successively propagated from the point A through an elastic medium. Conceive the pulses to be propagated by successive condensations and rarefactions of the medium, so that the densest part of every pulse may occupy a spherical surface described about the centre A, and that equal intervals intervene between the successive pulses. Let the lines de, fg, hi, kl, &c., represent the densest parts of the pulses, propagated through the hole BC; and because the medium is denser there than in the spaces on either side towards KL and NO, it will dilate itself as well towards those spaces KL, NO, on each hand, as towards the rare intervals between the pulses; and hence the medium, becoming always more rare next the intervals, and more dense next the pulses, will partake of their motion. And because the progressive motion of the pulses arises from the continual relaxation of the denser parts towards the antecedent rare intervals; and since the pulses will relax themselves on each hand towards the quiescent parts of the medium KL, NO with very near the same celerity; therefore the pulses will dilate themselves on all sides into the unmoved parts KL, NO with almost the same celerity with which they are propagated directly from the centre A; and therefore will fill up the whole space KLON. q.e.d. And we find the same by experience also in sounds which are heard through a mountain interposed; and, if they come into a chamber through the window, dilate themselves into all the parts of the room, and are heard in every corner; and not as reflected from the opposite walls, but directly propagated from the window, as far as our sense can judge. Case 3. Let us suppose, lastly, that a motion of any kind is propagated from A through the hole BC. Then since the cause of this propagation is that the parts of the medium that are near the centre A disturb and agitate those which lie farther from it; and since the parts which are urged are fluid, and therefore recede every way towards those spaces where they are less pressed: they mil by consequence recede towards all the parts of the quiescent medium, as well to the parts on each hand, as KL and NO, as to those right before, as PQ; and by this means all the motion, as soon as it has passed through the hole BC, vnW begin to dilate itself, and from thence, as from its principle and centre, q.e.d. will be propagated directly every way.

than in the unmoved parts

of the fluid

:

it will

Mathematical Principles

250

Proposition Every tremulous body in an

Theorem 34

medium propagates the motion of the pulses on in a nonelastic medium excites a circular motion.

elastic

every side straight forwards; but

Case

43.

The

parts of the tremulous body, alternately going and returning, do in going urge and drive before them those parts of the medium that lie nearest, and by that impulse compress and condense them; and in returning 1.

compressed parts to recede again, and expand themselves. Therefore the parts of the medium that lie nearest to the tremulous body move to and fro by turns, in like manner as the parts of the tremulous body itself do; suffer those

and for the same cause that the parts of this body agitate these parts of the medium, these parts, being agitated by like tremors, mil in their turn agitate others next to themselves; and these others, agitated in like manner, wdll agitate those that lie beyond them, and so on in infinitum. And in the same manner as the first parts of the medium were condensed in going, and relaxed in returning, so mil the other parts be condensed every time they go, and expand themselves every time they return. And therefore they mil not be all going and all returning at the same instant (for in that case they would always maintain determined distances from each other, and there could be no alternate condensation and rarefaction); but since, in the places where they are condensed, they approach to, and, in the places where they are rarefied, recede from each other, therefore some of them mil be going while others are returning; and so on in infinitum. The parts so going, and in their going condensed, are pulses, by reason of the progressive motion with which they strike obstacles in their way; and therefore the successive pulses produced by a tremulous body will be propagated in rectilinear directions; and that at nearly equal distances from each other, because of the equal intervals of time in which the body, by its several tremors, produces the several pulses. And though the parts of the tremulous body go and return in some certain and determinate direction, yet the pulses propagated from thence through the medium mil dilate themselves towards the sides, by the foregoing Proposition; and will be propagated on all sides from that tremulous body, as from a common centre, in surfaces nearly spherical and concentric, as in waves excited by shaking a finger in water, which proceed not only forwards and backwards agreeably to the motion of the finger, but spread themselves in the manner of concentric circles all round the finger, and are propagated on every side. For the gravity of the water supplies the place of elastic force.

Case 2. If the medium be not elastic, then, because its parts cannot be condensed by the pressure arising from the vibrating parts of the tremulous body, the motion mil be propagated in an instant towards the parts where the medium yields most easily, that is, to the parts which the tremulous body would othermse leave vacuous behind it. The case is the same with that of a body projected in any medium whatever. A medium yielding to projectiles does not recede in infinitum, but with a circular motion comes round to the spaces which the body leaves behind it. Therefore as often as a tremulous body tends to any part, the medium yielding to it comes round in a circle to the parts which the body leaves; and as often as the body returns to the first place, the medium will be driven from the place it came round to, and return to its original place. And though the tremulous body be not firm and hard, but

Book

II:

The Motion of Bodies

251

every way flexible, yet if it continue of a given magnitude, since it cannot impel the medium by its tremors anywhere without yielding to it somewhere the medium receding from the parts of the body where it is pressed will always come round in a circle to the parts that yield to it. q.e.d. Cor. Hence it is a mistake to think that the agitation of the parts of flame conduces to the propagation of a pressure in rectilinear directions through an ambient medium. Such a pressure must be derived not from the agitation only of the parts of flame, but from the dilatation of the whole. else,

Proposition 44. Theorem 35

MN

// water ascend and descend alternately in the erected legs KL, of a canal or pipe; and a pendulum be constructed whose length between the point of suspension and the centre of oscillation is equal to half the length of the water in the canal: I say, that the water will ascend and descend in the same times in which the pendu-

lum

oscillates.

I measure the length of the water along the axes of the canal and its legs, and make it equal to the sum of those axes; and take no notice of the resistance of the water arising from its attrition by the sides of the canal. Let, therefore, AB, CD represent the mean height of the water in both legs; and when the water in the leg KL ascends to the height EF, the water will descend in the

M

K

D H

p

leg

MN to the height GH.

Q

Let

P be

a pendulous body,

VP

the thread,

V

the

point of suspension, RPQS the cycloid which the pendulum describes, P its lowest point, PQ an arc equal to the height AE. The force with which the motion of the water is accelerated and retarded alternately is the excess of the weight of the water in one leg above the weight in the other; and, therefore, when the water in the leg ascends to EF, and in the other leg descends to GH, that force is double the weight of the water EABF, and therefore is to the weight of the whole water as AE or PQ to VP or PR. The force also with which the body P is accelerated or retarded in any place, as Q, of a cycloid, is (by Cor., Prop. 51, Book i) to its whole weight as its distance PQ from the lowest place P to the length PR of the cycloid. Therefore the motive forces of the water and pendulum, describing the equal spaces AE, PQ, are as the weights to be moved; and therefore if the water and pendulum are quiescent at first, those forces will move them in equal times, and will cause them to go and return together with a reciprocal motion. q.e.d. Cor. i. Therefore the reciprocations of the water in ascending and descend-

KL

Mathematical Principles

252 ing are

all

performed in equal times, whether the motion be more or

less intense

or remiss.

Cor. II. If the length of the whole water in the canal be of 6/^ feet of French measure, the water will descend in one second of time, and will ascend in another second, and so on by turns in infinitum; for a pendulum of S]/is such feet in length will oscillate in one second of time. Cor. III. But if the length of the water be increased or diminished, the time of the reciprocation vdW be increased or diminished as the square root of the length.

Proposition The

45.

Theorem 36

waves varies as the square root of the breadths. This follows from the construction of the following Proposition. velocity of

Proposition

To find

46.

Problem

10

the velocity of waves.

Let a pendulum be constructed, whose length between the point of suspension and the centre of oscillation is equal to the breadth of the Avaves, and in the time that the pendulum mil perform one single oscillation the waves will advance forwards nearly a space equal to their breadth. That w^hich I call the breadth of waves is the transverse measure lying between the deepest part of the hollows, or the tops of the ridges. Let ABCDEF represent the surface of stagnant water ascending and descending in successive

waves; and let A, C, E, &c., be the tops of the waves; and let B, D, F, &c., be the intermediate hollows. Because the motion of the waves is carried on by the successive ascent

and descent

of the water, so that the parts thereof, as A, C,

E, &c., which are highest at one time, become lowest immediately after; and because the motive force, by which the highest parts descend and the lowest ascend, is the weight of the elevated water, that alternate ascent and descent \vi\\ be analogous to the reciprocal motion of the water in the canal, and Avill observe the same laws as to the times of ascent and descent; and therefore (by Prop. 44) if the distances between the highest places of the waves A, C, E and the lowest B, D, F be equal to twice the length of any pendulum, the highest parts A, C, E mil become the lowest in the time of one oscillation, and in the time of another oscillation will ascend again. Therefore mth the passage of each wave, the time of two oscillations mil occur; that is, the wave will describe its breadth in the time that pendulum will oscillate tmce; but a pendulum of four times that length, and therefore equal to the breadth of the q.e.i. waves, will just oscillate once in that time. Cor. I. Therefore, waves whose breadth is equal to S^/is French feet, will advance through a space equal to their breadth in one second of time; and therefore in one minute mil go over a space of 1833^ feet; and in an hour, a space of 11,000 feet, nearly. Cor. II. And the velocity of greater or less waves mil be augmented or diminished as the square root of their breadth.

Book

The Motion of Bodies

II:

253

These things are true upon the supposition that the parts of water ascend or descend in a straight Une; but, in truth, that ascent and descent is rather performed in a circle; and therefore I give the time defined by this Proposition as only approximate.

Proposition

47.

Theorem

37

If pulses are propagated through a fluid, the several particles of the fluid, going and returning with the shortest reciprocal motion, are always accelerated or retarded

law of the oscillating pendulum. Let AB, BC, CD, &c., represent equal distances of successive pulses; ABC the line of direction of the motion of the successive pulses propagated from A to B; E, F, G three physical points of the quiescent medium situate in the right line AC at equal distances from each other; Ee, F/, Gg equal spaces of extreme shortness, through which those points go and return ^vith a reciprocal motion in each vibration; e, y any intermediate places of the same points; EF, FG physical short lines, or linear parts of the D medium lying between those points, and successively transferred into the places €0, 07, and ef, fg. Let there be drawn the right line PS equal to the right line Ee. Bisect the same in 0, and from the centre O, with the radius OP, describe the circle SIP*. Let the whole time of one vibration, with its proportional parts, be represented by the whole circumference of this circle and its parts, in such sort, that, when any time PH or PHS/i is completed, if there be let fall to PS the perpendicular HL or hi, and there be taken Ee equal to PL or Fl, the physical point E may be found in e. A point, as E, moving according to this law ^vith a reciprocal motion, in its going from E through e to e, and returning again through e to E, will perform its several vibrations with the same degrees of acceleration and retardation mth those of an oscillating pendulum. We are now to prove that the gf several physical points of the medium will be agitated mth such a kind of motion. Let us suppose, then, that a medium hath such a motion excited in it from any cause whatsoever, and consider what will r according

to the

,

liii

iiiii

follow from thence.

In the circumference PHS/i let there be taken the equal arcs, HI, IK, or /it, t A- having the same "J ratio to the whole circumference as the equal P right lines EF, FG have to BC, the whole interval of the pulses. Let fall the perpendiculars IM, KN, or im, kn; then because the points E, F, G are successively agitated with like motions, and perform their entire vibrations composed of their going and return, while the pulse is transferred from B to C if PH or PHS/i be the time elapsed since the beginning of the motion of the point E, then will PI or PHSt be the time elapsed since the beginning of the motion of the point F, and PK or PHSA; the time elapsed since the beginning of the motion of the point G; and therefore Ee, F0, Gt A\ill be respectively equal to PL, PM, PN, while the points ,

;

G ilil

B I

Mathematical Principles

254

Vm, Pw, when the points are returning. Therefore ey or EG+G7 — Ee will, when the points are going, be equal to EG — LN, and in their return equal to EG + Zn. But ey is the breadth or expansion of the part EG of the medium in the place ey and therefore the expansion of that part in its going is to its mean expansion as EG — LN to EG; and in its return, as EG -\- In or EG + LN is to EG. Therefore since LN is to KH as I to the radius OP, and KH to EG as the circumference PHS/iP to BC; that is, if we put V for the radius of a circle whose circumference is equal to BC, the interval of and to

are going,

PZ,

;

M

the pulses, as OP is to V; and, multipljdng together corresponding terms of the as IM to V; the expansion of the part EG, proportions, we obtain LN to or of the physical point F in the place ey, is to the mean expansion of the same part in its first place EG, as V — IM is to V in going, and as is to V in its return. Hence the elastic force of the point F in the place ey is to its

EG

V+m

mean

V+m

is

to

:rf

V

in its return.

the physical points

And by

E and G in

the difference of the forces

is

—^vr

EG

elastic force in the place

as :^

to the

VV-VHL-VKN+HLKN

.

IS

y_TTj and

mean

^^ ^^^ going,

V

y_t7^tst

elastic force of the

1 ^u ^to ^; that is, as ^

v

and as

the same reasoning the elastic forces of

going are as

HL-KN

^^ **^

'

HL-KN —

—

^7^7

VV

.

is

is

to

:^;

and

medium ,

to

as

1

V :rz,

or as

HL — KN is to V; if we suppose (by reason of the very short extent of the vibrations) HL and KN to be indefinitely less than the quantity V. Therefore since the quantity V is given, the difference of the forces is as HL-KN; that

HL-KN

OM

is proportional to HK, and (because to 01 or OP; and beand OP are given), as OM; that is, if F/ be bisected in Q, as ?« 8° 20' 00"; and of its ascending node Q 27° 24' 20"; and the difference of meridians between the Observatory at Greenwich and the Royal Observatory at Paris, 0*". 9"". 20^: but the mean motion of the moon and of its apogee are not yet obtained with

But the theory

of the

sufficient accuracy.

Proposition

To find

The

the force of the

sun's force

25), in the

and the

sun

to

move

36.

Problem

17

the sea.

ML or PT to disturb the motions of the moon was (by Prop.

moon's quadratures, to the force of gravity with us, as 1 to 638092.6; — LM or 2PK in the moon's syzygies is double that quantity.

force

TM

But, descending to the surface of the earth, these forces are diminished in proportion of the distances from the centre of the earth, that is, in the proportion of 603^ to 1 and therefore the former force on the earth's surface is to the force of gravity as 1 to 38,604,600; and by this force the sea is depressed in such places as are 90 degrees distant from the sun. But by the other force, which is twice as great, the sea is raised not only in the places directly under the sun, but in those also which are directly opposed to it; and the sum of these forces is to the force of gravity as 1 to 12,868,200. And because the same force excites the same motion, whether it depresses the waters in those places which are 90 degrees distant from the sun, or raises them in the places which are directly under and directly opposed to the sun, the aforesaid sum will be the total force of the sun to disturb the sea, and will have the same effect as if the whole was employed in raising the sea in the places directly under and directly opposed to the sun, and did not act at all in the places which are 90 degrees removed from the sun. And this is the force of the sun to disturb the sea in any given place, where the sun is at the same time both vertical, and in its mean distance from the earth. In other positions of the sun, its force to raise the sea is directly as the versed sine of double its altitude above the horizon of the place, and inversely as the cube of the distance from the earth. Cor. Since the centrifugal force of the parts of the earth, arising from the earth's diurnal motion, which is to the force of gravity as 1 is to 289, raises the waters under the equator to a height exceeding that under the poles by 85,472 Paris feet, as above, in Prop. 19, the force of the sun, which we have now shown to be to the force of gravity as 1 is to 12,868,200, and therefore is to that centrifugal force as 289 to 12,868,200, or as 1 to 44,527, mil be able to raise the waters in the places directly under and directly opposed to the sun to a height exceeding that in the places which are 90 degrees removed from the sun only by one Paris foot and 1 13/^0 inches for this measure is to the measure of 85,472 feet as ;

;

1

to 44,527.

Proposition

37.

Problem

18

To find the force of the moon to move the sea. The force of the moon to move the sea is to be deduced from its ratio to the force of the sun, and this ratio is to be determined from the ratio of the motions

Book

III:

The System of the World

325

of the sea, which are the effects of those forces. Before the mouth of the river Avon, three miles below Bristol, the height of the ascent of the water in the vernal and autumnal syzygies of the luminaries (by the observations of Samuel Sturmy) amounts to about 45 feet, but in the quadratures to 25 only. The former of those heights arises from the sum of the aforesaid forces, the latter from their difference. If, therefore, S and L are supposed to represent respectively the forces of the sun and moon while they are in the equator, as well as in their mean distances from the earth, we shall have L+S to L — S as 45 to 25,

or as 9 to

5.

At Plymouth (by the observations of Samuel Colepress) the tide in its mean height rises to about 16 feet, and in the spring and autumn the height thereof in the syzygies may exceed that in the quadratures by more than 7 or 8 feet. Suppose the greatest difference of those heights to be 9 feet, and L+S will be to L — S as 203^2 to 11^) or as 41 to 23; a proportion that agrees well enough vnih the former. But because of the great tide at Bristol, we are rather to depend upon the observations of Sturmy; and, therefore, till we procure something that is more certain, we shall use the proportion of 9 to 5. But because of the reciprocal motions of the waters, the greatest tides do not happen at the times of the syzygies of the luminaries, but, as we have said before, are the third in order after the syzygies; or (reckoning

follow next after the third approach of the

moon

from the syzygies)

to the meridian of the place

Sturmy observes) are the third after the day of rather nearly after the twelfth hour from the new or the new or full moon, or full moon, and therefore fall nearly upon the forty-third hour after the new or full moon. But in this port they come to pass about the seventh hour after the after the syzygies; or rather (as

moon

to the meridian of the place; and therefore follow next after the approach of the moon to the meridian, when the moon is distant from

approach

of the

the sun, or from opposition with the sun by about 18 or 19 degrees forwards. So the summer and winter seasons come not to their height in the solstices themselves, but when the sun is advanced beyond the solstices by about a tenth part of its whole course, that is, by about 36 or 37 degrees. In like manner, the greatest tide is raised after the approach of the moon to the meridian of the place, when the moon has passed by the sun, or the opposition thereof, by about a tenth part of the whole motion from one greatest tide to the next following greatest tide. Suppose that distance about 183^ degrees; and the sun's force in this distance of the moon from the syzygies and quadratures will be of less moment to augment and diminish that part of the motion of the sea which proceeds from the motion of the moon than in the syzygies and quadratures themselves in the proportion of the radius to the cosine of double this distance, or of an angle of 37 degrees; that is, in the ratio of 10,000,000 to 7,986,355; and, therefore, in the preceding analogy, in place of S we must put 0.7986355S. But further, the force of the moon in the quadratures must be diminished, on account of its declination from the equator; for the moon in those quadratures, or rather in 183^ degrees past the quadratures, declines from the equator by about 23° 13'; and the force of either luminary to move the sea is diminished as it declines from the equator nearly as the square of the cosine of the declination; and therefore the force of the moon in those quadratures is only 0.8570327L; hence we have L+0.7986355S to 0.8570327L-0.7986355S as 9 to

5.

Mathematical Principles

326

Further yet, the diameters of the orbit in which the moon should move, setting aside the consideration of eccentricity, are one to the other as 69 to 70; and therefore the moon's distance from the earth in the syzygies is to its distance in the quadratures, other things being equal, as 69 to 70; and its distances, when 183^ degrees advanced beyond the syzygies, where the greatest tide was excited, and when 18)^ degrees passed by the quadratures, where the least tide was produced, are to its mean distance as 69.098747 and 69.897345 to 693^. But the force of the moon to move the sea varies inversely as the cube of its distance; and therefore its forces, in the greatest and least of those distances, are to its force in its mean distance as 0.9830427 and 1.017522 is to 1. From this

we have 1.017522L -0.79863558 to 0.9830427 •0.8570327L-0.7986355S as 9 to 5; and S to L as 1 to 4.4815. Therefore, since the force of the sun is to the force of gravity as 1

1

to 12,868,200, the moon's force will be to the force of gravity as

to 2,871,400.

Cor. I. Since the waters attracted by the sun's force rise to the height of 1 foot and IIV30 inches, the moon's force will raise the same to the height of 8 feet and 7/^2 inches; and the joint forces of both Avill raise the same to the height of 103^ feet; and when the moon is in its perigee to the height of 123^ feet, and more, especially when the wind sets the same way as the tide. And a force of that amount is abundantly sufficient to produce all the motions of the sea, and agrees well with the ratio of those motions; for in such seas as lie free and open from east to west, as in the Pacific sea, and in those tracts of the Atlantic and Ethiopic seas which lie without the tropics, the waters commonly rise to 6, 9, 12, or 15 feet; but in the Pacific sea, which is of a greater depth, as well as of a larger extent, the tides are said to be greater than in the Atlantic and Ethiopic seas; for, to have a full tide raised, an extent of sea from east to west is required of no less than 90 degrees. In the Ethiopic sea, the waters rise to a less height within the tropics than in the temperate zones because of the narrowness of the sea between Africa and the southern parts of America. In the middle of the open sea the waters cannot rise without falling together, and at the same time, upon both the eastern and western shores, when, notwithstanding, in our narrow seas, they ought to fall on those shores by alternate turns; upon this account there is commonly but a small flood and ebb in such islands as lie far distant from the continent. On the contrary, in some ports, where to fill and empty the bays alternately the waters are with great violence forced in and out through shallow channels, the flood and ebb must be greater than ordinary; as at Plymouth and Chepstow Bridge in England, at the mountains of St. Michael, and the town of Avranches, in Normandy, and at Cambaia and Pegu in the East Indies. In these places the sea is hurried in and out with such violence as sometimes to lay the shores under water, sometimes to leave them dry for many miles. Nor is this force of the influx and efflux to be stopped till it has raised and depressed the waters to 30, 40, or 50 feet and above. And a like account is to be given of long and shallow channels or straits, such as the Magellanic straits, and those channels which environ England. The tide in such ports and straits, by the violence of the influx and efflux, is augmented greatly. But on such shores as lie towards the deep and open sea with a steep descent, where the waters may freely rise and faU mthout that precipitation of influx and efflux, the ratio of the tides agrees with the forces of the sun and moon. :

Book

III

:

The System of the World

327

Since the moon's force to move the sea is to the force of gravity as 1 to 2,871,400, it is evident that this force is inappreciable in statical or hydrostatical experiments, or even in those of pendulums. It is in the tides only that

Cor.

II.

shows itself by any sensible effect. CoR. III. Because the force of the moon for moving the sea is to the like force of the sun as 4.4815 to 1, and those forces (by Cor. xiv, Prop. 66, Book i) are as the densities of the bodies of the sun and moon and the cubes of their apparent diameters conjointly, the density of the moon udll be to the density of the sun directly as 4.4815 to 1, and inversely as the cube of the moon's diameter to the cube of the sun's diameter; that is (seeing the mean apparent diameters of the moon and sun are 31' IQ^i", and 32' 12"), as 4891 to 1000. But the density of the sun was to the density of the earth as 1000 to 4000; and therefore the density of the moon is to the density of the earth as 4891 is to 4000, or as 11 to 9. Therefore the body of the moon is more dense and more earthly than the earth itself. Cor. IV. And since the true diameter of the moon (from the observations of astronomers) is to the true diameter of the earth as 100 to 365, the mass of matter in the moon will be to the mass of matter in the earth as 1 to 39.788. CoR. V. And the accelerative gravity on the surface of the moon will be about three times less than the accelerative gravity on the surface of the earth. CoR. VI. And the distance of the moon's centre from the centre of the earth "will be to the distance of the moon's centre from the common centre of gravity this force

of the earth

and moon as 40.788 to 39.788.

And

mean

distance of the centre of the moon from the centre of the earth ^\^ll be (in the moon's octants) nearly 60% of the greatest semidiameters of the earth; for the greatest semidiameter of theearthwas 19,658,600

CoR.

VII.

Paris feet,

the

and the mean distance

of the centres of the earth

and moon,

consisting of 6034 such semidiameters, is equal to 1,187,379,440 feet. And this distance (by the preceding Cor.) is to the distance of the moon's centre from the common centre of gravity of the earth and moon as 40.788 to 39.788; w^hich latter

1,158,268,534 feet. And since the moon, in respect of the fixed stars, performs its revolution in 27*^. 7^. 43|™., the versed sine of that angle which the moon in a minute of time describes is 12,752,341 to the radius distance, therefore,

is

and as the radius is to this versed sine, so are 1, 158,268,534 The moon, therefore, falling towards the earth by that force which retains it in its orbit, would in one minute of time describe 14.7706353 feet; and, if we augment this force in the proportion of 178^/^o to 177^%o, we shall have the total force of gravity at the orbit of the moon, by Cor., Prop. 3; and the moon falling by this force, in one minute of time would describe 14.8538067 feet. And at the 60th part of the distance of the moon from the earth's centre, that is, at the distance of 197,896,573 feet from the centre of the earth, a body falling by its weight, would, in one second of time, like"\vise describe 14.8538067 feet. And, therefore, at the distance of 19,615,800, which compose one mean semidiameter of the earth, a heavy body would describe in falhng 15.11175, or 15 feet, 1 inch, and 43/ii Hnes, in the same time. This vnW be the descent of bodies in the latitude of 45 degrees. And by the foregoing table, to be found under Prop. 20, the descent in the latitude of Paris vAW be a little greater by an excess of about 34 parts of a line. Therefore, by this computation, heavy bodies in the latitude of Paris falling in a vacuum will describe 1,000,000,000,000,000;

feet to 14.7706353 feet.

Mathematical Principles

328 15 Paris feet,

1

inch, 4^/^3 lines, very nearly, in one second of time.

And

if

the

gravity be diminished by taking away a quantity equal to the centrifugal force arising in that latitude from the earth's diurnal motion, heavy bodies falling there will describe in one second of time 15 feet, 1 inch, and 13^ lines. And mth this velocity heavy bodies do really fall in the latitude of Paris, as we have shown above in Props. 4 and 19. Cor. VIII. The mean distance of the centres of the earth and moon in the syzygies of the moon is equal to 60 of the greatest semidiameters of the earth, subtracting only about one 30th part of a semidiameter; and in the moon's quadratures the mean distance of the same centres is 60/^ such semidiameters of the earth; for these two distances are to the mean distance of the moon in the octants as 69 and 70 to 693/^, by Prop. 28. Cor. IX. The mean distance of the centres of the earth and moon in the syzygies of the moon is 60 mean semidiameters of the earth, and a 10th part of one semidiameter and in the moon's quadratures the mean distance of the same centres is 61 mean semidiameters of the earth, subtracting one 30th part of one ;

semidiameter. Cor. X. In the moon's syzygies its mean horizontal parallax in the latitudes of 0, 30, 38, 45, 52, 60, 90 degrees is 57' 20", 57' 16", 57' 14", 57' 12", 57' 10", 57' 8", 57' 4", respectively.

In these computations I do not consider the magnetic attraction of the earth, whose quantity is very small and unknown: if this quantity should ever be found out, and the measures of degrees upon the meridian, the lengths of isochronous pendulums in different parallels, the laws of the motions of the sea, and the moon's parallax, with the apparent diameters of the sun and moon, should be more exactly determined from phenomena: we should then be enabled to bring this calculation to a greater accuracy.

Proposition

To

38.

Problem

19

find the figure of the moon's body.

the moon's body were fluid like our sea, the force of the earth to raise that fluid in the nearest and remotest parts would be to the force of the moon by which our sea is raised in the places under and opposite to the moon as the accelerative gravity of the moon towards the earth is to the accelerative gravity of the earth towards the moon, and the diameter of the moon is to the diameter of the earth conjointly; that is, as 39.788 to 1, and 100 to 365 conjointly, or as 1081 to 100. Therefore, since our sea, by the force of the moon, is raised feet, the lunar fluid would be raised by the force of the earth to 93 feet; to and upon this account the figure of the moon would be a spheroid, whose greatest diameter produced would pass through the centre of the earth, and If

8%

exceed the diameters perpendicular thereto by 186 feet. Such a figure, thereq.e.i. fore, the moon possesses, and must have had from the beginning. Cor. Hence it is that the same face of the moon always is turned toward the earth; nor can the body of the moon possibly rest in any other position, but would return always by a libratory motion to this situation; but those librations, however, must be exceedingly slow, because of the weakness of the forces which excite them; so that the face of the moon, which should be always directed to the earth, may, for the reason assigned in Prop. 17, be turned towards

Book

III

:

The System of the World

329

the other focus of the moon's orbit, mthout being immediately drawn back, and turned again towards the earth.

Lemma

1

// APEp represent the earth uniformly dense, marked with the centre C, the poles P, p, and the equator AE; and if about the centre C, with the radius CP, we suppose to denote the plane on which a right line, the sphere Pape to he described, and drawn from the centre of the sun to the centre of the earth, stands at right angles; and

QR

further suppose that the several particles of the whole exterior earth PapAPepE, without the height of the said sphere, endeavor to recede towards this side and that side from the plane QR, every particle by a force proportional to its distance from

and efficacy of all the particles that are situated in AE, the circle of the equator, and disposed uniformly without the globe, encompassing the same after the manner of a ring, to wheel the earth about its centre, is to the whole force and efficacy of as many particles in that point A of the equator which is at the greatest distance from the plane QR, to wheel that plane; I say, in the first place, that the whole force

the earth about its centre with

a

like circular

motion as

is 1 to 2.

And

that circular

motion will be performed about an axis lying in the common section of the equator and the plane QR. For let there be described from the centre K, with the diameter IL, the semicircle INL. Suppose the semicircumference INL to be divided into into the diameter IL let fall numerable equal parts, and from the several parts will be equal all the sines the sines NM. Then the sums of the squares of

N

NM

M KM,

L

and both sums together mil be semidiameters KN; and therefore equal to the sums of the squares of as many will be but half so great as the sum the sum of the squares of all the sines of the squares of as many semidiameters KN. Suppose now the circumference of the circle AE to be divided into the like number of little equal parts, and from every such part F a perpendicular FG to be let fall upon the plane QR, as well as the perpendicular AH from the point A. Then the force by which the particle F recedes from the plane QR will (by supposition) be as that perpendicular FG; and this force multiplied by the distance CG will represent the power of the particle F to turn the earth round its centre. And, therefore, the power of a particle in the place F will be to the power of a particle in the place A as FG GC is to AH HC that is, as FC^ to AC^ and therefore the whole power of all the particles F, in their proper places to the

sums

of the squares of the sines

NM

•

:

•

;

Mathematical Principles

330 F, will be to the of all the

FC^

power

is

to the

of the like

sum

number

of all the

A as the sum we have demon-

of particles in the place

AC^

strated before), as 1 to 2. And because the action of those particles

that

is

(by what

q.e.d.

exerted in the direction of lines perpendicularly receding from the plane QR, and that equally from each side of this plane, they will wheel about the circumference of the circle of the equator, together with the adherent body of the earth, round an axis which lies as as in that of the equator. well in the plane is

QR

Lemivl\ 2

The same things still supposed, I say, in the second place, that the total force or power of all the particles situated everywhere about the sphere to turn the earth about the said axis is to the whole force of the like number of particles, uniformly disposed round the whole circumference of the equator AE in the fashion of a ring, to turn the whole earth about with the like circidar motion as is 2 to 5. For let IK be any lesser circle parallel to the equator AE, and let LI be any two equal particles in this circle, situated ^^^thout the sphere Fape; and if upon the plane QR, which is at right angles vnth a radius dra\\Ti to the sun, we let fall the perpendiculars LM, Im, the total forces by which these particles recede will be proportional to the perpendiculars LM, Im. Let the from the plane right line hi be drawn parallel to the plane Fape, and bisect the same in X; and draw Nn parallel to through the point the plane QR, and meeting the perpendicand n; and upon the ulars LM, Im, in let fall the perpendicular XY. plane And the contrary forces of the particles L and I to wheel about the earth contrari\vise are as LM-MC, and Im-mC; that is, as

QR

X

N

QR

LN-MC+NM-MC, or

and In-mC-nm-mC; and

LNMC+NMMC,

LNmC-

NM-mC, andLN-Mm-NM-(MC+mC), the difference of the two, is the force of both taken together to turn the earth round. The positive part of this difference LN Mm, or 2LN-NX, is to 2AH-HC, the force of two particles of the same size situated in A, as LX^ to AC^; and the negative part NM- (MC-t-mC), or 2XY-CY, is to 2AH-HC, the force of the same two particles situated in A, as CX^ to AC^. And therefore the difference of the parts, that is, the force of the two particles L and I, taken together, to wheel the •

earth about, is to the force of two particles, equal to the former and situated in the place A, to turn in like manner the earth round, as LX^ — CX^ is to AC^. But if the circumference IK of the circle IK is supposed to be divided into an infinite number of little equal parts L, all the LX^ \\all be to the like number of IX^ as 1 to 2 (by Lem. 1) and to the same number of AC^ as IX^ is to 2AC2; and the same number of CX^ to as many AC^ as 2CX2 is to 2AC2. Therefore the united forces of all the particles in the circumference of the circle IK are to the joint forces of as many particles in the place A as IX^ — 2CX^ is to 2AC^; and therefore (by Lem. 1) to the united forces of as many particles in the circumfer;

ence of the

circle

AE as 1X^-20X2 is

to

ACl

Book

Now

III:

The System of the World

331

Pp, the diameter of the sphere, is conceived to be divided into an infinite number of equal parts, upon which a like number of circles IK are supposed to stand, the matter in the circumference of every circle IK will be as IX^; and therefore the force of that matter to turn the earth about mil be as IX^ into 1X^ — 20X2; ^^^^ ^j^g force of the same matter, if it was situated in the circumference of the circle AE, would be as IX^ into AC^. And therefore the force of all the particles of the whole matter situated without the sphere in the circumferences of all the circles is to the force of the like number of particles situated in the circumference of the greatest circle AE as all the IX^ into IX2-2CX2 is to as many IX^ into AC^; that is, as all the AC^-CX^ into AC2-3CX2 to as many AC^-CX^ into AC^; that is, as all the AC*-4AC2. CX^+SCX" to as many AC"- AC^-CX^; that is, as the whole fluent quantity whose fluxion is AC"-4AC2-CX2+3CX^ is to the whole fluent quantity, whose fluxion is AC^ — AC^-CX^; and, therefore, by the method of fluxions, as AC^-CX-^AC^-CX^+^sCX^ is to AC^-CX-MAC^-CX^; that is, if for we write the whole Cp, or AC, as MsAC* is to %AC^; that is, as 2 is to 5. q.e.d. if

CX

Lemma

3

supposed, I say, in the third place, that the motion of the whole earth about the axis above named arising from the motions of all the particles, will be to the motion of the aforesaid ring about the same axis in a ratio compounded of the ratio of the matter in the earth to the matter in the ring; and the ratio of three squares of the quadrantal arc of any circle to two squares of its diameter, that is, in the

The same things

still

ratio of the matter to the matter,

and

of the

number 925,275 to

the

number 1,000,000.

For the motion of a cylinder revolved about its quiescent axis is to the motion of the inscribed sphere revolved together with it as any four equal squares are to three circles inscribed in three of those squares, and the motion

an exceedingly thin ring surrounding both sphere and cylinder in their common contact as double the matter in the cylinder is to triple the matter in the ring; and this motion of the ring, uniformly continued about the axis of the cylinder, is to the uniform motion of the same about its own diameter performed in the same periodic time as is the of this cylinder is to the

motion

of

circumference of a circle to double

its

diameter.

HYPOTHESIS if

II

the other parts of the earth were taken away, and the remaining ring was carried alone about the sun in the orbit of the earth by the annual motion, while by the diurnal motion it was in the meantime revolved about its own axis inclined to the plane of the ecliptic by an angle of 233^ degrees, the motion of the equinoctial points would be the same, whether the ring were fluid, or whether it consisted of a hard and rigid matter. Proposition

39.

Problem 20

To find the precession of the equinoxes. The middle hourly motion of the moon's nodes in a circular orbit, when the nodes are in the quadratures, was 16" 35'" 16'^ 36^; the half of which, 8" 17'" 3giv igv (f Qj. ^]^g reasons above explained) is the mean hourly motion of the nodes in such an orbit, which motion in a whole sidereal year becomes 20° 11' 46". ,

Mathematical Principles

332

Since, therefore, the nodes of the moon in such an orbit would be yearly transferred 20° 11' 46" backwards, and, if there were more moons, the motion

nodes of every one (by Cor. xvi, Prop. 66, Book i) would be as its periif upon the surface of the earth a moon was revolved in the time of a sidereal day, the annual motion of the nodes of this moon would be to 20° 11' 46" as 23^. 56'"., the sidereal day, is to 27"^. 7^. 43"^., the periodic time of our moon, that is, as 1436 is to 39,343. And the same thing would happen to the nodes of a ring of moons encompassing the earth, whether these moons did not mutually touch each the other, or whether they were molten, and formed into a continued ring, or whether that ring should become rigid and inflexible. Let us, then, suppose that this ring is in quantity of matter equal to the whole exterior earth PapAPepE, which lies without the sphere Fape (see Fig., Lem. 2) and because this sphere is to that exterior earth as aC^ is to AC^ — aC^, that is (seeing PC or aC the least semidiameter of the earth is to AC the greatest semidiameter of the same as 229 is to 230), as 52,441 is to 459; if this ring encompassed the earth round the equator, and both together were revolved about the diameter of the ring, the motion of the ring (by Lem. 3) would be to the motion of the inner sphere as 459 to 52,441 and 1,000,000 to 925,275 conjointly, that is, as 4590 to 485,223; and therefore the motion of the ring would be to the sum of the motions of both ring and sphere as 4590 is to 489,813. Therefore, if the ring adheres to the sphere, and communicates its motion to the sphere, by which its nodes or equinoctial points recede, the motion remaining in the ring will be to its former motion as 4590 is to 489,813; on account of which the motion of the equinoctial points will be diminished in the same ratio. Therefore, the annual motion of the equinoctial points of the body, composed of both ring and sphere, will be to the motion 20° 11' 46" as 1436 to 39,343 and 4590 to 489,813 conjointly, that is, as 100 to 292,369. But the forces by which the nodes of a number of moons (as we explained above), and therefore by which the equinoctial points of the ring recede (that is, the forces 3IT, in Fig., Prop. 30), are in the several particles as the distances of those particles from the plane QR; and by these forces the particles recede from that plane: and therefore (by Lem. 2) if the matter of the ring was spread all over the surface of the sphere, after the fashion of the figure PapAPepE, in order to make up that exterior part of the earth, the total force or power of all the particles to wheel about the earth round any diameter of the equator, and therefore to move the equinoctial points, would become less than before in the proportion of 2 to 5. Therefore the annual regress of the equinoxes now would be to 20° 11' 46" as 10 is to 73,092; that is, would be 9" 56'" 50'^. of the

odic time,

;

But because the plane

of the

equator

is

inclined to that of the ecliptic, this

motion is to be diminished in the ratio of the sine 9 1 706 (which is the cosine of 23 3^ degrees) to the radius 100,000; and the remaining motion will now be 9" 7'" 20'^, which is the annual precession of the equinoxes arising from the force of the sun. But the force of the moon to move the sea was to the force of the sun nearly as 4.4815 to 1 and the force of the moon to move the equinoxes is to that of the sun in the same proportion. Whence the annual precession of the equinoxes proceeding from the force of the moon comes out 40" 52'" 52'"^, and the total annual precession arising from the united forces of both will be 50" 00'" 12'^, the amount of which motion agrees with the phenomena; for the precession of the equinoxes, by astronomical observations, is about 50" yearly. ,

;

Book

III:

The System of the World

333

the height of the earth at the equator exceeds its height at the poles by miles, the matter thereof \vill be more rare near the surface than at the centre; and the precession of the equinoxes will be augmented by the excess of height, and diminished by the greater rarity. If

more than 17/^

And now we have planets,

it

described the system of the sun, the earth, moon, and remains that we add something about the comets.

Lemma

4

The comets are more remote than the moon, and are in the regions of the planets. As the comets were placed by astronomers beyond the moon, because they were found to have no diurnal parallax, so their annual parallax is a convincing proof of their descending into the regions of the planets; for all the comets which move in a direct course according to the order of the signs, about the end of their appearance become more than ordinarily slow or retrograde, if the earth is between them and the sun; and more than ordinarily swift, if the earth is approaching to a heliocentric opposition with them; on the other hand, those which move against the order of the signs, towards the end of their appearance appear swifter than they ought to be, if the earth is between them and the sun; and slower, and perhaps retrograde, if the earth is in the other side of its orbit. And these appearances proceed chiefly from the diverse situations which the earth acquires in the course of its motion, after the same manner as it happens to the planets, which appear sometimes retrograde, sometimes more slowly, and sometimes more smftly, progressive, according as the motion of the earth falls in with that of the planet, or is directed in the contrary way. If the earth move the same way with the comet, but, by an angular motion about the sun, so much swifter that right lines drawn from the earth to the comet converge towards the parts beyond the comet, the comet seen from the earth, because of its slower motion, will appear retrograde; and even if the earth is slower than the comet, the motion of the earth being subtracted, the motion of the comet Avill at least appear retarded; but if the earth tends the contrary way to that of the comet, the motion of the comet will from thence appear accelerated; and from this apparent acceleration, or retardation, or regressive motion, the distance of the comet may be inferred in this manner. Let TQA, TQB, TQC be three observed longitudes of the comet about the time of its first appearing, and TQF its observed longitude before its disappearing. Draw the right line ABC,

last

whose parts AB, BC, intercepted between the right lines QA and QB, QB and QC, may be one to the other as the two times between the three first observations. Produce AC to G, so that AG may be to AB as the time between the first and last observations is to the time between the first and second; and join QG. Now if the comet did move uniformly in a right line, and the earth either stood still, or was likewise carried forwards in a right line by an uniform motion, the angle TQG would be the longitude of the comet at the time of the last observation. Therefore, the angle FQG, which is the difference of the longitude, proceeds from the inequality

334

Mathematical Principles

motions of the comet and the earth; and if the earth and comet move contrary ways, this angle is added to the angle TQG, and accelerates the apparent motion of the comet; but if the comet moves the same way with the earth, it is subtracted, and either retards the motion of the comet, or perhaps renders it retrograde, as we have just now explained. This angle, therefore, proceeding chiefly from the motion of the earth, is justly to be esteemed the parallax of the comet, there being neglected thereby some little increment or decrement that may arise from the unequal motion of the comet in its orbit. From this parallax we thus deduce the distance of the comet. Let S represent of the

the sun, acT the great orbit, a the earth's place in the first observation, c the place of the earth in the third observation, T the place of the earth in the last observation, and TT a right line drawn to the beginning of Aries. Set off the angle TTV

\\.

^^%

equal to the angle TQF, that is, equal to the longitude of the comet at the time when the earth is in T; join ac, and produce it to g, so that ag may be to ac as AG is to AC; and g will be the place at which the earth would have arrived in the time of the last observation, if it had continued to move uniformly in the right parallel line ac. Therefore, if we draw to TT, and make the angle TgrV equal to the angle TQG, this angle TgrV will be equal to the longitude of the comet seen from the place g, and the angle TVgr will be the parallax which arises from the earth's being transferred from the place g into the place T; and therefore V will be the place of the comet in the plane of the ecliptic. And this place V is commonly lower than the orbit of Jupiter. The same thing may be deduced from the incurvation of the way of the comets; for these bodies move almost in great circles, while their velocity is great; but about the end of their course, when that part of their apparent motion which arises from the parallax bears a greater proportion to their whole apparent motion, they commonly deviate from those circles, and when the earth goes to one side, they deviate to the other; and this deflection, because of its corresponding with the motion of the earth, must arise chiefly from the parallax; and the quantity thereof is so considerable, as, by my computation, to place the disappearing comets a good deal lower than Jupiter. Hence it follows that when they approach nearer to us in their perigees and perihelions they often descend below the orbits of Mars and the inferior planets. The near approach of the comets is further confirmed from the light of their heads; for the light of a celestial body, illuminated by the sun, and receding to remote parts, diminishes as the fourth power of the distance; namely, as the square, on account of the increase of the distance from the sun, and as another square, on account of the decrease of the apparent diameter. Therefore, if both the quantity of light and the apparent diameter of a comet are given, its distance will be given also, by taking the distance of the comet to the distance of a planet directly as their diameters and inversely as the square root of their lights. Thus, in the comet of the year 1682, Mr. Flamsteed observed with a

^

Book

III

:

The System of the World

335

and measured with a micrometer, the least diameter of its 00" but the nucleus or star in the middle of the head scarcely amounted head, to the tenth part of this measure, and therefore its diameter was only 11" or 12"; but in the Ught and splendor of its head it surpassed that of the comet in the year 1680, and might be compared with the stars of the first or second magnitude. Let us suppose that Saturn with its ring w^as about four times more lucid; and because the light of the ring was almost equal to the light of the globe mthin, and the apparent diameter of the globe is about 21", and therefore the united light of both globe and ring would be equal to the light of a globe whose diameter is 30", it follows that the distance of the comet was to the distance of Saturn inversely as 1 to a/4, and directly as 12" to 30"; that is, as 24 to 30, or 4 to 5. Again; the comet in the month of April, 1665, as Hewelcke informs us, excelled almost all the fixed stars in splendor, and even Saturn itself, as being of a much more vivid color; for this comet was more lucid than that other which had appeared about the end of the preceding year, and had been compared to the stars of the first magnitude. The diameter of its head was about 6' but the nucleus, compared A^ith the planets by means of a telescope, was plainly less than Jupiter; and sometimes judged less, sometimes judged equal, to the globe of Saturn within the ring. Since, then, the diameters of the heads of the comets seldom exceed 8' or 12', and the diameter of the nucleus or central star is but about a tenth or perhaps fifteenth part of the diameter of the head, it appears that these stars are generally of about the same apparent magnitude with the planets. But since their light may be often compared A\dth the light of Saturn, yea, and sometimes exceeds it, it is evident that all comets in their perihelions must either be placed below or not far above Saturn; and they are much mistaken w^ho remove them almost as far as the fixed stars; for if it were so, the comets could receive no more light from our sun than our planets do from the fixed stars. So far we have gone, mthout considering the obscuration which comets suffer from that plenty of thick smoke which encompasses their heads, and through which the heads always show dull, as through a cloud. But the more a body is obscured by this smoke, the nearer must it be allowed to come to the sun, that it may vie with the planets in the quantity of light which it reflects. Hence it is probable that the comets descend far below the orbit of Saturn, as we proved before from their parallax. But, above all, the thing is evinced from their tails, which must be due either to the sun's light reflected by a smoke arising from them, and dispersing itself through the ether, or to the light of their own heads. In the former case, we must shorten the distance of the comets, lest we be obliged to allow that the smoke arising from their heads is propagated through such a vast extent of space, and mth such a velocity and expansion as ^^^ll seem altogether incredible; in the latter case, the whole light of both head and tail is to be ascribed to the central nucleus. But, then, if we suppose all this light to be united and condensed within the disk of the nucleus, certainly the nucleus will by far exceed Jupiter itself in splendor, especially when it emits a very large and lucid tail. If, therefore, under a less apparent diameter, it reflects more light, it must be much more illuminated by the sun, and therefore much nearer to it; and the same argument mil bring down the heads of comets sometimes ^Wthin the orbit of Venus, viz., when, being hid under the sun's rays, they emit such huge and splendid tails, like beams of fire, as sometimes they do; for if all telescope of 16 feet, 2'

;

;

336

Mathematical Principles

that light was supposed to be gathered together into one star, it would sometimes exceed not one Venus only, but a great many such united into one. Lastly, the same thing is inferred from the light of the heads, which increases in the recess of the comets from the earth towards the sun, and decreases in their return from the sun towards the earth. Thus, the comet of the year 1665 (by the observations of Hewelcke), from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee; but yet the splendor of its head was daily increasing, till, being hid under the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observations of the same Hewelcke), about the end of July, when it first appeared, moved at a very slow rate, advancing only about 40 or 45 minutes in its orbit in a day's time; but from that time its diurnal motion was continually upon the increase, till September 4, when it arose to about 5 degrees; and therefore, in all this interval of time, the comet was approaching to the earth. This is Hkewise proved from the diameter of its head, measured with a micrometer; for, on August 6, Hewelcke found it only 6' 5", including the coma, which, on September 2, he observed to be 9' 7"; and therefore its head appeared far less about the beginning than towards the end of the motion, though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hewelcke declares. Therefore in all this interval of time, on account of its recess from the sun, it decreased in splendor, notwithstanding its approach towards the earth. The comet of the year 1618, about the middle of December, and that of the year 1680, about the end of the same month, did both move with their greatest velocity, and were therefore then in their perigees, but the greatest splendor of their heads was seen two weeks before, when they had just got clear of the sun's rays, and the greatest splendor of their tails a little earlier, when yet nearer to the sun. The head of the former comet (according to the observations of Cysat), on December 1, appeared greater than the stars of the first magnitude; and, on December 16 (then in the perigee), it was diminished but little in magnitude, but much diminished in the splendor and brightness of its Hght. On January 7, Kepler, being uncertain about the head, left off observing. On December 12, the head of the latter comet was seen and observed by Mr. Flamsteed, when but 9 degrees distant from the sun, which is scarcely to be done in a star of the third

On December

and 17, it appeared as a star of the third magnitude, its luster being diminished by the brightness of the clouds near the setting sun. On December 26, when it moved with the greatest velocity, being almost in its perigee, it was less than the mouth of Pegasus, a star of the third magnitude. On January 3, it appeared as a star of the fourth. On January 9, as one of the fifth. On January 13, it was hid by the splendor of the moon, then in her increase. On January 25, it was scarcely equal to the stars of the seventh magnitude. If we compare equal intervals of time, taken on one side of the perigee and then on the other, we shall find that the head of the comet, which at both intervals of time was far, but yet equally removed from the earth, and should therefore have shone with equal splendor, appeared brightest on the side of the perigee towards the sun, and disappeared on the other. Therefore, from the great difference of light in the one situation and in the other, we conclude the great vicinity of the sun and comet in the former, for the light of comets tends to be regular, and to appear greatest when the heads move fastmagnitude.

15

Book and are therefore

III

:

The System of the World

337

it is increased by their nearness to the sun. Cor. I. Therefore the comets shine by the sun's Ught, which they reflect. Cor. II. From what has been said, we may likewise understand why comets are so frequently seen in that region in which the sun is, and so seldom in the other. If they were visible in the regions far above Saturn, they would appear more frequently in the parts opposite to the sufi for such as were in those parts would be nearer to the earth, whereas the presence of the sun must obscure and hide those that appear in the region in which he is. Yet, looking over the history of comets, I find that four or five times more have been seen in the hemisphere towards the sun than in the opposite hemisphere; besides, mthout doubt, not a few, which have been hid by the fight of the sun: for comets descending into our parts neither emit tails, nor are so well illuminated by the sun as to reveal themselves to our naked eyes, until they have come nearer to us than Jupiter. But the far greater part of that spherical space, which is described about the sun -with so small a radius, lies on that side of the earth which faces the sun; and the comets in that greater part are commonly more strongly illuminated, for they are for the most part nearer to the sun. CoR. III. Hence also it is evident that the celestial spaces are void of resistance; for though the comets are carried in oblique paths, and sometimes contrary to the course of the planets, yet they move every way vnth. the greatest freedom, and preserve their motions for an exceeding long time, even where contrary to the course of the planets. I am out in my judgment, if they are not a sort of planets revolving in orbits returning into themselves \dth a continual motion; for the opinion of some writers, that they are no other than meteors, an opinion based on the continual changes that happen to their heads, seems to have no foundation; for the heads of comets are encompassed with huge atmospheres, and the lowermost parts of these atmospheres must be the densest; and therefore it is in the clouds only, not in the bodies of the comets themselves, that these changes are seen. Thus the earth, if it were viewed from the planets, would, without all doubt, shine by the fight of its clouds, and the sofid body would scarcely appear through the surrounding clouds. Thus also the belts of Jupiter are formed in the clouds of that planet, for they change their position to each other, and the solid body of Jupiter is hardly to be seen through them; and much more must the bodies of comets be hid under their atmospheres, which are both deeper and thicker.

est,

in their perigees, except so far as

;

Proposition move

40.

Theorem

20

some of the conic sections, having their foci in the centre of to the sun describe areas proportional to the times. This Proposition appears from Cor. i, Prop. 13, Book i, compared with

That

the comets

the sun,

Props.

and by

and Hence

8, 12,

Cor. I. orbits vdW be

in

radii

13,

drawn

Book

iii.

comets revolve in orbits returning into themselves, the elfipses; and their periodic times \^-ill be to the periodic times of the planets as the /^th power of their principal axes. And therefore the comets, which for the most part of their course are more remote than the planets, and upon that account describe orbits with greater axes, vdW require a longer time if

to finish their revolutions. Thus if the axis of a comet's orbit was four times greater than the axis of the orbit of Saturn, the time of the revolution of the

Mathematical Principles

338

comet would be to the time of the revolution of Saturn, that is, to 30 years, as 4V4 (or 8) is to 1, and would therefore be 240 years. Cor. II. But their orbits mil be so near to parabolas, that parabolas may be used for them Avithout sensible error. Cor. III. And, therefore, by Cor. vii. Prop. 16, Book i, the velocity of every comet will always be to the velocity of any planet, supposed to be revolved at the same distance in a circle about the sun, nearly as the square root of double the distance of the planet from the centre of the sun to the distance of the comet from the sun's centre. Let us suppose the radius of the great orbit, or the greatest semidiameter of the ellipse which the earth describes, to consist of 100,000,000 parts; and then the earth by its mean diurnal motion mil describe 1,720,212 of those parts, and 71,6753/2 by its hourly motion. And therefore the comet, at the same mean distance of the earth from the sun, with a velocity which is to the velocity of the earth as \/2 to 1, would by its diurnal motion describe 2,432,747 parts, and 101,3643^ parts by its hourly motion. But at greater or less distances both the diurnal and hourly motion mil be to this diurnal and hourly motion inversely as the square root of the distances, and is therefore given.

Cor.

the latus rectum of the parabola is four times the radius of the great orbit, and the square of that radius is supposed to consist of 100,000,000 parts, the area which the comet will daily describe by a radius drawn to the sun will be 1,216,3733^2 parts, and the hourly area will be 50,6823^ parts. But, if the latus rectum is greater or less in any ratio, the diurnal and hourly area will be less or greater inversely as the square root of that ratio. IV.

Therefore

if

Lemma To find a curved number of points.

line of the parabolic

5

kind which shall pass through any given

Let those points be A, B, C, D, E, F, &c., and from the same to any right line HN, given in position, let fall as many perpendiculars AH, BI, CK, DL,

EM, FN,

&c.

46

36

26 2c

d

3c

2d

56 4c

Sd 2e

f HI, IK, KL, &c., the intervals of the points H, I, K, L, M, N, &c., are equal,

Case

take

6,

1.

If

26, 36, 46, 56, &c., the first differences

&c.; their third,

CK = 26,

d,

AH,

CK,

&c. their second differences, c, 2c, 3c, 4c, 2d, 3d, &c., that is to say, so as — BI may be = 6, BI —

of the perpendiculars

BI,

;

AH

CK-DL = 36, DL+EM=46,

-EM+FN = 56, &c.;

then

6-26 = c,

and so on to the last difference, which is here /. Then, erecting any perpendicular RS, which may be considered as an ordinate of the curve required, &c.,

in order to find the length of this ordinate,

LM,

&c., to

+SK = ME,

r,

be units, and

}/ir

let

suppose the intervals HI, IK, KL,

AH = a, — HS = p,

into+SL = s, y^s nto-|-SM = ^;

3^p into — IS = g, }iq into proceeding in this manner, to

the last perpendicular but one, and prefixing negative signs before the terms HS, IS, &c., which lie from S towards A; and positive signs before the

Book

III

The System of the World

:

339

terms SK, SL, &c., which He on the other side of the point S; and, observing well the signs, RS will be =a-\-hp-\-cq-\-dr-\-es-\-ft, +&c. Case 2. But if HI, IK, &c., the intervals of the points H, I, K, L, &c., are unequal, take b, 26, 36, 46, 56, &c., the first differences of the perpendiculars AH, BI, CK, &c., divided by the intervals between those perpendiculars; c, 2c, 3c, 4c, &c., their second differences, divided by the intervals between every two; d, 2d, Sd, &c., their third differences, divided by the intervals between every three; e, 2e, &c., their fourth differences, divided by the intervals between A

every four; and so forth; that „,

26 =

is,

in such

manner, that

6

may

6-26 ^ 26-36 CK-DL „, = -^^, -j^-, 36 = -^^, &c., then c-^^,2c

BI-CK Q

„

2c

2c

,,

^

be

3c

=

XT

— TDT

=j=

HI 36-46 , =

-^^,

&c.,

3c

d= 2d^ &c. And those differences being found, let AH be liVl ML = a, — HS===p, pinto —l^ = q, q'm.io +SK = r, rinto +SL = s, sinto +SM = ^; proceeding in this manner to ME, the last perpendicular but one; and the ordinate RS will be = a+6p+cg+dr+es+/^+&c. Cor. Hence the areas of all curves may be nearly found; for if some number then

,

,

be squared are found, and a parabola be supposed to be dra^vn through those points, the area of this parabola will be nearly the same with the area of the curvilinear figure proposed to be squared but the parabola can be always squared geometrically by methods generally known. of points of the curve to

:

Lemma

6

Certain observed places of a comet being given, intermediate given time.

Let HI, IK, KL,

LM

to

find the place of the same at any

the preceding Fig.) represent the times between the observations; HA, IB, KC, LD, ME, five observed longitudes of the comet; and HS the given time between the first observation and the longitude required. Then if a regular curve is supposed to be drawn through the (in

ABCDE

points A, B, C, D, E, and the ordinate RS ma, RS will be the longitude required.

By the same method, from five

is

found out by the preceding Lem-

observed latitudes, we

may

find the latitude

at a given time.

the differences of the observed longitudes are small, let us say 4 or 5 degrees, then three or four observations will be sufficient to find a new longitude and latitude; but if the differences are greater, as of 10 or 20 degrees, five observations ought to be used. Lemma 7 If

Through a given point P

to

drawn a

BC, whose parts PB, PC, cut AB, AC, given in position, may

right line

off

by two right lines be one to the other in a given ratio. From the given point P suppose any right line to be drawn to either of the right fines given, as AB and produce the same towards AC, the other given right line, as far as E, so as PE may be to PD in the given ratio. Let EC be parallel to AD. Draw CPB,

PD

and

PC

will

be to

PB

as

PE

to

PD. q.e.f.

Mathematical Principles

340

Lemma

8

he a parabola, having its focus in S. By the chord AC bisected in I cut ABCI, whose diameter is I/x and vertex ix. In \^l produced take segment off the /iO equal to one-half of Im- Join OS, and produce it to |, so that S^ may be equal to 2S0. Now, supposing a comet to revolve in the arc CBA, draw ^B, cutting AC E / say, the point E will cut off from the chord AC the segment AE, nearly

Let

ABC

m

:

proportional

to the time.

we join EO, cutting the parabolic arc ABC in Y, and draw /xX touching the same arc in the vertex ti, and meeting EO in X, the curviUnear area AEXjuA will be to the curvilinear area ACY^A as AE to AC; and, therefore, For

if

since the triangle

ASEX/iA is

to

SO

will

ASE is

to the triangle

be to the whole area

as 3 to

1,

and

EO

to

ASC in

ASCY/xA

the same ratio, the whole area as AE is to AC. But, because |0

XO in the same ratio, SX will be parallel to EB;

and, therefore, joining BX, the triangle SEB will be equal to the triangle XEB. Therefore, if to the area ASEX//A we add the triangle EXB, and from the sum subtract the triangle SEB, there will remain the area ASBX/xA, equal to the area ASEX^A, and therefore in the ratio to the area ASCY^A as AE to AC. But the area ASBY/xA is nearly equal to the area ASBX^tA; and this area ASBY^tA is to the area ASCY^uA as the time of description of the arc AB is to the time of description of the whole arc AC; and, therefore, AE is to AC nearly in the proportion of the times. q.e.d. Cor. When the point B falls upon the vertex /i of the parabola, AE is to AC accurately in the proportion of the times.

Scholium ^u^ cutting AC in 5, and in it take ^n in proportion to /xB as 27MI and draw Bn, this Bn will cut the chord AC, in the ratio of the times, more accurately than before; but the point n is to be taken beyond or on this side the point ^, according as the point B is more or less distant from the prin-

If

we

join

to 16 Mju,

cipal vertex of the parabola

than the point

Lemma The

right lines \n

For

and ^M, and

4Sju is the latus

rectum

/x.

9

AP

the length

7a~, are equal among themselves.

of the parabola belonging to the vertex

/x.

Book

III

:

The System of the World Lemma /xN may he

341

10

one-third of lA, and SP may he to SN N and P, so that time would describe the arc AjuC, if it was supand in the that a comet as SN to S/Lt; posed to move always forwards with the velocity which it has in a height equal to SP, it would descrihe a length equal to the chord AC. For if the comet with the velocity which it hath in /x was in the said time supposed to move uniformly forwards in the right line which touches the parabola in ix, the area which it would describe by a radius drawn to the point S would be equal to the parabolic area ASC^uA; and therefore the space contained under the length described in the tangent and the length Sm would be to the space contained under the lengths AC and SM as the area ASC/xA is to

Produce Sm

to

the triangle ASC, that is, as SN to SM. Therefore AC is to the length described in the tangent as Sju to SN. But since the velocity of the comet in the height SP (by Cor. vi. Prop. 16, Book i) is to the velocity of the same in the height S/x, inversely as the square root of SP to Sm, that is, in the ratio of S/x to SN, it follows that the length described with this velocity will be to the length in the same time described in the tangent, as S/x to SN. Therefore, since AC, and the length described with this new velocity, are in the same proportion to the length described in the tangent, q.e.d. they must be equal between themselves. hath in the height velocity which it Cor. Therefore a comet, with that S/x+^I/x, would in the same time nearly describe the chord AC.

Lemma // a comet void of

all

motion was

let

11

fall from the height

SN,

or Sfx-\-}/^lfx, towards

and was still impelled to the sun hy the same force uniformly continued by which it was impelled at first, the same, in one-half of that time in which it might descrihe the arc AC in its own orhit, would in descending descrihe a space equal to the length I/x. For in the same time that the comet would require to describe the parabolic arc AC, it would (by the last Lemma), with that velocity which it hath in the height SP, describe the chord AC; and, therefore (by Cor. vii. Prop. 16, Book i), if it was in the same time supposed to revolve by the force of its own gravity in a circle whose semidiameter was SP, it would describe an arc of that circle, the length of which would be to the chord of the parabolic arc AC in the ratio of 1 to v'2. Therefore, if with that weight, which in the height SP it hath towards the sun, it should fall from that height towards the sun, it would (by Cor. ix, Prop. 16, Book i) in half the said time describe a space equal to the square of half the said chord, divided by four times the height SP, that is, it would describe the space the sun,

Mathematical Principles

342

-^^. But since the weight of the comet towards the sun in the height

SN

is

to the weight of the same towards the sun in the height SP as SP to S^, the comet, by the weight which it hath in the height SN, in falling from that

height towards the sun, would in the same time describe the space -^^ that •

4b/i'

is,

a space equal to the length I^ or /zM.

Proposition

From

41.

q.e.d.

Problem

21

three given observations to determine the orbit of a comet

moving in a

parabola.

This being a Problem of very great difficulty, I tried many methods of resolving it; and several of those Problems, the composition whereof I have given in the first book, tended to this purpose. But afterwards I contrived the

somewhat more simple. Select three observations distant one from another by intervals of time nearly equal; but let that interval of time in which the comet moves more slowly be somewhat greater than the other; namely, so that the difference of follo\ving solution,

which

is

the times may be to the sum of the times as the sum of the times is to about and may err therefrom 600 days; or that the point E may fall nearly upon rather towards I than towards A. If such direct observations are not at hand, a new place of the comet must be found, by Lemma 6. Let S represent the sun; T, t, r three places of the earth in the earth's orbit; TA, tB, rC three observed longitudes of the comet; V the time between the the time between the second and the third; first observation and the second; the comet might describe with the length which in the whole time that velocity which it has in the mean distance of the earth from the sun, which length is to be found by Cor. iii, Prop, xl. Book iii; and tY a perpendicular upon the chord Tr. In the mean observed longitude iB take at pleasure

M

W

X

n

-

V+W

...

the point B, for the place of the comet in the plane of the ecliptic; and from thence, towards the sun S, draw the Hne BE, which may be to the perpendicular ^V as the product of SB and S^^ is to the cube of the hypothenuse of the right-angled triangle whose sides are SB and the tangent of the latitude of the

comet in the second observation to the radius iB. And through the point E (by Lem. 7) draw the right hne AEC, whose parts AE and EC, terminating in the right lines TA and rC, may be one to the other as the times V and W: then A and C will be nearly the places of the comet in the plane of the ecUptic in the first and third observations, if B was its place rightly assumed in the second. Upon AC, bisected in I, erect the perpendicular \i. Through B imagine the

Book

III:

The System of the World

343

AC. Imagine the line St drawn, cutting AC in X, and complete the parallelogram ilXn. Take la- equal to SIX; and through the sun S imagine the line a^ drawn equal to 3S(T+3tX. Then, canceling the letters A, E, C, I, from the point B towards the point ^, imagine the new line BE drawn, line

Bi drawn

parallel to

which may be to the former BE as the square of the ratio of the distance BS to the quantity S/x+J/^iX. And through the point E draw again the right Hnc AEC by the same rule as before; that is, so that its parts AE and EC may be one to the other as the times V and between the observations. Thus A and C will be the places of the comet more accurately. Upon AC, bisected in I, erect the perpendiculars AM, CN, 10, of which and may be the tangents of the latitudes in the first and third observations, to the radii TA and tC. Join MN, cutting 10 and O. Draw the rectangular parallelogram HX/jl, as before. In lA produced take ID equal to S/i+ %iX. Then in MN, towards N, take MP, which may be to the above-found length as the square root of the ratio of the mean distance of the earth from the sun (or of the semidiameter of the earth's orbit) to the distance OD. If the point P fall upon the point N; A, B, and C will be three places of the comet, through which its orbit is to be described in the plane of the ecliptic. But if the point P falls not upon the point N, in the right line AC take CG equal to NP, so that the points G and P may lie on the same side of the line NC. By the same method as the points E, A, C, G were found from the assumed point B, from other points h and /3 assumed at pleasure, find out the new points e, a, c, g; and e, a, k, y. Then through G, g, and y draw the circumference of a circle Ggy, cutting the right line rC in Z: and Z will be one place of the comet in the plane of the ecliptic. And in AC, ac, aK, taking AF, af, a4>, equal respectively to CG, eg, Ky through the points F, /, and , draw the circumference of a circle F/0, cutting the right line AT in X; and the point will be another place of the comet in the plane of the ecliptic. And at the points and Z, erecting the tangents of the latitudes of the comet to the radii and rZ, two places of the comet in its own orbit will be determined. Lastly, if (by Prop. 19, Book i) to the focus S a parabola is described passing through those two places, this parabola will be the orbit of the comet. q.e.i. The demonstration of this construction follows from the preceding Lemmas, because the right line AC is cut in E in the proportion of the times, by Lemma

W

AM

CN

X

;

X

X

TX

Mathematical Principles

344 ought to be by

Lemma

and BE, by

Lemma

a portion of the right line BS or B^ in the plane of the ecliptic, intercepted between the arc (by Cor., Lem. 10) is the length of the ABC and the chord AEC; and chord of that arc, which the comet should describe in its proper orbit between the first and third observations, and therefore is equal to MN, providing B is a true place of the comet in the plane of the ecliptic. But it will be convenient to assume the points B, h, j3, not at random, but nearly true. If the angle AQt, at which the projection of the orbit in the plane of the ecliptic cuts the right line tB, is roughly known, at that angle mth B^ draw the line AC, which may be to iTr as the square root of the ratio of SQ to St; and, drawing the right line SEB so as its part EB may be equal to the length V^, the point B will be determined, which we are to use for the first time. Then, canceUng the right line AC and drawing anew AC according to the preceding construction, and, moreover, finding the length MP, in tB take the point b, by this rule, that, if TA and tC intersect each other in Y, the distance to Y6 may be to the distance YB in a ratio compounded of the ratio of MN, and the square root of the ratio of SB to S6. And by the same method you may find the third point /3, if you please to repeat the operation the third time; but if this method is followed, two operations generally will be sufficient; for if the distance B6 happens to be very small, after the points F, /, and G, g, are found, draw the right lines Ff and Gg, and they will cut TA and rC in the and Z. points required, 7,

as

it

8;

11, is

MP

MP

X

EXAMPLE Let the comet of the year 1680 be proposed. The following table shows the motion thereof, as observed by Flamsteed, and calculated afterwards by him from his observations, and corrected by Dr. Halley from the same observations.

Book

III:

The System of the World

345

Mathematical Principles

346

Mr. Pound has since observed a second time the positions of these fixed stars amongst themselves, and obtained their longitudes and latitudes according to the preceding table.

The

positions of the

comet to these

fixed stars

were observed to be as

follows

Friday, February 25,

o.s.,

at 8}^^. p.m., the distance of the

comet

in

p from

the star E was less than /^s AE, and- greater than Vs-'^E, and therefore nearly equal to ^/uAE; and the angle ApE was a little obtuse, but almost right. For from A, letting fall a perpendicular on pE, the distance of the comet from that perpendicular was /^pE. The same night, at 93^^., the distance of the comet in P from the star E was greater than 3^^^^^, and less than /^j^AE, and therefore nearly equal to /^j^ of AE, or 3/39 AE. But the distance of the comet from the perpendicular let fall from the star A upon the right line PE was %PE. Sunday, February 27, 8^^^. p.m., the distance of the comet in Q from the star and the right line QO proO was equal to the distance of the stars and and B. I could not, by reason of intervening duced passed between the stars clouds, determine the position of the star to greater accuracy. Tuesday, March 1, 11^. p.m., the comet in R lay exactly in a line between was a little greater the stars and C, so as the part CR of the right line than HCK, and a Httle less than 3^CK + ^CR, and therefore = ^CK+VieCR,

H

;

K

K

or

CRK

i%5CK.

Wednesday, March 2, 8^. p.m., the distance of the comet in S from the star C was nearly %FC; the distance of the star F from the right line CS produced was /^4FC; and the distance of the star B from the same right Une was five times greater than the distance of the star F; and the right line NS produced passed between the stars H and I five or six times nearer to the star H than to the star I. Saturday, March 5, ll}^,^- p-M., when the comet was in T, the right line was equal to 3^ML, and the right line LT produced passed between B and F four or five times nearer to F than to B, cutting off from BF a fifth or sixth part thereof towards F; and produced passed on the outside of the space BF towards the star B four times nearer to the star B than to the star F. was a very small star, scarcely to be seen by the telescope; but the star L was greater, and of about the eighth magnitude. Monday, March 7, 93^*^. p.m., the comet being in V, the right line Va produced did pass between B and F, cutting off, from BF towards F, Vio of BF, and was to the right line V/S as 5 to 4. And the distance of the comet from the

MT

MT

M

a/S was 3^VjS. Wednesday, March 9, 83^^. p.m., the comet being in X, the right line 7X was equal to 34t^; and the perpendicular let fall from the star 8 upon the right line 7X was of yd. The same night, at 12^^., the comet being in Y, the right line 7Y was equal to }/3 of yd, or a little less, as perhaps /^e of yd; and a perpendicular let fall from the star 8 on the right line 7Y was equal to about ]/e> or /^ y8. But the comet, being then extremely near the horizon, was scarcely discernible, and therefore its place could not be determined mth the same certainty as in the

right line

%

foregoing observations. From these observations,

by constructions

of figures

and

calculations, I

Book

III:

The System of the World

deduced the longitudes and latitudes

347

comet; and Mr. Pound, by correcting the places of the fixed stars, has determined more correctly the places of the comet, which correct places are set down above. Though my micrometer w^as none of the best, yet the errors in longitude and latitude (as derived from my observations) scarcely exceed one minute. The comet (according to my observations), about the end of its motion, began to decline sensibly towards the north, from the parallel which it described about the end of February. Now, in order to determine the orbit of the comet from the observations above described, I selected those three which Flamsteed made (Dec. 21, Jan. 5, and Jan. 25) from which I found St of 9842.1 parts, and V^ of 455, supposing the semidiameter of the earth's orbit contains 10,000. Then, for the first observation, assuming tB of 5657 of those parts, I found SB 9747, BE for the first time 412, S/x 9503, z'X413, BE for the second time 421, OD 10,186, 8528.4, 8450, 8475, NP 25; from this, by the second operation, I obtained the distance th 5640; and by this operation I at last deduced the distances TX 4775 and rZ 11,322. From these values, determining the orbit, I found its descending node in ^, and ascending node in v5 1° 53'; the inclination of its plane to the plane of the ecliptic 61° 20 J/^', the vertex thereof (or the perihelion of the comet) distant from the node 8° 38', and in ^ 27° 43', with latitude 7° 34' south; its latus rectum 236.8; and the diurnal area described by a radius drawn to the sun 93,585, supposing the square of the semidiameter of the earth's orbit 100,000,000; that the comet in this orbit moved directly according to the order of the signs, and on Dec. 8^^. OO*". 04™. p.m. was in the vertex or perihelion of its orbit. All this I determined by scale and compass, and the chords of angles, taken from the table of natural sines, in a pretty large figure, in which, to wit, the radius of the earth's orbit (consisting of 10,000 parts) was equal to 163^ inches of an English foot. Lastly, in order to discover whether the comet did truly move in the orbit so determined, I investigated its places in this orbit partly by arithmetical operations, and partly by scale and compass, to the times of some of the observations, as may be seen in the follomng table: of the

;

PM

X

MN

The Comet's

348

Mathematical Principles

This comet also appeared in the November before, and at Coburg, in Saxony, was observed by Mr. Gottfried Kirch, on the 4th of that month, on the 6th and 11th, O.S.; from its positions to the nearest fixed stars observed with sufficient accuracy, sometimes Avith a two-foot, and sometimes with a ten-foot telescope; from the difference of longitudes of Coburg and London, 11°; and from the places of the fixed stars observed by Mr. Pound, Dr. Halley has determined the places of the comet as follows:

Nov. 3, 17''. 2™., apparent time at London, the comet was in Q 29° 51', with 45" latitude north. Nov. 5, IS''. SS'"., the comet was in W 3° 23', with 1° 6' latitude north. Nov. 10, IQ^. 31°*., the comet was equally distant from two stars in q, which are designated a- and r in Bayer; but it had not quite touched the right line that joins them, but was very little distant from it. In Flamsteed's cat1° 17'

Book after Julius Caesar

was

III

:

The System of the World

killed; [in a.d.] 531, in the consulate of

month

349

Lampadius and

February: and at the end of the year 1680; and that with a long and remarkable tail, except when it was seen after Caesar's death, at which time, by reason of the inconvenient situation of the earth, the tail was not so conspicuous), set himself to find out an elliptic orbit whose greater axis should be 1,382,957 parts, the mean distance of the earth from the sun containing 10,000 such; in this orbit a comet might revolve in 575 years; and, placing the ascending node in a 2° 2', the inclination of the plane of the orbit to the plane of the ecliptic in an angle of 61° 6' 48", the perihelion of the comet in this plane in ^ 22° 44' 25", the equal time of the perihelion Dec. 7"^. 23^. 9™., the distance of the perihelion from the ascending node in the Orestes;

[in]

1106, in the

of

Mathematical Principles

350

morning at

Rome

(that

fixed stars, observed the

is,

5^. 10™. at

comet

in

^

London), by threads directed to the

8° 30', with latitude 0° 40' south. Their

observations may be seen in a treatise which Ponthio published concerning this comet. Cellio, who was present, and communicated his observations in a letter to Cassini, saw the comet at the same hour in :=: 8° 30', with latitude 0° 30' south. It was likewise seen by Gallet at the same hour at Avignon (that is, at 5^. 42"". morning at London) in = 8° without latitude. But by the theory the comet was at that time in ^ 8° 16' 45", and its latitude was 0° 53' 7" south. Nov. 18, at 6^. 30"". in the morning at Rome (that is, at 5^.40"". at London), Ponthio observed the comet in ^^ 13° 30', with latitude 1° 20' south; and Celho in ^ 13° 30', with latitude 1° 00' south. But at 5^. 30'". in the morning at Avignon, Gallet saw it in ^ 13° 00', with latitude 1° 00' south. In the University of La Fleche, in France, at 5^. in the morning (that is, at 5^. 9"". at London), it was seen by Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayer's ^; and the other is the outmost of the wing, Bayer's 6. Hence, the comet was then in := 12° 46' with latitude 50' south. And I was informed by Dr. Halley, that on the same day at Boston in New England, in the latitude of 423/2'; at 5''. in the morning (that is, at 9^. 44"". in the morning at London) the comet was seen near ^ 14°, with latitude 1° 30' south. Nov. 19, at 4}/^^. at Cambridge, the comet (by the observation of a young man) was distant from Spica ITP about 2° towards the northwest. Now the Spike was at that time in ^ 19° 23' 47", with latitude 2° 1' 59" south. The same day, at 5^. in the morning, at Boston in New England, the comet was

W

with the difference of 40' in latitude. The same day, in the island of Jamaica, it was about 1° distant from Spica iTP. The same day, Mr. Arthur Storer, at the river Patuxent, near Hunting Creek, in Maryland, in the confines of Virginia in latitude 383^°, at 5^. in the morning (that is, at 10^. at London), saw the comet above Spica iTP, and very nearly joined mth it, the of one degree. And from these observadistance between them being about 9''. 44"". at London the comet was in tions compared, I conclude, that at = 18° 50', with about 1° 25' latitude south. Now by the theory the comet was at that time in ^ 18° 52' 15", with 1° 26' 54" latitude south. distant from Spica

1°,

^

Montenari, Professor of Astronomy at Padua, at 6^. in the morning at Venice (that is, 5^. 10"". at London), saw the comet in ^ 23°, with latitude 1° 30' south. The same day, at Boston, it was distant from Spica ny by about

Nov.

20,

and therefore was in ^^ 23° 24', nearly. Ponthio and his companions, at 73^''. in the morning, observed the

4° of longitude east,

Nov. 21, comet in ^ 27°

south; Celho, in ^ 28°; Ango at 5^. in the morning, in :c= 27° 45'; Montenari in == 27° 51'. The same day, in the island of Jamaica, it was seen near the beginning of na, and of about the same latitude with Spica ny, that is, 2° 2'. The same day, at 5**. morning, at Ballasore, in the

50',

with latitude

East Indies (that

is,

1° 16'

at

ll"". 20"*.

of the night preceding at Lonwas taken 7° 35' to the east. It

don), the distance of the comet from Spica Tiy was in a right line between the Spike and the Balance, and therefore was then in ^ 26° 58', with about 1° 11' latitude south; and after 5"^. 40"". (that is, at

morning at London), it was in ^^ 28° 12' with 1° 16' latitude south. by the theory the comet was then in := 28° 10' 36", with 1° 53' 35"

5^. in the

Now

latitude south.

Book

III:

The System of the World

351

Nov. 22, the comet was seen by Montenari in ni 2° 33'; but at Boston in New England it was found in about m 3°, and mth almost the same latitude as before, that is, 1° 30'. The same day, at 5*^. morning at Ballasore, the comet was 1° 50'; and therefore at 5*". morning at London, the comet was observed in in TR 3° 5', nearly. The same day at 63^''. in the morning at London, Dr. Hooke observed it in about Tti 3° 30', and that in the right line which passeth through Spica H? and Cor Leonis; not, indeed, exactly, but deviating a little from that line towards the north. Montenari Ukewise observed, that this day, and some days after, a right line dra^vn from the comet through Spica passed by the south side of Cor Leonis at a very small distance therefrom. The right hne through Cor Leonis and Spica up did cut the ecliptic in w 3° 46' at an angle of 2° 51'; and, if the comet had been in this hne and in m 3°, its latitude would have been 2° 26'; but since Hooke and Montenari agree that the comet was at some small distance from this line towards the north, its latitude must have been somewhat less. On the 20th, by the observation of Montenari, its latitude was almost the same with that of Spica W, that is, about 1° 30'. But by the agreement of Hooke, Montenari, and Ango, the latitude was continually increasing, and therefore must now, on the 22d, be sensibly greater than 1° 30'; and, taking a mean between the extreme limits but now stated, 2° 26' and 1° 30', the latitude will be about 1° 58'. Hooke and Montenari agree that the tail of the comet was directed towards Spica tip, declining a little from that star towards the south according to Hooke, but towards the north according to Montenari; and, therefore, that declination was scarcely sensible; and the tail, lying nearly parallel to the equator, deviated a httle irom the opposition of the Til,

sun towards the north.

Nov. 23, O.S., at 5''. morning, at Nuremberg (that is, at 4^*". at London), Mr. Zimmerman saw the comet in in 8° 8', with 2° 31' south latitude, its place being obtained by taking its distances from fixed stars. Nov. 24, before sunrise, the comet was seen by Montenari in iu 12° 52' on the north side of the right line through Cor Leonis and Spica w, and therefore its latitude was somewhat less than 2° 38'; and since the latitude, as we said, by the concurring observations of Montenari, Ango, and Hooke, was continually increasing, therefore, it was now, on the 24th, somewhat greater than 1° 58'; and, taking the mean quantity, may be reckoned 2° 18', without any considerable error. Ponthio and Gallet will have it that the latitude was now decreasing; and Cellio, and the observer in New England, that it continued the same, viz., of about 1°, or 13^°. The observations of Ponthio and Cellio are rougher, especially those which were made by taking the azimuths and altitudes; as are also the observations of Gallet. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hooke, Ango, and the observer in New England, and sometimes by Ponthio and CelHo. The same day, at 5^. morning, at Ballasore, the comet was observed in TU 11° 45'; and, therefore, at 5^. morning at London, was in r([ 13°, nearly. And, by the theory, the comet was at that time in m 13° 22' 42". Nov. 25, before sunrise, Montenari observed the comet in Ttl 17^4°, nearly; and Cellio observed at the same time that the comet was in a right line between the bright star in the right thigh of Virgo and the southern scale of Libra; and this right line cuts the comet's way in ml 8° 36'. And, by the theory, the comet was in in 183^^°, nearly.

Mathematical Principles

352

From

all

this

it is

plain that these observations agree -with the theory, so far

as they agree with one another; and by this agreement it is made clear that it was one and the same comet that appeared all the time from Nov. 4 to Mar. 9. The path of this comet did twice cut the plane of the ecliptic, and therefore was not a right line. It did cut the ecUptic not in opposite parts of the heavens, but

Virgo and beginning of Capricorn, including an arc of about 98° and therefore the way of the comet did very much deviate from the path of a great circle; for in the month of Nov. it declined at least 3° from the ecliptic towards the south; and in the month of Dec. following it declined 29° from the ecliptic towards the north; the two parts of the orbit in which the comet descended towards the sun, and ascended again from the sun, declining one from the other by an apparent angle of above 30°, as observed by Montenari. This comet traveled over nine signs, namely, from the last degree of Q to the beginning of Jt beside the sign of £} through which it passed before it began to be seen and there is no other theory by which a comet can go over so great a part of the heavens with a regular motion. The motion of this comet was very unequable; for about the 20th of Nov. it described about 5° a day. Then its motion being retarded between Nov. 26 and Dec. 12, to wit, in the space of 153^ days, it described only 40°. But the motion thereof being afterwards accelerated, it described near 5° a day, till its motion began to be again retarded. And the theory which justly corresponds with a motion so unequable, and through so great a part of the heavens, which observes the same laws with the theory of the planets, and which accurately agrees with accurate astronomical observations, cannot be otherwise than true. And, thinking it would not be improper, I have given in the annexed figure, plotted in the plane of the curve, a true representation of the orbit which this comet described, and of the tail which it emitted in several places. In this drawing ABC represents the orbit of the comet, D the sun, DE the axis of the orbit, DF the line of the nodes, the intersection of the sphere of the earth's in the

end

of

,

,

;

GH

orbit with the plane of the comet's orbit, I the place of the

comet Nov.

4,

1680; K the place of the same Nov. 11; L the place of the same Nov. 19; M its place Dec. 12; N its place Dec. 21 its place Dec. 29; P its place Jan. 5 following; Q its place Jan. 25; R its place Feb. 5; S its place Feb. 25; T its place ;

March 5; and V its place March made the following observations:

9.

In determining the length of the

tail, I

Nov. 4 and 6, the tail did not appear; Nov. 11, the tail just began to show but did not appear above 3^ degree long through a 10-foot telescope; Nov. 17, the tail was seen by Ponthio more than 15° long; Nov. 18, in New England, the tail appeared 30° long, and directly opposite to the sun, extending itself,

Book

III

:

The System

of the

World

353

to the planet Mars, which was then in Tif, 9° 54'; Nov. 19, in Maryland, the tail was found 15° or 20° long; Dec. 10 (by the observation of ]\Ir. Flamsteed), the tail passed through the middle of the distance intercepted between itself

the serpent of Ophiuchus and the star 5 in the south wing of Aquila, and did terminate near the stars A, w, b in Bayer"s tables. Therefore the end of the tail was in v3 19^°, A\ith latitude about 343^° north; Dec. 11. it ascended to the head of Sagitta (Bayer's a, /S). terminating in u 26° 43', A\-ith latitude 38° 34' north; Dec. 12, it passed through the middle of Sagitta, nor did it reach the

tail of

much farther;

terminating in = 4°. ^^•ith latitude 423^^2° north, nearly. But these things are to be understood of the length of the brighter part of the tail; for, with a more faint light, observed, too, perhaps, in a serener sky, at Rome, Dec. 12. o^. 40"°., by the observation of Ponthio, the tail arose to 10° above the rump of the Swan, and the side thereof towards the west and towards the north was 45' distant from this star. But about that time the tail was 3° broad towards the upper end; and therefore the middle thereof was 2° 15' distant 22°, Anth latitude from that star towards the south, and the upper end Avas in 61° north; and thence the tail was about 70° long; Dec. 21. it extended almost to Cassiopeia's Chair, equally distant from /3 and from Schedir, so as its distance from either of the two was equal to the distance of the one from the other, and therefore did terminate in t 24°, ^Aith latitude 473^°; Dec. 29, it reached to a contact with. Scheat on its left, and exactly filled up the space between the two stars in the northern foot of Andromeda, being 54° in length; and therefore terminated in b 19°, with 35° of latitude; Jan. 5, it touched the star T in the breast of Andromeda on its right side, and the star of the girdle on its left; and, according to our observations, was 40° long; but it was curved, and the convex side thereof lay to the south: and near the head of the comet it made an angle of 4° ^A-ith the circle w^hich passed through the sun and the comet's head; but towards the other end it was inclined to that circle in an angle of about 10° or 11°; and the chord of the tail contained with that circle an angle of 8°. Jan. 13, the tail terminated between Alamech and Algol, ^-sith a hght that was sensible enough; but -with a faint light it ended over against the star K in Perseus' side. The distance of the end of the tail from the circle passing through the sun and the comet was 3° 50'; and the inclination of the chord of the tail to that circle was 83-2°- Jan. 25 and 26, it shone A\ith a faint light to the length of 6° or 7°; and, for a night or two after, when there was a very clear sky, it extended to the length of 12°, or someAvhat more, A\ith a hght that was very faint and very hardly to be seen; but the axis thereof was exactly directed to the bright star in the eastern shoulder of Auriga, and therefore deviated from the opposition of the sun towards the north by an angle of 10°. Lastly, Feb. 10, AATth a telescope I observed the tail 2° long; for that fainter hght which I spoke of did not appear through the glasses. But Ponthio writes that, on Feb. 7, he saw the tail 12° long. Feb. 25, the comet was AAithout a tail, and so continued till it disappeared. Now if one reflects upon the orbit described, and duly considers the other appearances of this comet, he will be easily satisfied that the bodies of comets are sohd, compact, fixed, and durable, like the bodies of the planets; for if they were nothing else but the vapors or exhalations of the earth, of the sun, and other planets, this comet, in its passage by the neighborhood of the sun, would have been immediately dissipated; for the heat of the sun is as the density of :-:

jj.

Mathematical Principles

354

that is, inversely as the square of the distance of the places from the since on Dec. 8, when the comet was in its perihelion, the disTherefore, sun. tance thereof from the centre of the sun was to the distance of the earth from its rays,

the same as about 6 to 1000, the sun's heat on the comet was at that time to the heat of the summer-sun with us as 1,000,000 to 36, or as 28.000 to 1. But the heat of boiling water is about three times greater than the heat which dry earth acquires from the summer-sun, as I have tried; and the heat of red-hot iron (if my conjecture is right) is about three or four times greater than the heat of boiling water. And therefore the heat which dry earth on the comet, while in

might have received from the rays of the sun, was about 2000 times greater than the heat of red-hot iron. But by so fierce a heat, vapors and exhalations, and every volatile matter, must have been immediately consumed and dissipated. This comet, therefore, must have received an immense heat from the sun, and retained that heat for an exceeding long time for a globe of iron of an inch in diameter, exposed red-hot to the open air, will scarcely lose all its heat in an hour's time; but a greater globe would retain its heat longer in the ratio of its diameter, because the surface (in proportion to which it is cooled by the contact of the ambient air) is in that ratio less in respect of the quantity of the included hot matter; and therefore a globe of red-hot iron equal to our earth, that is, about 40,000,000 feet in diameter, would scarcely cool in an equal number of days, or in above 50,000 years. But I suspect that the duration of heat may, on account of some latent causes, increase in a yet less ratio than that of the diameter; and I should be glad that the true ratio was investigated by its perihelion,

;

experiments. It is further to be observed, that the comet in the month of December, just after it had been heated by the sun, did emit a much longer tail, and much more splendid, than in the month of November before, when it had not yet arrived at its perihelion; and, universally, the greatest and most fulgent tails always arise from comets immediately after their passing by the neighborhood of the sun. Therefore the heat received by the comet conduces to the greatness of the tail from this, I think I may infer, that the tail is nothing else but a very fine vapor, which the head or nucleus of the comet emits by its heat. But we have had three several opinions about the tails of comets for some will have it that they are nothing else but the beams of the sun's light transmitted through the comets' heads, which they suppose to be transparent; others, that they proceed from the refraction which light suffers in passing from the comet's head to the earth; and, lastly, others, that they are a sort of cloud or vapor constantly rising from the comets' heads, and tending towards the parts opposite to the sun. The first is the opinion of such as are yet unacquainted with optics; for the beams of the sun are seen in a darkened room only in consequence of the light that is reflected from them by the little particles of dust and smoke which are always flying about in the air; and, for that reason, in air impregnated with thick smoke, those beams appear with great :

:

brightness,

more

faint,

and impress the eye more strongly; in a yet finer air they appear and are less easily discerned; but in the heavens, where there is no

matter to reflect the light, they can never be seen at all. Light is not seen as it is in the beam, but as it is thence reflected to our eyes; for vision can be produced in no other way than by rays falling upon the eyes; and, therefore, there

Book

III

:

The System of the World

355

reflecting matter in those parts where the tails of the comets are seen: for otherwise, since all the celestial spaces are equally illuminated by the sun's light, no part of the heavens could appear with more splendor than an-

must be some

other.

The second opinion

is liable

to

many

diflaculties.

The

tails of

comets are

never seen variegated with those colors which commonly are inseparable from refraction; and the distinct transmission of the light of the fixed stars and planets to us is a demonstration that the ether or celestial medium is not endowed with any refractive power: for, as to what is alleged, that the fixed stars have been sometimes seen by the Egyptians environed with a coma, because that has but rarely happened, it is rather to be ascribed to a casual refraction of clouds; and so the radiation and scintillation of the fixed stars to the refractions both of the eyes and air; for, upon laying a telescope to the eye, those radiations and scintillations immediately disappear. By the tremulous agitation of the air and ascending vapors, it happens that the rays of light are alternately turned aside from the narrow space of the pupil of the eye but no such thing can have place in the much wider aperture of the object glass of a telescope and hence it is that a scintillation is occasioned in the former case, which ceases in the latter; and this cessation in the latter case is a demonstration of the regular transmission of light through the heavens, without any perceptible refraction. But, to obviate an objection that may be made from the appearing of no tail in such comets as shine but with a faint light, as if the secondary rays were then too weak to affect the eyes, and for that reason it is that the tails of the fixed stars do not appear, we are to consider, that by the means of telescopes the light of the fixed stars may be augmented above an hundredfold, and yet no tails are seen; that the light of the planets is yet more copious without any tail; but that comets are seen sometimes with huge tails, when the light of their heads is but faint and dull. For so it happened in the comet of the year 1680, when in the month of December it was scarcely equal in light to the stars of the second magnitude, and yet emitted a notable tail, extending to the length of 40°, 50°, 60°, or 70°, and upwards; and afterwards, on the 27th and 28th of January, when the head appeared but as a star of the 7th magnitude, yet the tail (as we said above), with a light that was clearly perceptible, though faint, was stretched out to 6° or 7° in length, and with a languishing light that was more difficult to see, even to 12°, and upwards. But on the 9th and 10th of February, when to the naked eye the head appeared no more, through a telescope I viewed the tail of 2° in length. But further: if the tail was due to the crumbling of the celestial matter, and did deviate from the opposition of the sun, according to the figure of the heavens, that deviation in the same places of the heavens should be always directed towards the same parts. But the comet of the year 1680, December 28^^. 8^2^- p-m. at London, was seen in K 8° 41', with latitude north 28° 6'; while the sun was in v3 18° 26'. And the comet of the year 1577, December 29^., was in k 8° 41', with latitude north 28° 40', and the sun, as before, in about v3 18° 26'. In both cases the situation of the earth was the same, and the comet appeared in the same place of the heavens; yet in the former case the tail of the comet (as well by my observations as by the observations of others) deviated from the opposition of the sun towards the north by an angle of 43^2 degrees; whereas in the latter there was (according to the observations of Tycho) a deviation of 21 degrees towards the south. The crumbling, therefore, of the heavens being thus dis;

;

Mathematical Principles

356 proved,

it

from some

remains that the phenomena of the

tails of

comets must be derived

reflecting matter.

And

that the tails of comets do arise from their heads, and tend towards the parts opposite to the sun, is further confirmed from the laws which the tails observe: As that, lying in the planes of the comets' orbits which pass through the sun, they constantly deviate from the opposition of the sun towards the parts which the comets' heads in their progress along these orbits have left. That to a spectator, placed in those planes, they appear in the parts directly

opposite to the sun; but, as the spectator recedes from those planes, their deviation begins to appear, and daily becomes greater. That the deviation, other things being equal, appears less when the tail is more oblique to the orbit of the comet, as well as when the head of the comet approaches nearer to the sun, especially if the angle of deviation is estimated near the head of the comet. That the tails which have no deviation appear straight, but the tails which

deviate are like\\'ise bended into a certain curvature. That this curvature is greater when the deviation is greater; and is more sensible when the tail, other things being equal, is longer; for in the shorter tails the curvature is hardly to be perceived. That the angle of deviation is less near the comet's head, but greater towards the other end of the tail; and that because the convex side of the tail regards the parts from which the deviation is made, and which lie in a right line drawn out infinitely from the sun through the comet's head. And that the tails that are long and broad, and shine with a stronger light, appear more resplendent and more exactly defined on the convex than on the concave side. Upon these accounts it is plain that the phenomena of the tails of comets depend upon the motions of their heads, and by no means upon the places of the heavens in which their heads are seen; and that, therefore, the tails of comets do not proceed from the refraction of the heavens, but from their own heads, which furnish the matter that forms the tail. For, as in our air, the smoke of a heated body ascends either perpendicularly if the body is at rest, or obliquely if the body is moved obliquely, so in the heavens, where all bodies gravitate towards the sun, smoke and vapor must (as we have already said) ascend from the sun, and either rise perpendicularly if the smoking body is at rest, or obliquely if the body, in all the progress of its motion, is always leaving those places from which the upper or higher parts of the vapor had risen before and that obliquity will be least where the vapor ascends with most velocity, namely, near the smoking body, when that is near the sun. But, because the obliquity varies, the column of vapor will be incurvated; and because the vapor in the preceding side is something more recent, that is, has ascended something more late from the body, it will therefore be somewhat more dense on that side, and must on that account reflect more light, as well as be better defined. I add nothing concerning the sudden uncertain agitation of the tails of comets, and their irregular figures, which authors sometimes describe, because they may arise from the mutations of our air, and the motions of our clouds, in part

obscuring those

tails; or,

have been confounded

perhaps, from parts of the Milky Way which might and mistaken for parts of the tails of the comets

\vith

as they passed by. But that the atmospheres of comets

enough to our

own

fill

so

air; for

immense

spaces,

may furnish a supply of vapor great we may easily understand from the rarity of

the air near the surface of our earth possesses a space 850

Book

III

:

The System of the World

357

times greater than water of the same weight; and therefore a cyhnder of air 850 feet high is of equal weight ^\ith a cyhnder of water of the same breadth, and but one foot high. But a cyhnder of air reaching to the top of the atmosphere is of equal weight Anth a cylinder of water about 33 feet high: and, therefore, if from the whole cylinder of air the lower part of 850 feet high is taken away, the remaining upper part vn.ll be of equal weight with a cylinder of water 32 feet high: and from thence (and by the hypothesis, confirmed by many experiments, that the compression of air is as the weight of the incumbent atmosphere, and that the force of gravity is inversely as the square of the distance from the centre of the earth) proceeding by calculation, by Cor., Prop. 22, Book II, I found, that, at the height of one semidiameter of the earth, reckoned from the earth's surface, the air is more rare than mth us in a far greater ratio than that of the whole space ^^^thin the orbit of Saturn to a spherical space one inch in diameter; and therefore, if a sphere of our air of but one inch in thickness was equally rarefied with the air at the height of one semidiameter of the earth from the earth's surface, it would fiJl all the regions of the planets to the orb of Saturn, and far beyond it. Therefore, since the air at greater distances is immensely rarefied, and the coma or atmosphere of comets is ordinarily about ten times higher, reckoning from their centres, than the surface of the nucleus, and the tails rise yet higher, they must therefore be exceedingly rare; and though, on account of the much thicker atmospheres of comets, and the great gravitation of their bodies towards the sun, as well as of the particles of their air and vapors towards each other, it may happen that the air in the celestial spaces and in the tails of comets is not so vastly rarefied, yet from this computation it is plain that a very small quantity of air and vapor is abundantly sufficient to produce all the appearances of the tails of comets; for that they are, indeed, of a very notable rarity appears from the shining of the stars through them. The atmosphere of the earth, illuminated by the sun's light, though but of a few miles in thickness, quite obscures and extinguishes the light not only of all the stars, but even of the moon itself; whereas the smallest stars are seen to shine through the immense thickness of the tails of comets, likewse illuminated by the sun, without the least diminution of their splendor. Nor is the brightness of the tails of most comets ordinarily greater than that of our air, an inch or two in thickness, reflecting in a darkened room the light of the sunbeams let in by a hole of the Avindow shutter. And we may pretty nearly determine the time spent during the ascent of the vapor from the comet's head to the extremity of the tail, by dramng a right line from the extremity of the tail to the sun, and marking the place where that right line intersects the comet's orbit; for the vapor that is now in the extremity of the tail, if it has ascended in a right line from the sun, must have begun to rise from the head at the time when the head was in the point of intersection. It is true, the vapor does not rise in a right line from the sun, but, retaining the motion which it had from the comet before its ascent, and compounding that motion with its motion of ascent, arises obliquely; and, therefore, the solution of the Problem vnW be more exact, if we di'aw the line which intersects the orbit parallel to the length of the tail; or rather (because of the curvilinear motion of the comet) diverging a little from the line or length of the tail. And by means of this principle I found that the vapor which, January 25, was in the extremity of the tail, had begun to rise from the head before

Mathematical Principles

358

and therefore had spent in its whole ascent 45 days but that the whole tail which appeared on December 10 had finished its ascent in the space of the two days then elapsed from the time of the comet's being in its perihelion. The vapor, therefore, about the beginning and in the neighborhood of the sun rose with the greatest velocity, and afterwards continued to ascend with a motion constantly retarded by its own gravity; and the higher it ascended, the more it added to the length of the tail; and while the tail continued to be seen, it was made up of almost all that vapor which had risen since

December

11

;

,

the time of the comet's being in

its

perihelion; nor did that part of the vapor

which had risen

first, and which formed the extremity of the tail, cease to aptoo great distance, as well from the sun, from which it received its light, as from our eyes, rendered it invisible. Whence also it is that the tails of other comets which are short do not rise from their heads with a smft and continued motion, and soon after disappear, but are permanent and lasting columns of vapors and exhalations, which, ascending from the heads with a slow motion of many days, and partaking of the motion of the heads which they had from the beginning, continue to go along together -with them through the heavens. From this again we have another argument proving the celestial spaces to be free, and without resistance, since in them not only the solid bodies of the planets and comets, but also the extremely rare vapors of comets' tails, maintain their rapid motions with great freedom, and for an exceeding

pear,

till its

long time.

Kepler ascribes the ascent of the tails of the comets to the atmospheres of and their direction towards the parts opposite to the sun to the action of the rays of light carrying along with them the matter of the comets' tails; and without any great incongruity we may suppose that, in so free spaces, so fine a matter as that of the ether may yield to the action of the rays of the sun's light, though those rays are not able sensibly to move the gross substances in our parts, which are clogged vnth so palpable a resistance. Another author thinks that there may be a sort of particles of matter endowed their heads;

a principle of levity, as well as others are mth a power of gravity; that the matter of the tails of comets may be of the former sort, and that its ascent from the sun may be owing to its levity; but, considering that the gravity of terrestrial bodies is as the matter of the bodies, and therefore can be neither more nor less in the same quantity of matter, I am inclined to believe that this ascent may rather proceed from the rarefaction of the matter of the comets' tails. The ascent of smoke in a chimney is due to the impulse of the air with which it is entangled. The air rarefied by heat ascends, because its specific gravity is diminished, and in its ascent carries along with it the smoke which floats in it and why may not the tail of a comet rise from the sun after the same manner? For the sun's rays do not act upon the mediums which they pervade otherwise than by reflection and refraction; and those reflecting particles heated by this action, heat the matter of the ether which is involved ^\'ith them. That matter is rarefied by the heat which it acquires, and because, by this rarefaction, the specific gravity mth which it tended towards the sun before is diminished, it will ascend therefrom, and carry along with it the reflecting particles of which the tail of the comet is composed. But the ascent of the vapors is further promoted by their circumgyration about the sun; in consequence thereof they endeavor to recede from the sun, while the sun's atmos^\'ith

;

Book

III

:

The System of the World

359

phere and the other matter of the heavens are either altogether quiescent, or are only moved wdth a slower circumgyration derived from the rotation of the sun. And these are the causes of the ascent of the tails of the comets in the neighborhood of the sun, where their orbits are bent into a greater curvature, and the comets themselves are plunged into the denser and therefore heavier parts of the sun's atmosphere: upon which account they do then emit tails of an huge length; for the tails which then arise, retaining their own proper motion, and in the meantime gravitating towards the sun, must be revolved in ellipses about the sun in like manner as the heads are, and by that motion must always accompany the heads, and freely adhere to them. For the gravitation of the vapors towards the sun can no more force the tails to abandon the heads, and descend to the sun, than the gravitation of the heads can oblige them to fall from the tails. They must by their common gravity either fall together towards the sun, or be retarded together in their common ascent therefrom; and, therefore (whether from the causes already described, or from any others) the tails and heads of comets may easily acquire and freely retain any position one to the other, without disturbance or impediment from that ,

common The

gravitation.

comets will remote parts, and together with the heads

therefore, that rise in the perihelian positions of the

tails,

go along with their heads into far from thence to us, after a long course of years, or rather will be there rarefied, and by degrees quite vanish away; for afterwards, in the descent of the heads towards the sun, new^ short tails will be emitted from the heads wdth a slow motion; and those tails by degrees will be augmented immensely, especially in such comets as in their perihelian distances descend as low as the sun's atmosphere; for all vapor in those free spaces is in a perpetual state of rarefaction and dilatation; and from hence it is that the tails of all comets are broader at their upper extremity than near their heads. And it is not unlikely but that the vapor, thus continually rarefied and dilated, may be at last dissipated and scattered through the whole heavens, and by little and little be attracted towards the planets by its gravity, and mixed with their atmosphere for as the seas are absolutely necessary to the constitution of our earth, that from them, the sun, by its heat, may exhale a sufficient quantity of vapors, which, being gathered together into clouds, may drop down in rain, for watering of the earth, and for the production and nourishment of vegetables; or, being condensed with cold on the tops of mountains (as some philosophers mth reason judge), may run down in springs and rivers; so for the conservation of the seas, and fluids of the planets, comets seem to be required, that, from their exhalations and vapors condensed, the wastes of the planetary fluids spent upon vegetation and putrefaction, and converted into dry earth, may be continually supplied and made up; for all vegetables entirely derive their growths from fluids, and afterwards, in great measure, are turned into dry earth by putrefaction; and a sort of sfime is always found to settle at the bottom of putrefied fluids; and hence it is that the bulk of the solid earth is continually increased; and the fluids, if they are not supplied from without, must be in a continual decrease, and quite fail at last. I suspect, moreover, that it is chiefl}^ from the comets that spirit comes, which is indeed the smallest but the most subtle and useful part of our air, and so much required to sustain the life of all things with us. wall either return again

;

360

The atmospheres

Mathematical Principles of comets, in their descent

towards the sun, by running

out into the tails, are spent and diminished, and become narrower, at least on that side which regards the sun; and in receding from the sun, when they less run out into the tails, they are again enlarged, if Hewelcke has justly marked their appearances. But they are seen least of all just after they have been most heated by the sun, and on that account then emit the longest and most resplendent tails; and, perhaps, at the same time, the nuclei are environed with a denser and blacker smoke in the lowermost parts of their atmosphere; for smoke that is raised by a great and intense heat is commonly the denser and blacker. Thus the head of that comet which we have been describing, at equal distances both from the sun and from the earth, appeared darker after it had passed by its perihelion than it did before; for in the month of December it was commonly compared with the stars of the third magnitude, but in November with those of the first or second; and such as saw both appearances have described the first as of another and greater comet than the second. For, November 19, this comet appeared to a young man at Cambridge, though with a pale and dull light, yet equal to Spica Virginis; and at that time it shone with greater brightness than it did afterwards. And Montenari, November 20, o.s., observed it larger than the stars of the first magnitude, its tail being then 2 degrees long. And Mr. Storer (by letters which have come into my hands) writes that in the month of December, when the tail appeared of the greatest bulk and splendor, the head was but small, and far less than that which was seen in the month of November before sun rising; and, conjecturing at the cause of the appearance, he judged it to proceed from the existence of a greater quantity of matter in the head at first, which was afterwards gradually spent. And, for the same reason, I find, that the heads of other comets, which did put forth tails of the greatest bulk and splendor, have appeared but obscure and small. For in Brazil, March 5, 1668, n.s., 7''. p.m., Valentin Estancel saw a comet near the horizon, and towards the southwest, with a head so small as scarcely to be discerned, but with a tail above measure splendid, so that the reflection thereof from the sea was easily seen by those who stood on the shore; it looked like a fiery beam extended 23 degrees in length from the west to south, almost parallel to the horizon. But this excessive splendor continued only three days, decreasing apace afterwards; and while the splendor was decreasing, the bulk of the tail increased: also in Portugal it is said to have taken up one-quarter of the heavens, that is, 45 degrees, extending from west to east with a very notable splendor, though the whole tail was not seen in those parts, because the head was always hid under the horizon: and from the increase of the bulk and decrease of the splendor of the tail, it appears that the head was then in its recess from the sun, and had been very near to it in its perihelion, as the comet of 1680 was. And we read, in the Saxon Chronicle, of a like comet appearing in the year 1106, the star whereof was small and obscure (as that of 1680), but the splendor of its tail was very bright, and like a huge fiery beam stretched out in a direction between the east and north, as Hewelcke has it also from Simeon, the monk of Durham. This comet appeared in the beginning of February, about the evening, and towards the southwest part of heaven; from this, and from the position of the tail, we infer that the head was near the sun. Matthew Paris says, It was distant from the sun by about a cubit, from three o'clock (rather six) till nine, putting forth a long tail. Such also was that re-

Book

III:

The System of the World

361

splendent comet described by Aristotle, Meteorology, i, 6. The head whereof could not he seen, because it had set before the sun, or at least was hid under the spin's rays; hut next day it was seen as well as might he; for, having left the sun hut

immediately after it. And the scattered light of the head, obscured by the too great splendor (of the tail) did not yet appear. But afterwards (as Aristotle says) when the splendor (of the tail) had diminished, (the head of)

a very

little

way,

it set

the comet recovered its native brightness;

and

the splendor (of its tail) reached

now

a third part of the heavens (that is, to 60°). This appearance was in the winter season (the fourth year of the 101st Olympiad), and, rising to Orion's girdle, it there vanished away. It is true that the comet of 1618, which came out directly from under the sun's rays with a very large tail, seemed to equal, if not to exceed, the stars of the first magnitude; but then, abundance of other comets have appeared yet greater than this, that put forth shorter tails; some of which are said to have appeared as big as Jupiter, others as big as Venus, or even as to

the moon.

We

have said that comets are a sort of planets revolved in very eccentric orbits about the sun; and as, in the planets which are without tails, those are commonly less which are revolved in lesser orbits, and nearer to the sun, so in comets it is probable that those which in their perihelion approach nearer to the sun are generally of less magnitude, that they may not agitate the sun too

much by

their attractions.

But

as to the transverse diameters of their orbits,

and the periodic times of their revolutions, I leave them to be determined by comparing comets together which after long intervals of time return again in the same orbit. In the meantime, the following Proposition may give some light in that inquiry.

^

Proposition

.^ 42.

t^

Problem 22

To correct a comet's orbit found as above. Operation 1. Assume that position of the plane of the orbit which was determined according to the preceding Proposition; and select three places of the comet, deduced from very accurate observations, and at great distances one from the other. Then suppose A to represent the time between the first observation and the second, and B the time between the second and the third; but it will be convenient that in one of those times the comet be in its perigee, or at least not far from it. From those apparent places find, by trigonometric operations, the three true places of the comet in that assumed plane of the orbit; then through the places found, and about the centre of the sun as the focus, describe a conic section by arithmetical operations, according to Prop. 21, Book I. Let the areas of this figure which are terminated by radii drawn from the sun to the places found be D and E; namely, D the area between the first observation and the second, and E the area between the second and third; and let T represent the whole time in which the whole area D + E should be described with the velocity of the comet found by Prop. 16, Book i. Retaining the inclination of the plane of the orbit to the plane of the ecHptic, let the longitude of the nodes of the plane of the orbit be increased by the addition of 20' or 30', which call P. Then from the aforesaid three observed places of the comet let the three true places be found (as before) in this new plane; as also the orbit passing through those places, and the two areas of the same described between the two observations, which call d and e; and let t be the whole time in which the whole area d-\-e should be described.

Oper.

2.

Mathematical Principles

362

Oper.

3.

Retaining the longitude of the nodes in the

first

operation, let the

inclination of the plane of the orbit to the plane of the ecliptic be increased by adding thereto 20' or 30', which call Q. Then from the aforesaid three observed

the three true places be found in this new plane, as well as the orbit passing through them, and the two areas of the same described between the observation, which call 5 and e and let t be the whole time in which the whole area 5+e should be described.

apparent places of the comet

let

;

Then taking C

to

1

as

A

to B;

and

G

to

1

as

D

to E;

and g to

1

as

rf

to

e;

S be the true time between the first observation and the + and — let such numbers m and n be third — found out as vAW make 2G 2C = mG — mgr+wG — n7; and 2T — 2S = mT — m^

and 7 to ;

1

as

5

to

e

;

let

and, observing well the signs

+nT — nr. And

if,

,

in the first operation, I represents the inclination of the

K

the longitude of either plane of the orbit to the plane of the ecliptic, and node, then I+nQ ^^dll be the true inclination of the plane of the orbit to the the true longitude of the node. And, lastly, plane of the echptic, and if in the first, second, and third operations, the quantities R, r, and p, represent

K+mP

the parameters of the orbit, and the quantities T^l,~, the transverse diameters

then R+???r — mR+np — nR

of the same,

—

-^—,

—

^^-j

T

^vill

will

be the true parameter, and

be the true transverse diameter of the orbit which

L+rw/ — mL + nA — nL the comet describes; and from the transverse diameter given the periodic time of the comet is also given, q.e.i. But the periodic times of the revolutions of comets, and the transverse diameters of their orbits, cannot be accurately enough determined but by comparing comets together which appear at differs:

ent times. If, after equal intervals of time, several comets are found to have described the same orbit, we may thence conclude that they are all but one and the same comet revolved in the same orbit; and then from the times of their revolutions the transverse diameters of their orbits will be given, and from those diameters the elliptic orbits themselves wdll be determined. To this purpose the orbits of many comets ought to be computed, supposing those orbits to be parabolic for such orbits will always nearly agree with the phenomena, as appears not only from the parabolic orbit of the comet of the year 1680, which I compared above with the observations, but likemse from that of the notable comet which appeared in the year 1664 and 1665, and was observed by Hewelcke, who, from his own observations, calculated the longitudes and latitudes thereof, though with little accuracy. But from the same observations Dr. Halley did again compute its places; and from those new places determined its orbit, finding its ascending node in h 21° 13' 55"; the inclination of the orbit to the plane of the echptic 21° 18' 40"; the distance of its perihelion from the node, estimated in the comet's orbit, 49° 27' 30", its perihehon in « 8° 40' 30", with hehocentric latitude south 16° 01' 45"; the comet to have been in its perihelion November 24^^. ll*". 52™. p.m. equal time ;

rectum of the parabola was 410,286 of such parts as the sun's mean distance from the earth is supposed to contain 100,000. And how nearly the places of the comet computed in this orbit agree with the observations, ^\^ll appear from the table calculated at London, or 13^.

by Dr. Halley

S'".

at Danzig, o.s.;

and that the

latus

(p 364).

In February, the beginning of the year 1665, the

first

star of Aries,

which

I

Book

III:

The System of the World

363

T 8'

58" north latitude; the 17' 18". with 8° 28' 16" north latitude; in second star of another star of the seventh magnitude, which I call A, was in T 28° 24' 45", with 8° 28' 33" north latitude. The comet Feb. 7^. 7^^. 30"^. at Paris (that is, Feb. 1^. 8^. 37™. at Danzig), o.s., made a triangle A\dth those stars 7 and A, which was right-angled in 7 and the distance of the comet from the star 7 w^as equal to the distance of the stars 7 and A, that is, 1° 19' 46" of a great circle; and therefore in the parallel of the latitude of the star 7 it was 1° 20' 26". Therefore if from the longitude of the star 7 there be subtracted the longitude 1° 20' 26", there ^\111 remain the longitude of the comet T 27° 9' 49". M. Auzout, from this observation of his, placed the comet in T 27° 0', nearly; and, by the dramng in which Dr. Hooke delineated its motion, it w^as then in T 26° 59' 24". I place it in T 27° 4' 46", taking the middle between the tw^o

was Aries was

shall hereafter call 7,

in

T 28° T 29°

30' 15", vdth.

;

extremes. From the same observations, M. Auzout made the latitude of the comet at that time 7° and 4' or 5' to the north but he had done better to have made it 7° 3' 29", the difference of the latitudes of the comet and the star 7 being equal to the difference of the longitude of the stars 7 and A. February 22^^. 1^. 30"^. at London, that is, February 22"^. 8^. 46°^. at Danzig, the distance of the comet from the star A, according to Dr. Hooke's observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, dehneated in like manner by M. Petit, was a fifth part of the distance betw^een the star A and the first star of Aries, or 15' 57"; and the distance of the comet from a right line joining the star A and the first of Aries was a fourth part of the same fifth part, that is, 4'; and therefore the comet was in T 28° 29' 46", with 8° 12' 36" north latitude. March 1, 7^. O'". at London, that is, March 1, 8^. 16"^. at Danzig, the comet was observed near the second star in Aries, the distance between them being to the distance between the first and second stars in Aries, that is, to 1° 33', as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8' 16" according to Dr. Hooke, or 8' 5" according to M. Gottignies; or, taking a mean between both, 8' 10". But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1' 35" (in which he agrees very well vath M. Auzout); or, according to Dr. Hooke, not quite so much, as perhaps only 1'. Therefore if to the longitude of the first star in Aries we add 1', and 8' 10" to its latitude, we shall have the longitude of the comet T 29° 18', ;

with 8° 36' 26" north latitude. March 7, 7^. 30"^. at Paris, that is, March 7, 7^. 37°^. at Danzig, from the observations of M. Auzout, the distance of the comet from the second star in Aries was equal to the distance of that star from the star A, that is, 52' 29"; and the difference of the longitude of the comet and the second star in Aries was 45' or 46', or, taking a mean quantity, 45' 30"; and therefore the comet was 0°2'48". From the dra^ving constructed by M. Petit, based on the obserin vations of M. Auzout, Hewelcke determined the latitude of the comet 8° 54'. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion; and Hevelius, in the drawing of M. Auzout's observations which he constructed himself, corrected this irregular curvature, ts'

364 Apparent time at

Danzig

Mathematical Principles

Book

III:

The System of the World

365

made

the latitude of the comet 8° 55' 30". And, by further correcting this irregularity, the latitude may become 8° 56', or 8° 57'. This comet was also seen March 9, and at that time its place must have been in ^ 0° 18', \vith 9° 33^' north latitude, nearly. This comet appeared for three months, in which space of time it traveled over almost six signs, and in one of the days described almost 20 degrees. Its course did very much deviate from a great circle, bending towards the north, and its motion towards the end from retrograde became direct; and, notwithstanding that its course was so uncommon, yet by the table it appears that the theory, from beginning to end, agrees with the observations no less accurately than the theories of the planets usually do with the observations of them but we are to subtract about 2' when the comet was swiftest, which we may effect by taking off 12" from the angle between the ascending node and the perihelion, or by making that angle 49° 27' 18". The annual parallax of both these comets (this and the preceding) was very conspicuous, and by its quantity demonstrates the annual motion of the earth in the earth's orbit. This theory is likewise confirmed by the motion of that comet, which in the year 1683 appeared retrograde, in an orbit whose plane contained almost a right angle Avith the plane of the ecliptic, and whose ascending node (by the computation of Dr. Halley) was in w 23° 23'; the inclination of its orbit to the ecliptic 83° 11'; its perihelion in jt 25° 29' 30"; its perihelian distance from

and

so

the sun 56,020 of such parts as the radius of the earth's orbit contains 100,000; and the time of its perihelion was July 2^^. 3^. 50". And the places thereof, computed by Dr. Halley in this orbit, are compared with the places observed by Mr. Flamsteed, in the following table. This theory is yet further confirmed by the motion of that retrograde comet which appeared in the year 1682. The ascending node of this (by Dr. Halley's

1683

Mathematical Principles

366

computation) was in b' 21° 16' 30"; the inclination of its orbit to the plane of the ecUptic 17° 56' 00"; its perihelion in ~ 2° 52' 50"; its periheUan distance from the sun 58,328 parts, of which the radius of the earth's orbit contains 100,000; the equal time of the comet's being in its perihelion September 4"^. 7''. 39"". And its places determined from Mr. Flamsteed's observations, are compared with its places computed from our theory in the follomng table: 1682

App. time

Book

III

:

The System of the World

367

From

these examples it is abundantly evident that the motions of comets accurately represented by our theory than the motions of the planare no ets commonly are by the theories of them; and, therefore, by means of this theory, we may enumerate the orbits of comets, and so discover the periodic time of a comet's revolution in any orbit; hence, at last, we shall have the transverse diameters of their elliptic orbits and their aphelian distances. less

That retrograde comet which appeared in the year 1607 described an orbit whose ascending node (according to Dr. Halley's computation) was in b" 20° 21'; and the incHnation of the plane of the orbit to the plane of the ecliptic 17° 2'\ whose perihehon was in ~ 2° 16'; and its perihelian distance from the sun 58,680 of such parts as the radius of the earth's orbit contains 100,000; and the comet was in its perihehon October \^^. 3^. 50°".; which orbit agrees very nearly wAXh the orbit of the comet which was seen in 1682. If these were not two different comets, but one and the same, that comet mil finish one revolution in the space of 75 years; and the greater axis of its orbit will be to the greater axis of the earth's orbit as -v^75^ to 1, or as 1778 to 100, nearly. And the aphelian distance of this comet from the sun ^^dll be to the mean distance of

the earth from the sun as about 35 to 1 from these data it ^nll be no hard matter to determine the elliptic orbit of this comet. But these things are to be supposed on condition that, after the space of 75 years, the same comet shall return again in the same orbit. The other comets seem to ascend to greater heights, and to require a longer time to perform their revolutions. But, because of the great number of comets, of the great distance of their aphelions from the sun, and of the slowness of their motions in the aphelions, they will, by their mutual gravitations, disturb each other; so that their eccen;

and the times of their revolutions vfiSS. be sometimes a little increased, and sometimes diminished. Therefore, we are not to expect that the same comet will return exactly in the same orbit, and in the same periodic times it will be sufficient if we find the changes no greater than may arise from the causes just spoken of. And hence a reason may be assigned why comets are not comprehended within the limits of a zodiac, as the planets are; but, being confined to no bounds, are with various motions dispersed all over the heavens; namely, to this purpose, that in their aphelions, where their motions are exceedingly slow, receding to greater distances one from another, they may suffer less disturbance from their mutual gravitations: and hence it is that the comets which descend the lowest, and therefore move the slowest in their aphelions, ought tricities

:

also to ascend the highest.

The comet which appeared in the year 1680 was in its perihelion less distant from the sun than by a sixth part of the sun's diameter; and because of its extreme velocity in that proximity to the sun, and some density of the sun's atmosphere, it must have suffered some resistance and retardation; and therefore, being attracted somewhat nearer to the sun in every revolution, AAdll at last fall doA\Ti upon the body of the sun. Nay, in its aphelion, where it moves the slowest, it may sometimes happen to be yet further retarded by the attractions and in consequence of this retardation descend to the sun. So fixed stars, that have been gradually wasted by the light and vapors emitted from them for a long time, may be recruited by comets that fall upon them; and from this fresh supply of new fuel those old stars, acquiring new splendor, of other comets,

Mathematical Principles

368

pass for new stars. Of this kind are such fixed stars as appear on a sudden, and shine mth a wonderful brightness at first, and afterwards vanish by little and little. Such was that star which appeared in Cassiopeia's Chair; which Cornelis Gemma did not see upon the 8th of November, 1572, though he was observing that part of the heavens upon that very night, and the sky was

may

perfectly serene; but the next night (November 9) he saw it shining much brighter than any of the fixed stars, and scarcely inferior to Venus in splendor.

Tycho Brahe saw it upon the Uth of the same month, when it shone with the greatest lustre and from that time he observed it to decay by little and Uttle and in 16 months' time it entirely disappeared. In the month of November, when it first appeared, its light was equal to that of Venus. In the month of December, its light was a little diminished, and was now become equal to that of Jupiter. In January, 1573, it was less than Jupiter, and greater than Sirius, and about the end of February and the beginning of March became equal to that star. In the months of April and May it was equal to a star of the second magnitude; in June, July, and August, to a star of the third magnitude; in September, October, and November, to those of the fourth magnitude; in December and January, 1574, to those of the fifth; in February to those of the sixth magnitude; and in March it entirely vanished. Its color at the beginning was clear, bright, and inclining to white; afterwards it turned a little yellow; and in March, 1573, it became ruddy, like Mars or Aldebaran; in May it turned to a kind of dusky whiteness, like that we observe in Saturn and that color it retained ever after, but growing always more and more obscure. Such also was the star in the right foot of Serpentarius, which Kepler's scholars ;

;

observed September 30, o.s., 1604, with a light exceeding that of Jupiter, though the night before it was not to be seen; and from that time it decreased by little and Uttle, and in 15 or 16 months entirely disappeared. Such a new star appearing with an unusual splendor is said to have moved Hipparchus to observe, and make a catalogue of, the fixed stars. As to those fixed stars that appear and disappear by turns, and increase slowly and by degrees, and scarcely ever exceed the stars of the third magnitude, they seem to be of another kind, which revolve about their axes, and, having a light and a dark side, show those two different sides by turns. The vapors which arise from the sun, the fixed stars, and the tails of the comets, may meet at last with, and fall into, the atmospheres of the planets by their gravity, and there be condensed and turned into water and humid spirits; and from thence, by a slow heat, pass gradually into the form of salts, and sulphurs, and tinctures, and mud, and clay, and sand, and stones, and coral, and other terrestrial substances. first

GENERAL SCHOLIUM The hypothesis

of vortices is pressed

with

many difficulties. That every planet

by a radius drawn to the sun may describe areas proportional to the times of description, the periodic times of the several parts of the vortices should observe the square of their distances from the sun; but that the periodic times of the planets

may

obtain the /^th power of their distances from the sun, the

periodic times of the parts of the vortex ought to be as the /^th

power of their That the smaller vortices may maintain their lesser revolutions about Saturn, Jupiter, and other planets, and swim quietly and undisturbed distances.

in the greater vortex of the sun, the periodic times of the parts of the sun's

vortex should be equal; but the rotation of the sun and planets about their axes, which ought to correspond with the motions of their vortices, recede far from all these proportions. The motions of the comets are exceedingly regular, are governed by the same laws mth the motions of the planets, and can by no means be accounted for by the hypothesis of vortices; for comets are carried mth very eccentric motions through all parts of the heavens indifferently, with a freedom that is incompatible with the notion of a vortex. Bodies projected in our air suffer no resistance but from the air. Withdraw the air, as is done in Mr. Boyle's vacuum, and the resistance ceases; for in this void a bit of fine down and a piece of solid gold descend with equal velocity. And the same argument must apply to the celestial spaces above the earth's atmosphere; in these spaces, where there is no air to resist their motions, all bodies will move with the greatest freedom; and the planets and comets will constantly pursue their revolutions in orbits given in kind and position, according to the laws above explained; but though these bodies may, indeed, continue in their orbits by the mere laws of gravity, yet they could by no means have at first derived the regular position of the orbits themselves from those laws. The six primary planets are revolved about the sun in circles concentric with the sun, and with motions directed towards the same parts, and almost in the same plane. Ten moons are revolved about the earth, Jupiter, and Saturn, in circles concentric with them, with the same direction of motion, and nearly in the planes of the orbits of those planets; but it is not to be conceived that mere mechanical causes could give birth to so many regular motions, since the comets range over all parts of the heavens in very eccentric orbits; for by that kind of motion they pass easily through the orbs of the planets, and with great rapidity; and in their aphelions, where they move the slowest, and are detained the longest, they recede to the greatest distances from each other, and hence suffer the least disturbance from their mutual attractions. This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One; especially since the 369

370

Mathematical Principles

same nature with the Hght of the sun, and from every system Ught passes into all the other systems and lest the systems of the fixed stars should, by their gravity, fall on each other, he hath placed those systems at immense distances from one another. This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God -KavroKpaTwp, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do light of the fixed stars is of the

:

not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God^ usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a Hving, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity and infinity, but eternal and infinite; he is not duration or space,

but he endures and is present. He endures forever, and is everywhere present; and, by existing always and everywhere, he constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is everywhere, certainly the Maker and Lord of all things cannot be never and nowhere. Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, coexistent parts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and everywhere. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him^ are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he the oblique case di), which signifies Lord. And in this sense princes are called gods, Psalms, 82.6; and John, 10.35. And Moses is called a god to his brother Aaron, and a god to Pharaoh, Exodus, 4.16; and 7.1. And in the same sense the souls of dead princes were formerly, by the heathens, called gods, but ^Dr.

falsely,

Pocock derives the Latin word Deus from the Arabic du

(in

because of their want of dominion.

^This was the opinion of the ancients. So Pythagoras, in Cicero De natura deorum i. Thales, Anaxagoras, Virgil, in Georgics iv. 220; and Aeneid vi. 721. Philo, Allegories, at the beginning of Book I. Aratus, in his Phaenomena, at the beginning. So also the sacred writers: as St. Paul, in Acts, 17.27, 28. St. John's Gospel, 14.2. Moses, in Deuteronomy, 4.39; and 10.14. David, in Psalms, 139.7,8,9. Solomon, in I Kings, 8.27. Job, 22.12,13,14. Jeremiah, 23.23,24. The idolaters supposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped; but erroneously.

Book

III:

The System of the World

and everywhere. Whence also he is all similar, arm, all power to perceive, to understand, and to

exists always

371 all

eye, all ear, all

act; but in a mannot at all corporeal, in a manner utterly ner not at all human, in a manner unknowTi to us. As a blind man has no idea of colors, so have we no idea of the manner by which the all-^^ise God perceives and understands all things. He is utterly void of all body and bodilj^ figure, and can therefore neither be seen, nor heard, nor touched; nor ought he to be worshiped under the representation of any corporeal thing. We have ideas of his attributes, but what the real substance of anything is we know not. In bodies, we see only their figures and colors, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savors; but their inward substances are not to be known either by our senses, or by any reflex act of our minds: much less, than, have we any idea of the substance of God. We know him only by his most \^'ise and excellent contrivances of things, and final causes; we admire him for his perfections; but we reverence and adore him on account of his dominion: for we adore him as his servants; and a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and everywhere, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and \\dll of a Being necessarily existing. But, by way of allegory, God is said

brain, all

to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to

be angry, to fight, to frame, to work, to build; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to rejoice, to

natural philosophy. Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, T\dthout suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes used to do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun decreases accurately as the inverse square of the distances as far as the orbit of Saturn, as evidently appears from the quiescence of the aphelion of the planets; nay, and even to the remotest aphelion of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity

Mathematical Principles

372

and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and does really

exist,

of our sea.

concerning a certain most subtle spirit which pervades and lies hid in all gross bodies; by the force and action of which spirit the particles of bodies attract one another at near distances, and cohere, if contiguous; and electric bodies operate to greater distances, as well repelling as attracting the neighboring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the \\dll, namely, by the vibrations of this spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished mth that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and

And now we might add something

elastic spirit operates.

OPTICS

CONTENTS Advertisements to First and Second Editions

377

BOOK ONE PART I.

II.

Definitions I-VIII Axioms I-VIII Propositions I-VIII Propositions I-XI

379 380 386 424

BOOK TWO I.

II.

III.

Observations concerning the reflexions, refractions,

and

colours of thin,

transparent bodies

457

Remarks upon

470

Of

the

the foregoing observations

permanent colours of natural

them and

bodies,

and

the

analogy between

the colours of thin, transparent plates (Propositions

I-XX)

478

IV. Observations concerning the reflexions and colours of thick transparent

496

polished plates

BOOK THREE I.

Observations concerning the inflexions of the rays of colours

made

light,

and

the

507

thereby '

Queries

1-31

516

375

ADVERTISEMENT TO FIRST EDITION Part

about light was written at the desire of some gentlemen of the Royal Society, in the year 1675, and then sent to their Secretary, and read at their meetings, and the rest was added about twelve years after to complete the theory; except the third book, and the last proposition of the second, which were since put together out of scattered papers. To avoid being engaged in disputes about these matters, I have hitherto delayed the printing, and should still have delayed it, had not the importunity of friends prevailed upon me. If any other papers writ on this subject are got out of my hands they are imperfect, and were perhaps written before I had tried all the experiments here set down, and fully satisfied myself about the laws of refractions and composition of colours. I have here published what I think proper to come abroad, -wishing that it may not be translated into another language of the ensuing discourse

^vithout

my

consent.

The crowns

which sometimes appear about the Sun and Moon, I have endeavoured to give an account of; but for want of sufficient observations leave that matter to be further examined. The subject of the third book I have also left imperfect, not having tried all the experiments which I intended when I was about these matters, nor repeated some of those which I did try, until I had satisfied myself about all their circumstances. To communicate what I have tried, and leave the rest to others for further enquiry, is all my design in of colours,

publishing these papers.

In a letter written to Mr. Leibnitz in the year 1679, and published by Dr.

mentioned a method by which I had found some general theorems about squaring curvilinear figures, or comparing them with the conic sections, or other the simplest figures with which they may be compared. And some years ago I lent out a manuscript containing such theorems, and having since met with some things copied out of it, I have on this occasion made it public, prefixing to it an Introduction, and subjoining a Scholium concerning that method. And I have joined with it another small tract concerning the curvilinear figures of the second kind, which was also written many years ago, and made known to some friends, who have solicited the making it public. I. N. Wallis, I

April

1,

1704.

377

ADVERTISEMENT TO SECOND EDITION In this Second Edition of these Optics I have omitted the mathematical tracts pubUshed at the end of the former edition, as not belonging to the subject. And at the end of the third book I have added some questions. And to shew that I

do not take gravity

for

an essential property

of bodies, I

tion concerning its cause, choosing to propose I

am

not yet satisfied about

it

for

want

it

by way

have added one quesof a question,

because

of experiments. I.

July

16, 1717.

378

N.

BOOK ONE Part

I

My design in this book is not to explain the properties of hght by hypotheses, but to propose and prove them by reason and experiments in order to which shall premise the following definitions and axioms. :

I

DEFINITIONS DEFINITION By

I

I understand its least parts, and those as well successive in the same lines, as contemporary in several lines. For it is manifest that light consists of parts, both successive and contemporary; because in the same place you may stop that which comes one moment, and let pass that which comes presently after; and in the same time you may stop it in any one place, and let it pass in any other. For that part of light which is stopped cannot be the same with that which is let pass. The least light or part of light, which may be stopped alone without the rest of the light, or propagated alone, or do or suffer any thing alone, which the rest of the light doth not or suffers not, I call a ray of light. the rays of light

DEFINITION

II

Refrangihility of the rays of light, is their disposition their way in passing out of one transparent body or

to he

refracted or turned out of into another. And a

medium

greater or less refrangihility of rays is their disposition to be turned

more or

less out

way in like incidences on the same medium. Mathematicians usually consider the rays of light to be lines reaching from the luminous body to the body illuminated, and the refraction of those rays to be the bending or breaking of those lines in their passing out of one medium into another. And thus may rays and refractions be considered, if light be propagated in an instant. But by an argument taken from the equations of the times of the eclipses of Jupiter's satellites, it seems that light is propagated in time, spending in its passage from the Sun to us about seven minutes of time: and, therefore, I have chosen to define rays and refractions in such general terms as may agree to light in both cases. of their

DEFINITION

III

same rays are more

Reflexihility of rays is their disposition to be reflected or turned hack into the

medium from any or less reflexihle

As if the

other

medium upon whose

which are turned back more or

surface of the glass

And

less easily.

and by being inchned more and more to and air, begins at length to be totally reflected

light pass out of a glass into air,

common

surface they fall.

379

Optics

380

by that surface; those copiously, or

most

by

which at

sorts of rays

like incidences are reflected

most

inclining the rays begin soonest to be totally reflected, are

reflexible.

DEFINITION IV The angle of incidence

angle which the line described by the incident ray

is that

contains with the perpendicular

to the reflecting or

refracting surface at the point of

incidence.

DEFINITION V The angle of

which the line described by the ray containeth with the perpendicular to the reflecting or

reflexion or refraction is the angle

reflected or refracted

refracting surface at the point of incidence.

DEFINITION VI The sines of incidence, and

incidence, reflexion,

and

reflexion,

refraction are the sines of the angles of

refraction.

DEFINITION

VII

whose rays are all alike refrangible I call Simple, Homogeneal and Similar; and that whose rays are some more refrangible than others I call compound, heterogeneal and dissimilar. The former light I call homogeneal, not because I would affirm it so in all respects, but because the rays which agree in refrangibility agree at least in all

The

light

those their other properties Avhich

I

consider in the following discourse.

DEFINITION The colours of homogeneal

lights

VIII

I call primary, homogeneal and simple; and

those of heterogeneal lights, heterogeneal

For these are always compounded appear in the following discourse.

and compound. homogeneal

of the colours of

lights; as \vill

AXIOMS AXIOM The angles of reflexion and

I

refraction lie in one

and

the

same plane with

the angle

of incidence.

AXIOM The angle of

II

reflexion is equal to the angle of incidence.

AXIOM

III

// the refracted ray be returned directly back to the point of incidence, refracted into the line before described by the incident ray.

AXIOM

it

shall be

IV

Refraction out of the rarer medium into the denser is made towards the perpendicular; that is, so that the angle of refraction be less than the angle of incidence.

Book

I:

Part

381

1

AXIOM V The sine of incidence

is either accurately or very nearly in

a given ratio

to the

sine

of refraction.

that proportion be known in any one inclination of the incident ray, 'tis kno\Mi in all the inclinations, and thereby the refraction in all cases of incidence on the same refracting body may be determined. Thus, if the refraction be made out of air into water, the sine of incidence of the red light is to the sine of its refraction as 4 to 3. If out of air into glass, the sines are as 17 to 11. In light of other colours the sines have other proportions: but the difference is

Whence

if

so Httle that

it

need seldom be considered.

Suppose, therefore, that RS [Fig. 1] represents the surface of stagnating water, and that C is the point of incidence in which any ray coming in the air from A in the line AC is reflected or refracted, and I would knoAv whither this ray shall go after reflexion or refraction

:

I erect

upon

the surface of the water from the point of incidence the perpendicu-

CP and

produce it downwards to Q, and conclude by the first Axiom that the ray after reflexion and refraction shall be found some-

lar

w-here in the plane of the angle of

incidence ACP produced. I let fall, therefore, upon the perpendicular

CP the

sine of incidence

AD; and

the reflected ray be desired, I be to B so that produce

if

DB

AD

equal to AD, and draw CB. For this Hne CB shall be the reflected ray; the angle of reflexion BCP and its sine being equal to the angle and sine of incidence, as they ought to be by the to H, so that second Axiom. But if the refracted ray be desired, I produce as the sine of refraction to the sine of incidence, that is (if may be to the light be red) as 3 to 4; and about the centre C and in the plane ACP, with the radius CA describing a circle ABE, I draw a parallel to the perpendicu-

BD

AD

DH lar

AD

CPQ, the

line

HE cutting the circumference in E and joining CE; this line

be the line of the refracted ray. For if EF be let fall perpendicularly on the line PQ, this hne EF shall be the sine of refraction of the ray CE, the

CE

shall

angle of refraction being ECQ; and this sine quently in proportion to the sine of incidence

AD

In hke manner,

EF

is

equal to

DH, and

as 3 to 4. if

there be a prism

C

"^"->.

of glass (that is, a glass bounded with two equal and parallel trian-

EZ

gular ends, and three plain and well polished sides, which meet in three parallel lines running from the three angles of one end to the

DO^

1:-VF

Fig. 2

conse-

Optics

382

three angles of the other end) and if the refraction of the Ught in passing [Fig. 2] represent a plane cutting this prism cross this prism be desired: let

ACB

three parallel lines or edges there where the light passeth be the ray incident upon the first side of the prism AC through it, and let the glass; and by putting the proportion of the sine of into where the light goes

transversely to

its

DE

incidence to the sine of refraction as 17 to 11 find EF the first refracted ray. Then, taking this ray for the incident ray upon the second side of the glass BC where the light goes out, find the next refracted ray FG by putting the proportion of the sine of incidence to the sine of refraction as 11 to 17. For if the sine

be to the sine of refraction as 17 to 11, the sine of incidence out of glass into air must on the contrary be to the sine of refraction as 11 to 17, by the third Axiom. Much after the same manner, if ACBD [Fig. 3] represent a glass spherically convex on both sides (usually called a lens, such as is a burning-glass, or spectacle-glass, or an object-glass of a telescope) and it be required to know of incidence out of air into glass

upon it from any lucid point Q shall be refracted, let QM reof its first spherical surface ACB, and present a ray falling upon any point by erecting a perpendicular to the glass at the point M, find the first refracted ray by the proportion of the sines 17 to 11. Let that ray in going out of the glass be incident upon N, and then find the second refracted ray Ng by the proportion of the sines 11 to 17. And after the same manner may the refraction be found when the lens is convex on one side and plane or concave on the other,

how

light falling

M

MN

or concave on both sides.

AXIOM

VI any

and

fall perpenplane or spheridicularly or almost perpendicularly on any reflecting or refracting cal surface, shall afterwards diverge from so many other points, or be parallel to so many other lines, or converge to so many other points, either accurately or without

Homogeneal rays which flow from

And

several points of

object,

same thing will happen if the rays be reflected or refracted successively by two or three or more plane or spherical surfaces. The point from which rays diverge or to which they converge may be called

any

sensible error.

the

And the focus of the incident rays being given, that of the reflected refracted ones may be found by finding the refraction of any two rays, as

their focus.

or

above; or more readily thus: Case 1. Let ACB [Fig. 4] be a reflecting or refracting plane, and

Q

the focus

Book

I

:

Part

383

1

and QqC a perpendicular to that plane. And if this perpendicular be produced to q, so that qC be equal to QC, the point q shall be the focus of the reflected rays; or if qC be taken on the same side of the plane with of the incident rays,

A

Q

A

E

^

B

?T B

Fig. 5

Fig. 4

QC, and in proportion to QC as the sine of incidence to the sine of refraction, the point q shall be the focus of the refracted rays. Case 2. Let ACB [Fig. 5] be the reflecting surface of any sphere whose centre is E. Bisect any radius thereof, (suppose EC) in T, and if in that radius on the same side the point T you take the points Q and q, so that TQ, TE, and Tq be continual proportionals, and the point Q be the focus of the incident rays, the point q shall be the focus of the reflected ones. Case 3. Let ACB [Fig. 6] be the refracting surface of any sphere whose centre is E. In any radius thereof EC produced both ways take ET and C^ equal to one another and severally in such proportion to that radius as the lesser of the

A

B

«

Fig. 6

and refraction hath to the difference of those sines. And then if in the same line you find any two points Q and q, so that TQ be to ET as E^ to tq, taking tq the contrary way from t which TQ lieth from T, and if the point Q be the focus of any incident rays, the point q shall be the focus of the refracted sines of incidence

ones.

And by

the same means the focus of the rays after two or more reflexions or refractions may be found.

Case

ACBD

be any refracting lens, spherically convex or concave or plane on either side, and let CD be its axis (that is, the line which 4.

Let

[Fig. 7]

A

and passes through the centres of the spheres), and in this axis produced let F and / be the foci of the refracted rays found as above, when the incident rays on both sides the lens are parallel to the same axis; and upon the diameter F/ bisected in E, describe a circle. Suppose now that any point Q be the focus of any incident rays. Draw QE cutting the cuts both

its

surfaces perpendicularly,

Optics

384

such proportion to ^E as ^E or TE hath to TQ. Let tq he the contrary way from t which TQ doth from T, and q shall be the focus of the refracted rays without any sensible error, provided the point Q be not so remote from the axis, nor the lens so broad as to make any of the rays fall too obliquely on the refracting surfaces. And by the like operations may the reflecting or refracting surfaces be found when the two foci are given, and thereby a lens be formed, which shall make the rays flow towards or from what place you please. So then the meaning of this Axiom is that if rays fall upon any plane or spherical surface or lens, and before their incidence flow from or towards any point Q, they shall, after reflexion or refraction, flow from or towards the point q found by the foregoing rules. And if the incident rays flow from or towards several points Q, the reflected or refracted rays shall flow from or towards so many other points q found by the same rules. Whether the reflected and refracted rays flow from or towards the point q is easily known by the situation of that point. For if that point be on the same side of the reflecting or refracting surface or lens with the point Q, and the incident rays flow from the point Q, the reflected flow towards the point q and the refracted from it; and if the incident rays flow towards Q, the reflected flow from g, and the refracted towards it. And the contrary happens when q is on the other side of the surface. said circle in

T

and

t,

and therein take

tq in

AXIOM

VII

rays which come from all the points of any object meet again in so many points after they have been made to converge by reflection or refraction, there they will make a picture of the object upon any white body on which they fall. So if PR [Fig. 3] represent any object without doors, and AB be a lens placed at a hole in the window-shut of a dark chamber, whereby the rays that come from any point Q of that object are made to converge and meet again in the

Wherever

the

point q; and if a sheet of white paper be held at q for the light there to fall upon it, the picture of that object PR will appear upon the paper in its proper shape and colours. For as the light which comes from the point Q goes to the point q, of the object will go to so so the light which comes from other points P and many other correspondent points p and r (as is manifest by the sixth Axiom) so that every point of the object shall illuminate a correspondent point of the

R

;

the object in shape and colour, this only excepted, that the picture shall be inverted. And this is the reason of that vulgar experiment of casting the species of objects from abroad upon a wall or picture,

and thereby make a picture

like

sheet of white paper in a dark room. In like manner, when a man views any object PQR, [Fig. 8] the light which comes from the several points of the object is so refracted by the transparent skins and humours of the eye (that is, by the outward coat EFG, called the tunica cornea, and by the crystalline humour AB which is beyond the pupil mk) as to converge and meet again in so many points in the bottom of the eye, and there to paint the picture of the object upon that skin (called the tunica retina) Avith which the bottom of the eye is covered. For anatomists, when they have taken off from the bottom of the eye that outward and most thick coat called

the dura mater, can then see through the thinner coats the pictures of objects lively painted thereon. And these pictures, propagated by motion along the fibres of the optic nerves into the brain, are the cause of vision. For accordingly

Book

Part

I:

385

1

—

Fl

Fig. 8

as these pictures are perfect or imperfect, the object is seen perfectly or imbe tinged with any colour (as in the disease of the jaundice)

perfectly. If the eye

bottom of the eye with that colour, then all the same colour. If the humours of the eye by old age decay, so as by shrinking to make the cornea and coat of the crystalline humour grow flatter than before, the light mil not be refracted enough, and for want of a sufficient refraction will not converge to the bottom of the eye but to so as to tinge the pictures in the

objects appear tinged

mth

some place beyond it, and by consequence paint in the bottom of the eye a confused picture, and according to the indistinctness of this picture the object will appear confused. This is the reason of the decay of sight in old men, and shews why their sight is mended by spectacles. For those convex glasses supply the defect of plumpness in the eye, and by increasing the refraction make the rays converge sooner, so as to convene distinctly at the bottom of the eye if the glass have a due degree of convexity. And the contrary happens in short-sighted men whose eyes are too plump. For the refraction being now too great, the rays converge and convene in the eyes before they come at the bottom and therefore the picture made in the bottom and the vision caused thereby "will not be distinct, unless the object be brought so near the eye as that the place where the converging rays convene may be removed to the bottom, or that the plumpness of the eye be taken off and the refractions diminished by a concave-glass of a due degree of concavity, or lastly that by age the eye grow flatter till it come to a due figure: For short-sighted men see remote objects best in old age, and therefore they are accounted to have the most lasting eyes. ;

AXIOM

VIII

An object seen by reflexion or refraction appears in that place from whence the rays after their last reflexion or refraction diverge in falling If

t

by

'">>''

the spectator's eye.

the object

A

[Fig. 9]

be seen

reflexion of a looking-glass

mn,

appear, not in its proper place behind the glass at a, from whence any rays AB, AC, AD, which flow from one and the same point of the object, do, after their reflexion made in the points B, C, D, diverge in going from the glass to E, F, G, where they are incident on the spectator's eyes. For these rays do make the same picture in the bottom of

,--5x-#1'* ^^ shall ^'-'1, 77^, 773^, 773^, 77>^, 78,

GM

the sines of refraction of those rays out of glass into air, their common sine of incidence being 50. So, then, the sines of the incidences of all the red-making rays out of glass into air were to the sines of their refractions not greater than 50 to 77, nor less than 50 to 773/^, but they varied from one another according to all intermediate proportions. And the sines of the incidences of the green-making rays were to the sines of their refractions in all proportions from that of 50 to 773^, unto that of 50 to 773^. And by the Uke limits above-mentioned were the refractions of the rays belonging to the rest of the colours defined, the sines of the red-making rays extending from 77 to 773^, those of the orange-making from 773^ to 77^5, those of the yellow-making from 77^5 to 773^, those of the green-making from 77 3^ to 77 3^ those of the blue-making f rom77 3^^ ioll%, those of the indigo-making from 67% to 77/^, and those of the violet from 77/^ to 78. These are the laws of the refractions made out of glass into air, and thence, by the third Axiom of the first part of this book, the laws of the refractions made out of air into glass are easily derived. ExpER. 8. I found, moreover, that when light goes out of air through several contiguous refracting mediums as through water and glass, and thence goes out ,

Optics

430

again into air, whether the refracting superficies be parallel or inclined to one another, that light as often as by contrary refractions 'tis so corrected, that it emergeth in lines parallel to those in which it was incident, continues ever after to be white. But if the emergent rays be inclined to the incident, the whiteness of the emerging light will by degrees in passing on from the place of emergence, become tinged in its edges with colours. This I tried by refracting light with prisms of glass placed within a prismatic vessel of water. Now, those colours argue a diverging and separation of the heterogeneous rays from one another by means of their unequal refractions, as in what follows will more fully appear. And, on the contrary, the permanent whiteness argues that in like incidences of the rays there is no such separation of the emerging rays, and by consequence no inequality of their whole refractions. Whence I seem to gather the two following theorems: 1 The excesses of the sines of refraction of several sorts of rays above their common sine of incidence when the refractions are made out of divers denser mediums immediately into one and the same rarer medium (suppose of air) are to one another in a given proportion. 2. The proportion of the sine of incidence to the sine of refraction of one

one medium

into another,

composed

and

of the pro-

the same sort of rays out of portion of the sine of incidence to the sine of refraction out of the first medium into any third medium, and of the proportion of the sine of incidence to the sine of refraction out of that third medium into the second medium. By the first theorem, the refractions of the rays of every sort made out of any medium into air are known by having the refraction of the rays of any one sort. As, for instance, if the refractions of the rays of every sort out of rain-water into air be desired, let the common sine of incidence out of glass into air be

subducted from the sines

of refraction,

is

and the excesses

will

be 27,

273-^,

27^4,

Suppose, now, that the sine of incidence of the least 273^3, 273/^, 27%, refrangible rays be to their sine of refraction out of rain-water into air as 3 to 4, and say as 1 the difference of those sines is to 3 the sine of incidence, so is 27 the least of the excesses above-mentioned to a fourth number 81 and 81 will be the common sine of incidence out of rain-water into air, to which sine (if you add all the above-mentioned excesses) you will have the desired sines of the 27/^, 28.

;

108%, 1083^,

refractions 108, 1083^,

1083/^,

108%, 108%,

109.

By the latter theorem, the refraction out of one medium into another is gathered as often as you have the refractions out of them both into any third medium. As if the sine of incidence of any ray out of glass into air be to its sine of refraction as 20 to 31 and the sine of incidence of the same ray out of air into water be to its sine of refraction as 4 to 3 the sine of incidence of that ray out of glass into water will be to its sine of refraction as 20 to 31 and 4 to 3 jointly; that is, as the factum of 20 and 4 to the factum of 31 and 3, or as 80 to 93. And these theorems being admitted into Optics, there would be scope enough of handling that science voluminously after a new manner, not only by teaching those things which tend to the perfection of vision, but also by determining mathematically all kinds of phenomena of colours which could be produced by refractions. For to do this, there is nothing else requisite than to find out the separations of heterogeneous rays, and their various mixtures and proportions in every mixture. By this way of arguing I invented almost all the phenomena described in these books, beside some others less necessary to the argument; ,

;

Book

I:

Part

431

2

and by the successes I met with in the trials, I dare promise that to him who shall argue truly, and then try all things with good glasses and sufficient circumspection, the expected event will not be wanting. But he is first to know what colours will arise from any others mixed in any assigned proportion. Proposition

4.

Theorem

3

may

he produced by composition which shall be like to the colours of homogeneal light as to the appearance of colour, but not as to the immutability of

Colours

and constitution of light. And those colours by how much they are more compounded by so much are they less full and intense, and by too much composition they may he diluted and weakened till they cease, and the mixture becomes white or colour

grey. There

may

he also colours produced by composition,

which are not fully

like

any of the colours of homogeneal light. For a mixture of homogeneal red and yellow compounds an orange, like in appearance of colour to that orange which in the series of unmixed prismatic colours lies between them; but the light of one orange is homogeneal as to refrangibility, and that of the other is heterogeneal, and the colour of the one, if viewed through a prism, remains unchanged, that of the other is changed and resolved into its component colours, red and yellow. And after the same manner other neighbouring homogeneal colours may compound new colours, like the intermediate homogeneal ones, as yellow and green, the colour between them both; and afterwards, if blue be added, there Avill be made a green the middle colour of the three which enter the composition. For the yellow and blue on either hand, if they are equal in quantity they draw the intermediate green equally towards themselves in composition, and so keep it as it were in equilibrium, that it verge not more to the yellow on the one hand, and to the blue on the other, but by their mixed actions remain still a middle colour. To this mixed green there may be further added some red and violet, and yet the green will not presently cease, but only grow less full and vivid, and by increasing the red and violet, it will grow more and more dilute until, by the prevalence of the added colours, it be overcome and turned into whiteness or some other colour. So if to the colour of any homogeneal light the Sun's white light composed of all sorts of rays be added, that colour will not vanish or change its species, but be diluted; and by adding more and more white it will be diluted more and more, perpetually. Lastly, if red and violet be mingled, there ^\ill be generated according to their various proportions various purples, such as are not like in appearance to the colour of any homogeneal light, and of these purples mixed with yellow and blue may be made other new colours.

Proposition Whiteness, and colours,

colours

all

5.

Theorem

4

grey colours between white and black,

and the whiteness of the Sun's mixed in a due proportion.

light is

may

compounded

he

of

compounded of primary

all the

The Proof by Experiments Experiment

shining into a dark chamber through a little round hole in the window-shut, and his light being there refracted by a prism to cast his coloured image PT [Fig. 5] upon the opposite wall, I held a white paper V to that image in such manner that it might be illuminated by the coloured light 9.

The Sun

Optics

432

from thence, and yet not intercept any part of that light in its passage from the prism to the spectrum. And I found that when the paper was held nearer to any colour than to the rest, it appeared of that colour to which it approached nearest; but when it was equally or almost equally distant from all the colours, so that it might be equally illuminated by them all, it appeared white. And in this last situation of the paper, if some colours were intercepted the paper lost its white colour, and appeared of the colour of the rest of the light which was not intercepted. So, then, the paper was illuminated with lights of various colours (namely, red, yellow, green, blue and violet) and every part

reflected

Fig. 5

proper colour until it was incident on the paper, and became reflected thence to the eye so that if it had been either alone (the rest of the hght being intercepted) or if it had abounded most, and been predominant in the light reflected from the paper, it would have tinged the paper with its own colour; and yet, being mixed with the rest of the colours in a due proportion, it made the paper look white, and therefore by a composition with the rest produced that colour. The several parts of the coloured light reflected from the spectrum, whilst they are propagated from thence through the air, do perpetually retain their proper colours, because wherever they fall upon the eyes of any spectator they make the several parts of the spectrum to appear under their proper colours. They retain, therefore, their proper colours when they fall upon the Paper V, and so by the confusion and perfect mixture of of the

Hght retained

its

;

those colours

compound

the whiteness of the light reflected from thence.

Let that spectrum or solar image PT [Fig. 6] fall now upon the above four inches broad, and about six feet distant from the prism lens ABC and so figured that it may cause the coloured light which divergeth from the prism to converge and meet again at its focus G, about six or eight feet distant from the lens, and there to fall perpendicularly upon a white paper DE. And if you move this paper to and fro, you will perceive that near the lens, as at de, the whole solar image (suppose at pf) will appear upon it intensely

ExpER.

10.

MN

Fig. 6

Book

I

:

Part

2

433

coloured after the manner above-explained; and that by receding from the lens those colours will perpetually come towards one another, and, by mixing more and more, dilute one another continually until at length the paper come to the focus G, where by a perfect mixture they will wholly vanish and be converted into whiteness, the whole light appearing now upon the paper like a little white circle. And afterwards by receding farther from the lens, the rays which before converged will now cross one another in the focus G, and diverge from thence, and thereby make the colours to appear again, but yet in a contrary order; suppose at 8e, where the red t is now above which before was below, and the

below which before was above. Let us now stop the paper at the focus G, where the hght appears totally white and circular, and let us consider its whiteness. I say, that this is composed of the converging colours. For if any of those colours be intercepted at the lens, the whiteness will cease and degenerate into that colour which ariseth from the composition of the other colours which are not intercepted. And then if the intercepted colours be let pass and fall upon that compound colour, they mix with it, and by their mixture restore the whiteness. So if the violet, blue and green be intercepted, the remaining yellow, orange and red will compound upon the paper an orange, and then if the intercepted colours be let pass, they will fall upon this compounded orange, and together with it decompound a white. So also if the red and violet be intercepted, the remaining yellow, green and blue will compound a green upon the paper, and then the red and violet being let pass will fall upon this green, and together with it decompound a white. And that in this composition of white the several rays do not suffer any change in their colorific qualities by acting upon one another, but are only mixed, and by a mixture of their colours produce white, may further appear by these arguments. If the paper be placed beyond the focus G, suppose at de, and then the red colour at the lens be alternately intercepted, and let pass again, the violet colour on the paper will not suffer any change thereby, as it ought to do if the several sorts of rays acted upon one another in the focus G, where they cross. Neither mil the red upon the paper be changed by any alternate stopping, and letting pass the violet which crosseth it. And if the paper be placed at the focus G, and the white round image at G be viewed through the prism HIK, and by the refraction of that prism be translated to the place rv, and there appear tinged with various colours (namely, the violet at v and red at r, and others between) and then the red colours at the lens be often stopped and let pass by turns, the red at r will accordingly disappear, and return as often, but the violet at v mil not thereby suffer any change. And so, by stopping and letting pass alternately the blue at the lens, the blue at v will accordingly disappear and return, without any change made in the red at r. The red, therefore, depends on one sort of rays, and the blue on another sort, which in the focus G, where they are commixed, do not act on one another. And violet

p

is

the same reason of the other colours. I considered, further, that when the most refrangible rays Pp, and the least refrangible ones T^, are by converging inclined to one another, the paper, if held very oblique to those rays in the focus G, might reflect one sort of them more copiously than the other sort, and by that means the reflected light would be tinged in that focus with the colour of the predominant rays, provided those there

is

Optics

434

rays severally retained their colours, or colorific qualities in the composition of white made by them in that focus. But if they did not retain them in that white, but became all of them severally endued there with a disposition to strike the sense vnth the perception of white, then they could never lose their whiteness by such reflexions. I inclined, therefore, the paper to the rays very obliquely, as in the second experiment of this second part of the first book, that the most refrangible rays might be more copiously reflected than the rest, and the whiteness at length changed successively into blue, indigo, and violet. Then I inclined it the contrary way, that the least refrangible rays might be more copious in the reflected hght than the rest, and the whiteness turned successively to yellow, orange,

and

red.

made an instrument XY in

fashion of a comb whose teeth, being in number sixteen, were about an inch and a half broad, and the intervals of the teeth about two inches wide. Then by interposing successively the teeth of this instrument near the lens, I intercepted part of the colours by the interposed Lastly,

I

them went on through the interval of the teeth to the paper DE, and there painted a round solar image. But the paper I had first placed so that the image might appear white as often as the comb was taken away; and then the Comb being as was said interposed, the whiteness by reason of the intercepted part of the colours at the lens did always change into the colour compounded of those colours which were not intercepted, and that tooth, whilst the rest of

colour was by the motion of the comb perpetually varied so that in the passing of every tooth over the lens all these colours (red, yellow, green, blue, and purple) did always succeed one another. I caused, therefore, all the teeth to pass successively over the lens, and when the motion was slow there appeared

a perpetual succession of the colours upon the paper; but if I so much accelerated the motion that the colours by reason of their quick succession could not be distinguished from one another, the appearance of the single colours ceased. There was no red, no yellow, no green, no blue, nor purple to be seen any longer, but from a confusion of them all there arose one uniform white colour. Of the light which now by the mixture of all the colours appeared white, there was no part really white. One part was red, another yellow, a third green, a fourth blue, a fifth purple, and every part retains its proper colour till it strikes the sensorium. If the impressions follow one another slowly, so that they may be severally perceived, there is made a distinct sensation of all the colours one after another in a continual succession. But if the impressions follow one another so quickly that they cannot be severally perceived, there ariseth out of them all one common sensation, which is neither of this colour alone nor of that alone, but hath itself indifferently to them all, and this is a sensation of whiteness. By the quickness of the successions, the impressions of the several colours are confounded in the sensorium, and out of that confusion ariseth a mixed sensation. If a burning coal be nimbly moved round in a circle with gyrations continually repeated, the whole circle will apear like fire; the reason of which is that the sensation of the coal in the several places of that circle remains impressed on the sensorium until the coal return again to the same place. And so in a quick consecution of the colours the impression of every colour remains in the sensorium, until a revolution of all the colours be completed, and that first colour return again. The impressions, therefore, of all the successive colours are at once in the sensorium, and jointly stir up a sensation of them all; and so it is

Book

I

:

Part

435

2

manifest by this experiment that the commixed impressions of all the colours do stir up and beget a sensation of white, that is, that whiteness is compounded of all the colours. And if the comb be now taken away, that all the colours may at once pass from the lens to the paper, and be there intermixed, and together reflected thence to the spectator's eyes, their impressions on the sensorium being now more subtly and perfectly commixed there, ought much more to stir up a sensation of whiteness. which, by reYou may instead of the lens use two prisms HIK and fracting the coloured light the contrary way to that of the first refraction, may make the diverging rays converge and meet again in G, as you see represented in the seventh Figure. For where they meet and mix, they will compose a white light, as when a lens is used. ExpER. 11. Let the Sun's coloured image PT [Fig. 8] fall upon the wall of a dark chamber, as in the third experiment of the first book, and let the same be viewed through a prism ahc, held parallel to the prism ABC, by whose refraction that image was made, and let it now appear lower than before, suppose in the place S over against the red colour T. And if you go near to the image PT,

LMN

S,"'

Fig. 8

the spectrum S will appear oblong and coloured like the image PT; but if you recede from it the colours of the spectrum S will be contracted more and more, and at length vanish, that spectrum S becoming perfectly round and white and if you recede yet farther, the colours will emerge again, but in a contrary order. Now that spectrum S appears white in that case, when the rays of several sorts which converge from the several parts of the image PT, to the prism ahc, ;

Optics

436

are so refracted unequally by it that in their passage from the prism to the eye they may diverge from one and the same point of the spectrum S, and so fall afterwards upon one and the same point in the bottom of the eye, and there be

mingled.

And, further, if the comb be here made use of, by whose teeth the colours at the image PT may be successively intercepted, the spectrum S, when the comb is moved slowly, will be perpetually tinged with successive colours. But when, by accelerating the motion of the comb, the succession of the colours is so quick that they cannot be severally seen, that spectrum S, by a confused and mixed sensation of them all, will appear white. ExPER. 12. The Sun shining through a large prism ABC [Fig. 9] upon a comb XY, placed immediately behind the prism, his light which passed through the interstices of the teeth fell upon a white paper DE. The breadths of the teeth were equal to their interstices, and seven teeth together with their interstices took up an inch in breadth. Now, when the paper was about two or three inches

IE Fig. 9

distant from the comb, the light which passed through its several interstices painted so many ranges of colours, kl, mn, op, qr, &c. which were parallel to one another, and contiguous, and mthout any mixture of white. And these ranges

continually up and down with a reciprocal motion, ascended and descended in the paper, and when the motion of the comb was so quick that the colours could not be distinguished from one another, the whole paper by their confusion and mixture in the sensorium apof colours,

if

the

peared white. Let the comb

comb was moved

now rest, and let the paper be removed farther from the

prism,

be dilated and expanded into one another and the several ranges more and more, and by mixing their colours will dilute one another, and at length, when the distance of the paper from the comb is about a foot, or a little more (suppose in the place 2D 2E) they will so far dilute one another as to of colours will

become white. With any obstacle,

the light be now stopped which passes through any one interval of the teeth, so that the range of colours which comes from thence may be taken away, and you will see the light of the rest of the ranges to be let all

Book expanded into the place

of the range

I

:

Part

2

437

taken away, and there to be coloured. Let

the intercepted range pass on as before, and its colours falhng upon the colours of the other ranges, and mixing with them, \\dll restore the whiteness. Let the paper 2D 2E be now very much inclined to the rays, so that the most refrangible rays may be more copiously reflected than the rest, and the white colour of the paper through the excess of those rays will be changed into blue and violet. Let the paper be as much inclined the contrary way, that the least refrangible rays may be now more copiously reflected than the rest, and by their excess the whiteness mil be changed into yellow and red. The several rays, therefore, in that white light do retain their colorific quahties, by which those

whenever they become more copious than the rest, do by their excess and predominance cause their proper colour to appear. And by the same way of arguing, applied to the third experiment of this of

any

sort,

book, it may be concluded that the white colour of all refracted light at its very first emergence, where it appears as white as before its incidence, is compounded of various colours. ExpER. 13. In the foregoing experiment the several intervals of the teeth of the comb do the office of so many prisms, every interval producing the phenomenon of one prism. Whence instead of those intervals using several prisms, I tried to compound whiteness by mixing their colours, and did it by using only three prisms, as also by using only two as follows: Let two prisms ABC and abc, [Fig. 10] whose refracting angles B and h are equal, be so placed parallel to one

second part of the

first

Fig. 10

B

one may touch the angle c at the base of the other, and their planes CB and cb, at which the rays emerge, may lie in directum. Then let the light trajected through them fall upon the paper MN, distant about 8 or 12 inches from the prisms. And the colours generated by the interior fimits B and c of the two prisms will be mingled at PT, and there compound white. For if either prism be taken away, the colours made by the other mil appear in that place PT, and when the prism is restored to its place again, so that its colours may there fall upon the colours of the other, the mixture of them both will restore the whiteness. This experiment succeeds also, as I have tried, when the angle h of the lower prism is a little greater than the angle B of the upper, and between the interior another that the refracting angle

of the

Optics

438

angles B and c there intercedes some space Be, as is represented in the figure, and the refracting planes BC and he are neither in directum nor parallel to one another. For there is nothing more requisite to the success of this experiment than that the rays of all sorts may be uniformly mixed upon the paper in the

most refrangible rays coming from the superior prism take up to P, the rays of the same sort which come from the inall the space from ferior prism ought to begin at P, and take up all the rest of the space from thence towards N. If the least refrangible rays coming from the superior prism take up the space MT, the rays of the same kind which come from the other prism ought to begin at T, and take up the remaining space TN. If one sort of the rays, which have intermediate degrees of refrangibility and come from the superior prism, be extended through the space MQ, and another sort of those rays through the space MR, and a third sort of them through the space MS, the same sorts of rays coming from the lower prism ought to illuminate the remaining spaces QN, RN, SN, respectively. And the same is to be understood of all the other sorts of rays. For thus the rays of every sort will be scattered uniformly and evenly through the whole space MN, and so, being everywhere mixed in the same proportion, they must everywhere produce the same colour. And, therefore, since by this mixture they produce white in the exterior spaces MP and TN, they must also produce white in the interior space PT. This is the reason of the composition by which whiteness was produced in this experiment, and by what other way soever I made the like composition, the result was place PT. If the

M

whiteness.

wth the teeth of a comb of a due size the coloured lights of the two prisms which fall upon the space PT be alternately intercepted, that space PT, Lastly,

if

the motion of the comb is slow, will always appear coloured, but by accelerating the motion of the comb so much that the successive colours cannot be distinguished from one another, it will apear white. ExpER. 14. Hitherto I have produced whiteness by mixing the colours of prisms. If, now, the colours of natural bodies are to be mingled, let water a little thickened with soap be agitated to raise a froth, and after that froth has

when

stood a Kttle there will appear to one that shall view it mtently various colours everywhere in the surfaces of the several bubbles; but to one that shall go so far off that he cannot distinguish the colours from one another, the whole froth will grow white with a perfect whiteness. ExPER. 15. Lastly, in attempting to compound a white, by mixing the coloured powders which painters use, I considered that all coloured powders do suppress and stop in them a very considerable part of the light by which they

become coloured by reflecting the light of their own and that of all other colours more sparingly, and yet

are illuminated. For they

colours

more

they do not

copiously,

reflect the light of their

own colours so copiously as white bodies do.

red lead, for instance, and a white paper be placed in the red light of the coloured spectrum made in a dark chamber by the refraction of a prism, as is described in the third experiment of the first part of this book, the paper will appear more lucid than the red lead, and therefore reflects the red-making rays more copiously than red lead doth. And if they be held in the light of any other colour, the light reflected by the paper will exceed the light reflected by the red lead in a much greater proportion. And the hke happens in powders of other colours. And, therefore, by mixing such powders we are not to expect a strong

If

Book

I:

Part

439

2

and full white, such as is that of paper, but some dusky obscure one, such as might arise from a mixture of light and darkness, or from white and black that is, a grey, or dun, or russet brown, such as are the colours of a man's nail, of a mouse, of ashes, of ordinary stones, of mortar, of dust and dirt in highways, and the like. And such a dark white I have often produced by mixing coloured powders. For thus one part of red lead, and five parts of viride ceris composed a dun colour Hke that of a mouse. For these two colours were severally so compounded of others that in both together were a mixture of all colours; and there was less red lead used than viride ceris, because of the fulness of its colour. Again, one part of red lead and four parts of blue bice composed a dun colour verging a little to purple, and by adding to this a certain mixture of orpiment and viride ceris in a due proportion, the mixture lost its purple tincture and became perfectly dun. But the experiment succeeded best without minium thus To orpiment I added by little and little a certain full bright purple, which painters use, until the orpiment ceased to be yellow, and became of a pale red. Then I diluted that red by adding a little viride ceris, and a little more blue bice than viride ceris, until it became of such a grey or pale white as verged to no one of the colours more than to another. For thus it became of a colour equal in whiteness to that of ashes, or of wood newly cut, or of a man's skin. The orpiment reflected more light than did any other of the powders, and therefore conduced more to the whiteness of the compounded colour than they. To assign ;

:

the proportions accurately may be difficult, by reason of the different goodness of powders of the same kind. Accordingly, as the colour of any powder is more or less full and luminous, it ought to be used in a less or greater proportion. Now, considering that these grey and dun colours may be also produced by

mixing whites and blacks, and by consequence differ from perfect whites, not in species of colours but only in degree of luminousness, it is manifest that there is nothing more requisite to make them perfectly white than to increase their light sufficiently; and, on the contrary, if by increasing their light they can be brought to perfect whiteness, it will thence also follow that they are of the same species of colour with the best whites, and differ from them only in the quantity of light. And this 1 tried as follows: I took the third of the above-mentioned grey mixtures, (that which was compounded of orpiment, purple, bice, and viride ceris) and rubbed it thickly upon the floor of my chamber where the Sun shone upon it through the opened casement and by it, in the shadow, I laid a piece of white paper of the same bigness. Then, going from them to the distance of 12 or 18 feet, so that I could not discern the unevenness of the surface of the powder, nor the little shadows let fall from the gritty particles thereof, the powder appeared intensely white, so as to transcend even the paper itself in whiteness, especially if the paper were a little shaded from the light of the clouds, and then the paper compared with the powder appeared of such a grey colour as the powder had done before. But by laying the paper where the Sun shines through the glass of the window, or by shutting the wndow that the Sun might shine through the glass upon the powder, and by such other fit means of increasing or decreasing the lights wherewith the powder and paper were illuminated, the light wherewith the powder is illuminated may be made stronger in such a due proportion than the light wherewith the paper is illuminated that they shall both appear exactly alike in whiteness. For when I was trying this, a friend coming to visit me, I stopped him at the door, and before I ;

Optics

440

him what the colours were, or what I was doing, I asked him which of the two whites was the best, and wherein they differed. And after he had at that distance viewed them well, he answered that they were both good whites, and that he could not say which was best, nor wherein their colours differed. Now, if you consider that this white of the powder in the sunshine was compounded of the colours which the component powders (orpiment, purple, bice, and viride ceris) have in the same sunshine, you must acknowledge by this experiment, as well as by the former, that perfect whiteness may be compounded of colours. From what has been said it is also evident that the whiteness of the Sun's light is compounded of all the colours wheremth the several sorts of rays whereof that light consists, when by their several refrangibilities they are separated from one another, do tinge paper or any other white body whereon they fall. For those colours (by Prop. II. Part 2.) are unchangeable, and whenever all those rays \vith those their colours are mixed again, they reproduce the same told

white light as before.

Proposition In a mixture of primary colours, know the colour of the compound.

With the

centre

O

[Fig.

11]

6.

Problem

the quantity

and radius

2

and quality of each being

OD

given, to

ADF, and DE, EF, FG, GA, AB, BC, CD, describe a circle

distinguish its circumference into seven parts proportional to the seven musical tones or intervals of the eight sounds, Sol, la, fa, sol, la, mi, fa, sol, contained in an eight; that is, proportional to the number

DE

represent a red colour, the part fourth CA green, the fifth AB the second EF orange, the third FG yellow, violet. And conceive that these blue, the sixth BC indigo, and the seventh are all the colours of uncompounded light gradually passing into one another, as they do when made by prisms; the circumference DEFGABCD, representing the whole series of colours from one end of the Sun's coloured image to the other, so that from to E be all degrees of red, at E the mean colour between red and orange, from E to F all degrees of orange, at F the mean between orange and yellow, from F to G all degrees of yellow, and so on. Let p be the centre of gravity of the arch DE, and q, r, s, t, u, x, the centres of gravity of the

y9, Me,

Mo,

M, Me, Me, M-

Let the

first

CD

D

FG, GA, AB, BC, and CD, respectively, and about those centres of arches EF,

gravity let circles proportional to the number of rays of each colour in the given mixture be described that is, the circle p proportional to the number of the red-making rays in the mixture, the circle q proportional to the number of the orange-making rays in the mixture, and so of the rest. Find the common centre :

of gravity of all those circles, p,

q, r, s,

t,

Let that centre be Z; and from the centre of the circle ADF, through Z to the circumference, drawing the right line

u, X.

^*^digo

Fig. 11

OY,

Y

in the cirthe place of the point colour ariscumference shall shew the

Book

I

:

Part

2

441

ing from the composition of all the colours in the given mixture, and the line OZ shall be proportional to the fulness or intenseness of the colour; that is, to its distance from whiteness. As if fall in the middle between F and G, the

Y

compounded colour shall be the best yellow; if Y verge from the middle towards F or G, the compound colour shall accordingly be a yellow, verging towards orange or green. If Z fall upon the circumference, the colour shall be intense and florid in the highest degree; if it fall in the midway between the circumference and centre, it shall be but half so intense; that is, it shall be such a colour as would be made by diluting the intensest yellow with an equal quantity of whiteness; and if it fall upon the centre O, the colour shall have lost all its intenseness, and become a white. But it is to be noted that if the point Z fall in or near the line OD, the main ingredients being the red and violet, the colour compounded shall not be any of the prismatic colours, but a purple, inclining to red or violet, accordingly as the point Z lieth on the side of the line DO towards E or towards C, and in general the compounded violet is more bright and more fiery than the uncompounded. Also, if only two of the primary colours which in the circle are opposite to one another be mixed in an equal proportion, the point Z shall fall upon the centre 0, and yet the colour compounded of those two shall not be perfectly white, but some faint anonymous colour. For I could never yet by mixing only two primary colours produce a perfect white. Whether it may be compounded of a mixture of three taken at equal distances in the circumference I do not know, but of four or five I do not much question but it may. But these are curiosities of little or no moment to the understanding the phenomena of Nature. For in all whites produced by Nature, there uses to be a mixture of all sorts of rays, and by consequence a composition of all colours.

To

give an instance of this rule, suppose a colour is compounded of these homogeneal colours: of violet one part, of indigo one part, of blue two parts, of green three parts, of yellow five parts, of orange six parts, and of red ten parts. Proportional to these parts describe the circles x,

v,

t,

s, r, q,

p, respec-

the circle x be one, the circle v may be one, the circle t two, the circle s three, and the circles r, q and p, five, six and ten. Then I find Z the common centre of gravity of these circles, and through Z drawing the line OY, the point falls upon the circumference between E and F, something nearer to E than to F; and thence I conclude that the colour compounded of these ingredients will be an orange, verging a little more to red than to yellow. Also I find that OZ is a little less than one half of OY, and thence I conclude that this orange hath a little less than half the fulness or intenseness of an uncompounded orange; that is to say, that it is such an orange as may be made tively, that

is,

so that

if

Y

by mixing an homogeneal orange with a good white in the proportion of the Line OZ to the Line ZY, this proportion being not of the quantities of mixed orange and white powders, but of the quantities of the lights reflected from them. This rule I conceive accurate enough for practice, though not mathematically accurate; and the truth of it may be sufficiently proved to sense by stopping any of the colours at the lens in the tenth experiment of this book. For the rest of the colours which are not stopped, but pass on to the focus of the lens, will there compound either accurately or very nearly such a colour as by this rule ought to result from their mixture.

Optics

442

Proposition

7.

Theorem

All the colours in the universe which are made by

5

and depend not on the of homogeneal lights, or compounded of light,

power ofi^nagination, are either the colours these, and thai either accurately or very nearly, according

to the rule

of the foregoing

problem.

has been proved (Prop. 1, Part 2) that the changes of colours made by refractions do not arise from any new modifications of the rays impressed by those refractions, and by the various terminations of Ught and shadow, as has been the constant and general opinion of philosophers. It has also been proved that the several colours of the homogeneal rays do constantly answer to their degrees of refrangibility (Prop. 1, Part 1 and Prop. 2, Part 2) and that their degrees of refrangibility cannot be changed by refractions and reflexions (Prop. 2, Part 1) and by consequence that those their colours are likewise immutable. It has also been proved directly by refracting and reflecting homogeneal lights apart, that their colours cannot be changed (Prop. 2, Part 2). It has been proved, also, that when the several sorts of rays are mixed, and in crossing pass through the same space, they do not act on one another so as to change each other's colorific qualities (Exper. 10, Part 2) but by mixing their actions in the sensorium beget a sensation differing from what either would do apart (that is, a sensation of a mean colour between their proper colours) and particularly when by the concourse and mixtures of all sorts of rays a white colour is produced, the white is a mixture of all the colours which the rays would have apart (Prop. 5, Part 2). The rays in that mixture do not lose or alter their several colorific qualities, but by all their various kinds of actions, mixed in the sensorium, beget a sensation of a middling colour between all their colours, which is whiteness. For whiteness is a mean between all colours, having itself indifferently to them all, so as with equal facility to be tinged with any of them. A red powder mixed with a little blue, or a blue with a little red, doth not presently lose its colour, but a white powder mixed \vith any colour is presently tinged with that colour, and is equally capable of being tinged with any colour whatever. It has been shewed, also, that as the Sun's light is mixed of all sorts of rays, so its whiteness is a mixture of the colours of all sorts of rays; those rays having from the beginning their several colorific qualities as well as their several refrangibihties, and retaining them perpetually unchanged notwithstanding any refractions or reflexions they may at any time suffer, and that whenever any sort of the Sun's rays is by any means (as by reflexion in Expers. 9 and 10, Part 1 or by refraction as happens in all refractions) separated from the rest, they then manifest their proper colours. These things have been proved, and the sum of all this amounts to the proposition here to be proved. For if the Sun's light is mixed of several sorts of rays, each of which have originally their several refrangibihties and colorific qualities, and notwithstand-

For

it

;

ing their refractions

and

reflexions,

and

their various separations or mixtures,

keep those their original properties perpetually the same without alteration; then all the colours in the world must be such as constantly ought to arise from the original colorific quahties of the rays whereof the lights consist by which those colours are seen. And, therefore, if the reason of any colour whatever be required, we have nothing else to do than to consider how the rays in the Sun's light have by reflexions or refractions, or other causes, been parted from one

Book

I

:

Part 2

443

another, or mixed together; or otherwise to find out what sorts of rays are in the hght by which that colour is made, and in what proportion; and then, by the last problem, to learn the colour which ought to arise by mixing those rays (or their colours) in that proportion. I speak here of colours so far as they arise from light. For they appear sometimes by other causes, as when by the power

phantasy we see colours in a dream, or a madman sees things before him which are not there; or when we see fire by striking the eye, or see colours like the eye of a peacock's feather by pressing our eyes in either corner whilst we look the other way. Where these and such like causes interpose not, the colour always answers to the sort or sorts of the rays whereof the light consists, as I have constantly found in whatever phenomena of colours I have hitherto been able to examine. I shall in the following Propositions give instances of this in of

the

phenomena

of chief est note.

Proposition

By

8.

Problem

3

made by prisms. a prism refracting the hght of the Sun, which comes into a dark chamber through a hole F^ almost as broad as the prism, and let represent a white paper on which the refracted light is cast, and suppose the most refrangible or deepest violet-making rays fall upon the space Ptt, the least refrangible or deepest red-making rays upon the space Tr, the middle sort between the indigo-making and blue-making rays upon the space Qx, the middle sort of the green-making rays upon the space R, the middle sort between the yellow-making and orange-making rays upon the space So-, and the discovered properties of light, to explain the colours

Let

ABC

[Fig. 12] represent

MN

other intermediate sorts upon intermediate spaces. For so the spaces upon which the several sorts adequately fall will, by reason of the different refrangibe so bility of those sorts, be one lower than another. Now, if the paper near the prism that the spaces PT and ttt do not interfere with one another, the distance between them Ttt will be illuminated by all the sorts of rays in that proportion to one another which they have at their very first coming out of the prism, and consequently be white. But the spaces PT and ttt on either hand will not be illuminated by them all, and, therefore, will appear coloured. And particularly at P, where the outmost violet-making rays fall alone, the colour must be the deepest violet. At Q where the violet-making and indigo-making rays are mixed, it must be a violet inclining much to indigo. At R where the

MN

444

Optics

violet-making, indigo-making, blue-making, and one half of the green-making rays are mixed, their colours must (by the construction of the second problem) compound a middle colour between indigo and blue. At S where all the rays are

mixed, except the red-making and orange-making, their colours ought by the same rule to compound a faint blue, verging more to green than indigo. And in the progress from S to T, this blue will grow more and more faint and dilute, till at T, where all the colours begin to be mixed, it ends in whiteness. So again, on the other side of the white at t, where the least refrangible or utmost red-making rays are alone, the colour must be the deepest red. At a the mixture of red and orange will compound a red inclining to orange. At p the mixture of red, orange, yellow, and one half of the green must compound a middle colour between orange and yellow. At x the mixture of all colours but violet and indigo will compound a faint yellow, verging more to green than to orange. And this yellow will grow more faint and dilute continually in its progress from X to TT. where by a mixture of all sorts of rays it will become white. These colours ought to appear were the Sun's light perfectly white; but because it inclines to yellow, the excess of the yellow-making rays whereby 'tis tinged with that colour, being mixed with the faint blue between S and T, will draw it to a faint green. And so the colours in order from P to r ought to be violet, indigo, blue, very faint green, white, faint yellow, orange, red. Thus it is by the computation and they that please to view the colours made by a prism will find it so in Nature. These are the colours on both sides the white when the paper isheldbetween the prism and the point where the colours meet, and the interj acent white vanishes. For if the paper be held still farther off from the prism, the most refrangible and least refrangible rays will be wanting in the middle of the light, and the rest of the rays which are found there will by mixture produce a fuller green than before. Also, the yellow and blue will now become less compounded, and by consequence more intense then before. And this also agrees with experience. And if one look through a prism upon a white object encompassed with blackness or darkness, the reason of the colours arising on the edges is much the same, as will appear to one that shall a little consider it. If a black object be encompassed with a white one, the colours which appear through the prism are to be derived from the light of the white one, spreading into the regions of the black, and therefore they appear in a contrary order to that, when a white object is surrounded with black. And the same is to be understood when an object is viewed, whose parts are some of them less luminous than others. For, in the borders of the more and less luminous parts, colours ought always by the same principles to arise from the excess of the light of the more luminous, and to be of the same kind as if the darker parts were black, but yet to be more ;

X

faint

and

What

dilute.

is

made by prisms may be easily applied of telescopes or microscopes, or by the humours

said of colours

made by

to colours

the glasses of the eye. the object-glass of a telescope be thicker on one side than on the other, or if one-half of the glass, or one-half of the pupil of the eye be covered mth any opaque substance, the object-glass, or that part of it or of the eye which is not

For

if

may be

considered as a wedge with crooked sides, and every wedge of glass or other pellucid substance has the effect of a prism in refracting the light which passes through it. covered,

Book

I

:

Part 2

445

How

the colours in the ninth and tenth experiments of the first part arise from the different refiexibihty of light, is evident by what was there said. But it is observable in the ninth experiment that whilst the Sun's direct light is yellow, the excess of the blue-making rays in the reflected beam of light suffices only to bring that yellow to a pale white inclining to blue, and not to

MN

with a manifestl}^ blue colour. To obtain, therefore, a better blue, I used instead of the yellow light of the Sun the white light of the clouds, by varying a little the experiment, as follows: ExpER. 16. Let HFG [Fig. 13] represent a prism in the open air, and S the eye of the spectator \ae\nng the clouds by their light coming into the prism at the plane side FIGK, and reflected in it by its base HEIG, and thence going out through its plane side HEFK to the eye. And when the prism and eye are conveniently placed, so that the angles of incidence and reflexion at the base may be about 40 degrees, the spectator A^•ill see a bow ]MN of a blue colour running from one end of the base to the other, with the concave side towards him, and the part of the base IMNG beyond this bow vdW be brighter than the other part on the other side of it. This blue colour MN, being made by nothing else than by reflexion of a specular superficies, seems so odd a phenomenon, and so difficult to be explained by the \ailgar hypothesis of philosophers, that I could not but think it deserved to be taken notice of. Now, for understanding the reason of it, suppose the plane ABC to cut the plane sides and base of the prism perpendicularly. From the eye to the line BC, wherein that plane cuts the base, draw the lines Sp and Sf, in the angles Spc 50 degrees/^, and S/c 49 degrees /^, and the point p ^^ill be the limit beyond tinge

it

EMNH

which none of the most refrangible rays can pass through the base of the prism, and be refracted, whose incidence is such that they may be reflected to the eye; and the point t

will

be the like limit for the least

beyond which none of them can pass through the base) whose incidence is such that by reflexion they may come to the eye. And the point r, taken in the middle way between p and t, will be the like limit for the meanly refrangible rays. And, therefore, all the least refrangible rays which fall upon the base beyond (that is, between t and B) and can come from thence to the eye, ^\ill be reflected thither; but on this side (that is, between t and c) many of these rays -will be transmitted through the base. And all the most refrangible rays which fall upon the base beyond p, (that is, between p and B) and can by reflexion come from thence to the eye ^^'ill be reflected thither, but everywhere between p and c many of these rays ^\'ill get through the base, and be refracted; and the same is to be understood of the meanly refrangible rays on either side of the point r. Whence it follows that the base of the prism must everywhere between t and B, by a total reflexion of all sorts of rays to the eye, look white and bright; and everywhere between p and C, by reason of the transmission refrangible rays (that

t,

t

is,

Optics

446

many

rays of every sort, look more pale, obscure, and dark. But at r, and between p and t, where all the more refrangible rays are reflected to the eye, and many of the less refrangible are transmitted, the excess of the most refrangible in the reflected light will tinge that light with their colour, which is violet and blue. And this happens by taking the line C 'prt B and EI. anywhere between the ends of the prism of

in other places

HG

Proposition

By

9.

Problem 4

the discovered properties of light, to explain the colours of the rainbow.

This

bow never appears but where

it

rains in the sunshine,

and may be made

by spouting up water which may break aloft, and scatter into drops, down like rain. For the Sun shining upon these drops certainly causes

artificially

and

fall

bow to appear to a spectator standing in a due position to the rain and Sun. And hence it is now agreed upon, that this bow is made by refraction of the the

Sun's light in drops of falling rain. This was understood by some of the ancients, and of late more fully discovered and explained by the famous Antonius de Dominis, Archbishop of Spalato in his book De Radiis Visus & Lucis, published by his friend Bartolus at Venice, in the year 1611, and written above 20 years before. For he teaches there how the interior bow is made in round drops of rain by two refractions of the Sun's light, and one reflexion between them, and

by two refractions, and two sorts of reflexions between them each drop of water, and proves his exg \ plications by experiments made with >j^^^ a phial full of water, and with globes ^ YV /^^^^ \ / of glass filled with water, and placed in the Sun to make the colours of the iX ^^ l\ / lU ^ two bows appear in them. The same B\\ C 7/'' the exterior

explication Descartes hath pursued in his Meteors, and mended that of

\

\

/

in

^v

''

\ \ /// ^Ov^ll'^ y/^

J

the exterior bow. But whilst they understood not the true origin of colours, it's necessary to pursue it here a little farther. For understanding, ^' therefore, how the bow is made, let a drop of rain, or any other spherical transparent body, be represented by the sphere BNFG, [Fig. 14] described with the centre C, and semi-diameter be one of the Sun's rays incident upon it at N, and thence CN. And let refracted to F, where let it either go out of the sphere by refraction towards V, or be reflected to G and at G let it either go out by refraction to R, or be reand at let it go out by refraction towards S, cutting the inciflected to

^^

X

AN

;

H

H

;

and RG, till they meet in X, and upon AX and NF, let fall the perpendiculars CD and CE, and produce CD till it fall upon the circumference at L. Parallel to the incident ray AN draw the diameter BQ, and dent ray in Y.

Produce

AN

the sine of incidence out of air into water be to the sine of refraction as I to to move from the point B R. Now, if you suppose the point of incidence continually till it come to L, the arch QF will first increase and then decrease, which the rays and co ntain; an d the arch and so Avill the angle to V3RR, will be biggest when is to as QF and angle will be to in which case as 2R to I. Also the angle AYS, which the rays let

N

AXR

AXR NE

AN

ND

ND

GR CN VH-RR

Book

I:

Part 2

447

AN and HS contain, will first decrease and then increase and grow least when ND is to CN as VH-RR to V8RR, in which case NE will be to ND as 3R And

so the angle which the next emergent ray (that is, the emergent ray will come to its limit after three reflexions) contains with the incident ray to

I.

AN

when

ND is to CN as VII-RR to V15RR, in which case NE will be to ND as

4R to I. And the angle which

the ray next after that emergent (that is, the ray emergent after four reflexions) contains with the incident will come to its limit is to to V'24RR, in which case will be to when as VII as 5R to I; and so on infinitely, the numbers 3, 8, 15, 24, &c. being gathered by continual addition of the terms of the arithmetical progression 3, 5, 7, 9, &c.

ND

CN

NE

-RR

ND

The truth of all this mathematicians will easily examine. Now, it is to be observed that, as when the sun comes to his tropics, days increase and decrease but a very little for a great while together; so when by increasing the distance CD, these angles come to their limits, they vary their quantity but very little for some time together; and, therefore, a far greater number of the rays which fall upon all the points N in the quadrant BL shall than in any other inclinations. And further it is to be observed that the rays which differ in refrangibility will have different limits of their angles of emergence, and by consequence according to their different degrees of refrangibility emerge most copiously in different angles, and being separated from one another appear each in their proper colours. And what those angles are may be easily gathered from the foregoing theorem by computation. For in the least refrangible rays the sines I and R (as was found above) are 108 and 81, and thence by computation the greatest angle AXR will be found 42 degrees and 2 minutes, and the least angle AYS, 50 degrees and 57 minutes. And in the most refrangible rays the sines I and R are 109 and 81, and thence by computation the greatest angle AXR will be found 40 degrees and 17 minutes, and the least angle AYS 54 degrees and 7 minutes. [Fig. 15] is the spectator's eye, and OP a line drawn Suppose, now, that parallel to the Sun's rays; and let POE, POF, POG, POH, be angles of 40 degrees 17 minutes, 42 degrees 2 minutes, 50 degrees 57 minutes, and 54 degrees 7 minutes, respectively, and these angles turned about their common side OP, shall with their other sides OE, OF, OG, OH, describe the verges of two rain-

emerge

in the limits of these angles

H

^ o -—

^tTi '

--P

/ '

r

' 1

'

1

r

'

/ '

111' 1

1

'

r

1

'

'

'

Wtil III \

^

I

V

t

I

1

1

1

III

Optics

448

bows AF, BE, and

CHDG.

For

E, F, G, H, be drops placed anywhere in the conical superficies described by OE, OF, OG, OH, and be illuminated by the Sun's rays SE, SF, SG, SH the angle SEO being equal to the angle POE, or 40 degrees 17 minutes, shall be the greatest angle in which the most refrangible rays can after one reflexion be refracted to the eye; and, therefore, all the drops in the line OE shall send the most refrangible rays most copiously to the eye, and thereby strike the senses with the deepest violet colour in that region. And in like manner the angle SFO being equal to the angle POF, or 42 degrees 2 minutes, shall be the greatest in which the least refrangible rays after one reflexion can emerge out of the drops; and, therefore, those rays shall come most copiously to the eye from the drops in the line OF, and strike the senses with the deepest red colour in that region. And, by the same argument, the rays which have intermediate degrees of refrangibility shall come most copiously from drops between E and F, and strike the senses with the intermediate colours in the order which their degrees of refrangibility require; that is, in the progress from E to F, or from the inside of the bow to the outside, in this order violet, indigo, blue, green, yellow, orange, red. But the violet, by the mixture of the white light of the clouds, will appear faint and incline to purple. Again, the angle SGO being equal to the angle POG, or 50° 51', shall be the least angle in which the least refrangible rays can after two reflexions emerge out of the drops; and, therefore, the least refrangible rays shall come most copiously to the eye from the drops in the line OG, and strike the sense with the deepest red in that region. And the angle SHO being equal to the angle POH, or 54 degrees 7 minutes, shall be the least angle in which the most refrangible rays after two reflexions can emerge out of the drops; and, therefore, those rays shall come most copiously to the eye from the drops in the line OH, and strike the senses with the deepest violet in that region. And by the same shall strike the sense with argument, the drops in the regions between G and the intermediate colours in the order which their degrees of refrangibility require; that is, in the progress from G to H, or from the inside of the bow to the outside, in this order: red, orange, yellow, green, blue, indigo, violet. And since these four lines OE, OF, OG, OH, may be situated anywhere in the abovementioned conical superficies, what is said of the drops and colours in these lines is to be understood of the drops and colours everywhere in those superficies. if

;

H

Thus

shall there

be

made two bows

of colours,

an interior and stronger, by

one reflexion in the drops, and an exterior and fainter by two; for the light becomes fainter by every reflexion. And their colours shall lie in a contrary order to one another, the red of both bows bordering upon the space GF, which is between the bows. The breadth of the interior bow EOF measured across the colours shall be 1 degree 45 minutes, and the breadth of the exterior GOH shall be 3 degrees 10 minutes, and the distance between them GOF shall be 8 degrees 15 minutes, the greatest semi-diameter of the innermost; that is, the angle POF being 42 degrees 2 minutes, and the least semi-diameter of the outermost POG, being 50 degrees 57 minutes. These are the measures of the bows as they would be were the Sun but a point; for by the breadth of his body the breadth of the bows ^vill be increased, and their distance decreased by half a degree, and so the breadth of the interior iris will be 2 degrees 15 minutes, that of the exterior 3 degrees 40 minutes, their distance 8 degrees 25 minutes, the greatest semi-diameter of the interior bow 42 degrees 17 minutes, and the

Book

I

:

Part 2

449

50 degrees 42 minutes. And such are the dimensions of the bows in the heavens found to be very nearly, when their colours appear strong and perfect. For once, by such means as I then had, I measured the greatest semi-diameter of the interior iris about 42 degrees, and the breadth of the red, yellow and green in that iris 63 or 64 minutes, besides the outmost faint red obscured by the brightness of the clouds, for which we may allow 3 or 4 minutes more. The breadth of the blue was about 40 minutes more besides the violet, which was so much obscured by the brightness of the clouds that I could not measure its breadth. But supposing the breadth of the blue and violet together to equal that of the red, yellow and green together, the whole breadth of this iris will be about 23^ degrees, as above. The least distance between this iris and the exterior iris was about 8 degrees and 30 minutes. The exterior iris was broader than the interior, but so faint, especially on the blue side, that I could not measure its breadth distinctly. At another time when both bows appeared more distinct, I measured the breadth of the interior iris 2 degrees 10 minutes, and the breadth of the red, yellow and green in the exterior iris was to the breadth of the same colours in the interior as 3 to 2. This explication of the rainbow is yet further confirmed by the known experiment (made by Antonius de Dominis and Descartes) of hanging up anywhere in the sunshine a glass globe filled mth water, and viewing it in such a posture that the rays which come from the globe to the eye may contain with the Sun's rays an angle of either 42 or 50 degrees. For if the angle be about 42 or 43 degrees, the spectator (suppose at 0) shall see a full red colour in that side of the globe opposed to the Sun as 'tis represented at F, and if that angle become less (suppose by depressing the globe to E) there will appear other colours, yellow, green and blue successive in the same side of the globe. But if the angle be made about 50 degrees (suppose by lifting up the globe to G) there \vill appear a red colour in that side of the globe towards the Sun, and if the angle be made greater (suppose by lifting up the globe to H) the red will turn successively to the other colours, yellow, green and blue. The same thing I have tried by letting a globe rest, and raising or depressing the eye, or otherwise moving it to make the angle of a just magnitude. I have heard it represented that if the light of a candle be refracted by a prism to the eye, when the blue colour falls upon the eye the spectator shall see red in the prism, and when the red falls upon the eye he shall see blue; and if this were certain the colours of the globe and rainbow ought to appear in a contrary order to what we find. But the colours of the candle being very faint, the mistake seems to arise from the difficulty of discerning what colours fall on the eye. For, on the contrary, I have sometimes had occasion to observe, in the Sun's light refracted by a prism, that the spectator always sees that colour in the prism which falls upon his eye. And the same I have found true also in candle-light. For when the prism is moved slowly from the line which is drawn directly from the candle to the eye, the red appears first in the prism and then the blue; and, therefore, each of them is seen when it falls upon the eye. For the red passes over the eye first, and then the blue. The light which comes through drops of rain by two refractions without any reflexion ought to appear strongest at the distance of about 26 degrees from the Sun, and to decay gradually both ways as the distance from him increases and decreases. And the same is to be understood of light transmitted through least of the exterior

450

Optics

And

the hail be a little flatted, as it often is, the light transmitted may grow so strong at a little less distance than that of 26 degrees, as to form a halo about the Sun or Moon; which halo, as often as the hailstones are duly figured, may be coloured, and then it must be red \\dthin by the least spherical hailstones.

if

and blue without by the most refrangible ones, especially if the snow in their centre to intercept the light within the halo (as Huygens has observed) and make the inside thereof more distinctly defined than it would otherwise be. For such hailstones, though spherical, by terminating the light by the snow, may make a halo red within and colourless without, and darker in the red than without, as halos used to be. For of those rays which pass close by the snow the rubriform will be least refracted, and so come to the eye in the directest lines. The light which passes through a drop of rain after two refractions, and three or more reflexions, is scarce strong enough to cause a sensible bow; but in those cylinders of ice by which Huygens explains the parhelia, it may perhaps refrangible rays,

hailstones have opaque globules of

be sensible.

Proposition

By

10.

Problem

the discovered 'properties of light, to explain the

5

permanent colours of natural

bodies.

These colours

from hence, that some natural bodies reflect some sorts of rays, others other sorts more copiously than the rest. Minium reflects the least refrangible or red-making rays most copiously, and thence appears red. Violets reflect the most refrangible most copiously, and thence have their colour, and arise

Every body reflects the rays of its own colour more copiously and from their excess and predominance in the reflected light has

so of other bodies.

than the

rest,

its colour.

ExpER. 17. For problem proposed

if,

in the

homogeneal

lights obtained

by the

solution of the

in the fourth Proposition of the first part of this book, you place bodies of several colours, you will find, as I have done, that every body looks most splendid and luminous in the light of its own colour. Cinnabar in

the homogeneal red light is most resplendent, in the green light it is manifestly less resplendent, and in the blue light still less. Indigo in the violet blue light is most resplendent, and its splendour is gradually diminished as it is removed thence by degrees through the green and yellow light to the red. By a leek the green light, and next that the blue and yellow which compound green, are more strongly reflected than the other colours red and violet, and so of the rest. But to make these experiments the more manifest, such bodies ought to be chosen as have the fullest and most vivid colours, and two of those bodies are to be

compared together. Thus, for instance, if cinnabar and ultra-marine blue, or some other full blue be held together in the red homogeneal light, they will both appear red, but the cinnabar will appear of a strongly luminous and resplendent red, and the ultra-marine blue of a faint obscure and dark red and if they be held together in the blue homogeneal light, they vnW both appear blue, but the ultra-marine will appear of a strongly luminous and resplendent blue, and the cinnabar of a faint and dark blue. Which puts it out of dispute that the cinnabar reflects the red light much more copiously than the ultra-marine doth, and the ultra-marine reflects the blue light much more copiously than the cinnabar doth. The same experiment may be tried successfully with red lead and indigo ;

Book

I:

Part

2

451

or ^\ith any other two coloured bodies, if due allowance be made for the different strength or weakness of their colour and light. And as the reason of the colours of natural bodies is evident by these experiments, so it is further confirmed and put past dispute by the two first experiments of the first part, whereby 'twas proved in such bodies that the reflected lights which differ in colours do differ also in degrees of refrangibility. For thence it's certain that some bodies reflect the more refrangible, others the less refrangible rays more copiously. And that this is not only a true reason of these colours, but even the only reason, may appear further from this consideration that the colour of homogeneal light cannot be changed by the reflexion of natural bodies. For if bodies by reflexion cannot in the least change the colour of any one sort of rays, they cannot appear coloured by any other means than by reflecting those which either are of their own colour, or which by mixture must produce it. But in trying experiments of this kind, care must be had that the light be sufficiently homogeneal. For if bodies be illuminated by the ordinary prismatic colours, they will appear neither of their own daylight colours, nor of the colour of the light cast on them, but of some middle colour between both, as I have found by experience. Thus, red lead (for instance) illuminated wth the ordinary prismatic green, will not appear either red or green, but orange or yellow, or between yellow and green, accordingly as the green light by w^hich 'tis illuminated is more or less compounded. For because red lead appears red when illuminated with white light, wherein all sorts of rays are equally mixed, and in the green light all sorts of rays are not equally mixed, the excess of the yellowmaking, green-making and blue-making rays in the incident green fight wdll cause those rays to abound so much in the reflected light as to draw the colour from red towards their colour. And because the red lead reflects the red-making rays most copiously in proportion to their number, and next after them the orange-making and yellow-making rays, these rays in the reflected light will be more in proportion to the light than they were in the incident green light, and thereby will draw the reflected light from green towards their colour. And, therefore, the red lead will appear neither red nor green, but of a colour between both. In transparently coloured liquors, 'tis observable that their colour uses to

vary

Thus, for instance, a red liquor in a conical glass, held between the fight and the eye, looks of a pale and dilute yellow at the bottom where 'tis thin, and a little higher where 'tis thicker grows orange, and \vith their thickness.

where 'tis still thicker becomes red, and where 'tis thickest the red is deepest and darkest. For it is to be conceived that such a liquor stops the indigomaking and \T.olet-making rays most easily, the blue-making rays more difficultly, the green-making rays still more difficultly, and the red-making most difficultly; and that if the thickness of the liquor be only so much as suffices to stop a competent number of the violet-making and indigo-making rays, Avithout diminishing much the number of the rest, the rest must (by Prop. 6, Part 2) compound a pale yellow. But if the liquor be so much thicker as to stop also a great number of the blue-making rays, and some of the green-making, the rest must compound an orange; and where it is so thick as to stop also a great number of the green-making and a considerable number of the yellow-making, the rest must begin to compound a red, and this red must grow deeper and

452

Optics

darker as the yellow-making and orange-making rays are more and more stopped by increasing the thickness of the liquor, so that few rays besides the red-making can get through. Of this kind is an experiment lately related to me by Mr. Halley, who, in diving deep into the sea in a diving vessel, found in a clear sunshine day that when he was sunk many fathoms deep into the water the upper part of his hand on which the Sun shone directly through the water and through a small glass window in the vessel appeared of a red colour, like that of a damask rose, and the water below and the under part of his hand illuminated by light reflected from the water below looked green. For thence it may be gathered, that the sea water reflects back the violet and blue-making rays most easily, and lets the red-making rays pass most freely and copiously to great depths. For thereby the Sun's direct light at all great depths, by reason of the predominating red-making rays, must appear red; and the greater the depth is, the fuller and intenser must that red be. And at such depths as the violet-making rays scarce penetrate unto, the blue-making, green-making, and yellow-making rays, being reflected from below more copiously than the red-making ones, must compound a green. Now, if there be two liquors of full colours, (suppose a red and blue) and both of them so thick as suffices to make their colours sufficiently full, though either liquor be sufficiently transparent apart, yet will you not be able to see through both together. For, if only the red-making rays pass through one liquor, and only the blue-making through the other, no rays can pass through both. This Mr. Hook tried casually with glass wedges filled with red and blue liquors, and was surprised at the unexpected event, the reason of it being then unknown; which makes me trust the more to his experiment, though I have not tried it myself. But he that would repeat it must take care the liquors be of very good and full colours. Now, whilst bodies become coloured by reflecting or transmitting this or that sort of rays more copiously than the rest, it is to be conceived that they stop and stifle in themselves the rays which they do not reflect or transmit. For, if gold be foliated and held between your eye and the light, the Hght looks of a greenish-blue, and therefore massy gold lets into its body the blue-making rays to be reflected to and fro within it till they be stopped and stifled, whilst it reflects the yellow-making outwards, and thereby looks yellow, and much after the same manner that leaf gold is yellow by reflected, and blue by transmitted light, and massy gold is yellow in all positions of the eye there are some liquors, as the tincture of lignum nephriticum, and some sorts of glass, which transmit one sort of light most copiously, and reflect another sort, and thereby look of several colours, according to the position of the eye to the light. But, if these liquors or glasses were so thick and massy that no light could get through them, I question not but they would, like all other opaque bodies, appear of one and the same colour in all positions of the eye, though this I cannot yet affirm by experience. For all coloured bodies, so far as my observation reaches, may be seen through if made sufficiently thin, and, therefore, are in some measure transparent, and differ only in degrees of transparency from tinged transparent liquors, these liquors as well as those bodies by a sufficient thickness becoming opaque. A transparent body which looks of any colour by transmitted light may also look of the same colour by reflected light, the light of that colour being ;

Book

I

:

Part 2

453

farther surface of the body, or by the air beyond it. And then the reflected colour mil be diminished, and perhaps cease, by making the body very thick, and pitching it on the backside to diminish the reflexion of its reflected

by the

farther surface, so that the Hght reflected from the tinging particles may predominate. In such cases, the colour of the reflected light will be apt to vary from

that tinged bodies and hquors reflect some sort of rays, and intromit or transmit other sorts, shall be said in the next book. In this Proposition I content myself to have put it past dispute

that of the Hght transmitted.

But whence

it is

that bodies have such properties, and thence appear coloured.

Proposition

By mixing

11.

Problem

6

compound a beam of light of the same Sun's direct light, and therein to experience

coloured lights,

to

colour

and

the truth of nature with a beam of the the foregoing Propositions. Let ABC abc [Fig. 16] represent a prism, by which the Sun's Hght let into a dark chamber through the hole F, may be refracted towards the lens MN, and paint upon it at p, q, r, s, and t, the usual colours (violet, blue, green, yellow, and red) and let the diverging rays by the refraction of this lens converge again towards X, and there, by the mixture of all those their colours, compound a

DEG

deg, white according to what was shewn above. Then let another prism parallel to the former, be placed at X, to refract that white Ught upwards towards Y. Let the refracting angles of the prisms and their distances from the lens be equal so that the rays, which converged from the lens towards X, and without refraction, would there have crossed and diverged again, may by the refraction of the second prism be reduced into parallelism and diverge no more. For then those rays will recompose a beam of white light XY. If the refracting angle of either prism be the bigger, that prism must be so much the nearer to the lens. You mil know when the prisms and the lens are well set together, by observing if the beam of Hght XY, which comes out of the second prism, be perfectly white to the very edges of the Hght, and at all distances from the prism continue perfectly and totally white Hke a beam of the Sun's light. For tlH this happens, the position of the prisms and lens to one another must be corrected; and then if by the help of a long beam of wood, as is represented in the Figure, or by a tube, or some other such instrument, made for that purpose, they be made fast in that situation, you may try all the same experiments in this which have been made in the Sun's direct Hght. compounded beam of Hght

XY

Fig. 16

Optics

454

For this compounded beam of light has the same appearance, and is endowed with all the same properties, with a direct beam of the Sun's light, so far as my observation reaches. And in trying experiments in this beam you may by stopping any of the colours, p, q, r, s, and t, at the lens, see how the colours produced in the experiments are no other than those which the rays had at the lens before they entered the composition of this beam; and, by consequence, that they arise not from any new modifications of the light by refractions and reflexions, but from the various separations and mixtures of the rays originally endowed with their colour-making qualities. So, for instance, having with a lens 43'^K

.^y-vx:rV/,.-

them began

to be transmitted, there arose in

>'::';:',.

/yy^

so little inclined to the

incident rays that

',>>'.

o"""'

.^>:;;
vs;% '.W' '',.

•

,o>

"

/ I \

/"^^

^—^ Fig. 1

'',

Optics

458

and bended more and more about the said transparent spot, till they were completed into circles or rings encompassing it, and afterwards continually grew more and more contracted. These arcs at their first appearance were of a violet and blue colour, and between them were white arcs of circles, which presently by continuing the motion of the prisms became a little tinged in their inward limbs with red and yellow, and to their outward limbs the blue was adjacent. So that the order of these colours from the central dark spot, was at that time white, blue, violet; black, red, orange, yellow, white, blue, violet, &c. But the yellow and red were much fainter than the blue and violet. The motion of the prisms about their axis being continued, these colours contracted more and more, shrinking towards the whiteness on either side of it, motion

of the prisms, these arcs increased

until they totally vanished into

it.

And then the

circles in

those parts appeared

any other colours intermixed. But by further moving black and the prisms about, the colours again emerged out of the whiteness, the violet and blue at its inward limb, and at its outward limb the red and yellow. So that now their order from the central spot was white, yellow, red; black; violet, blue, white, yellow, red, &c., contrary to what it was before. Obs. 3. When the rings or some parts of them appeared only black and white, they were very distinct and well-defined, and the blackness seemed as intense white, without

as that of the central spot. Also in the borders of the rings, where the colours began to emerge out of the whiteness, they were pretty distinct, which made

a very great multitude. I have sometimes numbered above thirty successions (reckoning every black and white ring for one succession) and seen more of them, which by reason of their smallness I could not number. But in other positions of the prisms, at which the rings appeared of many colours, I could not distinguish above eight or nine of them, and the exterior of those were very confused and dilute. In these two Observations to see the rings distinct, and without any other colour than black and white, I found it necessary to hold my eye at a good distance from them. For by approaching nearer, although in the same inclination of my eye to the plane of the rings, there emerged a bluish colour out of the white, which by dilating itself more and more into the black rendered the circles less distinct, and left the white a little tinged with red and yellow. I found also by looking through a slit or oblong hole, which was narrower than the pupil of my eye, and held close to it parallel to the prisms, I could see the circles much distincter and visible to a far greater number than otherwise. Obs. 4. To observe more nicely the order of the colours which arose out of the white circles as the rays became less and less inclined to the plate of air, I took two object-glasses (the one a plano-convex for a fourteen-foot telescope, and the other a large double convex for one of about fifty-feet) and upon this, laying the other with its plane side downwards, I pressed them slowly together, to make the colours successively emerge in the middle of the circles, and then slowly Hfted the upper glass from the lower to make them successively vanish again in the same place. The colour, which by pressing the glasses together, emerged last in the middle of the other colours, would upon its first appearance look like a circle of a colour almost uniform from the circumference to the centre and by compressing the glasses still more, grow continually broader until a new colour emerged in its centre, and thereby it became a ring en-

them

visible to

Book II: Part 1 And by compressing

459

the glasses still more, the compassing that new colour. diameter of this ring would increase, and the breadth of its orbit or perimeter decrease until another new colour emerged in the centre of the last. And so on until a third, a fourth, a fifth, and other following new colours successively emerged there, and became rings encompassing the innermost colour, the last of which was the black spot. And, on the contrary, by lifting up the upper glass from the lower, the diameter of the rings would decrease, and the breadth of their orbit increase, until their colours reached successively to the centre and then, they being of a considerable breadth, I could more easily discern and distinguish their species than before. And by this means I observed their succession and quantity to be as followeth: Next to the pellucid central spot made by the contact of the glasses succeeded blue, white, yellow, and red. The blue was so little in quantity that I could not ;

made by the prisms, nor could I well distinguish any but the yellow and red were pretty copious, and seemed about as much in extent as the white, and four or five times more than the blue. The next circuit in order of colours immediately encompassing these were violet, blue, green, yellow, and red; and these were all of them copious and vivid, excepting the green, which was very little in quantity, and seemed much more faint and dilute than the other colours. Of the other four, the violet was the least in extent, and the blue less than the yellow or red. The third circuit or order was purple, blue, green, yellow, and red; in which the purple seemed more reddish than the violet in the former circuit, and the green was much more discern

it

violet in

in the circles

it,

conspicuous, being as brisk and copious as any of the other colours, except the yellow, but the red began to be a little faded, inclining very much to purple. After this succeeded the fourth circuit of green and red. The green was very copious and lively, inclining on the one side to blue, and on the other side to yellow. But in this fourth circuit there was neither violet, blue, nor yellow, and the red was very imperfect and dirty. Also the succeeding colours became more and more imperfect and dilute, till after three or four revolutions they ended

'"">,

.^.^ -^

# zyxut

s

r

q po n

#

m Ik ih g f

e

d

c

=cdefghtklmnopqrstuxyz

^^

Fig. 2

Optics

460

Their form, when the glasses were most compressed so as to make the black spot appear in the centre, is deUneated in the second figure; where a, b, c, d, e: f, g, h, i, k: I, m, n, o, p: q, r: s, t: v, x: y, z, denote the colours reckoned in order from the center: black, blue, white, yellow, red; in perfect whiteness.

violet, blue, green, yellow, red; purple, blue, green, yellow, red; green, red;

greenish blue, red; greenish blue, pale red; greenish blue, reddish white. Obs. 5. To determine the interval of the glasses, or thickness of the intercolour was produced, I measured the diameters of the first six rings at the most lucid part of their orbits, and, squaring them, I found their squares to be in the arithmetical progression of the odd numbers, 1, 3, 5, 7, 9, 11. And since one of these glasses was plane, and the other spherical, their

jacent

air,

by which each

must be in the same progression. I measured also the dark or faint rings between the more lucid colours, and found diameters their squares to be in the arithmetical progression of the even numbers, 2, 4, 6, 8, 10, 12. And it being very nice and difficult to take these measures exactly, I intervals at those rings of the

repeated them divers times at divers parts of the glasses, that by their agreement I might be confirmed in them. And the same method I used in determining some others of the follomng observations. Obs. 6. The diameter of the sixth ring at the most lucid part of its orbit was 1^ parts of an inch, and the diameter of the sphere on which the double convex object-glass was ground was about 102 feet, and hence I gathered the thickness of the air or aereal interval of the glasses at that ring. But some time after, suspecting that in making this observation I had not determined the diameter of the sphere with sufiicient accurateness, and being uncertain whether the plano-convex glass was truly plane, and not something concave or convex on that side which I accounted plane; and whether I had not pressed the glasses together, as I often did, to make them touch (for by pressing such glasses together their parts easily yield inwards, and the rings thereby become sensibly broader than they would be, did the glasses keep their figures), I repeated the parts of experiment, and found the diameter of the sixth lucid ring about an inch. I repeated the experiment also with, such an object-glass of another telescope as I had at hand. This was a double convex ground on both sides to one and the same sphere, and its focus was distant from it 83% inches. And thence, if the sines of incidence and refraction of the bright yellow light be assumed in proportion as 11 to 17, the diameter of the sphere to which the glass was figured will by computation be found 182 inches. This glass I laid upon a flat one, so that the black spot appeared in the middle of the rings of colours mthout any other pressure than that of the weight of the glass. And now, measuring the diameter of the fifth dark circle as accurately as I could, I found it the fifth part of an inch precisely. This measure was taken with the points of a pair of compasses on the upper surface on the upper glass, and my eye was about eight or nine inches distance from the glass, almost perpendicularly over it, and the glass was /^ of an inch thick, and thence it is easy to collect that the true diameter of the ring between the glasses w^as greater than its measured diameter above the glasses in the proportion of 80 to 79, or thereabouts, and by consequence equal to ^/^g parts of an inch, and its true semidiameter equal to /^g parts. Now, as the diameter of the sphere (182 inches) is to the semi diameter of this fifth dark ring (/^g parts of an inch) so is this semidiameter to the thickness of the air at this fifth dark ring; which is, therefore,

t^

,

Book

II:

Part

461

1

or i.T^l^ysi parts of an inch; and the fifth part thereof (viz., the gg^ 39-th part of an inch) is the thickness of the air at the first of these dark g

^^Q 3 1

rings.

repeated ^^ith another double convex object-glass ground on both sides to one and the same sphere. Its focus was distant from it 1683^2 inches, and, therefore, the diameter of that sphere was 184 inches. This glass being laid upon the same plain glass, the diameter of the fifth of the dark rings, when the black spot in their centre appeared plainly without pressing the glasses, was by the measure of the compasses upon the upper glass i%^ parts of an inch, and by consequence between the glasses it was -iT^-f-l; for the upper glass was i of an inch thick, and my eye was distant from it 8 inches. And a third proportional to half this from the diameter of the sphere is gg^^gso P^rts of an inch. This is, therefore, the thickness of the air at this ring, and a fifth part thereof (viz., the gs.sso th part of an inch) is the thickness thereof at the first of the rings, as above. I tried the same thing by laying these object-glasses upon flat pieces of a broken looking-glass, and found the same measures of the rings; which makes me rely upon them till they can be determined more accurately by glasses ground to larger spheres, though in such glasses greater care must be taken of a

The same experiment

I

true plane.

These dimensions were taken when my eye was placed almost perpendicuabout an inch, or an inch and a quarter, distant from the incident rays, and eight inches distant from the glass; so that the rays were inclined to the glass in an angle of about four degrees. Whence, by the following Observation, you will understand that had the rays been perpendicular to the glasses, the thickness of the air at these rings would have been larly over the glasses, being

the proportion of the radius to the secant of four degrees (that is, of 10,000 to 10,024). Let the thicknesses found be, therefore, diminished in this proportion, and they will become gg_95g and -qq^-^, or (to use the nearest round number) the g 9.0 00 th part of an inch. This is the thickness of the air at the darkless in

made by

perpendicular rays; and half this thickness multiplied by the progression, 1, 3, 5, 7, 9, 11, &c. gives the thicknesses of the air at the most luminous parts of all the brightest rings, viz., YTaho^i 178,000' 1 78,000' 178,000' "'*"•' their arithmetical means ytYToVo' s.ooc thicknesses the darkest all the dark being its at parts of ones. &c. ng'^,0 Obs. 7. The rings were least when my eye was placed perpendicularly over the glasses in the axis of the rings; and when I viewed them obliquely they became bigger, continually swelling as I removed my eye farther from the axis. And partly by measuring the diameter of the same circle at several obliquities of my eye, partly by other means, as also by making use of the two prisms for very great obliquities, I found its diameter, and consequently the thickness of the air, at its perimeter in all those obliquities to be very nearly in the proportions expressed in the f ollomng Table

est part of the first dark ring

n

,

462

Book

II

:

Part

1

463

the interjacent air exhibited rings of colours, as well by transmitting light as by reflecting it. The central spot was now white, and from it the order of the colours were yello^^'ish red; black, violet, blue, white, yellow, red; violet, blue, green, yellow, red, &c. But these colours were very faint and dilute, unless when the hght was trajected very obliquely through the glasses; for b}^ that

means they became pretty vivid. Only the first yellowish red, like the blue in the fourth Observation, was so httle and faint as scarcely to be discerned. Comparing the coloured rings made by reflexion, \\ith these made by transmission of the light, I found that white was opposite to black, red to blue, yellow to violet, and green to a compound of red and ^aolet. That is, those parts of the glass were black when looked through, which when looked upon appeared white, and on the contrary. And so those which in one case exhibited blue, did in the other case exhibit red. And the like of the other colours. The manner you have represented in the third Figure, where AB, CD, are the surfaces of the glasses contiguous at E, and the black lines between them are their distances in arithmetical progression, and the colours written above are seen by reflected hght, and those below by light transmitted. Obs. 10. Wetting the object-glasses a httle at their edges, the water crept in slowly between them, and the circles thereby became less and the colours more faint, insomuch that as the water crept along, one half of them at which it first arrived would appear broken off from the other half, and contracted into a less room. By measuring them I found the proportions of their diameters to the diameters of the hke circles made by air to be about seven to eight, and consequently the intervals of the glasses at hke circles, caused by those two mediums (water and air) are as about three to four. Perhaps it may be a general rule that, if any other medium more or less dense than water be compressed between the glasses, their intervals at the rings caused thereby ^^dll be to their intervals caused by interjacent air, as the sines are which measure the refraction

made out

medium

of that

into air.

When

the water was between the glasses, if I pressed the upper edges to make the rings move nimbly from one place to anglass variously at other, a httle white spot would immediately follow the centre of them, which upon creeping in of the ambient water into that place would presently vanish. Its appearance was such as interjacent air would have caused, and it exhibited the same colours. But it was not air, for where any bubbles of air were in the water they would not vanish. The reflexion must have rather been caused by a subtler medium which could recede through the glasses at the creeping in of the water. Obs. 12. These observations were made in the open air. But farther to examine the effects of coloured light falling on the glasses, I darkened the room, and viewed them by reflexion of the colours of a prism cast on a sheet of white paper, my eye being so placed that I could see the coloured paper by reflexion in the glasses, as in a looking-glass. And by this means the rings became distincter and visible to a far greater number than in the open air. I have sometimes seen more than twenty of them, whereas in the open air I could not

Obs.

11.

its

discern above eight or nine.

Obs. 13. Appointing an assistant to move the prism to and fro about its axis, that all the colours might successively fall on that part of the paper which I saw by reflexion from that part of the glasses, where the circles appeared, so that aU the colours might be successively reflected from the circles to my eye,

Optics

464

held it immovable, I found the circles which the red light made to be manifestly bigger than those which were made by the blue and violet. And it was very pleasant to see them gradually swell or contract D B accordingly as the colour of the light was changed. The paieRed interval of the glasses at any of the rings, when they were Greenish Red «"^ made by the utmost red light, was to their interval at the Red same ring when made by the utmost violet, greater than Gr«"«* Biue Bluish Green Red as 3 to 2, and less than as 13 to 8. By the most of my Green Red Observations it was as 14 to 9. And this proportion seemed Red iBluish Green very nearly the same in all obliquities of my eye, unless Yellow Green Red when two prisms were made use of instead of the objectYel/ou, Blue jGreen Purple glasses. For then at a certain great obliquity of my eye, JBlue Red Yellow IViolet the rings made by the several colours seemed equal, and Green iRed at a greater obliquity those made by the violet would be Blue Violet \white greater than the same rings made by the red, the refracRed iBlue Yellow IViolet tion of the prism in this case causing the most refrangible rays to fall more obliquely on that plate of the air than the least refrangible ones. Thus the experiment succeeded White Black in the coloured light, which was sufficiently strong and Blue Yellouish Red copious to make the rings sensible. And thence it may be Black White gathered that, if the most refrangible and least refrangible Blue Yellouish Red rays had been copious enough to make the rings sensible White Black without the mixture of other rays, the proportion which here was 14 to 9 would have been a little greater, suppose Yello \Violet 141^ or 143/^ to 9. Red [Blue \White Violet Obs. 14. Whilst the prism was turned about its axis Blue Wellow Green \Red with a uniform motion, to make all the several colours fall successively upon the object-glasses, and thereby to Yello \Violet Red IBlue make the rings contract and dilate, the contraction or Purple \Green Blue Wellow dilatation of each ring thus made by the variation of its \Red Green Yello colour w^as swiftest in the red, and slowest in the violet, \Bluish Green Red and in the intermediate colours it had intermediate deGreen Wed grees of celerity. Comparing the quantity of contraction Red Bluish Green and dilatation made by all the degrees of each colour, I Greenish Blue Red found that it was greatest in the red, less in the yellow, Red still less in the blue, and least in the violet. And to make as just an estimation as I could of the proportions of their Fig. 3 contractions or dilatations, I observed that the whole contraction or dilatation of the diameter of any ring made by all the degrees of red was to that of the diameter of the same ring made by all the degrees of violet, as about four to three, or five to four, and that when the light was of the middle colour, between yellow and green, the diameter of the ring was very nearly an arithmetical mean between the greatest diameter of the same ring made by the outmost red, and the least diameter thereof made by the outmost violet contrary to what happens in the colours of the oblong spectrum made by the refraction of a prism, where the red is most contracted, the violet most expanded, and in the midst of all the colours is the confine of green and blue. And hence I seem to collect that the thicknesses of the air between the glasses there, where the ring is successively made by the Hmits of the five whilst

I

•

—

Book

II:

Part

1

principal colours (red, yellow, green, blue, violet) in order (that is, extreme red, by the limit of red and yellow in the middle of the orange,

465

by the by the

and blue, by the Hmit of blue the extreme violet) are to one and by and another very nearly as the sixth lengths of a chord which sound the notes in a sixth major, sol, la, mi,ja, sol, la. But it agrees something better with the Observation to say that the thicknesses of the air between the glasses there, where the rings are successively made by the limits of the seven colours (red, orange, limit of yellow

and

green,

by the

limit of green

\'iolet in the middle of the indigo,

yellow, green, blue, indigo, violet in order) are to one another as the cube roots of the squares of the eight lengths of a chord, which sound the notes in an eighth,

sol, la, fa. sol, la,

mi, fa, sol; that

is,

as the cube roots of the squares of the

numbers, 1, %, %, H, %, %, 'A,, 3^. Obs. 15. These rings were not of various colours like those made in the open air, but appeared all over of that prismatic colour only with which they were illuminated. And by projecting the prismatic colours immediately upon the glasses, I found that the light which fell on the dark spaces which were between the coloured rings was transmitted through the glasses without any variation

Fig. 4

For on a white paper placed behind, it would paint rings of the same colour mth those which were reflected, and of the bigness of their immediate spaces. And from thence the origin of these rings is manifest; namely, that the air between the glasses, according to its various thickness, is disposed in some places to reflect, and in others to transmit, the light of any one colour (as you may see represented in the fourth Figure) and in the same place to reflect that of one colour where it transmits that of another. Obs. 16. The squares of the diameters of these rings made by any prismatic colour were in arithmetical progression, as in the fifth Observation. And the diameter of the sixth circle, when made by the citrine yellow, and viewed almost perpendicularly, was about tot parts of an inch, or a little less, agreeof colour.

able to the sixth Observation.

The precedent Observations were made with a rarer thin medium, terminated by a denser, such as was air or water compressed between two glasses. In those that follow are set down the appearances of a denser medium thinned within a rarer, such as are plates of Muscovy glass, bubbles of water, and some other thin substances terminated on all sides wAth air. Obs. 17. If a bubble be blown with water first made tenacious by dissolving a little soap in it, 'tis a common observation that after a while it ^\"ill appear

Optics

466

tinged with a great variety of colours. To defend these bubbles from being agitated by the external air (whereby their colours are irregularly moved one among another, so that no accurate observation can be made of them), as soon as I had blown any of them I covered it with a clear glass, and by that means its colours emerged in a very regular order, like so many concentric rings encompassing the top of the bubble. And as the bubble grew thinner by the continual subsiding of the water, these rings dilated slowly and overspread the whole bubble, descending in order to the bottom of it, where they vanished successively. In the meanwhile, after all the colours were emerged at the top, there grew in the centre of the rings a small round black spot, like that in the

Observation, which continually dilated itself till it became sometimes more of an inch in breadth before the bubble broke. At first I thought than 3^ or there had been no light reflected from the water in that place, but, observing it more curiously, I saw within it several smaller round spots which appeared much blacker and darker than the rest, whereby I knew that there was some reflexion at the other places which were not so dark as those spots. And by further trial I found that I could see the images of some things (as of a candle or the Sun) very faintly reflected, not only from the great black spot, but also from the little darker spots which were within it. Besides the aforesaid coloured rings there would often appear small spots of colours, ascending and descending up and down the sides of the bubble, by reason of some inequalities in the subsiding of the water. And sometimes small black spots generated at the sides would ascend up to the larger black spot at the top of the bubble, and unite with it. Obs. 18. Because the colours of these bubbles were more extended and lively than those of the air thinned between two glasses, and so more easy to be distinguished, I shall here give you a further description of their order, as they were observed in viewing them by reflexion of the skies when of a white colour, whilst a black substance was placed behind the bubble. And they were these: red, blue; red, blue; red, blue; red, green; red, yellow, green, blue, purple; red, yellow, green, blue, violet; red, yellow, white, blue, black. The three first successions of red and blue were very dilute and dirty, especially the first, where the red seemed in a manner to be white. Among these there was scarce any other colour sensible besides red and blue, only the blues first

%

(and principally the second blue) inclined a little to green. The fourth red was also dilute and dirty, but not so much as the former three after that succeeded little or no yellow, but a copious green, which at first inclined a little to yellow, and then became a pretty brisk and good willow green, and afterwards changed to a bluish colour; but there succeeded neither blue nor violet. The fifth red at first inclined very much to purple, and afterwards became more bright and brisk, but yet not very pure. This was succeeded with a very bright and intense yellow, which was but little in quantity, and soon changed to green; but that green was copious and something more pure, deep and lively, than the former green. After that followed an excellent blue of a bright sky colour, and then a purple, which was less in quantity than the blue, and much inclined to red.

The

a very fair and lively scarlet, and soon after of a brighter colour, being very pure and brisk, and the best of all the reds. Then sixth red

was at

first of

Book

II:

Part

1

467

an intense bright and copious yellow, which was also the best of all the yellows, and this changed first to a greenish yellow, and then to a greenish blue; but the green between the yellow and the blue was very little and dilute, seeming rather a greenish white than a green. The blue which succeeded became very good, and of a very bright sky colour, but yet something inferior to the former blue and the violet was intense and deep with little or no redness in it, and less in quantity than the blue. In the last red appeared a tincture of scarlet next to violet, which soon changed to a brighter colour, inclining to an orange; and the yellow which followed was at first pretty good and lively, but afterwards it grew more dilute until by degrees it ended in perfect whiteness. And this whiteness, if the water was very tenacious and well-tempered, would slowly spread and dilate itself over the greater part of the bubble; continually gro\ving paler at the top, where at length it would crack in many places, and those cracks, as they dilated, would appear of a pretty good, but yet obscure and dark, sky colour; the white between the blue spots diminishing, until it resembled the threads of an irregular network, and soon after vanished, and left all the upper part of the bubble of the said dark blue colour. And this colour, after the aforesaid manner, dilated itself downwards, until sometimes it hath overspread the whole bubble. In the mean while at the top, which was of a darker blue than the bottom, and appeared also full of many round blue spots (something darker than the rest) there would emerge one or more very black spots, and within those, other spots of an intenser blackness, which I mentioned in the former Observation; and these continually dilated themselves until the bubble broke. If the water was not very tenacious, the black spots would break forth in the white, without any sensible intervention of the blue. And sometimes they would break forth within the precedent yellow, or red, or perhaps within the blue of the second order, before the intermediate colours had time to display after a lively orange followed

;

themselves. By this description you may perceive how great an affinity these colours have mth those of air described in the fourth Observation, although set down in a contrary order, by reason that they begin to appear when the bubble is thickest, and are most conveniently reckoned from the lowest and thickest part of the bubble upwards. Obs. 19. Viewing in several oblique positions of my eye the rings of colours emerging on the top of the bubble, I found that they were sensibly dilated by increasing the obliquity, but yet not so much by far as those made by thinned air in the seventh Observation. For there they were dilated so much as, when

viewed most obliquely, to arrive at a part of the plate more than twelve times thicker than that where they appeared when viewed perpendicularly; whereas, in this case, the thickness of the water at which they arrived when viewed most obliquely was, to that thickness which exhibited them by perpendicular rays, something less than as 8 to 5. By the best of my observations it was between 15 and 153^ to 10; an increase about 24 times less than in the other case. Sometimes the bubble would become of an uniform thickness all over, except at the top of it near the black spot, as I knew, because it would exhibit the same appearance of colours in all positions of the eye. And then the colours which were seen at its apparent circumference by the obliquest rays would be different from those that were seen in other places, by rays less oblique to it.

Optics

468

And

divers spectators

might

see the

same part

of

it

of differing colours,

by

viewing it at very differing obliquities. Now, observing how much the colours at the same places of the bubble, or at divers places of equal thickness, were

by the several obliquities of the rays, by the assistance of the 4th, 14th, 16th, and 18th Observations, as they are hereafter explained, I collect the thickness of the water requisite to exhibit any one and the same colour, at varied

several obliquities, to be very nearly in the proportion expressed in this Table. Incidence

Book

II:

Part

would be blue. And, on the contrary, when by it would appear red by transmitted light.

By wetting very thin

469

1

reflected light

it

appeared blue,

Muscovy glass, whose thinness made the like colours appear, the colours became more faint and languid, especially by wetting the plates on that side opposite to the eye but I could not perceive any variation of their species. So, then, the thickness of a plate requisite to produce any colour depends only on the density of the plate, and not on that of the ambient medium. And hence, by the 10th and 16th Observations, may be known the thickness which bubbles of water, or plates of Muscovy glass, or Obs. 21.

plates of

;

other substances, have at any colour produced by them. Obs. 22. A thin transparent body, which is denser than its ambient medium, exhibits more brisk and vivid colours than that which is so much rarer; as I have particularly observed in the air and glass. For blowing glass very thin at a lamp furnace, those plates encompassed with air did exhibit colours much more vivid than those of air made thin between two glasses. Obs. 23. Comparing the quantity of light reflected from the several rings, I found that it was most copious from the first or inmost, and in the exterior rings became gradually less and less. Also the whiteness of the first ring was stronger than that reflected from those parts of the thin medium or plate which were without the rings; as I could manifestly perceive by viewing at a distance the rings made by the two object-glasses; or by comparing two bubbles of water blown at distant times, in the first of which the whiteness appeared, which succeeded all the colours, and, in the other, the whiteness which preceded them all. Obs. 24. When the two object-glasses were laid upon one another, so as to make the rings of the colours appear, though with my naked eye I could not discern above eight or nine of those rings, yet by viewing them through a prism I have seen a far greater multitude, insomuch that I could number more than forty, besides many others, that were so very small and close together that I could not keep my eye steady on them severally so as to number them, but by their extent I have sometimes estimated them to be more than a hundred. And I believe the experiment may be improved to the discovery of far greater numbers. For they seem to be really unlimited, though visible only so far as they can be separated by the refraction of the prism, as I shall hereafter explain. But it was but one side of these rings (namely, that towards which the refraction was made) which by that refraction was rendered distinct, and the other side became more confused than when viewed by the naked eye, insomuch that there I could not discern above one or two, and sometimes none of those rings, of which I could discern eight or nine

my naked eye. And their segments or arcs, which on the other side appeared so numerous, for the most part exceeded not the third part of a circle. If the refraction was very great, or the prism very distant from the object-glasses, the middle part of those arcs became also confused, so as to disappear and constitute an even whiteness, whilst on either side their ends, as also the whole arcs farthest from the centre, became distincter than before, appearing in the form as you see with

them designed

in the fifth Figure.

Optics

470

where they seemed distinctest, were only white and black successively, without any other colours intermixed. But in other places there appeared colours whose order was inverted by the refraction in such manner that, if I first held the prism very near the object-glasses and then gradually removed it farther off towards my eye, the colours of the 2d, 3d, 4th and following rings, shrunk towards the white that emerged between them, until they wholly vanished into it at the middle of the arcs, and afterwards emerged again in a contrary order. But at the ends of the arcs they retained their order unchanged. I have sometimes so laid one object-glass upon the other that, to the naked eye, they have all over seemed uniformly white, without the least appearance of any of the coloured rings; and yet, by viewing them through a prism, great multitudes of those rings have discovered themselves. And in like manner plates of Muscovy glass, and bubbles of glass blown at a lamp-furnace, which were not so thin as to exhibit any colours to the naked eye, have through the prism exhibited a great variety of them ranged irregularly up and down in the form of waves. And so bubbles of water, before they began to exhibit their colours to the naked eye of a bystander, have appeared through a prism, girded about with many parallel and horizontal rings; to produce which effect it was necessary to hold the prism parallel, or very nearly parallel, to the horizon, and to dispose it so that the rays might be refracted upwards.

The

arcs,

Part II Remarks upon

Having given my Observations

the foregoing Observations.

make

use of them to unfold the causes of the colours of natural bodies it is convenient that, by the simplest of them (such as are the 2d, 3d, 4th, 9th, 12th, 18th, 20th, and 24th) I first explain the more compounded. And first, to shew how the colours in the fourth and eighteenth Observations are produced, let there be taken in any of these colours, before I

the lengths YA, YB, YC, YD, YE, YF, YG, YH, in proportion to one another, as the cube roots of the squares of the numbers, 3^, /^e, %, 34, %, /^, /^, 1, whereby the lengths of a musical chord to sound all the notes in an eighth are represented; that is, in the proportion of the numbers 6,300, 6,814, 7,114, 7,631, 8,255, 8,855, 9,243, 10,000. And at the right line

from the point Y,

[Fig. 6]

points A, B, C, D, E, F, G, H, let perpendiculars Aa, B(S, &c. be erected, by whose intervals the extent of the several colours, set underneath against them, is to be represented. Then divide the line Aa in such proportion as the numbers 1, 2, 3, 5, 6, 7, 9, 10, 11, &c. set at the points of division denote. And through those divisions from draw lines II, 2K, 3L, 5M, 6N, 70, &c. Now, if A2 be supposed to represent the thickness of any thin transparent

Y

body, at which the outmost violet is most copiously reflected in the first ring, will represent its thickor series of colours, then by the 13th Observation, in the same series. reflected which the copiously ness, at utmost red is most will denote the thicknesses Also, by the 5th and 16th Observations, A6 and at which those extreme colours are most copiously reflected in the second series, and AlO and HQ the thicknesses at which they are most copiously reflected in the third series, and so on. And the thickness at which any of the intermediate

HK

HN

Book

II

:

Part

471

2

colours are reflected most copiously, will, according to the 14th Observation, from the intermediate parts of the be defined by the distance of the line lines 2K, 6N, lOQ, &c. against

AH

c

which the names 43 42 41

But

39 38 37

of these

35

34 33 31

30 29

27 26 25

23 22 21 19 18 17 15

14 13

of those colours

are written below. farther, to define the latitude

colours in each ring or

Al design the least thickness, and A3 the greatest thickness, series, let

at which the extreme violet in the first series is reflected,

and

let

HI

HL

and

design the like limits for the extreme red, and let the intermediate colours be limited by the intermediate parts of the lines II

and 3L, against which the names of those colours are written, and so on; but yet vnth this caution: that the reflexions be supposed strongest at the intermediate spaces, 2K, 6N, lOQ, &c. and from thence to de-

crease gradually towards these limits, II, 3L, 5M, 70, &c. on either side; where you must not conceive them to be precisely limited, but to decay indefinitely. And whereas I

have assigned the same latitude to every

series,

I

did

it

because, al-

though the colours in the first series seem to be a little broader than the rest

,

by reason of a stronger reflexion

there, yet that inequality

is

sensible as scarcely to be

so indeter-

mined by observation.

Now, according

to this descrip-

Optics

472

AH

through all disObservations. For if you move the ruler gradually from no reflexion to little or which denotes the first space tances, having passed over be made by thinnest substances, it will first arrive at 1 the violet, and then very quickly at the blue and green, which together with that violet compound blue, and then at the yellow and red, by whose further addition that blue is converted into whiteness, which whiteness continues during the transit of the edge of the ruler

from

I

to

3,

after that

by the

successive deficience of its

compound

yellow,

and then to

and

colours, turns first to

red,

and

component

last of all the red

ceaseth at L. Then begin the colours of the second series, which succeed in order during the transit of the edge of the ruler from 5 to 0, and are more lively than before, because more expanded and severed. And, for the same reason, instead of the former white there intercedes between the blue and yellow a mixture of orange, yellow, green, blue and indigo, all which together ought to exhibit a dilute and imperfect green. So the colours of the third series all succeed in order; first, the violet, which a Uttle interferes with the red of the second order, and is thereby incHned to a reddish purple; then the blue and green, which are

and consequently more lively than before, especially the green; then follows the yellow, some of which towards the green is distinct and good, but that part of it towards the succeeding red, as also that red is mixed with the violet and blue of the fourth series, whereby various degrees of red very much inclining to purple are compounded. This violet and blue, which should succeed this red, being mixed with, and hidden in it, there succeeds a green. And this at first is much inclined to blue, but soon becomes a good green, the only unmixed and Uvely colour in this fourth series. For as it less

mixed with other

colours,

verges towards the yellow, it begins to interfere with the colours of the fifth series, by whose mixture the succeeding yellow and red are very much diluted and made dirty, especially the yellow, which being the weaker colour is scarce able to shew itself. After this the several series interfere more and more, and their colours become more and more intermixed, till after three or four more revolutions (in which the red and blue predominate by turns) all sorts of colours are in all places pretty equally blended, and compound an even whiteness. And since, by the 15th Observation, the rays endued with one colour are transmitted, where those of another colour are reflected, the reason of the colours

made by the transmitted light in the 9th and 20th Observations is from

hence evident. If not only the order and species of these colours, but also the precise thickness of the plate, or thin body at which they are exhibited, be desired in parts of an inch, that may be also obtained by assistance of the 6th or 16th Observations. For according to those Observations the thickness of the thinned air, which between two glasses exhibited the most luminous parts of the first six rings were TTiW^> i78^ooo i78^ooo its'ooo i78^oo iTs'.ooo- parts of an inch. Suppose the light reflected most copiously at these thicknesses be the bright citrine yellow, or confine of yellow and orange, and these thicknesses will be FX, F/JL, Fv, F^. Fo, Ft. And this being known, it is easy to determine what thick>

>

^

>

represented by G^, or by any other distance of the ruler from AH. But further, since by the 10th Observation the thickness of air was to the thickness of water, which between the same glasses exhibited the same colour, as 4 to 3, and by the 21st Observation the colours of thin bodies are not varied by varying the ambient medium, the thickness of a bubble of water, exhibiting

ness of air

is

Book any colour

^^•ill

be

II

:

Part

473

2

^ of the thickness of air producing the same colour. And so,

according to the same 10th and 21st Observations, the thickness of a plate of glass, whose refraction of the mean refrangible ray, is measured by the proportion of the sines 31 to 20, may be fy of the thickness of air producing the same colours; and the like of other mediums. I do not affirm that this proportion of 20 to 31 holds in all the rays; for the sines of other sorts of rays have other proportions. But the differences of those proportions are so little that I do not here consider them. On these grounds I have composed the follo^\^ng Table, wherein the thickness of air, water, and glass, at which each colour is most intense and specific, is expressed in parts of an inch divided into ten

hundred thousand equal parts. The thickness of coloured

plates

and

particles

of

Very black

Their colours of the

first

Black Beginning Blue

order

of black

White Yellow Orange

Red Violet

Indigo Blue

Green Yellow Orange

Of the second order

Bright red Scarlet

Purple Indigo Blue

Of the third order

Green Yellow

\

Red Bluish-red

Bluish-green

Of the fourth order

Green

J

Yellowish-green

Red f

Of the

fifth

order

Greenish-blue

[Red /Greenish-blue

Of the sixth order.

iRed Greenish-blue \ Ruddy white f

Of the seventh order

Air 1

Water

Glass

Optics

474

be further desired to delineate the manner how the colours appear when the two object-glasses are laid upon one another. To do which, let there be described a large arc of a circle, and a straight line which may touch that arc, and parallel to that tangent several occult lines, at such distances from it as the numbers set against the several colours in the Table denote. For the arc and its tangent will represent the superficies of the glasses terminating the interjacent air; and the places where the occult lines cut the arc mil show at what distances from the centre, or point of contact, each colour is reflected. There are also other uses of this Table. For by its assistance the thickness of the bubble in the 19th Observation was determined by the colours which it exhibited. And so the bigness of the parts of natural bodies may be conjectured by their colours, as shall be hereafter shewn. Also, if two or more very thin plates be laid one upon another, so as to compose one plate equalling them all in thickness, the resulting colour may be hereby determined. For instance, Mr. Hook observed, as is mentioned in his Micrographia, that a faint yellow plate of Muscovy glass laid upon a blue one, constituted a very deep purple. The yellow of the first order is a faint one, and the thickness of the plate exhibiting it, according to the Table, is 4%, to which add 9, the thickness exhibiting blue of the second order, and the sum will be 13/^, which is the thickness exhibiting the purple of the third order. To explain, in the next place, the circumstances of the 2d and 3d Observations; that is, how the rings of the colours may (by turning the prisms about their common axis the contrary way to that expressed in those observations) be converted into white and black rings, and afterwards into rings of colours again, the colours of each ring lying now in an inverted order: it must be remembered that those rings of colours are dilated by the obliquation of the rays to the air which intercedes the glasses, and that according to the Table in the 7th Observation their dilatation or increase of their diameter is most manifest and speedy when they are obliquest. Now, the rays of yellow being more refracted by the first superficies of the said air than those of red, are thereby made more oblique to the second superficies, at which they ai^e reflected to produce the coloured rings, and consequently the yellow circle in each ring will be more dilated than the red; and the excess of its dilatation will be so much the greater, by how much the greater is the obliquity of the rays, until at last it become of equal extent with the red of the same ring. And for the same reason the green, blue, and violet will be also so much dilated by the still greater obliquity of their rays, as to become all very nearly of equal extent ^\ith the red; that is, equally distant from the centre of the rings. And then all the colours of the same ring must be coincident, and by their mixture exhibit a white ring. And these white rings must have black and dark rings between them, because they do not spread and interfere with one another, as before. And for that reason also they must become distincter, and visible to far greater numbers. But yet the violet being obliquest mil be something more dilated, in proportion to its extent, than the other colours, and so very apt to appear at the exterior verges of the white. Afterwards, by a greater obliquity of the rays, the violet and blue become more sensibly dilated than the red and yellow^, and so, being farther removed from the centre of the rings, the colours must emerge out of the white in an order contrary to that which they had before the violet and blue at the exterior unless

it

;

Book

II:

Part 2

475

limbs of each ring, and the red and yellow at the interior. And the violet, by reason of the greatest obliquity of its rays, being in proportion most of all expanded, ^^'ill soonest appear at the exterior limb of each white ring, and become more conspicuous than the rest. And the several series of colours belonging to the several rings, will, by their unfolding and spreading, begin again to interfere, and thereby render the rings less distinct, and not visible to so great numbers. If instead of the prisms the object-glasses be made use of, the rings which they exhibit become not white and distinct by the obUquity of the eye, by reason that the rays in their passage through that air which intercedes the glasses are very nearly parallel to those Hues in which they were first incident on the glasses, and consequently the rays endued with several colours are not inchned one more than another to that air, as it happens in the prisms. There is yet another circumstance of these experiments to be considered, and that is why the black and white rings which, Avhen viewed at a distance appear distinct, should not only become confused by viewing them near at hand, but also yield a violet colour at both the edges of every white ring. And the reason is that the rays which enter the eye at several parts of the pupil have several obhquities to the glasses, and those which are most oblique, if considered apart, would represent the rings bigger than those which are the least oblique. Whence the breadth of the perimeter of every white ring is expanded outwards by the obliquest rays, and inwards by the least obhque. And this expansion is so much the greater by how much the greater is the difference of the obhquity; that is, by how^ much the pupil is wider, or the eye nearer to the glasses. And the breadth of the violet must be most expanded, because the rays apt to excite a sensation of that colour are most obhque to a second or farther superficies of the thinned air at which they are reflected, and have also the greatest variation of obliquity, which makes that colour soonest emerge out of the edges of the white. And as the breadth of every ring is thus augmented, the dark intervals must be diminished, until the neighbouring rings become continuous, and are blended, the exterior first, and then those nearer the centre so that they can no longer be ;

distinguished apart, but seem to constitute an even and uniform whiteness. Among all the Observations there is none accompanied Avith so odd circumstances as the twenty-fourth. Of those the principal are, that in thin plates, which to the naked eye seem of an even and uniform transparent whiteness,

without any terminations of shadows, the refraction of a prism should make rings of colours appear, whereas it usually makes objects appear coloured only there where they are terminated \\dth shadows, or have parts unequally luminous; and that it should make those rings exceedingly distinct and white, although it usually renders objects confused and coloured. The cause of these things you will understand by considering that all the rings of colours are really in the plate, when viewed with the naked eye, although by reason of the great breadth of their circumferences they so much interfere and are blended together that they seem to constitute an uniform whiteness. But when the rays pass through the prism to the eye, the orbits of the several colours in every ring are refracted, some more than others, according to their degrees of refrangibility; by which means the colours on one side of the ring (that is, in the circumference on one side of its centre) become more unfolded and dilated, and those on the other side more compHcated and contracted. And where by a due refraction they are so much contracted that the several rings become narrower than

Optics

476

to interfere with one another, they must appear distinct, and also white, if the constituent colours be so much contracted as to be wholly coincident. But

where the orbit

on the other

side,

unfolding of

its colours, it

and

so

become

must

of

every ring

interfere

is

made broader by

more with other

the farther

rings than before,

less distinct.

explain this a little further, suppose the concentric circles AV, and BX, [Fig. 7] represent the red and violet of any order, which, together with the intermediate colours, constitute any one of these rings. Now, these being viewed through a prism, the violet circle BX, will, by a greater refraction, be farther translated from its place than the red AV, and so approach nearer to it on that side of the circles, towards which the refractions are made. For instance,

To

the red be translated to av, the violet may be translated to hx, so as to approach nearer to it at x than before and if the red be farther translated to av, the violet may be so much farther translated to bx as to convene with it at x; and if the red be yet farther translated to aT, the violet may be still so much farther translated to /3| as to pass beyond it at ^, and convene \\dth it at e and/. And this being understood not only of the red and violet, but of all the other intermediate colours, and also of every revolution of those colours, you will if

;

easily perceive

how

those of the same revolution or order,

by

their nearness at

Fig. 7

XV and T|, and their coincidence at xv, e and/, ought to constitute pretty distinct arcs of circles, especially at xv, or at e and /; and that they will appear severally at xv, and at xv exhibit whiteness by their coincidence, and again

appear severally at T^, but yet in a contrary order to that which they had

beyond e and /. But on the other side, at ah, ab, or a^, these colours must become much more confused by being dilated and spread so as to interfere mth those of other orders. And the same confusion will happen at T^ between e and /, if the refraction be very great, or the prism very distant from the object-glasses; in which case no parts of the rings will be seen, save only two little arcs at e and /, w^hose distance from one another will be augmented by removing the prism still farther from the object-glasses. And these little arcs must be distinctest and Avhitest at their middle, and at their ends, where they begin to grow confused, they must be coloured. And the colours at one end of every arc must be in a contrary order to those at the other end, by reason that they cross in the intermediate white; namely, their ends, which verge towards T|, \n\\ be red and yellow on that side next the centre, and blue and violet on the other side. But their other ends which verge from T|, mil on the contrary be blue and violet on that side towards the centre, and on the other side red and yellow. Now, as all these things follow from the properties of light by a mathematical way of reasoning, so the truth of them may be manifested by experiments. For in a dark room, by viemng these rings through a prism, by reflexion of the several prismatic colours, which an assistant causes to move to and fro upon a wall or paper from whence they are reflected, whilst the spectator's eye, the prism, and the object-glasses (as in the 13th Observation) are placed steady; before,

and

still

retain

Book

II:

Part

2

477

the position of the circles made successively by the several colours, will be found such, in respect of one another, as I have described in the Figures ahxv, or abxv, or aj3^T. And by the same method the truth of the explications of

other Observations

may

be examined.

By what hath been said, the like phenomena of water and thin plates of glass may be understood. But in small fragments of those plates there is this further observable, that where they lie flat upon a table, and are turned about their centres w^hilst they are viewed through a prism, they ^\'ill in some postures exhibit waves of various colours; and some of them exhibit these waves in one

but the most of them do in all positions exhibit them, and most part appear almost all over the plates. The reason is that the superficies of such plates are not even, but have many cavities and swellings, which, how shallow soever, do a little vary the thickness of the plate. For at the several sides of those cavities, for the reasons newly described, there ought to be produced waves in several postures of the prism. Now, though it be but some very small and narrower parts of the glass by which these waves for the most part are caused, yet they may seem to extend themselves over the whole glass, because from the narrowest of those parts there are colours of several orders; that is, of several rings, confusedly reflected, which by refraction of the or

two positions

make them

only,

for the

prism are unfolded, separated, and, according to their degrees of refraction, dispersed to several places, so as to constitute so many several waves, as there were divers orders of colours promiscuously reflected from that part of the glass. These are the principal phenomena of thin plates or bubbles, whose explications depend on the properties of light, which I have heretofore delivered. And these you see do necessarily follow from them, and agree with them, even to their very least circumstances; and not only so, but do very much tend to their proof. Thus, by the 24th Observation it appears that the rays of several colours, made as well by thin plates or bubbles as by refractions of a prism, have several degrees of refrangibility; whereby those of each order, which at the reflexion from the plate or bubble are intermixed ^\dth those of other orders, are separated from them by refraction, and associated together so as to become visible by themselves like arcs of circles. For if the rays were all alike refrangible, 'tis impossible that the whiteness, which to the naked sense appears uniform, should by refraction have its parts transposed and ranged into those black and white arcs. It appears also that the unequal refractions of difform rays proceed not from any contingent irregularities; such as are veins, an uneven polish, or fortuitous position of the pores of glass; unequal and casual motions in the air or ether, the spreading, breaking, or dividing the same ray into many diverging parts; or the like. For, admitting any such irregularities, it would be impossible for refractions to render those rings so very distinct and well defined as they do in the 24th Observation. It is necessary, therefore, that every ray have its proper and constant degree of refrangibility connate mth it, according to which its refraction is ever justly and regularly performed; and that several rays have several of those degrees.

And what

may

be also undersood of their some at a greater and reflexibility; that is, of their dispositions to be others at a less thickness of thin plates or bubbles; namely, that those dispositions are also connate mth the rays, and immutable; as may appear by the is

said of their refrangibility

reflected,

Optics

478

and 15th Observations, compared with the fourth and eighteenth. Observations, it appears also that whiteness is a dissimilar precedent By the mixture of all colours, and that light is a mixture of rays endued with all those colours. For, considering the multitude of the rings of colours in the 3d, 12th, and 24th Observations, it is manifest that although in the 4th and 18th Observations there appear no more than eight or nine of those rings, yet there are really a far greater number, which so much interfere and mingle with one another as, after those eight or nine revolutions, to dilute one another wholly, and constitute an even and sensibly uniform whiteness. And, consequently, that whiteness must be allowed a mixture of all colours, and the light which conveys it to the eye must be a mixture of rays endued with all those colours. But further; by the 24th Observation it appears that there is a constant relation between colours and ref rangibility the most refrangible rays being violet, the least refrangible red, and those of intermediate colours having proportionably intermediate degrees of refrangibility. And by the 13th, 14th, and 15th Observations, compared \vith the 4th or 18th, there appears to be the same constant relation between colour and refiexibility the violet being in like circumstances reflected at least thicknesses of any thin plate or bubble, the red at greatest thicknesses, and the intermediate colours at intermediate thicknesses. Whence it follows that the colorific dispositions of rays are also connate with them, and immutable; and, by consequence, that all the productions and appearances of colours in the world are derived, not from any physical change caused in light by refraction or reflexion, but only from the various mixtures or 13th, 1-ith,

;

;

separations of rays, by virtue of their different refrangibility or refiexibility. And in this respect the science of colours becomes a speculation as truly mathematical as any other part of Optics. I mean, so far as they depend on the nature of light, and are not produced or altered by the power of imagination, or by striking or pressing the eye.

Part III 0/ the permanent

colours of natural bodies, and the analogy between them colours of thin transparent plates.

and

the

to another part of this design, which is to consider how the phenomena of thin transparent plates stand related to those of all other natural bodies. Of these bodies I have already told you that they appear of divers I

AM now come

they are disposed to reflect most copiously the rays endued with, those colours. But their constitutions, whereby they reflect some rays more copiously than others, remain to be discovered; and these I shall endeavour to manifest in the follo^\^ng Propositions. colours, accordingly as

originally

Proposition

1

Those superficies of transparent bodies reflect the greatest quantity of light, which have the greatest refracting power; that is, which intercede mediums that differ most in their refractive densities. And in the confines of equally refro,cting mediums there is no reflexion. The analogy between reflexion and refraction \\dll appear by considering

Book

11:

Fart 3

479

obliquely out of one medium into another which refracts from the perpendicular, the greater is the difference of their refractive density, the less obliquity of incidence is requisite to cause a total reflexion. For as the sines are which measure the refraction, so is the sine of incidence at which the total reflexion begins, to the radius of the circle; and, consequently, that angle of incidence is least where there is the greatest difference of the sines. Thus, in the passing of light out of water into air, where the refraction that,

when hght passeth

when the In passing degrees minutes. out of 35 glass into angle of incidence is about 48 air, where the refraction is measured by the ratio of the sines 20 to 31, the total reflexion begins when the angle of incidence is 40 degrees 10 minutes; and so in is

measured by the

ratio of the sines 3 to 4, the total reflexion begins

passing out of crystal, or more strongly refracting mediums into air, there is still a less obliquity requisite to cause a total reflexion. Superficies therefore v-hich refract most do soonest reflect all the light which is incident on them, and so must be allowed most strongly reflexive. But the truth of this Proposition will further appear by observing that, in the superficies interceding two transparent mediums (such as are air, water, oil, common glass, crystal, metalline glasses, island glasses, white transparent arsenic, diamonds, &c.), the reflexion is stronger or weaker accordingly as the superficies hath a greater or less refracting power. For in the confine of air and sal-gem 'tis stronger than in the confine of air and water, and still stronger in the confine of air and common glass or crystal, and stronger in the confine of air and a diamond. If any of these, and such like transparent solids, be im-

becomes much weaker than before; and still w^eaker if they be immerged in the more strongly refracting liquors of wellrectified oil of vitriol or spirit of turpentine. If water be distinguished into two parts by any imaginary surface, the reflexion in the confine of those two parts is none at all. In the confine of water and ice 'tis very little; in that of water and oil 'tis something greater: in that of water and sal-gem still greater; and in that of water and glass, or crystal or other denser substances still greater, accordingly as those mediums differ more or less in their refracting powers. Hence, in the confine of common glass and crystal, there ought to be a weak reflexion, and a stronger reflexion in the confine of common and metalline glass; though I have not yet tried this. But in the confine of two glasses of equal density there is not any sensible reflexion, as was shewn in the first Observation. And the same may be understood of the superficies separating two crystals, or two liquors, or any other substances in which no refraction is caused. So, then, the reason why uniform pellucid mediums (such as water, glass, or crystal) have no sensible reflexion but in their external superficies, where they are adjacent to other mediums of a different density, is because all their contiguous parts have one and the same degree of density.

merged

in water, its reflexion

Proposition 2 The

least parts of

almost

all

natural bodies are in some measure transparent:

the opacity of those bodies ariseth

from

the multitude of reflexions

And

caused in their

internal parts.

That this is so has been observed by others, and will easily be granted by them that have been conversant with microscopes. And it may be also tried by applying any substance to a hole through which some light is immitted

Optics

480

For how opaque soever that substance may seem in the open air, it will by that means appear very manifestly transparent if it be of a sufficient thinness. Only white metalline bodies must be excepted, which by reason of their excessive density seem to reflect almost all the light incident on their first superficies; unless by solution in menstruums they be reduced into very small particles, and then they become transparent. into a dark room.

Proposition 3 Between

the parts of

replenished with

wherewith any

opaque and coloured bodies are

many

spaces, either empty, or

mediums of other densities; as water between

the tinging corpuscles

liquor is impregnated, air between the aqueous globules that consti-

most part spaces void of both air and water, but yet perhaps not wholly void of all substance, between the parts of hard bodies. The truth of this is evinced by the two precedent Propositions. For, by the second Proposition, there are many reflexions made by the internal parts of bodies, which, by the first Proposition, would not happen if the parts of those bodies were continued without any such interstices between them; because reflexions are caused only in superficies, which separate mediums of a differing tute clouds or mists;

density (Prop.

But

and for

the

1).

the principal cause of the opacity of bodies will appear by considering that opaque substances become transparent by filling their pores with any substance of equal or almost equal density with their parts. Thus, paper dipped in water or oil, the Oculus mundi further, that this discontinuity of parts

is

stone steeped in water, linen cloth oiled or varnished, and many other substances soaked in such liquors as will intimately pervade their little pores, become by that means more transparent than otherwise; so, on the contrary, the most transparent substances may, by evacuating their pores, or separating their parts, be rendered sufficiently opaque as salts or wet paper, or the Oculus mundi stone by being dried, horn by being scraped, glass by being reduced to ;

powder, or otherwise flawed; turpentine by being stirred about with water till they mix imperfectly, and water by being formed into many small bubbles, either alone in the form of froth, or by shaking it together with oil of turpentine, or olive oil, or with some other convenient liquor with which it will not perfectly incorporate. And to the increase of the opacity of these bodies, it conduces something, that by the 23d Observation the reflexions of very thin transparent substances are considerably stronger than those made by the same substances of a greater thickness.

Proposition 4 The parts of bodies and their interstices must not be less than of some definite bigthem opaque and coloured. For the opaquest bodies, if their parts be subtly divided (as metals, by being dissolved in acid menstruums, &c.), become perfectly transparent. And you may also remember that in the eighth Observation there was no sensible reflexion at the superficies of the object-glasses, where they were very near one another, though they did not absolutely touch. And in the 17th Observation the reflexion of the water-bubble where it became thinnest was almost insensible, so as to cause very black spots to appear on the top of the bubble, by the ness, to render

want

of reflected light.

Book

II

:

Part

3

481

On

these grounds I perceive it is that water, salt, glass, stones, and such like substances are transparent. For, upon divers considerations, they seem to be as full of pores or interstices between their parts as other bodies are, but yet their parts and interstices to be too small to cause reflexions in their common surfaces.

Proposition 5

The transparent parts of bodies, according to their several sizes, reflect rays of one and transmit those of another, on the same grounds that thin plates or bubbles do reflect or transmit those rays. And this I take to be the ground of all colour,

their colours.

a thinned or plated body, which being of an even thickness appears all over of one uniform colour, should be slit into threads, or broken into fragments, of the same thickness with the plate, I see no reason why every thread or fragment should not keep its colour, and by consequence why a heap of those threads or fragments should not constitute a mass or powder of the same colour, which the plate exhibited before it was broken. And the parts of all natural bodies, being like so many fragments of a plate, must on the same grounds exhibit the same colours. Now, that they do so will appear by the aflfinity of their properties. The finely coloured feathers of some birds, and particularly those of peacocks' tails, do, in the very same part of the feather, appear of several colours in several positions of the eye, after the very same manner that thin plates were found to do in the 7th and 19th Observations; and, therefore, their colours arise from the thinness of the transparent parts of the feathers; that is, from the slenderness of the very fine hairs, or capillamenta, which grow out of the sides of the grosser lateral branches or fibres of those feathers. And to the same purpose it is that the webs of some spiders, by being spun very fine, have appeared coloured, as some have observed, and that the coloured fibres of some silks, by varying the position of the eye, do vary their colour. Also the colours of silks, cloths, and other substances, which water or oil can intimately penetrate, become more faint and obscure by being immerged in those liquors, and recover their vigour again by being dried; much after the manner declared of thin bodies in the 10th and 21st Observations. Leaf-gold, some sorts of painted glass, the infusion of lignum nephriticum, and some other substances, reflect one colour, and transmit another, like thin bodies in the 9th and 20th Observations. And some of those coloured powders which painters use may have their colours a little changed by being very elaborately and finely ground. Where I see not what can be justly pretended for those changes, besides the breaking of their parts into less parts by that contrition, after the same manner that the colour of a thin plate is changed by varying its thickness. For which reason also it is that the coloured flowers of plants and vegetables, by being bruised, usually become more transparent than before, or at least in some degree or other change their colours. Nor is it much less to my purpose that, by mixing divers liquors, very odd and remarkable productions and changes of colours may be effected, of which no cause can be more obvious and rational than that the saline corpuscles of one liquor do variously act upon or unite with the tinging corpuscles of another, so as to make them swell, or shrink (whereby not only their bulk but their density also may be changed) or to divide them

For

if

Optics

482

into smaller corpuscles (whereby a coloured liquor may become transparent), or to make many of them associate into one cluster, whereby two transparent liquors may compose a coloured one. For we see how apt those saline men-

struums are to penetrate and dissolve substances to which they are applied, and some of them to precipitate what others dissolve. In like manner, if we consider the various phenomena of the atmosphere, we may observe that when vapours are first raised they hinder not the transparency of the air, being divided into parts too small to cause any reflexion in their superficies. But when in order to compose drops of rain they begin to coalesce and constitute globules of all intermediate sizes, those globules, when they become of convenient size to reflect some colours and transmit others, may constitute clouds of various colours, according to their sizes. And I see not what can be rationally conceived in so transparent a substance as water for the production of these colours, besides the various sizes of its fluid

and globular

parcels.

Proposition 6 The parts of bodies on which

their colours depend, are denser

than the

medium

which pervades their interstices. This Avill appear by considering that the colour of a body depends not only on the rays which are incident perpendicularly on its parts, but on those also which are incident at all other angles. And that, according to the 7th Observation, a very little variation of obliquity will change the reflected colour, where the thin body or small particles is rarer than the ambient medium, insomuch that such a small particle mil at diversely oblique incidences reflect all sorts of colours, in so great a variety that the colour resulting from them all, confusedly reflected from a heap of such particles, must rather be a white or grey than any other colour, or at best it must be but a very imperfect and dirty colour. Whereas if the thin body or small particle be much denser than the ambient medium, the colours, according to the 19th Observation, are so little changed by the variation of obliquity, that the rays which are reflected least obliquely may predominate over the rest, so much as to cause a heap of such particles to appear very intensely of their colour. It conduces also something to the confirmation of this Proposition that, according to the 22d Observation, the colours exhibited by the denser thin body within the rarer are more brisk than those exhibited by the rarer within the denser.

Proposition 7 The bigness of

the

component parts of natural bodies

may

be conjectured by their

colours.

For since the parts of these bodies (by Prop. 5), do most probably exhibit the same colours with a plate of equal thickness, provided they have the same refractive density; and since their parts seem for the most part to have much the same density with water or glass, as by many circumstances is obvious to collect; to determine the sizes of those parts, you need only have recourse to the precedent Tables, in which the thickness of water or glass exhibiting any colour

is

expressed. Thus,

which being the

if it

of equal density

number 16^ shews

it

be desired to

mth

know

the diameter of a corpuscle, green of the third order,

glass shall reflect

to be il%^oo parts of an inch.

Book II: Part 3 here to know of what

483

order the colour of any body 4th and 18th Observations; the is. And for this end we must have recourse to from whence may be collected these particulars. Scarlets, and other reds, oranges, and yellows, if they be pure and intense, are most probably of the second order. Those of the first and third order also may be pretty good; only the yellow of the first order is faint, and the orange and

The

greatest dijSiculty

is

red of the third order have a great mixture of violet and blue. There may be good greens of the fourth order, but the purest are of the third. And of this order the green of all vegetables seems to be, partly by reason of the intenseness of their colours, and partly because when they ^^ither some of them turn to a greenish yellow, and others to a more perfect yellow or orange, or perhaps to red, passing first through all the aforesaid intermediate colours. Which changes seem to be effected by the exhaling of the moisture which may leave the tinging corpuscles more dense, and something augmented by the accretion of the oily and earthy part of that moisture. Now the green, ^\ithout doubt, is of the same order \nth those colours into which it changeth, because the changes are gradual, and those colours, though usually not very full, yet are often too full and lively to be of the fourth order. Blues and purples may be either of the second or third order, but the best are of the third. Thus the colour of violets seems to be of that order, because their syrup by acid Hquors turns red, and by urinous and alkalizate turns

and of alkalies to precipitate or incrassate, if the purple colour of the syrup was of the second order an acid hquor, by attenuating its tinging corpuscles, would change it to a red of the first order, and an alkali by incrassating them would change it to a green of the second order; which red and green, especially the green, seem too imperfect to be the colours produced by these changes. But if the said purple be supposed of the third order, its change to red of the second, and green of the third, may \vithout any inconvenience be allowed. If there be found any body of a deeper and less reddish purple than that of the violets, its colour most probably is of the second order. But yet there being no body commonly known whose colour is constantly more deep than theirs, I have made use of their name to denote the deepest and least reddish purples, green.

For

since

it is

of the nature of acids to dissolve or attenuate,

such as manifestly transcend their colour in purity. The blue of the first order, though very faint and little, may possibly be the colour of some substances; and particularly the azure colour of the skies seems to be of this order. For all vapours when they begin to condense and coalesce into small parcels become first of that bigness, whereby such an azure must be reflected before they can constitute clouds of other colours. And so, this being the first colour which vapours begin to reflect, it ought to be the colour of the finest and most transparent skies, in which vapours are not arrived to that grossness requisite to reflect other colours, as we find it is by experience. Whiteness, if most intense and luminous, is that of the first order, if less strong and luminous, a mixture of the colours of several orders. Of this last kind is the whiteness of froth, paper, linen, and most white substances; of the former I reckon that of white metals to be. For whilst the densest of metals,

metals become transparent if dissolved in menstruums or vitrified, the opacity of white metals ariseth not from their density alone. They, being less dense than gold, would be more transparent gold,

if

foliated, is transparent,

and

all

Optics

484

than it, did not some other cause concur with their density to make them opaque. And this cause I take to be such a bigness of their particles as fits them to reflect the white of the first order. For, if they be of other thicknesses, they may reflect other colours, as is manifest by the colours which appear upon hot steel in tempering it, and sometimes upon the surface of melted metals in the skin or scoria which arises upon them in their cooling. And as the white of the first order is the strongest which can be made by plates of transparent substances, so it ought to be stronger in the denser substances of metals than in the rarer of air, water, and glass. Nor do I see but that metallic substances of such a thickness as may fit them to reflect the white of the first order, may,

by reason

of their great density (according to the tenor of the first of these

Propositions) reflect all the light incident upon them, and so be as opaque and it's possible for any body to be. Gold or copper mixed \\ith less than half their weight of silver, or tin, or regulus of antimony, in fusion, or

splendent as

amalgamed with a very

little

mercury, become white; which shews both that

the particles of white metals have much more superficies, and so are smaller, than those of gold and copper, and also that they are so opaque as not to suffer the particles of gold or copper to shine through them. Now, it is scarce to be doubted but that the colours of gold and copper are of the second and third order; and, therefore, the particles of white metals cannot be much bigger than is requisite to make them reflect the white of the first order. The volatility of

mercury argues that they are not much bigger, nor may they be much less, lest they lose their opacity, and become either transparent as they do when attenuated by vitrification, or by solution in menstruums, or black as they do when ground smaller, by rubbing silver, or tin, or lead upon other substances to draw black lines. The first and only colour which white metals take by grinding their particles smaller is black, and therefore their white ought to be that which borders upon the black spot in the centre of the rings of colours; that is, the white of the first order. But, if you would hence gather the bigness of metallic particles, you must allow for their density. For were mercury transparent, its density is such that the sine of incidence upon it (by my computation) would be to the sine of its refraction as 71 to 20, or 7 to 2. And, therefore, the thickness of its particles, that they may exhibit the same colours with those of bubbles of water, ought to be less than the thickness of the skin of those bubbles in the proportion of 2 to 7. Whence it's possible that the particles of mercury may be as little as the particles of some transparent and volatile fluids, and yet reflect the white of the first order. Lastly, for the production of black, the corpuscles must be less than any of those which exhibit colours. For at all greater sizes there is too much light reflected to constitute this colour. But if they be supposed a little less than is requisite to reflect the white and very faint blue of the first order, they will, according to the 4th, 8th, 17th and 18th Observations, reflect so very little light as to appear intensely black, and yet may perhaps variously refract it to and fro within themselves so long, until it happen to be stifled and lost, by w^hich means they will appear black in all positions of the eye without any transparency. And from hence may be understood why fire, and the more subtle dissolver putrefaction, by dividing the particles of substances, turn them to black; why small quantities of black substances impart their colour very freely and intensely to other substances to which they are applied, the

Book

II

:

Part

485

3

minute particles of these, by reason of their very great number, easily overspreading the gross particles of others; why glass ground very elaborately vnth. sand on a copper plate, till it be well polished, makes the sand, together with what is worn off from the glass and copper, become very black: why black substances do soonest of all others become hot in the Sun's light and burn (which effect may proceed partly from the multitude of refractions in a little room, and partly from the easy commotion of so very small corpuscles;) and why blacks are usually a little inclined to a bluish colour. For that they are so may be seen by illuminating white paper by light reflected from black substances. For the paper ^^ill usually appear of a bluish- white and the reason is that black borders in the obscure blue of the order described in the 18th Observation, and, therefore, reflects more rays of that colour than of any other. In these descriptions I have been the more particular, because it is not impossible but that miscroscopes may at length be improved to the discovery of the particles of bodies on which their colours depend, if they are not already in some measure arrived to that degree of perfection. For if those instruments are or can be so far improved as mth sufficient distinctness to represent objects five or six hundred times bigger than at a foot distance they appear to our naked eyes, I should hope that we might be able to discover some of the greatest of those corpuscles. And by one that would magnify three or four thousand times perhaps they might all be discovered, but those which produce blackness. In the meanwhile I see nothing material in this discourse that may rationally be doubted of, excepting this position That transparent corpuscles of the same thickness and density with, a plate do exhibit the same colour. And this I would have understood not ^\ithout some latitude, as well because those corpuscles may be of irregular figures, and many rays must be obliquely incident on them, and so have a shorter way through them than the length of their diameters, as because the straitness of the medium put in on all sides Avithin such corpuscles may a Httle alter its motions or other qualities on which the reflexion depends. But yet I cannot much suspect the last, because I have observed of some small plates of Muscovy glass, which were of an even thickness, that through a microscope they have appeared of the same colour at their edges and corners where the included medium was terminated, which they appeared of in other places. However, it "will add much to our satisfaction if those corpuscles can be discovered A\ith microscopes; which, if we shall at length attain to, I fear it mil be the utmost improvement of this sense. For it seems impossible to see the more secret and noble works of Nature within the corpuscles by reason of their transparency. Proposition 8 ;

:

The cause of reflexion is not the impinging of light on the solid or impervious parts of bodies, as is commonly believed. This ^\dll appear by the f ollo\\dng considerations First, that in the passage :

of fight out of glass into air there is

a reflexion as strong as in

air into glass, or rather a little stronger,

its

passage out of

and by many degrees stronger than

in

passage out of glass into water. And it seems not probable that air should have more strongly reflecting parts than water or glass. But if that should possibly be supposed, yet it A\dll avail nothing; for the reflexion is as strong or stronger when the air is drawn away from the glass (suppose by the air-pump invented by Otto Gueriet, and improved and made useful by Mr. Boyle) as its

486

Optics

when it is adjacent to it. Secondly, if light in its passage out of glass into air be incident more obliquely than at an angle of 40 or 41 degrees it is Avholly reflected, if less obliquely it is in great measure transmitted. Now, it is not to be imagined that light at one degree of obliquity should meet ^^•ith pores enough in the air to transmit the greater part of it, and at another degree of obliquity should meet A\'ith nothing but parts to reflect it wholly, especially considering that in its passage out of air into glass, how oblique soever be its incidence, it finds pores enough in the glass to transmit a great part of it. If any man suppose that it is not reflected by the air, but by the outmost superficial parts of the glass, there is still the same difficulty; besides that, such a supposition is unintelligible, and will also appear to be false by applying water behind some part of the glass instead of air. For so in a convenient obliquity of the rays, (suppose of 45 or 46 degrees) at which they are all reflected where the air is adjacent to the glass, they shall be in great measure transmitted where the water is adjacent to it; which argues that their reflexion or transmission depends on the constitution of the air and water behind the glass, and not on the upon the parts of the glass. Thirdly, if the colours made by a prism placed at the entrance of a beam of light into a darkened room be successively cast on a second prism placed at a greater distance from the former, in such manner that they are all alike incident upon it, the second prism may be so inclined to the incident rays that those which are of a blue colour shall be all reflected by it, and yet those of a red colour pretty copiously transmitted. Now, if the reflexion be caused by the parts of air or glass, I would ask why at the same obliquity of incidence the blue should wholly impinge on those parts, so as to be all reflected, and yet the red find pores enough to be in a great measure transmitted. Fourthly, where two glasses touch one another, there is no sensible reflexion, as was declared in the first Observation; and yet I see no reason why the rays should not impinge on the parts of glass, as much when contiguous to other glass as when contiguous to air. Fifthly, when the top of a water-bubble (in the 17th Observation) by the continual subsiding and exhaling of the water grew very thin, there was such a little and almost insensible quantity of light reflected from it that it appeared intensely black; whereas round about that black spot, where the water was thicker, the reflexion was so strong as to make the water seem very white. Nor is it only at the least thickness of thin plates or bubbles that there is no manifest reflexion, but at many other thicknesses continually greater and greater. For in the 15th Observation the rays of the same colour were by turns transmitted at one thickness, and reflected at another thickness, for an indeterminate number of successions. And yet in the superficies of the thinned body, where it is of any one thickness, there are as many parts for the rays to impinge on as where it is of any other thickness. Sixthly, if reflexion were caused by the parts of reflecting bodies, it would be impossible for thin plates or bubbles, at one and the same place, to reflect the rays of one colour, and transmit those of another, as they do according to the 13th and 15th Observations. For it is not to be imagined that at one place the rays which, for instance, exhibit a blue colour, should have the fortune to dash upon the parts, and those which exhibit a red to hit upon the pores of the body; and then at another place, where the body is either a little thicker or a little thinner, that on the contrary the blue should hit upon its pores, and the red upon its parts. Lastly, were the rays of light reflected by impinging on striking of the rays

Book

II

:

Part

3

487

the solid parts of bodies, their reflexions from polished bodies could not be so regular as they are. For in polishing glass A\ith sand, putty, or tripoli, it is not to be imagined that those substances can, by grating and fretting the glass, bring all its least particles to an accurate polish; so that all their surfaces shall

be truly plane or truly spherical, and look all the same way, so as together to compose one even surface. The smaller the particles of those substances are, the smaller ^nll be the scratches by which they continually fret and wear away the glass until it be polished; but be they never so small they can wear away the glass no other\nse than by grating and scratching it, and breaking the protuberances; and, therefore, polish it no otherwise than by bringing its roughness to a very fine grain, so that the scratches and frettings of the surface become too small to be visible. And, therefore, if light were reflected by impinging upon the solid parts of the glass, it would be scattered as much by the most polished glass as by the roughest. So, then, it remains a problem how glass polished by fretting substances can reflect light so regularly as it does. And this problem is scarce otherwise to be solved than by saying that the reflexion of a ray is effected, not by a single point of the reflecting body, but by some power of the body which is evenly diffused all over its surface, and by which it acts upon the ray without immediate contact. For that the parts of bodies do act upon light at a distance shall be shewn hereafter. Now^, if light be reflected, not by impinging on the solid parts of bodies but by some other principle, it's probable that as many of its rays as impinge on the solid parts of bodies are not reflected but stifled and lost in the bodies. For otherwise we must allow two sorts of reflexions. Should all the rays be reflected which impinge on the internal parts of clear water or crystal, those substances would rather have a cloudy colour than a clear transparency. To make bodies look black, it's necessary that many rays be stopped, retained, and lost in them; and it seems not probable that any rays can be stopped and stifled in them which do not impinge on their parts. And hence w^e may understand that bodies are much more rare and porous than is commonly believed. Water is nineteen times lighter, and by consequence nineteen times rarer, than gold; and gold is so rare as very readily and without the least opposition to transmit the magnetic effluvia, and easily to admit quick-silver into its pores, and to let water pass through it. For a concave sphere of gold filled with water, and soldered up, has, upon pressing the sphere with great force, let the water squeeze through it, and stand all over its outside in multitudes of small drops, like dew, without bursting or cracking the body of the gold, as I have been informed by an eye-witness. From all which we may conclude that gold has more pores than solid parts, and by consequence that water has above forty times more pores than parts. And he that shall find out an hypothesis by w^hich water may be so rare, and yet not be capable of compression by force, may doubtless by the same hypothesis make gold, and water, and all other bodies, as much rarer as he pleases so that light may find a ready passage through transparent substances. The magnet acts upon iron through all dense bodies not magnetic nor red hot, without any diminution of its virtue; as for instance, through gold, silver, lead, glass, water. The gravitating power of the Sun is transmitted through the vast bodies of the planets without any diminution, so as to act upon all their parts to their very centres with the same force and according to the same laws, ;

488

Optics

the part upon which it acts were not surrounded with the body of the planet. The rays of Hght, whether they be very small bodies projected, or only motion or force propagated, are moved in right lines; and whenever a ray of light is by any obstacle turned out of its rectilinear way, it ^^'ill never return as

if

way, unless perhaps by very great accident. And yet through pellucid solid bodies in right lines to very great distances. How bodies can have a sufficient quantity of pores for producing these effects is very difficult to conceive, but perhaps not altogether impossible. For the colours of bodies arise from the magnitudes of the particles which reflect them, as was explained above. Now, if we conceive these particles of bodies to be so disposed amongst themselves that the intervals or empty spaces between them may be equal in magnitude to them all; and that these particles may be composed of other particles much smaller, which have as much empty space between them as equals all the magnitudes of these smaller particles; and that in like manner these smaller particles are again composed of others much smaller, all which together are equal to all the pores or empty spaces between them; and so on perpetually till you come to solid particles, such as have no pores or empty spaces within them; and if in any gross body there be, for instance, three such degrees of particles, the least of which are solid, this body ^^^ll have seven times more pores than solid parts. But if there be four such degrees of particles, the least of w^hich are solid, the body mil have fifteen times more pores than solid parts. If there be five degrees, the body will have one and thirty times more pores than sohd parts. If six degrees, the body will have sixty and three times more pores than solid parts. And so on perpetually. And there are other ways of conceiving how bodies may be exceeding porous. But what is really their inward frame is not yet known to us. into the

same

rectilinear

light is transmitted

Proposition 9 Bodies

reflect

and

refract light by one

and

the

same power, variously exercised in

various circumstances.

This appears by several considerations. First, because when light goes out of glass into air, as obliquely as it can possibly do, if its incidence be made still more oblique, it becomes totally reflected. For the power of the glass after it has refracted the light as obliquely as is possible, if the incidence be still made more oblique, becomes too strong to let any of its rays go through, and by consequence causes total reflexions. Secondly, because light is alternately reflected and transmitted by thin plates of glass for many successions, accordingly as the thickness of the plate increases in an arithmetical progression. For here the thickness of the glass determines whether that power by which glass acts upon light shall cause it to be reflected, or suffer it to be transmitted. And, thirdly, because those surfaces of transparent bodies which have the greatest refracting power reflect the greatest quantity of light, as was shewn in the first Proposition.

Proposition 10 // light he swifter in bodies than in vacuo, in the proportion of the sines which

measure

the refraction of the bodies the forces of the bodies to reflect

and

refract light

are very nearly proportional to the densities of the same bodies; excepting that unctuous and sulphureous bodies refract more than others of this same density. Let represent the refracting plane surface of any body, and IC a ray

AB

Book incident very obliquely infinitely little,

and

let

II:

Part

489

3

in C, so that the angle ACI may be be the refracted ray. From a given point B perpendicular to the refracting surface erect BR meeting with the refracting ray CR in R, and if CR represent the motion of the refracted ray, and this motion be distinguished into two motions CB and BR, whereof CB is parallel to the refracting plane, and BR perpendicular to it CB shall represent the motion of the incident ray, and BR the motion generated by the

upon the body

CR

:

Pig 8

have of late explained. Now, if any body or thing, in moving through any space of a given breadth terminated on both sides by two parallel planes, be urged forward in all parts of that space by forces tending directly forwards towards the last plane, and, before its incidence on the first plane, had no motion towards it, or but an infinitely little one; and if the forces in all parts of that space, between the planes, be at equal distances from the planes equal to one another, but at several distances be bigger or less in any given proportion, the motion generated by the forces in the whole passage of the body or thing through that space shall be in a subduplicate proportion of the forces, as mathematicians ^vill easily understand. And, therefore, if the space of activity of the refracting superficies of the body be considered as such a space, the motion of the ray generated by the refracting force of the body, during its passage through that space (that is, the motion BR) must be in subduplicate proportion of that refracting force. I say, therefore, that the square of the line BR, and by consequence the refracting force of the body, is very nearly as the density of the same body. For this will appear by the foUomng Table, wherein the proportion of the sines which measure the refractions of several bodies, the square of BR, supposing CB an refraction, as opticians

unit, the densities of the bodies estimated

by

their specific gravities,

and

their

power in respect of their densities are set down in several columns. The refraction of the air in this Table is determined by that of the atmosphere observed by astronomers. For, if light pass through many refracting substances or mediums gradually denser and denser, and terminated with parallel surfaces, the sum of all the refractions mil be equal to the single refraction which it would have suffered in passing immediately out of the first medium into the last. And this holds true, though the number of the refracting substances be refractive

increased to infinity, and the distances from one another as much decreased, so that the light may be refracted in every point of its passage, and by continual

And, therefore, the whole refraction of light, from the highest and rarest part thereof down to the lowest and densest part, must be equal to the refraction which it would suffer in passing at like obliquity out of a vacuum immediately into air

refractions bent into a curve-line.

in passing through the atmosphere

of equal density \\ith that in the lowest part of the atmosphere.

Now, although a pseudo-topaz, a selenitis, rock crystal,

island crystal, vulgar

sand melted together) and glass of antimony, which are terrestrial stony alkalizate concretes, and air which probably arises from such substances by fermentation, be substances very differing from one another in density, yet by this Table, they have their refractive powers almost in the same proportion to one another as their densities are, excepting that the refraction of that strange substance, island crystal, is a little bigger than the rest. And

glass (that

is,

Optics

490

3,500 times rarer than the pseudo-topaz, and 4,400 antimony, and 2,000 times rarer than the selenitis, glass times rarer than glass of vulgar, or crystal of the rock, has notwithstanding its rarity the same refractive particularly

air,

which

is

The square

The proportion The

of the sines of incidence and refraction of

refracting bodies

of

BR,

which

the

refracting force of the

body

yelloiv light

to

The

The density and specific gravity of the

is

body

refractive

power of the body in respect of its

proportionate

density

A

pseudo-topazius, being a natural pellucid brittle, hairy stone, of a yellow colour. Air Glass of antimony ,

,

.

A

.

23 to 14 3201 to 3200 17 to

10 to

7

529 to

396

0.7845

1.

1.375 0.866 0.996 0.913 0.932 0.874 1.04 3.4

Glass vulgar Crystal of the rock Island crystal Sal

gemmse

vitriol

Oil of vitriol

Rain water

Gum

arable

Spirit of

wine well

rectified.

Camphor Olive oil Linseed oil Spirit of turpentine

Amber

A

diamond

power

20

5 to

3

17 to

Nitre

Danzig

31 to 25 to

35 22 32 303

Borax

4.27 0.0012 5.28 2.252

1.213 1 4025 1.445 1.778 1.388 1 1267 1.1511 1.345 1.295 1.041

61 to

selenitis

Alum

9 41

1.699 0.000625 2.568

31 100 3 22

16 11

to

24

to

15

to

21

to

200

.

.

to

21

1.179

to

73 2 15 27

0.8765

to to

40 to 25 to 14 to 100 to

17

9 41

1.25 1.1511 1 1948 1 1626 1.42 4.949 .

.

2.,58

2..65 2..72 2. 143 1

.714

1

.714

1.9 1.715 1.7

3979 5208 4864 5386 5436 5450 6536 6477 6570 6716 7079 7551 6124 7845 8574 10121 12551 12607 12819 13222 13654 14556

which those very dense substances have in respect of theirs, excepting so far as those differ from one another. Again, the refraction of camphor, olive oil, linseed oil, spirit of turpentine and amber, which are fat sulphureous unctuous bodies, and a diamond, which probably is an unctuous substance coagulated, have their refractive powers in proportion to one another as their densities without any considerable variation. But the refractive powers of these unctuous substances are two or three times greater in respect of their densities than the refractive powers of the former in respect of its density

substances in respect of theirs. Water has a refractive power in a middle degree between those two sorts of substances, and probably is of a middle nature. For out of it grow all vegetable and animal substances, which consist as well of sulphureous fat and inflammable parts, as of earthy lean and alkalizate ones. Salts and vitriols have refractive powers in a middle degree between those of earthy substances and water, and accordingly are composed of those two sorts of substances. For by distillation and rectification of their spirits a great part of

Book

II

:

Part

491

3

them goes into water, and a great part remains behind in the form

of

a dry fixed

earth capable of vitrification. Spirit of A\ine has a refractive power in a middle degree between those of water and oily substances, and accordingly seems to be composed of both, united by fermentation; the water, by means of some saline spirits with which 'tis impregnated, dissolving the oil, and volatizing it by the action. For spirit of wane is inflammable by means of its oily parts, and being distilled often from salt of tartar, grows by every distillation more and more aqueous and phlegmatic. And chemists observe that vegetables (as lavender, rue, marjoram, &c.)

fermentation yield oils without any burning spirits, but after fermentation yield ardent spirits without oils; which shews that their oil is by fermentation converted into spirit. They find also that if oils be poured in a small quantity upon fermentating vegetables, they distil over after ferdistilled per se, before

mentation in the form of spirits. So then, by the foregoing Table, all bodies seem to have their refractive powers proportional to their densities (or very nearly) excepting so far as they partake more or less of sulphureous oily particles, and thereby have their refractive power made greater or less. Whence it seems rational to attribute the refractive power of all bodies chiefly, if not wholly, to the sulphureous parts with which they abound. For it's probable that all bodies abound more or less with sulphurs. And as light congregated by a burning-glass acts most upon sulphureous bodies, to turn them into fire and flame, so, since all action is mutual, sulphurs ought to act most upon light. For that the action between light and bodies is mutual may appear from this consideration that the densest bodies which refract and reflect light most strongly grow hottest in the summer Sun, ;

:

by the I

action of the refracted or reflected light. have hitherto explained the power of bodies to reflect and refract,

and

shewed that thin transparent plates, fibres, and particles do, according to their several thicknesses and densities, reflect several sorts of rays, and thereby appear of several colours; and by consequence that nothing more is requisite for producing all the colours of natural bodies than the several sizes and densities of their transparent particles. But whence it is that these plates, fibres, and particles do, according to their several thicknesses and densities, reflect several sorts of rays, I have not yet explained. To give some insight into this matter, and make way for understanding the next part of this book, I shall conclude this part mth a few more Propositions. Those which preceded respect the nature of bodies, these the nature of light; for both must be understood before the reason of their actions upon one another can be known. And because the last Proposition depended upon the velocity of light, I will begin with a Proposition of that kind.

Proposition

11

Light is propagated from luminous bodies in time, and spends about seven or eight minutes of an hour in passing from the Sun to the Earth. This was observed first by Romer, and then by others, by means of the eclipses of the satellites of Jupiter. For these eclipses, when the Earth is between the Sun and Jupiter, happen about seven or eight minutes sooner than they ought to do by the Tables, and when the Earth is beyond the Sun they happen about seven or eight minutes later than they ought to do the reason being that the light of ;

Optics

492

the satellites has farther to go in the latter case than in the former by the diameter of the Earth's orbit. Some inequalities of time may arise from the eccentricities of the orbs of the satellites; but those cannot answer in all the satellites, and at all times to the position and distance of the Earth from the Sun. The

mean motions

of Jupiter's satellites is also swifter in his descent from his aphelium to his perihelium, than in his ascent in the other half of his orb. But this inequality has no respect to the position of the Earth, and in the three interior satellites is insensible, as I find by computation from the theory of

their gravity.

Proposition 12 passage through any refracting surface is put into a certain transient constitution or state, which in the progress of the ray returns at equal intervals, and disposes the ray at every return to be easily transmitted through the next refracting surface, and between the returns to be easily reflected by it. This is manifest by the 5th, 9th, 12th, and 15th Observations. For by those Observations it appears that one and the same sort of rays at equal angles of incidence on any thin transparent plate, is alternately reflected and transmitted for many successions accordingly as the thickness of the plate increases in arithmetical progression of the numbers, 0, 1,2, 3, 4, 5, 6, 7, 8, &c. so that if the first reflexion (that which makes the first or innermost of the rings of colours there described) be made at the thickness 1, the rays shall be transmitted at the thicknesses 0, 2, 4, 6, 8, 10, 12, &c. and thereby make the central spot and rings of light, which appear by transmission, and be reflected at the thickness 1, 3, 5, 7, 9, 11, &c. and thereby make the rings which appear by

Every ray of light in

its

And this alternate

and transmission, as I gather by the 24th Observation, continues for above a hundred vicissitudes, and by the Observations in the next part of this book, for many thousands, being propagated from one surface of a glass plate to the other, though the thickness of the plate be a quarter of an inch or above; so that this alternation seems to be propagated from every refracting surface to all distances without end or limitation. This alternate reflexion and refraction depends on both the surfaces of every reflexion.

reflexion

depends on their distance. By the 21st Observation, if Muscovy glass be wetted, the colours caused by the alternate reflexion and refraction grow faint; and, therefore, it depends on thin plate, because

it

either surface of a thin plate of

them both. It is therefore

performed at the second surface for

were performed at the first, before the rays arrive at the second, it would not depend on the second. It is also influenced by some action or disposition propagated from the first to the second, because otherwise at the second it would not depend on the first. ;

if it

And this action or disposition, in its propagation, intermits and returns by equal intervals, because in all its progress it inclines the ray at one distance from the first surface to be reflected by the second, at another to be transmitted by it, and that by equal intervals for innumerable vicissitudes. And because the ray is disposed to reflexion at the distances 1, 3, 5, 7, 9, &c. and to transmission at the distances 0, 2, 4, 6, 8, 10, &c. (for its transmission through the first surface is at the distance 0, and it is transmitted through both together, if their distance be infinitely little or

transmitted at the distances

much

2, 4, 6, 8, 10,

less

&c.

is

than 1) the disposition to be to be accounted a return of the

Book same disposition which the ray

first

II:

Part

493

3

had at the distance

0; that

is,

at its trans-

mission through the first refracting surface. All which is the thing I would prove. What kind of action or disposition this is; whether it consists in a circulating or a vibrating motion of the ray, or of the medium, or something else, I do not here enquire. Those that are averse from assenting to any new discoveries, but such as they can explain by an hypothesis, may for the present suppose that as stones by falling upon water put the water into an undulating motion, and all bodies by percussion excite vibrations in the air, so the rays of hght, by impinging on any refracting or reflecting surface, excite vibrations in the refracting or reflecting medium or substance, and by exciting them agitate the solid parts of the refracting or reflecting body, and by agitating them cause the body to grow warm or hot; that the vibrations thus excited are propagated in the refracting or reflecting medium or substance, much after the manner that vibrations are propagated in the air for causing sound, and move faster than the rays so as to overtake them; and that when any ray is in that part of the vibration which conspires with its motion,

when

it is

it

easily breaks through a refracting surface,

but

motion

it is

in the contrary part of the vibration

which impedes

its

by consequence, that every ray is successively disposed to or easily transmitted, by every vibration which overtakes it.

easily reflected; and,

be easily reflected, But whether this hypothesis be true or false I do not here consider. I content myself with the bare discovery that the rays of light are, by some cause or other, alternately disposed to be reflected or refracted for many vicissitudes.

DEFINITION disposition of any ray to he reflected I will call its fits of easy reflexion, and those of its disposition to he transmitted its fits of easy transmisson, and the space it passes hetween every return and the next return,

The returns of

the

the interval of its fits.

Proposition 13 The reason why the surfaces of all thick transparent bodies reflect part of the light incident on them, and refract the rest, is that some rays at their incidence are in fits of easy refiexion, and others in fits of easy transmission. This may be gathered from the 24th Observation, where the light reflected by thin plates of air and glass, which to the naked eye appeared evenly white all over the plate, did through a prism appear waved with many successions of light and darkness made by alternate fits of easy reflexion and easy transmission, the prism severing and distinguishing the waves of which the white reflected light was composed, as was explained above. And hence light is in fits of easy reflexion and easy transmission before its incidence on transparent bodies. And probably it is put into such fits at its first emission from luminous bodies, and continues in them during all its progress. For these fits are of a lasting nature, as will appear by the next part of this book. In this Proposition I suppose the transparent bodies to be thick; because if the thickness of the body be much less than the interval of the fits of easy reflexion and transmission of the rays, the body loseth its reflecting power. For if the rays, which at their entering into the body are put into fits of easy transmission, arrive at the farthest surface of the body before they be out of those

Optics

494

they must be transmitted. And this is the reason why bubbles of water lose their reflecting power when they grow very thin; and why all opaque bodies, when reduced into very small parts, become transparent.

fits,

Proposition 14 Those surfaces of transparent bodies, which if the ray he in a fit of refraction do refract it most strongly, if the ray be in a fit of reflexion do reflect it most easily. For we shewed above, in Prop. 8, that the cause of reflexion is not the impinging of light on the solid impervious parts of bodies, but some other power by which those solid parts act on light at a distance. We shewed also, in Prop. 9, that bodies reflect and refract light by one and the same power, variously exercised in various circumstances; and in Prop. 1 that the most strongly refracting surfaces reflect the most light. All which compared together evince and rarify both this and the last Proposition.

Proposition 15

same sort of rays, emerging in any angle out of any refracting one and the same medium, the interval of the following fits of easy

In any one and surface into

the

reflexion and transmission are either accurately or very nearly as the rectangle of the secant of the angle of refraction, and of the secant of another angle, whose sine is the first of 106 arithmetical mean proportionals, between the sines of incidence and refraction, counted

This

is

from

the sine of refraction.

manifest by the 7th and 19th Observations.

Proposition 16 emerging in equal angles out of any refracting surface into the same medium, the intervals of the following fits of easy reflexion and easy transmission are either accurately, or very nearly, as the cube roots of the squares of the lengths of a chord, which found the notes in an eight, sol, la, fa, sol, la, mi, fa, sol, with all their intermediate degrees answering to the colours of those rays, according to the analogy described in the seventh experiment of the second part of the first book. This is manifest by the 13th and 14th Observations.

In

several sorts of rays

Proposition 17 If rays of any sort pass perpendicularly into several mediums, the intervals of the fits of easy refiexion and transmission in any one medium are to those intervals in any other, as the sine of incidence to the sine of refraction, when the rays pass out of the first of those two mediums into the second. This is manifest by the 10th Observation.

Proposition 18 If the rays which paint the colour in the confine of yellow and orange pass perpendicularly out of any medium into air, the intervals of their fits of easy reflexion are the

^^ P^^^ ^/

8 9,0

^^

inch.

And

of the

same

length are the intervals of their fits

of easy transmission.

manifest by the 6th Observation. it is easy to collect the intervals of the fits of easy reflexion and easy transmission of any sort of rays refracted in any angle into any medium; and thence to know whether the rays shall be reflected or trans-

This

is

From

these Propositions

Book

II

:

Part

3

495

mitted at their subsequent incidence upon any other peUucid medium. Which thing, being useful for understanding the next part of this book, was here to be set doAvn. And for the same reason I add the two following Propositions.

Proposition 19 // any sort of rays falling on the polite surface of any pellucid medium he reflected back, the fits of easy reflexion, which they have at the point of reflexion, shall still continue to return; and the returns shall be at distances from the point of reflexion in the arithmetical progression of the numbers 2, 4, 6, 8, 10, 12, &c. and between these flts the rays shall be in flts of easy transmission.

and easy transmission are of a returning nature, there is no reason why these fits, which continued till the ray arrived at the reflecting medium and there inclined the ray to reflexion, should there cease. And if the ray at the point of reflexion was in a fit of easy reflexion, the progression of the distances of these fits from that point must begin from 0, and so be of the numbers 0, 2, 4, 6, 8, &c. And, therefore, the progression of the distances of the intermediate fits of easy transmission, reckoned from the same point, must be in the progression of the odd numbers 1, 3, 5, 7, 9, &c. contrary to what happens when the fits are propagated from points of refraction. For since the

fits

of easy reflexion

Proposition 20 The

intervals of the flts of easy reflexion

and easy transmission, propagated from

points of reflexion into any medium, are equal to the intervals of the like flts which the same rays would have if refracted into the same medium in angles of refraction

equal

to their

angles of reflexion. Hght is reflected by the second surface of thin plates,

For when it goes out afterwards freely at the first surface to make the rings of colours which appear by reflexion; and, by the freedom of its egress, makes the colours of these rings more vivid and strong than those which appear on the other side of the plates by the transmitted light. The reflected rays are, therefore, in fits of easy transmission at their egress; which would not always happen if the intervals of the fits Avithin the plate after reflexion were not equal, both in length and number, to their intervals before it. And this confirms also the proportions set down in the former Proposition. For if the rays both in going in and out at the first surface be in fits of easy transmission, and the intervals and numbers of those fits between the first and second surface, before and after reflexion, be equal, the distances of the fits of easy transmission from either surface must be in the same progression after reflexion as before; that is, from the first surface which transmitted them in the progression of the even numbers 0, 2, 4, 6, 8, &c. and from the second which reflected them, in that of the odd numbers 1, 3, 5, 7, &c. But these two Propositions will become much more evident by the Observations in the following part of this book.

Optics

496

Part

IV

Observations concerning the reflexions and colours of thick transparent polished plates.

There which

no glass or speculum how well soever polished but, besides the

is

it

refracts or reflects regularly, scatters every

way

light

irregularly a faint

which the polished surface, when illuminated in a dark room by a beam of the Sun's light, may be easily seen in all positions of the eye. There are certain phenomena of this scattered light, which when I first observed them, seemed very strange and surprising to me. My Observations were light,

by means

of

as follows.

Obs. 1. The Sun shining into my darkened chamber through a hole one-third of an inch wide, I let the intromitted beam of light fall perpendicularly upon a glass speculum ground concave on one side and convex on the other, to a sphere of five feet and eleven inches radius, and quick-silvered over on the convex side. And holding a white opaque chart or a quire of paper at the centre of the spheres to which the speculum was ground (that is, at the distance of about five feet and eleven inches from the speculum, in such manner that the beam of light might pass through a little hole made in the middle of the chart to the speculum, and thence be reflected back to the same hole) I observed upon the chart four or five concentric irises or rings of colours, like rainbows, encompassing the hole much after the manner that those, which in the fourth and following Observations of the first part of this book appeared between the object-glasses, encompassed the black spot, but yet larger and fainter than those. These rings as they grew larger and larger became diluter and fainter, so that the fifth was scarce visible. Yet sometimes, when the Sun shone very clear, there appeared faint lineaments of a sixth and seventh. If the distance of the chart from the speculum was much greater or much less than that of six feet, the rings became dilute and vanished. And if the distance of the speculum from the mndow was much greater than that of six feet, the reflected beam of light would be so broad, at the distance of six feet from the speculum where the rings appeared, as to obscure one or two of the innermost rings. And, therefore, I usually placed the speculum at about six feet from the window, so that its focus might there fall in with the centre of its concavity at the rings upon the chart. And this posture is always to be understood in the following Observations where no other is expressed.

Obs. 2. The colours of these rainbows succeeded one another from the centre outwards, in the same form and order with those which were made in the ninth Observation of the first part of this book, by light not reflected but transmitted through the two object-glasses. For, first, there was in their common centre a white round spot of faint light, something broader than the reflected beam of light, which beam sometimes fell upon the middle of the spot, and sometimes by a little inclination of the speculum receded from the middle, and left the spot white to the centre. This white spot was immediately encompassed with a dark grey or russet, and that dark grey \\ath the colours of the first iris; which colours on the inside

Book

II

:

Part

497

4

next the dark grey were a little violet and indigo, and next to that a blue, which on the outside grew pale, and then succeeded a little greenish yellow, and after that a brighter yellow, and then on the outward edge of the iris a red which on the outside inclined to purple. This iris was immediately encompassed with a second, whose colours were, in order from the inside outwards: purple, blue, green, yellow, light red, a red

mixed

mth

purple.

Then immediately followed the

colours of the third

iris,

which were in order

outwards a green inclining to purple, a good green, and a red more bright than that of the former iris. The fourth and fifth iris seemed of a bluish-green Avithin, and red without, but so faintly that it was difficult to discern the colours. Obs. 3. Measuring the diameters of these rings upon the chart as accurately as I could, I found them also in the same proportion to one another mth the rings made by light transmitted through the two object-glasses. For the diameters of the four first of the bright rings measured between the brightest parts of their orbits, at the distance of six feet from the speculum, were l^/^e, 2^, 2^M2. 33^ inches, whose squares are in arithmetical progression of the numbers 1, 2, 3, 4. If the white circular spot in the middle be reckoned amongst the rings, and its central light, where it seems to be most luminous, be put equipollent to an infinitely little ring, the squares of the diameters of the rings will be in the progression 0, 1,2, 3, 4, &c. I measured also the diameters of the dark circles between these luminous ones, and found their squares in the progression

numbers

&c. the diameters of the first four, at the distance of six feet from the speculum, being l/^ie, 2/^6, 2%, 3/^o inches. If the distance of the chart from the speculum was increased or diminished, the diameters of the circles were increased or diminished proportionally. Obs. 4. By the analogy between these rings and those described in the Observations of the first part of this book, I suspected that there were many more of them which spread into one another, and by interfering mixed their colours, and diluted one another so that they could not be seen apart. I viewed them, therefore, through a prism, as I did those in the 24th Observation of the first part of this book. And when the prism was so placed as by refracting the light of their mixed colours to separate them, and distinguish the rings from one another, as it did those in that Observation, I could then see them distincter than before, and easily number eight or nine of them, and sometimes twelve or thirteen. And had not their light been so very faint, I question not but that I might have seen many more. Obs. 5. Placing a prism at the mndow to refract the intromitted beam of light, and cast the oblong spectrum of colours on the speculum, I covered the speculum \vith a black paper which had in the middle of it a hole to let any one of the colours pass through to the speculum, whilst the rest were intercepted by the paper. And now I found rings of that colour only which fell upon the speculum. If the speculum was illuminated with red, the rings were totally red with dark intervals; if with blue they were totally blue; and so of the other colours. And when they were illuminated with any one colour, the squares of their diameters, measured between their most luminous parts, were in the arithmetical progression of the numbers, 0, 1, 2, 3, 4 and the squares of the diameters of their dark intervals in the progression of the intermediate numbers of the

3^, 1}/^, 2}^, 33^,

Optics

498

/^» I3^j 23^2' 33^. But if the colour was varied, they varied their magnitude. In the red they were largest, in the indigo and violet least, and in the intermediate colours (yellow, green, and blue) they were of several intermediate bignesses answering to the colour; that is, greater in yellow than in green, and greater in green than in blue. And hence I knew that when the speculum was illuminated with white light, the red and yellow on the outside of the rings were produced

by the least refrangible rays, and the blue and violet by the most refrangible, and that the colours of each ring spread into the colours of the neighbouring rings on either side, after the manner explained in the first and second parts of this book, and by mixing diluted one another so that they could not be distinguished, unless near the centre where they were least mixed. For in this Observation I could see the rings more distinctly, and to a greater number than before, being able in the yellow light to faint

shadow

of a tenth.

To

number eight

or nine of them, besides a

how much

the colours of the several

satisfy myself

measured the diameters of the second and third rings, and found them, when made by the confine of the red and orange, to be to the same diameters when made by the confine of blue and indigo, as 9 to 8, or thereabouts. For it was hard to determine this proportion accurately. Also the circles made successively by the red, yellow, and green differed more from one another than those made successively by the green, blue and indigo. For the circle made by the violet was too dark to be seen. To carry on the rings spread into one another,

I

us therefore suppose that the differences of the diameters of the outmost red, the confine of red and orange, the confine of orange and yellow, the confine of yellow and green, the confine of green and blue, the confine of blue and indigo, the confine of indigo and violet, and outmost violet, are in proportion as the differences of the lengths of a monochord which sound the tones in an eight sol, la, fa, sol, la, mi, fa, sol; that is, as the numbers /^, Ms- M2, M2> /^i, V21, lAs,- And if the diameter of the circle made by the confine of red and orange be 9A, and that of the circle made by the confine of blue and indigo be 8A as above, their difference (9A — 8A) will be to the difference of the diameters of the circles made by the outmost red, and by the confine of red and orange, as M8+M2+M2+/^7 to ]/^ (that is, as /^y to /^, or 8 to 3) and to the difference of the circles made by the outmost violet, and by the confine of blue and indigo, as M84-M2+M2+/^7 to V27+M8 (that is, as ^7 to %4, or as 16 to 5). And, therefore, these differences mil be and /^eA. Add the first to 9A and subduct the last from 8A, and you will have the diameters of the circles made by the least and most refrangible rays ^^A and ^^^/^A. These diameters are, therefore, to one another as 75 to 613^ or 50 to 41, and their squares as 2,500 to 1,681; that is, as 3 to 2 very nearly. Which proportion differs not much from the proportion of the diameters of the circles made by the outmost red and outmost violet, in the 13th Observation of the first part of this book. Obs. 6. Placing my eye where these rings appeared plainest, I saw the speculum tinged all over with waves of colours (red, yellow, green, blue) Hke those which in the Observations of the first part of this book appeared between the object-glasses, and upon bubbles of water, but much larger. And after the manner of those, they were of various magnitudes in various positions of the eye, swelling and shrinking as I moved my eye this way and that way. They were formed like arcs of concentric circles, as those were; and when my eye was

computation, circles

let

made by

:

^A

Book

II

:

Part 4

499

over against the centre of the concavity of the speculum, (that is, 5 feet and 10 inches distant from the speculum) their common centre was in a right line with that centre of concavity, and with the hole in the window. But in other postures of my eye their centre had other positions. They appeared by the hght of the clouds propagated to the speculum through the hole in the ^^^ndow; and when the Sun shone through that hole upon the speculum, his light upon it was of the colour of the ring whereon it fell, but by its splendor obscured the rings made by the light of the clouds, unless when the speculum was removed to a great distance from the A\-indow so that his light upon it might be broad and faint. By varying the position of my eye, and moving it nearer to or farther from the direct beam of the Sun's Hght, the colour of the Sun's reflected Hght constantly varied upon the speculum, as it did upon my eye, the same colour always appearing to a bystander upon my eye which to me appeared upon the speculum. And thence I knew that the rings of colours upon the chart were made by these reflected colours, propagated thither from the speculum in several angles, and that their production depended not upon the termination of light

and shadow.

Obs.

7.

By the analogy of all these phenomena ^\ith those

of the

Hke rings

of

part of this book, it seemed to me that these colours were produced by this thick plate of glass, much after the manner that those were produced by very thin plates. For, upon trial, I found that if the quick-silver were rubbed off from the backside of the speculum, the glass alone would cause the same rings of colours, but much more faint than before; and, therefore, the phenomenon depends not upon the quick-silver, unless so far as the quick-silver, by increasing the reflexion of the backside of the glass, in-

colours described in the

first

creases the Hght of the rings of colours. I found also that a speculum of metal without glass made some years since for optical uses, and very well wrought, produced none of those rings; and thence I understood that these rings arise

not from one specular surface alone, but depend upon the two surfaces of the plate of glass whereof the speculum was made, and upon the thickness of the glass between them. For as in the 7th and 19th Observations of the first part of this book a thin plate of air, water, or glass of an even thickness appeared of one colour when the rays were perpendicular to it, of another when they were a little oblique, of another when more obHque, of another when still more obHque, and so on; so here, in the sixth Observation, the Hght which emerged out of the glass in several obliquities made the glass appear of several colours, and being propagated in those obliquities to the chart, there painted rings of those colours. And as the reason why a thin plate appeared of several colours in several obliquities of the rays was that the rays of one and the same sort are reflected by the thin plate at one obHquity and transmitted at another, and those of other sorts transmitted where these are reflected, and reflected where these are transmitted; so the reason why the thick plate of glass whereof the speculum was made did appear of various colours in various obHquities, and in those obHquities propagated those colours to the chart, was that the rays of one and the same sort did at one obHquity emerge out of the glass, at another did not emerge, but were reflected back towards the quick-silver by the hither

and accordingly as the obHquity became greater and greater, emerged and were reflected alternately for many successions; and that in one and the same obliquity the rays of one sort were reflected, and those of surface of the glass,

Optics

500

another transmitted. This is manifest by the fifth Observation of this part of this book. For in that Observation, when the speculum was illuminated by any one of the prismatic colours, that light made many rings of the same colour upon the chart with dark intervals, and, therefore, at its emergence out of the speculum was alternately transmitted and not transmitted from the speculum to the chart for many successions, according to the various obliquities of its emergence. And when the colour cast on the speculum by the prism was varied, the rings became of the colour cast on it, and varied their bigness with their colour; and, therefore, the light was now alternately transmitted and not transmitted from the speculum to the chart at other obliquities than before. It seemed to me, therefore, that these rings were of one and the same original with those of thin plates, but yet Avith this difference that those of thin plates are made by the alternate reflexions and transmissions of the rays at the second surface of the plate, after one passage through it; but here the rays go twice through the plate before they are alternately reflected and transmitted. First, they go through it from the first surface to the quick-silver, and then return through it from the quick-silver to the first surface, and there are either transmitted to the chart or reflected back to the quick-silver, accordingly as they are in their fits of easy reflexion or transmission when they arrive at that surface. For the intervals of the fits of the rays which fall perpendicularly on the speculum, and are reflected back in the same perpendicular lines, by reason of the equality of these angles and lines, are of the same length and number wthin the glass after reflexion as before, by the 19th Proposition of the third part of this book. And, therefore, since all the rays that enter through the first surface are in their fits of easy transmission at their entrance, and as many of these as are reflected by the second are in their fits of easy reflexion there, all these must be again in their fits of easy transmission at their return to the first, and by consequence there go out of the glass to the chart, and form upon it the white spot of light in the centre of the rings. For the reason holds good in all sorts of rays, and, therefore, all sorts must go out promiscuously to that spot, and by their mixture cause it to be white. But the intervals of the fits of those rays which are reflected more obliquely than they enter, must be greater after reflexion than before, by the 15th and 20th Propositions. And thence it may happen that the rays at their return to the first surface may in certain obliquities be in fits of easy reflexion, and return back to the quick-silver, and in other intermediate obliquities be again in fits of easy transmission, and so go out to the chart, and paint on it the rings of colours about the white spot. And because the intervals of the fits at equal obliquities are greater and fewer in the less refrangible rays, and less and more numerous in the more refrangible, therefore the less refrangible at equal obliquities shall make fewer rings than the more refrangible, and the rings made by those shall be larger than the like number of rings made by these; that is, the red rings shall be larger than the yellow, the yellow than the green, the green than the blue, and the blue than the violet, as they were really found to be in the fifth Observation. And, therefore, the first ring of all colours encompassing the white spot of light shall be red without any violet within, and yellow, and green, and blue in the middle, as it was found in the second Observation; and these colours in the second ring, and those that follow, shall be more expanded, till they spread into one another, and blend one an:

other

by

interfering.

Book

II

:

Part 4

501

and this put me upon observing the thickness of the glass, and considering whether the dimensions and proportions of the rings may be truly derived from it by computation. These seem to be the reasons

of these rings in general

;

measured, therefore, the thickness of this concavo-convex plate of glass, and found it everywhere one-quarter of an inch precisely. Now, by the sixth Observation of the first part of this book, a thin plate of air transmits the brightest Hght of the first ring (that is, the bright yellow) when its thickness is the ggooo th part of an inch; and by the tenth Observation of the same part, a thin plate of glass transmits the same light of the same ring when its thickness is less in proportion of the sine of refraction to the sine of incidence (that is, when its thickness is the 1.513^000 ^^ or 13 7,^5 4 th part of an inch, supposing the sines are as 11 to 17). And if this thickness be doubled, it transmits the same bright light of the second ring; if tripled, it transmits that of the third, and so on; the bright yellow light in all these cases being in its fits of transmission. And, therefore, if its thickness be multiplied 34,386 times, so as to become one-quarter of an inch, it transmits the same bright light of the 34,386th ring. Suppose this be the bright yellow light transmitted perpendicularly from the reflecting convex side of the glass through the concave side to the white spot in the centre of the rings of colours on the chart; and by a rule in the 7th and 19th Observations in the first part of this book, and by the 15th and 20th Propositions of the third part of this book, if the rays be made oblique to the glass, the thickness of the glass requisite to transmit the same bright light of the same ring in any obliquity is to this thickness of one-quarter of an inch, as the secant of a certain angle to the radius, the sine of which angle is the first of a hundred Obs.

8. I

means between the sines of incidence and refraction, counted from the sine of incidence when the refraction is made out of any plated body into any medium encompassing it; that is, in this case, out of glass into air. Now, if the thickness of the glass be increased by degrees, so as to bear to its first thickness (viz., that of a quarter of an inch), the proportions which 34,386 (the number of fits of the perpendicular rays in going through the glass towards the white spot in the centre of the rings) hath to 34,385, 34,384, 34,383, and 34,382 (the numbers of the fits of the obhque rays in going through the glass towards the first, second, third, and fourth rings of colours) and if the first thickness be divided into 100,000,000 equal parts, the increased thicknesses will be 100,002,908, 100,005,816, 100,008,725, and 100,011,633, and the angles of which these thicknesses are secants will be 26' 13", 37' 5", 45' 6", and 52' and

six arithmetical

;

26", the radius being 100,000,000; and the sines of these angles are 762, 1,079, 1,321, and 1,525, and the proportional sines of refraction 1,172, 1,659, 2,031, and 2,345, the radius being 100,000. For since the sines of incidence out of glass into air are to the sines of refraction as 11 to 17, and to the above-mentioned

secants as 11 to the first of 106 arithmetical means between 11 and 17 (that is, as 11 to 11 rg-e), those secants will be to the sines of refraction as 11 to 17, and by this analogy will give these sines. So, then, if the obUquities of the rays to the concave surface of the glass be such that the sines of their refraction in passing out of the glass through that surface into the air be 1,172, 1,659,

t^

2,031, 2,345, the bright light of the 34,386th ring shall

emerge at the thicknesses

which are to one-quarter of an inch as 34,386 to 34,385, 34,384, 34,383, 34,382, respectively. And, therefore, if the thickness in all these cases be one-quarter of an inch (as it is in the glass of which the speculum was made) of the glass,

502

Optics

the bright light of the 34,385th ring shall emerge where the sine of refraction is 1,172. and that of the 34,384th, 34,383rd, and 34,382nd ring where the sine is 1,659, 2,031, and 2,345, respectively. And in these angles of refraction the light of these rings shall be propagated from the speculum to the chart, and there paint rings about the white central round spot of light which, we said, was the light of the 34,386th ring. And the semidiameters of these rings shall subtend

the angles of refraction made at the concave surface of the speculum, and by consequence their diameters shall be to the distance of the chart from the speculum as those sines of refraction doubled are to the radius (that is, as 1,172, 1,659, 2,031, and 2,345, doubled, are to 100,000). And, therefore, if the distance of the chart from the concave surface of the speculum be six feet (as it was in the third of these Observations), the diameters of the rings of this bright yellow light upon the chart shall be 1.688, 2.389, 2.925, 3.375 inches; for these diameters are to six feet as the above-mentioned sines doubled are to the radius. Now, these diameters of the bright yellow rings, thus found by computation, are the very same \\ith those found in the third of these Observations by inches; and, therefore, measuring them, viz., with l^Vie, 2^, 2^/^2, and the theory of deriving these rings from the thickness of the plate of glass of which the speculum was made, and from the obliquity of the emerging rays, agrees with the Observation. In this computation I have equalled the diameters of the bright rings made by light of all colours, to the diameters of the rings made by the bright yellow. For this yellow makes the brightest part of the rings of all colours. If you desire the diameters of the rings made by the light of any other unmixed colour, you may find them readily by putting them to the diameters of the bright yellow ones in a subduplicate proportion of the intervals of the fits of the rays of those colours when equally inclined to the refracting or reflecting surface which caused those fits; that is, by putting the diameters of the rings made by the rays in the extremities and limits of the seven colours (red, orange, yellow, green, blue, indigo, violet) proportional to the cube roots of the numbers, 1, %, 34, M, %, 34, Me, M, which express the lengths of a monochord sounding the notes in an eighth. For by this means the diameters of the rings of these colours Anil be found pretty nearly in the same proportion to one another which they ought to have by the fifth of these Observations. And thus I satisfied myself that these rings were of the same kind and original with those of thin plates, and by consequence that the fits or alternate dispositions of the rays to be reflected and transmitted are propagated to great distances from every reflecting and refracting surface. But yet to put the matter out of doubt, I added the following Observation. Obs. 9. If these rings thus depend on the thickness of the plate of glass, their

3%

diameters at equal distances from several speculums made of such concavoconvex plates of glass as are ground on the same sphere ought to be reciprocally in a subduplicate proportion of the thicknesses of the plates of glass. And if this proportion be found true by experience it A\-ill amount to a demonstration that these rings (like those formed in thin plates) do depend on the thickness of the glass. I procured, therefore, another concavo-convex plate of glass ground on both sides to the same sphere Anth the former plate. Its thickness was parts of an inch; and the diameters of the three first bright rings measured between the brightest parts of their orbits at the distance of six feet from the glass were 3, 4/^, 53^ inches. Now, the thickness of the other glass being one-

^

Book

II

:

Part

503

4

quarter of an inch was to the thickness of this glass as 3^ to 'y^2', that is, as 31 to 10, or 310,000,000 to 100,000,000; and the roots of these numbers are 17,607 and 10,000, and in the proportion of the first of these roots to the second are the diameters of the bright rings made in this Observation by the thinner glass, 3, 4/^, 53^, to the diameters of the same rings made in the third of these Observations by the thicker glass l^Me, 2^, 2^/^2; that is, the diameters of the rings are reciprocally in a subduplicate proportion of the thicknesses of the plates of glass.

which are alike concave on one side, and alike and alike quick-silvered on the convex sides, and

So, then, in plates of glass

convex on the other side, differ in nothing but their thickness, the diameters

of the rings are reciprocally

in a subduplicate proportion of the thicknesses of the plates.

And

shews They depend this

depend on both the surfaces of the glass. on the convex surface because they are more luminous when that surface is quick-silvered over than when it is mthout quick-silver. They depend also upon the concave surface, because without that surface a speculum makes them not. They depend on both surfaces, and on the distances between them, because their bigness is varied by varying only that distance. And this dependence is of the same kind mth that which the colours of thin plates have on the distance of the surfaces of those plates, because the bigness of the rings, and their proportion to one another, and the variation of their bigness arising from the variation of the thickness of the glass, and the orders of their colours, is such as ought to result from the Propositions in the end of the third part of this book, derived from the phenomena of the colours of thin plates set down in the first sufficiently that the rings

part.

There are yet other phenomena of these rings of colours, but such as follow from the same Propositions, and therefore confirm both the truth of those Propositions, and the analogy between these rings and the rings of colours made by very thin plates. I shall subjoin some of them. Obs. 10. When the beam of the Sun's light was reflected back from the speculum, not directly to the hole in the mndow but to a place a little distant from it, the common centre of that spot, and of all the rings of colours, fell in the middle way between the beam of the incident light and the beam of the reflected light, and by consequence in the centre of the spherical concavity of the speculum, whenever the chart on which the rings of colours fell was placed at that centre. And as the beam of reflected light by inclining the speculum receded more and more from the beam of incident light and from the common centre of the coloured rings between them, those rings grew bigger and bigger, and so also did the w^hite round spot, and new rings of colours emerged succes-

common

and the white spot became a white ring encompassing them; and the incident and reflected beams of light always fell upon the opposite parts of this white ring, illuminating its perimeter like two mock Suns in the opposite parts of an iris. So, then, the diameter of this ring, measured from the middle of its light on one side to the middle of its light on the other side, was always equal to the distance between the middle of the incident beam of light, and the middle of the reflected beam measured at the chart on which the rings appeared. And the rays which formed this ring were reflected by the speculum in angles equal to their angles of incidence, and by sively out of their

centre,

consequence to their angles of refraction at their entrance into the

glass,

but

Optics

504

yet their angles of reflexion were not in the same planes with their angles of incidence.

Obs.

1 1

.

The

colours of the

new

rings were in a contrary order to those of the

former, and arose after this manner: the white round spot of light in the middle of the rings continued white to the centre till the distance of the incident and reflected beams at the chart was about J^ parts of an inch, and then it began to

And when

that distance was about l/^e of an inch, the white spot was become a ring encompassing a dark round spot which in the middle inclined to violet and indigo. And the luminous rings encompassing it were grown equal to those dark ones which in the four first Observations encompassed them; that is to say, the white spot was grown a white ring equal to the first of those dark rings, and the first of those luminous rings was now gro\vn equal to the second of those dark ones, and the second of those luminous ones to the third of those dark ones, and so on. For the diameters of the lu-

grow dark

minous

in the middle.

rings were

When the

now

distance between

2%,

3/^o,