10.1 Velocity and Tangents

10.1   Notation. If $E_1(x,y)$ and $E_2(x,y)$ denote equations or inequalities in $x$ and $y$, we will use the notation

$$\begin{eqnarray*} \{E_1(x,y)\} &=& \{(x,y):\in\mbox{{\bf R}}^2\colon E_1(x,y)\} ... ...x,y)\in\mbox{{\bf R}}^2\colon E_1(x,y) \mbox{ and } E_2(x,y)\}.\end{eqnarray*}$$

In this section we will discuss the idea of tangent to a curve and the related idea of velocity of a moving point.

You probably have a pretty good intuitive idea of what is meant by the tangent to a curve, and you can see that the straight lines in figure a below represent tangent lines to curves.

It may not be quite so clear what you would mean by the tangents to the curves in figure b at the point $(0,0)$.

Euclid (fl. c. 300 B.C.) defined a tangent to a circle to be a line which touches the circle in exactly one point. This is a satisfactory definition of tangent to a circle, but it does not generalize to more complicated curves.

For example, every vertical line intersects the parabola $\{y=x^2\}$ in just one point, but no such line should be considered to be a tangent.

Apollonius (c 260-170 B.C.) defined a tangent to a conic section (i.e., an ellipse or hyperbola or parabola) to be a line that touches the section, but lies outside of the section. Apollonius considered these sections to be obtained by intersecting a cone with a plane, and points inside of the section were points in the cone.

This definition works well for conic sections, but for general curves, we have no notion of what points lie inside or outside a curve.

In the figure, the line $\bf {ab}$ ought to be tangent to the curve at $\bf c$, but there is no reasonable sense in which the line lies outside the curve. On the other hand, it may not be clear whether $\bf {p q}$ (which lies outside the curve $\{x^{2/3}+y^{2/3}=1\}$ is more of a tangent than the line $\bf {rs}$ which does not lie outside of it. Leibniz [33, page 276] said that

to find a tangent means to draw a line that connects two points of the curve at an infinitely small distance, or the continued side of a polygon with an infinite number of angles, which for us takes the place of the curve.

From a modern point of view it is hard to make any sense out of this.

Here is a seventeenth century sort of argument for finding a tangent to the parabola whose equation is $y=x^2$.

Imagine a point $\bf p$ that is moving along the parabola $y=x^2$, so that at time $t$, $\bf p$ is at $(x,y)$. (Here $x$ and $y$ are functions of $t$, but in the seventeenth century they were just flowing quantities.) Imagine a point $\bf q$ that moves along the $x$-axis so that $\bf q$ always lies under $\mbox{{\bf p}}$ and a point $\bf r$ moving along the $y$-axis so that $\bf r$ is always at the same height as $\mbox{{\bf p}}$. Let ${\dot x}$ denote the velocity of $\bf q$ when $\mbox{{\bf p}}$ is at $(x,y)$ and let ${\dot y}$ denote the velocity of $\bf r$ when $\mbox{{\bf p}}$ is at $(x,y)$. Let $o$ be a very small moment of time. At time $o$ after $\mbox{{\bf p}}$ is at $(x,y)$, $\mbox{{\bf p}}$ will be at $(x+o{\dot x},y+o{\dot y})$ (i.e., $x$ will have increased by an amount equal to the product of the time interval $o$ and its velocity $\dot x$). Since $\mbox{{\bf p}}$ stays on the curve, we have

$$y+o{\dot y}=(x+o{\dot x})^2$$

or

$$y+o{\dot y}=x^2+2xo{\dot x}+o^2{\dot x}^2.$$

Since $y=x^2$, we get

$$o{\dot y}=2xo{\dot x}+o^2\dot x^2$$ (10.2)

or

$$\dot y = 2x\dot x +o\dot x^2$$ (10.3)

Since we are interested in the velocities at the instant that $\mbox{{\bf p}}$ is at $(x,y)$, we take $o=0$, so

$$\dot y=2x\dot x.$$

Hence when $p$ is at $(x,y)$, the vertical part of its velocity (i.e., $\dot y$) is $2x$ times the horizontal component of its velocity. Now the velocity should point in the direction of the curve; i.e., in the direction of the tangent, so the direction of the tangent at $(x,y)$ should be in the direction of the diagonal of a box with

$$\mbox{ vertical side } = 2x \times \mbox{ horizontal side. }$$

The tangent to the parabola at $(x,y)=(x,x^2)$ is the line joining $(x,y)$ to $(0,-y)$, since in the figure the vertical component of the box is

$$2y=2x^2=(2x)x;$$

i.e., the vertical component is $2x$ times the horizontal component.

In The Analyst: A Discourse Addressed to an Infidel Mathematician[7, page 73], George Berkeley (1685-1753) criticizes the argument above, pointing out that when we divide by $o$ in line (10.3) we must assume $o$ is not zero, and then at the end we set $o$ equal to $0$.

All which seems a most inconsistent way of arguing, and such as would not be allowed of in Divinity.

The technical concept of velocity is not a simple one. The idea of uniform velocity causes no problems: to quote Galileo (1564-1642):

By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal[21, page 154].

This definition applies to points moving in a straight line, or points moving on a circle, and it goes back to the Greek scientists. The problem of what is meant by velocity for a non-uniform motion, however, is not at all clear. The Greeks certainly realized that a freely falling body moves faster as it falls, but they had no language to describe the way in which velocity changes. Aristotle (384-322 B.C.) says

there cannot be motion of motion or becoming of becoming or in general change of change[11, page 168].

It may not be clear what this means, but S. Bochner interprets this as saying that the notion of a second derivative (this is a technical term for the mathematical concept used to describe acceleration which we will discuss later) is a meaningless idea[11, page 167]. Even though we are in constant contact with non-uniformly moving bodies, our intuition about the way they move is not very good. In the Dialogues Concerning Two New Sciences, Salviati (representing Galileo) proposes the hypothesis that if a stone falls from rest, then it falls in such a way that `` in any equal intervals of time whatever, equal increments of speed are given to it"[21, page 161].

In our language, the hypothesis is that the velocity $v(t)$ at time $t$ satisfies

$$v(t)=kt \mbox{ for some constant } k.$$

Sagredo objects to this on the grounds that this would mean that the object begins to fall with zero speed `` while our senses show us that a heavy falling body suddenly acquires great speed." (I believe Sagredo is right. Try dropping some bodies and observe how they begin to fall.) Salviati replies that this is what he thought at first, and explains how he came to change his mind.

Earlier, in 1604, Galileo had supposed that

$$v(x)=kx \mbox{ for some constant } k;$$

i.e., that in equal increments of distance the object gains equal increments of speed (which is false), and Descartes made the same error in 1618 [13, page 165]. Casual observation doesn't tell you much about falling stones.

10.4   Entertainment (Falling bodies.) Try to devise an experiment to support (or refute) Galileo's hypothesis that $v(t)=kt$, using materials available to Galileo; e.g., no stop watch. Galileo describes his experiments in [21, pages 160-180], and it makes very good reading.