CHAP, xxi] IMPOSSIBILITY OF z 3 +?/ 3 = 2 3 . 547 



A. G. Kastner 13 checked the construction by use of trigonometric 

 functions and logarithmic tables. 



I. K. Hagner 14 set a = BC,b = GH,c~ GF. Then 



GR = , FA* = FR-HF=(b- c) 

 Having GA~, we see that BA and AC are 



Equating this to a 3 , and writing 46c+a 2 = (a+2/) 2 , which gives c = (a +/)//&, 

 we get 



62 = 4{2a 2 +(a+/)(a+2/)r 



By the expression for BA, we must have b>c, whence /<a(0.29- ) 

 The value /=a/4 gives Glenie's solution. Taking /=(& 3/2)a, we see 

 that the expression for 6 2 is the square of {3 6&+5A; 2 /2ja/(4/c 2 6&+6) if 

 A; = 24/23, whence 6 = 5a/38. If in Euler's 8 equation* 2p(p 2 +3q*) =z\ we 

 set 2p = rz, we obtain q, whence 



rz z^4:r 3 



and see why the cubic equation is solved by use of a curve of order 2. For 

 r = 3/2, we get Glenie's case. 



C. F. Hauber 15 proved Glenie's construction and solved 



, P * m 



x 3 +2r=-i x-\-y = a 

 q n 



for re, ?/ and discussed their geometric constructibility, but made no dis- 

 cussion as to rationality. 



J. W. Becker 16 gave a construction simpler than Glenie's, as he avoided 

 irrationals. Take a circle of radius 772 = 152, lay off #G=124, #F = 279, 

 draw perpendiculars FA and BC to IR to cut the circle at the vertices 

 A, B, C of the required triangle (see above figure). For Prob. 2, take 

 772 = 639, 72G = 198, 7^ = 550. For Prob. 3, take 772 = 5, RG = 1, 72F = 4. 

 In general, let the sum of the cubes of the sides equal e times the cube of 

 the base a. Denote the sum of the sides by as, the difference by ad. Thus 



He asked if s can be chosen to make d rational, stating it to be impossible 



13 Archiv der reinen u. angewandten Math, (ed., Hindenburg), 1, 1795, 352-6, 481-7. 



14 Ibid., 2, 1798, 448-457. 

 16 Ibid., pp. 458-70. 



16 Ibid., 471-80. 



