106 THE SQUARING OF THE CIRCLE. 



tion is not possil)]e by means of the i)Ostuljites enumerated in the 

 preceding section. And therefore the sohition of the quadrature of the 

 circle founded on the construction of the quadrat rix is not an elementary 

 solution in the sense discussed in the last section. We can, it is true, 

 couceive a mechanism that will draw this curve as well as compasses 

 draw a circle; and with the assistance of a mechanism of this descrip- 

 tion the squaring of the circle is solvable with exactitude. But if it be 

 allowed to employ in a solution an apparatus especially adapted thereto, 

 every problem may be said to be solvable. Strictly taken, the invention 

 of the curve of Hippias substitutes for one insuperable difficulty another 

 equally insuperable. Sometime afterwards, about the year 350, the 

 mathematician Dinostratus showed that the qiiadratrix could also be 

 used to solve the problem of rectification, and from that time on this 

 problem plays almost the same role in Grecian mathematics as the 

 related problem of quadrature. 



The Sophists' solution. — As these problems gradually became known to 

 the non-mathematicians of Greece, attempts at solution at once sprang 

 up that are worthy of a place by the side of the solutions of modern ama- 

 teur circle-squarers. The Sophists, especially, believed themselves com- 

 petent by seductive dialectic to take a stronghold that had defied the 

 intellectual onslaughts of the greatest mathematicians. With verbal 

 nicety, amounting to puerility, it was said that the squaring of the circle 

 depended upon the finding of a number which represented in itself both 

 a square and a circle ; a square by being a square number, a circle in 

 that it ended with the same number as the root number from which, by 

 multiplication with itself, it was produced. The number 36, accord- 

 ingly, was, as they thought, the one that embodied the solution of the 

 famous problem. 



Contrasted with this twisting of words the speculations of Bryson and 

 Antiphon, both contemporaries of Socrates, though inexact, appear in 

 high degree intelligent. 



Antiphon's attempt. — Antiphon divided the circle into four equal arcs, 

 and by joining the points of division obtained a square; he then divided 

 each arc again into two equal parts and thus obtained an inscribed octa- 

 gon; thence he constructed an inscribed dodecagon, and perceived that 

 the figure so inscribed more and more approached the shape of a circle. 

 In this way, he said, one should proceed, until there was inscribed in 

 the circle a polygon whose sides by reason of their smallness should 

 coincide with the circle. Now this polygon could, by methods already 

 taught by the Pythagoreans, be converted into a square of equal area; 

 and upon the basis of this fact Antiphon regarded the squaring of the 

 circle as solved. 



Nothing can be said against this method except that, however far 

 the bisection of the arcs is carried, the result must still remain an ap- 

 proximate one. 



Bryson of Heraklea .—The attempt of Bryson of Heraklea was better 

 still ; for this scholar did not rest content with finding a square that was 



