SIMSON. 139 



y are so general that they express any numbers, any 

 lines, nay, any ideas, any rewards or punishments, any 

 thoughts of the mind, it is manifest that the square of 

 the differential of a thought, or the differential of the 

 differential of a reward or punishment, has no meaning; 

 and so of every thing else but of the very tangent or 

 the osculating circle's radius : consequently the gen- 

 erality of the symbols is wholly useless ; the particular 

 case of two lines being the only thing to which the 

 expressions can possibly be meant to apply. Why, 

 then, all geometrical symbols should be so carefully 

 avoided when we are really treating of geometrical 

 examples and geometrical ideas, and of these alone, 

 seems hard to understand. 



As the exclusive lovers of modern analysis have 

 frequently and very erroneously suspected the ancients 

 of possessing some such instrument, and concealing 

 the use of it by giving their demonstrations syntheti- 

 cally after reaching their conclusions analytically, so 

 some lovers of ancient analysis have supposed that 

 Sir Isaac Newton obtained his solutions by algebraic 

 investigations, and then covered them with a synthetic 

 dress. Among others, Dr. Simson leant to this opin- 

 ion respecting the ' Principia.' He used to say that 

 he knew this from Halley, by whose urgent advice 

 Sir Isaac was induced to adopt the synthetic form of 

 demonstration, after having discovered the truths ana- 

 lytically. Machin is known to have held the same 

 language; he said that the 'Principia' was algebra in 

 disguise. Assuredly, the probability of this is far 

 greater than that of the ancients having possessed and 

 kept secret the analytical process of modern times. 

 In the preface to his ' Loci Plani,' Dr. Simson fully re- 

 futes this notion respecting the ancients : a notion 

 which, among others, no less a writer than Wallis had 

 strongly maintained.* 



* Algebra Prsef. " Hanc Grsecos olim habuisse non est quod dubite- 

 mus ; sed studio celatam, nee temere propalandam. Ejus effectus (utut 



