Page:A Source Book in Mathematics.djvu/662

 Since, furthermore the method of investigating indefinite quadratures or their impossibilities is with me only a special case (and indeed an easier one) of the far greater problem which I call the inverse method of tangents, in which is included the greatest part of all transcendental geometry; and because it could always be solved algebraically, all things were looked upon as discovered; and nevertheless up to the present time I see no satisfactory result from it; therefore I shall show how it can be solved no less than the indefinite quadrature itself. Therefore, inasmuch as algebraists formerly assumed letters or general numbers for the quantities sought, in such transcendental problems I have assumed general or indefinite equations for the lines sought, for example, the abscissa and the ordinate [being represented] by the usual $$x$$ and $$y$$, my equation for the line sought is,

$0 =a+ bx + cy + exy + fx^2 + gy^2$, etc.;

by the use of this indefinitely stated equation, I seek the tangent to a really definite line (for it can always be determined, as far as need be), and comparing what I find with the given property of the tangents, I obtain the value[s] of the assumed letters, $$a$$, $$b$$, $$c$$, etc., and even establish the equation of the line sought, wherein occasionally certain [things] still remain arbitrary; in which case innumerable lines may be found satisfying the question, which was so involved that many, considering the problem as not sufficiently defined at last, believed it impossible. The same things are also established by means of series. But, according to the calculation to be effected, I use many things, concerning which [I shall speak] elsewhere. And if the comparison does not succeed, I decide that the line sought is not algebraic but transcendental. Which being done, in order that I may discover the species of the transcendence (for some transcendentals depend upon the general section of a ratio, or upon logarithms, others upon the general section of an angle, or upon the arcs of a circle, others upon other more complex indefinite questions); therefore, besides the letters $$x$$ and $$y$$, I assume still a third, as $$v$$, which signifies a transcendental quantity, and from these three I form the general