TRANSCENDENTAL, or TRANSCEN DANT, something elevated or raised above other things; which passes and transcends the nature of other, inferior things. Tran scendental quantities, among geometri cians, are indeterminate ones, or such as °winot be fixed or expressed by any con staM equation: such are all transcendental curves, which cannot be defined by any algebraic equation ; or which, when ex pressed by all equation, one of the terms thereof is a variable quantity. Now whereas algebraists use to assume some general letters or numbers, for the quan tity sought in these transcendental prob lems, Mr. Leibnitz assumes general or indefinite equations, for the lines sought ; e. sr. putting x and y for the absciss and ordinate; the equation he uses for a line sought, is a+bx+.cy+ex y +fxx +g y y, by the help of which in definite equation he seeks the tangent ; and by comparing the result with the given property of tangents, he finds the value of the assumed letters, a, b, c, he. and thus defines. the equation of the line sought. _ , If the comparison above-mentioned do not proceed, be pronounces the line sought, not to be an algebraical, hut a transcendental one. This supposed, he goes on to find the species of transcen dency : for some trabscendentals depend on the general division or section of a ra tio, or upon the logarithms ;. others upon the arcs of a circle ; and others on more indefinite and compound enquiries. He therefore, besides the symbols,. a- and y, assumes a third, as v, which denotes the transcendental quantity ; and of these three, forrnsa general equation thr the line sought, from which lie finds the tangent, according to the differential method, which succeeds even in transcendental quantities. The result he compares with the given properties of the tangent, and so discovers, not only the values of a, b, c, d, &c. but also the particular nature of the transcendental quantity. And though it may sometimes happen, that the several transcendent& are so to be made use of, and those of different natures too, one from another ; also, though there be transcendents of transcendentals, and a of these in infinitum ; yet we may he satisfied with the most easy and useful one ; and for the most part, may have. recourse to some peculiar artifices
for shortening the calculus, and reducing the problem to as simple terms as may be, , This method being applied to the busi ness of quadratures, or to the invention of quadratics, in which the property of the tangent is always given, it is manifest, not only how it may be discovered, whether the indefinite quadrature may be alge braically impossible ; but also, how, when this impossibility is discovered, a trans cendental qnadratrix may be found, which is a thing not before shown. So that it seems, that geometry, by this method, is carried infinitely beyond the bounds to which Vieta and Des Cartes brought it ; since, by this means, a certain and general analysis is established, which extends to all problems of no certain degree, and consequently not comprehended within algebraical equations.
Again, in order to manage transcen dental problems, wherever the business of tangents or quadratures occurs, by a calculus, there is hardly any that can be imagined shorter, more advantageous, or more universal, than the differential cal culus, or analysis of indivisibles and infi Mtes. By this method we may explain the nature of transcendental lines, by an equation ; e. gr. let a be the arch of a cir cle, and x the versed sine ; then will S d x a = ; and if the ordinate of q/ the cycloid be y, then will y 2 x — S d x x x ± ; which equation per 2 x—xx fectly expresses the relation between the ordinate, p, and the absciss, x, and from itall the properties of the cycloid may be demonstrated.
• Thus is the analytical calculus extend. ed to those lines which have hitherto been excluded ; for no other reason, but that they were thought incapable of it. *