66 GREEK GEOMETRY. 



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 synthetically after reaching their con- 

 clusions 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 analytically. Machin is 

 known to have held the same language; he said that the 

 ' Principia ' was algebra in disguise. Assuredly, the pro- 

 bability 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 this notion respecting the ancients : a notion which, 

 among others, no less a writer than Wallis had strongly 

 maintained.* 



That he did not undervalue algebra and the calculus to 



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

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

 clam celatse) satis conspicui apud Archimedem, Apollonium, aliosque." It 

 is strange that any one of ordinary reflection should have overlooked the 

 utter impossibility of all the geometricians in ancient times keeping the 

 secret of an art which must, if it existed, have been universally known in 

 the mathematical schools, and at a time when every man of the least 

 learning, or even of the most ordinary education, was taught geometry. 

 Montucla touches on this subject, but not with his wonted accuracy, 

 (1. 166). Indeed, he seems here to confound ancient witli modern analysis, 

 although no one has more accurately described and illustrated the ancient 

 method, (I. 164, 275). He adopts the erroneous notion of Plato having 

 discovered this method ; but he does not fall into the other error of ascrib- 

 ing to Jiim the discovery of Conic Sections, (ib. 168). 

 



