THE SQUARING OF THE CIRCLE. 



H3 



problems of the rectification and squaring of the circle with the 

 help only of algebraical instruments like straight edge and com- 

 passes are insolvable. Lindemann's proof appeared successively 

 in the Reports of the Berlin Academy (June, 1882), in the Comptes 

 Rendus of the French Academy (Vol. 115, pp. 72 to 74), and in the 

 Mathematische Annalen (Vol. 20, pp. 213 to 225). 



"It is impossible with straight edge and compasses to con- 

 struct a square equal in area to a given circle." These are the 

 words of the final determination of a controversy which is as old as 

 the history of the human mind. But the race of circle-squarers, 

 unmindful of the verdict of mathematics, the most infallible of 

 arbiters, will never die out as long as ignorance and the thirst for 

 glory remain united. 



Professor Lindemann first demonstrates the following proposition : If the coeffi- 

 cients of an equation of the wth degree are all real or complex whole numbers and 

 the n roots of this equation z, z t , . . ., Z M are different from zero and from each 

 other it is impossible for 



to be equal to -f, where a and b are real or complex whole numbers. It is then 

 shown that also between the functions 



<ri _j_ ^** + # 4. . . . . f " , 



where r denotes an integer, no linear equation can exist with rational coefficients 

 different from zero. Finally the beautiful theorem results : If z is the root of an 

 irreducible algebraic equation the coefficients of which are real or complex whole 

 numbers, then e* cannot be equal to a rational number. Now in reality e*Y~ l is 

 equal to a rational number, namely, i. Consequently, TTV i, and therefore TT 

 itself, cannot be the root of an equation of the nth degree having whole numbers 

 for coefficients, and therefore also not of such an equation having rational coeffi- 

 cients. The property last mentioned, however, TT would have if the squaring of the 

 circle with straight edge and compasses were possible. [The questions involved 

 in the discussions of the last three pages have been excellently treated by Klein in 

 Famous Problems of Elementary Geometry recently translated by Beman and 

 Smith (Ginn & Co., Boston). Lindemann's proof is here presented in a simplified 

 form, and so brought within the comprehension of students conversant only with 

 algebra. 7r.] 





