. Carnegie Institution of Washington publication. CHAPTER I. POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. The formation of triangular numbers 1, 1 + 2, 1 + 2 + 3, ••-, and of square numbers 1, 1 + 3, 1 + 3 + 5, • • •, by the successive addition of numbers in arithmetical progression, called gnomons, is of geometric origin and goes back to Pythagoras1 (570-501 ): 9 • • • •—1 If the gnomons added are 4, 7, 10, • • • (of common difference 3), the resulting numbers 1, 5, 12, 22, • • • are pentagonal. If the common differ- ence of the gnomons is m — 2, we obtain m-gonal numbers or polygonal numbe
. Carnegie Institution of Washington publication. CHAPTER I. POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. The formation of triangular numbers 1, 1 + 2, 1 + 2 + 3, ••-, and of square numbers 1, 1 + 3, 1 + 3 + 5, • • •, by the successive addition of numbers in arithmetical progression, called gnomons, is of geometric origin and goes back to Pythagoras1 (570-501 ): 9 • • • •—1 If the gnomons added are 4, 7, 10, • • • (of common difference 3), the resulting numbers 1, 5, 12, 22, • • • are pentagonal. If the common differ- ence of the gnomons is m — 2, we obtain m-gonal numbers or polygonal numbers with m In the cattle problem of Archimedes (third century ), the sum of two of the eight unknowns is to be a triangular number (see Ch. XII). Speusippus,2 nephew of Plato, mentioned polygonal and pyramidal numbers: 1 is point, 2 is line, 3 triangle, 4 pyramid, and each of these numbers is the first of its kind; also, 1 + 2 + 3 + 4 = 10. About 175 , Hypsicles gave a definition of polygonal numbers which was quoted by Diophantus8 in his Polygonal Numbers, "If there are as many numbers as we please beginning with one and increasing by the same common difference, then when the common difference is 1, the sum of all the terms is a triangular number; when 2, a square; when 3, a pentagonal number. And the number of the angles is called after the number exceeding the common difference by 2, and the side after the number of terms including ; Given therefore an arithmetical progres- sion with the first term 1 and common difference m — 2, the sum of r terms is the r-th m-gonal number3 prm. The arithmetic of Theon of Smyrna4 (about 100 or 130 ) contains 32 chapters. In Ch. 15, p. 41, the squares are obtained from 1 + 3 = 4, XF. Hoefer, Histoire des mathematiques, Paris, ed. 2, 1879, ed. 5, 1902, 96-121; W. W. R. Ball, Math. Gazette, 8, 1915, 5-12; M. Cantor, Geschichte Math., 1, ed. 3, 1907, 160-3, 252. 2 Theologumena a
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