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GREECE 31
was neglected. Of this period only two names deserve mention,
Eratosthenes (about 275-194 b.c) and Hypsicles (between
200 and 100 b.c). To the latter we owe researches on polygonal
numbers and arithmetical progressions. Eratosthenes
invented the celebrated "sieve" for finding prime numbers.
Write down in succession all odd numbers from 3 up. By
erasing every third number after 3, sift out all multiples of 3 ;
by erasing every fifth number after 5, sift out all multiples
of 5, and so on. The numbers left 3fter this sifting 3re all
prime. While the invention of the " sieve" called for no
great mental powers, it is remarkable that after Er3tosthenes
no 3dvance was made in the mode of finding the primes, nor in
the determiuation of the number of primes which exist in the
numerical series 1, 2, 3, ... n, until the nineteenth century,
when Gauss, Legendre, Dirichlet, Riemann, and Chebichev
enriched the subject with investig3tions mostly of great difficulty
and complexity.
The study of 3rithmetic W3S revived about 100 a.d. by
Nicomachus,^ 3 native of Gerasa (perhaps a town in Arabi3)
and known 3S a Pythagorean. He wrote in Greek a work
entitled Introductio Arithmetica. The historical importance of
this work is great, not so much on accoimt of origin3l matter
therein contained, but because it is (so far as we know) the
earliest systematic text-book on arithmetic, and because for
over 1000 years it set the fashion for the treatment of this
subject in Europe. In a small measure, Nicomachus did for
arithmetic what Euclid did for geometry. His arithmetic was
as famous in his day as was, later, Adam Riese's in Germany,
3nd Cocker's in England. Wishing to compliment a computer,
Lucian says, " Ton reckon like Nicomachus of Gerasa."' The
1 See Nesselmann, pp. 191-216 ; Gow, pp. 88-95; Cantoe, Vol. I.,
pp. 400-404.
2 Quoted by Gow (p. 89) from Philopatris, 12.