Encyclopaedia Britannica [13, 4 ed.]

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ENCYCLOPEDIA

BRITANNICA.

Cntpclopaefcta

Jkrttanmca;

OR, A

DICTIONARY

OF

ARTS, SCIENCES, AND MISCELLANEOUS

LITERATURE;

ENLARGED AND IMPROVED.

THE FOURTH EDITION.

UitlustrateO toitl) nearly sir ininoreti cEngrabtngs.

VOL. xm.

INDOCTI DISCANT ; AMENT MEMINISSE PERITI.

EDINBURGH: Printed by Andrew Bell, the Proprietor^ FOR ARCHIBALD CONSTABLE AND COMPANY, EDINBURGH J AND FOR VERNOR, HOOD, AND SHARPE, LONDON. 1810.

ENCYCLOPEDIA BRITANNICA.

Material, TV /jf ATERIAL, denotes MaterialV | matter. In which

tfts. ^

1

oppofed to immaterial.

fometliing compofed of fenfe the word ftands See MATTER and META-

PHYSICS.

MATERIALISTS, a fe£l in the ancient church, compofed of perfons who, being prepoffeffed with that maxim in the ancient philofophy, Ex nihilo nihil Jit, “ Out of nothing nothing can arife,” had recourfe to an internal matter, on which they fuppofed God wrought in the creation 5 inflead of, admitting God alone as the l#le caufe of the exiftence of all things. Tertullian vi-

goroufly oppofes the doarine of the materialirts in his treatife againft Hermogenes, who was one of their number. PTaterialifls is alfo a name given to thofe who maintain that the foul of man is material} or that the principle of perception and thought is not a fubdance diflinft from the body, but the refult of corporeal organization : See METAPHYSICS. There are others, called by this name, who have maintained that there is nothing but matter in the univerfe ; and that the Deity himfelf is material. See SPINOZA.

Material-

ifts.

MATHEMATICS. Definition

of mathematics.

A /TATH EMATICS is divided into two kinds, pure -t-VJL anc) mixed. In pure mathematics magnitude is confidered in the abftradl ; and as they are founded on the fimpleft notions of quantity, the conclufions to which they lead have the fame evidence and certainty as the elementary principles from which thefe conclufions are deduced. This branch of mathematics comprehends, 1. Arithmetic, which treats of the properties of numbers. 2. Geometry, which treats of extenlion as endowed with three dimennons, length, breadth, and thicknefs, without confidering the phyfical qualities infeparable from bodies in their natural date. 3. Algebra, fometimes called univerfal arithmetic, wTich compares together all kinds of quantities, whatever be their value. 4. The dircB and inverfe method of Fluxfo/; r,(called on the continent, the differential and integral calculi'), which confider magnitudes as divided into two kinds, condant and variable, the variable magnitudes being generated by motion ; and which determines the value of quantities from the velocities of the motions with which they are generated. Mixed Mathematics is the application of pure mathematics to certain edablidied phyfical principles, and comprehends all the phyfico-mathematical fciences, namely, 1. Mechanics ; 1. Hydrodynamics; 3. Optics; 4. AJlronomy; 5. Acoujlics; 6. EleBricity, and 7. Maghetifm. The hidory of thefe various branches of fcience having been given at full length, w7e lhall at prefent dired! the attention of the reader to the origin and progrefs of pure mathematics. 2. In attempting to difeover the origin of arithmetic VOL. XIII. Part I.

and geometry, it would be a fruitlefs talk to conduA the reader into thofe ages of fable which preceded the records of authentic hidory. Our means of information upon this fubjedl are extremely limited and imperfeft 5 ^ud it w7ould but ill accord with the dignity of a fcience whofe principles and conclufions are alike irrefidible, to found its hidory upon conjedlure and fable. But notwithdanding this obfeurity in which XIie the early hidory of the fciences is enveloped, one thing fciences crlappears certain that arithmetic and geometry, and fome ginated in of the phyfical fciences, had made confiderable progrefs in Egypt, when the myderies and the theology of that favoured kingdom were tranfplanted into Greece. It is highly probable that much natural and moral knowledge was taught in the Eleufinian and Dionyfian myderies, which the Greeks borrowed from the Egyptians and that feveral of. the Grecian philofophers were induced by this circumdance to travel into Egypt, in fearch of thofe higher degrees of knowledge, which an acquaintance with the Egyptian myderies had taught them to anticipate. We accordingly find Thales andA c 6lPythagoras fuccedively under the tuition of the Egyp-A! C. slo. tian prieds, and returning into Greece loaded with the intellectual treafures of Egypt. By the eftabliOiment of the Ionian fchool at Miletus, Thales indru&ed his E'fcoverics countrymen in the knowledge which he had received ofT1,al®s‘ and gave birth to that fpirit of invedigation and difcovery with which his followers were °infpired. He taught them the method of afeertaining the height of the pyramids of Memphis by the length of their fhadowsj and there is reafon to believe that he was the A firii

MATHEMATICS.

2

who employed the circumfererice of a circle for the menfuration of angles. That he was the author of greater difcoveries, which have been either loft or afcribed to others, there can be little doubt ; but thefe are the only fa£ts in the hiftory of Thales which time has fpared. Difcoveries 3fcience of arithmetic was one of the chief ofPythago-branches of the Pythagorean difcipline. Pythagoras' las attached feveral myfterious virtues to certain combinations of numbers. He fwore by four, which he regarded as the chief of numbers. In the number three he fuppoled many wonderful properties to exift; and he regarded a knowledge'of arithmetic as the chief good. But of all Pythagoras’s difcoveries in arithmetic, none have reached our times but his multiplication table. In geometry, however, the philofopher of Samos feems to have been more fuctefsful. The difcovery of the celebrated propofition which forms the 47th of the firft book of Euclid’s Elements, that in every right-angled triangle the fquare of the fide fubtending the right angle is equal to the fum of the fquares of the other two lides, has immortalized his name ; and whether we confider the inherent beauty of the propofition, or the extent of its application in the mathematical fciences, we cannot fail to clafs it among the moft important truths in geometry. From this propofition its author concluded that the diagonal of a fquare is incommenturate to its fide ; and thus gave occafion to the difcovery of feveral general properties of other incommenfurate lines and numbers. 4. In the time which elapfed between the birth of Pythagoras and the deftruftion of the Alexandrian fchool, the mathematical fciences were cultivated with great ardour and fuccefs. Many of the elementary propofitions of geometry were difcovered during this period •, but hiftory does not enable us to refer each difcovery to its proper author. The method of letting fall a perpendicular upon a right line from a given point (Euclid, B. 1. prop, xi.) ;—of dividing an angle into two equal parts, (Euclid, B. I. prop. ix.)j and of making an angle equal to a given angle, (Euclid B. I. prop, xxiii.} were inofOerioY8

vente

^ k)7 Oenopidus of Chios. About the fame time fome of whofe writings have been preferved nodorus. by 1 heon in his commentary on Ptolemy, demonftrated, in oppofition to the opinion then entertained, that ifoperimetrical figures have equal areas. Coeval with this difcovery was the theory of regular bodies, for which we are indebted to the Pythagorean fchool. The cele5* About this time the celebrated problem of the dubrated pro- plication of the cube began to occupy the attention of Idem of the the Greek geometers. In this problem it was required dus and Ze- ^eno(Iorus>

onheTubeto conftru& a cube whofe folid content fliould be propofed " double that of a given cube; and the afliftance of no and invefti-other inftrument but the rule and compaffes wms to be gated. employed. Ehe origin of this problem has been aferibed by tradition to a demand of one of the Grecian deities. The Athenians having offered fome affront to Apollo, were affliifted wdth a dreadful peftilence ; and upon conlulting the oracle at Delos, received for anlwTer, Double the altar of Apollo. The altar alluded to happened to be cubical; and the problem, fuppofed to be of divine origin, was inveftigated with ardour by the Greek geometers,^ though it afterwards baffled all their acutenefs. I he lolution of this difficulty was attemptA C 450. ed by Hippocrates of Chios. He difcovered, that if

two mean proportionals could be found between the fide of the given cube, and the double of that fide, the firft of thefe proportionals would be the fide of the cube fought. In order to effedt this, Plato invented an inftrument compofed of twm rules, one of which moved in grooves cut in two arms at right angles to the other, fo as always to continue parallel with it ; but as this method wras mechanical, and likewife fuppofed the defeription of a curve of the third order, it did not fatisfy the ancient geometers. The doctrine of conic Conic feefedlions, which wTas at this time introduced into geo-tions difeometry by Plato, and which w’as fo widely extended as vered by to receive the name of the higher geometry, was fuccefs- ^a^)' ^ fully employed in the problem of doubling the cube. Menechmus found that the two mean proportionals mentioned by Hippocrates, might be coniidered as the ordinates of two conic fedfions, wfflich being conftructed according to the conditions of the problem, would interfeff one another in two points proper for the folution of the problem. The queftion having affumed this form, gave rife to the theory of geometrical loci, of which fo many important applications have been made. In doubling the cube, therefore, we have only to employ the inttruments which' have been invented for deferibing the conic fedtions by one continued motion. It was afterwards found, that inftead of employing two conic fections, the problem could be folved by the interfeclion of the circle of the parabola. Succeeding geometers employed other curves for this purpofe, fuch as the conchoid of Nicomedes and the ciffoid of Diodes, &c. ^ 2^‘ An ingenious method of finding the two mean proper•4 tionals, without the aid of the conic fedfions, was after- A. D. 400. wards given by Pappus in his mathematical colledlions. 6. Another celebrated problem, to trifedl an angle, The trifecwas agitated in the fchool of Plato. It was found that this tion of aa problem depended upon principles analogous to thofe ofan§le* the duplication of the cube, and that it could be conftrudted either by the interfedlion of two conic fedlions, or by the interfedlion of a circle with a parabola. Without the aid of the conic fedtions, it was reduced to this fimple propofition :—To draw a line to a femicircle from a given point, which line lhall cut its circumference, and the prolongation of the diameter that forms: its bafe, fo that the part of the line comprehended between the two points of interfedlion fliall be equal to the radius. From this propofition feveral eafy conftructions may be derived. Dinoftratus of the Platonic fchool, and the cotemporary of Menechmus, invented a curve by which the preceding problem might be folved. It had the advantage alfo of giving the multiplication of an angle, and the quadrature of the circle, from which it derived the name of quadratrix. 7. While Hippocrates of Chios was paving the way for Hippethe method of doubling the cube, which was afterwards crates’s given by Pappus, he diftinguiffled himfelf by the qua-nuladrature of the lunulae of the circle ; and had from this A'‘ C' circumftance the honour of being the firft who found a curvilineal area equal tq a fpace bounded by right lines. He was likewife the author of Elements of Geometry, a work, which, though highly approved of by his cotemporaries, has ftiared the fame fate with fome of the moft valuable produsftions of antiquity. 8. After the conic feilions had been introduced into geometry by Plato, they received many important additions from Eudoxus, Menechmus, and Arifteus. The latter

M A T H E M A T 1 .C S. latter of thefe pRIlofophers wrote five books on conic factions, which, unfortunately for fcience, have not reached A. C. 300. our times. Elements of 9* About this time appeared Euclid’s Elements of GeoEuclid. metry, a work which has been employed for 20C0 years in teaching the principles of mathematics, and which is ftill reckoned the molt complete work upon the fubjeft. Peter Ramus has afcribed to Theon both the propolitions and the demonftrations in Euclid. It has been the opinion of others that the propofitions belong to Euclid, and the demonftrations to Theon, while others have given to Euclid the honour of both. It feems moft probable, however, that Euclid merely collected and arranged the geometrical knowledge of the ancients, and that he fupplied many new propolitions in order to form that chain of reafoning which runs through his elements. This great work of the Greek geometer conftfts of fifteen books : the eleven firft books contain the elements of pure geometry, and the reft contain the general theory of ratios, and the leading properties of commenfurate and incommenfurate numbers. Bifcoveries 10. Archimedes, the greateft geometer among the anofArchi- cients, flourilhed about half a century after Euclid. A. C. 380.

W /TcTz2 o '^'e Was t^ie ^° ^oun^ t^ie rati° between the dia‘ ^°' meter of a circle and its circumference ; and, by a method of approximation, he determined this ratio to be as 7 to 22. This refult was obtained by taking an arithmetical mean between the perimeters of the inferibed and circumfcribed polygon, and is fufficiently accurate for every praftical purpofe. Many attempts have fince been made to affign the precife ratio of the circumference of a circle to its diameter ; but in the prefent ftate of geometry this problem does not feem to admit of a folution. The limits of this article will not permit us to enlarge upon the difeoveries of the philofopher of Syracufe. We can only ftate, that he difeovered the fuperficies of a fphere to be equal to the convex fur face of the circumfcribed cylinder, or to the area of four of its great circles, and that the folidity of the fphere is to that of the cylinder as 3 to 2. He difeovered that the folidity of the paraboloid is one half that of the circumfcribed cylinder, and that the area of the parabola is two thirds that of the circumfcribed re&angle ; and he was the firft who pointed out the method of drawing tangents and forming fpirals. Thefe difeoveries are contained in his works on the dimenfion of the circle, on the fphere and cylinder, on conoids and fpheroids, and on fpiral lines. Archimedes was fo fond of his difeovery of the proportion between the folidity of the fphere and that of the cylinder, that he ordered to be placed upon his tomb a fphere inferibed in_ a cylinder, and likewife the numbers which exprefs the ratio of thefe folids. 1 Bifcoveries While geometry was thus advancing with fuch raof Apollo- pid fteps, Apollonius Pergaeus, fo called from being nms. born at Perga in Pamphylia, followed in the fteps of A. C zoo. .Archimedes, and w-idely extended the boundaries of the fcience. In addition to leveral mathematical works, which are now loft, Apollonius wrote a treatife on the theory of the conic ledftions, which contains all their properties with relation to their axes, their diameters, and their tangents. He demonftrated the celebrated theorem, that the parallelogram deferibed about the two conjugate diameters of an ellipfe or hyperbola is

equal to the rectangle deferibed round the tw’o axes, and that the fum or difference of the fquaves of the two conjugate diameters are equal to the fum or difference of the fquares of the two axes. In his fifth book he determines the greateft and the leaft lines that can be drarvn to the circumferences of the conic feflions from a given point, whether this point is fituated in or out of the axis. This work, which contains every where the deepeft marks of an inventive genius, procured for its author the appellation of the Great Geometer. 12. There is fome reafon to believe, that the Egyptians Meneluus were a little acquainted with plane trigonometry ; and writes on there can be no doubt that it was known to the Greeks.^ fiei'kal Spherical trigonometry, which is a more difficult part of geometry", does not feem to have made any progrefs till ‘J the time of Menelaus, an excellent geometrician and aftronomer. In his work on fpherical triangles, he gives the method of conftru&ing them, and of refolving moft of the cafes which were neceffary in the ancient aftronomy. An introdudlion to fpherical trigonometry had Theoiloalready been given to the world by Theodofius in his fius’s fpheTreatife on Spherics, where he examines the relative pro- ncsperties of different circles formed by cutting a fphere in A" C‘ all diredlions. 13. Though the Greeks had made great progrefs in the pr0grefs 0f fcience of geometry, they do not feem to have hitherto ana.yfis. confidered quantity in its general or abftrafl ftate. In the waitings of Plato wTe can difeover fomething like traces of geometrical analyfis j and in the feventh propofition of Archimedes’s work on the fphere and the cylinder, thefe traces are more diftinftly marked. He reafons about unknown magnitudes as if they were known, and he finally arrives at an analogy, whicl, when put into the language of algebra, gives an equation of the third degree, which leads to the folution of the problem. x 4. It was referved, however, for Diophantus to lay the 7-^ anaiy, foundation of the modern analyfis, by his invention offisofiudethe analyfis of indeterminate problems ; for the method 'erminate which he employed in the refolution of thefe problems Pro^lerris has a ftriking analogy to the prefent mode of refolving ^ equations of the lit and 2d degrees. He w^as likewifetiis.an' the author of thirteen books on arithmetic, feveral of A. D. 350. which are now loft. The works of Diophantus were honoured wfith a commentary by the beautiful and learned Hypatia, the daughter of Theon. The fame A -p fanaticifm which led to the murder of this accomplifhed ^I