BY SIR THOMAS HEATH

NOT ONLY did the early Pythagoreans make many contributions in the realm of philosophy, but their mathematical studies laid the foundation for the development of Greek geometry, and many portions of Euclid's Elements can be traced back to mathematical discoveries of the Pythagorean school.

This listing of early Pythagorean mathematical discoveries is excerpted from Thomas Heath's History of Greek Mathematics, vol. I, pp, 166-169.

A SUMMARY OF PYTHAGOREAN MATHEMATICAL DISCOVERIES

1. They were acquainted with the properties of parallel lines, which they used for the purpose of establishing by a general proof the proposition that the sum of the three angles of any triangle is equal to two right angles. This latter proposition they again used to establish the well-known theorems about The sums of the exterior and interior angles, respectively, of any polygon.

2. They originated the subject of equivalent areas, the transformation of an area of one form into another of different form and, in particular, the whole method of application of areas, constituting a geometrical algebra, whereby they effect the equivalent of the algebraical processes of addition, subtraction, multiplication, division, squaring, extraction of the square root, and finally the complete solution to the mixed quadratic equation x² ± pxq = 0 so far as its roots are real. Expressed in terms of Euclid, this means the whole content of Book I. 35-48 and Book II. The method of application of areas is one of the most fundamental in the whole of later Greek geometry; it takes its place by the side of the powerful method of proportion; moreover, it is the starting point of Apollonius' theory of conics, and the three fundamental terms, parabole, ellipsis, and hyperbole used to describe the three separate problems in 'application' were actually employed by Apollonius to denote the three conics, names which, of course, are those which we use to-day. Nor was the use of the geometrical algebra for solving numerical problems unknown to the Pythagoreans; this is proved by the fact that the theorems of Eucl. II. 9, 10 were invented for the purpose of finding successive integral solutions of the indeterminate equations

2x² - y² = ± 1

3. They had a theory of proportion pretty fully developed. We know nothing of the form in which it was expounded; all we know is that it took no account of incommensurable magnitudes. Hence we conclude that it was a numerical  theory, a theory on the same lines as that contained in Book VII of Euclid's  Elements.

They were aware of the properties of similar figures. This is clear from the fact that they must be assumed to have solved the problem, which was, according to Plutarch, attributed to Pythagoras himself, of describing a figure which shall be similar to one given figure and equal in area to another given figure. This implies a knowledge of the proposition that similar figures (triangles or polygons) are to one another in the duplicate ratio of corresponding sides (Eucl. VI. 19, 20). As the problem is solved in Eucl. VI. 25, we assume that, subject to the qualification that their theorems about similarity, &c., were only established of figures in which corresponding elements are commensurable, they had theorems corresponding to a great part of Eucl., Book VI.

Again, they knew how to cut a straight line in extreme and mean ratio (Eucl. VI. 30); [1] this problem was presumably solved by the method used in Eucl. II. 11, rather than by that of Eucl. VI. 30, which depends on the solution of a problem in the application of areas more general than the methods of Book II enable us to solve, the problem namely of Eucl. VI. 29.

4. They had discovered, or were aware of the existence of, the five regular solids. These they may have constructed empirically by putting together squares, equilateral triangles, and pentagons. This implies that they could construct a regular pentagon and, as this construction depends upon the construction of an isosceles triangle in which each of the base angles is double of the vertical angle, and this again on the cutting of a line in extreme and mean ratio, we may fairly assume that this was the way in which the construction of the regular pentagon was actually evolved. It would follow that the solution of problems by analysis was already practised by the Pythagoreans, notwithstanding that the discovery of the analytical method is attributed by Proclus to Plato. As the particular construction is practically given in Eucl. IV. 10, 11, we may assume that the content of Eucl. IV was also partly Pythagorean.

5. They discovered the existence of the irrational in the sense that they proved the incommensurability of the diagonal of a square with reference to its side; in other words, they proved the irrationality of √2. At; a proof of this is referred to by Aristotle in terms which correspond to the method used in a proposition interpolated in Euclid, Book X, we may conclude that this proof is ancient, and therefore that it was probably the proof used by the discoverers of the proposition. The method is to prove that, if the diagonal of a square is commensurable with the side, then the same number must be both odd and even; here then we probably have of early Pythagorean use of the method of reductio ad absurdum.

Not only did the Pythagoreans discover the irrationality of √2; they showed, as we have seen, how to approximate as closely as we please to its numerical value. After the discovery of this one case of irrationality, it would be obvious that  propositions theretofore proven by means of the numerical theory of proportion, which was inapplicable to incommensurable magnitudes, were only partially proved. Accordingly, pending the discovery of a theory of proportion applicable to incommensurable as well as commensurable magnitudes, there would be an inducement to substitute, where possible, for proofs employing the theory of proportion other proofs independent of that theory. This substitution is carried rather far in Euclid, Books I-IV; it does not follow that the Pythagoreans remodelled their proofs to the same extent as Euclid felt bound to do.

_______________

Notes:

1. The extreme and mean division of the line is an important mathematical and geometrical ratio which underlies various universal forms. This is the so-called "Divine Proportion," "Golden Section," or Phi ratio. On the properties and significance of this principle see Ghyka, The Geometry of Art and Life, and other titles on sacred geometry listed in the bibliography.