A History of Greek Mathematics and Astronomy

Page 10

The Theory of Numbers then began with Pythagoras (about 572-497 B. C.). It included definitions of the unit and of number, and the classification and definitions of the various classes of numbers, odd, even, prime, composite, and sub-divisions of these such as odd-even, even-times-even, &c. Again there were figured numbers, namely, triangular numbers, squares, oblong numbers, polygonal numbers (pentagons, hexagons, &c.) corresponding respectively to plane figures, and pyramidal numbers, cubes, parallelepipeds, &c., corresponding to solid figures in geometry. The treatment was mostly geometrical, the numbers being represented by dots filling up geometrical figures of the various kinds. The laws of formation of the various figured numbers were established. In this investigation the gnomon played an important part. Originally meaning the upright needle of a sun-dial, the term was next used for a figure like a carpenter's square, and then was applied to a figure of that shape put round two sides of a square and making up a larger square. The arithmetical application of the term was similar. If we represent a unit by one dot and put round it three dots in such a way that the four form the corners of a square, three is the first gnomon. Five dots put at equal distances round two sides of the square containing four dots make up the next square (3²), and five is the second gnomon. Generally, if we have n² dots so arranged as to fill up a square with n for its side, the gnomon to be put round it to make up the next square, (n+1)², has 2n+1 dots. In the formation of squares, therefore, the successive gnomons are the series of odd numbers following 1 (the first square), namely 3, 5, 7, ... In the formation of oblong numbers (numbers of the form n(n+1)), the first of which is 1. 2, the successive gnomons are the terms after 2 in the series of even numbers 2, 4, 6.... Triangular numbers are formed by adding to 1 (the first triangle) the terms after 1 in the series of natural numbers 1, 2, 3 ...; these are therefore the gnomons (by analogy) for triangles. The gnomons for pentagonal numbers are the terms after 1 in the arithmetical progression 1, 4, 7, 10 ... (with 3, or 5-2, as the common difference) and so on; the common difference of the successive gnomons for an a-gonal number is a-2.

From the series of gnomons for squares we easily deduce a formula for finding square numbers which are the sum of two squares. For, the gnomon 2n+1 being the difference between the successive squares n² and (n+1)², we have only to make 2n+1 a square. Suppose that 2n+1=m²; therefore n=½(m²-1), and {½(m²-1)}²+m²={½(m²+1)}², where m is any odd number. This is the formula actually attributed to Pythagoras.

Pythagoras is said to have discovered the theory of proportionals or proportion. This was a numerical theory and therefore was applicable to commensurable magnitudes only; it was no doubt somewhat on the lines of Euclid, Book VII. Connected with the theory of proportion was that of means, and Pythagoras was acquainted with three of these, the arithmetic, geometric, and sub-contrary (afterwards called harmonic). In particular Pythagoras is said to have introduced from Babylon into Greece the 'most perfect' proportion, namely:

a:(a+b)/2=2ab/(a+b):b,

where the second and third terms are respectively the arithmetic and harmonic mean between a and b. A particular case is 12:9=8:6.

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T. L. Heath
A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath
A History of Greek Mathematics and Astronomy