Historical considerations

Perhaps one the earliest presentations of parametric functions in history can be contributed to Euler in his 1748 Introductio in Analysin Infinitorum (Dubinsky et al., 1995). Book II of this text is where we can find Euler’s extensive presentation on the study of curves. Many of Euler’s descriptions of curves are comparable to ideas discussed in the literature review, such as the reports by Keene (2007) and Oehrtman et al. (2008).

Book II, Chapter 1 of Euler’s text was devoted to the discussion of curves in general. In Section 7, he described how a curve is defined by a function:

In this way the curve which results from the function y is completely known, since each of its points is determined by the function y. At each point P, the perpendicular P M is determined, and the point M lies on the curve. Indeed, each point on the curve is found in this way. In whatever way the curve may be viewed, for each point on the curve there is one point on the straight line RS, which is on the perpendicular to the line through the point on the curve. In this way we obtain the intervalAP, which gives the value of x, and the length of the perpendicular P M, which represents the value of the function y. It follows that there is no point of the curve which is not defined by the function y. (Euler, 1990, p. 5)

Immediately following, in Section 8, Euler acknowledged the dynamic aspect of curves: Although many different curves can be described mechanically as a continuously moving point, and when this is done the whole curve can be seen by the eye, still 4_{What is interesting is that none of the textbooks that I reviewed explicitly described parametric functions}

in this way. In fact, none of these texts, includingCalculus, Concepts & Computersby Dubinsky et al. (1995), mention the phrase parametric function. Instead, they use phrases like parametric equation or parametric form.

we will consider these curves as having their origin in functions, since then they will more apt for analytic treatment and more adapted to calculus. (Euler, 1990, p. 5)

In Section 14 of Chapter 1, Euler stated the following:

Since y is a function of X, either y is equal to an explicit function of x, or we have an equation in x and y where y is defined by x. In either case we have an equation which is said to repress the nature of the curve. For this reason every curve is expressed by an equation in two variables x and y. (Euler, 1990, p. 8)

In the previous excerpt Euler expressed that every curve (presumably plane curve) can be expressed by an equation between between two variables. Although Euler initially referred to these variables as x and y, in later chapters he described using other coordinate systems to represent curves. It was not until his discussion of three-dimensional objects that Euler used three variables in one representation. For example, in Chapter VI, Section 133 of the appendix, Euler gave a presentation on the intersection of two surfaces by introducing the notion of a non-planar curve. This is where it appears that Euler was describing what is considered today to be the concept of parametric function:

The nature of any non-planar curve is most conveniently expressed by two equa- tions in three variables, for example, x, y, and z which represent mutually per- pendicular coordinates. By means of the two equations, two of the variables can be determined by the third. For instance, y and z are equal to some functions of x. We can even choose one of the variables arbitrarily for elimination, so that we can find three equations in only two variables: one in x and y, another in x andz, and a third iny andz. Of these three equations, any one is automatically determined by the other two; given equations in x and y and in x and z, the third can be found by eliminating x from these two. (Euler, 1990, p. 453)

In the last line of the previous excerpt, Euler described two parametric equations with independent variable (parameter) x and the process of eliminating the parameter to obtain an equation iny and z. This is precisely the idea of converting from parametric to standard form as it appears in modern calculus textbooks today.

In Section 134, Euler described constructing a plane curve from a non-planar curve by dropping perpendiculars from each point on the curve to a plane:

Suppose that some non-planar curve is given. In figure 148 we let M represent any point on the curve. We arbitrarily choose three mutually perpendicular axes,AB, AC,AD, by means of which the three mutually perpendicular planes BAC, BAD, and CAD are determined. From the point M on the curve we drop the perpendicular M Q to the plane BAC from the point Q we draw QP perpendicular to the axisAD. ThenAP,P Q, andQM are the three coordinates in which the two equations will determine the nature of the curve. We letAP =x, P Q=y, and QM =z. From the two given equations in x,y, and z we eliminate z to obtain an equation in only the two variables x and y. This will determine the position of the pointsQin the planeBAC. All of the pointsQcorresponding to points M produce a curve EQF, whose nature is given by the equation in x and y. (Euler, 1990, p. 453)

In the next section, Euler went on to explain that this method can be used to “discover the nature of the curve EQF” (p. 453). Furthermore, he called the plane curve EQF the projection of the non-planar curve onto the plane BAC.

It is worth mentioning that Euler’s description of curves being generated by a “con- tinuously moving point” is comparable with Keene’s (2007) observation of students using the fictive motion metaphor, while Euler’s description of non-planar curves is exactly how Oehrtman et al. (2008) described imagining a t-axis perpendicular to the x- andy-axes.

Based on the characterization by Dubinsky et al. (1995) of curves in _R2 as functions from_Rto_R2_{, what follows is my own presentation of what I call a function approach to the} study of curves defined parametrically.