478 THE PRINCIPLES OF SCIENCE. [CHAP. 



and the polygonal line becomes undistinguishable from a 

 perfect curve. 1 



In abstract mathematical theorems the approximation 

 to absolute truth is perfect, because we can treat of in- 

 finitesimals. In physical science, on the contrary, we 

 treat of the least quantities which are perceptible. Never- 

 theless, while carefully distinguishing between these two 

 different cases, we may fearlessly apply to both the prin- 

 ciple of the superposition of small effects. In physical 

 science we have only to take care that the effects really 

 are so small that any joint effect will be unquestionably 

 imperceptible. Suppose, for instance, that there is some 

 cause which alters the dimensions of a body in the ratio 

 of I to I + a, and another cause which produces an alter- 

 ation in the ratio of I to I + ft. If they both act at once 

 the change will be in the ratio of I to (i + a)(i + ft), 

 or as i to I + a + ft + aft. But if a and ft be both very 

 small fractions of the total dimensions, aft will be yet far 

 smaller arid may be disregarded ; the ratio of change is 

 then approximately that of i to i + a + ft, or the joint 

 effect is the sum of the separate effects. Thus if a body 

 were subjected to three strains, at right angles to each 

 other, the total change in the volume of the body would 

 be approximately equal to the sum of the changes pro- 

 duced by the separate strains, provided that these are very 

 small. In like manner not only is the expansion of every 

 solid and liquid substance by heat approximately propor- 

 tional to the change of temperature, when this change is 

 very small in amount, but the cubic expansion may also 

 be considered as being three times as great as the linear 

 expansion. For if the increase of temperature expands 

 a bar of metal in the ratio of i to i + a, and the expansion 

 be equal in all directions, then a cube of the same metal 

 would expand as i to (i + a) 1 , or as I to I + 3a + 30? + a 3 . 

 When a is a very small quantity the third term 3 a 2 will 

 be imperceptible, and still more so the fourth term a 3 . 

 The coefficients of expansion of solids are in fact so 

 small, and so imperfectly determined, that physicists 

 seldom take into account their second and higher powers. 



1 Cballis, Notes on the Principles of Pure and Applied Calculation 

 1869, P. 83. 



