I4 2 THE SQUARING OF THE CIRCLE. 



Proceeding slowly but surely it was next sought to discover 

 the essential properties which distinguish problems solvable with 

 straight edge and compasses from problems the construction of 

 which is elementarily impossible, that is by employing the postu- 

 lates only. Slight reflection showed, that a problem, to be elemen- 

 tarily solvable, must always be such that the unknown lines of its 

 figure are connected with the known lines by an equation for the 

 solution of which equations of the first and second degree only are 

 requisite, and which can be so arranged that the measures of the 

 known lines will appear as integers only. The conclusion to be 

 drawn from this was that if the quadrature of the circle and conse- 

 quently its rectification were solvable elementarily, the number n, 

 which represents the ratio of the unknown circumference to the 

 known diameter, must be the root of a certain equation, of a very 

 high degree perhaps, but in which all the numbers are whole num- 

 bers; that is, there would have to exist an equation, made up en- 

 tirely of whole numbers, which would be correct if its unknown 

 quantity were made equal to n. 



Since the beginning of this century, consequently, the efforts 

 of a number of mathematicians have been bent upon proving that 

 n generally is not algebraical, that is, that it cannot be the root of 

 an equation having whole numbers for coefficients. But mathe- 

 matics had to make tremendous strides forward before the means 

 were at hand to accomplish this demonstration. After the French 

 Academician, Professor Hermite, had furnished important prepara- 

 tory assistance in his treatise Sur la Fonction Exponenticllc, pub- 

 lished in the seventy-seventh volume of the Comptes Rendus, Pro- 

 fessor Lindemann, at that time of Freiburg, now of Munich, finally 

 succeeded, in June 1882, in rigorously demonstrating that the num- 

 ber n is not algebraical,* and so supplied the first proof that the 



* For the benefit of my mathematical readers I shall present here the most 

 important steps of Lindemann's demonstration. M. Hermite in order to prove the 

 transcendental character of 



, _!_ , J_ , Jt , _1 , 



*~ 1 ~*~ fa "*" 1.2.8 "** 1.2.8.4 "*" ' * 



developed relations between certain definite integrals (ComQtes Rendus of the 

 Paris Academy, Vol. 77, 1873). Proceeding from the relations thus established, 



