qj' certain Bjanc/ies of Analysis. SiS 



quiry. This may perhaps arise from a desire to develope the 

 subject in an entirely elementary manner ; but it should be re- 

 collected, that the arithmetic of sines, or any portion of geo- 

 metry, when treated analytically, is not properly an elementary 

 subject. It is decidedly transcendental, and the whole re- 

 sources of the calculus may be legitimately applied in its de- 

 velopment. If it be conceived that such a mode would render 

 one important part of practical mathematics of difficult ac- 

 quisition, I answer, that practical subjects, considered as such, 

 must always be developed in a manner correspondingly ele- 

 mentar}'; laut if they are considered as portions of science, there 

 is no reason why they should not take the places which of 

 right belong to them in the course of its regular develojiment. 

 The fact that mechanics must be taught in as elementary a 

 manner as possible, in many institutions in this kingdom for 

 instance, is no argument against the importance of those re- 

 fined analytical investigations of the doctrines of equilibrium 

 and motion : and neither should a similar necessity deprive 

 geometry of a similar treatment. If then two grand transcen- 

 dental subjects were placed side by side, as they ought to be, 

 and alike subjected to the operation of the most recondite as 

 well as the most elementary relations of abstract magnitude, 

 then might we expect a renovation in geometry similar to what 

 has already been effected in mechanics, and a final exclusion of 

 every particular or tentative process. It is my intention to 

 attempt, in the course of a few papers, to exhibit the renovf),- 

 ting and extending effect of general methods upon the arith- 

 metic of sines. I hope to reduce every formula now known in 

 this branch of analysis to a particular case of some extensive 

 class of expressions, derivable by the mere mechanical modi- 

 fication of a still more general method; and thus to enable the 

 humblest calculator to develope into series of any form, an in- 

 finite variety of finite trigonometric functions, almost as rapidly 

 as he can write them down. 



The arithmetic of sines occurs very early in the course of a 

 purely analytical system of geometry. It was while delinea- 

 ting such a system that it came before me, in the form in which' 

 I would present it ; and I allow it to retain the marks of its 

 origin, by giving it as a portion of an extensive inquiry. Se- 

 veral liicts are of course involved, with which the analysis of 

 the previous equations easily furnished me : and the omission 

 of these previous iiuiuiries has compelled me to conform to 

 the coMUJion modes of writing, more closely then I would other- 

 wise have done. 



The liuidamenlal experiment which led to the definition of 

 Vol. 6.7. No. 325. May IK2,'). X x ^ in 



