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

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http://dx.doi.org/10.1090/mbk/057/01