Page:Proceedings of the Royal Society of London Vol 1.djvu/90

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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 oftenbeen 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 adapted by geometriciws, 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; lst, 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; 3rdly, of the resolution of the series .232 i 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 are. 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 geometry 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 exercise of our mental faculties, our author admits that the ﬁnest examples of reasoning are to be found in the works of the ancient