MACLAURIN'S TREATISE, 1742 187 



never arrive at it. . . . The Author insists on these 

 Subjects,the rather that they are commonly described 

 in very mysterious Terms, and have the most fertile 

 of Paradoxes of any Parts of the higher Geometry.'* 



The ideal of mathematica! rigour, as entertained 

 by eighteenth-century writers, was reached by the 

 Greek geometricians, Euclid and Archimedes. To 

 derive the rules of fluxions by the rigorous methods 

 of the ancients was the ambition of Maclaurin. 

 Barring some obvious slips that are easily remedied, 

 Maclaurin certainly reached the ideal he had set. 

 Nor is this so very strange. Fluxions involve 

 questions concerning limits ; the ancients overcame 

 the difficulties of such questions by their method of 

 exhaustion. It was a rigorous method, but dread- 

 fully tedious. Maclaurin secured his aim at a 

 tremendous sacrifice. His work on fluxions consists 

 of 763 good-sized pages ; the first 590 pages do not 

 contain the notation of fluxions at ali ; they deal 

 with the derivation of the fluxions of different 

 geometrie figures, of logarithms, of trigonometrie 

 functions, also with the discussions of maxima and 

 minima, asymptotes, curvature, and mechanics, in 

 a manner that the ancients might have followed, 

 and with the verbosity of which the ancients are 

 guilty. The consequence was that the work was 

 not attractive reading. 



Maclaurin was fuUy aware of the value of a good 

 notation and case of operation, for he says of the 

 doctrine (p. 575): ''The improvements that have 

 been made by it, either in geometry or in philo- 



