BABYLONIAN MATHEMATICS with Special Reference to Recent Discoveries

Abstract

In a vice-presidential address before Section A of the American Association for the Advancement of Science just six years ago, I made a somewhat detailed survey1 of our knowledge of Egyptian and Babylonian Mathematics before the Greeks. This survey set forth considerable material not then found in any general history of mathematics. During the six years since that time announcements of new discoveries in connection with Egyptian mathematics have been comparatively insignificant, and all known documents have probably been more or less definitively studied and interpreted. But the case of Babylonian mathematics is entirely different; most extraordinary discoveries have been made concerning their knowledge and use of algebra four thousand years ago. So far as anything in print is concerned, nothing of the kind was suspected even as late as 1928. Most of these recent discoveries have been due to the brilliant and able young Austrian scholar Otto Neugebauer who now at the age of 36 has a truly remarkable record of achievement during the past decade. It was only in 1926 that he received his doctor's degree in mathematics at Göttingen, for an interesting piece of research in Egyptian mathematics; but very soon he had taken up the study of Babylonian cuneiform writing. He acquired a mastery of book and periodical literature of the past fifty years, dealing with Sumerian, Akkadian, Babylonian, and Assyrian grammar, literature, metrology, and inscriptions; he discovered mathematical terminology, and translations the accuracy of which he thoroughly proved. He scoured museums of Europe and America for all possible mathematical texts, and translated and interpreted them. By 1929 he bad founded periodicals called Quellen und Studien zur Geschichte der mathematik2 and from the first, the latter contained remarkable new information concerning Babylonian mathematics. A trip to Russia resulted in securing for the Quellen section, Struve's edition of the first complete publication of the Golenishchev mathematical papyrus of about 1850 B.C. The third and latest volume of the Quellen, appearing only about three months ago, is a monumental work by Neugebauer himself, the first part containing over five hundred pages of text, and the second part in large quarto format, with over 60 pages of text and about 70 plates. This work was designed to discuss most known texts in mathematics and mathematical astronomy in cuneiform writing. And thus we find that by far the largest number of such tablets is in the Museum of Antiquities at Istanbul, that the State Museum in Berlin made the next larger contribution, Yale University next, then the British Museum, and the University of Jena, followed by the University of Pennsylvania, where Hilprecht, some thirty years ago, published a work containing some mathematical tables. In the Museum of the Louvre are 16 tablets; and then there are less than 8 in each of the following: the Strasbourg University and Library, the Musec Royaux du Cinquantenaire in Brussels, the J. Pierpont Morgan Library Collection (temporarily deposited at Yale) the Royal On tario Museum of Archaeology at Toronto, the Ashmolean Museum at Oxford, and the Böhl collection at Leyden. Most of the tablets thus referred to date from the period 2000 to 1200 B.C. It is a satisfaction to us to know that the composition of this wonderful reference work was in part made possible by The Rockefeller Foundation. Some two years ago it cooperated in enabling Neugebauer to transfer his work to the Mathematical Institute of the University of Copenhagen, after Nazi intolerance had rendered it impossible to preserve his self respect while pursuing the in tellectual life. This new position offered the opportunity for lecturing on the History of Ancient Mathematical Science. The first volume of these lectures3 on “Mathematics before the Greeks,” was published last year, and in it are many references to results, the exact setting of which are only found in his great source work referred to a moment ago. In these two works, then, we find not only a summing up of Neugebauer's wholly original work, but also a critical summary of the work of other scholars such as Frank, Gadd, Genouillac, Hilprecht, Lenormant, Rawlinson, Thureau-Dangin, Weidner, Zimmern, and many others.4 Hence my selection of material to be presented to you to-night will be mainly from these two works. Before turning to this it may not be wholly inappropriatp to interpolateoneremarkregarding Neugebauer's service to mathematics in general. Since 1931 his notable organizing ability has been partially occupied in editing and directing two other periodicals, (1) Zentralblatt fur Mathematik (of which 11 volumes have already appeared), and (2) Zentralblatt fur Mechanik, (3 volumes) a job which of itself would keep many a person fully employed. Mais, revenons à nos moutons!