Babylonian Mathematics with Special Reference to Recent Discoveries*
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Babylonian Mathematics with Special Reference to Recent Discoveries* RAYMOND CLARE ARCHIBALD IN A VICE-PRESIDENTIAL ADDREss before Sec fifty years, dealing with Sumerian, Akkadian, tion A of the American Association for the Ad Babylonian, and Assyrian grammar, literature, vancement of Science just six years ago, I made a metrology, and inscriptions; he discov~red math somewhat detailed survey' of our knowledge of ematical terminology, and translations the accu Egyptian and Babylonian Mathematics before the racy of which he thoroughly proved. He scoured Greeks. This suJVey set forth considerable mate museums of Europe and America for all possible rial not then found in any general history of math mathematical texts, and tr~nslated and interpreted ematics. During the six years since that time an them. By 1929 he had founded periodicals called nouncements ofnew discoveries in connection with Quellen und Studien zur Gefchichte der MathematiF Egyptian mathematics have been comparatively and from the first, the latter contained remarkable insignificant, and all known documents have prob new information concerning Babylonian math ably been more or less definitely studied and inter ematics. A trip to Russiafesulted in securing for preted. But the case ofBabylonian mathematics is the Que/len section, Struve's edition of the first entirely different; most extraordinary discoveries complete publication of the Gnlenishchev math have been made concerning their knowledge and ematical papyrus of about 1850 B.c. The third and use of algebra four thousand years ago. So far as latest volume of the Que/len, appearing only about anything in print is concerned, nothing of the kind three months ago, is a monumental work by was suspected even as late at 1928. Most of these Neugebauer himself; the first part containing over recent discoveries have been due to the brilliant five hundred pages of text, and the second part in and able young Austrian scholar Otto Neugebauer large quarto format, with over 60 pages of text and who now at the age of 36 has a truly remarkable about 70 plates. This work was designed to discuss record of achievement during the past decade. It most known texts in mathematics and mathemati was only in 1926 that he received his doctor's cal astronomy in-cuneiform writing. And thus we degree in mathematics at GOttingen, for an inter find that by far the largest number of such tablets esting piece of research in Egyptian mathematics; is in the Museum of Antiquities at lstanbnl, that but very soon he had taken up the study of the State Museum in Berlin made the next larger Babylonian cuneiform writing. He acqnired a mas contribution, Yale University next, then the Brit tery of book and periodical literature of the past ish Museum, and the University ofJena, followed by the University ofPennsylvania, where Hilprecht, some thitty years ago, published a work containing Reprinted from Mathematits Teacher 29 (May, 1936): 209-19; with permission of the National Council of Teachers of some mathematical tables. In the Museum of the Mathematics. Louvre are 16 tablets; and then there are less than 113 PART III/HUMAN IMPACT AND THE SOCIETAL STRUCTURING OF MATHEMATICS 8 in each of the following: the Strasbourg Univer ~e Tigris and Euphrates Rivers, the non-semitic sity and Library, the Musee Royaux du Cinquan Sumerians, south of the semitic Akkadians, were teraire in Brussels, the J. Pierpont Morgan Library generally predominant in Babylonia. By 2000 B.C. Collection (temporarily deposited at Yale) the Royal they were absorbed in a larger political group. One Ontario Museum of Archaeology at Toronto, the of the greatest of the Sumerian inventions was the Ashmolean Museum at Oxford, and the Bohl col adoption of cuneiform script; notable engineering lection at Leyden. Most of the tablets thus referred works of the Babylonians, by means of which to date from the period 2000 to 1200 B.C. It is a marshes were drained and the overflow of the satisfaction to us to know that the composition of rivers regulated by canals, went back to Sumerian this wonderful reference work was in part made times, like also a considerable part of their religion possible by The Rockefeller Foundation. Some and law, and their system of mathematics, except, two years ago it cooperated in enabling Neuge possibly, for certain details. As to mathematical bauer to transfer his work to the Mathematical transactions we find that long before coins were in Institute of the University of Copenhagen, after use the custom of paying interest for the loan of Nazi intolerance had rendered it impossible to produce, or of a certain weight of a precious metal, preserve his self respect while pursuing the was common. Sumerian tablets indicate that the intellectual life. This new position offered the rate of interest varied from 20 per cent to 30 per opportunity for lecturing on the History ofAncient cent, the higher rate being charged for produce. At Mathematical Science. The first volume of these a later period the rate was 51h per cent to 33'h per lectures,3 on "Mathematics before the Greeks," cent for produce. 5 An extraordinary number of was published last year, and in it are many references tablets show that the Sumerian merchant of 2500 to results, the exact setting ofwhich are only found B.c. was familiar with such things as weights and in his great source work referred to a moment ago. measures, bills, receipts, notes, and accounts. In these two works, then, we find not only a Sumerian mathematics was essentially sexages summing up of Neugebauer's wholly original imal and while a special symbol for 10 was con work, but also a critical summary of the work of stantly used it occupied a subordinate position; other scholars such as Frank, Gadd, Genouillac, there were no special symbols for 100 or for 1000. Hilprecht, Lenormant, RawlinSon, Thureau One hundred was thought of as 60 + 40 and 1000 Dangin, Weidner, Zimmern, and many others.4 as 16 · 60 + 40. But in these cases the Sumerian Hence my selection of material to be presented to would write simply 1,40 and 16,40; you tonight will be mainly from these two works. Before turning to this it may not be wholly <(YY<(<( Vl «<YYY inappropriate to interpolate one remark regard = 12 x 60 2 + 25 x 60 + 33 = 44733. Hence the ing Neugebauer's service to mathematics in Sumerians had a relative positional notation for general. Since 1931 his notable organizing ability the numbers. The word cuneiform means has been partially occupied in editing and direct wedgeshaped and the numbers from one to nine ing two other periodicals, (1) Zentra/blatt fur were denoted by the corresponding number of Mathematik {of which 11 volumes have already wedges, where the Egyptian simply employed appeared), and {2) Zentra/b/att fur Mechanik, (3 strokes. For 10, as we have seen, an angle-shaped volumes)-ajob which of itself would keep many sign was used. Practically all other integers were a person fully employed. Mais, revenons a nos made up of combinations of these in various moutons! ways. There is great ambiguity because, for ex From about 3500 to 2500 years before Christ, ample, a single upright wedge may stand for 1 or in the country north of the Persian Gulf between 60 or any positive or negative integral multiple 114 BABYLONIAN MATHEMATICS of 60. Hence there was a special sign D for 60; the purpose of the tablet in question was to solve ()> or 19 or 'lo for 600, the last of which suggests cUbic equations in this "normal form." He con 60 X 10; 0 for 3600; and @ for 36000, again tended that it was within the power of the suggesting a product. No special sign for zero in Babylonians, by a linear transformation z = x + c, to 3 2 Sumerian times, other than an empty space, has reduce a four-term cubic equation x + a1x + a2x 3 2 yet been discovered. But by the time of the Greeks + a3 = 0 to z + b1 z + b2 = 0. Multiplying this 3 equation by 1/b1 we have at once (on setting z = 2 3 <{. YY ~ '(~<( YYY =12x60 b1w, and a= -b2 /b1 ) the normal form + 0 + 33. = 43233, the sign ~ being fur zero. But to matters of numeral notation we shall make no Neugebauer's theory as to the possibiliry of such a further reference, except to remark that the reduction is in part based on problems to which I Babylonians thought of any positive integer shall later refer. Up to the present, however, a = Ic 60", and in the form Neugebauer has found no four-term cubic equa " tion solved in this way. And indeed in these same British Museum tablets are two problems which lead naturally to such equations but are solved by a This may not, of course, correspond to what we different method.' call integers. By means of negative values of n, Neugebauer feels that tables are the foundation fractions were introduced. ofall discussion ofBabylonian mathematics, that more Babylonian multiplication tables are very nu tables, such as the one to which we have just merous and are often the products of a certain referred, are likely to be discovered, and to illumi number, successively, by 1, 2, 3 · · · 20, then 30, 40, nate other mathematical operations. There are and 50. For example, on tablets of about 1500 B.c. many tables of parallel columns of integers such as at Brussels are tables of7, 10, 12 112, 16, 24, each 2 30 multiplied into such a series of numbers. There are 3 20 various tablets giving the squares of numbers from 4 15 1 to 50, and also the cubes, square roots, and cube 5 12 roots of numbers.