THE SQUARING OF THE CIRCLE. 119 



quadrature of the circle could not be solved; but it also could uot be 

 proved that the problem was insolvable with ruler and compasses, 

 although everybody was conviuced of its insolv^ability. In mathemat- 

 ics, however, a conviction is only justified when supported by incon- 

 trovertible proof; and in the place of endeavors to solve the quadra- 

 ture there accordingly now come endeavors to prove the impossibility 

 of solving the celebrated problem, 



LamberVs contribution. — The first step in this direction, small as it 

 was, was made by the French matliematician Lambert, who proved in 

 the year 1761 that r was neither a rational number nor even the square 

 root of a rational number; that is, that neither r nor the square of - 

 can be exactly represented by a Iraction the denominator and nume- 

 rator of which are whole numbers, however great the numbers be taken. 

 Lambert's proof showed, indeed, that the rectification and the quadra- 

 ture of the circle could not be possibly accomplished in the particular 

 way in which its impossibility was demonstrated, but it still did not 

 exclude the possibility of the problem being solvable in some other more 

 complicated way, and without requiring lurther aids than ruler and 

 compasses. 



The conditions of the demonstration. — Proceeding slowly but surely it 

 was next sought to discover the essential distinguishing properties that 

 separate problems solvable with ruler and compasses, from problems 

 the construction of which is elementarily impossible, that is, by solely 

 employing the postulates. Slight reflection showed that a problem ele- 

 mentarily solvable, must always possess the property of having the 

 unknown lines in the figure relating to it connected with the known 

 lines of the figure by an equation for the solution of which equations of 

 the first and second degree alone are requisite, and which may be so 

 disposed that the common measures of the known lines will appear only 

 as integers. The conclusion was to be drawn from this, that if the 

 quadrature of the circle and consequently its rectification were ele- 

 mentarily solvable, the number -, 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 that appear are whole numbers ; that is, there would have to 

 exist an equation, made up entirely of whole numbers, which would be 

 correct if its unknowu quantity were made equal to n. 



Final success of Professor TAndemann. — Since the beginning of this 

 century, consequently, the efiorts of a number of mathematicians have 

 been bent upon proving that n generally is not algebraical, that is, 

 that it can not be the root of any equation having whole numbers for 

 coefficients. But mathematics had to make tremendous strides for- 

 ward before the means were at hand to accomi)lish this demonstration. 

 After the French academician, Professor Hermite, had furnished im- 

 portant preparatory assistance in his treatise Snr la Fonction Expo- 

 7ientieUe, published in the seventy-seventh volume of the Comptes 



