114 LIMITS AND FLUXIONS 



yet not arrived at any magnitude, like the infini- 

 tesimals of differential calculus. But this is con- 

 trary to the express words of Sir Isaac Newton, 

 who after he had shewn how to assign by his 

 method of prime and ultimate ratios the proportion, 

 that difìerent fluxions have to one another, he thus 

 concludes. In finitis autem quantitatilms Analysin 

 sic institueì'e et finitarum nascentium vel evafiescentiufn 

 rationes p7'imas vel ultimas investigare consonum est 

 geo7netrice veteì-uni: et volui estendere^ quod in methodo 

 fluxionum non opus sit figuras infinite pai-vas in 

 geometriam introducei-e.'' (See our §§ 33, 41.) 



130. Robins proceeds to explain that the method 

 of prime and ultimate ratios is ''no other.than the 

 abbreviation and improvement of the form of 

 demonstrating used by the ancients on the like 

 occasions. " It has nothing to do with infinitely 

 small quantities, which have led into error not only 

 Leibniz in studying the resistance of fluids and the 

 motion of heavenly bodies, but also Bernoulli like- 

 wise in the resistance of fluids and in the study of 

 isoperimetrical curves. Such infinitely small quanti- 

 ties led Parent to make wrong deductions. It was 

 argued that because a heavy body descends through 

 the chord of a circle terminating at its lowest point 

 in the same time as along a vertical diameter, " the 

 time of the fall through the smallest arches must be 

 equal to the time of the fall through the diameter." 

 To relieve Newton of the suspicion of not being 

 free from the obscure methods of indivisibles, 

 Robins says he [Robins] defined an ** ultimate 



