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The Babylonian Quadratic Equation

Published online by Cambridge University Press:  03 November 2016

A. E. Berriman
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Extract

The purpose of this note is to show at a glance the significance of successive steps in the solutions to some of the quadratic equations that have come down in the cuneiform texts as examples of the mathematical instruction given to Babylonian students c. 1600 B.C. It is due to the translations made by O. Neugebauer in Germany and later, with A. Sachs, in the U.S.A. and to those published by Thureau-Dangin in France that this ancient material is available for general study, and, of course, it is from these sources that the following examples are taken. The texts are rhetorical, they instruct the pupil to perform arithmetical operations with specific numbers derived from the problems; in the translations these numbers retain their sexagesimal form f but here they are in our customary notation although expressed in a somewhat unusual way to facilitate, in particular, ready recognition of the re-entry of a number that has been temporarily left behind; in the text the pupil is reminded that this is the number that “your head held”.

Type
Research Article
Information
The Mathematical Gazette , Volume 40 , Issue 333 , October 1956 , pp. 185 - 192
Copyright
Copyright © Mathematical Association 1956 

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References

page 185 note * Neugebauer, O., Mathematische Keilschrift-Texte (1935)CrossRefGoogle Scholar.

Thureau-Dangin, F., Textes Mathématiques Babyloniens (1938)Google Scholar.

Neugebauer, O. and Sachs, A., Mathematical Cuneiform Texts (1945)Google Scholar.

page 185 note † Neugebauer uses a semi-colon to separate the whole number from the fraction, and commas to separate the other powers of 60. Thureau-Dangin’s numbers are accented as in angular measure, the units being marked as degrees. For example:

In the original cuneiform there is nothing to distinguish a fraction from its associated whole number, the figure sequence must be interpreted in the light of the context. Moreover, most of the problem texts belong to the Old-Babylonian period c. 1800 to 1600 B.C. when there was no sign for zero, and the scribe did not always indicate a numerical void by a blank space. In the Seleucid period beginning c. 300 B.C. (which covers the important astronomical texts) a zero was indicated by the period mark used to separate sentences.

page 186 note * It was during the reign and possibly at the command of the caliph Al Mamum (A.D. 813-833) that Mohammed ben Musa, a scholar from Khowarism on the eastern border of the Islamic empire, wrote the book Hisāb al-Diābr wa’l-Mukabāla that introduced western Europe to the method of calculation subsequently known (from al-Diābr in the title) as algebra. An Arabic manuscript copy (A.D. 1342) in the Bodleian, translated by F Rosen (1831), contains a preface in which the author refers to his treatise as “a short work on calculating by completion and reduction, confined to what is easiest and most useful in arithmetic such as men constantly require in cases of inheritance, legacies, partition, law suits and trade the measuring of land, the digging of canals, geometrical computations.”

page 186 note † The Sanskrit texts of the twelfth and eighteenth chapters were found and translated by Colebrooke : they are included in the volume containing his translations of Bhascara’s Viga-ganita and Lilavati (c. A.D. 1120).

page 187 note * The reciprocal of (13/9) is recorded as unknown because (1/13) in sexagesimal notation is the recurring fraction 0; 4, 36, 55, 23 and was omitted from the standard tables of reciprocals with which the pupils would be familiar. All recurring fractions were omitted from such tables but this does not necessarily mean that mathematicians were unaware of abbreviated values for them.

page 188 note * Historical Metrology by A. E. Berriman (Dent. 1953).

page 190 note * The text gives the instruction to add and subtract.

page 190 note † In the Babylonian linear scale, 1 gar=12kus = 60 gin =360 shusi. Volume sar = gar2kus = 3602 × 30 cu. shusi. ar of bricks = 720 × 2400 cu. shusi. On other evidence Neugebauer interprets the sar of bricks numerically as 60 dozen; a representative brick, therefore, could be 2400 cu. shusi in volume or say, a foot (20 shusi) square by a gin (6 shusi) thick, but there were bricks of many different sizes. The Sumerian shusi represented by the average length of the divisions of the linear scale on Gudea’s statue (c. 2175 B.C.) in the Louvre is 0-66 inch; surprisingly, therefore, the English pole =10 Sumerian cubits and the medieval foot of 13-2 inches (which Pétrie called the most usual English unit) reflects the Sumerian foot, it measures (1/15) pole. Equally surprising is the fact that the principal unit of the linear scale engraved on a fragment of shell found at Mohenjo-daro in the Indus Valley measures 1-32 inches = 2 Sumerian shusi; I call this the Indus inch.