288 THIRD REPORT — 1833. 



exist in other languages : and the labours of Gauss, Bessel, and 

 Jacobi, and the numerous and important memoirs which appear 

 in their public Journals and Transactions upon the most difficult 

 questions of analysis and the physical sciences, sufficiently show 

 that the mathematical literature of this most learned nation is 

 not less diligently and successfully cultivated than that which 

 belongs to every other department of human knowledge. 



The combinatorial analysis, which Hindenburg first intro- 

 duced, has been cultivated in Germany with a singular and 

 perfectly national predilection *; and it must be allowed that it 

 is well calculated to compress into the smallest possible space 

 the greatest possible quantity of meaning. In the doctrine of 

 series it is also frequently of great use, and enables us to ex- 

 hibit and to perceive relations which would not otherwise be 

 easily discoverable. Without denying, however, the advantages 

 which may attend either the study or the use of the notation of 

 the combinatorial analysis, it may be very reasonably doubted 

 whether those advantages form a sufficient compensation for 

 the labour of acquiring an habitual command over the use and 

 interpretation of a conventional symbolical language, which is 

 necessarily more or less at variance with the ordinary usage and ' 

 meaning of the symbols employed and of the laws of their com- 

 binations. These objections would apply, if such a conven- 

 tional use of symbolical language was universally adopted and 

 understood ; but they acquire a double force and authority, 

 when it appears that they are only partially used in the only 

 country f in which the combinatoi'ial analysis is extensively 

 cultivated, and that, consequently, those works in which it is 

 adopted are excluded from general perusal, in consequence 

 of their not being written in that peculiar form of symbolical 

 language with which our mathematical associations ai'e indis- 

 solubly connected. 



Trigonometry. — The term Trigonometry sufficiently indicates 

 the primitive object of this science, which was the determina- 

 tion, from the requisite data, of the sides and angles of trian- 

 gles : it was in fact considered in a great degree as an inde- 



* See Eytelwein's Grundlehre der hohern Analysis, avery voluminous work, 

 which contains the principal results of modern analysis and of the theory of 

 series exhibited in the language and notation of this analysis. 



\ Professor Jarrett, of Catherine Hal], Cambridge, in some papers in the 

 Transactions of the Philosojihical Society of Cambridge, and in a Treatise on 

 Algebraical Developement, has attempted to introduce the use of the lan- 

 guage of the combinatorial analysis. The great neglect, however, which has 

 attended those speculations, which are very general and in some respects 

 extremely ingenious, is a sufficient proof of the difficulty of overcoming those 

 mathematical habits which a long practice has generated and confirmed. 



