VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 67I 



from the centre to a right line given in position, increasing or diminishing the 

 logarithms of those rays always in a given ratio, and increasing or diminishing 

 the angles contained by them and the perpendicular in the same ratio. From 

 any figure of this kind a series of figures is derived by determining the inter- 

 sections of the tangents of the figure with the perpendiculars from the centre. 

 Every series of this kind gives 2 distinct sort of fluents; and any one fluent 

 being given, all the other fluents taken alternately from it in the series depend 

 on it, or are measured by it ; but it does not appear that ihe fluents of one 

 sort can be compared with those of the other sort, or with those of any dif- 

 ferent series of this kind. 



The inverse method is prosecuted farther in the 4th chapter, by various 

 theorems concerning the area when the ordinate is expressed by a fluent, or 

 when the ordinate and base are both expressed by fluents. The first is the J 1th 

 proposition of Sir Isaac Newton's Treatise of Quadratures. In art. 81 9, 820, 

 &c. the author supposes the ordinate and base to be both expressed by fluents, 

 and shows, in many cases, that the area may be assigned by the product of two 

 simple fluents, as of two circular arcs, or of a circular arc and a logarithm. 

 This subject deserves to be prosecuted, because the resolution of problems is 

 rendered more accurate and simple, by reducing fluents to the products of 

 fluents already known, than by having immediately recourse to infinite series. 

 One of the examples in art. 822 may be easily applied for demonstrating, that 

 the sum of the fractions which have unit for their common numerator, and the 

 squares of the numbers 1, 2, 3, 4, 5, 6, &c. in their natural order, for their 

 successive denominators, is -^ part of the number, which expresses the ratio of 

 the square of the periphery of a circle to the square of its diameter; which is 

 deduced by Mr. Euler, Comment. Petropol. tom. 7, in a different manner, 

 and other theorems of this kind may be demonstrated from the same or like 

 principles. 



The series that is deduced by the usual methods for computing the area or 

 fluent, converge in some cases at so slow a rate, as to be of little or no use 

 without some farther artifice. For example: the sum of the first 1000 terms 

 of Lord Brounker's series, for the logarithm of 2, is deficient in the 3th deci- 

 mal. In order therefore to render the account of the inverse method more 

 complete, the author shows how this may be remedied, in many cases, by 

 theorems derived from the method of fluxions itself, which likewise serve for 

 approximating readily to the values of progressions, and for resolving problems 

 that are commonly referred to other methods. Those theorems had been de- 

 scribed in the first book, art. 352, &c. but the demonstration and examples 

 were referred to this place, as requiring a good deal of computation. The base 



