100 THE PRINCIPLES OF SCIENCE. 



may even be doubtful whether he could treat it in any other 

 manner. Nevertheless there is no error in the final results, 

 because having obtained the formulae flowing from this 

 supposition, each straight line is then regarded as be- 

 coming infinitely small, and the polygonal line becomes 

 ^indistinguishable from a perfect curve z . 



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. Neverthe- 

 less, while carefully distinguishing between these two dif- 

 ferent cases, we may fearlessly apply to both the principle 

 of the superposition of small motions or effects. In 

 physical science we have only to take care that the effects 

 really are so small that any joint effect will be unquestion- 

 ably 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 

 alteration 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 -f 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 and 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 of 

 very small amount. In like manner not only is the ex- 

 pansion of every solid and liquid substance by heat 

 approximately proportional to the change of temperature, 

 when this change is very small in amount, but the cubic 



z Challis, ' Notes on the Principles of Pure and Applied Calculation/ 

 1869, p. 83. 



