By Florian Cajori

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**Example text**

Hardly any general rules of procedure. writer treats in succession several problems of the same kind. From them rules, it but this would be easy, by induction, to obtain general not done. When we remember that only one is hundred years ago it was the practice of many English arith metical writers to postpone the discussion of fractions to the end of their books, it is surprising to find that this hand-book of 4000 years ago begins with exercises in fractions, and pays but little attention to whole numbers.

For a German translation, see G. WEETHEIM, Die Arithmetik und die Schrift uber PolygonalzahUn des Diophantus von Alexandria, Leipzig, 1890, pp. 330-344. 2 WERTHEIM, op. , p. 337. Whether it originated at the time of Archimedes or later is discussed by T. L. HEATH, Diophantos of Alexandria, Cambridge, 1885, pp. 1428 147. A HISTORY 34 Find the number of bulls OF MATHEMATICS and cows. high numbers, but to add to its This leads to excessively complexity, a second series of superadded, leading to an indeterminate equa tion of the second degree.

Tastic suggested by them. numbers into odd and even, and observed that the sum of the odd numbers from 1 to 2 n -f- 1 was always a perfect square. Of no particular value were their classifications of numbers into heteromecic, trian The Pythagoreans classified gular, perfect, excessive, defective, amicable. 1 2 Gow, p. 69. For their definitions see Gqw, 2 p. 70, or CAJOEI, op. The Pythagcit,, p. 68. A HISTOEY OF MATHEMATICS 30 oreans paid much attention to the subject of proportion. Tha,, quantities a, b, c, d, were said to be in arithmetical proportion, when a b=cd; in geometrical proportion, when a b = c d : : ; in har/nojiic proportion, when a "proportion, : -| when a (a -f &) &:& c=a:c; = 2 ab/(a + b):b.