0)1 the Duplication of the Cube. 121 



point H will not coincide with I ; for CF : CE : : CD : CG 

 by similar triangles, also CD : CG : : CB : CH, and by 

 division CF : C£ : : FB : EH ; but CF : FB : : CD : 



DK by construction ; and consequently CD : DK : : CE : 

 EH ; whence by (Euc. 2. 6.) KH is parallel to DE. 

 Draw Gh, making the angle EGh equal to the angle 

 GEH, then wdil the triangle EGh, be similar and equal 

 to the triangle GEH, and the line Hhg passing through 

 the point h, will be parallel to EG ; (Euc. 39. 1.) whence 

 the series of parallels to FD, DE, EG, commencing at 

 B, Avill terminate at the point g. Now^ if, according to 

 the conditions of the problem, the terms of the propor- 

 tionals be in a ratio, mir.oris aut majoris inequalitatis, or 

 any otherwise to each other, than in a ratio of equality, 

 the point g will not coincide with A, the extremity of the 

 given line CA. For when CG is greater than C£, then 

 the angle CEG is greater than the angle CGE, and 

 GEH (EGh) the supplement of CEG is less than AGE 

 the supplement of CGE, by the angle AGh ; also Eh is 

 less than E A bv the line Ah subtending the angle AGh : 

 but Eh : EA : :' Gg : GA : : EH : EI, and therefore Gg 

 is less than GA, and EH is less than EI. In the same 

 manner, it m.ay be proved that when CG is less than 

 CE, Gg and EH are respectively greater than GA and 

 EI ; as therefore in no condition of the problem, does 

 the point li coincide with the point I (for we consider 

 its conditions to vanish, when the extreme proportionals 

 are equal) and as KH has been proved to be parallel to 

 ED, it is manifest that IK cannot be parallel to P^D, and 

 that the four proportionals made by similar triangles com- 

 mencing with CB, are BC, CK, CH, andCg. This last 

 term not being equal to CA the given extreme, it is evi- 

 dent that this process by similar triangles fails to produce 

 a just solution of the problem. 



The same conclusions might have been deduced from 

 I other principles, which in their application, would have 

 illustrated somethini?; of that relation and harmonv be- 

 tween algebraic and geometrical quantities, which con- 

 I stitute one of the most beautiful theories of the mathe- 

 matics, and from which the laws of geometrical con- 

 1 structions are derived ; but this would lead to an exten- 



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