THEORY OF APPROXIMATION. 93 



can measure. Unless any of the quantities D, E, &c., should 

 happen to be very great, it is evident that all the suc- 

 ceeding terms will also be inappreciable, because the 

 powers of x become rapidly smaller in geometrical ratio. 

 Thus when x is made small enough the quantity y seems 



to obey the equation 



y A + B#. 



If x should be made still less, if it should become so 

 small, for instance, as If00 o.ooo of an inch, and B should 

 not be very great, then y would appear to be the fixed 

 quantity A, and would not seem to vary with x at all. 

 On the other hand, were x to grow greater, say equal to 

 YO inch, and C not be very small, the term C x 2 would 

 become appreciable, and the law would now be more 

 complicated. 



We can invert the mode of viewing this question, and 

 suppose that while the quantity y undergoes variations 

 depending on many powers of x, that our power of de- 

 tecting the changes of value is more or less acute. While 

 our powers of observation remain very rude and imperfect 

 we may even be unable to detect any change in the 

 quantity at all, that is to say B x may always be smaller 

 than to come within our notice, just as in former days 

 the fixed stars were so called because they remained at 

 apparently fixed distances from each other. With the 

 use of telescopes and micrometers we become able to de- 

 tect the existence of some motion, so that the distance of 

 one star from another may be expressed by A + B x, the 

 term including x 2 being still inappreciable. Under these 

 circumstances the star will seem to move uniformly, or in 

 simple proportion to the time, x. With much improved 

 means of measurement it will probably be found that this 

 uniformity of motion is only apparent, and that there 

 exists some acceleration or retardation due to the next 

 term. More and more careful investigation will show 



