1843.] Treatment of Geometry as a branch of Analysis. 113 



and hence all three to two right angles. The proof is then extended 

 to triangles in general, by dividing them into right angled triangles. 



4. The publication of this train of reasoning excited a discussion un- 

 precedented in the cold calm regions of science, and one which assumed 

 a character of acrimony, that can only be accounted for by the politi- 

 cal antipathies which extended even to the schools of mathematics. 

 Ivory, Leslie, Playfair, Brewster, Maurice, Nieuport, and the great 

 author himself, took prominent parts in the controversy. It is not my 

 intention to raise, or lay the ghosts of departed objections. Stated in 

 the manner I have done, divested of the appalling formalities of a func- 

 tional investigation, there are only two points in Legendre's proof over 

 which the reader will pause for an instant. 



The first is, why will geometric determination afford any grounds 

 for numerical calculation? This is easily answered. The remaining 

 elements being geometrically given, their proportions to the data are 

 given, that is, a series of numbers being assumed for these last, a series 

 of numbers for the rest are found. Hence the necessity of supposing a 

 numerical process connecting the consequent numbers with the as- 

 sumed ones. 



The second is of a graver character. It is suggested at the place 

 where, having settled that the calculation of the third angle involves 

 only the magnitudes of the other two, we conclude that two triangles, 

 having two angles equal each to each, will also have the third angles 

 equal. This conclusion is evidently founded on the assumption that 

 there is an invariable formula of calculation for all triangles, connecting 

 the third angle with the other two. The question is, having assured 

 ourselves that the triangle ABC has a formula connecting the angle 

 C with A and B, what grounds have we to suppose that the same 

 formula will be applicable to A'B'C ? The fairest mode of meeting 

 the query I conceive to be this. When a base is laid down and lines 

 are drawn making given angles with it, we perceive intuitively that the 

 system is fixed. The magnitude of the base and base angles is not a 

 constituent of this fixity. They may vary, but the conception of deter- 

 mination remains not the less distinct. To express this fact analyti- 

 cally, we must say that the magnitudes in the triangular system vary 

 inter se, but the laws which connect their respective variations are 

 invariable and universal. Hence we conclude that every geometrical 



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