xviii FOURTH REPORT 1834. 



undertaking it, we must consider the production of a work of 

 so much labour as the present, which, as yet, is incomplete, but 

 which the author has promised to resume, as the best trophy 

 to which we can refer in proof of the entire efficiency of the As- 

 sociation according to its original plan, — as a proof of the 

 ability and the indefatigable industry which it has enlisted in 

 its service, — as a proof that its aim is not the dissemination of 

 superficial literature, stamped with the effigy of science, and 

 lowered for the demand of the indolent and the careless, — but 

 that it is intended to refine the precious metal until it reaches 

 a state of chemical purity, not to alloy and coin it for the pur- 

 poses of a promiscuous and debased currency. Mr. Peacock 

 undertook his report in the early days of the Association, when 

 its friends were yet few and its success dubious ; its execution 

 has been delayed by the extent of the subject and labour of the 

 task. The report on the Dilferential and Integral Calculus, which 

 v.as intended to form the basis of it, is delayed, and the present 

 one is devoted to a discussion chiefly of algebraic methods, and 

 a close examination of the metaphysical principles upon which 

 this interpretation of analysis is founded. The author has thus 

 been led to extend the views which, in his recent systematic 

 treatise, he had developed in regard to the signs of affection of 

 algebraic quantities, including those of imaginary quantities, of 

 discontinuous functions, and the interpretations of zero and infi- 

 nity. The author has then treated of Series, as regards their 

 fitness for giving directly conclusive results, particularly when 

 such series are divergent, leaving to the other part of the report 

 a detail of the progress in the application of series, which is 

 more practical than metaphysical. The author then treats hi- 

 storically of the elementary works in use on Algebra and Trigo- 

 nometry ; and devotes the last part of the report, consisting of 

 above fifty pages, to the Theory of Equations, in which he has 

 minutely analysed some of the most remarkable papers on this 

 abstruse subject. Altogether this report (especially when com- 

 pleted,) cannot fail to fulfill, in a striking manner, the two great 

 objects of such works : first, to supply those engaged in colla- 

 teral branches of science with the means of referring to and ob- 

 taining the information they may require upon methods which 

 perhaps are of daily utility in physico-mathematical inquiries, 

 but with which, from the vast extent of the science of pure ma- 

 thematics, the shortness of human life prevents the possibility 

 of a complete and systematic acquaintance, unless it be made 

 the special object of study ; and, in the second place, to point 

 out, where chasms of reasoning occur, what mathematical me- 

 thods are impregnable, and what rest upon a still dubious basis, 



