122 Oh the Duplication of the Cube. 



sive field of speculative science, not much frequented 

 even by mathematicians. I would only observe, that 

 geometricians have, long since, demonstrated, generally,, 

 the impos^bility of solving geometrical problems of the 

 second degree, or order, by any lines of the first order, 

 since these cannot be so combined, as to involve the 

 more complicated conditions and relations, necessarily 

 implied in such problems. 



The other problem, which is the subject of Mr. Win- 

 throp's second paper, is that of the trisection of an an- 

 gle, to which though equally capable of a solution by right 

 lines, as that of the duplication of the cube, a dissimilar 

 one has been given from the consideration of lines of a 

 superior order, in a manner consonant to the strict prin- 

 ciples of geometrical constructions, and which appears 

 to be not less novel than ingenious. The author, how- 

 ever, has omitted to investigate the nature and specific 

 properties of the curve, called by him, the trisecting 

 curve ; but it is easily shown, that it is no other than the 

 common hyperbola: For, using his scheme, suppose 

 VC,*^ the curve, AB the directrix, V the vertex, P 

 any point in the curve ; from P, draw Pd perpendicular 

 to F V ; and VA, PI perpendicular to the directrix AB ; 

 draw the line FP from the focus F to the point P ; put 

 VA=a, Vd=x, then PI=a + x, FP=2a+2x, and 

 Fd=2a- — x; let Pd^y. Nov/ in the right-angled tri- 

 angle PdF, Pd^ (y^)=FF^~Fd^- (2a4-2x^-^2t=x^) = 

 12 a x -f 3 x"" ; or y=v''l2ax-f 3 x''. This equation is 

 that of the common hyperbola, whose axes have a given 

 ratio commensurable in power, and therefore js very 

 easily constructed in the following manner. 



From a given point C, in a right line CD, draw two 

 right lines CK, CL, indefinitely on each side, making 

 with CD an angle of 60°, or with one another, an angle 

 of 120°: Set off from C, the distance CV =2 a ; then 

 between the two assymptotes CK, CL, and through the 

 vertex V, construct an hyperbola, and this will ht. the 

 curve required, or what is called by Mr. Winthrop, the 

 trisecting curve. 



* See Fig. 2, Plate 1. 



