The Babylonian Quadratic Equation

Abstract

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”.