Introduction to Arithmetic

CHAPTER I

THE SOURCES OF GREEK MATHEMATICS

ARITHMETIC is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after Pythagoras, however, arithmetic was primarily a philosophical study, having no necessary connection with practical affairs. Indeed the Greeks gave a separate name to the arithmetic of business, logistike the art of calculation; of this division of the science no Greek treatise has been transmitted to us. In general the philosophers and mathematicians of Greece undoubtedly considered it beneath their dignity to treat of this branch, which probably formed a part of the elementary instruction of children. The evidence for the existence of treatises on the fundamental operations is very insecure and vague, resting upon a passage of Diogenes Laertiusoriginal: "Vitae Philosophorum," VIII. 12, where a certain Apollodorus is designated as ὁ λογιστικός, which may mean, as Cantor thinks, that he was a Rechenmeister (master of calculation). and a citation by Eutocius.original: In the Commentary on the Measurement of the Circle by Archimedes (in Heiberg, Archimedis Opera Omnia cum Commentariis Euctocii, Leipzig, 1881, vol. III, p. 302, line 4), he mentions the λογιστικά of a certain Magnus or Magnes.

So far as the content of the logistic is concerned, our main source of information is the scholiumoriginal: "scholium," a marginal annotation or commentary on Plato's Charmides, 163 E. This scholium is undoubtedly based on the lost work of Geminus, although it may be through the medium of Anatolius.original: Tannery, La Géométrie Grecque, Paris, 1887, pp. 48-49. A passage in Proclusoriginal: Proclus, In Primum Euclidis Elementorum Librum Commentarii, pp. 38, 1—42, 8 (ed. Friedlein). which explicitly mentions Geminus touches analogous points.

The scholium is as follows: "Logistic is the theory which deals with numerable objects and not with numbers;original: Compare the similar distinction made by Aristotle. it does not, indeed, consider number in the proper sense of the term, but assumes 1 to be unity, and anything which can be numbered to be number (thus in place of the triad, it employs 3; in place of the decad, 10), and discusses with these the theorems of arithmetic."