MERITS AND DEFECTS 283 



that a limit actually exists. In this respect the 

 treatment vvas purely intuitive. 



240. Another defect in the British exposition of 



fluxions was in the use of the word ''quantity. " 



No definition of it was given, yet quantities were 



added, subtracted, multiplied, and divided. It is 



possible to treat quantities or magnitudes without the 



use of number. The fifth and tenth books of Euclid's 



Elenients contain such treatment. We may speak of 



the ratio of one magnitude to another magnitude, or 



we may speak of the ratio of one number to another 



number. Which was meant in the treatment of 



fluxions ? Straight Hnes were drawn and the ratios 



of parts of these lines were written down. What 



were these the ratios of? Were they the ratios 



of the Hne-segments themselves, or the ratios of 



the numbers measuring the lengths of these line- 



segments ? No explicit anewer to this was given. 



Our understanding of authors like Maclaurin, Rowe, 



and others is that in initial discussions such phrases 



as *' fluxions of curvih'neal figures," ''fluxion of a 



rectangle," are used in a non-arithmetical sense ; 



the idea is purely geometrical. When later the 



finding of the fluxions of terms in the equations of 



curves is taken up, the arithmetical or algebraical 



conception is predominant. Rarely does a writer 



speak of the difference between the two. Perhaps 



" His notions fitted things so well, 

 That which was which he could not teli." 



241. Analytical geometry practically identìfied 

 geometry with arithmetic. It was tacitly assumed 



