Functions of a Curve: Leibniz's Original Notion of Functions and Its Meaning for the Parabola

Summary

When the notion of a function evolved in the mathematics of the late seventeenth century, the meaning of the term was quite different from our modern set theoretic definition, and also different from the algebraic notions of the nineteenth century. The main conceptual difference was that curves were thought of as having a primary existence apart from any analysis of their numeric or algebraic properties. Equations did not create curves, curves gave rise to equations. When Descartes published his Geometry [10] in 1637, he derived for the first time the algebraic equations of many curves, but never once did he create a curve by plotting points from an equation. Geometrical methods for drawing each curve were always given first, and then by analyzing the geometrical actions involved in the curve drawing apparatus he would arrive at an equation that related pairs of coordinates (not necessarily at right angles to each other) [20]. Descartes used equations to create a taxonomy of curves [17].

This tradition of seeing curves as the result of geometrical actions continued in the work of Roberval, Pascal, Newton, and Leibniz. Descartes used letters to represent various lengths but did not create any specific system of names. Leibniz, who introduced the term function into mathematics [2], considered six different functions associated with a curve, i.e., line segments or lengths that could be determined from each point on a curve relating it to a given line or axis.