116 THE SQUARING OF THE CIRCLE. 



into many geometrical text books, is that published by Kochansky in 

 1685, in the Leipziger Berichte. It is as follows : 



Erect upou the diameter of a circle at its extremities perpendiculars ; with the center 

 as vertex, mark oflf npou the diameter au angle of 30° ; find the point of intersection 

 with the perpendicular of the line last drawn, and join this point of intersection with 

 that point upon the other perpendicular, which is at a distance of three radii from the 

 base of the perpendicular. The line of junction thus obtained is then very approxi- 

 mately equal to one-half of the circumference of the given circle. 



Calculation shows that the difference between the true length of the 

 circumference and the line thus constructed is less than yoo^o-oo o^ t^® 

 diameter. 



Inutility of constructive approximations.— Although such construc- 

 tions of approximation are very interesting in themselves, they never- 

 theless play but a subordinate role in the history of the squaring of the 

 circle; for on the one hand they can never furnish greater exactness 

 for circle computation than the thirty-five decimal places which Ludolf 

 found, and on the other hand they are not adapted to advance in any 

 way the question whether the exact quadrature of the circle with ruler 

 and compasses is possible. 



The researches of Newton, Leibnitz, Wallis, and Brounclcer. — The 

 numerical side of the problem, however, was considerably advanced 

 by the new mathematical methods perfected by Newton and Leibnitz, 

 commonly called the differential and the integral calculus. And 

 about the middle of the seventeenth century, some time before New- 

 ton and Leibnitz represented iz by series of powers, the English mathema- 

 ticians Wallis and Lord Brouncker, Newton's predecessors in a certain 

 sense, succeeded in representing - by an infinite series ot figures com- 

 bined by the first four rules of arithmetic. A new method of computa- 

 tion was thus opened. Wallis found that the fourth part of - is repre- 

 sented more exactly by the regularly formed product 



|x^xixix^x|x|x, etc., 



the farther the multiplication is continued, and that the result always 

 comes out too small if we stop at a proper fraction, but too large if we 

 stop at an improper fraction. Lord Brouncker, on the other hand, rep- 

 resents the value in question by a continued fraction in which all the 

 denominators are equal to 2 and the numerators are odd square num- 

 bers. Wallis, to whom Brouncker had communicated his elegant result 

 without proof, demonstrated the same in his "Arithmetic of Infinites." 



The computation of - could hardly be farther advanced by these re- 

 sults than Ludolf and others had carried it, though of course in a more 

 laborious way. However, the series of powers derived by the assistance 

 of the differential calculus of Newton and Leibnitz furnished a means 

 of computing tt to hundreds of decimal places. 



Other calculations.— Gregory, Newton, and Leibnitz next found that 

 the fourth part of t: was equal exactly to 



