THE SQUARING OP THE CIRCLE. 105 



many i one later employed. It is not known how Ahmes or his pre- 

 decessors arrived at tliis approximate quadrature; but it is certain 

 that it was handed down in Egypt from century to century, and in late 

 Egyptian times it repeatedly appears. 



The Biblical and Babylonian quadratures. — Besides among the Egyp- 

 tians we also find in pre-Grecian antiquity an attempt at circle-compu- 

 tation among the Babylonians. This is not a quadrature ; but aims at 

 the rectification of the circumference. The Bablyonian mathematicians 

 had discovered that if the radius of a circle be successively inscribed 

 as chord within its circumference, after the sixth inscription we arrive 

 at the point of departure, and they concluded from this that the circum- 

 ference of a circle must be a little larger than a line which is six times 

 as long as the radius, that is, three times as long as the diameter. A 

 trace of this Babylonian method of computation may even be found in 

 the Bible; for in i Kings vii, 23, and ii Chrou. iv, 2, the great laver is 

 described, which under the name of the "molten sea" constituted an 

 ornament of the Temple of Solomon; and it is said of this vessel that it 

 measured 10 cubits from brim to brim, and 30 cubits roundabout. The 

 number 3 as the ratio between the circumference and the diameter is 

 still more plainly given in the Talmud, where we read that "that which 

 measures three lengths in circumference is one length across." 



Among the Greel's. — With regard to the earlier Greek mathemati- 

 cians — as Thales and Pythagoras — we know that they acquired the 

 foundations of their mathematical knowledge in Egypt. But nothing 

 has been handed down to us which shows that they knew of the old 

 Egyptian quadrature, or that they dealt with the problem at all. But 

 tradition says that subsequently the teacher of Euripides and Pericles, 

 the great philosopher and mathematician Anaxagoras, whom Plato 

 80 highly praised, "drew the quadrature of the circle" in prison, in the 

 year 434. This is the account of Plutarch in the seventeenth chapter 

 of his work "De Exilio." 



Anaxagoras. — The method is not told as in which Anaxagoras had 

 supposably solved the i^roblem, and it is not said whether knowingly 

 or unknowingly he accomplished an approximate solution after the 

 manner of Ahmes. But at any rate, to Anaxagoras belongs the merit 

 of having called attention to a problem that bore great fruit, in having 

 incited Grecian scholars to busy themselves with geometry, and thus 

 more and more to advance that science. 



The quadratrix of Eippias of Illis. — Again, it is reported that the 

 mathematician Hippias of Elis invented a curved line that could be 

 made to serve a double purpose; first, to trisect an angle, and, second, 

 to square the circle. This curved line is the rer payiovi^ooaa so often men- 

 tioned by the later Greek mathematicians, and by the Romans, callod 

 " quadratrix." Regarding the nature of this curve we have exact knowl- 

 edge from Pappus. But it will be sufficient, here, to state that the 

 quadratrix is not a circle nor a portion of a circle, so that its construe- 



