338 THREE GEOMETRICAL PROBLEMS [CH. XVI 



we have to find a line of length so, such that a? = 2a s . Again, 

 to trisect a given angle, we may proceed to find the sine of the 

 angle, say a, then, if a; is the sine of an angle equal to one-third 

 of the given angle, we have 4a? = 3a; — a. Thus the first and 

 second problems, when considered analytically, require the solu- 

 tion of a cubic equation ; and since a construction by means of 

 circles (whose equations are of the form a? + y" + ax + by + c = 0) 

 and straight lines (whose equations are of the form ax+fiy + 7= 0) 

 cannot be equivalent to the solution of a cubic equation, it is 

 inferred that the problems are insoluble if in our constructions 

 we are restricted to the use of circles and straight lines. If the 

 use of the conic sections is permitted, both of these questions 

 can be solved in many ways. The third problem is different in 

 character, but under the same restrictions it also is insoluble. 



I propose to give some of the constructions which have 

 been proposed for solving the first two of these problems. To 

 save space I shall not draw the necessary diagrams, and in 

 most cases I shall not add the proofs: the latter present but 

 little difficulty. I shall conclude with some historical notes on 

 approximate solutions of the quadrature of the circle. 



The Duplication of the Cube*. 



The problem of the duplication of the cube was known in 

 ancient times as the Delian problem, in consequence of a 

 legend that the Delians had consulted Plato on the subject. 

 In one form of the story, which is related by Philoponusf, it 

 is asserted that the Athenians in 430 B.C., when suffering from 

 the plague of eruptive typhoid fever, consulted the oracle at 

 Delos as to how they could stop it. Apollo replied that they 

 must double the size of his altar which was in the form of a 

 cube. To the unlearned suppliants nothing seemed more easy, 

 and a new altar was constructed either having each of its edges 



* See Historia Problematis de Cuhi Duplication by N. T. Reimer, Gottingen, 

 1798 ; and Historia Problematis Cubi Duplicandi by 0. H. Biering, Copenhagen, 

 1844 : also Das Delische Problem, by A. Sturm, Linz, 1895-7. Some notes on 

 the subject are given in my History of Mathematics. 



t Philojoonus ai Aristotelis Analytica Posteriora, bk. i, ohap. vii. 



