sict. i.] DISSERTATION SECOND. 37 



Of the works that appeared in the early stages of the 

 calculus, none is more entitled to notice than the Harmo- 

 nia Mensurarum of Cotes. The idea of reducing the areas 

 of curves to those of the circle and hyperbola, in those cases 

 which did not admit of an accurate comparison with rectilineal 

 spaces, had early occurred to Newton, and was very fully 

 exemplified in his Quadrature of Curves. Cotes extended 

 this method : — his work appeared in 1722, and gave the 

 rules for finding the fluents of fractional expressions, whe- 

 ther rational or irrational, greatly generalized and highly 

 improved by means of a property of the circle discovered 

 by himself, and justly reckoned among the most remarka- 

 ble propositions in geometry. It is singular that a work 

 so profound, and so useful as the Harmonia Mensurarum, 

 should never have acquired, even among the mathemati- 

 cians of England, the popularity which it deserves ; and 

 that, on the Continent, it should be very little known, 

 even after the excellent commentary and additions of 

 Bishop Walmsley. The reasons, perhaps, are, that, in ma- 

 ny parts, the work is obscure ; that it does not explain 

 the analysis which must have led to the formula contain- 

 ed in the tables ; and that it employs an unusual language 

 and notation, which, though calculated to keep in view the 

 analogy between circular and hyperbolic areas, or between 

 the measures of angles and of ratios, do not so readily ac- 

 commodate themselves to the business of calculation as those 

 which are commonly in use. Demoivre, a very skilful and 

 able mathematician, improved the method of Cotes ; and ex- 

 plained many things in a manner much more clear and ana- 

 ly tical than had hitherto been done. 1 



1 Demoivre, Miscellanea Analytica. See also thn work of 

 an anonymous author. Epistola ad Amicum <lc Colcsii Inventis. 



