120 THE SQUARING OV THE CIRCLE. 



L'enilKs, Professor Lindeinann, at tbat time of Freiburg, now of Konigs- 

 iHMji, liiiall.v succeeded, in dime, 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 comi)asses are 

 insolvable. Lindemann's i)roof appeared successively in the Reports 

 of the Berlin Academy (June, 1882), in the Comptcft licndns of the 

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

 nalen (vol. xx, pp. 2i;i-22o). 



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 infallible of arbiters, 

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

 united. 



• For tbe benefit of my inatheinatical readers I hIkiII prewoiit here the most imjior- 

 tant steps of Lindemaini's demoustratiou, M. Hermite iu order to prove the transcen- 

 dental character of 



^ - ^ + i + l.t> + 1.2.\i + 1.2.3.4 + • • • • 



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

 Academy, 1H73, vol. LXXVli). Proceeding from the relations thus established, Pro- 

 fessor Lindemanii first demonstrates the following proposition: If tl»e coefficients of 

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

 equation Zi, z-i, . . ., 2„ are difiereut from zero and from each other it is impossible for 



e^' + e ^■- + «'='.... +e^» 



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

 that also between the functions 



e '■'' + , '-'-^ + , '--^ + . . . . c '''", 



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

 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 r^ can not be equal to a rational number. Now, in reality e"^' is equal 

 to a rational number, namely, — i. Consequently, 7r-\/ — 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, n would have the 

 property last mentioned. 



