Introduction to the Analysis of the Infinite

arise: yet all these equations are so constituted that from any one of them one is able to derive all the others. For if an equation between the coordinates is known, one thereby also knows the curve; but if this is known, one can also find the equation between the rectangular coordinates by taking any straight line as the axis and any point therein as the origin of the abscissae. In this chapter, therefore, the manner and method shall be taught by which one can find, from an equation given for a curve, another equation that expresses the nature of this same curve for every other axis and for every other origin of the abscissae. In this way, we shall become acquainted with all the equations that represent the nature of one and the same curve, and thereby be enabled to judge more easily the diversity of curved lines from the diversity of their equations.

Therefore, let an equation between $x$ and $y$ be given from which—if, in Fig. 7, the straight line $RS$ is taken as the axis, the point $A$ as the origin of the abscissae, and $x$ denotes the abscissa $AP$ while $y$ denotes the applicate $PM$—the curved line $CBM$ arises, and consequently the nature of this curved line is expressed by the given equation. If, first of all, the axis is retained, but another point $D$ therein is taken as the origin of the abscissae, so that now the abscissa $DP = t$ corresponds to the point $M$ of the curve, while the applicate $MP = y$ remains the same: an equation between $t$ and $y$ can be found in the following way, which