234 THE PRINCIPLES OF SCIENCE. [CIUP. 



yet there may be no general reason why they should be 

 so. The results of deductive geometrical reasoning are 

 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, less than what the 

 senses can discern, would falsify them. 



There are geometrical relations again which we cannot 

 assign exactly, but can carry to any desirable degree of ap- 

 proximation. The ratio of the circumference to the dia- 

 meter of a circle is that of yi4iS9 2 ^>S3S^9793 2 3^4^- 

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

 tent by the expenditure of sufficient labour. Mr. W. 

 Shanks has given the value of this natural constant, known 

 as -TT, to the extent of 707 places of decimals. 1 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 nearer than I part in 540. We might imagine mea- 

 surements so accurately executed as to give us eight or 

 ten places correctly. But the power of the hands and 



1 Proceedings of the Royal Society (1872-3), vol. xxi. p. 319. 



