DISCOVERIES OE SIB ISAAC NEWTON. 461 



— what gigantic strides are these ! It was the fashion of the age to 

 hide processes, and offer results without demonstration : the proposi- 

 tions in the Principia are all geometrical (indeed they would other- 

 wise not have been understood for a century,) but there is little 

 doubt most of them were obtained originally by analysis — singu- 

 larly unfortunate both for Newton's fame and for the sake of us who 

 should reap the benefit of his labours. One proposition given with- 

 out demonstration proves that he had mastered the calculus of vari- 

 ations, the invention of which afterwards became the centre-stone 

 of Lagrange's chaplet : in his "rectification of curves" he must have 

 passed through the integrals which now bear Euler's name : a single 

 construction for conic sections would seem to shew that he had anti- 

 cipated one of the most recent and beautiful processes in analytical 

 geometry invented by M. Chasles. Nothing can be more startling 

 than thus, in the apparently impenetrated forest, to come across a 

 mighty tree felled, with "Newton — his mark" plain upon it: some 

 of his propositions remain undemonstrated to this day ; for instance, 

 the general properties he asserts of curves of the third order, (the 

 classificatiou of which is not the least remarkable of his labours,) 

 and also some strange properties of the roots of algebraic equations. 

 In other cases no one has even guessed at the methods by which he 

 obtained his results ; as in the case of that ratio of the oval axes of 

 the moon's orbit, and of the axes of the earth's figure, where he 

 boldly contradicted the then universal opinion that the equatorial 

 was shorter than the polar ; or again, consider this sentence from 

 the 23rd proposition of the third book, when speaking of the pro- 

 gression of the moou's perigee : " Diminui tamen debet motus 

 augis sic inventus in ratione 5 ad 9 vel. 1 ad 2 circiter, ob causam 

 quam hie exponere non vacat" — " for a cause which here I have not 

 leisure to explain;" — this very inequality nearly drove subsequent 

 calculators to reject altogether the Newtonian theory of gravitation, 

 and it was not till the third trial that Clairaut in despair carried his 

 process to a closer approximation and found the next term give him 

 the required result. Equally wonderful is the way in which Newton 

 sets about doing things that would seem to require a century of 

 preparation to solve : nothing seems to stop him — his tread is that 

 of a lion: — " Ex uugue leonem," as Leibnitz said: if he wants an 

 equation solved, he invents a method of approximation for it ; if lie 

 wants an algorithm for annuities, he makes one ; if he wants to 

 explain the precession of the equinoxes, and suspects it to arise 

 from solar and lunar action on the earth's equatorial protuberance, 

 he considers this latter a belt of satellites, and docs it ; if he wants 



