Add the next odd number, i.e. 2n + 1, and we have n² + (2n + 1) = (n + 1)². In order, then, to get two square numbers such that their sum is a square we have only to see that 2n + 1 is a square. Suppose that 2n + 1 = m²; then n = ½(m² − 1), and we have {½ (m² − 1) }² + m² = {½ (m² + 1) }², where m is any odd number; and this is the general formula attributed to Pythagoras.

Proclus also attributes to Pythagoras the theory of proportionals and the construction of the five “cosmic figures,” the five regular solids.

One of the said solids, the dodecahedron, has twelve pentagonal 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. How much of this was due to Pythagoras himself we do not know; but the whole method was at all events fully worked out by the Pythagoreans and proved one of the most powerful of geometrical methods.

The most elementary case appears in Euclid, I., 44, 45, where it is shown how to apply to a given straight line as base a parallelogram having a given angle (say a rectangle) and equal in area to any 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 another given parallelogram (Eucl. VI., 28, 29). This is the geometrical equivalent of the most general form of quadratic equation ax ± mx² = C, so far as it has real roots; while the condition that the roots may be real was also worked out (= Eucl. VI., 27).