GEOMETRY: ANCIENT AND MODERN. 259 



The pursuit of the impossible seems to have an irresistible attraction for 

 some minds. This remark applies only to the modern devotees of the 

 subject, however. The Greeks did not know that the thing they sought 

 was an impossibility. To square the circle, to trisect an angle and to 

 duplicate the cube were three problems upon which the Greeks lavished 

 more attention probably than upon any others. It was not labor 

 wasted, because it led incidentally to many theorems, which otherwise 

 might have remained unknown, but the principal object sought was not 

 attained. To make matters clear it should be stated that to meet the 

 requirements of Greek geometry the instruments used in the solution 

 must be only the compasses and the unmarked straight edge. So that 

 to square the circle meant to construct by these means the side of a 

 square whose area should equal that of a given circle. The Greeks 

 eventually succeeded in solving the last two problems by the aid of 

 curves other than the circle, but this, of course, was unsatisfactory. As 

 we know now they were pursuing ignes fatui. Nevertheless it is 

 brought to the knowledge of mathematicians with painful frequency 

 that a vast amount of energy is still wasted upon these problems, espe- 

 cially the first. Let me, therefore, take the space here to repeat that 

 squaring the circle is not simply one of the unsolved problems of 

 mathematics which is awaiting the happy inspiration of some genius, 

 but that it has been ably demonstrated to be incapable of solution in 

 the manner proposed. 



When Euclid compiled his 'Elements' the knowledge of geometry 

 current amongst the Greeks was about the same as that which we have 

 to-day under the name of elementary geometry. The term Euclidean 

 geometry has a somewhat different signification, which will be ex- 

 plained below. 



About a century before Euclid's time the Greeks discovered the 

 conic sections, and Apollonius of Perga, who lived about a century after 

 Euclid, brought the geometry of these curves to a high degree of per- 

 fection. Archimedes, whose time was intermediate between that of 

 Euclid and of Apollonius, had a more practical turn of mind and applied 

 his mathematical knowledge to useful purposes. Amongst other things 



he showed that the value of n lies between 3- and 3=^ that is, 



between 3.1429 and 3,1408, a closer approximation than the Egyptian. 

 We see, therefore, that in the few centuries during which the Greeks 

 occupied themselves with the study of geometry the knowledge of the 

 right line, circle and conic sections reached about as high a state of de- 

 velopment as it was possible to attain until the invention of more pow- 

 erful methods of research, and many centuries were destined to elapse 

 before this was to occur. I do not overlook the fact that the beautiful 

 and extensive modern geometry of the triangle and the systems of re- 



