486 SIMSON. 



siderations ; for although x and 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 osculating circle's radius : 

 consequently the generality 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 synthe- 

 tically 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 

 opinion 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 refutes 



