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 (Sn) of n terms of the series 1, 2, 2², 2³ ... is prime, then Sn.2n-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 (‘bloom’) of Thymaridas, and amounting to the solution of any number of simultaneous equations of the following form:
x+x1 + x2 + ... + xn-1 = s,
x + x1 = a1,
x + x2 = a2,
....
x+xn-1 = an-1,
the solution being x=((a1+a2+...+an-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.
The Pythagorean contributions to geometry were even more remarkable. The most famous proposition attributed to Pythagoras himself is of course the theorem of Eucl. I. 47 that the square on the hypotenuse of any right-angled triangle is equal to the sum of the squares on the other two sides. But Proclus also attributes to him, besides the theory of proportionals, the construction of the ‘cosmic figures’, the five regular solids.
One of the said solids, the dodecahedron, has twelve regular pentagons for faces, and the construction of a regular pentagon involves the cutting of a straight line ‘in extreme and mean ratio’ (Eucl. II. 11 and VI. 30), which is a particular case of the method known as the application of areas. This method was fully worked out by the Pythagoreans and proved one of the most powerful in all Greek geometry. The most elementary case appears in Eucl. I. 44, 45, where it is shown how to apply to a given straight line as base a parallelogram with one angle equal to a given angle and equal in area to any given rectilineal figure; this construction is the geometrical equivalent of arithmetical division. The general case is that in which the parallelogram, though applied to the straight line, overlaps it or falls short of it in such a way that the part of the parallelogram which extends beyond or falls short of the parallelogram of the same angle and breadth on the given straight line itself (exactly) as base is similar to any given parallelogram (Eucl. VI. 28, 29). This is the geometrical equivalent of the solution of the most general form of quadratic equation ax±mx²=C, so far as it has real roots; the condition that the roots may be real was also worked out (=Eucl. VI. 27). It is in the form of ‘application of areas’ that Apollonius obtains the fundamental property of each of the conic sections, and, as we shall see, it is from the terminology of application of areas that Apollonius took the three names parabola, hyperbola, and ellipse which he was the first to give to the three curves.