First Edition of the First Printed Systematic Work on Trigonometry and an Influence on Copernicus's Work "De Revolutionibus"
REGIOMONTANUS, Johannes. Triangulis omnímodis libri quinque. quibus explicantur res necessariae cognitu, uolentibus ad scientiarum astronomicarum perfectionem deuenire: quae cum nusqua[m] alibi hoc tempore expositae habeantur, frustra sine harum instructione ad illam quisquam aspirarit : accesserunt huc in calce plera[que] D. Nicolai Cusani De quadratura circuli, De[que] recti ac curui commensuratione: item[que] Io. de monte Regio eadem de re elenktika hactenus à nemine publicata : omnia recens in lucem edita, fide & diligentia singulari. Nuremberg: Johann Petreius , 1533.
First edition. Completed in 1464 and published for the first time posthumously in 1533. Two parts in one folio volume (11 1/2 x 7 3/4; 293 x 195 mm). 137, [3, blank], 93, [1, errata], [2, blank] pp. With the Blank at R6 and a blank at l4, but not clear if l4 is the original blank. With numerous woodcut diagrams and a woodcut vignette on title-page.

Full seventeenth or eighteen-century speckled calf. Boards ruled in blind with gilt corner devices and central gilt lozenge. Spine stamped and lettered in gilt. Board edges tooled in gilt. Some professional restoration to outer joints. Newer endpapers. Internally very clean. Overall a very good to about fine copy.

Regiomontanus (Johann Müller of Königsberg, 1436-1476), a pupil and later colleague of Georg von Peuerbach, is one of the great figures in the history of mathematics and astronomy.

Although Islamic mathematical writings had treated trigonometry as a separate branch of mathematics as early as the thirteenth century, Regiomontanus's De Triangulis omnímodi (on all classes of triangles) was the first work published in Latin-speaking Europe to divorce trigonometry from astronomy. Regiomontanus had recognized the need for such a work while preparing the epitome of Ptolemy's Almagest, and set about writing De Triangulis shortly after completing the Epitome. De Triangulis was completed in 1464, but remained in manuscript for nearly seventy years. The work contains the first formulation of the cosine law (although Regiomontanus used the versed sine rather than the cosine itself), states the proportionality of the sides of a plane triangle to the sines of the opposite angle (the sine law), and solves, for the first time in the non-Islamic west. a trigonometric problem by means of algebra." (Norman Library of Science and Medicine, 1556).

"In 1539 Georg Joachim Rheticus presented a copy of the [De Triangulis'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' remark that Regiomontanus began the reconstruction of astronomy that Copernicus completed takes on a fuller meaning" (Rose, The Italian Renaissance of Mathematics, pp. 99-100).

"It contains the earliest statement of the cosine law for spherical triangles, stating the proportionality of the sides of a plane triangle to the sines of the opposite angle. This fundamental proposition of spherical trigonometry appears as theorem 2 in book V of the treatise. In the second part, Regiomontanus proves the errors of Nicolaus de Cusa's theory of squaring the circle... In 1594, Thomas Blundeville would reprint the tables, together with explanations for their use, and thus give Englishmen the first complete canon of trigonometrical functions printed in England. The work of Regiomontanus and Peurbach therefore constitutes one of the monumental breakthroughs in the practice of navigation." (Christies)

Norman Library 1556. Stillwell, The Awakening Interest in Science, 218.

HBS # 68291 $22,500