Introduction to the Analysis of the Infinite

likewise expresses the nature of the curve C B M. One sets AD = f, whereby, since AD lies toward the left, DP = t = f + x, and consequently x = t — f, and then introduces this value of x into the equation given between x and y. Since the magnitude of? AD = f is left to our discretion, one already obtains in this way innumerable equations, which all express precisely the same curved line.

§. 26.

If the curve intersects the axis R S at any point, e.g., at C, one obtains, by taking this point C as the origin of the abscissae, an equation which, if the abscissa C P = 0, also gives the applicate P M = 0, provided that no more than one applicate belongs to the point C of the axis. The point C, however—whether it be the only intersection point or whether there be several of them—is found from the first given equation between x and y by setting y = 0 and then developing the value or values of x therefrom. For just as y becomes 0 where the curve intersects the axis, so also, if one sets? y = 0, one must find from the equation all those abscissae or values of x where the curve meets the axis.

§. 27.

The origin of the abscissae is therefore changed, while retaining the axis, if one increases or diminishes the abscissa x by a given magnitude, i.e., if one sets t — f for x; and in this case f is a positive magnitude if one takes the new origin of the abscissae D on the left side of A, and a negative one if