CH. xxi.] THEORY OF APPROXIMATION. 457 



or retarding forces, we could never prove that it moved 

 in a perfect ellipse. To prove the elliptical form we 

 should have to measure infinitely small angles, and in- 

 finitely small fractions of a second ; we should have to 

 perform impossibilities. All we can do is to show that 

 the motion of an unperturbed planet approaches very 

 nearly to the form of an ellipse, and more nearly the 

 more accurately our observations are made. But if we go 

 on to assert that the path is an ellipse we pass beyond 

 our data, and make an assumption which cannot be veri- 

 fied by observation. 



But, secondly, as a matter of fact no planet does move 

 in a perfect ellipse, or manifest the truth of Kepler's laws 

 exactly. The law of gravity prevents its own results 

 from being clearly exhibited, because the mutual pertur- 

 bations of the planets distort the elliptical paths. Those 

 laws, again, hold exactly true only of infinitely small 

 bodies, and when two great globes, like the sun and 

 Jupiter, attract each other, the law must be modified. 

 The periodic time is then shortened in the ratio of the 

 square root of the number expressing the sun's mass, to 

 that of the sum of the numbers expressing the masses of 

 the sun and planet, as was shown by Newton. 1 Even at 

 the present day discrepancies exist between the observed 

 dimensions of the planetary orbits and their theoretical 

 magnitudes, after making allowance for all disturbing 

 causes. 2 Nothing is more certain in scientific method 

 than that approximate coincidence alone can be expected. 

 In the measurement of continuous quantity perfect corre- 

 spondence must be accidental, and should give rise to 

 suspicion rather than to satisfaction. 



One remarkable result of the approximate character of 

 our observations is that we could never prove the existence 

 of perfectly circular or parabolic movement, even if it 

 existed. The circle is a singular case of the ellipse, for 

 which the eccentricity is zero ; it is infinitely improbable 

 that any planet, even if undisturbed by other bodies, 

 would have a circle for its orbit; but if the orbit were a 

 circle we could never prove the entire absence of ecceu- 



1 Principia, bk. iii. Prop. 15. 



2 Lockyer's Lessons in Elementary Astronomy, p. 301. 



