30 ON THE NATURE OF MATHEMATICAL KNOWLEDGE, 



possibility of solving this problem exactly by the use of straight 

 edge and compasses alone, (See pp, 116, 141-143.) 



It is, of course, the insoluble problems that have the longest 

 history; partly because it is harder to show that a thing is impos- 

 sible than that it is possible, and, on the other hand, because prob- 

 lems that have long defied solution are ever evoking anew the spirit 

 of inquiry and the ambition of mathematicians, and because the un- 

 certainty of insolubility lends to such problems a peculiar charm. 

 Of the geometrical problems that have occupied competent and in- 

 competent minds from the time of the ancient Greeks to the present 

 may be mentioned in addition to the squaring of the circle two 

 others that are also perhaps well-known to educated readers, at least 

 by name : the trisection of the angle and the Delic problem of the 

 duplication of the cube. All three problems involve the condition, 

 which is often overlooked by lay readers, that only straight edge 

 and compasses shall be employed in the constructions. In the tri- 

 section of the angle any angle is assigned, and it is required to find 

 the two straight lines which divide the angle into three equal parts. 

 In the Delic problem the edge of a cube is given and the edge of a 

 second cube is sought, containing twice the volume of the first cube. 

 In Greece, in the golden age of the sciences, when all scholars had 

 to understand mathematics, it was a fashionable requisite almost to 

 have employed oneself on these famous problems. 



Fortunately for us, these problems were insoluble. For in their 

 ambition to conquer them it came to pass that men busied themselves 

 more and more with geometry, and in this way kept constantly dis- 

 covering new truths and developing new theories, all of which per- 

 haps might never have been done if the problems had been soluble 

 and had early received their solutions. Thus is the struggle after 

 truth often more fruitful than the actual discovery of truth. So, too, 

 although in a slightly different sense, the apophthegm of Lessing is 

 confirmed here, that the search for truth is to be preferred to its 

 possession. 



Whilst the three above-named problems are now acknowledged 

 to be insoluble, and have ceased, therefore, to stimulate mathemat- 

 ical inquiry, there are of course other problems in mathematics 



