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 .
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.
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.
In the figure, the line ought to be tangent to the curve at , but there is no reasonable sense in which the line lies outside the curve. On the other hand, it may not be clear whether (which lies outside the curve is more of a tangent than the line 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 .
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 in line (10.3) we must assume is not zero, and then at the end we set equal to .
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 at time satisfies
Earlier, in 1604, Galileo
had supposed that