Thales also solved two problems of a practical kind: (1) he showed how to measure the distance of a ship at sea, and (2) he found the heights of pyramids by means of the shadows thrown on the ground by the pyramid and by a stick of known length at the same moment; one account says that he chose the time when the lengths of the stick and of its shadow were equal, but in either case he argued by similarity of triangles.
In astronomy Thales predicted a solar eclipse which was probably that of the 28th May 585 B. C. Now the Babylonians, as the result of observations continued through centuries, had discovered the period of 223 lunations after which eclipses recur. It is most likely therefore that Thales had heard of this period, and that his prediction was based upon it. He is further said to have used the Little Bear for finding the pole, to have discovered the inequality of the four astronomical seasons, and to have written works On the Equinox and On the Solstice.
After Thales come the Pythagoreans. Of the Pythagoreans Aristotle says that they applied themselves to the study of mathematics and were the first to advance that science, going so far as to find in the principles of mathematics the principles of all existing things. Of Pythagoras himself we are told that he attached supreme importance to the study of arithmetic, advancing it and taking it out of the region of practical utility, and again that he transformed the study of geometry into a liberal education, examining the principles of the science from the beginning.
The very word μαθηματα, which originally meant ‘subjects of instruction’ generally, is said to have been first appropriated to mathematics by the Pythagoreans.
In saying that arithmetic began with Pythagoras we have to distinguish between the uses of that word then and now. Αριθμητικη with the Greeks was distinguished from λογιστικη, the science of calculation. It is the latter word which would cover arithmetic in our sense, or practical calculation; the term αριθμητικη was restricted to the science of numbers considered in themselves, or, as we should say, the Theory of Numbers. Another way of putting the distinction was to say that αριθμητικη dealt with absolute numbers or numbers in the abstract, and λογιστικη with numbered things or concrete numbers; thus λογιστικη included simple problems about numbers of apples, bowls, or objects generally, such as are found in the Greek Anthology and sometimes involve simple algebraical equations.
The Theory of Numbers then began with Pythagoras (about 572-497 B. C.). It included definitions of the unit and of number, and the classification and definitions of the various classes of numbers, odd, even, prime, composite, and sub-divisions of these such as odd-even, even-times-even, &c. Again there were figured numbers, namely, triangular numbers, squares, oblong numbers, polygonal numbers (pentagons, hexagons, &c.) corresponding respectively to plane figures, and pyramidal numbers, cubes, parallelepipeds, &c., corresponding to solid figures in geometry. The treatment was mostly geometrical, the numbers being represented by dots filling up geometrical figures of the various kinds. The laws of formation of the various figured numbers were established. In this investigation the gnomon played an important part. Originally meaning the upright needle of a sun-dial, the term was next used for a figure like a carpenter’s square, and then was applied to a figure of that shape put round two sides of a square and making up a larger square. The arithmetical application of the term was similar. If we represent a unit by one dot and put round it three dots in such a way that the four form the corners of a square, three is the first gnomon. Five dots put at equal distances round two sides of the square containing four dots make up the next square (3²), and five is the second gnomon. Generally, if we have n² dots so arranged as to fill up a square with n for its side, the gnomon to be put round it to make up the next square, (n+1)², has 2n+1 dots. In the formation of squares, therefore, the successive gnomons are the series of odd numbers following 1 (the first square), namely 3, 5, 7, ... In the formation of oblong numbers (numbers of the form n(n+1)), the first of which is 1. 2, the successive gnomons are the terms after 2 in the series of even numbers 2, 4, 6.... Triangular numbers are formed by adding to 1 (the first triangle) the terms after 1 in the series of natural numbers 1, 2, 3 ...; these are therefore the gnomons (by analogy) for triangles. The gnomons for pentagonal numbers are the terms after 1 in the arithmetical progression 1, 4, 7, 10 ... (with 3, or 5-2, as the common difference) and so on; the common difference of the successive gnomons for an a-gonal number is a-2.
From the series of gnomons for squares we easily deduce a formula for finding square numbers which are the sum of two squares. For, the gnomon 2n+1 being the difference between the successive squares n² and (n+1)², we have only to make 2n+1 a square. Suppose that 2n+1=m²; therefore n=½(m²-1), and {½(m²-1)}²+m²={½(m²+1)}², where m is any odd number. This is the formula actually attributed to Pythagoras.
Pythagoras is said to have discovered the theory of proportionals or proportion. This was a numerical theory and therefore was applicable to commensurable magnitudes only; it was no doubt somewhat on the lines of Euclid, Book VII. Connected with the theory of proportion was that of means, and Pythagoras was acquainted with three of these, the arithmetic, geometric, and sub-contrary (afterwards called harmonic). In particular Pythagoras is said to have introduced from Babylon into Greece the ‘most perfect’ proportion, namely:
a:(a+b)/2=2ab/(a+b):b,
where the second and third terms are respectively the arithmetic and harmonic mean between a and b. A particular case is 12:9=8:6.