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REGIOMONTANUS ON TRIANGLES
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“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 •
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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