in Mathematical Reasoning’. 377 
cot 5 E3 Jab og 
in which the numerator is instantly perceived to be the square 
root of the sum of the products of the radii, two by two. In as 
far as regards its form, it does not admit of any further improve- 
ment; but if this expression were to be employed in any enquiry 
into the properties of a triangle, it would be much more convenient 
to use the letters 4, B, C, and a, b, c, to denote respectively the 
angles and sides, and to employ for the radii of these circles, the 
mr 
letters p’, p”, p’”, thus, 
A _ SEER EEE 
2 p jj 
the convenience of so employing the letters 4, B, C, a, b, c, has 
been already noticed, and the letters p’, p’, p’”’, cannot fail, after 
a very short use, to recal the idea of radii, as well as fix the par- 
ticular side, which each circle touches. 
cot 
I have now enumerated what appear to me to be the princi- 
ple causes which exert an influence on the success of mathematical 
reasoning, and have illustrated, with examples, those which were 
susceptible of it. They may be recapitulated in few words. The 
nature of the quantities which form the subject of the science, 
together with the distinctness of its definitions —the power of 
placing in a prominent light, the particular point on which the 
reasoning turns— the quantity of meaning condensed into small 
space—the possibility of separating difficulties, and of combining 
innumerable cases,—together with the symmetry, which may be 
made to pervade the reasoning, both in the choice, and in the 
position of the symbols, are the grounds of that pre-eminence, 
which has invariably been allowed to the accuracy of the conclu- 
sions deduced by mathematical reasoning. 
