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Second Chapter.

On the Transformation of Coordinates.

§. 23.

Just as a curve, given by an equation between the coordinates $x$ and $y$ (of which the former signifies the abscissae and the latter the ordinates), can be described upon the axis $RS$ (Fig. 2) if the origin of the abscissae $A$ is assumed at will on this axis: so conversely, every curve already described can be expressed by an equation between the coordinates. However, although in this case the curve itself is given, two matters remain left to our discretion, namely the position of the axis $RS$ and the origin of the abscissae $A$. Since these things can be varied in infinitely many ways, infinitely many equations can also be found for one and the same curve; one must not, therefore, immediately conclude from a difference in the equations that there is a difference in the curves expressed by them, although conversely, different curves always yield different equations.

§. 24.

Thus, although by changing the axis and the origin of the abscissae infinitely many equations arise, all of which nevertheless express the nature of one and the same curve: