A History of Greek Mathematics and Astronomy

Page 11

This bears upon what was probably Pythagoras's greatest discovery, namely that the musical intervals correspond to certain arithmetical ratios between lengths of string at the same tension, the octave corresponding to the ratio 2:1, the fifth to 3:2 and the fourth to 4:3. These ratios being the same as those of 12 to 6, 8, 9 respectively, we can understand how the third term, 8, in the above proportion came to be called the 'harmonic' mean between 12 and 6.

The Pythagorean arithmetic as a whole, with the developments made after the time of Pythagoras himself, is mainly known to us through Nicomachus's Introductio arithmetica, Iamblichus's commentary on the same, and Theon of Smyrna's work Expositio rerum mathematicarum ad legendum Platonem utilium. The things in these books most deserving of notice are the following.

First, there is the description of a 'perfect' number (a number which is equal to the sum of all its parts, i.e. all its integral divisors including 1 but excluding the number itself), with a statement of the property that all such numbers end in 6 or 8. Four such numbers, namely 6, 28, 496, 8128, were known to Nicomachus. The law of formation for such numbers is first found in Eucl. IX. 36 proving that, if the sum (S{n}) of n terms of the series 1, 2, 2², 2³ ... is prime, then S{n}.2^{n-1} is a perfect number.

Secondly, Theon of Smyrna gives the law of formation of the series of 'side-' and 'diameter-' numbers which satisfy the equations 2x²-y²=±1. The law depends on the proposition proved in Eucl. II. 10 to the effect that (2x+y)²-2(x+y)²=2x²-y², whence it follows that, if x, y satisfy either of the above equations, then 2x+y, x+y is a solution in higher numbers of the other equation. The successive solutions give values for y/x, namely 1/1, 3/2, 7/5, 17/12, 41/29, ..., which are successive approximations to the value of √2 (the ratio of the diagonal of a square to its side). The occasion for this method of approximation to √2 (which can be carried as far as we please) was the discovery by the Pythagoreans of the incommensurable or irrational in this particular case.

Thirdly, Iamblichus mentions a discovery by Thymaridas, a Pythagorean not later than Plato's time, called the επανθημα {epanthêma} ('bloom') of Thymaridas, and amounting to the solution of any number of simultaneous equations of the following form:

x+x₁+x₂+ ... +x{n-1} = s, x+x₁ = a₁, x+x₂ = a₂, ... x+x{n-1} = a{n-1},

the solution being x=((a₁+a₂+ ... +a{n-1})-s)/(n-2).

The rule is stated in general terms, but the above representation of its effect shows that it is a piece of pure algebra.

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

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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