348 THREE GEOMETRICAL PROBLEMS [CH. XVI 



conclusion as to the quadrature of the circle is to be drawn, 

 nor did Newton draw any. In the earlier editions of this work 

 I expressed an opinion that the result presupposed a particular 

 definition of the word oval, but on more careful reflection I think 

 that the conclusion is valid without restriction. 



With the aid of the quadratrix, or the conchoid, or the 

 cissoid, the quadrature of the circle is easy, but the construction 

 of those curves assumes a knowledge of the value of ir, and thus 

 the question is begged. 



I need hardly add that, if ir represented merely the ratio 

 of the circumference of a circle to its diameter, the deter- 

 mination of its numerical value would have but slight interest. 

 It is however a mere accident that ir is defined usually in 

 that way, and it really represents a certain number which 

 would enter into analysis from whatever side the subject was 

 approached. 



I recollect a distinguished professor explaining how different 

 would be the ordinary life of a race of beings born, as easily 

 they might be, so that the fundamental processes of arithmetic, 

 algebra and geometry were different to those which seem to us 

 so evident, but, he added, it is impossible to conceive of a 

 universe in which e and ir should not exist. 



I have quoted elsewhere an anecdote, which perhaps will 

 bear repetition, that illustrates how little the usual definition 

 of ir suggests its properties. De Morgan was explaining to an 

 actuary what was the chance that a certain proportion of some 

 group of people would at the end of a given time be alive ; and 

 quoted the actuarial formula, involving ir, which, in answer to 

 a question, he explained stood for the ratio of the circumference 

 of a circle to its diameter. His acquaintance, who had so far 

 listened to the explanation with interest, interrupted him and 

 exclaimed, " My dear friend, that must be a delusion, what can 

 a circle have to do with the number of people alive at the end 

 of a given time ? " In reality the fact that the ratio of the 

 length of the circumference of a circle to its diameter is the 

 number denoted by ir does not afford the best analytical defini- 

 tion of ir, and is only one of its properties. 



