Page:The Rhind Mathematical Papyrus, Volume I.pdf/194

1917

Karpinski, L. C., "Algebraical developments among the Egyptians and the Babylonians," American Mathematical Monthly, vol. 24, 1917, pp.257–265. "Algebraical ideas in Egypt," pp. 258-263.

Turaev^[1], B. A., "The volume of the truncated pyramid in Egyptian mathematics," Ancient Egypt, London, 1917, pp. 100–102.

This paper exhibits one of "nineteen"^[2] problems (four of which are geometrical) occurring in a hieratic mathematical papyrus, written about 1800 B.C., which was probably also the date of the original of the Rhind mathematical papyrus, the copy we possess having been written one or two hundred years later. This papyrus,formerly the property of Golenishchev, compare Cantor 1894 [1880], now professor of Egyptian philology at the Egyptian University, Cairo, was acquired about 1916 by the Museum of Fine Arts in Moscow. It appears to indicate a familiarity with the formula for the volume of the frustum of a square pyramid, $V=(h/3)(a_{1}^{2}+a_{2}^{2}+a_{1}a_{2})$, where h is the altitude of the frustum, the sides of whose bases are ^[3] a₁ and a₂.

This extraordinary result, and the facts here revealed by the late Professor Turaev, would suggest that accounts of Egyptian mathematics may have to be rewritten so soon as all of the contents of this Mosoow papyrus are generally known. But Professor Peet who had access to a photograph of the papyrus has written, Feet (1923, 2), p. 6: "though the papyrus is of the highest interest owing to its early date and admirable state of preservation (in part at least) it contains nothing, with the exception of the problem of the truncated pyramid, which will greatly modify the conception of Egyptian mathematics given to us by the already published papyri and fragments.

1. The name in connection with this article was transliterated into the form Touraeff.
2. This is the number of the problems in the main body of the papyrus without any reference to what is to be found in the fragments. We refer later to a fifth geometrical problem on one of the fragments; the total number of problems is 26 since there are 7 problems on the fragments, as Professor Struve has kindly informed me.
3. Brahmagupta (about 628 A.D.) gave the equivalent of a formula which reduces to this (Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, ed. by Colebrooke, London, 1817, pp. 312-313). So also Mahāvirācārya (c. 850 A.D.). The Ganita-Sāra-Sangraha . . ., with English translation and notes by M. Rangācārya, Madras, 1912, p. 260 of the translation. In the Moscow papyrus a; = 4, a. = 2, and h = 6. In al-Khowarizml's algebra, the same problem is solved for a₁ = 4, a₂ = 2, and h = 10. (The Algebra of Mohammed Ben Musa. Edited and translated by F. Rosen, London, 1831, pp. 83-84; see also Annali di Matematica, vol. 7, pp. 279–280, 1866); the method of solution is: determine the height (20), and hence the volume, of the completed pyramid, from which is subtracted the volume of the pyramid of height 10 on the upper base. To Democritus (who flourished about 400 B.C.) is due the discovery that the volume of a pyramid is one-third that of the prism having the same base and equal height; the first rigorous proof of this result was given by Eudoxus. The general formula for the volume of a frustum of a pyramid $(h/3)(A_{1}+A_{2}+{\sqrt {A_{1}+A_{2}}})$,