462 THE PRINCIPLES OF SCIENCE. [CHAP. 



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 one ten-thousandth part of an inch, a thick- 

 ness which would be taken off in a few rubs of the polisher. 

 Watts' ingenious parallel motion was intended to produce 

 rectilinear movement of the piston-rod. In reality the 

 motion was always curvilinear, but for his purposes a 

 certain part of the curve approximated sufficiently to a 

 straight line. 



Approximation to Exact Laws. 



Though we can not prove numerical laws with perfect 

 accuracy, it would be a great mistake to suppose that 

 there is any inexactness in the laws of nature. We 

 may even discover a law which we believe to represent 

 the action of forces with perfect exactness. The mind 

 may seem to pass in advance of its data, and choose out 

 certain numerical results as absolutely true. We can 

 never really pass beyond our data, and so far as assump- 

 tion enters in, so far want of certainty will attach to our 

 conclusions ; nevertheless we may sometimes rightly prefer 

 a probable assumption of a precise law to numerical results, 

 which are at the best only approximate. We must accord- 

 ingly draw a strong distinction between the laws of nature 

 which we believe to be accurately stated in our formulas, 

 and those to which our statements only make an approxi- 

 mation, so that at a future time the law will be differently 

 stated. 



The law of gravitation is expressed in the form 



F = -p> meaning that gravity is proportional directly to 



the product of the gravitating masses, and indirectly to the 

 square of their distance. The latent heat of steam is ex- 

 pressed by the equation log F = a + &' + c/3', in which are 

 five quantities a, I, c, a, ft, to be determined by experiment. 

 There is every reason to believe that in the progress of 

 science the law of gravity will remain entirely unaltered, 

 and the only effect of further inquiry will be to render it a 

 more and more probable expression of the absolute truth. 

 The law of the latent heat of steam on the other hand, will 



