x. Proceedings. [December gth, 1920. 



copy of an older document of the Xllth Dynasty, roughly 

 1900 B.C. 



The notation was cumbrous, there being separate signs only 

 for one and for the various powers of ten, the numbers in 

 between being represented by repetitions of these signs. Thus 

 37 was written with 3 tens and 7 units. 



Fractions were employed, but only those whose numerator 

 was unity. The only exception to this rule was two-thirds, 

 which was regarded as one over one and a half. 



Tables for multiplication by 2 existed, and all multiplication 

 by larger numbers had to be achieved indirectly. Thus, to 

 multiply by nine the Egyptian multiplied first by 2, then by 2 

 again, then by 2 once more thus giving 8 times, and then 

 added on the number itself. 



Division by 2 was naturally done by means of the 2-times 

 multiplication table. Division by larger or more complicated 

 numbers was done by trial. 



Simple problems, such as the division of a certain number 

 of loaves between a number of persons, gave little trouble, 

 despite the fractions which they sometimes involved. 



Many of the problems concerned areas. The area of the 

 rectangle was correctly determined as the product of its two 

 sides, and the circle was said to be the square of eight-ninths 

 of its diameter, a very close approximation to the truth. In 

 the case of the triangle, carelessness of the scribe in the draw- 

 ing of a figure and our own ignorance of the exact meaning of 

 an Egyptian mathematical term leaves us uncertain whether 

 they had reached the correct solution or not. 



Among solid figures the parallelopiped was correctly cubed, 

 and the volume of a cylinder given as the product of the area 

 of its base and its height. The Egyptians showed themselves 

 perfectly clear and competent in dealing with the units of 

 different dimensions. Several problems dealt with a method 

 of determining in a form useful to the stonecutter the batter 

 or slope of a pyramid whose base and height are given, a 

 measurement needed in the dressing of the outer facing-blocks. 

 A problem in the Moscow papyrus sets out to determine the 

 volume of a truncated pyramid, but an ambiguity in one of 

 the measurements prevents us from gauging the nearness of 

 the approximation. 



Other practical problems dealt with the exchange of loaves 

 of various sizes against one another or against beer. The 

 basis of calculation was the " cooking strength," or number 

 ol loaves or jugs of beer of a fixed size which could be obtained 

 from a bushel of grain. 



