120 THE SQUARING OF TFIE CIRCLE. 



Rendns, Professor Lindemann, at tbat time of Freiburg, now of Konigs- 

 berg, finally succeeded, in .Tune, 1882, in rigorously demonstrating that 

 the number ;: is not algebraical,* thus supplying the first proof that 

 the problems of the rectification and the squaring of the circle, with 

 the help only of algebraical instruments like ruler and compasses are 

 insolvable. Lindemann's proof appeared successively in the Reports 

 of the Berlin Academy (June, 1882), in the Comptcs Rendus of the 

 French Academy (vol. cxv, pp. 72-74), ard in the Mathematuclien An- 

 nalen (vol. xx, pp. 213-22.5). 



The verdict of mathematics. — "It is impossible with ruler and com- 

 passes to construct 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, un- 

 mindful of the verdict of mathematics, that most infalin)le of arbiters, 

 will never die out so long as ignorance and the thirst ior glory shall be 

 united. 



* For the benefit of ray mathematical readers I shall present here the most impor- 

 tant steps of Lindemann's demonstration, M. Hermite in order to jirove the transcen- 

 dental character of 



^-^ + 1+172 + iT^ + L2.3.4 + • • • • 



developed relations between certain definite integrals (Comptes Rendus of the Paris 

 Academy, 1873, vol. Lxxvii). Proceeding from the relations thus established, Pro- 

 fessor Lindemann first demonstrates the following proposition: If the coetficients of 

 an equation of wth degree are all real or complex whole numbers and the n roots of this 

 equation Zi, z^, . . ., ^„ are ditferent from zero and from each other it is impossible for 



to be equal to r, where a and h are real or complex whole numbers. It is then shown 

 that also between the functions 



,rz,j^^rz,^^rz,_^ . . . . e »'^", 



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

 variant from zero. Finally the beautiful theorem results: If z is the root of an irre- 

 ducible algebraic equation the coefficients of which are real or complex whole num- 

 bers, then f^ can not be equal to a rational number. Now, in reality e^ '^ ~ is equal 

 to a rational number, namely, — i. Consequently, it-\/ — 1, and therefore itself, can- 

 not be the root of an equation of nth degree having whole numbers for coefficients, 

 and therefore also not of such an equation having rational coefficients. If the squar- 

 ing of the circle with ruler and compasses were possible, however, it would have the 

 property last mentioned. 



