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On the Independence of the analytical and geometrical Methods of In- 

 vestigation ; and on the Advantages to be derived from their Sepa- 

 ration. By Robert Woodhouse, A.M. Fellow of Caius College, 

 Cambridge. Communicated by Joseph Planta, Esq. Sec. R.S. 

 Read January 14, 1802. [Phil Trans. 1802, p. 85.] 



The author, in the prefatory part of this paper, points out the dif- 

 ference between the two methods of solving problems, the one using 

 lines and diagrams as the signs of quantity, and making an individual 

 to represent a genus ; and the other employing generic terms and 

 signs, which bear no resemblance to the things signified : and insists 

 that, in order to make the process of deduction distinct, exact, and 

 luminous, only one of the two methods ought to be adhered to. This, 

 he says, has not been sufficiently attended to, expressions and formu- 

 las of the two methods having often been blended together, the con- 

 sequence of which has been much ambiguity and paradox ; since the 

 true method of combining algebraical formulas cannot be well under- 

 stood, unless we duly attend to their true analytical source and com- 

 bination. To show that the language of algebra need not be infected 

 with the mode of expression adopted by geometricians, and that it is 

 of itself an adequate instrument of argumentation, is the principal 

 object of Mr. Woodhouse's paper. And he declares that he has 

 entered on this inquiry, not merely for the sake of gratifying specu- 

 lative curiosity, being firmly of opinion that the process of calculation 

 will be much more direct, sure, and expeditious, if it be duly freed 

 from all foreign encumbrances. 



In order to illustrate and confirm this opinion, he has selected a 

 few cases from those expressions and formulas which are supposed 

 to require for their solution the aid of geometrical theorems, and of 

 the properties of curves. 



From purely analytical principles he has given demonstrations ; 

 1st, of the integrals of a series for the sine of an arc in terms of the 

 arc ; 2ndly, of the expression for the root of a cubic equation in the 



irreducible case ; Srdly, of the resolution of the series x + a , &c., 



into quadratic factors ; and, 4thly, of the series for the chord, sine, 

 cosine, &c. of a multiple arc, in terms of the chord, sine, &c. of the 

 simple arc. These demonstrations the author presumes to be direct 

 and rigorous, which advantages, he asserts, are in a great measure 

 owing to the deductions being expressed in algebraical language, and 

 effected throughout by analytical processes. 



The paper concludes with a brief comparison of the ancient geo- 

 metry and modern analysis respecting the advantages of perspicuity 

 and commodious calculation. The result of this comparison is, that 

 some of the excellencies of the former science have been exaggerated, 

 and others deemed essential, which in fact are only accidental. If 

 the object of mathematical study be chiefly recreation, and the ex- 

 ercise of our mental faculties, our author admits that the finest 

 examples of reasoning are to be found in the works of the ancient 



