§ 1. THE ELEMENTS OF GEOMETRY AND THE FIVE GROUPS OF AXIOMS.

Let us consider three distinct systems of things.
The things composing the first system, we will
call points and designate them by the letters A, B,
C,…; those of the second, we will call straight
lines and designate them by the letters a, b, c,…;
and those of the third system, we will call planes and
designate them by the Greek letters α, β, γ,…
The points are called the elements of linear geometry;
the points and straight lines, the elements of plane geometry;
and the points, lines, and planes, the elements
of the geometry of space or the elements of space.

We think of these points, straight lines, and planes
as having certain mutual relations, which we indicate
by means of such words as “are situated,” “between,”
“parallel,” “congruent,” “continuous,” etc.
The complete and exact description of these relations
follows as a consequence of the axioms of geometry.
These axioms may be arranged in five groups. Each
of these groups expresses, by itself, certain related
fundamental facts of our intuition. We will name
these groups as follows:

I, 1–7. Axioms of connection.

II, 1–5. Axioms of order.

III. Axiom of parallels (Euclid's axiom).

IV, 1–6. Axioms of congruence.

V. Axiom of continuity (Archimedes's axiom).

§ 2. GROUP I: AXIOMS OF CONNECTION.

The axioms of this group establish a connection
between the concepts indicated above; namely, points,
straight lines, and planes. These axioms are as follows:

I, 1.
Two distinct points A and B always completely
determine a straight line a. We write AB = a
or BA = a.

Instead of “determine,” we may also employ other
forms of expression; for example, we may say A
“lies upon” a, A “is a point of” a, a “goes through”
A “and through” B, a “joins” A “and” or “with”
B, etc. If A lies upon a and at the same time upon
another straight line b, we make use also of the expression:
“The straight lines” a “and” b “have the
point A in common,” etc.

I, 2.
Any two distinct points of a straight line completely
determine that line; that is, if AB = a and
AC=a, where B ≠ C, then is also BC=a.

I, 3.
Three points A, B, C not situated in the same
straight line always completely determine a plane
α. We write ABC=α.

We employ also the expressions: A, B, C, “lie
in” α; A, B, C “are points of” α, etc.

I, 4.
Any three points A, B, C of a plane α, which
do not lie in the same straight line, completely determine
that plane.

I, 5.
If two points A, B of a straight line a lie in
a plane α, then every point of a lies in a.

In this case we say: “The straight line a lies in
the plane α,” etc.

I, 6.
If two planes α, β have a point A in common,
then they have at least a second point B in common.

I, 7.
Upon every straight line there exist at least two
points, in every plane at least three points not
lying in the same straight line, and in space there
exist at least four points not lying in a plane.

Axioms I, 1–2 contain statements concerning points
and straight lines only; that is, concerning the elements
of plane geometry. We will call them, therefore,
the plane axioms of group I, in order to distinguish
them from the axioms I, 3–7, which we will
designate briefly as the space axioms of this group.

Of the theorems which follow from the axioms
I, 3–7, we shall mention only the following:

Theorem 1. Two straight lines of a plane have
either one point or no point in common; two
planes have no point in common or a straight
line in common; a plane and a straight line
not lying in it have no point or one point in
common.

Theorem 2. Through a straight line and a point
not lying in it, or through two distinct straight
lines having a common point, one and only one
plane may be made to pass.

§ 3. GROUP II: AXIOMS OF ORDER.

Footnote — These
axioms were first studied in detail by M.~Pasch in his Vorlesungen
über neuere Geometrie, Leipsic, 1882. Axiom II, 5 is in particular due to him.

The axioms of this group define the idea expressed
by the word “between,” and make possible, upon the
basis of this idea, an order of sequence of the points
upon a straight line, in a plane, and in space. The
points of a straight line have a certain relation to one
another which the word “between” serves to describe.
The axioms of this group are as follows:

II, 1.
If A, B, C are points of a straight line and
B lies between A and C, then B lies also between
C and A.

II, 2.
If A and C are two points of a straight line,
then there exists at least one point B lying between
A and C and at least one point D so situated that
C lies between A and D.

II, 3.
Of any three points situated on a straight line,
there is always one and only one which lies between
the other two.

II, 4.
Any four points A, B, C, D of a straight line
can always be so arranged that B shall lie between
A and C and also between A and D, and, furthermore,
that C shall lie between A and D and also
between B and D.

Definition. We will call the system of two points
A and B, lying upon a straight line, a segment and
denote it by AB or BA. The points lying between A
and B are called the points of the segment AB or the
points lying within the segment AB. All other points
of the straight line are referred to as the points lying
outside the segment AB. The points A and B are called
the extremities of the segment AB.

II, 5.
Let A, B, C be three points not lying in the
same straight line and
let a be a straight line lying in the plane ABC and not passing through any of the
points A, B, C. Then, if the straight line a passes through a point
of the segment AB, it will also pass through either a point of the segment
BC or a point of the segment AC.
Axioms II, 1–4 contain statements concerning the
points of a straight line only, and, hence, we will call
them the linear axioms of group II. Axiom II, 5 relates
to the elements of plane geometry and, consequently,
shall be called the plane axiom of group II.

§ 4. CONSEQUENCES OF THE AXIOMS OF CONNECTION AND ORDER.

By the aid of the four linear axioms II, 1–4, we
can easily deduce the following theorems:

Theorem 3. Between any two points of a straight
line, there always exists an unlimited number of
points.

Theorem 4. If we have given any finite number
of points situated upon a straight line, we can
always arrange them in a sequence A, B, C,
D, E,…, K so that B shall lie between A
and C, D, E,…, K; C between A, B and D,
E,…, K; D between A, B, C and E,…K,
etc. Aside from this order of sequence, there
exists but one other possessing this property
namely, the reverse order K,…, E, D, C,
B, A.

Theorem 5. Every straight line a, which lies in
a plane α, divides the remaining points of this
plane into two regions having the following
properties: Every point A of the one region determines
with each point B of the other region
a segment AB containing a point of the straight
line a. On the other hand, any two points A,
A' of the same region determine a segment
AA' containing no point of a.

If A, A', O, B are four points of a straight line a,
where O lies between A and B but not between A and
A', then we may say: The points A, A' are situated
on the line a upon one and the same side of the point O,
and the points A, B are situated on the straight line a
upon different sides of the point O.

All of the points of
a which lie upon the same side of O, when taken
together, are called the half-ray emanating from O.
Hence, each point of a straight line divides it into
two half-rays.

Making use of the notation of theorem 5, we say:
The points A, A' lie in the plane α upon one and the
same side of the straight line a, and the points A, B lie
in the plane α upon different sides of the straight line a.

Definitions. A system of segments AB, BC,
CD, …, KL is called a broken line joining A with L
and is designated, briefly, as the broken line ABCDE …
KL. The points lying within the segments AB,
BC, CD, …, KL, as also the points A, B, C, D, …,
K, L, are called the points of the broken line. In
particular, if the point A coincides with L, the broken
line is called a polygon and is designated as the polygon
ABCD … K. The segments AB, BC, CD, …, KA
are called the sides of the polygon and the points A, B,
C, D, …, K the vertices. Polygons having 3, 4,
5, …, n vertices are called, respectively, triangles,
quadrangles, pentagons, …, n-gons. If the vertices of
a polygon are all distinct and none of them lie within
the segments composing the sides of the polygon,
and, furthermore, if no two sides have a point in common,
then the polygon is called a simple polygon.

With the aid of theorem 5, we may now obtain,
without serious difficulty, the following theorems:

Theorem 6. Every simple polygon, whose vertices
all lie in a plane α, divides the points of
this plane, not belonging to the broken line
constituting the sides of the polygon, into two
regions, an interior and an exterior, having the
following properties: If A is a point of the interior
region (interior point) and B a point of
the exterior region (exterior point), then any
broken line joining A and B must have at least
one point in common with the polygon. If, on
the other hand, A, A' are two points of the interior
and B, B' two points of the exterior region,
then there are always broken lines to be
found joining A with A' and B with B' without
having a point in common with the polygon.
There exist straight lines in the plane α which
lie entirely outside of the given polygon, but
there are none which lie entirely within it.

Theorem 7. Every plane α divides the remaining
points of space into two regions having the
following properties: Every point A of the one
region determines with each point B of the
other region a segment AB, within which lies
a point of α. On the other hand, any two points
A, A' lying within the same region determine a
segment AA' containing no point of α.

Making use of the notation of theorem 7, we may
now say: The points A, A' are situated in space upon
one and the same side of the plane α, and the points A, B
are situated in space upon different sides of the plane α.

Theorem 7 gives us the most important facts relating
to the order of sequence of the elements of
space. These facts are the results, exclusively, of the
axioms already considered, and, hence, no new space
axioms are required in group II.

§ 5. GROUP III: AXIOM OF PARALLELS. (EUCLID'S AXIOM.)

The introduction of this axiom simplifies greatly
the fundamental principles of geometry and facilitates
in no small degree its development. This axiom may
be expressed as follows:

III. In a plane α there can be drawn through any
point A, lying outside of a straight line a, one and
only one straight line which does not intersect the
line a. This straight line is called the parallel to
a through the given point A.

This statement of the axiom of parallels contains
two assertions. The first of these is that, in the plane
α, there is always a straight line passing through A
which does not intersect the given line a. The second
states that only one such line is possible. The latter
of these statements is the essential one, and it may
also be expressed as follows:

Theorem 8. If two straight lines a, b of a plane
do not meet a third straight line c of the same
plane, then they do not meet each other.

For, if a, b had a point A in common, there would
then exist in the same plane with c two straight lines
a and b each passing through the point A and not
meeting the straight line c. This condition of affairs
is, however, contradictory to the second assertion contained
in the axiom of parallels as originally stated.
Conversely, the second part of the axiom of parallels,
in its original form, follows as a consequence of theorem
8.

The axiom of parallels is a plane axiom.

§ 6. GROUP IV. AXIOMS OF CONGRUENCE.

The axioms of this group define the idea of congruence
or displacement.

Segments stand in a certain relation to one another
which is described by the word “congruent.”

IV, I. If A, B are two points on a straight line a,
and if A' is a point upon the same or another
straight line a', then, upon a given side of A' on
the straight line a', we can always find one and
only one point B' so that the segment AB (or BA)
is congruent to the segment A'B'. We indicate
this relation by writing

AB≡ A'B'.

Every segment is congruent to itself; that is, we
always have

AB≡ AB.

We can state the above axiom briefly by saying
that every segment can be laid off upon a given side
of a given point of a given straight line in one and
and only one way.

IV, 2. If a segment AB is congruent to the segment
A'B' and also to the segment A”B”, then the segment
A'B' is congruent to the segment A”B”; that
is, if AB ≡ A'B and AB ≡ A”B”, then
A'B' ≡ A”B”.

IV, 3. Let AB and BC be two segments of a straight
line a which have no points in common aside from
the point B, and, furthermore, let A'B' and B'C'
be two segments of the same or of another straight
line a' having, likewise, no point other than B' in
common. Then, if AB ≡ A'B' and BC ≡ B'C',
we have AC ≡ A'C'.

Definitions. Let α be any arbitrary plane and h,
k any two distinct half-rays lying in α and emanating
from the point O so as to form a part of two different
straight lines. We call the system formed by these
two half-rays h, k an angle and represent it by the
symbol ∠(h, k) or ∠(k, h). From axioms II, 1–5, it
follows readily that the half-rays h and k, taken together
with the point O, divide the remaining points
of the plane a into two regions having the following
property: If A is a point of one region and B a point
of the other, then every broken line joining A and B
either passes through O or has a point in common
with one of the half-rays h, k. If, however, A, A'
both lie within the same region, then it is always possible
to join these two points by a broken line which
neither passes through O nor has a point in common
with either of the half-rays h, k. One of these two
regions is distinguished from the other in that the segment
joining any two points of this region lies entirely
within the region. The region so characterised is
called the interior of the angle (h,k). To distinguish
the other region from this, we call it the exterior of
the angle (h,k). The half rays h and k are called the
sides of the angle, and the point O is called the vertex
of the angle.

IV, 4.
Let an angle (h,k) be given in the plane
α and let a straight line a' be given in a plane α'.
Suppose also that, in the plane α', a definite side
of the straight line a' be assigned. Denote by h' a
half-ray of the straight line a' emanating from a
point O' of this line. Then in the plane α' there
is one and only one half-ray k' such that the angle
(h,k), or (k,h), is congruent to the angle (h',k')
and at the same time all interior points of the angle
(h',k') lie upon the given side of a'. We express
this relation by means of the notation

 ∠ (h,k) ≡ ∠ (h',k')

Every angle is congruent to itself; that is,

∠ (h,k) ≡ ∠ (h,k)

or

∠ (h,k) ≡ ∠ (k,h)

We say, briefly, that every angle in a given plane
can be laid off upon a given side of a given half-ray in
one and only one way.

IV, 5.
If the angle (h,k) is congruent to the angle
(h',k') and to the angle (h”,k”), then the angle
(h',k') is congruent to the angle (h”,k”); that is
to say, if ∠ (h, k) ≡ ∠ (h', k') and
∠ (h, k) ≡ ∠ (h”,k”), then
∠ (h',k') ≡ ∠ (h”,k”).

Suppose we have given a triangle ABC. Denote
by h, k the two half-rays emanating from A and passing
respectively through B and C. The angle (h,k)
is then said to be the angle included by the sides AB
and AC, or the one opposite to the side BC in the
triangle ABC. It contains all of the interior points
of the triangle ABC and is represented by the symbol
∠ BAC, or by ∠ A.

IV, 6.
If, in the two triangles ABC and A'B'C'
the congruences

AB ≡ A'B', AC ≡ A'C', ∠ BAC ≡ ∠ B'A'C'

hold, then the congruences

∠ ABC ≡ ∠ A'B'C' and ∠ ACB ≡ ∠ A'C'B'

also hold.

Axioms IV, 1–3 contain statements concerning the
congruence of segments of a straight line only. They
may, therefore, be called the linear axioms of group
IV. Axioms IV, 4, 5 contain statements relating to
the congruence of angles. Axiom IV, 6 gives the connection
between the congruence of segments and the
congruence of angles. Axioms IV, 4–6 contain statements
regarding the elements of plane geometry and
may be called the plane axioms of group IV.

§ 7. CONSEQUENCES OF THE AXIOMS OF CONGRUENCE.

Suppose the segment AB is congruent to the segment
A'B'. Since, according to axiom IV, 1, the segment
AB is congruent to itself, it follows from axiom
IV, 2 that A'B' is congruent to AB; that is to say, if
AB ≡ A'B', then A'B' ≡ AB. We say, then, that the
two segments are congruent to one another.

Let A, B, C, D,…, K, L and A', B', C', D',…,
K', L' be two series of points on the straight lines a
and a', respectively, so that all the corresponding segments
AB and A'B', AC and A'C', BC and B'C',…,
KL and K'L' are respectively congruent, then the two
series of points are said to be congruent to one another.
A and A', B and B',…, L and L' are called corresponding
points of the two congruent series of points.

From the linear axioms IV, 1–3, we can easily deduce
the following theorems:

Theorem 9. If the first of two congruent series
of points A, B, C, D,…, K, L and A', B',
C', D',…, K', L' is so arranged that B lies
between A and C, D,…, K, L, and C between
A, B and D,…, K, L, etc., then the points A',
B', C', D',…, K', L' of the second series are
arranged in a similar way; that is to say, B'
lies between A' and C', D',…, K', L', and C'
lies between A', B' and D',…, K', L', etc.

Let the angle (h,k) be congruent to the angle
(h',k'). Since, according to axiom IV, 4, the angle
(h,k) is congruent to itself, it follows from axiom IV,
5 that the angle (h',k') is congruent to the angle
(h,k). We say, then, that the angles (h,k) and (h',k')
are congruent to one another.

Definitions. Two angles having the same vertex
and one side in common, while the sides not common
form a straight line, are called supplementary angles.
Two angles having a common vertex and whose sides
form straight lines are called vertical angles. An angle
which is congruent to its supplementary angle is called
a right angle.

Two triangles ABC and A'B'C' are said to be congruent
to one another when all of the following congruences
are fulfilled:

AB ≡ A'B', AC ≡ A'C', BC ≡ B'C',

∠ A ≡ ∠ A', ∠ B ≡ ∠ B', ∠ C ≡ ∠ C'.

Theorem 10. (First theorem of congruence for
triangles). If, for the two triangles ABC and
A'B'C', the congruences

 AB ≡ A'B', AC ≡ A'C', ∠ A ≡ ∠ A'

hold, then the two triangles are congruent to
each other.

Proof. From axiom IV, 6, it follows that the
two congruences

∠ B ≡ ∠ B' and ∠ C ≡ ∠ C'

are fulfilled, and it is, therefore, sufficient to show that
the two sides BC and B'C' are congruent. We will
assume the contrary to be true, namely, that BC and
B'C' are not congruent, and show that this leads to a
contradiction. We take upon B'C' a point D' such that
BC ≡ B'D'. The two triangles ABC and A'B'D' have,
then, two sides and the included angle of the one
agreeing, respectively, to two sides and the included
angle of the other. It follows from axiom IV, 6 that
the two angles BAC and B'A'D' are also congruent to
each other. Consequently, by aid of axiom IV, 5,
the two angles B'A'C' and B'A'D' must be congruent.

This, however, is impossible, since, by axiom IV, 4,
an angle can be laid off in one and only one way on a
given side of a given half-ray of a plane. From this
contradiction the theorem follows.

We can also easily demonstrate the following theorem:

Theorem 11. (Second theorem of congruence
for triangles). If in any two triangles one side
and the two adjacent angles are respectively
congruent, the triangles are congruent.

We are now in a position to demonstrate the following
important proposition.

Theorem 12. If two angles ABC and A'B'C' are
congruent to each other, their supplementary
angles CBD and C'B'D' are also congruent.

Proof. Take the points A', C', D' upon the sides
passing through B' in such a way that

A'B' ≡ AB, C'B' ≡ CB, D'B' ≡ DB.

Then, in the two triangles ABC and A'B'C', the sides
AB and BC are respectively congruent to A'B' and
C'B'. Moreover, since the angles included by these
sides are congruent to each other by hypothesis, it
follows from theorem 10 that these triangles are congruent;
that is to say, we have the congruences

AC ≡ A'C, ∠ BAC ≡ ∠ B'A'C'.

On the other hand, since by axiom IV, 3 the segments
AD and A'D' are congruent to each other, it follows
again from theorem~10 that the triangles CAD and
C'A'D' are congruent, and, consequently, we have the
congruences:

 CD ≡ C'D', ∠ ADC ≡ ∠ A'D'C'.

From these congruences and the consideration of the
triangles BCD and B'C'D', it follows by virtue of
axiom~IV, 6 that the angles CBD and C'B'D' are congruent.

As an immediate consequence of theorem~12, we
have a similar theorem concerning the congruence of
vertical angles.

Theorem 13. Let the angle (h, k) of the plane α
be congruent to the angle (h', k') of the plane
α', and, furthermore, let l be a half-ray in the
plane α emanating from the vertex of the angle
(h, k) and lying within this angle. Then, there
always exists in the plane α' a half-ray l' emanating
from the vertex of the angle (h', k') and
lying within this angle so that we have

∠ (h, l) ≡ ∠ (h', l'), ∠ (k, l) ≡ ∠ (k', l').

Proof. We will represent the vertices of the angles
(h, k) and (h', k') by O and O', respectively, and
so select upon the sides h, k, h', k' the points A, B,
A', B' that the congruences

 OA ≡ O'A', OB ≡ O'B'

are fulfilled. Because of the congruence of the triangles
OAB and O'A'B', we have at once

AB ≡ A'B', ∠ OAB ≡ O'A'B', ∠ OBA ≡ ∠ O'B'A'.

Let the straight line AB intersect l in C. Take the
point C' upon the segment A'B' so that A'C' ≡ AC.
Then, O'C' is the required half-ray. In fact, it follows
directly from these congruences, by aid of axiom
IV, 3, that BC ≡ B'C'. Furthermore, the triangles
OAC and O'A'C' are congruent to each other, and the
same is true also of the triangles OCB and O'B'C'.
With this our proposition is demonstrated.

In a similar manner, we obtain the following proposition.

Theorem 14. Let h, k, l and h', k', l' be two sets
of three half-rays, where those of each set emanate
from the same point and lie in the same
plane. Then, if the congruences

∠ (h, l) ≡ ∠ (h', l'), ∠ (k, l) ≡ ∠ (k', l')

are fulfilled, the following congruence is also
valid; viz.:

∠ (h,k) ≡ ∠ (h', k').

By aid of theorems 12 and 13, it is possible to deduce
the following simple theorem, which Euclid held–although
it seems to me wrongly–to be an axiom.

Theorem 15. All right angles are congruent to
one another.

Proof. Let the angle BAD be congruent to its
supplementary angle CAD, and, likewise, let the angle
B'A'D' be congruent to its supplementary angle
C'A'D'. Hence the angles BAD, CAD, B'A'D', and
C'A'D' are all right angles. We will assume that the
contrary of our proposition is true, namely, that the
right angle B'A'D' is not congruent to the right angle
BAD, and will show that this assumption leads to a
contradiction. We lay off the angle B'A'D' upon the
half-ray AB in such a manner that the side AD” arising
from this operation falls either within the angle
BAD or within the angle CAD. Suppose, for example,
the first of these possibilities to be true. Because
of the congruence of the angles B'A'D' and
BAD”, it follows from theorem 12 that angle C'A'D'
is congruent to angle CAD”, and, as the angles B'A'D'
and C'A'D' are congruent to each other, then, by
IV, 5, the angle BAD” must be congruent to CAD”.

Furthermore, since the angle BAD is congruent to the
angle CAD, it is possible, by theorem 13, to find within
the angle CAD a half-ray AD”' emanating from A, so
that the angle BAD” will be congruent to the angle
CAD”', and also the angle DAD” will be congruent
to the angle DAD”'. The angle BAD” was shown
to be congruent to the angle CAD” and, hence, by
axiom IV, 5, the angle CAD”, is congruent to the
angle CAD”'. This, however, is not possible; for,
according to axiom IV, 4, an angle can be laid off in
a plane upon a given side of a given half-ray in only
one way. With this our proposition is demonstrated.
We can now introduce, in accordance with common
usage, the terms “acute angle” and “obtuse angle.”

The theorem relating to the congruence of the
base angles A and B of an equilateral triangle ABC
follows immediately by the application of axiom IV,
6 to the triangles ABC and BAC. By aid of this theorem,
in addition to theorem 14, we can easily demonstrate
the following proposition.

Theorem 16. (Third theorem of congruence for
triangles.) If two triangles have the three sides
of one congruent respectively to the corresponding
three sides of the other, the triangles are
congruent.

Any finite number of points is called a figure. If
all of the points lie in a plane, the figure is called a
plane figure.

Two figures are said to be congruent if their points
can be arranged in a one-to-one correspondence so
that the corresponding segments and the corresponding
angles of the two figures are in every case congruent
to each other.

Congruent figures have, as may be seen from theorems
9 and 12, the following properties: Three points
of a figure lying in a straight line are likewise in a
straight line in every figure congruent to it. In congruent
figures, the arrangement of the points in corresponding
planes with respect to corresponding lines
is always the same. The same is true of the sequence
of corresponding points situated on corresponding
lines.

The most general theorems relating to congruences
in a plane and in space may be expressed as follows:

Theorem 17. If (A, B, C,…) and (A', B', C',…)
are congruent plane figures and P is a
point in the plane of the first, then it is always
possible to find a point P in the plane of the
second figure so that (A, B, C,… , P) and (A', B', C', … , P')
shall likewise be congruent figures.
If the two figures have at least three
points not lying in a straight line, then the selection
of P' can be made in only one way.

Theorem 18. If (A, B, C, …) and (A', B', C',…=
are congruent figures and P represents
any arbitrary point, then there can always be
found a point P' so that the two figures (A, B, C, …, P)
and (A', B', C', …, P') shall
likewise be congruent. If the figure (A, B, C, … , P)
contains at least four points not lying
in the same plane, then the determination of
P' can be made in but one way.

This theorem contains an important result; namely,
that all the facts concerning space which have reference
to congruence, that is to say, to displacements
in space, are (by the addition of the axioms of groups
I and II) exclusively the consequences of the six
linear and plane axioms mentioned above. Hence, it
is not necessary to assume the axiom of parallels in
order to establish these facts.

If we take, in addition to the axioms of congruence,
the axiom of parallels, we can then easily establish
the following propositions:

Theorem 19. If two parallel lines are cut by a
third straight line, the alternate-interior angles
and also the exterior-interior angles are congruent.

Conversely, if the alternate-interior or
the exterior-interior angles are congruent, the
given lines are parallel.

Theorem 20. The sum of the angles of a triangle
is two right angles.

Definitions. If M is an arbitrary point in the
plane a, the totality of all points A, for which the segments
MA are congruent to one another, is called a
circle. M is called the centre of the circle.

From this definition can be easily deduced, with
the help of the axioms of groups III and IV, the
known properties of the circle; in particular, the possibility
of constructing a circle through any three
points not lying in a straight line, as also the congruence
of all angles inscribed in the same segment of
a circle, and the theorem relating to the angles of an
inscribed quadrilateral.

§ 8. GROUP V. AXIOM OF CONTINUITY. (ARCHIMEDEAN AXIOM.)

This axiom makes possible the introduction into
geometry of the idea of continuity. In order to state
this axiom, we must first establish a convention concerning
the equality of two segments. For this purpose,
we can either base our idea of equality upon the
axioms relating to the congruence of segments and
define as “equal” the correspondingly congruent segments,
or upon the basis of groups I and II, we may
determine how, by suitable constructions (see Chap.
V, §~24), a segment is to be laid off from a point of a
given straight line so that a new, definite segment is
obtained “equal” to it. In conformity with such a

convention, the axiom of Archimedes may be stated
as follows:

V. Let A₁ be any point upon a straight line between
the arbitrarily chosen points A and B. Take the
points A₂, A₃, A₄,… so that A₁ lies between A
and A₂, A₂ between A₁ and A₃, A₃ between A₂ and
A₄ etc. Moreover, let the segments

 A A₁, A₁ A₂, A₂ A₃, A₃ A₄, …

be equal to one another. Then, among this series
of points, there always exists a certain point Aₙ
such that B lies between A and Aₙ.

The axiom of Archimedes is a linear axiom.

Remark.
Footnote — Added
by Professor Hilbert in the French translation.—Tr.
To the preceeding five groups of axioms,
we may add the following one, which, although
not of a purely geometrical nature, merits particular
attention from a theoretical point of view. It may be
expressed in the following form:

Axiom of Completeness.
Footnote — See Hilbert,
“Ueber den Zahlenbegriff,” Berichte der deutschen Mathematiker-Vereinigung,
1900.
(Vollständigkeit): To a
system of points, straight lines, and planes, it is
impossible to add other elements in such a manner
that the system thus generalized shall form a new
geometry obeying all of the five groups of axioms.
In other words, the elements of geometry form a
system which is not susceptible of extension, if we
regard the five groups of axioms as valid.

This axiom gives us nothing directly concerning
the existence of limiting points, or of the idea of convergence.
Nevertheless, it enables us to demonstrate
Bolzano's theorem by virtue of which, for all sets of
points situated upon a straight line between two definite
points of the same line, there exists necessarily
a point of condensation, that is to say, a limiting point.
From a theoretical point of view, the value of this
axiom is that it leads indirectly to the introduction
of limiting points, and, hence, renders it possible to
establish a one-to-one correspondence between the
points of a segment and the system of real numbers.
However, in what is to follow, no use will be made of
the “axiom of completeness.”

COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.
