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
The results of this study are formulated in statements which are called propo-
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