Page:A History of Mathematics (1893).djvu/90

 Without doubt, much is original with Euclid. The seventh book begins with twenty-one definitions. All except that for 'prime' numbers are known to have been given by the Pythagoreans. Next follows a process for finding the G.C.D. of two or more numbers. The eighth book deals with numbers in continued proportion, and with the mutual relations of squares, cubes, and plane numbers. Thus, XXII., if three numbers are in continued proportion, and the first is a square, so is the third. In the ninth book, the same subject is continued. It contains the proposition that the number of primes is greater than any given number.

After the death of Euclid, the theory of numbers remained almost stationary for 400 years. Geometry monopolised the attention of all Greek mathematicians. Only two are known to have done work in arithmetic worthy of mention. Eratosthenes (275-194 B.C.) invented a 'sieve' for finding prime numbers. All composite numbers are 'sifted' out in the following manner: Write down the odd numbers from 3 up, in succession. By striking out every third number after the 3, we remove all multiples of 3. By striking out every fifth number after the 5, we remove all multiples of 5. In this way, by rejecting multiples of 7, 11, 13, etc., we have left prime numbers only. Hypsicles (between 200 and 100 B.C.) worked at the subjects of polygonal numbers and arithmetical progressions, which Euclid entirely neglected. In his work on 'risings of the stars,' he showed (1) that in an arithmetical series of 2 n terms, the sum of the last n terms exceeds the sum of the first n by a multiple of $$\scriptstyle{n^2}$$; (2) that in such a series of $$\scriptstyle{2n+1}$$ terms, the sum of the series is the number of terms multiplied by the middle term; (3) that in such a series of 2n terms, the sum is half the number of terms multiplied by the two middle terms.[6]

For two centuries after the time of Hypsicles, arithmetic