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 

 denning 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 

 contrivances for the solution of 

 transcendental problems, or of those 

 where the use of the compass and 

 the ruler do not suffice. Although 

 accurate constructions with a ruler 

 and compass, or with either alone, 

 were known to the ancients only in 

 comparatively small numbers, ap- 

 proximations, and sometimes very 

 close ones, seem to have been 

 known. A very interesting exam- 

 ple is Rb'ber's construction of the 

 regular heptagon, of which we read 

 in the correspondence of Sir W. R. 

 Hamilton with De Morgan (Life of 

 Hamilton, by Graves, vol. iii. pp. 

 141, 534), and which was described 

 by him in the 'Phil. Mag.,' Feb- 

 ruary 1864. The approximation to 

 the correctly calculated figure of 



the true septisection of the circle 

 was so close that he could not 

 discover, up to the 7th decimal, 

 whether the error was in the direc- 

 tion of more or less. On carrying 

 the calculation further, he found the 

 approximation to be such that a 

 heptagon stepped round a circle 

 equal in size to the equator would 

 reach the starting-point within 50 

 feet. The inventor or discoverer 

 of this method Rober, an archi- 

 tect of Dresden supposed that it 

 was known to the ancient Egyptians, 

 and in some form or other con- 

 nected with the plans of the temple 

 at Edfu, but on this point I have 

 obtained no information. The ques- 

 tion is not referred to in Prof. 

 Cantor's ' History of Mathematics." 



