458 THE PRINCIPLES OF SCIENCE. [CHAP. 



tricity. All that we could do would be to declare the 

 divergence from the circular form to be inappreciable. 

 Delambre was unable to detect the slightest ellipticity 

 in the orbit of Jupiter's first satellite, but he could only 

 infer that the orbit was nearly circular. The parabola is 

 the singular limit between the ellipse and the hyperbola. 

 As there are elliptic and hyperbolic comets, so we might 

 conceive the existence of a parabolic comet. Indeed if an 

 undisturbed comet fell towards the sun from an infinite 

 distance it would move in a parabola ; but we could never 

 prove that it so moved. 



Substitution of Simple Hypotheses, 



In truth -men never can solve problems fulfilling the 

 complex circumstances of nature. All laws and explana- 

 tions are in a certain sense hypothetical, and apply exactly 

 to nothing which we can know to exist. In place of the 

 actual objects which we see and feel, the mathematician 

 substitutes imaginary objects, only partially resembling 

 those represented, but so devised that the discrepancies 

 are not of an amount to alter seriously the character of 

 the solution. When we probe the matter to the bottom 

 physical astronomy is as hypothetical as Euclid's elements. 

 There may exist in nature perfect straight lines, triangles, 

 circles, and other regular geometrical figures ; to our 

 science it is a matter of indifference whether they do or 

 do not exist, because in any case they must be beyond 

 our powers of perception. If we submitted a perfect 

 circle to the most rigorous scrutiny, it is impossible that 

 we should discover whether it were perfect or not. 

 Nevertheless in geometry we argue concerning perfect 

 curves, and rectilinear figures, and the conclusions apply 

 to existing objects so far as we can assure ourselves that 

 they agree with the hypothetical conditions of our 

 reasoning. This is in reality all that we can do in the 

 most perfect of the sciences. 



Doubtless in astronomy we meet with the nearest ap- 

 proximation to actual conditions. The law of gravity is 

 not a complex one in itself, and we believe it with much 

 probability to be exactly true; but we cannot calculate 

 out in any real case its accurate results. The law asserts 



