THEORY OF APPROXIMATION. 73 



theoretical point of view, the planets never move according 

 to those laws. Even if we could observe the motions of a 

 planet, of a perfect globular form, free from all perturbing 

 or retarding forces, we could never perfectly prove that it 

 moved in an 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 the 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 may be more or 

 less probable, but cannot be proved, in the strict sense of 

 that term. 



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 very 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 

 planetary 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 a . Even at 

 the present day discrepancies exist between the observed 

 dimensions of the planet's orbits and their theoretical 

 magnitudes, after making allowance for all disturbing 

 causes b . Nothing, in fact, is more certain in scientific 

 method than that approximate coincidence can alone be 

 expected. In the measurement of continuous quantity 



a 'Principia/ bk. III. Prop. 15. 



b See Lockyer's 'Lessons in Elementary Astronomy/ p. 301. 



