Friedrich Schur' 1899 Analytic Geometry Book

Ching-Kuang Shene (冼鏡光), Professor Emeritus
Department of Computer Science
Michigan Technological University
Houghton, MI 49931
USA
Created September 17, 2025

We saw the similarity between the diagrams in the proofs of the angle different and angle sum identities for the sine and cosine functions and coordinate axes rotation. Because coordinate axes rotation is not a new topic in coordinate/analytic geometry, searching for old to very old textbooks would provide us with interesting insights. The first analytic geometry textbook I found in German was the one by Friedrich Schur, Lehrbuch Der Analytischen Geometrie in 1899. Actually, I found this book when I was a graduate student at the Johns Hopkins Univertiy doing geometric modeling; however, earlier version may be available. The left diagram below shows the cover of my 1912 edition and the right diagram has the page 22 on which it is clearly shown that the Pythagorean Idxentity is proved using the angle sum identity for the sine function. Note that this same proof was also mentioned in Versluys' 1914 book.

This page (i.e., p. 22) appears in the section of Drehung der Koordinatenachsen (i.e., rotation of the coordinate axes). Therefore, a trigonometric proof of the Pythagorean Identity appeared in a 1912 textbook on analytic geometry. It is important to note that Schur's book was published long before Loomis' book and, of course, long before Zimba's 2009 paper. Please continue to Versluys' 1914 Book because Versluys cited Schur's book, or Coordinate Rotation for more information.

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