REGIOMONTANUS (Johann MÜLLER 1436-1476). De triangulis omnimodis libri quinque... accesserunt huc in calce pleraque D. Nicolai Cusani de Quadratura circuli. Edited by J. Schner. Nuremberg: Johann Petri, 1533.
Two parts in one volume, 2o (297 x 194 mm). Woodcut diagram on main title, numerous woodcut diagrams in text, with blank R6. (Lacks blank l4, F1 and F4 with marginal repairs, some staining at beginning and end, light browning to text.) Modern half vellum. Provenance: Robert Honeyman (his sale part IV, Sotheby's New York, 20 May 1981, lot 3299).
FIRST EDITION OF THE FIRST PRINTED SYSTEMATIC WORK ON TRIGONOMETRY. De triangulis was Regiomontanus's most important scientific contribution. Completed in 1464, De triangulis remained in manuscript for nearly seventy years before being published in this edition, edited by J. Schner (1477-1547). 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.
"When, in the middle of the fifteenth century [Georg] Peurbach and his pupil Regiomontanus, Professors of Mathematics at Vienna University, prepared the first modern sine tables, they kept the 360o division of the circle used by the Greeks but, in order to avoid awkward fractions, they divided the diameter much more minutely, into 20,000 parts... These tables, being printed in folio-size books, were essentially works for the scholar's desk and astronomer's observatory..." (D.W. Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times, New Haven, 1958, p.353). 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. Adams R-280; BM/STC German, p. 631; Norman 1556; Stillwell Science 218.