Pythagorean thought was scientific as well as metaphysical and included specific developments in arithmetic and geometry, in the science of musical tones and harmonies, and in astronomy.

Early Pythagorean achievements in mathematics are unclear and largely disputable, and the following is, therefore, a compromise between the widely divergent views of modern scholars. In the speculation on odd and even numbers, the early Pythagoreans used so-called gnomones (Greek: "carpenter's squares"). Judging from Aristotle's account, gnomon numbers, represented by dots or pebbles, were arranged in the manner shown in Figure 2. If a series of odd numbers is put around the unit as gnomons, they always produce squares; thus, the members of the series 4, 9, 16, 25, . . . are "square" numbers. If even numbers are depicted in a similar way, the resulting figures (which offer infinite variations) represent "oblong" numbers, such as those of the series 2, 6, 12, 20 . . . . On the other hand, a triangle represented by three dots (as in the upper part of the tetraktys) can be extended by a series of natural numbers to form the "triangular" numbers 6, 10 (the tetraktys), 15, 21. . . . This procedure, which was, so far, Pythagorean, led later, perhaps in the Platonic Academy, to a speculation on "polygonal" numbers.

Gnomon for Pythagorean theorem. The marked off "carpenter's square"--comprising 3 groups of 3 dots each (3 3)--thus represents 32, which when added to 42 yields 52 (the total gnomon).

Probably the square numbers of the gnomons were early associated with the Pythagorean theorem (likely to have been used in practice in Greece, however, before Pythagoras), which holds that for a right triangle a square drawn on the hypotenuse is equal in area to the sum of the squares drawn on its sides; in the gnomons it can easily be seen, in the case of a 3,4,5-triangle for example (see Figure 3), that the addition of a square gnomon number to a square makes a new square: 32 + 42 = 52, and this gives a method for finding two square numbers the sum of which is also a square.

Some 5th-century Pythagoreans seem to have been puzzled by apparent arithmetical anomalies: the mutual relationships of triangular and square numbers; the anomalous properties of the regular pentagon; the fact that the length of the diagonal of a square is incommensurable with its sides--i.e., that no fraction composed of integers can express this ratio exactly (the resulting decimal is thus defined as irrational); and the irrationality of the mathematical proportions in musical scales. The discovery of such irrationality was disquieting because it had fatal consequences for the naive view that the universe is expressible in whole numbers; the Pythagorean Hippasus is said to have been expelled from the brotherhood, according to some sources even drowned, because he made a point of the irrationality.

In the 4th century, Pythagorizing mathematicians made a significant advance in the theory of irrational numbers, such as the-square-root-of-n ({sqroot n}), n being any rational number, when they developed a method for finding progressive approximations to {sqroot 2} by forming sets of so-called diagonal numbers.

It is notable that the properties of the circle seem not to have interested the early Pythagoreans. But perhaps the tradition that Pythagoras himself discovered that the sum of the three angles of any triangle is equal to two right angles may be trusted. The idea of geometric proportions is probably Pythagorean in origin; but the so-called golden section--which divides a line at a point such that the smaller part is to the greater as the greater is to the whole--is hardly an early Pythagorean contribution. Some advance in geometry was made at a later date, by 4th-century Pythagoreans; e.g., Archytas offered an interesting solution to the problem of the duplication of the cube--in which a cube twice the volume of a given cube is constructed--by an essentially geometrical construction in three dimensions; and the conception of geometry as a "flow" of points into lines, of lines into surfaces, and so on, may have been contributed by Archytas; but on the whole the numerous achievements of non-Pythagorean mathematicians were in fact more conspicuous than those of the Pythagoreans. (see also Index: Pythagorean theorem)

*Excerpt from the Encyclopedia Britannica without permission.*