PHILOSOPHY OF INDUCTIVE INFERENCE. 269 



absolutely certain, and are either exactly true or capable 

 of being carried to any required degree of approximation. 

 In a perfect triangle, the angles must be equal to one half- 

 revolution precisely ; even an infinitesimal divergence 

 would be impossible ; and I believe with equal confidence, 

 that however many are the angles of a figure, provided 

 there are no re-entrant angles, the sum of the angles will 

 be precisely and absolutely equal to twice as many right- 

 angles as the figure has sides, less by four right-angles. 

 In such cases, the deductive proof is absolute and com- 

 plete ; empirical verification can at the most guard against 

 accidental oversights. 



There is a second class of geometrical truths which can 

 only be proved by approximation ; but, as the mind sees 

 no reason why that approximation should not always go 

 on, we arrive at complete conviction. We thus learn that 

 the surface of a sphere is equal exactly to two-thirds of 

 the whole surface of the circumscribing cylinder, or to four 

 times the area of the generating circle. The area of a 

 parabola is exactly two-thirds of that of the circumscribing 

 parallelogram. The area of the cycloid is exactly three 

 times that of the generating circle. These are truths that 

 we could never ascertain, nor even verify by observation ; 

 for any finite amount of difference, vastly less than what 

 the senses can discern, would falsify them. There are 

 again geometrical relations which we cannot assign ex- 

 actly, but can carry to any desirable degree of approxi- 

 mation. Thus, the ratio of the circumference to the 



diameter of a circle is that of 3* 141 59265358979323846 



to i, and the approximation may be carried to any ex- 

 tent by the expenditure of sufficient labour, as many as 607 

 places of figures having been calculated. 11 Some years since, 

 I amused myself by trying how near I could get to this 

 ratio, by the careful use of compasses, and I did not come 



" ' English Cyclopaedia/ art. Tables. 





