Regiomontanus On triangles

Citation preview

NUNC COGNOSCO EX PARTE

TRENT UNIVERSITY LIBRARY

Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation

https://archive.org/details/regiomontanusontOOOOregi

REGIOMONTANUS ON TRIANGLES

iTli

B§> itl' */K.

1

jlCUIANNE vNV’TlEfiioJVio^TE./^ j/wj' 3ivjtijavvs

fK-srrm,

“John of Regiomont, otherwise known as Muller,’’ unknown artist, 1726.

REGIOMONTANUS ON TRIANGLES De triangulis omnimodis by Johann Muller, otherwise known as Regiomontanus, translated by Barnabas Hughes, O.F.M., with an Introduction and Notes

THE UNIVERSITY OF WISCONSIN

PRESS

MADISON • MILWAUKEE • LONDON

1967

Published by the University of Wisconsin Press Madison, Milwaukee, and London U.S.A.: Box

/379,

Madison, Wisconsin 53701

U.K.: 26-28 Hallam Street, London, W. 1 Copyright

(c) /967

by

the Regents of the University of Wisconsin Printed in the United States of America by North Central Publishing Company, St. Paul, Minn. Library of Congress Catalog Card Number 66-22861

Samueli Barchas Urbano in Lege Peritissimo Astronomico Doctissimo Atque Amico Carissimo Gratissime D.D.D. Interpres

73414

PREFACE Five hundred years ago, in June of

1464,

Sufficient thanks cannot be expressed to the

John Muller, Regiomontanus, completed On

translator’s good friend, Samuel I. Barchas,

Triangles. In its one hundred and thirty-seven

who spontaneously and at great expense pur¬

pages he gathered together the trigonometric

chased a first edition of De Triangulis omni¬

knowledge of his predecessors and enriched

modis simply that it might be translated by

it with his own improvements. As described

this high school teacher. Mr. Barchas is in¬

by A. Wolf,

deed a mathematical “Good Samaritan.” Par¬

Regiomontanus

systematically

summed

up the work of both the Greek and Arab pioneers in plane and spherical trigo¬ nometry. His own special contribution was the application, to the solution of special

triangles, of algebraic methods

of reasoning derived from Diophantus,

ticular appreciation is due also to Universal Microfilm Services, Inc., of Phoenix, Arizona, for xeroxographing the De Triangulis and the vitae of Adam and Gassendus, which made the translator’s work easier. In addi¬ tion, assistance was gladly given by the New York Public Library, the Stadtbibliothek of Niirnberg, and particularly by the Stanford

though without the use of abbreviations.1

University Library. Finally, gratitude is due

In tribute to the man who laid the founda¬

Professor Ernst Zinner, Reverend Joseph T.

tions2 of modern trigonometry, this trans¬

Clark, S.J., and Sister Mary Claudia Zeller, O.S.F.,

lation has been prepared.

for invaluable

information,

to

my

There is a further reason for this transla¬

superiors for their encouragement and ap¬

tion. It is the belief of the translator that in¬

proval, and to the University of Wisconsin

structors in the mathematical sciences should

Press for preparing the publication of this

be familiar with the sources upon which their

work. Fr. Barnabas Hughes, O.F.M.

subjects have been founded. Fifteenth-cen¬ tury Latin, however, is not an easy diet for

Saint Mary’s High School

those with but several years of a classical

Phoenix, Arizona

Latin background that has, perhaps, decayed

June 1965

with the passage of time. And so in transla¬

1 A. Wolf, A History of Science, Technology,

tion the Triangles is offered here for those

and Philosophy in the 16th and ryth Centuries

who wish a deeper appreciation of the source

(1950), p. 189. 2 George Sarton, Six Wings: Men of Science in the Renaissance (1957), p. 25. 3 David Eugene Smith, History of Mathematics (1958), I, 260.

of modern trigonometry, of “the first work that may be said to have been devoted solely to trigonometry.” 3

•

Vll

•

_

CONTENTS Preface

vii

Introduction

g

Trigonometry before 1464

5

The Contribution of Regiomontanus

7

The Text and Plates

9

The Biography Text and Translation

10 20

Bibliography

295

Index

297

Frontispiece “John of Regiomont, otherwise known as Muller,” unknown artist, 1726 Plate I A sketch of Regiomontanus, possibly by Kepler Plate II

The earliest likeness of Regiomontanus

6 12

.

INTRODUCTION

M

INTRODUCTION The reputation of Regiomontanus and his

vites his readers to compare his work, Dis¬

influence upon his contemporaries and those

course on the Variation of the Cumpas (1581),

who followed him in the next hundred years

with that of Regiomontanus, evidently his

have been subjected to close attention.1 But

accepted authority.9

there is still room for further study. Five men in particular knew of him and his writings:

Tycho

Brahe

established

his

reputation

with the publication in 1573 of De nova stella.

Columbus, Beheim, Novara, Rhaeticus, and

In this short work of a little more than one

Copernicus. Columbus was born in the same

hundred

pages

(and

only

twenty-seven

of

year as Regiomontanus and used the astro¬

these are on the appearance of the new star

nomical tables of Regiomontanus even after

in Cassiopea), he uses Regiomontanus’ Tri¬

he reached the West Indies.2 Martin Beheim

angles as his nearly exclusive authority for

was in Lisbon in

1484, where “by passing

himself off as a pupil of [Regiomontanus] he managed

to

enter

the

most

learned

and

1Lynn Thorndike in A History of Magic and Experimental Science (1932), V, 332-77, offers a broad analysis of the impact of Regiomontanus.

courtly circles.”3 Rhaeticus had mastered the

A more detailed survey has been prepared by

Triangles before he allied himself with Co¬

Sister Mary Claudia Zeller, O.S.F., in

pernicus.4 Copernicus, previous to this, had studied in Bologna under Dominicus Maria Novara, himself a pupil of Regiomontanus.5

to Pitiscus (1944). 2 Charles Singer, A Short History of Scientific Ideas to 1900 (1959), p. 196. 3Samuel Eliot Morison, Admiral of the Ocean

Moreover, there is some indication that Regio¬ montanus contributed to the foundation of The

testimony of scholars about

Sea (1942), I, 70. 1 Zeller,

the heliocentric theory.6

try

Regio¬

montanus and his Triangles is indeed inter¬

5 John

Charles Hutton wrote, “. . . he enriched trig¬

The

Development

of

Trigonome¬

p. 55. Kepler,

Tabulae Rudolphinae

(1627),

preface, p. 3. 6 Leonardo

esting. The opinions of Wolf and Smith have already been cited in the Preface. In 1795,

The De¬

velopment of Trigonometry from Regiomontanus

Olschki

in

The

Genius

of Italy

(1949), p. 374, states “. . . he went so far as to express

his

doubts about

the

validity

of

the

Ptolemaic cosmology as a whole and of the geo¬

onometry with many theorems and precepts.

centric system in particular.” This author does

Indeed, excepting for the use of logarithms,

not give any primary source for his statement. Cf.

the trigonometry of Regiomontanus is but little inferior to that of our own time.”7 A century earlier, Edward Sherburne wrote,

also: Arthur Koestler, The Sleepwalkers (1959) p. 209. Ernst Zinner in Entstehung und Ausbreitung der Copernicanischen Lehre (1943), pp. 13536, indicates that this question is still open.

“[The Triangles] is still a Book of good accompt, as containing in it divers extraordi¬

7 Charles Hutton, A Mathematical and Philo sophical Dictionary (1795), II, 132. 8 Edward

nary Cases about plain Triangles.”8 In the sixteenth century, too, came signifi¬

“Catalogue of Astrono¬

Marcus Manilius.

cant testimony from England, France, and Denmark. William Borough, for instance, in¬

Sherburne,

mers” (1675), p. 41, appendix to his Sphere of 0 Marie Boas, The Scientific Renaissance, 1450-

1630 (1962), p. 235.

• 3 •

Regiomontanus On Triangles establishing the location of the new star. Six

only with the most diligent study, the

times he cites the

greatest work and the most adroit efforts

Triangles by book and

theorem. Finally he writes,

of the mind. . . . They ought to admire

We have found the longitude and lati

in John his proficiency in each of the

tude of this new star with the help of the

languages [Latin and Greek] and his most

infallible method of the doctrine of tri¬

rigid demonstrations of his statements.16

angles. Exactly how we went about doing

Despite the laudatory remarks from the past

this in finding the sides and angles of the

five centuries, it must be pointed out that

triangles needed is indicated in the ref¬

some experts in the fields of mathematics and

erences. There is no place here for fur¬

astronomy do not agree with the opinions

ther explanation of them, for they are

presented. Among the moderns, it is note¬

quite long and would only enlarge this

worthy that Hooper17 does not even mention

work too much. A good part of the prop¬

Regiomontanus by name in his chapter on

ositions are from the Fourth Book of

the development of trigonometry. In the last

[the Triangles by] Regiomontanus. This

century, while Delambre13 does call him “le

work

was

used

because

everything

is

plus savant astronome qu’eut encore produit

closely tied together geometrically.10

1’Europe,” he feels that the stature of Regio¬

It would certainly seem that Tycho Brahe

montanus as a mathematician is perhaps ex-

held Regiomontanus in the highest esteem.

aggerated. In particular he questions whether

Only a single side reference to Copernicus

or not Regiomontanus introduced the tangent

kept the full credit for the method of locating

concept, and offers good reason for supposing

the position of the new star from going to the

that he did not. At least, Delambre concludes,

Elsewhere11

the mathematical reputation of Regiomon¬

Brahe refers to him as “Clarissimus & prae-

tanus cannot be determined at the time of writing, in 1819.

Triangles

of

Regiomontanus.

stantissimus Germanorum Mathematicus Jo¬

Whether or not Regiomontanus understood

hannes de Monteregio Francus . . . .” Fifty years

before

Brahe,

the renowned

the tangent concept at the time he wrote the

French mathematician Oronce Fine indicated

Triangles deserves attention here. There is

his high regard for Regiomontanus by refer¬

clear evidence that he knew, used, and ap¬

ring to him as “Matematico accuratissimo.” 12

preciated

the

tangent when

he wrote

his

Early laudatory testimony can be found in

Tabulae directionum in 1467, which he de¬

many places. Peter Apian (1495-1552) in his

scribed as a “fruitful table” (tabula fecun¬

Cosmographia (1574 printing) refers to Regio¬

da).™ But did he have the tangent in 1462

montanus’

vul.

when he began the Triangles? Zinner remarks,

Kunigsperg, the birthplace of John Regio¬

“Tycho Brahe, De nova stella (1573), B4V. Also in Opera omnia (1648), p. 357.

birthplace:

“Mons

Regius,

montanus who restored the science of mathe¬ matics.”

None

of

the

other

famous

11 Brahe, Opera omnia, p. 35.

men

mentioned in this work is given such praise.13 In a similar vein speak Philip Melanchton 14 and Erasmus Rheinhold (1511—

553)-15

!

The 1492 edition of the Alfonsine Tables was prompted by the correspondence between

12 Oronce Fine, Della geometria, I, Ch. xiii, 16, in Opere divise (1587), which is the Italian trans¬ lation of his collected works originally known as Protomathesis (1532). 18 Peter Apian, Cosmographia (1574; ed. Gem¬ ma Frisius [1508-1555]), fol. 33. 14 John of Sacrobosco, De sphaera (1563; ed. in 1531 by Philip Melanchton), A-4.

Santritter and Moravus which was printed along with the manual. In a letter to Moravus, Santritter wrote,

10 George tarum verso.

Peurbach,

(1542;

ed.

Theoricae

Erasmus

novae

plane¬

Rheinhold), p.

102

10 Tabule astronomice Alfonsi regis (1492), A-3.

Who can be patient with those unlearned persons who do not know the first things about mathematics. They carp at and condemn the astronomical works of John of Monte Regio, things that he found

17 Alfred (1948).

Hooper,

Makers

of

Mathematics

J. B. J. Delambre, Histoire de I’Astronomie du Moyen Age (1819), pp. 292-323, 347-65. Zeller, The Development . . . , p. 34.

* 4 *

of

Trigonometry

Introduction “There is no application of the tangent in the Triangles," but notes that Regiomontanus was using the tangent concept the next year (1465) in Rome and afterward in Hungary (1467)- Finally, Zinner observes that Peurbach had outlined a tangent table in

rived.-1 Abu-1 Wafa' (940-998) was the first to generalize the sine law to spherical tri¬ angles. He used the umbra (tangent) as a real trigonometric line and arrived at the re¬ lation tan a:l —sin a:cos a. For this he saw

1455.20 It

the utility of setting the radius equal to unity.

seems likely, then, that Regiomontanus knew

Subsequently al-Blrunl (973-1048) wrote the

of the tangent function when he wrote his

sine law for plane triangles. All of these ad¬

Triangles. Why he did not use it is another

vances were brought to Europe by the transla¬ tors.

question.

From the considerable work of the Euro¬ pean translators, the following are the most

TRIGONOMETRY BEFORE 1464

significant to the development of mathematics It is quite difficult to describe with certainty

in the West. Adelard of Bath (ca. 1116-1142)

the beginnings of trigonometry.21 There is

translated

just not enough evidence. In general, one

Khowarizmi (ca. 845) into Latin, thereby in¬

may say that the emphasis was placed first on

troducing the sine and tangent functions into

astronomy,

part of Europe. John of Seville (ca. 1135-

nometry,

then shifted to spherical trigo¬

and

finally

moved

on

to

plane

trigonometry.

J153)

the

translated

astronomical

tables

al-Farghanl’s

of

al-

Elements

of

Astronomy (ca. 861); Regiomontanus had a

In particular, it seems certain22 that the

copy of this. He also had a copy of al-Battanl’s

Babylonians of the old period (before 1600

The Motion of the Stars (ca. 920), which was

b.c.) had some knowledge of chords for astro¬

translated

nomical

of

Regiomontanus must have had a Greek trans¬

chords employed the ratio of a chord of a cir¬

lation of the Almagest. There is also evi¬

purposes.

The

trigonometry

by Plato of Tivoli

(1134-1145).

cle to the diameter of that circle to determine

dence24 that he had some of the works of

the central angle. Hipparchus (ca.

Jabir (or Geber, ca.

180-125

1225), who improved

b.c.) formulated a table of chords. But it was

Menelaus’ theorem concerning segments of

Menelaus of Alexandria (ca.

the sides of a triangle by showing it applicable

first

formulated

triangles:

the

theorem

100 a.d.) who basic

to

all

The product of the three ratios

to four quantities as well as six, and that he had something25 of al-Zarqall (ca. 1075).

of the consecutive segments of the sides of 20 Ernst Zinner, Leben

a plane triangle made by any rectilinear trans¬ versal equals unity. His successor Ptolemy (ca.

und

Wirken

des Jo¬

hannes Milller von Konigsberg genannt Regio¬ montanus (1939), pp. 29-30, 64, 101, 107, 115.

150 a.d.) developed the heritage of the past

21 This resume is based principally upon the

and left his masterpiece, Syntaxis, or Alma¬

following: John David Bond, “The Development

gest, for the future. In this work he brought

of Trigonometric Methods down to the Close of

a significant measure of perfection to deter¬ mining his tables of chords. These tables were

the XVth Century,” Isis, IV (October 1921), 295323; George Sarton, Introduction to the History of Science (1927-1948), Vol. I, II (Pt. 1), III (Pt.

used throughout Europe without substantial

i); J. F. Scott, A Histoiy of Mathematics (1958),

improvement until Regiomontanus published

Ch. 3; David Eugene Smith, History of Mathe¬ matics (1958; 2 vols.).

his Tabula fecunda.

“Otto

The next advance in trigonometry was in

schichte

Neugebauer, der Antihen

Vorlesungen

ilber

Mathematischen

Ge-

Wissen-

the East. The Hindus had the works of Mene¬

schaften (1934), I, 168. See also O. Neugebauer

laus and Ptolemy, but they improved upon

and A. Sachs, eds., Mathematical Texts (1945), pp. 38-41.

their predecessors by considering the half¬ chord and the radius of the circle. Thus, they discovered the sine ratio upon which modern

23 Louis C. Karpinski, “The Unity of Hindu Contributions to Mathematical entia, XLIII (June 1928), 382. 24 Pedro

trigonometry is based. They took another step forward and performed calculations on a new ratio,

based on

from which

the

function was de¬ ■

Nunes,

Tratado

da

Sciences,” Sphera

Sci¬

(1537),

fob 39, in Obras, I, 62: “. . . Joannis de Monte Regio qui Gebrum imitatus est . . . .”

measurements of shadows, tangent

Cuneiform

25 This was probably Peurbach’s copy of alZarqali’s

5 •

Canons

or

Rules

on

the

Tables

of

Plate I. A sketch of Regiomontanus, possibly by Kepler.

Introduction While Regiomontanus had access to many

the remaining theorems (20 through 57) of

of the translated works, he did not have them

Book I propose geometric solutions for right,

all; and this is significant. The tangent func¬

isosceles,

tion was brought into Europe by Adelard of

seven exceptions to this method of solution.

and

scalene

triangles.

There

are

Bath, but it was slow to find its way into cen¬

I heorems 20, 27» and 28 mention or use the

tral Europe. Rather it stayed north in Paris

sine function explicitly. The solutions for the

and England. There is no doubt that in Par¬

four cases of oblique triangles are handled in

is the tangent functions (umbra versa, umbra

theorems 49, 50, 52, and 53. While the sine

recta) were known, at least to Dominicus de

function is not mentioned in any of these

Clavasio

four theorems, reference is made to theorem

(/?.

1346). And in England John

Manduith (fl. 1310), whom Sarton calls “the

27 where it is used.

real initiator of western trigonometry,” knew

The systematic ordering of trigonometric

and used these functions in his Small Tract.

knowledge may be said to begin with theorem

Moreover,

Richard Wallingford (ca.

1292—

1335) wrote Quadripartitum de sinibus dem¬

1 of Book II. Here Regiomontanus states the Law of Sines:

onstratis, using these functions. But appar¬

In every rectilinear triangle the ratio of

ently none of these works penetrated into

[one] side to [another] side is as that of

central and south-central Europe. In this re¬

the right sine of the angle opposite one

gard, one might say that a wall separated the

of [the sides] to the right sine of the angle

Paris-English schools from the rest of Europe.

opposite the other side.29

There is, however, one unexplained excep¬ tion to the previous observation. Campanus

He uses this law to solve two cases of the

of Novara (ca. 1260-1280) wrote a true table

oblique triangle problem in theorems 4 and

of tangents26 for each degree, o to 45. In

5 of Book II: Wherrtwo angles and any side

view of the meanderings of Regiomontanus

or two sides and the angle opposite one of

about Italy and central Europe where he took

them are given, the remaining parts can be

every opportunity to hunt out scientific works,

found.

it is surprising that he apparently never came Toledo. See M. Curtze, “Urkunde zur Geschichte

across this work. Thus, when Regiomontanus organized his material for On Triangles, he was well famil¬

der

Trigonometric . . . ,”

Bibliotheca

Mathe¬

matica (1900), ser. 3, I, 338. If this is the case, it is difficult to understand why Regiomontanus did

iar with the heritage of Ptolemy and the works

not utilize the umbra (cotangent, here) in his own

of some of the Hindu-Arabic scholars. He

work. For al-Zarqall devoted several paragraphs

knew of the tables of chords and their de¬ termination,

the

trigonometric

ratios

to its determination and use (Curtze, pp. 342-43, 352). In fact, John de Lineriis (1300-1350), pro¬

(his

fessor of mathematics at Paris and a follower of

knowledge of the tangent function is ques¬

al-Zarqall, defined clearly the umbra recta and

tionable, at best), and the sine and cosine

umbra versa in his own Canons on the Tables

laws, all of this for both plane and spherical

of the Primum Mobile (Curtze, p. 399). Another question is, why was Regiomontanus not familiar

trigonometry.

with this work of John de Lineriis? 20 Neither Braunmiihl nor Zinner could find any evidence of the use of this table in Regio¬

THE CONTRIBUTION

montanus’ writings. See Zinner, Leben und Wir-

OF REGIOMONTANUS

ken . . . , p. 107.

What Nasir ad-Din had done two centuries previously for the East, Regiomontanus27 did for the West:

“He constructed a uniform

27 Only the contributions of the Triangles are considered here. 28 Zeller,

foundation and a systematic ordering of trig¬ onometric knowledge.” 28 geometric

method — definitions,

postulates,

and theorems — with Euclid his major author¬ ity. The first part of Book I

(theorems

1

through 19) treats magnitudes and ratios, and

Development

of

Trigonometry

^Zinner (Leben und Wirken . . . , pp. 65—66) shows that

The foundation is laid in geometry and the

The

. . . , p. 19. Regiomontanus was most

probably

not familiar with the trigonometric work of Levi ben Gerson, and that the statement of the Sine Law may be attributed to Regiomontanus. Zeller (The Development of Trigonometry . . . , p. 25) suggests that Regiomontanus was dependent up¬ on al-BIrunl for the statement of this law.

• 7 *

Regiomontanus On Triangles Book II is particularly noteworthy for two

Apparently Regiomontanus first found the

things. First, in theorems 12 and 13 Regio¬

cosine law when, as a young man in Vienna,

montanus offers algebraic solutions for find¬

he was studying the Astronomy of al-Bat-

ing the lengths of the sides of a triangle.

tanl.33

Both of these solutions employ quadratic

thus reworked the law into its first practical

equations

formulation.

whose

solutions

Regiomontanus

He recognized

its importance,

and

assumes are quite familiar to the reader. The

In summary, Regiomontanus laid a solid

algebra is literary rather than syncopated or

foundation in plane and spherical geometry

symbolic. The second noteworthy aspect of

for a complete trigonometry. Besides offering

Book II is theorem 26:

a number of original

If the area of a triangle is given together with the rectangular product of the two

theorems (including

among them an implication of the trigono¬ metric formula for the area of a triangle),

sides, then either the angle opposite the

he used algebra

base becomes known or [that angle] to¬

problems, and he presented the first practical

twice

to solve geometric

gether with its known [exterior] angle

theorem for the Law of Cosines in spherical trigonometry. From his work, his great suc¬

equals two right angles.

cessors,

Copernicus

and

Rhaeticus,

sought

This is the first implicit statement of the

assistance and inspiration for their own trigo¬

trigonometric formula for the area of a tri¬

nometries.

angle.30

Just what influence the Triangles had on

Finally, scattered throughout Book II are

the mathematicians of the sixteenth century

a number of theorems which a modern trigo¬

is explored in some detail by Braunmiihl.34

nometry text would classify as exercises, such

After the untimely death of Regiomontanus in

as theorem 8: ‘‘If the ratios of three sides are

1474, Bernard Walter took control of his pos¬

given and if the perpendicular is known,

sessions, books, manuscripts, and instruments.

each side can be measured.”

When Walter died in 1504, these were scat¬

Book III again is an elementary founda¬

tered. Willibald

Pirkheimer (1470-1530), a

tion, for it is a spherical geometry developing

leading

much detail for what will come in Book IV.

retrieve some of the works of Regiomontanus,

(One must remember that Regiomontanus

notably the Triangles. Pirkheimer made his

citizen

of

Niirnberg,

managed

to

was primarily an astronomer and that while

home a center of learned activity where what¬

the Triangles is a work on trigonometry, the

ever was available, including the Regiomon¬

author in his own preface considers it a neces¬

tanus manuscripts, was common property. In

sary tool for astronomy.31) By theorem 16 he is ready for the Law of Sines for spherical triangles which he carries into theorem 17. Theorems 25, 26, and 27 treat right-angled spherical triangles. And theorems 28 through 34 give the six cases for solving oblique spher¬ ical triangles. Among these is theorem

29

his circle was John Werner (1468-1528), who wrote

book

on

spherical

cleric of Werner, George Hartmann (14801545)- Thence it went to George Rhaeticus (1514-1576).

From this, one may reasonably conclude

Book V continues the solution of problems of spherical triangles. Here in theorem 2 is

that, after Pirkheimer made available to his friends

contained the Law of Cosines for spherical triangles, disguised in the terminology of the versed sine. In modern notation32 this theo¬

the

manuscript

of

which can be reduced to

32 Zeller, The Development .... p. 30.

Trigonometry

p.

of

27 below.

Trigonometry

33 Zinner, Leben und Wirken . . . , p. 66.

cos a — cos b cos c sin b sin c

of

31 See “Text and Translation,”

~ sin b sin c ’

Triangles,

book, he may have borrowed from Regio-

1

vers sin a - vers sin (b - c)

the

erner probably saw it. In writing his own

30 Zeller, The Development . . . , p. 25.

rem states vers sin A

trigonometry.

it did find its way into the hands of a fellow

which deals with the ambiguous case.

cos A =

a

While Werner's book was never published,

34 A. von

Braunmiihl,

Vorlesungen

iiber Ge-

schichte der Trigonometrie (1900), I, 133, 1 p.

• 8 •

Introduction montanus whatever was helpful. Eventually

1661). That translation is in the city library

Rhaeticus had Werner’s book. There is then

of Reutlingen (Mss, No. 1873 and 1879).

a direct line from Regiomontanus to Rhaeti¬ cus

and,

consequently,

to Copernicus,

for

Rhaeticus instructed Copernicus.

and the Charybdis of liberality. An effort was

While this line of speculation is interest¬ ing,

Braunmiihl

notes

that

In the translation, an effort was made to steer a path between the Scylla of literalness made to utilize the words and expressions

Rhaeticus did

that Regiomontanus used. Occasionally, how¬

personally inscribe a copy of the Triangles

ever, this was not possible. For instance, on

for Copernicus. And Copernicus did study

page 38, line 44, the text reads

the work thoroughly. This copy has been

ex gb in bh.” Literally, this translates into

preserved and it shows numerous marginal

“• • • that which becomes from GB in BH.”

notations in Copernicus’ handwriting.

When the reference to Euclid is consulted, the

Other testimony to the influence of the Triangles in wanting.35

the sixteenth century is

Francis

Maurolyco

. . quod fit

expression becomes

. . the product of GB

not

and BH.” Consonant with the tenor of the

(1494-1575)

age, the author from time to time would omit

took some of the definitions and propositions

the subject of the sentence or the verb or the

from Regiomontanus’ work for his transla¬

object.

tion and commentary on Euclid’s Elements.

phrase not justified by translation seemed

John Blagrave (d. 1611) cited Regiomontanus

desirable, it was added in brackets. Finally,

as the source for sections of his The Mathe¬

his various expressions to close the proofs of

matical Jewel. And Adrian Metrius (1571 —

theorems (quod intendebamus-, qui est quod

1635) took from the Triangles the figure and

libuit

proof for the sine law for spherical triangles,

theorema

as Book V of his Universal Astrolobe shows.

translated simply as “Q.E.D.” The symbols

Wherever

absoluere;

an

additional

verum

proposuit;

igitur

etc.)

are

word

est,

or

quod

frequently

Z, A. and 0 have been used in the transla¬ tion for angle, triangle, and degree; and the THE TEXT AND PLATES

capitalization

The text36 used for the translation, as shown

and

punctuation

have

been

modernized.

by the title page and colophon, is the edition

Paragraphing is nearly nonexistent in the

published posthumously in octavo by John

text. To make the reading of the translation

Petreus of Niirnberg for John Schoner, in

easier, the English text has been arranged

1533. Peter Gassendus testifies that this is the first edition.37 Its title is, “Doctissimi viri et

35 Zeller,

The Development

of

Trigonometry

mathematicarum disciplinarum eximii profes¬

■ . ■ , pp. 72, 90, 104. The works cited here were

soris Joannis de Regio Monte de Triangulis

not available to the translator.

Omnimodis

libri

quinque.” A second,

en¬

36 Zinner notes (see Leben und Wirken . . . , p. 230) that the original manuscript consists of

larged edition was published by Daniel Sant-

106 pages in quarto, that Werner wrote the title

bech

“De Triangulis omnimodis quinque volumina,”

at

montani,

Basil

in

1561,

mathematici

“Joannis

Regio-

praestantissimi

de

Triangulis planis et sphaericis libri quinque,

but that another hand wrote the dedicatory, “To the Most Reverend Father in Christ and Lord Bessarion,

una cum tabulis sinuum.”

Bishop of Frascati, Cardinal of the

Holy Roman Church and Patriarch of Constan¬

Despite the fact that the second edition

tinople, John the German of Regiomont offers

has eleven more pages plus the table of sines

himself, a most devoted servant.” The Appendix

and chords which the first edition refers to

described on the title page has not been trans¬

but does not contain, a careful comparison of the two editions reveals no striking dis¬

lated here. 37 Peter Gassendus, Tychonis Brahei vita (1655), pp. 368-69.

similarities.38

^Zeller,

There were no more editions of the Tri¬ seventeenth

century.39

Zinner,40

how¬

ever, notes that it was translated into Ger¬

Trigonometry

30 Louis C. Karpinski, “Bibliographical Check

angles published in either the sixteenth or the

The Develop7nent of

. . . , p. 19. List of All Works on Trigonometry Published up to 1700 a.d.,” Scripta Mathematica, XII (1946), 268.

man by Matthew Beger of Reutlingen (d.

• 9 *

40 Zinner, Leben und Wirken . . . , p. 231.

Regiomontanus On Triangles thus: the first paragraph following the state¬

theorem 29, page 68. The key word is "huius”

ment of the theorem indicates what is given

(translated throughout as “above”), for this

and what is to be proved. Each major part of

is a clear reference to theorem 6 and theorem

the proof receives a separate paragraph. Illus¬

7 of the present book.

However,

as John

trative examples, which often show the me¬

Schoner observes in his dedication, it is some¬

chanics for using the theorem, receive one or

what annoying that Regiomontanus did not

more paragraphs, depending on their length.

complete his references in Books II through

The footnotes in this translation fall un¬ der four categories. The first group clarifies certain difficult or archaic terminology. The second

offers

a

few

definitions

of

Latin

phrases. The third indicates corrections to the figures (except where Regiomontanus has specifically directed the student, "When [it] is drawn”). And finally, the fourth encom¬ passes corrections to the Latin text where

V. Very ungraciously he left a blank;

example, in Book II, theorem 8, page 114, he

proof. Hence, there are no footnotes for in¬

wrote

"ex

processu

igitur

primi

hu¬

ius . . . .” This reference, of course, is to some theorem of the First Book ("above”). Such omissions have been supplied in translation

within

brackets,

for

this

example,

“Th. I-[44] above.” It is unfortunate that there is no contempo¬

the correction is directly pertinent to the verted letters, misspellings, and trivia such

for

rary picture of Regiomontanus.43 The three plates included here are representative

of

as the misnumbering of theorems. The refer¬

posthumous work. The Frontispiece is from

ences to the figures in the Latin

the collection of Frederick Roth-Scholtz, en¬

consistent in

are not

their order of notation

(i.e.,

graved on copper by an unknown artist in

line AD alternately appears as line BA with

1726. The inscription reads “John of Reg-

no

iomont,

geometric

distinction intended);

these

otherwise

outstanding

remain as they appear in the Latin text.

known

mathematician

as

Muller.

and

An

renowned

Finally mention must be made of the au¬

publisher of Nurn’oerg. Born June 6, 1436.

thorities Regiomontanus used.41 His proofs

Died July 6, 1476. In his 41st year.” Plate I

of the theorems rested upon three sources.

is a sketch very possibly by Kepler for the

The first is that of axiom or definition, many

title page of his Rudolfine Tables. Plate II is

of which are enumerated at the beginning of

the earliest likeness. It appears as the fron¬

the text. The second source is Euclid’s Ele¬

tispiece to Regiomontanus' Epitome which

ments.

was published in Venice in 1496.

Gassendus

tells us

that Regiomon¬

tanus was quite familiar with the Campanus edition of Euclid,42 so it may be assumed that THE BIOGRAPHY

that was the text he used. References to Euclid are expressed in one of two ways: "per 29. &

Something should be known about the man

34. primi elementorum Euclidis,” as in theo¬

who wrote the Triangles. His life has been

rem 1, page 32; or “per primam sexti” from

commented on, more or less, by many of the

the same place. Since the first example given

historians of mathematics and astronomy. Al¬

is the first reference Regiomontanus made

though biographical articles had been writ¬

to any author and the second example fol¬

ten before, the first definitive biography of

lows in the same paragraph, it was concluded that the second example together with all other similar unidentified references must be to Euclid. Occasionally Regiomontanus gave an incomplete reference to Euclid: either the theorem number or the book number was omitted. This is recognized in the translation by the phrase “[not given].” The third au¬ thority is Regiomontanus himself. When he referred to a previous theorem, it was always in this fashion: “per 6 aut 7 huius,” as in •

11 While Regiomontanus referred only to Eu¬ clid and himself as the authorities in the Tri¬ angles, he did mention in a letter to Bianchini his dependence on Menelaus, Theodosius, and Jabir (or Geber). See Zinner, Leben und Wirken ■■■>?■ 65. 42 Gassendus, Tycho Brahei vita, p. 364. 43 See Zinner, Leben und Wirken .... pp. '92 T the information in this paragraph. The Frontispiece is reprinted here through the courtesy of Prof. Ernst Zinner. Plate I is from the Stadtbibliothek Niirnberg, and Plate II from Samuel I. Barchas.

IO •

Introduction Regiomontanus was written by Peter Gassen-

final

dus in

Equitis

1651, as an appendix to his much

pages

(345-373)

Dani,

of

Tychonis Brahei,

Astronomorum

Coryphaei,

larger biography of Tycho Brahe.44 He used,

Vita.

besides the books and letters of Regiomon¬

Peurbachii, & Joannis Regiomontani, Astro¬

Accessit

Nicolai

Copernici,

Georgii

tanus with their dates and contents, the inci¬

nomorum

dental remarks of others whom he quoted

Gassendo, Regio Matheseos Professore. Edi¬

by name: Starovolsius (p. 345), Cardanus (p.

tio

celebrium,

Secunda

auctior

Vita. 8c

Authore

correctior.

Petro Hagae-

361), Ramus (p. 361), Kepler (p. 367), Schoner

Comitum. Ex Typographaei Adriani Vlacq,

(p. 368), and others. Briefly, Gassendus con¬

M.DC.LV.

structed his biography from

the works of

With the exception of the more specific

Regiomontanus and occasional comments by

localization

other astronomers and literary luminaries.45

birth, the few additional contributions on

of

Regiomontanus’

place

of

What follows is an abridged translation of the

his life from more modern writers have been

Gassendus biography.46 It was taken from the

assigned to footnotes.

John Miiller, Regiomontanus John Miiller was born at

Unfinden, near

His studies began with the more developed

Konigsberg in Lower Franconia,4^ on June 6,

theory

1436. Earlier writers sometimes refer to him

trained in the theory of spheres. Using the

as Johannes Germanus or Johannes Francus, the one because he was a German, the other because

Franconia

was

sometimes

called

Eastern France. He learned his grammar at home until the age of twelve when his par¬ ents sent him to Leipzig for his formal edu¬ cation. Here both dialectics and the theory of spheres were favorites. These led him to the study of astronomy and whatever arith¬ metic and geometry were necessary for a bet¬ ter understanding of this science. These he mastered

quickly;

in

particular,

whatever

was wrapped in theory drew his avid study. From Leipzig he graduated to the Academy at Vienna where he came under the influence of George Peurbach. This was in

1451

or

influence

of

1452. Peurbach

had

the

greatest

anyone over Regiomontanus. And this was acknowledged with gratitude on many occa¬ sions. Peurbach recognized the remarkable genius of this young man and realized that this was the one student destined for great things. In particular he respected the enthusiasm of Regiomontanus for astronomy, and saw in this ingenious youth an opportunity to re¬ juvenate that science. Since the respect was reciprocated, nothing

that

Peurbach would

promised

bring

the

to

omit

desires

of

Regiomontanus to fruition. From this time onward, the one was likened to a father, the other to a son.

of

planets,

since

he

was

already

44 Melchior Adam included a seven column biography of “Joannes Mullerus Regiomontanus” in his Vitae Germanorum philosophorum (1615), using material which he obtained from Philip Melanchton. Melanchton, in turn, borrowed from Erasmus Rheinhold’s Oratio de Joanne Regiomontano mathematico (1549) for his own Selectarum declamationum (1551). Another bio¬ graphical article was included by Paul Jovius in his Elogia doctorum virorum (1556). And there are others. 45 The Reverend Joseph T. Clark, S.J., has courteously supplied the translator with the in¬ formation on the Gassendus biography. 40 Several weeks after digesting the Gassendus biography, the translator happened upon a copy of Johann Friedrich Weidler’s Historia astrono¬ miae (1741). And in this on pp. 304-13 is a digest in Latin of the Gassendus together with Regio¬ montanus’ Index and a catalogue of Regiomon¬ tanus’ printed works. It was altogether satisfying to find that the biography as condensed here is practically parallel to the abstract by Weidler. 47 This location is given by Zinner, Entstehung . . . , p. 101. D. E. Smith (History of Mathe¬ matics, I, 259) identifies the birthplace as Unfied. Florian Cajori in A History of Mathematical Notations (1928), I, 97, has an illustration from Regiomontanus’ Almanac with the title, “Cal¬ ender des Magister Johann von Kunsperk (Jo¬ hannes Regiomontanus).” Charles Hutton in A Mathematical and Philosophical Dictionary, II, 130, names the birthplace as Koningsberg. Gas¬ sendus merely states, “Natus est Joannes ... in oppido . . . cui Regius Mons nomen. . . . ap¬ pellitatus potius fuerit Joannes de Monte Regio, vel de Regio Monte, ac Regiomontanus” (Tycho Brahei vita, p. 345).

Plate II. The earliest likeness of Regiomontanus.

Introduction former as the central idea for all of astron¬

angles to each other and one of these was

omy, he found it easier to begin the study

extended at both ends, the end of the other

of the Ptolemaic doctrine. While studying

diameter could

Ptolemy,

or

another circle of radius equal to a third of

practice in calculations. In these fields he

the circumference of the original circle, and

built an enviable reputation for speed and

that the line segment included between the

nicety of demonstration. To reach an even

points of intersection of the new circle and

he did

not

neglect geometry

be

used as

the

center of

greater perfection he studied all the mathe¬

the extended diameter of the original circle

matical works written in Latin that he could

wotdd be equal to half the circumference of

find. For this he had James of Cremona's

the original circle. Regiomontanus showed

translation

of Archimedes,

as well

as

the

that this line segment would in fact be less

works in translation of Apollonius and Dio-

than what the Cardinal claimed.

phantus. Peurbach never had to encourage

After Peurbach, perhaps the greatest influ¬

him. Indeed he was like a straining horse,

ence in the life of Regiomontanus was Cardi¬

eager for the finish line. Perseverance was a

nal Bessarion. Apart from being a successful

bit of a problem at the start, and Peurbach

diplomat and trouble-shooter for the Pope

had to remind him that what was begun with

he was a scholar in his own right, particularly

great ardor must be persevered in.

in astronomy. A Greek by birth, he mastered

Regiomontanus

to

the Latin language so that he could produce

master what was available regarding astron¬

a definitive translation of Ptolemy. Unfor¬

omy: for there was much previous knowledge

tunately his ecclesiastical duties kept him too

to

be harvested

made

in

every

attempt

the science of astron¬

occupied for this

sort of work;

hence,

it

omy; and it all had to be gathered in were

was fortunate that he met Peurbach. Peur¬

the art to be entirely rebuilt. In particular

bach had spent much time in the study of

he had to make himself familiar with the

Ptolemy, attempting to correct a Latin trans¬

points of the Zodiac, the hinges of the Eclip¬

lation simply by analyzing the translation as

tic, just under the Aplanes or Firmament.

such, for he knew no Greek. Nor, with Regio¬

While he did not learn all of the fixed stars,

montanus in mind, did he feel it necessary

he knew those which he could compare with

to master the language. For he had intro¬

the

planets. He became familiar with the

duced his protege to the Cardinal, and from

instruments of Hipparchus and Ptolemy, to¬

this time onward Regiomontanus began his

gether with other instruments

that would

study of the Greek language. In a compara¬

assist him in observing the celestial bodies.

tively short time he became proficient in this

One of his earliest observations was that in

new tongue. Not only as a reward but also

comparing the

of Mars with the

as an incentive for further study, the Cardi¬

nearby fixed stars, he discovered that the

nal made available to Regiomontanus other

tables of the time were two degrees off. Reg¬

scientific works written in Greek.

position

three

The preparation of an Epitome of Ptol¬

lunar eclipses which he observed in his early

emy’s Almagest was but half completed when

iomontanus

left

written

records

of

years. On all three occasions he was able

Peurbach died, April 8, 1461, at the age of

to correct the time predicted for the eclipse

37. This was, perhaps, the greatest loss in

according to the Alfonsine Tables, some of

the life of Regiomontanus. Of his teacher he

his corrections being as small as one minute

wrote, “He was a man of the first caliber in

and others as great as seventy minutes.

habit and integrity of life, a scholar in every

About this time Peurbach received a copy

subject and superior to all in mathematics.”

of Cardinal Nicolas of Cusa’s work, On the

On his death bed, Peurbach committed the

Quadrature of the Circle. While there is no

Ptolemaic translation and its completion to

written record of Peurbach’s criticism, Regio¬

Regiomontanus. This became a sacred trust

montanus gave his

for the fatherless student.

mentor credit for

the

criticism he himself developed. The essence

Leaving the remains of Peurbach in Vien¬

of the Cardinal’s idea was that, when two

na, he accompanied Cardinal Bessarion to

diameters in a circle were drawn at right

Italy, where in Rome he brought the prepa-

*

13

*

Regiomontanus On Triangles ration of the Epitome closer to completion.

Jews celebrating the Pasch at different times.

It was at this time that he first met George Trebizond,

an

authority

to

Rome

by

December,

he

sought to improve his library of rare works,

his commentator Theon. Moreover, he cul¬

either by purchasing the books outright or

tivated

the

by copying them. For he was anticipating a

learned

man,

of

particularly

Ptolemy

Returning

and

friendship

on

every those

available versed

in

return to

Germany from where he would

Greek. At the same time he kept busy with

have little opportunity to obtain the books

his astronomical observations, spending much

he would want. It was at this time he en¬

time on

these between December of

1461

and the following March. Many of these ob¬ servations were made at Rome, and for the others he went to Viterbio particularly in the summer and fall of other

occupations

was

1462. Among his

the

collecting

and

copying of rare books, both in Greek and in Latin. And among these was a New Testa¬ ment, which was his constant companion. About this time Cardinal Bessarion was sent to Greece on Church affairs and Regio¬ montanus left for Ferrari. This place ap¬ pealed to him, for John Blanchinus, of whom Peurbach had spoken so highly, was teaching here. He sought out the company of Theo¬ dore Gaza, among others, for further instruc¬ the

philosophers

and

poets.

From Venice he left for Budapest to accept the invitation

torious from war with the Turks, and part of the

works in Greek which he had seized in Con¬ stantinople, Athens,

was imminent. So it happened, and the king

Another of his Hungarian friends was the Archbishop of Strigonium, to whom Peurbach many

all of the ancient writings both in Latin and

he indicated that he was about to learn it.

1464, he left for Venice to await Cardinal Bessarion. It was here during May and June that he finished his work, On Triangles. The remainder of June and most of July were the

Quadrature of Nicolas of Cusa. He also de¬ voted some time attempting to rectify the had

Christians

and •

had

sent

a

geometric time

4SIn a written criticism of Trebizond's work, Regiomontanus in direct address calls him, “im¬ pudentissime atque perversissime blatorator.” (You are the most impudently perverse blabber¬ mouth!) Cantor notes that such remarks were not uncommon among men of this caliber and were not considered serious insults, ordinarily. Con¬ sidering what may have been the cause of Regio¬ montanus’ death, however, one wonders how seriously Trebizond’s sons took his remark. (See Moritz Cantor, Vorlesungen ilber Geschichte der Mathematik (1900), II, 257.)

After the lunar eclipse at Padua in April 2,

that

before

Regiomontanus spent some

with this ecclesiastic, instructing him in as-

wrapped in a language unknown to him, but

of

years

gnomon.

The Arabic writings were still

current calendar

the

showered him with further gifts.

told his audience that he himself had read

refutation

in

on during a recent eclipse and that recovery

lecture. In his initial lecture on al-Fargham

the

and elsewhere

Turkish domains. Regiomontanus accepted

he sang the praises of Peurbach, and then he

preparing

of many scientific

was quite adept) that the king was merely

Padua where the Academy invited him to

in

booty consisted

suffering from weakness of the heart brought

The following year saw Regiomontanus in

spent

Hun-

tanus showed from astrology (in which he

and

translator forget the mistakes.48

Greek.

Mathias

had despaired of the king’s life, Regiomon¬

seriously. And he would not let the poor

in

of the king,

niades Corvinus. The king had returned vic¬

king who had become quite ill. While others

the Theonic commentary. Here he discovered frequently

Regiomontanus left Rome for Venice again where he taught mathematics for a while.

Library of Buda. Shortly after arriving in the

and finish the entire text of Ptolemy and erred

pushed his criticisms too far. At any rate

capital city, he offered further services to the

that enabled Regiomontanus to go through

had

taries of Theon. He made much about point¬ ing out the most serious errors. Perhaps he

him to be the librarian of the new Royal

These gave to his mastery of Greek the polish

that Trebizond

for his poor interpretation of the Commen¬

the opportunity and handsome salary offered

tion in Greek. Here he read the orators and historians,

countered Trebizond and took him to task

14

•

Introduction trology49 and the use of a set of tables which

described in his Index, it “contains the true

he had written.

conjunctions and oppositions of the stars

The King of Hungary unfortunately em¬

together with their eclipses, the place of the

broiled himself in a war over Bohemia, and

stars from day to day, descriptions of the

Regiomontanus found it to his advantage to

equinoctial and seasonal hours,

leave for Niirnberg. Before he departed, he

useful pieces of information.” His next book,

took time to observe that Jupiter was in the

the Almanac, enjoyed wide popularity, be¬

constellation Virgo on March 15, 1471. He

ing distributed

reached Niirnberg by June 2nd for a lunar

many, France, and England. With these four

eclipse.

books

Regiomontanus had a particular affinity

and other

throughout Hungary,

Ger¬

in print, Regiomontanus then pub¬

lished his Index of Books.

for Niirnberg. It was close to his home town.

Nowhere,

perhaps,

is

the enthusiasm of

It had become a center of learning, and his

Regiomontanus for the sciences better seen

library was appreciated. But two things in

than in the Index. For he announced the

particular captured his enthusiasm. First was

titles of the books he intended to publish

the printing press which had been set up.

and, in some instances, the reasons for pub¬

He saw its

lishing the books. These books fall within

possibilities and was eager to

spread the printed word of science. The sec¬

two groups.

ond was that the city had become a center of the

practical

arts,

and

their

The first group consisted of the works of

practitioners

the ancients, particularly the Greeks. In pre¬

could manufacture the astronomical instru¬

paring these for publication he would rely

ments he desired. The

financial means to

not only on his own abilities in the languages

take advantage of these two boons was soon

and sciences but also on the learning of his

provided

associates, as he wrote in a letter to M. Chris¬

from

the

close

friendship

that

sprung up between Regiomontanus and Ber¬

tian of Erfort. The following list is taken

nard Walter. This latter, besides being one

from the Index itself. “(1) A new translation

of the most influential citizens of Niirnberg

of the Cosmography of Ptolemy, since that

and quite wealthy, was a scholar in his own

old translation of the Florentine James Angel

right, a patron of the arts, and an amateur

is vicious. Although he meant well, he was

astronomer.

quite weak in his knowledge of both Greek

Among

the

instruments

for

and mathematics. This opinion is concurred

Regiomontanus were astronomical staffs for

with by the pre-eminent Latin and Greek

measuring the altitudes of the sun, moon,

scholar Theodore Gaza and the scholar in

and stars,

to

mathematics and Greek, Paul Florentine. (2)

determine the distances of the stars. Next was

Ptolemy’s Great Composition (or Almagest)

an armilla such as Ptolemy and Hipparchus

in a new translation. (3) The Campanus edi¬

had used to note the location and movements

tion

of the stars. Finally he constructed other in¬

Ascensions, which must be corrected in many

struments, such as a torquet and a Ptolemaic

places and to which a commentary of sorts will

meteoroscope.

it

be added. (4) The Commentary on the Alma¬

possible for Regiomontanus to make further

gest of the eminent mathematician of Alex-

then

constructed

an astronomical radius

These

instruments

made

of Euclid’s Elements with

Hypsicles’

corrections in the Alfonsine Tables. He wrote, in fact, “What discrepancy there is between

40 Lynn Thorndike claims to have seen two edi¬

Alphonse and the Heavens. His tables are

tions of Regiomontanus’ Almanac which show the extent to which astrology could enter into a

frivolous.’’ It was about this time that Regio¬

person’s private life. Both of these editions give

montanus divided the whole sine into one

the favorable hours for having one’s hair cut

hundred thousand parts to facilitate his com¬

and for taking a bath! (A History of Magic . . . , III-IV, 442). Astrology was in widespread

putations.50

use

among the people of the times.

The first books he printed were the New

50 Hutton says that Regiomontanus made the

Theories of Peurbach and the Astronomy of

radius or whole sine into one million parts (A

Manilius. Then he published the first of his

Mathematical and Philosophical Dictionary, II,

own printed works, the New Calendar. As

132)-

•

15

'

Regiomontanus On Triangles andria,

Theon.

(5)

Proclus’

Astronomical

armilla

together with

the

entire

heavens,

Hypotheses. (6) A new translation of Ptol¬

described on such a level that anyone can

emy’s Tetrabiblos and the one hundred fruits

learn what never before was available to those

thereof. (7) The works of Julius Firmicus that

understanding only Latin, because of the ex¬

are available, together with the writings of

tremely poor translations. (2) A small Com¬

Leopold of Austria, the fragments —which are

mentary

against

the

translation

of

the

quite useful —of Anthony of Montulmo, and

Florentine James Angel, which will be sent to

any other worthy writers on astrology. (8)

arbitrators

The geometric works of Archimedes, namely,

above). (3) A Defense of Theon of Alexandria

(Gaza

and

Paul,

mentioned

The Sphere and Cylinder, The Measurement

in six books, against George Trebizond, in

of the Circle, Conoids and Spheroids, The

which anyone will be able to see how super¬

Spirals, Equilibrium, The Square of the Par¬

ficial was his commentary on the Almagest

abola, The Sand Reckoner, together with the

and how poor his translation of that work

commentaries of Eutocius Ascalon on The

of Ptolemy. (4) A small Commentary on the

Sphere and Cylinder, The Measurement of

Campanus edition of Euclid’s Elements, in

the

transla¬

which certain gratuitous statements will be

tion will be that of James of Cremona with

disproved. (5) The Five Equilateral Solids —

some corrections. (9) The Perspective (Optics)

which have a place in nature and which do

of Vitello, an immense and noble work. (10)

not — and this will be against the commen¬

Circle,

Ptolemy’s

and Equilibrium. The

Perspective.

(11)

The

Music

of

tator on Aristotle, Averroes. (6) A Commen¬

Ptolemy together with Porphyry’s exposition.

tary on those Archimedian books which lack

(12) A new edition of The Sphere of Mene¬

Eutocius’ comments.

laus. (13) Theodosius’ Spherics, Habitations,

cle, against Nicolas of Cusa. (8) Directions,

and a new translation of his Days and Nights.

against the Archdeacon of Parma. (9) On the

(7) Squaring the

Cir¬

(14) The Conics of Apollonius of Perga, to¬

Distinction

gether with his Serene Cylindrics. (15) The

against Campanus and John Gazulus Ragu-

Spiritalia of Heron, a truly pleasureful work

sinus, and in this work some statements from

of the Places in

the Heavens,

on mechanics. (16) The Elements of Arithme¬

The Hours will be exposed. (10) The Motion

tic of Jordan and his Data Arithmetica. (17)

of the Eighth Sphere, against Thebit and

The

Quadripartitum

his followers. (11) A Reformed Ecclesiastical

that

abounds

with

of Numbers, various

a work

insights.

(18)

Aristotle’s Mechanics. (19) The Astronomy of

Calendar. (12) A Short Almanac. (13)

Tri¬

angles of Every Kind, in

(14)

five books.

Hyginus with his chart of the heavenly bodies.

Astronomical

(20) The Rhetoric of Tulliana. (21) Maps of

everything

the known

their size, distance from the earth, and true

world,

of Italy,

Spain,

all

of

Problems

in

the

with

Almagest.

reference (15)

to

Comets,

France, Greece, and Germany, together with

position. (16) Geometric Problems of Every

histories collected from various sources that

Kind, a particularly fruitful work. (17) The

will discuss the geographies of the mountains

Pannoman

and seas, the lakes and rivers, and of other

written at Strigonium and dedicated to the

particular places.”

Archbishop.

The second group of works would be his own

books.

Besides the

Primum

Game or (18)

Mobile,

Tables of Directions,

The Great with

Table of the

many

uses

and

Calendar and Al¬

grounded on solid reasons. (ig) Balances and

manac already mentioned, these would in¬

Aqueducts, with illustrations of the instru¬

clude the following. “(1) His Great Commen¬

ments necessary for these. (20) Burning Mir¬

tary upon Ptolemy’s Cosmography, in which

rors and many other things together with

will be explained the construction and use

their wonderful uses. (21) The Astronomer’s

of

Ptolemy

Workshop, in which many instruments for

himself estimated almost all the numbers in

observing the heavens as well as instruments

the

meteoroscope

with

which

his works (despite the beliefs of some that

for a more earthly use will be described.” He

the numbers of the longitude and latitude

concludes the Index with the comment that

were reached by observation of the stars),

if he can print all of this before he dies,

and also will be included a description of the

death will not have any sting to it; for he

•

16

Introduction wishes to leave all this to posterity that it not

ing year. Whatever the cause, Regiomontanus

be without the necessary books.

died July 6, 1476, at the age of forty years

The

publication

of

the

new

Calendar

and one month. His premature death was

the reigning

greatly mourned, even more so since he had

Pope, Sixtus IV, to come to Rome and revise

accomplished so much in such a short time.

the Julian Calendar. As an inducement, the

It is reported that he was buried in the Pan¬

brought him a request from

Pope appointed Regiomontanus the Bishop

theon. Typical of the eulogies sung in his

of Ratisbon. The esteem in which he was

praise was that of Jovius: “A man of wonder¬

held at this time was recorded by Ramus.

ful skill, divine ingeniousness, he was certainly

Regiomontanus,

the most outstanding of all Astronomers, even

his mathematics, his studies, his works.

of those who preceded him.” Latomus com¬

Tarento

posed the following in his memory.

Niirnberg

gloried had

Archimedes,

in

its Archyta, Byzantium

Syracuse

Proclus,

its

Alex¬

When Jove beheld the spinning Spheres

andria Ctesibius, and Niirnberg Regio¬ montanus. Proclus

Archytas

and

and

Ctesibius

mathematicians

of

increas’d,

Archimedes,

are

Tarento

dead.

The

and

Syra¬

And a bright Tenth just added to the rest, Began to dread his Palace wou’d be seen, Naked, and open to the View of Men.

cuse, of Byzantium and Alexandria are

In vain, said he, we mov’d that Race

gone. But among the Masters of Niirn-

from hence,

berg, the joy of the scholars is the mathe¬

Since

matician Regiomontanus.

Can

It would be hard for him to leave Niirnberg,

for he had established himself as

a

In

are

no

none of us

these bold Attempts

vain,

said

Hermes, you

the Fates

defy;

machine, he made a mechanical fly that could person’s hand and,

Battlements

Or shall we sit content, invaded thus?

that, besides constructing a perpetual motion a

high

oppose?

scientist of no mean ability. Ramus records

leave

these

Defence.

’Tis their Decree, this Man shall never

after buzzing

die.

about the room, return to one’s hand. Larger

If you’d restrain him, I advise you thus —

than this was an eagle that could leave the

This Instant canonize him one of us.52

city, fly to meet an approaching dignitary (the Emperor is mentioned), and accompany

With the news of his death, Niirnberg went

him to the city of Niirnberg.51 The weight

into public mourning. No one mourned more

and responsibility of the pastoral office was

faithfully

not particularly attractive to Regiomontanus,

tinued the work of Regiomontanus until his

than

Bernard

Walter

who

for he much preferred the joys of study and

own death, sometime after

disliked leaving unfinished the works he had

made his last observation in the skies.

1491

con¬

when he

begun. But at the insistence of Cardinal Bes-

Mention has already been made of what

sarion, he was willing to leave the unfinished

Regiomontanus himself published while he

projects in the capable hands of his close

had the use of the printing press in Niirn-

friend

Walter.

berg: the New Calendar, the Almanac (both

And so toward the end of July, 1475, he left

and

collaborator,

Bernard

of these in 1464), and the Index of Books

for Rome.

(in the following year or so). But most of

And in the Eternal City everything came to

his completed works were published by John

an abrupt end. He died within a year. For he

Schoner of Niirnberg. The list of these books

had incurred the vehement hatred of Treb-

follows. (1) Triangles of Every Kind. (2) On

izond’s sons and they took the earliest oppor¬

51 Zinner

discounts

these

stories

as

legends

tunity to poison him. Paul Jovius, however,

passed on from overzealous Niirnbergers to an

says that he died during a plague. Kepler

unsuspecting

agrees with him and at the same time takes

•

Ramus,

who

hastened

to

161-62.

issue with Joachim Camerarius who lays his death to a comet which appeared the follow¬

Peter

record them. See Leben und Wirken . . . , pp. “Translated by Benjamin Martin in his Biographia philosophica (1765), p. 156.

17 •

Regiomontanus On Triangles Sines and Chords, a joint work of Regiomon¬

metric Problems. (9) The Great Table of the

tanus and Peurbach. (3) The Rejection of

Primum

the Manner of Erecting Themas as proposed

against the Theories of Cremona. (11) The

Mobile.

(10)

The

Disputations

by Campanus, (j) The Genethliacum, which

Tables of Directions. (12) The letter to John

was enlarged

Driandrus on the Composition of the Me-

by Schoner.

(5) Descriptions

and Explanations of various astronomical in¬

teoroscope. (13) The letter to Cardinal Bes-

struments including the

the Ar¬

sarion in which Regiomontanus severely criti¬

milla, the Great Rule of Ptolemy, the Staff,

cizes the work of Trebizond. His work on the

and

Commentaries

the

Astronomical

Torquet, Radius.

(6)

The

Thesaurus, a joint work of Regiomontanus

on

Ptolemy

was

published by Joachim Camerarius.

and Walter. (7) Dialogue against Cusa and

Thus

his Quadrature. (8) The Solutions of all Geo¬

•

of Theon

was

Regiomontanus:

work, his reputation.

18

•

his

life,

his

TEXT AND TRANSLATION

DOCTISSIMI VIRI ET MATH& matiorum difciplinarum eximrj proTelioris

IO ANNIS DE RE GIO

MONTE MODIS

DE

TriANGVLIS

LIBRI

OMNI*

Q_V I N QJV Et

Quibus explicantur res neceflariae cognitu,uolentibus ad fcientiarum Aftronomicarum perfedHonem deueni* reiquae cum nufqua alibi Iioc tempore expolitae habeantur,fruftra line harum inftrudione ad illam quifquam afpirarit.

Acceflerunt huc in calce pleracp D.Nicolai Culani de Qua dratura circuli, Deqjrefti ac curui commenfuratione; itemcp Io.de monte Regio eadem de re itoyfa* >u4 ha&enus i nemine publicata.

Omnia recens in lucem edita, fide & diligentia lingulari. Norimbcrga in ardibus Io. Petrei» ANNO M.

CHRISTI

D. XXXIII.

ON TRIANGLES OF EVERY KIND JOHN, REGIOMONTANUS In five books the author explains all those things necessary for one who wishes to perfect his knowledge of astronomy. Since these matters have never been developed before anywhere, one cannot aspire to learn this science without these ideas.

An Appendix contains the works of Nicolas of Cusa on the Squar¬ ing of the Circle and the measuring of straight and curved lines, together with their heretofore unpublished refutation by John, Regiomontanus.

All of these have been thoroughly edited with singular diligence and fidelity at the Publishing House of John Petreus, in Ntirnberg. a.d.

1533.

IOANNES SCHONE'' RVS

CAROLOSTADIVS

AMPLISS, SENA prudentiis, S, P, d,

torum Ordini duitatis Noricae Dominis

TINAM prudenrifltmiDomini,itaDeouifumfuiflet,utqua occafione nunc ego lum ulus ad celebrandam rempubl.ueftra, dum uobis op rim is (eruptis do relique omnes notae ’ i 4 > occurrent.Erit enim per 13.huius proportio a quantitas 7’ ' ' tis ad lingulas alias data,quare per foetam huius a quan titate nota exifirente,lingulae reliquat innotefcent,quod erat concludendu. V Ope rationem ex tredecima dC foeta huius facile comparabis. XVI.

Quod fub duabus inter fe datis reftis lineis continetur, parallelo* gramum recflangulum latere non poterit. Sit parallelogrammu retfangulum a b c d, duabus inter Ce datis contentum lineis a b & a d.Dico,9 ipium prodibit cognitum.Quonia enim dux lineat ab & a d inter icnota;fi.int,meniurabit eas communiter una quantitas quat fit h. menfurct itartj lineam a b fecundum numerum k lineam a d fecundum nu* meru 1, &abicindant ex duobus lateribus galleIogrami\ppofiti:duatlinex ag8i a e,quarum utracprnenfursecommuni h acqualisexiftat,producflis lineis e n q dem acquediftante ipfi a b & g f,arquediftantelateri a d,eritcgper i?.34*pri* mi & diffinitionem quadrati fuperficies a f qua ^ drata.cunqt h fiue a g fibi aequalis mcniiiret la* tus a b fecundum numerum k,crit a g in a b quoties unitas in k,& ideo proportio a g ad a b,ficutunitatis ad k numerum, quarep prima fexti proportio quadratelli a f ad parallegram* mum a n,ficutunitatis ad k numenim.Vnita* tis autem ad k numerum proportio data eft per animiconceptionem,quare8(! ,pportio quadra* telli a f ad parallelogrammum a n nota perhibebitur. Item quonia a e aequae lisipfih menfuratlarus a d fecundumnumera l,erit a e in a d,quoties unitas in 1 numero.quare proportio a d ad a e eft ut numeri 1 adunitatem.proportio autem a d ad a e per prima foeti eft tancjj parallelogrami a c ad parallelogram mum a n .parallelogramum ergo a c ad parallelogramum a n ficutnum crus ad unitatem.fed numerus 1 ad unitatem proportionem habet datam ex commits ni animi conceptione,unde proportio a c ad a n fcita ueniet.Iam igitur dua* tum quantitatum (uperficialium a f & a c,utracp ad parallclogramu a n pro* portionem habet datam,quare per 1 ijhuius earum inter fe conflabit proportio.

C

•

5°

*

&quo

Book I of On Triangles of these forms a given ratio with C, then

Now on these two sides of the given paral¬

their ratio to each other is known by Th. 12

lelogram, measure off two lines AG and AE,

above. Hence, because by Th. 6 above quan¬

each of which is equal to the common meas¬

tity E is known, quantity B becomes known,

ure H. When line EN, paralleling AB, and

as was expected.

line GF, paralleling side AD, are drawn, then

The mechanics for this are taken from Ths.

by [Euclid] 1.29 and 1.34 and the definition

12 and 6 above. For after the ratio of the two

of a square, the area AF is a square. Since

quantities is found by Th. 12 above, Th. 6

H, or its equal AG, measures AB according

gives the value of the unknown quantity.

to the number K, then AG will be in AB as many times as unity is in K. Hence, the ratio

Theorem /5

of AG to AB is as that of unity to K. There¬

If any number of quantities form given ratios with another particular quantity, and

fore

by

[Euclid]

VI. 1,

the ratio

of small

square AF to parallelogram AN is as that of

if any one of these quantities is known, then

unity to the number K. The ratio of unity

all the others may be found.

to the number K is given by axiom, and

If any three quantities A, B, and C are

therefore the ratio of small square AF to

forms a given

parallelogram AN is made known. Similarly,

ratio with quantity C,* and if some one of

since AE, or its equal H, measures side AD

them — A, for example — is known, then all

according to the number L, then AE will be

the other quantities may be found.

in AD as often as the unit measure is in L.

given

[such that] each one

By Th. 13 above, the ratio of quantity A

Whence the ratio of AD to AE is as that of

to each of the others individually is known.

number L to unity. Moreover, the ratio of AD to AE, by [Euclid] VI.1, is the same as

And because quantity Af is known by Th. 6 above,

the remaining quantities

are

indi¬

the ratio of parallelogram AC to parallelo¬ gram AN. Therefore parallelogram AC is to

vidually found. One may easily provide the mechanics here

parallelogram AN as number L§ is to unity. But the ratio of L to unity is given by axiom.

from those of Ths. 13 and 6 above.

Hence,

Theorem 16 The product]; of two straight lines given in the same units reveals [the area of] a rec¬ rectangular

parallelogram

ratio

of AC

to

AN becomes

ties AF and AC, each has a given ratio to parallelogram AN; hence, by Th. 12 above, their ratio to each other is established.

tangular parallelogram. If

the

known. Now then, of the two areal quanti¬

ABCD

*For C read D. -f-For A read E.

is

given, bounded by two lines AB and AD expressed in the same units, then [their prod¬ uct] will determine the [area of the rectan¬ gular parallelogram] itself. Since the two lines AB and AD are ex¬ pressed in terms of each other, one common quantity H can measure them. Line AB is thus measured by H according to the number K and line AD according to the number L.

•

51

XQuod sub duabus lineis continetur, some¬ times with rectangulum added (e.g., on pages 92 and 130), refers to the rectangular area con¬ tained under two sides; hence, the product. Again it should be recalled that this is a text primarily for applied trigonometry rather than for the more contemporary theoretical trigonom¬ etry, and physical qualities and measurements are apparent throughout. §In Latin, for numerus read numerus l.

*

tS

IjQH*

DE

JIONTEREGIO

& quoniam quantitas a f notaefWfl omnibus ncp menfurationibus notam (tip poni oportet menluram,per foetam huius parallelograms a c notum e nuncia* bitur quod erat peragendum. Conftat autem hoc in ^ppofito quadratellum at el fc menfura Tifperfidalem,q> coftam eius a e menfurar lineali h aquale initio fta* tuerimus. V Idem alio tramite confequemur.Prolongetur utraq? linearum c b # d a uerfusiiniftia,doncc duae lineae b p & a q fibi,&hnrae a b aequales ueni ent.continuatis'cp termini's earum p & q ,per lineam p q claudetur quadratum q b per 29. 3 3 .primi,& diffinitionem quadrati,cuius cum hyp^hefis nota dederit coftam a b ,ipiiim qc£ g prima huius notum habebitur.eft aute ex pri mafexti,pportio quadrati q p ad parallelogra mum a c,tanqp q a fiue a b ad a d . propor.* tioautem a b linere notae per hypo theftm ad a d notam per diffinitionem data, unde dC proportio quadrati q b ad parallelograms a c data erit, per (exta igit huiusCquadratoq b noto exiftente)ueritatem theorematis inferemus, V Ad* huc aliter dC ad operationem aptius.Refumpta figuratione prima,numerus k in numeru l^dudus, efficiat numeru 1 m .Gu itaq^ut fupra memoratu eft, prop or tio quadratdli a f ad parallelograms a n eft,ficut unitatis ad k numerum, a n aute ad ac 1L* cut unitatis ad 1 numerum. fuperius enim erat a c ad a n tancp numeri 1 adunitatem,unitatis dcmS ad 1 numerum (icut numeri k ad numerS m.eftenim k in m,quoties unitas in lex diffini done multiplicationis, per re qua igit ^pportiona litatemerit a f quadratelli ad a c parallelogra mum,ficut unitatis ad m numerum.quare a tin c . . . a c,quoties unitas in m numero reperitur, g dif linitioncmitaq? notae quantitatis parallelogramum a c notum effecimus, in eo emmmeimiralfuperfidalis a f continetur fecundum numerum no tum,quieftm quod libuit abloluere. V Opus autem docebimus unicum, tametfi demonftra* tione freti limus uaria.Duos numeros duorum laterum parallelogra mi notorum in lemultiplicabis,alterum uidelicet in alterum,^ducetur enim numerus paralie .°Sr^rm iecundu quem meniura fuperficialis,quadratu fcilicet men fur re linealis impio continebitur parallelogramo.Vtfilatus a b 5-latus a d 7 ,pedes co* p ectatur Iineales,dudis y in 7,creabuntur 3 5-.tot igitur pedes quadrati parallelo gramum a c confnttient.Ita in canteris operabere.

Ex dato latere quolibet parallelogrami re&anguli cogniti reliquu latus emerget notum» Sit parallelogramum redangulu a b c d cognitum,aiius etiam latus unu quodcunqj fuerit notum habeatur, fii%(uerbi gratia) a b .Dico, q> reliquu latus eius a d fcitumerit. Edudis namcp lineis d a & c b ad punda q & p,donec utraq? linearu a q & b p aequabitur lineae a b datae, comple atur quadratum q b protrada linea p q,erit itacp per prima fexti pportio q b quadrati ad a c parallelogramum ficut lineae q a ad linea

a d,eft • 52 •

Book I of On Triangles Since, by Th. 6 above, the areal quantity AF

nition of multiplication, then, by simulta¬

is known — and a known measure need not

neous solution of these proportions, small

be given in all measurements — parallelogram

square AF is to parallelogram AC as unity is

AC is declared* known. Q.E.D. Moreover, it

to number M. By definition of a known quan¬

is established by the proposition

that the

tity, therefore, we have made parallelogram

small square AF is the areal measure because its side AE was initially set equal to the linear

AC known, for the areal measure is contained in it M number of times. This ends the

measure H.

solution.

This can be proven in another way. Extend

Despite the varying proofs, only one meth¬

each of the lines CB and DA toward the left

od will be presented for the mechanics. Multi¬

until the two lines BP and AQ are equal to

ply the two numbers together that correspond

each other and to line AB. When their ends

to the two known sides of the parallelogram

P and Q are joined by line PQ, a square QB

to find the number of times the areal measure

is

— namely, the square of the linear measure —

formed,

by

[Euclid]

1.29,

1.33,

and

the

definition of a square. Since the hypothesis

is contained in the parallelogram.

gave the side of square AB, then the [area of

ample, if side AB is 5 linear feet and AD is

the square] itself will also be known by Th.

7 linear feet, then multiply 5 by 7 to get 35;

For ex¬

1 above. Now by [Euclid] VI.i, the ratio of

thus that many square feet make up the area

square QPf to parallelogram AC is the same

of the parallelogram. And so on in similar

as that of AQ, or AB, to AD. Moreover, the

problems.

ratio of line AB, known by hypothesis, to

Theorem /7

AD, known by definition, is given. Hence the ratio of square QB to parallelogram AC will be given.

Therefore, by Th. 6 above

(because square QB is now known), we may

From any given side of a rectangular paral¬ lelogram [of] known [area], one can deter¬ mine the other side. If [the area of] rectangular parallelogram

infer the truth of the theorem. Another proof is offered that is more perti¬ nent to the practical mechanics. Referring to the first figure,

multiply number K by

ABCD is known [and] any one side is also known —for example, AB — then its other side AD can be found.

number L to obtain the number LM\. There¬

Extend line AD to Q and line CB to P,

fore, since, as was mentioned above, the ratio

until each of the lines AQ and BP is equal

of the small square AF to parallelogram AN

to the given line AB. Complete the square

is as that of unity to number K, and further¬

QB by constructing line PQ. Now by [Euclid]

more since AN is to AC as unity is to L, or,

VI. 1 the ratio of square QB to parallelogram

as before, AC is to AN as number L is to

AC will be as that of line QA to line AD.

unity, and finally since unity is to number

L as number K is to number M, for K is in M as many times as unity is in L from the defi¬

• 53 •

*For e nuntiabitur read enuntiabitur. fFor QP read QB. tFor LM read M.

DE

TRIANGVLIS

MB*

U

15»

a d.eft autem proportio quadrati q b ad ipfum parallelogramum a c data per diffinitionem,q> utracp fiiperficierum qbaa c data fit. q b quidem p primam huius.eft enim quadratum lines a b datae,parallelogramum autem a c notum ftibiecithypothefis.Proportio igitur S>C lineae q a ad lineam a d nota redditur, led q a & a b funtcofts quadrati q b aequales, unde Si proportionem a b ad a d notam efleoportet.aincp altera illarum/cilicet a b nota (upponatur,erit per (betam huius reliqua linea fdlicet a d nota,fiecp reliquum parallelogrami latus a d notumexegimus,quodlibuitattingere. V Idem aliter Si ad operationem accomodatius.Quonialatus a b notufupponit,menfiiretipfiim hfamofaquan titas fecundum numenim k,fitcp utraq? linearum a g Si a e ex parallelogrami noftri lateribus abfiimp tarum aqualis menfurs h,Si ducatur e n quide squedi* ftans lateri a b,g fuero lateri a d squedifta ns,eri t cp fuperficies a f quadrata per zp.Si 34.primi 8i diffinitionem quadrati,quae quidem fuperficies menfurabit parallelograms a c fecundum numeru notum, qui fit m .quonia parallelograms fupponitur co gnitum ,hunc numerum in poftremo per nume^ rum k partiamur,ut exeat 1 numerus. Quia l'tacp ^portio a b ad h ,fiue ad a g eft, ut numeri k ad unitatem, h menfiirantelineam a b fecundu k numerum:erit per prima fexti parallelogrami a n adquadratellum a f,ficut numeri k ad uni¬ AYl tatem .quadratelli autem a f ad parallelograms a c,ficutunitatisadnumerum m,q> parallelogramsipfiimquadratello menfu* retur fecundum numerS m.quare per aequam proportio parallelogrami a n ad parallelogramum a c eft,ut numeri k ad numerum m. eft autem a n ad a c• tancp a e fiue h fibi aequalis ad linea a d per prima fexti. Vnde Si h ad a d eft ut k numeri ad numerum m,fed k ad m ficut unitatis ad 1 numenim per diffrs nitionemdiuifionis.quareproportio h adlmea a d eft,ficut unitatis ad 1 nume rum.menfiira igitur h in linea a d,quoties unitas in numero 1 continetur, quas: re linea a d nota concluditur,quonia menfiira h famofa continetur in ea (earns: dum numerum 1 notum.reliquum ergo parallelogrami latus effecimus cognitu, quod intendebamus. V Opus breue.Numemm parallelogrami noti in numess rum lateris noti partiaris,Sc' exibit numenis lateris reliqui qusfitus.Vt fi paralie* logramum a c offeratur 3 6.pedum fuperficialium,habens latus a b 4 pedum li¬ nealium, diuida numenim 3 6 in numerum 4.S! exibunt 5» .Latus igitur reliquum a d ,nouem pedes complcdctur lineales. XVIII.

Ex data proportione laterum parallelogrami redanguli cogniti, utriufcp lateris pendebit noticia. Sit parallelogramum redangulum abed cognitum,cuius latera ab&a d proportionem habeant adinuicem nota.Dico,q> utruncp iplorum notum habe bitur.Refiimpta emm prima figuratione prscedentis,erit per prima fexti propor tio a c parallelogrami ad a p quadratum,ficut lineas d a ad linea a q.eft aute proportio lines d a ad linea a q data,q> a q squalis habeatur lineas a b.quas: re dC proportio parallelogrami a c ad quadratum a p nota redditur.cunc^ no* tumfubiecerimusparalldogramiim a c,eritper fexta huius S: quadratum a p

C 2.

• 54 •

cognitum

Book I of On T riangles Therefore the ratio of square QB to paral¬

as that of number K to number M. Further¬

lelogram AC is given by definition, since each

more, AN is to AC as AE, or H, is to line AD

of these areas QB and AC is given; for indeed,

by [Euclid] VI. 1. Hence H is to AD as num¬

by Th. t above, QB is the square of the given

ber K is to number M. But K is to M as unity

line AB, and the hypothesis has given paral¬

is to number L by the definition of division.

lelogram AC. Therefore, the ratio of line QA

Therefore, the ratio of H to line AD is as

to line AD becomes known. But QA and AB

that of unity to number L. Thus measure H

are the equal sides of square QB, and hence

is contained in line AD as many times as unity

the ratio of AB to AD is necessarily known.

is contained in number L. Therefore line AD

Therefore since one of the lines — namely AB

is inferred known because the known meas¬

— is given, by Th. 6 above the other line — namely AD — will be known, and thus the remaining side of the parallelogram has been

ure H is contained in it L times. And so the other side of the found.

determined. Q.E.D. Another,

more

parallelogram has been

The mechanics in brief. Divide the area practical

proof

follows.

of the parallelogram by the given side to find

Since side AB is given, let the known quan¬

the unknown side. For example, if the area

tity H measure AB according to the number

of parallelogram AC is given as 36 square

K, and let each of the lines AG and AE,

feet with side AB as 4 linear feet, divide 36

measured off on the sides of our parallelo¬

by 4 to get 9. The unknown side AD is there¬

gram, be equal to measure H. Then, when

fore 9 feet long.

EN is drawn parallel to side AB, and GF parallel to side AD, the area AF is a square

Theorem 18

by [Euclid] 1.29, 1.34, and the definition of

From the given ratio of the sides of a rec¬

a square. This area measures parallelogram

tangular parallelogram with known area, the

AC according to the known number M. Since

length of each of the sides can be found.

the parallelogram was given

If the area of rectangular parallelogram

known, then this number [M] is divided by

ABCD is known and its sides AB and AD

[the area of]

number K to get number L. Because AB is

have a known ratio to each other, then each

to H, or AG, as number K is to unity (H

of these [sides] can be found. When

measuring line AB according to number K),

the

first

figure

of

the

previous

then by [Euclid] VI. 1 parallelogram AN is

theorem is used again, the ratio of parallelo¬

to the small square AF as number K is to

gram AC to square AP will be as that of line

unity. Moreover, small square AF is to paral¬

DA to line AQ by [Euclid] VI. 1. Moreover,

lelogram AC as unity is to number M, be¬

the ratio of line DA to line AQ is given, be¬

cause the parallelogram itself is measured by

cause AQ is equal to line AB. Therefore the

the little square according to the number M.

ratio of parallelogram AC to square AP is

Hence, by simultaneous solution, the ratio

known. Since parallelogram AC was given,

of parallelogram AN to parallelogram AC is

square AP is found by Th. 6 above.

»

• 55 *

10

10 H#

DE

M0NTEREGIO

-cognitum, inde quoq? per fecunda huius cofta fuaab non ignorabitur: quae quideeft altera ex lateribus parallelogrami propofiti.data au* tem tradidit hypothecs proportionem laterum didi parallelogrami.ex latere igitur a b ia co* gnitofexta huius,reliquum latus a d fufcitabit motum.utninq? ergo parallelogrami latus effe¬ cimus menluratum,quod pollicebatur praeiens theorema, V Opus ita compa rabis.Si proportio laterum data eft per denominationern,diuide numemm paral lelogrami per denominatorem proportfonis, & exibit numerus quadrati lateris confequentis,cuius radix quadrata latus ipfum confequens notificabit,poftca ad operationem foeta: huiusaut praecedentis confugias,quae reliquu latus elidet co* gniturn.Vt fi parallelogramum a c contineat 48 quadratos pedes,latus aute a d lateri a b triplum faerit,ecce denominato rem proportionis 3. per quem diuido . numerum parallelogrami 48 ,&C exeunt 1 ^.numerus qui debetur quadrato lateris a b confequentis,cuiusradixquadrata4,latus a b notumfadet.4autem tripli cans,quonia proportionem triplaelegimus^aut 48 diuifis per 4, exurget latus a d reliquum 1 z.QcP fi proportio lateris ad latus data fuerit,'non per denominatio nem,fed per fibi aequalem proportionem,ut fi diceretur, proportio lateris a d aru tecedentis ad latus a b confequentis eft,ut 5- ad 3 :multiplicabis numerum paralie logrami dati per terminum confequentem,& produdtu partieris in numerum an tecedentem,exibitenimnumerusaffignandus quadrato lateris confequentis: c3 quo ut antehac pcedendum erit,Vt fi parallelogramu a c fuerit 60,latus autem a d ad latus a b fe habeat,ut y ad 3 .multiplicabo 60 per 3 .producutur»80. quae diuifa per 5-,eliciunt 3 ^.quadratumfcilicet lateris a b,cuius radix quadrata 6, ipfum lalrlis a b notificabit.reliqua autem per operatione praecedetis abfoluent, XIX,

Si quatuor quantitatum proportionaliumtresqiraelibet datae fue* *int,& quarta reliqua innotefeee. Sint quatuor quatitates a b c d proportionales,quarumtres notae fint quae** quarta reliqua nota ueniet.Quauis aute ipfa ignota quatitas nunc primum,nunc fecundum,interdu uero reliqua foleat occupare loca,tamen ne ope ris uarietas,quaenecelTario hanc mutationem confequinir,le(f?ore perturbetpla* cuit femper ignotae quatitati poftremu deputare locum.Praefens igitur theorema facile confirmabimus,!? prius quo paeffo quatitas ignota,quocunqj nobis offerat loco^poftrcma fiat docebimus,Conftat autem huiufmodi quatuor quatitatum p Pornonafitas ex duabus proportionibus,quara unius ambo termini funt cogni** faciemus prima.fecund* autem pportionis unus duntaxat notus eft ter* f, minus,qui fi fuerit antecedens,ia ordinate funt quam** or ill* quantitates,ut uolumus, habebit enim ignota quatitas tertia c, fcilicet nota fit per menfuram e fecundu nu* memm h .reperiaturt# numerus k,ad quem fe habeat h ,ficut f ad g. quod fiet, fiprodudumcx h in g,per numerum t partiemur,quemadmodum ex uigefima leptimi elementori! trahitur .erit autem numerus k , fecundu quem &>ta habebit, poftrema quatitatu p menfura quide e. Na ex quinta huius ^portio a quatitatis ad b erit,ficut numen'f ad numeru g.fed a ad b ficut c ad d exhypothefi,&,f adg ficut h ad k, quare c ad d ficut h ad K, & couerfim d ad c tancjj k ad h, dCc ad menfura e,ficut h numeri! ad unitate,q> e mefuretc fecundu numeru h.p aequa igitur ,p portio d quatitatis poftremce ad e menfura, tancf? numeru k ad uni tatem.eftitacp e menfura in d,quotiesu*itasin k numero, d ergo quatitas no ta reddi tur per menfura e fecundu numeru k ,quod libuit explanare» V Opus, multiplica numerum fecundae quatitatis per numerum tertiae,Qi produdu in nu* merumprimae quatitatis diuide,exibit enim numerus poftremae quati tatis quae fi* tus.Vtfi a fuerit 4. & b 5>,c ueto 1 2. multiplico 1 2 per 9 ,producumr ioS,quae diuifa per 4,eliciunt 27 numeru uidelicet quantitatis poftremae»

XX. In omni triangulo redangulo,fi fuperuertice acuti angulf,fecundu quantitatem lateris maximi circulum defcripferis,erit latus ipfum acu tum,fubtendens angulum finus reiffus conterminalis fibi arcus dicftu angulum refpicientisdateri autem tertio finus complementi arcus di* fupficies a g b c arquediftatibus continent lineis.angulus uero c a g,fiue e a k redus eft per z^.primi,propter xquediftan tiam linearum b c ad a g,quare per ultima fexti arcus e k circumfermtiae fax quadras pro&bitur.arcus itacp b k complementum ipfiusarcus b e dittmietur, cuiuslinus b g lateri a c aequalis nuperrime concludebaturaKrancg igitur pro* portionis partem fatis oftendifle uidemur* XXI.

Omnem angulum redum notum fcflfe oportet. Vnus enim redus angidus ad quatuor redos nota habet proportionem, fum «na autem quatuor redo rum nota eft, cum gradus unus ftilicet 360. pars quatuor redorum;qua tancp menfura famofa utuntur umuerfi Geometraeilecundu nil* merum notum 3 60.contineatur in quatuor redis.quare per fexta huius oC unus angulus redus notus habebitur,quod erat lucubradum. Quarta aute pars ex 3 60 eft 90 Jufto igitur computo 90 gradus angulo redo uedicabimus. Miraberis lor fitan,quo pado diuerfi generis quatitates menlura gradualis metiatur. Dicimus enim circumferentia circuli uel arcum tot uel totcompledi gradus,item quatuor redos angulos ,uel alium angulum quemcunq; aliquot continere gradus. Quid igituruocabulo gradus fignificareuelimus,paucis habeto.Mcnlura famofa arcu um eft gradus circumferentialis fcilicet 360.pars circumferentiae circuli, menfu* ra autem famola angulorum eft gradus angularis ,uidelicet 360.pars quatuor redorum angulorum,id eft,fpacrj plangquod circa pundum quodlibet exiftit.Ima gtnado enim duas iemidiametros circuli liiper pundo quocunqj tancp centro de a icriptftgradum circumferentiae ciraili didi intercipientes,angulus quem ipfe le midiameyri ambiunt gradus uocabitur angularis,quonia angulus ille 3 60,in qua tuor redis,fiue toto fpacio centrum circuli ambiente continetur; ficut & gradus circumferentialis in tota circum ferentia,huius enim anguli ad quator redos,& il lius arcus ad tota circumferentia,eandem efle proportionem,ex ultima lexti faci* le comprobabit. TTrahimus poftremo ex iam recitatis,qj cuilibet angulo & ar* cui le refpicienti de circuferentia ciraili luper uertice ipfius anguli dclcripti, unus & idem (eruit numerus.uerbi gratia,fi angulum quemlibet 3 6 graduu ftatuimus, erit 8C arcus fe refpiciens 3 6 graduum,& econtra.quod quidem ex identitate mu meri totius circumferentiae circuli & quatuor redonim,ultima lexti ratiocinante pendere dinolcitur. XXII.

Altero duorum acutorum,quos habet triangulus redanguhis^da* to,reliquus non latebit. Duo enim acuti angul i,quos habet triangulus redangulus ,per 3 z.primi,ua lent unum redum,q> tertius angulus fit redus,aggregatu itacp ex duobus didis acutis angulis notum eft,quonia ex praemifla redus angulus notus eft,led dC alter acutoruex hypothefi datus eft,quare per quartam huius reliquum cognofcemus. V Opus,numeri anguli acuti dati,ex numero unius redi minuas, & relinque* tur quantitas alterius.Vt fi angulus b fuerit 20 , minuo 2.0 ex _^o, relinquuntur 70,tantum igitur habebo angulum c reliquum.

Si duo

• 60 •

Book I of On T riangles side [AC] of A ABC by [Euclid] 1.34 because the area AGBC is bounded by parallel lines. But

/

CAG, or EAK, is a right angle by

[Euclid] I.29 because line BC is parallel to AG. Hence by the last theorem of [Euclid] VI, arc EK is shown to be a quadrant of the cir¬ cumference. Consequently arc BK is defined as the complement of arc BE, and the sine BG [of arc BK] was just shown to be equal to side AC.

And

thus

both

parts of

the

theorem* have been proven.

enclose

is called an angular degree, since

that angle is contained 360 times in four right angles or the total space surrounding the center of the circle, just as a circumfer¬ ential degree [is contained 360 times] in the total circumference. Then the fact that the ratio of this angle to four right angles is the same as the ratio of that arc to the total circumference is easily confirmed by the last theorem of [Euclid] VI. Finally, we conclude from the above dis¬ courses that one and the same number de¬

Theorem 21

scribes an angle and the arc opposite it on

Every right angle is necessarily known.

the circumference of the circle that has been

One right angle has a known ratio to four

described around the vertex of that same

right angles. Furthermore, the sum of four

angle. For example, if we take any angle

right angles is known, because one degree —

of 36°, it and the arc determined by it will

namely, a 360th part of four right angles, [such a part being] used as an accepted meas¬ ure by geometry throughout the world — is contained

in

four right angles

360 times.

Hence, by Th. 6 above, one right angle can be found, as was to be determined. A fourth

be of 36°, and the converse. Indeed, this is understood, by the reasoning of the last theo¬ rem of [Euclid] VI, as a consequence of the exact equality of the numerical values of the total circumference of the circle and of four right angles.

part of 360 is 90. Thus, by accurate calcula¬ Theorem 22

tion, we may assert that there are 90° in a right angle.

If one of the two acute angles of a right

You may perhaps wonder how degrees can

triangle is given, the other can be found.

measure quantities of diverse kinds, for we

By [Euclid] 1.32 the two acute angles of

say that the circumference of a circle or an

a right triangle have the value of one right

arc encompasses so many degrees [and] simi¬

angle. Because the third angle [of the given

larly that four right angles or any other angle contains several degrees. What we wish to indicate by the word “degree” can be ex¬ pressed briefly.

triangle] is a right angle, the sum of the two given acute angles is known. Since from the preceding [theorem] any right angle is known, and since

The accepted measure of arcs is the cir¬ cumferential degree, namely, a 360th part

acute angles is given, then, by Th. 4 above, the other acute angle is found.

of the circumference of a circle. The accepted measure of angles is the angular degree, name¬ ly, a 360th part of four right angles — that is, of the planar space that exists around any point. For if it is imagined that two radii of a circle, described around any point as center, intercept a degree of the circumference of

The mechanics: Subtract the value of the given acute angle from the value of one right angle, and the quantity of the other [angle] remains. For example, if / B is 20, then 90 minus 20 leaves 70, the size of the other

Z c.

this circle, then that angle which the radii

• 6l

from the hypothesis one of the

*In Latin, for proportionis read propositionis.

•

DE

TRIANGVLIS

LIB*

I,

XXIII*

Si duo latera trianguli,redhim continentia angulum, fuerint «equa ha,duo acuti anguli eis oppofi ti reddentur noti* Duo latera a c,c b trianguli a b c redanguli, re* dumangulum c ambienria,fint aequalia.Dico, q>uterqj angulorum b & c notus prodibit,Erunt enim per hypos thelim & quinta primi duo anguli a & b aequales, cunqj per 31.primi ipfiualeant unum redum. angulo c redo exiftente,erit uterqj eorum medietas per 11 huius cogni* ti.quare per angulus b a c duplus erit angulo a b c.Extenda turenim a c ufepadpundum d,donec c d habeat aeqlislateri a c,dudalinea b d.erititacp a b linea aequalisipfi a d,q>utriuicpmedietas fit a c.fed per quarta primi duae bales a b,b d triangulorum a b c,& b c d funt aequales, anguli quoq? abc&db „ a c aequales .totus igitur angulus a b d duplus eft ad angulum a b c.eft autem to tus angulus a b d aequalis angulo b a d,fiue b a c per quinta primi triangulo a b d aequilateroexiftente.unde8>Cangulus b a c dupluserit adangulu abc,qd* oportuit dcmonftrare. V Ex hoc patebit corollarium. Quonia enim in trian* gulo noftro angulus a duplus iam declaratus habeatur ad angulum b,ideft,ficut zad 1 ,eritconiundimaggregatuexduobusangulis a &C b ad angulum b,ficut 3 ad 1.Illud autem aggregatum aequipollet angulo redo, hypothefi & 3 z .prim i docentibus.proportio igitur anguli redi ad angulum b nota eft,uidelicet ficut 3 ad 1 .quare per fexta huius angulus b notus eriqredo per % 1 huius noto exiften* te.pouremo etia refiduus ex redo angulus,fcflicet a g 4 huius notus declarabit. XXV.

Duobus trianguli cuiufcuncp cognitis angulis, tertium reliquum datum iri* Trianguli a b c,duo anguli a & b fint cogniti,Dico,q> & angulus c notus emerget.Tres enim anguli a b c ,duobus redis aequantur. 3 z .primi id confirma* te.duo aut redi funt noti per x 1 dC 3 huius,quare QC aggregatum ex tribus angu lis tri*

• 62

•

Book I of On Triangles Theorem 25

is equal to AD because half of each is AC. But

If, in a [right] triangle, the two sides con¬

by [Euclid] I.4 the two bases AB and BD of

taining the right angle are equal, the two

triangles ABC and BCD

acute angles opposite the sides may be found.

equal, and the angles ABC and DBC are also

If in a right A ABC the two sides AC and CB enclosing the right / C are equal, then each of the angles B and C* can be found. The two angles A and B are equal, by the hypothesis and [Euclid] 1.5. Since by [Euclid]

equal. Therefore the total z ABD is double to z BAD, or BAC, because by [Euclid] 1.5

I.32 they have the value of one right angle

From this a corollary arises. Since, in our

[respectively]

are

Z ABC. Moreover, the total z ABD is equal A ABD is equilateral. Therefore z BAC is double z ABC. Q.E.D.

when C is a right angle, then, by Th. 21 above,

triangle, z A has already been determined to

each of them will be half of that known angle.

be double Z B — that is, as 2 to 1 — then by

Therefore by Th. 6 above, each of the angles

addition the sum of the two angles A and B

will be known.

to z B is as 3 to 1. Furthermore, this sum is

The mechanics: Halve the quantity of a

equal to a right angle, by the teachings of

right angle, and the quantity of each of the

the hypothesis and [Euclid] 1.32. Thus the

acute angles appears. For example, since the

ratio of a right angle to z B is known to be

right angle is 90°, take half of 90 and you will

as 3 to 1. Therefore, by Th. 6 above, z B

have 45 for half the right angle; each of the

will be known because by Th. 21 above the

angles A and B will be declared [to be] that

right angle is known. And finally by Th. 4

much.

above, the other angle is found.

Theorem 24 Theorem 25 If the [hypotenuse] of a right triangle is If two angles of any triangle are known,

double the length of one of the sides adjacent to the right angle, then the acute angle in¬

the third may be found. If, in A ABC, the two angles A and B are

cluded by that side and the hypotenuse is double the other acute angle. Hence geome¬

known, then z C can be found. The three angles A, B, and C equal two

try also reveals each of the angles. If right A ABC is given with right /_ C

right angles, a fact confirmed by [Euclid] 1.32.

subtended by hypotenuse AB which is double

Now the [value of] two right angles is known

side AC, then / BAC is double £ ABC.

by Ths. 2i and 3 above; hence the sum of

Extend AC all the way to point D until CD

the three angles

is equal to side AC. Draw line BD. Then AB

• 63 •

*For C read A.

IOH»

DE

MONTEREGIO

[is 1trianguli propofiti notum habebitur. cuncp duos eoru lis datos fubiecerit hypothefis,p 4 .huius tertius reliquus no •datos ignorabitur,quod libuit inferre. V Opus.Summa duo* rum angulorum,qui dati font ex quatitate duorum redo* rum minuas,& relinquetur terti} anguli quantitas dcfide* rata.Vt fi angulus a fuerit zo,& angulus b 35- graduu, colledos zo3s gradus, quireddunt yy.ex 180 minuo» tdidienim n? gradus,angulo c adnumerabuntur, XXVI.

Omnis trianguli re&anguli daobus lateribus cognitis, tertium ex; templo manifeftari. Triangulus a b c angulum c redum habeat,cuius duo latera quaelibet^ fine nota.Dico,q> reliquum eius latus notum habebitur.Si enim duo latera reducon tinentia angulum offerantur nota,erunt per prima huius quadrata eorum nota,' aggregatum quocjj ex eis per tertia huius notum, quod acquipollet quadrato a b per penult ima primi, unde ipfum notu,& ideo per lecun^ dam huius cofta fua, lams fcilicet a b non ignorabitur. Si ueroaltera coram fit datum cum latere redum fubtera dente angulum,quadratu minoris demptum ex quadra*: to maioris,per penulrima primi 6C quarta huius relinquet quadratu reliqui lateris notum,& ideo per fecunda huius coTfareius cognita o tier, quae fvzere lucubrada. V Opus uulgare.Si latera redum ambientia angulum fuerint da ~ ta,quadrabis ea,quadratacp congregabis, Qt colledi ex eis radix quadrata quatitatem lateris quaditi manifefta* bit.Si uero altera eorum fit datum aim latere redum fubtendente angulum, qua¬ dratum minoris ex quadrato maioris demas,& relidi quadrata radix tertium la= tus notificabit.Vt fi lanis a c fiierit 1 z,& b c j-,quadrabo 1 z,exurgunt 144 .ite quadrabo jqueniunt z 5-. coi ligo 1448c zy,fiunt 16'p.quomm radicem quadratam inuerrio 13 .tantumcp fore didici latus a b .Sed ponatur latus a b zp, dd lanis a c zo.duco zp in fe,ueniunt 841 .fimiliter zo in ie,faciunt 400.aufero400 ex 841« relinquutur44J,quorumradrcequadrata z 1 lateri b c deputabo. Ita in cae teris. XXVII.

Trianguliredanguliduobus lateribus cognitis, omnes angulos datum iri» Si alterum datonim laterum redo opponatur angulo,fatis eft.fi uero no, per praecedentem ipfum addifcemus,nam abfcp eo propofitum attingendi no erit po* teftas.Sit itaq? triangulus a b c,angulum c redum habens,cuius duo latera ab &ac fint data.Dico,q? oin nes anguli ipfius notierunt.Super uertice enim an* guli acuti b ,quem uidelicet latus Fubtendit datum tanefj centro .feamdum quan* titatem lateris b a circulodefcripto,eritper zo huius a c finus arcus fibi conteri fninalis,qui refpondet angulo a b c quem inquirimus, cucp duae lineae ab&ac tnterfedataefintexhypothefi'permenfur5ueterem,a b autem femidiameter cirs culi deferipti data fit per menfura noua,quae quidem eft una partium finus totius^ ertcOd a c nota per eandem menfura docente feptima huius , dum igitur a b eft

finus tOa»

• 64 •

Book I of On Triangles of the given triangle will be known. Since

if one of the given sides subtends the right

the hypothesis gave two of these angles, the

angle, then subtract the square of the smaller

third will be found by Th. 4 above. Q.E.D.

[side] from the square of the larger, and the

The mechanics: Subtract the sum of the

square root of the remainder will identify

two given angles from the quantity of two

the third side. For instance, if side AC is 12

right angles, and the desired quantity of the

and BC 5, then square 12 to produce 144.

third angle remains. For example, if / A

Similarly, square 5 to find 25. Add 144 and

is 20° and / B is 350, then subtract their

25 to make 169, of which the square root is

sum, which is 55, from 180. The

found to be 13, the length of side AB. On the

125° left

are the value of / C.

other hand, let side AB be 29 and side AC 20. Then 29 squared is 841 and similarly 20 squared is 400; 400 subtracted from 841 leaves

Theorem 26 If two sides of a right triangle are known, the third is directly apparent.

441, of which the square root is 21, the length of side BC. And so on.

If A ABC has a right Z C [and] any two

Theorem 27

of its sides are known, then the third side When

can be found. Now if the two given sides include the

two sides of a right triangle are

known, all the angles can be found.

their

If one of the given sides is opposite the

squares will be known. The sum of these

right angle, that is sufficient; if not, however,

right

angle,

then,

by Th.

1

above,

[squares] will also be known by Th. 3 above, this

[sum]

being equivalent to the square

of AB by the penultimate theorem of [Euclid] I. Hence [the square] itself is known, and therefore, by Th. 2 above, its root — namely, side AB — will be found. But if one of the given sides subtends the right angle, then, when the square of the smaller [side] is subtracted from the square of the larger [side], by the penultimate theo¬ rem of

[Euclid]

I

and

Th.

4

above,

the

we will find it, also, by the preceding theorem, for without it,

it will not be possible

to

handle the theorem. Thus, if A ABC is given with C a right angle and sides AB and AC known, then all the angles can be found. When a circle is described with / B, which the given side [AC] subtends, as center and side BA as radius, then, by Th. 20 above, AC will be the sine of its adjacent arc, which is opposite the angle, ABC, that we seek. Now since from the hypothesis the two lines AB and AC are expressed in the same

terms

square of the third side is found. Therefore

through the old measure, and furthermore

the side itself is found by Th. 2 above. Q.E.D.

since AB, the radius of the described circle,

The mechanics: If the two given sides in¬

is known by the new measure — that of one

clude the right angle, then square the sides

of the parts of the whole sine — then by the

and add the squares. The square root of this

teaching of Th. 7 above, AC will be known

sum is the length of the sought side. However,

by that same measure. Since AB is

• 65 •

DE

TRIANGVUS

LIB.

Anus totusuelfinus quadrantis,erit a c finus notus,8C per tabula finus,qua negleda.hoc in propfito nihil effice re poflamus,arcu eius addifcemus, cognito aute arcudi $ifinus,dat&angulus querefpicit arais ille,naarcus ipie & angulus fecundu eunde numera mefurant,queadmodu tota circuferetia & qtuor redi anguli fecundu ain de numera coi ter mefuratur,c)d in z i huius comemoraui mus.per zzitaqjhuius reliquu acutum angulum b a c cognofcemus.Redum autem angulum acb,u huius notum demonftrabat.Vniuerfbs igitur ti^anguli noftri angulos reddidimus notos, quod decuit explanare. V Opus.Numera lateris redum fubtendentis angulum conftitue primum,& numerum lateris refpicientis angu lum quae fi'tum pro fecundo ponas,numera uero finus totius tertium. Multiplica igitur fecundum per tertium,& produdum diuide per primum,exibit enim finus arcus refpicientis angulum quariitum,cui per tabula finus araim fuum elicias,cu* ius etia numeras angulum quariitum manifeftabit.hunc fi ex anguli redi quanti* tatedempferis,relidumnumerabisfecundumanguluacutum.Vtfi a b fuerit zo a c iz,& b c i 6, finus autem totus quemadmodu in tabula noftra fuppofiiimus 60000,multiplicabo izper 60000,producuntur7zoooo,quaediuidoper zo,exe* unt 3 6000,huius finus arcum tabula praebet gradus 3 6,SC minuta j-z fere. tantu igiturpronunciabimus angulum a b c,qui tandem fublatus ex s>o,relinquet 5-3 gradus dC 8 minuta fere,& tantus habebitur angulus reliquus acutus, XXVIII.

Data proportione duorum laterum trianguli redtanguli, angulos eius percontari. Aut enim alteram duoru laterum opponitur redo angulo,aut non. Si primu fitlatus a b redo angulo a c b oppofi tum,cuius proportio ad latus a c fit no* ta,Dico,qj anguli huius trianguli inno telcent.Eft: enim a c finusarcusangulia b c per huius,dum a b eftiemidiameter circuli fcilicet finus totus, proportio era go finus totius ad finum anguli a b c nota eft, hinc finus ille notificabitur, & tan de angulus a b c non Iatebit.Si uero proportio duorum laterum bc&ac data fuerit ,erit proportio quadratora notum data,& coniundim proportio aggregatiexquadrato b c cum quadrato a c , hoc eft quadrati a b propter angulum c redum ad quadratum a c nota crit;unde Si linearam proportio non ignorabit, reliquant ante, V Operatio.Si alterum duorum laterum redo angulo oppo* natur,multiplica terminu minorem proportionis datae per finum totum, & pro* dudum diuide per terminu maiorem,exibit enim finus anguli,cuius latus breuius opponitur.Si uero duorum laterum redo circumftantiu data fuerit proportio, duc utrunqj terminoru in fe,£C colledi ex produdis radice accipe quadrata, ipfa enim erit terminus lateri,quod redum fubtendit angulum accomodandus , per* duceris ergo ad iter priftinu.Vt fi proportio a b ad a c fuerit ficut 9 ad 7. multi plico 7 in finum redum totum 60000,fiunt 4Z0000 .qux diuido per 9, exeunt 46667 fere.arcus autem refpondens huic finui redo eft gr.5- 1. minuta 3 fere, 8C tantus habebitur angulus a b c .Sed ponatur proportio a c ad c b ficut iz ad j-.duco i zinfe,fiunt 144.1'tem 5- in fe,reddunt zj\haxconiungo,faciunt i65>,ho rum radix eft 13, attribuenda lateri a b,ficproportio ab ad a c erit ut 13 ad 1 z.

D

• 66 •

unde

Book I of On Triangles the whole sine, or [in other words] the sine of the quadrant, sine AC therefore becomes

One of the two sides is opposite the right angle or else none is. First, if side AB, whose

known. Through the table of sines, by whose

ratio to side AC is known, is opposite right

neglect we can accomplish nothing in this

Z ACB, then the angles of this triangle be¬ come known.

theorem, we also ascertain the arc. Moreover, when

for this sine is known, the

AC is the sine of the arc of z ABC by Th.

angle opposite this arc is also given, for the

the arc

20 above, provided that AB is the radius —

arc itself and the angle are measured accord¬

that is, the whole sine — of the circle. There¬

ing to the same number, just as the total

fore, the ratio of the whole sine to the sine

circumference and the four right angles are

of z ABC is known. Hence the latter sine can

conjointly measured according to the same

be found, and finally z ABC will be known.

number, as was mentioned in Th. 21 above.

However, if the ratio of the two sides BC

Thus we know the other acute z BAC by

and AC is given,

Th. 22 above. Th. 21 above showed the right

squares will be known. And by addition the

Z ABC to be known. Therefore all the angles

ratio of the sum of the square of BC plus the

of our triangle have been found. Q.E.D.

square of AC — that sum being the square of

The mechanics: Take the value of the side

then

the ratio of

their

subtending the right angle as the first num¬

AB because of right z C — to the square of AC will be known. Hence the ratio of the

ber, and take the value of the side opposite

lines is known. And what remains is as before.

the desired

angle for the second number,

The mechanics: If one of the two sides is

while the value of the whole sine is the third

opposite the right angle, multiply the smaller

number. Then multiply the second by the

term of the given ratio by the whole sine and

third and divide the product by the first, for

divide the product by the larger term. This

the sine of the arc opposite the desired angle

will yield the sine of the angle that is opposite

will result. From the table of sines you may

the shorter side. However, if the ratio of the

determine that arc, whose value equals the

two sides which include the right angle is

desired

angle. If you subtract this [angle]

given, square each of the terms and take the

from the value of a right angle, the number

square root of the sum of the squares. This

angle.

[square root] will be the appropriate term

For example, if AB is 20, AC 12, and BC 16

for the side which subtends the right angle.

that

remains

is

the

second

acute

and the whole sine found in our table is

Then continue in the previous way. For ex¬

60000,

ample, if the ratio of AB to AC is as 9 to 7,

then

multiply

60000 by

12

to get

720000, which, divided by 20, leaves 36000.

multiply 7 by 60000, the whole right sine, to

For this sine the table gives an arc of approxi¬

get 420000. This, divided by 9, yields about

mately 36° 52 min. This amount is z ABC,

46667. The arc corresponding to this right

which, subtracted from 90, finally yields about 53° 8 min., the size of the other acute angle.

sine is about 51° 3 min., and that amount is Z ABC. But if the ratio of AC to CB is given as 12 to 5, then square 12 to get 144, and similarly square 5 to find 25. These, when

Theorem 28

added, make 169, of which the root is 13, this When the ratio of two sides of a right triangle is given, its angles can be ascertained.

being assigned to side AB. Thus the ratio of AB to AC will be as that of 13 to 12.

• 67 •

l6

io K*

DE

MONTBREGIO

unde ut prius angulo a b c cognolccndo uia parati eft# XXIX#

Afrerc^uorum acutorum angulorum, quos habet triangulus rc« Angulus,cognito, cum uno eius latere dt angulos cunftos & latera metiemur#

„

f f_ .

Trianguli a b c angulum c redum habetis ,angU lus b fit cognitus cu latere uno quocuq$(ucrbi gratia)a c# Dico,q> omneseius anguli cum lateribus omnibus inno tcfcent, Anglii profedo cognoicentur ex z»Si ^ buius. reftat igitur inuenirelatera.Per zo autem huius Si tabtu lam finus hypothefi iuuante,erit utruncp laterum a c Si b c cognitum,ut a b eft finus totus,duo itacp latera quae libet trianguli propofid datam inuicem habebunt ,ppor* c~ tionem,arnq? ex hypothefi unu eorum datum fit per me* lura nouam,enmt per 6 aut 7 huius reliqua data,quod li¬ buit attingere, V Opus pulchru & perutile.Sinum arcus anguli dati,Sieius co plementi addiicas,habebisc^ tria latera nota per menfuram ueterem,quae eft pars una finus totius ,nam latus redum fubtendens angulum eft finus totus. Si igitur latus,quod redum (ubtendit angulum,fuerit datum per menfuram nouam, pone finum totum pro primo,Si finum arcus angui i, cui opponitur latus quajfitum,pro lecundo,numeru autem nouae dationis tertiu.muldplicatoq? fecundo per tertium produdum diuide per primum,Si exibit numerus lateris qutefiti.Si uero altenim duorum laterum redo fiibiacentium derur,uolendo menlurare latus redum lub* tendes angulum,pone finum arcus anguli,cui oppommr ipfum lanis datum pro primo,Setinum totum pro fecundo,numerum autem dationis noua: tertium,abs lolutofcp opere uulgari qua tuor numero ru proportionaliu ad metam perduceris cupitam. Quod fi reliquu latus redo lubftratu angulo inueftigaueris, pone finu arcusangulfcui opponitur,latus datum pro primo,Si finum complementi eius ro lecundo,numeru uero dationis nous terthim,reliqua ut antehac exeaiturus* iexemplo,Detur angulus a b c 36 graduu,Si latus a b zo pedu,(ubtraho 3 61 90,manebunt 74 gradus,qui determinant quantitatem anguli b a c.inuenioau tem lineam a c 35-267 ex tabula finus,b c uero 485-41,dum a b eft finus totus 60 000,Multiplico igitur 35-267 per zo, producuntur 7o5 34o,qusediutfa per 6oooo,eliciut 11 |£fere.Iatus itaqi a c habebit pedes 11 Sif£,idefttres quartas pedis unius,Similiter multiplico 485-41 per 20,producuntur 970S zo,quat diui latus trianguli a b c potentialiter fefquitertiu fit ppendiailari a d .ipfa enim perpendiailaris balim in aequas partitur fe d tones, pertult ima primi & comunibus animi conceptio nibus id concludentibus.b c igitur aut a c fibi aequalis, dupla eft ad lineam d c,quare per quarta fecundi, aut /8 fexti quadratu a c quadruplu erit quadrato d c.penultima aute primi quadratu a C duobus quadratis linearunt a d & d c aequiualere docuit.Duo igitur huiufi modi quadrata coniuda quadruplu efficiunt quadrato d c,habent itaq? propor tionem ad quadratu d c hCut4ad i .quare diuifim ,p portio quadrati a d ad qua dratum d c eritficut 3 ad 1 ,tripla uidelicet.erat autem quadratu a c ad quadra tum d c quadmpla.maioris itacp hanim pportionudenominator eft 4 ,minoris ueto 3 ,quare per corollariu 1 z huius quadrati a c ad quadratu a d ficut4 ad 3 (efquitertia fcilicet concludetur.potentia igitur lateris a c,quauocant quadrati* eius potentia: perpendicularis a d fefquitertia conuincitur,quod intendebat pro nofitio.Quod autem Corollariu huius pollicebatur,fexta 8C prima huius enitent. Nam facta huius ex ^portione iam data quadrati a c ,per primam huius noti ad quadratu a d,ipfiint quadratu a d fiifdtabit notum,cuius demum cofta ppendi cularis fdlicet a d per fecudam huius emerget cogru'ta,fimiliter ex a d perpen* diculari data latus a c notum expIicabirmis.Poteris etiam, fi libeat,ex latere a c D

•

74

•

3

noto

Book I of On Triangles D and £ is a right angle by the definition of a right angle, and therefore by [Euclid] 1.32

is four-thirds of the [second] power of perpen¬ dicular AD.

the other two angles are equal. Then by [Euclid] VI.4 the ratio of side AC to side

CB will be as that of perpendicular AD to perpendicular BE. Now the hypothesis gave the

first

three

quantities. fourth

of

Hence,

these by

four proportional

Th.

19

above,

the

[term] — perpendicular BE — can

be

found. We will determine the value of the other perpendicular no differently. Q.E.D. From this we may draw a general conclu¬ sion that any two perpendiculars have the same ratio as that of the adjacent sides form¬ ing the angle opposite these perpendiculars. The mechanics are the same as those for Th.

Now the penultimate theorem of [Euclid] I and the axioms show that the perpendicular divides the base into two equal parts. There fore BC, or its equal AC, is twice line DC. I hus by [Euclid] II.4 or VI.18 the square of

AC is four times the square of DC. Further¬ more, the penultimate [theorem] of [Euclid] I shows the square of AC to be equal to the square of line AD plus the square of line

DC. A herefore the sum of these two squares is four times the square of DC. Hence the ratio of their sum to the square of DC is as that of 4 to 1. Then by division, the ratio of the square of AD to the square of DC is as

19 above.

that of 3 to 1 — namely, threefold. Moreover,

Theorem 33

the square of AC was fourfold the square of

I he three angles of every equilateral tri¬

DC. Then the designator of this larger ratio

angle can be proven known; whence it is

is four while that of the smaller [ratio] is

agreed that any one of [the angles] is acute.

three. Therefore, by the corollary of Th. 12

Every equilateral triangle has three equal angles by [Euclid] 1.5, and by [Euclid] 1.32 they are equivalent to two right angles. Hence, any one of them will be a third of two right angles. We already know the sum of two right angles from Ths. 21 and 3 above. Therefore, by Th. 6 above, any one of them can be found.

above, [the ratio of] the square of AC to the square of AD is as that of 4 to 3 — namely, four-thirds. Hence the [second] power (which [is another name for] the square) of side AC is proven to be four-thirds of the [second] power of its perpendicular AD, as the propo¬ sition indicated.

Q.E.D. Moreover, since the sum of two right angles is thrice any one of these [angles] but twice one right angle, each of these [three angles] will be less than a right angle by [Euclid] V.10, and thus by definition [each will be] an

What the corollary of this promised Ths. 6 and 1 above will show. For since the ratio of the square of AC, which is known by Th. 1 above, to the square of AD is already found [to be %], the square of AD itself will be known by Th. 6 above. Then its root — name¬

acute [angle]. This is offered as a corollary.

ly,

the

perpendicular

AD — will

become

known by Th. 2 above. Similarly when the

Theorem 34

perpendicular AD is given, side AC may be

In every equilateral triangle the [second]

found. Aho, if it is desired, since side AC

power of a side is four-thirds of the [second] *The reference numbers to theorems subse¬ quent to Th. 25 should be one number greater, e.g., 28 should read 29, 34 should read 35, etc. The frequent occurrence of this error might sug¬ gest that Regiomontanus inserted a theorem, possibly Th. 26, belatedly. Henceforth, when necessary, the correct theorem number will ap¬ pear in brackets; however, in several places the numbers had already been corrected prior to publication of the Latin text.

power of its perpendicular. Hence, if the side is known, the perpendicular can be found, and vice versa. In equilateral A ABC, drop a perpendic¬ ular AD from / A to base BC. By the pre¬ ceding theorem and Th. 3o[3i]* above, this [perpendicular]

falls

within

the

triangle.

Then the [second] power of a side of A ABC

•

75

•

I OH*

$o

DE

MONTEREGIO

notoperhypothefim corolhrij pradentis,& angulo a c b per praecedentem co¬ gnito 28 hu/us dirigente perpendiculare a d menfurare,& uiceueria ex perpen¬ diculari datalatusredderecognitum. VOperatio.Dimklinlatusnotum infe dudlum triplibiSjtripJatfqj radicem extrahes quadrata,ut libet ,ppinquifEme, •qua aflcribcs perpendiculari a d, Autex quadrato lateris cogniti quarta fin par¬ tem demas,& relidi radicem elicias quadrata ,ppinqua,ut placeqquae ppendicu* [arem a d norificabit.InexernpIo.Sitlatus a c 12 A'medietas eius 6.quarduda in fe reddunt j6,hxc triplata faciunt i o 8 .quoru radix quadrata ^ppinqua eft 10 7§^,tantam igitur feri ppendiculare a d praedicabis,Aut multiplicatis 12. in fe fiunt 144.quomm quarta pars 3 ^.dempta ex ipfis 144.relinquet 108 «quadratu uidelicet perpendicularis a d,cum quo ut antehac operabimur.Qd* fi ex perpen¬ diculari data libeat elicere Iatus,ppendicularem in fe multiplica, produdocp tertiam fiu partem adqcias3& refill tabit numerus quadrato latens de putandus, cuius radix ppinqua latus ipfiim manifeftabit. Vt fi perpendicularis fuerit 6. multipli co 6 in fe,fi‘unt 3 reliqua duo latera nota fient cum perpendiculari* ~ bus.Detur enim primo alterum duorn latera 8Cfit a c «eminaq?perpendiculari a d adbafim b c,erit trian¬ gulus a d c redangulus.aiiusangulus c aaitus exeo ro lano 3 4.I1UIUS notus habebitur,fiue per hypothefim folam fiue per hypothefim & 3^.huius,quare per 28 hu .Ia?5 3 ° noto ex