De triangulis planis et sphaericis libri quinque : una cum tabulis sinuum, in quibus tota ipsorum triangulorum scientia ex primis fundamentis geometricarum ... continentur, quam multiplicem usum haec triangulorum doctrina omnibus ... adferat ... qui sana rerum intelligentia sunt instructi, in sequenti opere, quod complectitur ordinata astronomicorum et geometricorum problematum descriptionem (autore D. Santbech) ... deprehendere poterunt. (Tractatus Georgii Peurbachii super propositiones Ptolomaei de sinubus et chrodis) omnia nunc simul in lucem edita in gratiam matheseos studiosorum per Danielem Santbech.

Basel: [colophon:] Heinrich Petri & Peter Perna, 1561.

Second, much augmented, edition of “the first systematic treatise on plane and spherical trigonometry to be published in Europe” (Rose, p. 99), completed by 1464 but first published in 1533 at Nuremberg by Johann Schöner. “Regiomontanus’s monumental work on Triangles, the first publication of which was delayed until 12 August 1533, attracted many important readers and thereby exerted an enormous influence on the later development of trigonometry because it was the first printed systematization of that subject as a branch of mathematics independent of astronomy” (DSB XI: 350). “Although it drew heavily on Arabic sources, those earlier treatises had been either lost or forgotten by 1533 when Regiomontanus’s work was first printed. Among the notable contents of this work are the sine law and perhaps the first European application of algebra to trigonometry. [Another notable contribution is theorem 2 in book V, which contains the earliest statement of the cosine law for spherical triangles.] Indeed with De triangulis trigonometry was established as an independent discipline. Regiomontanus' original purpose, however, had been to furnish astronomers with a mathematical technique essential for their studies, and in this De triangulis had a success perhaps greater than its author could have dreamed of. For in 1539 Georg Joachim Rheticus presented a copy of the work’s 1533 edition as a gift to Copernicus. The great astronomer had already written the trigonometrically-based portion of his De revolutionibus without knowledge of his predecessor’s treatise. After reading the new book, Copernicus modified the presentation of several of his own indispensable theorems by inserting two leaves in the manuscript of the De revolutionibus. Hence, Rheticus’s remark that Regiomontanus began the reconstruction of astronomy that Copernicus completed takes on a fuller meaning” (ibid., pp. 99-100). In this edition, the main text of De triangulis is essentially unchanged, except that an appendix in the first edition in which Regiomontanus proves the errors of Nicolaus de Cusa’s theory of squaring the circle has been omitted and replaced by a letter of Schöner’s dated 1541. It is augmented by two allied treatises, first published in 1541: Tabula sinuum ad 6000000 partes per I. de Regiomonte computata, Regiomontanus’s tables of sines and chords based on a radius of 6,000,000 – only an abbreviated table with r = 60,000 was included in the 1533 edition (this means that the tables were of r sin α, for example, rather than of sin α, a device that was intended to avoid the use of fractions); and Tractatus super propositiones Ptolemaei de sinubus et chordis by Peurbach. It is also immeasurably improved by the addition of Santbech’ treatise, Problematum astronomicorum et geometricorum sectiones septem, appearing here for the first time, which contains a wealth of information about astronomy in the first years after Copernicus, occasionally citing him (e.g. pp. 46, 52). As well as astronomy, the work treats sundials, surveying, levelling for water-courses, and the subject of gunnery and ballistics, both theoretically as well as for the practical application of war. The trajectories of cannonballs “were depicted with abruptly acute angles and straight lines, allowing him to create a right-angled triangle from which ranges were computed with the help of a table of sines. Santbech was of course fully aware that a cannonball’s true trajectory would not consist of a straight line and a sudden drop, but these depictions were meant to assist with mathematical calculations” (Bennet & Johnston, pp. 46-47). Editions of De triangulis dated 1541 and 1546 are mentioned by some sources, but we have been unable to locate any copies of editions with these dates and believe they are ghosts.

Johannes Müller (1436-76), named Regiomontanus (‘king’s mountain’) after his birthplace Königsberg, enrolled at the University of Vienna on 14 April 1450 and was awarded the bachelor’s degree on 16 January 1452 at the age of 15; but because of the regulations of the university, he could not receive the master’s degree until he was 21. On 11 November 1457 he was appointed to the faculty, thereby becoming a colleague of Georg Peuerbach (1423-61), with whom he had studied astronomy. The two men worked closely together as observers of the heavens. On 5 May 1460, Cardinal Bessarion, the papal legate to the Holy Roman Empire, arrived in Vienna. He met both Peurbach and Regiomontanus, and persuaded Peurbach to undertake a new translation of Ptolemy’s Almagest from the Greek. Further plans were made for Peurbach and Regiomontanus to accompany Bessarion to Rome and to work with him there, using Bessarion’s Greek manuscripts as the basis of the new translation. Peurbach, however, had completed only the first six books of the abridgment when he died, not yet 38 years old. On his deathbed he made Regiomontanus promise to complete the work, which the latter did in Italy during the next year or two. The completed work was dedicated to Bessarion by Regiomontanus, probably in 1463, but was not published until 1496, as the Epitome in Almagestum Ptolemaei.

“At the end of his dedication to the Epitome, Regiomontanus referred to a book he planned to write on triangles. In putting the Epitome together it became clear that a survey of all rules on triangles would be useful to readers of the Epitome. Peuerbach had thought of composing such a trigonometry, but this, like the completion of the Epitome, was curtailed by his untimely death. Regiomontanus proceeded to finish the Epitome and then to write a book on triangles. This book is a landmark of the new era, and attracted more and more attention as time went by. Unfortunately, it was never finished: Book V, with its application of the Law of Cosines, presented here for the first time, is just a fragment. This incomplete work was published by Schöner in 1533. Regiomontanus had thought of publishing it himself but his untimely death prevented this. The book was dedicated to Bessarion.

“He first completed the spherical trigonometry, which later formed Books III and IV of the whole work. As in the Epitome, the De triangulis contains the Law of Sines, which relates the sides and angles of a spherical triangle. Regiomontanus left both of these books behind in Rome when he went to Venice in the cardinal’s court. As he wrote in a letter to Bianchini in late 1463, he wanted to send for the books from Rome and relay them on to him. Presumably he soon got the books and began to expand his work. He wrote a treatment of planar triangles, calling this Book II, although it appears as Book I of De triangulis, and the previously finished material on spherical triangles became Books II and III.

“Regiomontanus wrote two books on plane triangles. The first treated proportions and plane triangles in general. He then taught the solution of right triangles by using his table of sines. This table, called his own in the dedication, was calculated for a radius of r = 6,000,000, at intervals of one minute; seconds were found by means of an auxiliary table. For this he wrote a treatise which the handwriting dates at about 1462 and which was published by Schöner in 1541. [This is the work by Regiomontanus that was added to the 1561 edition of De triangulis.] However, only an abbreviated table with r = 60,000 was used in De triangulis

“The second book contains the Law of Sines and its application to the solution of plane triangles. Paragraphs 12 and 23 are of interest, in which the sides of a triangle are found by using quadratic equations. Paragraph 26 also deserves comment: it contains the exercise ‘to find the angle opposite the base of a triangle, given the area and the product of the two including sides.’ This is an implicit formula for the area, although none is given explicitly. The spherical Law of Sines plays the same role for spherical triangles as for plane triangles. Regiomontanus gives two theorems that hold only for right spherical triangles,

sin α cos b = cos β and cos a cos b = cos c

[where α is the angle opposite the side of length a, etc., and the right angle is opposite the side of length c], as well as the theorem that for all spherical triangles, the three angles determine the three sides, and conversely. His trigonometry is based on these three theorems.

“In Book V he presents the sinus versus (versin α = 1 – cos α) and the spherical Law of Cosines. And so this remarkable book ends; despite its rough spots (often found in incomplete works), it became crucial for European mathematics. According to Braunmühl,

‘His De triangulis had the potential of being developed and completed in many different ways; this potential was realized by numerous scholars at later times, all of whom followed his ideas. Thus, the entire subsequent development of trigonometry in the West showed the influence of his work’” (Zinner, pp. 55-56).

“Peurbach wrote a short work on the computation of sines and chords, Tractatus super propositiones Ptolemaei de sinubus et chordis [the second work appended to the 1561 edition of De triangulis]; the work was twice printed (Nuremberg, 1541; Basel, 1561) along with Regiomontanus’s Compositio tabularum sinuum rectorum and sine tables [the first appended work]. He first explains the computation using kardagas (arcs of 15°) according, he says, to the method of al-Zarqali [fl. 11th century], and then, at somewhat greater length, sets out Ptolemy’s derivation from the first book of the Almagest” (DSB).

This 1561 edition of De triangulis was edited by the Dutch mathematician and astronomer Daniel Santbech, about whom little is known. He is also called ‘Noviomagus’, which may indicate that he was from Nijmegen. Together with the works of Peurbach and Regiomontanus he included his own treatise, Problematum astronomicorum et geometricorum sectiones septem, which is published here for the first time. “The book, despite its treatment of up-to-date topics like cannon shots, is much indebted to Ptolemy, beginning with a long philosophical introduction … Santbech’s book couldn’t just dismiss practical questions as Ptolemy’s did, though, because his book was necessarily practical … after the philosophical introduction Santbech’s first topic is how to build a quadrant for sighting and measuring angles … Santbech’s suggestions for things you could see, measure, and compute for yourself included practical astronomical examples, like telling time by stars, but also deeply philosophical matters like the discrepancies in the calendar that had accumulated since Ptolemy’s time. He was writing after Copernicus but before the Gregorian calendar reform, so these discrepancies were very lively questions. He refers quite impartially to Ptolemy and Copernicus and does not at all consider them to be in conflict. Copernicus had tried to incorporate some modern observations into his system, and that was a recommendation in his favour, but for many purposes his system was just an alternative digest of the same data that were already summarized in Ptolemy’s scheme, and Ptolemy’s theory was readily available in Regiomontanus’s Epitome, so for theoretical questions Santbech refers to Regiomontanus. Tycho Brahe, as a young student at the protestant University of Wittenberg, would certainly have seen Santbech’s book. It would have contributed to his conviction that astronomy was not on a sound foundation and that it needed a systematic modern data set, and it would have helped him to think how best to make new observations.

“The earthly sections of Santbech’s book tell you such things as how to determine the distances and heights of towers and, what is truly eye-catching in the charming woodcut illustrations [e.g., p. 293], how to drop cannonballs onto them. Santbech anticipates a lot of criticism from people who will say that he is ‘mixing heaven and earth’ in treating this topic in a mathematical book, but he aggressively insists that he will do it anyway, without even making much of a philosophical argument about it …

“The main virtue of Santbech’s new theory of cannon shots, from his point of view, is apparently that it uses trigonometry … Here is the theory: you should draw a straight line extending the direction of the cannon barrel as far as it takes for the cannonball to lose its ‘impetus’, from which point the cannonball then drops straight down. Its trajectory is thus two legs of a triangle, a hypotenuse going up, followed by a vertical drop going down. The goal of the cannoneer is to get the cannonball to drop on the tower. Santbech knows that cannonballs follow curved trajectories, but he still argues that although cannonballs don’t actually follow the straight hypotenuse of his constructed triangle, falling rather a bit below the hypotenuse, still, they stay very close to it, so that the geometrical theory is correct for the practical purpose of dropping the ball onto the tower” (Peterson, pp. 243-247).

Adams R-281; see Stillwell, Awakening 218; Cockle, Military Books, p. 23; see PMM 40; DSB 11: 348-52 & 15:478. Benner & Johnston, The geometry of war, 1500-1750: catalogue of the exhibition, 1996. Peterson, Galileo's Muse: Renaissance Mathematics and the Arts, 2011. Rose, The Italian Renaissance of Mathematics, 1975. Zinner, Regiomontanus: His Life and Work, 1990.



Folio, pp. [xvi], 146, [38]; [20], 294, [2], with engraved historiated initials and numerous woodcut diagrams in text. Contemporary vellum (hinge of title and final leaf mended, some light occasional browning).

Item #2852

Price: $13,500.00