Dr. Adamson on Geometrical Demonstrations. 297 



on each conductor increases gradually from the point of contact 

 to the remotest points of the two surfaces. 



P.S. The calculation by the method shown in the preceding 

 paper,, of the various quantities required for determining the 

 force between two spheres of equal radii (each unity), insulated 

 with their centres at distances 2*1, 2*2, 2*3, &c. up to 4, has 

 been undertaken, and is now nearly complete. I hope to be 

 able to communicate the results for publication in the next or in 

 an early number of the Magazine. 



Glasgow College, March 21, 1853. 



XL VI. On a proposed Test of the Necessity of Indirect Proof in 

 Geometrical Demonstrations , with Remarks on Methods of 

 Demonstration. By James Adamson, D.D. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen^ 



AS you have in some late numbers of the Philosophical Maga- 

 zine admitted the discussion of matters which are elemen- 

 tary in regard to geometry, but are of some interest in regard to 

 the nature of geometrical argument, and as I have been called 

 upon by peculiar circumstances to analyse that subject as a mode 

 of intellectual education, I trust that you w411 permit me to offer 

 to your notice a few of the conclusions to which I have been led. 



Sir William Hamilton, Professor of Logic in the University of 

 Edinburgh, has expressed himself as estimating mathematical 

 science at no high value for such an end. It may perhaps be 

 shown that less benefit than is attainable has been derived from 

 it, not so much on account of the nature of the subject itself, as 

 from the mode in which it has been treated. 



In the Number of the Philosophical Magazine for December 

 1852, Mr. Sylvester has noticed a theorem regarding isosceles 

 triangles, the discussion of which appears to have been attended 

 with circumstances somewhat singular; and from his examina- 

 tion of it he has deduced a principle offered as a test for distin- 

 guishing those cases of geometrical argument in which the pro- 

 cess is necessarily indirect. The circumstances remarked as sin- 

 gular are, that mathematicians of eminence had found the demon- 

 stration to be difficult, and that they had been able to establish 

 its truth only by modes which are indirect. It was therefore 

 taken by Mr. Sylvester as fitted to illustrate a suggestion by him, 

 that the nature of the equations connected with such geometrical 

 truths may determine the question whether the demonstrations 

 admit, or do not admit, of direct modes of argument. 



In regard to these positions, I would beg leave to observe, 

 that, to a mind trained to correct modes of geometrical reasoning, 



Phil Mag, S. 4. Vol. 5. No. 32. April 1853. X 



