THEORY OF ArPJlOXIMATlOX. 79 



torsion, which is again only true for infinitely small 

 angles . 



We need to take great care that in simplifying a 

 problem we do not overlook some circumstance which 

 from peculiar mathematical conditions is of importance. 

 Thus in experiments upon the density of the earth we 

 may treat irregularities of its contour as producing in 

 considerable effects. But a like assumption must not be 

 made concerning irregularities in the strata of the earth 

 at a short distance below the point of experiment . 



Such is the purely approximate character of all our 

 operations that it is not uncommon to find the theo 

 retically worse method giving truer results than the theo 

 retically perfect method. The common pendulum which 

 is not isochronous is better for practical purposes than the 

 cycloidal pendulum which is isochronous in theory, but 

 subject to mechanical difficulties. The spherical form is 

 not the correct form for a speculum or lense, but it differs 

 so slightly from the true form, and is so much more easily 

 produced mechanically, that it is generally best to rest 

 content with the spherical surface. Even in a six-feet 

 mirror the difference between the parabola and the sphere 



is only about of an inch, a thickness which would 



f 10,000 



be taken off in a few rubs of the polisher. Watts 

 ingenious parallel motion was intended to produce recti 

 linear movement of the piston rod. In reality the motion 

 was always curvilinear, but a certain part of the curve 

 approximated sufficiently for his purposes to a straight 

 line. 



Approximation to Exact Lairs. 



Though we can never prove any numerical law with 

 perfect accuracy, it would be a great mistake to suppose 



* Biiiiy, Memoirs of the Royal Astronomical Society, vol. xiv, p. 99. 

 ? Airy, Philosophical Transactions, vol. cxlvi, p. 334. 



