94 THE PRINCIPLES OF SCIENCE. 



the law to be more and more complicated than was pre 

 viously supposed. 



There is yet another way of explaining the apparent 

 results of a complicated law. If we take any curve and 

 regard only a portion of it free from any kind of discon 

 tinuity, we may represent the character of such portion 

 by an equation of the form 



?/ = A + B # + C x 2 + D x* + 



Restrict the attention to a very small portion of the curve, 

 and the eye will be unable to distinguish its difference 

 from a straight line, which amounts to saying that in the 

 portion examined the term 0.x- 2 has no value appreciable 

 by the eye. Take a larger portion of the curve and it wiU 

 be apparent that it possesses curvature, but it will be 

 possible to draw a parabola or ellipse so that the curve 

 shall be apparently coincident with a portion of that 

 parabola or ellipse. In the same way if we take larger 

 and larger arcs of the curve it will assume the character 

 successively of a curve of the third and fourth degrees ; 

 that is to say, it corresponds to equations involving the 

 third and fourth powers of the variable quantity. 



We have arrived then at the conclusion that every phe 

 nomenon, when its amount can only be rudely measured, 

 will either be of fixed amount, or will seem to vary uni 

 formly like the distance between two inclined straight 

 lines. More exact measurement may show the error of 

 this first assumption, and the variation will then appear 

 to be like that of the distance between a straight line 

 and a parabola or ellipse. We may afterwards find that 

 a curve of the third or higher degrees is really required 

 to represent the variation. I propose to call the variation 

 of a quantity linear, elliptic, cubic, quartic, quintic, &c., 

 according as it is discovered to involve the first, second, 

 third, fourth, fifth or higher powers of the variable. It .is 

 a general rule in quantitative investigation that we com- 



