OQO PHILOSOPHICAL TRANSACTIONS. [aNNO 1722. 



Qumber of degrees in the angle, and then by finding a fourth proportional to 

 three given quantities; for that will be the measure required. The simplest 

 hyperbolic area may indeed be squared by the same operation taught in the 

 author's 4th proposition; but the simplest elliptic area requires somewhat more: 

 those that are more complex in both kinds, which generally happens, require 

 an additional trouble to reduce them to the simplest; to square them by infinite 

 series is still more operose, and does not answer the end of geometry. On the 

 whole therefore it may deserve to be considered, for what purposes should pro- 

 blems be always constructed by conic areas, unless it be to please or assist the 

 imagination. The design of theoretical geometry differs from problematical ; 

 the former consists in the discovery and contemplation of the properties and 

 relations of figures for the sake of naked truth ; but the design of the latter is 

 to do something proposed, and is best executed by the least apparatus of the 

 former. 



The logometria was first published by the author himself, in the Philos. 

 Trans, of the year 1714, N** 338. But his logometrical and trigonometrical 

 theorems abovementioned, were not published till after his decease. These 

 theorems make the 'id part of the book, and are calculated to give the fluents 

 of fluxions, reduced to 18 forms, by measures of ratios and angles; in such a 

 manner, that any person may perfectly comprehend their construction and use, 

 though altogether unacquainted with curvilinear figures, as expressed by equa 

 tions. And this circumstance also renders their application to the analysis and 

 construction of problems extremely easy. Of this kind the author has given a 

 great many choice examples, both in abstract and physical problems; which 

 make up the third and last part of the book. 



The author, a little before his decease has informed us (in a letter of May 5, 

 J 716, written to his friend Mr. Jones, " That geometers had not yet promoted 

 the inverse method of fluxions, by conic areas, or by measures of ratios and 

 angles, so far as it is capable of being promoted by those methods. There is 

 an infinite field (says he) still reserved, which it has been my fortune to find an 

 entrance into. Not to keep you longer in suspense, I have found out a general 

 and beautiful method by measures of ratios and angles, for the fluent of any 



quantity which can come under this form ^r^ , in which d^ej' 



are any constant quantities, z the variable, n any index, 6 any whole number 



J* 

 affirmative or negative, - any fraction whatever. The fluents of this form, 



Sn — I 6n + ^— 1 



which have hitherto been considered, are -. — & ^ : these vou 



