THEORY OF APPROXIMATION. 77 



are indefinitely move numerous and complex than the 

 few larger terms which are retained. All then is merely 

 approximate. 



Concerning other branches of physical science the same 

 general statements are even more evidently true. We 

 speak and calculate about inflexible bars, inextcnsible 

 lines, heavy points, homogeneous substances, uniform 

 spheres, perfect fluids and gases, and we deduce an infinite 

 number of beautiful theorems; but all is hypothetical. 

 There is no such thing as an inflexible bar, an inextensible 

 line, nor any one of the other perfect objects of mechanical 

 science ; they are to be classed with those other almost 

 mythical existences, the straight line, triangle, circle, 

 rectangle, &c., about which Euclid so freely discoursed. 

 Take the simplest operation considered in statics the use 

 of a crowbar in raising a heavy stone, and we shall find, 

 as Thomson and Tait have pointed out, that we neglect 

 far more than we observe d . If we suppose the bar to be- 

 quite rigid, the fulcrum and stone perfectly hard, and the 

 points of contact real points, we might give the true re 

 lation of the forces. But in reality the bar must bend, 

 and the extension and compression of different parts in 

 volve us in difficulties. Even if the bar be homogeneous 

 in all its parts, there is no mathematical theory capable of 

 determining with accuracy all that goes on ; if, as is in 

 finitely more probable, the bar is not homogeneous, the 

 complete solution will be indefinitely more complicated, 

 but hardly more hopeless. No sooner had we determined 

 the change of form according to simple mechanical prin 

 ciples, than we should discover the interference of thermo- 

 dynamic principles. Compression produces heat and 

 extension cold, and thus the conditions of the problem 

 are modified throughout. In attempting a fourth ap 

 proximation we should have to allow for the conduction 

 d Tmiti.sc on Natural Philosophy. vol. i. pp. 337- &c - 



