By Prof. Dr. Bartel Leenert van der Waerden (auth.)

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**Additional resources for A History of Algebra: From al-Khwārizmī to Emmy Noether**

**Example text**

Warum haben die Griechen die Existenz der vierten Proportionale angenommen? Quellen und Studien Gesch. Math. B 2, p. 369-387. Omar Khayyam also raises the quest ion whether ratios can be regarded as a kind of "number" in a larger sense. He writes: Then there is the question about the ratio of the magnitudes: is it inherent the number according to her nature, or a logical consequence of the number, or is it connected with the number by something that follows from its nature without the need of any extemal factor?

Descartes formulated Tabit's rule explicitely and presented a third example: 9363584=2 7 x 191 x 383 9437056=2 7 x 73727 (Rene Descartes, Oeuvres II, p. 93-94 and p. 148). Now the question arises: How did Tabit find his rule? The weIl known pair 220 and 284 has a factorization of the form in which p, q, and rare primes. So let us see whether we can find a pair such that M is the sum of the proper divisors of N and conversely. I suppose that Tabit knew that the sum of all divisors of N (including N itself) is (1+2+ ...

Clagett, Archimedes in the Middle Ages I, p. 224 and 658-660. 141818 .... For the surveyor who does not understand the Ptolemaic proeedure of determining halfehords from given ares, appropriate instruetions and a table of chords are provided. This is the only plaee where the term sinus versus arcus, certainly borrowed from Arabie trigonometry, appears. The fourth chapter is devoted to the division of surfaees; it is a reworking of the Liber embadorum, whieh ultimately derives from Euclid's lost Book on Divisions of Figures; the latter ean be reconstructed (see Arehibald) from the texts of Plato of Tivoli and of Leonardo and from that of an Arabie version.