THE SQUARING OF THE CIRCLE. 14! 



actly equal in length, to the side of each of these squares, is cast 

 haphazard upon the floor. If we calculate, now, the probabilities 

 of the needle so falling as to lie wholly within one of the squares, 

 that is so that it does not cross any of the parallel lines forming the 

 squares, the result of the calculation for this probability will be 

 found to be exactly equal to n 3. Consequently, a sufficient 

 number of casts of the needle according to the law of large num- 

 bers must give the value of it approximately. As a matter of fact, 

 Professor Wolff, after 10000 trials, obtained the value of n correctly 

 to 3 decimal places. 



Fruitful as the calculus of Newton and Leibnitz was for the 

 evaluation of n, the problem of converting a circle into a square 

 having exactly the same area was in no wise advanced thereby. 

 Wallis, Newton, Leibnitz, and their immediate followers distinctly 

 recognised this. The quadrature of the circle could not be solved ; 

 but it also could not be proved that the problem was insolvable 

 with straight edge and compasses, although everybody was con- 

 vinced of its insolvability. In mathematics, however, a conviction 

 is only justified when supported by incontrovertible proof ; and in 

 the place of endeavors to solve the quadrature there accordingly 

 now come endeavors to prove the impossibility of solving the cele- 

 brated problem. 



The first step in this direction, small as it was, was made by 

 the French mathematician Lambert, who proved in the year 1761 

 that n was neither a rational number nor even the square root of a 

 rational number ; that is, that neither it nor the square of it could 

 be exactly represented by a fraction the denominator and numera- 

 tor of which are whole numbers, however great the numbers be 

 taken. Lambert's proof* showed, indeed, that the rectification and 

 the quadrature of the circle could not be accomplished in one par- 

 ticular simple way, but it still did not exclude the possibility of the 

 problem being solvable in some other more complicated way, and 

 without requiring further aids than straight edge and compasses. 



* Given in Legendre's Geometry, in the Appendix to De Morgan, op. cit., p 



495, and in Rudio, op. cit. Tr. 



