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Francis Borceux A Differential Approach to Geometry 2014 Geometric Trilogy III 10.1007/978-3-319-01736-5_1

Springer International Publishing Switzerland 2014

1. The Genesis of Differential Methods

Francis Borceux 1

(1)

Universit catholique de Louvain, Louvain-la-Neuve, Belgium

Abstract

The study of arbitrary curvesnot just conicsbecomes possible during the 17th century, first via the consideration of polynomial equations. The development of differential calculus allows next the study of very general curves. We describe the first historical attempts for handling questions like the tangent to a curve, its length or its curvature. We pay a special attention to some curves, like the cycloid, which have played an important role in the development of these notions.

This first chapter is intentionally provocative, and useless! By useless (besides being at once provocative) we mean: this first chapter is not formally needed to follow the systematic treatment of the theory of curves and surfaces developed in the subsequent chapters.

So what is this chapter about? Usually, when you open a book onlet us saythe theory of curves in the real plane, you expect to find first the precise definition of a plane curve, followed by a careful study of the properties of such a notion. We all have an intuitive idea of what a plane curve is. Everybody knows that the straight line, the circle or the parabola are curves, but a single point or the empty set are not curves! Nevertheless, all these figures can be described by an equation F ( x , y )=0, with F a polynomial: for example, x 2+ y 2=0 is an equation of the origin in while x 2+ y 2=1 is an equation of the empty set. Thus a curve cannot simply be defined via an equation F ( x , y )=0, even when F is a very good function! For example, consider the picture comprised of 7 hyperbolas, thus 14 branches. Is this one curve, or seven curves, or fourteen curves? After all, it is not so clear what a curve should be!

Starting at once with a precise definition of a curve would give the false impression that this is the definition of a curve. Instead it should be stressed that such a definition is a possible definition. Discussing the advantages and disadvantages of the various possible definitions, in order to make a sensible choice, is an important aspect of every mathematical approach.

There is also a second aspect that we want to stress. For Euclid , a straight line was What has a length and no width and is well-balanced at each of its points (see Definition 3.1.1 in [], Trilogy I ). The intuition behind such a sentence is clear, but such a definition assumes that before beginning to develop geometry, we know what a length is. Of course what we want to do concerning a length is then to find a formula to compute it, such as 2 R for a circle of radius R .

With more than two thousand years of further mathematical developments and experience, we now feel quite uneasy about such an approach. How can we establish a formula to compute the length of a curve if we did not define first what the length of a curve is?

For many centuriesessentially up to the 17th centurymathematicians could hardly handle problems of length for curves other than the straight line and the circle. Differential calculus, with the full power of the theories of derivatives and integrals, opened the door to the study of arbitrary curves. However, in some sense, one was still taking the notion of length (or surface or volume) as something which exists and that one wants to calculate.

Like many authors today, we adopt in the following chapters a completely different approach: the theory of integration is a well-established part of analysis and we use it to define a length. Analogously the theory of derivatives is a well-established part of analysis and we use it to define a tangent. And so on.

This first chapter is intended to be a bridge between the historical and the contemporary approaches. We present typical arguments developed in the past (and sometimes, still today) to master some geometrical notions (like length, or tangent), but we do that in particular to develop an intuition for the contemporary definitions of these notions. In this introductory chapter, we refer freely to [], Trilogy I and II , when the historical arguments that we have in mind have been developed there.

Various arguments in this chapter can appear quite disconcerting. We often rely on our intuition, without trying to formalize the argument. We freely apply many results borrowed from a first calculus course, taking as a blanket assumption that when we apply a theorem, the necessary assumptions for its validity should be satisfied, even if we have not tried to determine the precise context in which this is the case! This is not a very rigorous attitude, however our point in this chapter is not to prove results, but to guess what possible good definitions should be.

1.1 The Static Approach to Curves

Originally, Greek geometry (see [], Trilogy I ) was essentially concerned with the study of two curves: the line and the circle.

The line is what has length and no width and is well-balanced around each of its points .

The circle is the locus of those points of the plane which are at a fixed distance R from a fixed point O of the plane .

Passing analogously to three dimensional space, using a circle in a plane and a point not belonging to the plane of the circle, you can thenusing linesconstruct the cone on this circle with vertex the given point. Cutting this cone by another plane then yields new curves that, according to the position of the cutting plane, you call ellipse , hyperbola or parabola . This is the origin of the theory of curves.

It is common practice to describe a curve by giving its equation with respect to some basis. In this book, we are interested in the study of curves in the real plane . For example a circle of radius R centered at the origin admits the equation (see Chap. 1 in [], Trilogy II )

which we can equivalently write as

One might be tempted to introduce a general theory of curves by allowing equations of the form

where

is an arbitrary function. But it does not take long to realize that:

• choosing F ( x , y )= x 2+ y 2, we get the equation of a single point: the origin;
• choosing F ( x , y )= x 2+ y 2+1, we even get the equation of the empty set!

In both cases the function F ( x , y ) is certainly a very good one: it is even a polynomial, but we do not want a point or the empty set to be considered as a curve.

For more food for thought, look at the picture in Fig. : should this be considered as one curve, or as six curves?

Fig. 1.1

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