114 THE SQUARING OF THE CIRCLE. 



of mathematical demonstration ; he takes it for granted that things 

 are so because they seem so to him. Errors of logic, also, abound 

 in his book. But, minor fallacies apart, wherein does the real error 

 of this "unmasker" of Archimedes consist? It requires consider- 

 able labor to extricate the kernel of the demonstration from the 

 turgid language and bombastic style in which the author has buried 

 his conclusions. But it is this. The author inscribes a square in a 

 circle, circumscribes another about it, then points out that the inside 

 square is made up of four congruent triangles, whereas the circum- 

 scribed square is made up of eight such triangles ; from which fact, 

 seeing that the circle is larger than the one square and smaller than 

 the other, he draws the bold conclusion that the circle is equal in 

 area to six such triangles. It is hardly conceivable that a rational 

 being could infer that something which is greater than 4 and less 

 than 8 must necessarily be 6. But with a man that attempts the 

 squaring of the circle this kind of ratiocination is possible. 



It is the same with all the other attempted solutions of the 

 problem ; in all of them either logical fallacies or violations of ele- 

 mentary arithmetical or geometrical truths can be pointed out. 

 Only they are not always of such a trivial nature as in the book 

 just mentioned. 



Let us now inquire into the origin of this propensity which 

 leads people to occupy themselves with the quadrature of the circle. 



Attention must first be called to the antiquity of the problem. 

 A quadrature was attempted in Egypt 500 years before the exodus 

 of the Israelites. Among the Greeks the problem never ceased to 

 play a part that greatly influenced the progress of mathematics. 

 And in the middle ages also the squaring of the circle sporadically 

 appears as the philosopher's stone of mathematics. The problem 

 has thus never ceased to be dealt with and considered. But it 

 is not by the antiquity of the problem that circle-squarers are en- 

 ticed, but by the allurement which everything exerts that is cal- 

 culated to raise the individual above the mass of ordinary human- 

 ity, and to bind about his temples the laurel crown of celebrity 

 Ambition spurred men on in ancient Greece and still spurs them 

 on in modern times to crack this primeval mathematical nut. 



