Introduction to Arithmetic

notable advance in astronomy in Babylon was undoubtedly made during the period in which the science was making real progress in Greece; indebtedness was mutual, but independent scientific progress on both sides is incontestably established.

To deny to Babylon, to Egypt, and to India their part in the development of science and scientific thinking is to defy the testimony of the ancients,¹ Hipparchus and Ptolemy, Theon of Smyrna, and Diodorus, as cited above; Herodotus, II. 109; Berosus, fragments 17 ff. in C. Müller, Fragmenta Historicorum Graecorum, vol. II, pp. 509 f.; Clemens Alexandrinus, Stromata, II. 4; Pliny, Natural History, VI. 121; VII. 193. supported by the discoveries of modern authorities.² Heath, Berthelot, Boll, Cumont, as cited above; Kugler, The Babylonian Moon-Calculation (Freiburg, 1900), pp. 50-51; 203-211; Epping, Astronomicals from Babylon (Freiburg, 1889), pp. 183-190. The efforts which have been made to ascribe to Greek influence the science of Egypt, of later Babylon, of India, and later of the Arabs, do not add to the glory that was Greece. How could the Babylonians of the golden age of Greece have taken over the developments of Greek astronomy? This would have been possible only if they had arrived at a stage of development in astronomy which would have enabled them properly to estimate and appreciate the work which was to be absorbed. There has never been any question concerning the nature and origin of such feeble beginnings of science as are found among the American Indians. As regards the Babylonians, the Hindus, and the civilization of Europe in the time of Alexander the Great and up to 600 A.D., the problem is entirely different. These are peoples who had reached approximately the same stage of development. The admission that Greek astronomy immediately affected the astronomical theories of Babylon and India carries with it the implication that this science had attained somewhat the same level in these countries as in Greece. Without serious questioning we may assume that a significant part of the science of Babylon and Egypt that was developed during the times which we think of as Greek was indigenous. Nor do we thereby detract from the real greatness of Greece. The Hellenic civilization remains as an integral and vital part of all civilization, and not as something apart.

Turning to the arithmetica theory of numbers proper, we may first inquire as to the Egyptian attempts at systematization of the science. The Ahmes manual in itself is evidence of a noteworthy step in this direction, since it establishes the fact that the body of ideas which we now group under the name ‘mathematics’ was recognized as a separate field by