The third, book of the geometry treats of the construction of equations by geometric curves, and it also contains a new method of resolving biquadratic equations.
The leading principles of algebra were now unfolded, and the notation was brought, from a mere contrivance for abridging common language, to a system of symbolical writ ing, admirably fitted. to assist the mind in the exercise of thought.
The happy idea, indeed, of expressing quantity, and the operations on quantity, by conven tional symbols, instead of representing the first by real magnitudes, and enunciating the second in words, could not but make a great change on the nature of mathematical investigation. The language of mathematics, whatever may be its form, must always consist of two parts ; the one denoting quantities simply, and the other denoting the manner in which the quan tities are combined, or the operations understood to be performed on them. Geometry expresses the first of these by real magnitudes, or by what may be called natural signs ; a line by a line, an angle by an angle, an area by an area, &c. ; and it describes the latter by words. Algebra, on the other hand, denotes both quantity, and the operations on quantity, by the same system of conventional symbols. Thus, in the expression rt--a + = o, the letters a, b, denote quantities, but the terms a .e, &c. de note certain operations performed on those quantities, as well as the quantities themselves; is the ,quantity .v raised to the cube ; and ax' the same quantity .r raised to the square, and then multiplied into a, &c. ; the combination, by addition or subtraction, being also expressed by the signs + And Now, it is when applied to this latter purpose that the algebriic language possesses such exclusive excellence. The mere magnitudes themselves might be represented by figures, as in geometry, as well as in any way whatever ; but the operations they are to be subjected to, if described in words, must be set before the mind slowly, and in suc- _ cession, so that the impression is weakened, and the clear apprehension rendered difficult.
In the algebraic expression, on the other hand, so much meaning is concentrated into a narrow space, and the impression made by all the parts is so simultaneous, that nothing can be more favourable to the exertion of the reasoning powers, to the continuance of their action, and their security against error. Another advantage resulting from the use of the
same notation, consists in the reduction of all the different relations among quantities to the simplest of those relations, that of equality, and the expression of it by equations. This gives a great facility of generalization, and of comparing quantities with one another. A third arises from the substitution of the arithmetical operations of multiplication and division, for the geometrical method of the composition and resolution of ratios. Of the first of these, the idea is so clear, and the work so simple ; of the second, the idea is comparatively so obscure, and the process so complex, that the substitution of the former for the latter could not but be accompanied with great advantage. This is, indeed, what constitutes the great difference in practice between the algebraic and the geometric me thod of treating quantity. When the quantities are of a complex nature, so as to go be yond what in algebra is called the third power, the geometrical expression is so circuitous and involved, that it renders the reasoning most laborious and intricate. The great faci lity of generalization in algebra, of deducing one thing from another, and of adapting the analysis to every kind of research, whether the quantities be constant or variable, finite or infinite, depends on this principle more than any other. Few of the early algebraists seem to have been aware of these advantages.