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Report on the Recent Progress and Present State of certain 

 Branches of Analysis. By George Peacock, M.A., F.R.S., 

 F.G.S., F.Z.S., F.R.A.S., F.C.P.S., Fellow and Tutor of 

 Trinity College, Cambridge. 



The present Report was intended in the first instance to have 

 comprehended some notice of the recent progress and present 

 state of analytical science in general, including algebra, the 

 apphcation of algebra to geometry, the differential and integral 

 calculus, and the theory of series : a very httle progress, how- 

 ever, in the inquiries which were required for the execution of 

 this undertaking convinced me of the necessity of confining 

 them within much narrower limits, unless I should have ven- 

 tured to occupy a much larger space in the annual pubhcation 

 of the Proceedings and Reports of the British Association than 

 could be properly or conveniently allotted to one department 

 of science, when so many others were required to be noticed. 

 It is for these reasons that I shall restrict my observations, 

 in the following Report, to Algebra, Trigonometry, and the 

 Arithmetic of Sines ; at the same time I venture to indulge a 

 hope of being allowed, upon some future occasion, to bring 

 before the Members of the Association some notice of those 

 higher branches of analysis which at present I feel myself 

 compelled, though reluctantly, to omit. 



Algebra. — The science of algebra may be considered under 

 two points of view, the one having reference to its principles, 

 and the other to its apphcations : the first regards its complete- 

 ness as an independent science ; the second its usefulness and 

 power as an instrument of investigation and discovery, whether 

 as respects the merely symbolical results which are deducible 

 from the systematic developement of its principles, or the ap- 

 plication of those results, by interpretation, to the physical 

 sciences. 



Algebra, considered with reference to its principles, has re- 

 ceived very little attention, and consequently very little im- 

 provement, during the last century; whilst its applications, 

 using that term in its largest sense, have been in a state of 

 continued advancement. Many causes have contributed to this 

 comparative neglect of the accurat^e and logical examination of 

 the first principles of algebra : in the first place, the proper 



