xxi.] THEORY OF APPROXIMATION. 473 



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 micro- 

 meters we become able to detect 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. More careful investigation will show the 

 law to be more and more complicated than was previously 



There is yet another way of explaining the apparent 

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

 regard a portion of it free from any kind of discontinuity, 

 we may represent the character of such portion by an 

 equation of the form 



y = A + Ba; -f Ca; 2 + Dx 3 + 



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 C x 2 has no value appreciable 

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

 be apparent that it possesses curvature, but it will be 

 possible to draw a parabola or ellipse so that the curve 

 shall apparently coincide 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, fourth, and perhaps higher degrees ; 

 that is to say, it corresponds to equations involving the 

 third, fourth, and higher 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 



