474 THE PRINCIPLES OF SCIENCE. [CHAP. 



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- 

 mence by discovering linear, and afterwards proceed to 

 elliptic or more complicated laws of variation. The ap- 

 proximate curves which we employ are all, according to 

 De Morgan's use of the name, parabolas of some order 

 or other ; and since the common parabola of the second 

 order is approximately the same as a very elongated 

 ellipse, and is in fact an infinitely elongated ellipse, 

 it is convenient and proper to call variation of the 

 second order elliptic. It might also be called quadric 

 variation. 



As regards many important phenomena we are yet only 

 in the first stage of approximation. We know that the 

 sun and many so-called fixed stars, especially 61 Cygni, 

 have a proper motion through space, and the direction of 

 this motion at the present time is known with some degree 

 of accuracy. But it is hardly consistent with the theory 

 of gravity that the path of any body should really be a 

 straight line. Hence, we must regard a rectilinear path 

 as only a provisional description of the motion, and look 

 forward to the time when its curvature will be detected, 

 though centuries perhaps must first elapse. 



We are accustomed to assume that on the surface of the 

 earth the force of gravity is uniform, because the variation 

 is of so slight an amount that we are scarcely able to 

 detect it. But supposing we could measure the variation, 

 we should find it simply proportional to the height. 

 Taking the earth's radius to be unity, let h be the height 

 at which we measure the force of gravity. Then by the 

 well-known law of the inverse square, that force will be 

 proportional to 



(T+l)' or to g(i-2h + 3h z -4^ + ). 



But at all heights to which we can attain h will be 

 so small a fraction of the earth's radius that 3 h z will 

 be inappreciable, and the force of gravity will seem 

 to follow the law of linear variation, being proportional 

 to I - 2 h. 



