114 THE SQUARING OF THE CIRCLE. 



matical jtoiiitof view tlian tlio work of Lndolf. In bis book Cychmetri- 

 cus iSiK'Utook the position tliat the method of comparison of polygons, 

 whirh originated with Archimedes and was employed by Lndolf, need 

 by no means be the best method of attaining tho end songht; and he 

 sneceedeil, by the employment of propositions wliich state that certain 

 arcs of a circle are greater or smaller than certain straight lines con- 

 nected with the circle, in obtaining metiiods that make it possible to 

 reach resnlts like the Lndoltian with much less labor of calcnlation. 

 The beantifnl theorems of Snell were proved a second time, and better 

 proved, by the celebrated Dntcli promoter of the science of optics, 

 n ny gens (0/)t'm Varia, pp. 3G5 et seq. ; Thcoremata J>e Circnli et Hy- 

 perbolae Quadraturaj 1651), as well as perfected in many ways. Snell 

 and Ilnygens were fnlly aware that they had advanced only the prob- 

 lem of nnmerical qnadratnre, and not that of the constrnctive qnadra- 

 tnre. This, in Hnygens's case, plainly appeared from the vehement 

 (lispnte he conducted with the English mathematician, James Gregory. 

 Tliis controversy has some significance for the history of our problem, 

 from the fact that Gregory made the first attempt to prove that the 

 squaring of the circle with ruler and comi)asses must be impossible. 



The controversy heticeen Huygens and Gregory. — The result of the con- 

 troversy, to which we owe many valuable treatises, was that Huygens 

 finally demonstrated in an incontrovertible manner the incorrectness of 

 Gregory's proof of impossibility, adding that he also was of opinion 

 that the solution of the problem with ruler and compasses was impossi- 

 ble, but nevertheless was not himself able to demonstrate this fact. 

 And Newton, later, expressed himself to a similar effect. As a matter 

 of fact it took till the most recent period, that is over 200 years, until 

 higher mathematics was far enough advanced, to furnish a rigid dem- 

 onstration of impossibility. 



Before we proceed to consider the promotive influence which the in- 

 vention of the differential and the integral calculus had upon our prob- 

 lem, we shall enumerate a few at least of that never-ending line of 

 mistaken quadrators who have delighted the world by the fruits of their 

 ingenuity from the time of Newton to the present period ; and out of a 

 pious and sincere consideration for the contemporary world, we shall 

 entirely omit in this to speak of the circlesquarers of our own time. 



Hohbcs\s quadrature. — First to be mentioned is the celebrated English 

 philosopher Hobbes. In his book, De Problematis Physicis, in which he 

 chietly proposes to explain the phenomena of gravity and of ocean 

 tides, he also takes up the quadrature of the circle and gives a very 

 trivial construction that in his opinion definitively solved the problem, 

 making -=3!!. In view of Hobbes's importance as a philosopher, two 

 mathematicians, lluygeus and Wallis, thought it i)roper to refute 



