123 THE SQUARING OF THE CIRCLE. 



III. 

 THE EGYPTIANS, BABYLONIANS, AND GREEKS. 



In the oldest mathematical work that we possess we find a rule 

 telling us how to construct a square which is equal in area to a 

 given circle. This celebrated book, the Rhind Papyrus of the Brit- 

 ish Museum, translated and explained by Eisenlohr (Leipsic, 1877), 

 was written, as stated in the work itself, in the thirty-third year of 

 the re'ign of King Ra-a-us, by a scribe of that monarch, named 

 Ahmes. The composition of the work falls accordingly in the period 

 of the two Hyksos dynasties, that is, in the period between 2000 

 and 1700 B. C. But there is another important circumstance to be 

 noted. Ahmes mentions in his introduction that he composed his 

 work after the model of old treatises, written in the time of King 

 Raenmat; whence it appears that the originals of the mathematical 

 expositions of Ahmes are half a thousand years older still than the 

 Rhind Papyrus. 



The rule given in this papyrus for obtaining a square equal to 

 a circle specifies that the diameter of the circle shall be shortened 

 one-ninth of its length and upon the shortened line thus obtained 

 a square erected. Of course, the area of a square of this construc- 

 tion is only approximately equal to the area of the circle. An idea 

 may be obtained of the degree of exactness of this original, primi- 

 tive quadrature by remarking, that if the diameter of the circle in 

 question is one metre in length, the square that is supposed to be 

 equal to the circle is a little less than half a square decimetre too 

 large ; an approximation not so accurate as that made by Archi- 

 medes, yet much more correct than many a one later employed. It 

 is not known how Ahmes or his predecessors arrived at this ap- 

 proximate quadrature ; but it is certain that it was handed down in 

 Egypt from century to century, and in late Egyptian times it ap- 

 pears repeatedly. 



In addition to the effort of the Egyptians, we also find in pre- 

 Grecian antiquity an attempt at circle-computation among the Ba- 

 bylonians. This is not a quadrature, but is intended as a rectifica- 



