** Appendix A - Hilbert's Axioms for Euclidean Geometry
***Printout*

*Mathematics is a game played according to
certain rules with meaningless marks on paper.
—*

** Introductory Note.** Hilbert's Axiom set is an example of what is
called a

**Undefined Terms** *point,
line, plane,
lie, between, *and
*congruence.*

*(Some web browsers display some characters incorrectly, an angle shows as
*
∠,* congruent
shows as * ≅*,
and the Greek characters alpha and beta show as *
*α**, *
*β**.)*

I.1. For every two points

A, B,there exists a linemthat contains each of the pointsA, B.

I.2. For every two pointsA, B,there is not more than one linemthat contain each of the pointsA,B.

I.3. There exist at least two points on a line. There exist at least three points that do not lie on a line.

I.4. For any three pointsA, B, Cthat do not lie on the same line, there exists a planeathat contains each of the pointsA, B, C.For every plane there exists a point which it contains.

I.5. For any three pointsA, B, Cthat do not lie on the same line, there exists no more than one plane that contains each of the three pointsA, B, C.

I.6. If two pointsA, Bof a linemlie in a planeα, then every point ofmlies in the planeα.

I.7. If two planesα,βhave a pointAin common, then they have at least one more pointBin common.

I.8. There exist at least four points which do not lie in a plane.

*Group II. Axioms of Order*

II.1. If point

Blies between pointsAandC, thenA, B, Care three distinct points of a line, andBalso lies betweenCandA.II.2. For any two distinct points

AandC,there exists at least one pointBon the lineACsuch thatClies betweenAandB.

II.3. Of any three points on a line there exists no more than one that lies between the other two.

II.4. LetA, B, Cbe three points that do not lie on a line and letmbe a line in the planeABCwhich does not meet any of the pointsA, B, C. If the linempasses through a point of the segmentAB, it also passes through a point of the segmentACor segmentBC.

*Group III. Axioms of Congruence*

III.1. If

A, Bare two points on a linem, andA'is a point on the same or on another linem'then it is always possible to find a pointB'on a given side of the linem'throughA'such that the segmentABis congruent to the segmentA'B'. In symbolsAB≅A'B'.

III.2. If two segments are congruent to a third one, they are congruent to each other.

III.3. On the linemletABandBCbe two segments which except forBhave no point in common. Furthermore, on the same or on another linem'letA'B'andB'C'be two segments which except forB'also have no point in common. In that case, ifAB≅A'B'andBC≅B'C',thenAC≅A'C'.III.4. Let ∠(

h,k)be an angle in a planeαandm'a line in a planeα'andlet a definite side ofm'inα'be given. Leth'be a ray on the linem'that emanates from the pointO'. Then there exists in the planeα'one and only one rayk'such that the angle ∠(h,k)is congruent to the angle ∠(h',k')and at the same time all interior points of the angle ∠(h',k')lie on the given side ofm'.Symbolically ∠(h, k)≅∠(h',k').Every angle is congruent to itself.

III.5. If for two trianglesABCandA'B'C'the congruencesAB≅A'B', AC≅A'C',∠BAC≅ ∠B'A'C'hold, then the congruence ∠ABC≅ ∠A'B'C'is also satisfied.

*Group IV. Axiom of Parallels*

IV. Let

mbe any line andAbe a point not on it. Then there is at most one line in the plane, determined bymandA, that passes throughAand does not intersectm.

*Group V. Axioms of Continuity*

V.1. (

Axiom of measureorArchimedes' Axiom) IfABandCDare any segments, then there exists a numbernsuch thatnsegmentsCDconstructed contiguously fromA, along the ray fromAthroughB, will pass beyond the pointB.

V.2. (Axiom of line completeness) An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III, and form V.1 is impossible.

**Defined Terms**

- Consider two points
*A*and*B*on a line*m*. The set of the two points*A*and*B*is called a*segment*. The points between*A*and*B*are called the points of the segment*AB*, or are also said to lie*inside*the segment*AB*. - Let
*A, A', O, B*be four points of a line*m*such that*O*lies between*A*and*B*, but not between*A*and*A'*. The points*A, A'*are then said to lie*on the line m**on one and the same side of the point O*and the points*A, B*are said to lie*on the line m on different sides of the point O*. The totality of the points of the line*m*that lie on the same side of*O*is called a*ray*emanating from*O.* - Let
*α*be a plane and*h, k*any two distinct rays emanating from*O*in*α*and lying on distinct lines. The pair of rays*h, k*is called an*angle*and is denoted by ∠(*h,k)*or ∠(*k,h)*. - Let the ray
*h*lie on the line*h'*and the ray*k*on the line*k'*. The rays*h*and*k*together with the point*O*partition the points of the plane into two regions. All points that lie on the same side of*k'*as those of*h*, and also those that lie on the same side of*h'*as those on*k*, are said to lie in the*interior*of the angle ∠(*h,k)*. - If
*A, B, C*are three points which do not lie on the same line, then the system of three segments*AB, BC, CA,*and their endpoints is called the*triangle ABC.*

_____________

Hilbert, David, *Foundations of Geometry (Grundlagen der Geometrie)*,
Second English Edition trans. by Unger,L. LaSalle: Open Court Publishing
Company, 1971 (1899).

© Copyright 2005, 2006 - Timothy Peil |