By Prof. Dr. Bartel Leenert van der Waerden (auth.)
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Additional info for A History of Algebra: From al-Khwārizmī to Emmy Noether
Omar first constructs a square c 2 equal to the given number b, and next a block with base c 2 and height h equal to the given number b. This means, as he has explained earlier, that the block with sides c, c, and h is made equal to a block with sides e, e, and be, where e is the unity of length and b be the given number on the right hand side of equation (5). Thus, the equation (5) can be written in the homogeneous form (11) in which c=AB and h=BC are given line segments. 9) having its summit at B, its axis being BZ and its "parameter" AB=c.
My "Science Awakening" I, p. 98). Tabit ben Qurra has written a "Book on the Determination of Amicable Numbers". He proved: If p=3·2"-1_1 and q=3·2"-1 and r=9·2 2 "-1_1 are prime, then M=2"pq and N=2 n r are amicable numbers. Tabit's book has been commented upon and partly translated by F. Woepcke: Notice sur une theorie ajoutee par Thäbit ben Korra a l'arithmetique speculative des Grecs, Journal asiatique (4) 20, p. 420-429 (1852). Tabit's rule for obtaining amicable pairs was rediscovered by Pierre de Fermat and Rene Descartes.
Juschkewitseh has drawn the attention to two very remarkable passages on ratios in Omar Khayyam's eommentary "Discussion of Difficulties of Euclid" (edited by Erani, Teheran 1936). Mrs. Yvonne Dold had the kindness to translate the two passages directIy from Arabie into English. In what folIows, I shall reproduee her translation. Chapter 1. Three MusliInic Authors 30 As luschkewitsch notes, Omar Khayyam states that Euclid's definition of proportion is correct, but that it is not a true definition of the notion ratio.
A History of Algebra: From al-Khwārizmī to Emmy Noether by Prof. Dr. Bartel Leenert van der Waerden (auth.)