In saying that arithmetic began with Pythagoras we have to distinguish between the uses of that word then and now. Αριθμητικη {Arithmêtikê} with the Greeks was distinguished from λογιστικη {logistikê}, the science of calculation. It is the latter word which would cover arithmetic in our sense, or practical calculation; the term αριθμητικη {arithmêtikê} 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 {arithmêtikê} dealt with absolute numbers or numbers in the abstract, and λογιστικη {logistikê} with numbered _things_ or concrete numbers; thus λογιστικη {logistikê} 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.