Introduction to Arithmetic

nection between arithmetic and music accounts, in some measure, for the complete and even tedious discussion of ratios in the Greek treatises on arithmetic. In consequence, our consideration of the origins of Greek arithmetic will necessarily touch incidentally not only the processes of computation of the Greeks, but also geometry, music, and even other sciences, as related to the sciences of the older civilizations.For more complete discussion of arithmetic and logistic, see Heath, A History of Greek Mathematics (Oxford, 1921), vol. I, pp. 13-16.

For the sources of the early Greek arithmetical sciences we must look to Egypt and to Babylon, possibly even beyond to India and China. Evidence of the exchange of ideas between Greece and Egypt, and between Greece and Babylon, has accumulated so much in recent years as to show a degree of intimacy long unsuspected.F. Cumont, The Oriental Religions in Roman Paganism (Chicago, 1911); and Astrology and Religion among the Greeks and Romans (New York, 1912); Milhaud, Nouvelles Études sur l'Histoire de la Pensée Scientifique (Paris, 1911), pp. 41-133. In the early centuries of the Christian era, knowledge of Greek astronomy was carried to India; traces of reciprocal influence in ancient times are not wanting, although any detailed statement must await more accurate information of the historical development of Hindu learning. The sciences, biological, physical, and mathematical, as well as the fine arts and technical arts, are involved in the interchange of ideas between Orient and Occident, but our interest is centered upon the mathematical sciences. In this field the Oriental science served primarily as a directive force, determining the topics which for centuries occupied the attention of Greek mathematicians.

In mathematics and astronomy the early traces of Oriental influence cover a wide range of ideas, touching at the lower point the simplest operations of computation and at the upper point the development of complicated astronomical theories. At the outset we may say that one extraordinary achievement in mathematics remains undisputedly Greek in its origin, namely, the development of logical, demonstrative geometry. WritersLike John Burnet, Greek Philosophy, Part I, Thales to Plato (London, 1920), pp. 4 ff. who confound with the whole of science the systematization of the sciences achieved by the Greeks, together with this process of logical demonstration, entirely mistake the nature of science and the processes of its progress. Science is concerned with the problems involved in comprehending the universe in which we live. Science involves inevitably the knowledge of numbers and form, or

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