Introduction 1. A region of space which is bounded in all directions is called a volume. The common part of two contiguous regions of space is called a surface. A sheet of paper can give us an approximate idea of a surface. Indeed, it bounds two regions of space, the ones situated on the two sides of the sheet. But more rigorously, such a sheet is not a surface, because these two regions are separated by an intermediate region — the thickness of the paper. We can arrive at the notion of a surface by considering a sheet of paper whose thickness decreases indefinitely. The common part of two contiguous portions of a surface is called a line. This definition is equivalent to the following: a line is the intersection of two surfaces. The lines which we draw give us an idea of geometric lines the idea is approx- imate because, no matter how thin, they always have some width, which geometric lines do not have. Finally, the common part of two contiguous portions of a line, or the intersection of two lines which meet, is called a point. A point has no dimension. Any collection of points, lines, surfaces, and volumes is called a figure. 1b. Geometric loci. Every line contains infinitely many points. It can be viewed as being generated by a point which moves along it. This is what happens when we trace a line on paper with the point of a pencil or pen (these points are similar to geometric points when they are suﬃciently fine). In the same way, a surface can be generated by a moving line. Definition. If a point can occupy infinitely many positions (generally, a line or a surface), we call the figure formed by the set of these positions the geometric locus of the point. In the same way, we can view a surface as the geometric locus of a moving line. 2. Geometry is the study of the properties of figures and of the relations between them. The results of this study are formulated in statements which are called propo- sitions. A proposition consists of two parts: the first, called the hypothesis, indicates the conditions which we impose on ourselves the other, the conclusion, expresses a fact which, under these conditions, must necessarily be true. Thus, in the proposition: Two quantities A, B equal to a third C are themselves equal, the hypothesis is The quantities A, B are both equal to C the conclusion is These two quantities A, B are equal. Among propositions, there are some which we consider obvious without proof. These are called axioms. One of these is the proposition just considered: “Two quantities equal to a third are themselves equal.” All other propositions are called 1 http://dx.doi.org/10.1090/mbk/057/01

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