722 SCIENTIFIC THOUGHT. 



properties of the number e, the basis of the Napierian or 

 natural logarithms, this number having been shown by 

 Euler to stand in a remarkable arithmetical relation to 

 the number TT a relation which could be very simply 

 expressed if one had the courage to make use of the 

 imaginary unit. As in the instance referred to above, 

 when I dealt with the problem of the solution of the 

 higher order of equations, so also in the case of the three 

 celebrated problems now under review, the reasoning of 

 the mathematicians of the nineteenth century lay largely 

 in proving why these problems were insoluble or in 

 defining those special cases in which they were soluble. 

 Moreover, the labours of Gauss and the class of mathe- 

 maticians who followed or read him were directed 

 towards the defining and fixing of general conceptions, 

 the study and elaboration of which embraced these single 

 problems as special cases. Prime numbers had always 

 been the object of special attention. Division and par- 



an account of several mechanical ' the true septisection of the circle 

 contrivances for the solution of I was so close that he could not 

 transcendental problems, or of those . discover, up to the 7th decimal, 

 where the use of the compass and ! whether the error was in the direc- 

 the ruler do not suffice. Although I tion of more or less. On carrying 

 accurate constructions with a ruler j the calculation further, he found the 

 and compass, or with either alone, | approximation to be such that a 

 were known to the ancients only in j heptagon stepped round a circle 

 comparatively small numbers, ap- i equal in size to the equator would 

 proximations, and sometimes very ' reach the starting-point within 50 

 close ones, seem to have been I feet. The inventor or discoverer 

 known. A very interesting exam- j of this method Rober, an archi- 

 ple is Rober's construction of the . tect of Dresden supposed that it 

 regular heptagon, of which we read was known to the ancient Egyptians, 

 in the correspondence of Sir W. R. and in some form or other con- 

 Hamilton with De Morgan (Life of j nected with the plans of the temple 

 Hamilton, by Graves, vol. iii. pp. j at Edfu, but on this point I have 

 141, 534), and which was described obtained no information. Theques- 

 by him in the 'Phil. Mag..' Feb- : tion is not referred to in Prof, 

 ruary 1864. The approximation to Cantor's ' History of Mathematics.' 

 the correctly calculated figure of 



