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
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