CRITICISMS BY BRTTISH WRITERS 267 



expressly proposes, to lay down the principles of the 

 differential calculus, independently of ali considera- 

 tion of infinitely small, or vanishing quantities, of 

 limits, or of fluxions " (p. xviii). While Wood- 

 house considers Lagrange's discussion as very valu- 

 able, he does not find it free from logicai faults. 



William Hales, 1804 



228. As a protest against the new movement and 

 a vindication of Newton from the attacks upon 

 fluxions in the Monthly Review^ William Hales pre- 

 pared a hook, the Analysis F/uxionum, which was 

 published in Maseres' Scriptores Logarithniici^ voi. v, 

 London, 1804. Hales endeavours to show that the 

 doctrine of prime and ultimate ratios is really the 

 same as the doctrine of the limits of the ratios. 

 Hales's fundamental definitions are : 



'' Rationes ultimai sunt limites, ad quos quanti- 

 tatum sine fine decrescentium rationes, i, semper 

 appropinquant ; et, 2, quas propiùs assequi possunt 

 quàm prò data quàvis differentià ; 3, nunquam vero 

 transgredi ; 4, nec priùs attingere, quam quantitates 

 ipsai diminuuntur in infinitum." 



"Momentum est fluentis augmentum aut decre- 

 meniuiìi iìwinentaneuiii\ id est, tempore quam minimo 

 genitum. Estque fluxioni proportionale." 



After Hales's work had gone to press, he became 

 acquainted with Benjamin Robins's Discourse oi 1735, 

 and published in appendices^ numerous extracts from 



^ Maseres, Scriptores Logarif limici, voi. v, pp. 848, 854, 856. 



