By Sir Thomas Heath
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Additional info for A History of Greek Mathematics, Vol. 1: From Thales to Euclid
It should be added that to the distinction between arithmetic and logistic there corresponded (up to the time of Nicomachus) different methods of treatment. With rare exceptions, such as Eratosthenes’s κόσκινον, or sieve, a device for separating out the successive prime numbers, the theory of numbers was only treated in connexion with geometry, and for that reason only the geometrical form of proof was used, whether the figures took the form of dots marking out squares, triangles, gnomons, &c. (as with the early Pythagoreans), or of straight lines (as in Euclid VII–IX); even Nicomachus did not entirely banish geometrical considerations from his work, and in Diophantus’s treatise on Polygonal Numbers, of which a fragment survives, the geometrical form of proof is used.
21 But it is later writers on the classification of mathematics who alone go into any detail of what λογιστική included. Geminus in Proclus, Anatolius in the Variae Collectiones included in Hultsch’s Heron, and the scholiast to Plato’s Charmides are our authorities. Arithmetic, says Geminus,22 is divided into the theory of linear numbers, the theory of plane numbers, and the theory of solid numbers. It investigates, in and by themselves, the species of number as they are successively evolved from the unit, the formation of plane numbers, similar and dissimilar, and the further progression to the third dimension.
He adds: ‘The reasons for this are two. First, it is because no State holds them in honour that these problems, which are difficult, are feebly investigated; and, secondly, those who do investigate them are in need of a superintendent, without whose guidance they are not likely to make discoveries. e. so long as the director has not the authority of the State behind him: this seems to be the best interpretation in view of the whole context; but it is possible, as a matter of construction, to connect the phrase with the preceding words, in which case the meaning would be ‘and, even when such a superintendent has been found, as is the case at present’, and Plato would be pointing to some distinguished geometer among his contemporaries as being actually available for the post.