Here below is the axioms presented for comparison for the different sources that I have been citing in the last few posts. You may be interested in comparing them. Similar Axioms are aligned on the same row. In some instances I put multiple in the same box, where an axiom was related to more than one. Columns on the left were given priority over ones on the right, so that the axioms in the leftmost column had the fewest gaps between axioms. This system was useful but a couple relationships were hidden. For example, Hilbert describes how on a line, that between any two points, there is a point between them. This is a property called "density" by mathematicians. He also says that there is also only one way to mark off a segment length. Both of these ideas are encapsulated by Holt's Ruler Axiom, which states that the points of a line can be placed in a one-to-one correspondence with the real numbers. The set of so-called real numbers has the property of density.
Related Articles
| Euclid's Axioms | Playfair's axioms | Hilbert's axioms | Holt's axioms |
| 1. A straight line[segment] may be drawn from any point to any point. 2. A finite straight line may be produced continually in a straight line. | -If a straight line[segment] be drawn joining two points, then any other segment drawn between the two points will coincide with the first segment. -A straight line[segment] may be supposed to be drawn from any point, to any distance, and in any direction. | Two points always completely determine a straight line. Two distinct points on a line always completely determine that line. | Through any two points there is exactly one line. |
| 3.A circle may be constructed using any point as the center with any given radius. | . | ||
| 4. All right angles are equal to each other. | |||
| 5. If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines if produced indefinitely meet on that side which are the angles less than two right angles. | Through for any line and point not on the line, there exists exactly one line parallel to the given line passing through the point. | Parallel Axiom: Given a line and a point not on the line, there exists exactly one line parallel to the given line passing through the point. Corresponding Angles Axiom: If two parallel lines are crossed by a transversal, then corresponding angles are congruent. Converse Corresponding Angles Axiom: If two lines are crossed by a transversal and corresponding angles are congruent, then the two lines are parallel. | |
| The intersection of two lines is a point. | If two lines intersect, then they interesect in exactly one point. | ||
| A plane may be supposed to be described on any two straight lines which form an angle and enlarged to any extent. | -Three distinct non-collinear points completely determine a plane. -Three distinct non-collinear points that lie on the same plane always determines completely that plane. | Through any three non-collinear points there is exactly one plane containing them. | |
| If one straight line[segment] be applied to another, then they will coincide except inasmuch as one is longer than the other, if either is produced [extended] they will still coincide. | |||
| A straight line[segment] is the shortest distance that can be drawn between two points. | |||
| If any two points be taken in a plane, then the line between the two points is also on the plane. | If two points lie on a plane, then the line containing those two points also lies on the plane. | If two points lie on a plane, then the line containing those two points also lies on the plane. | |
| If any two planes have a point in common, then there exists atleast one other point that they also have in common. | If two planes intersect, then they intersect in exactly one line. | ||
| A line contains atleast 2 points, a plane contains atleast 3 non-collinear points, and space contains atleast 4 non-coplanar points. | |||
| If points A, B, and C lie on the same line such that B is between A and C, then B is between C and A. | |||
| If A and B lie on a straight line, then there exists point C between A and B, and a point D such that B is between C and D. | |||
| For any three points on a straight line, exactly one of those points lies between the other two. | |||
| For any three non-collinear points A, B, and C. Any line passing through AB, but not A or B, must pass through BC or AC also. | |||
| For any given segment AB on line AB, and point A' either on line AB or not, there exists exactly one B' on a given side of A' on the line containing A' such that segment AB is congruent to A'B'. | Ruler Axiom: The points on a line can be put on a one-to-one correspondance to the real numbers. | ||
| The congruence of segments is reflexive and symmetric. | |||
| If B lies between A and C on line AC, and B' lies between A' and C' on line A'C', and segment AB is congruent to segment A'B' while segment BC is congruent to segment B'C', then segment AC is congruent to segment A'C'. | |||
| The congruence of angles is reflexive and symmetric. | |||
| Given angle AOB, for any ray A'O' there exists exactly one ray on a given side of line A'O' such that A'O'B' is congruent to angle AOB. | Protractor Axiom: Given a line and a point on the line, all of the possible rays that can be drawn from the point can be placed in a one to one correspondence witht the real numbers from zero to 180. | ||
| Archimedes Axiom: Let A1 be a point between arbitrarly chosen point A and B. Take points A1, A2, A3, A4,.....such that A1 is between A and A2, A2 is between A1 and A3, A4 is between A3 and A5, and so on. Moreover, let segments AA1, A1A2, A2A3,... be congruent. Then, always in such a serie of points there exists a An such that B is between A and An. | |||
| Axiom of Completeness: to the system of points, lines, and planes established by the axioms it is not possible to add elements to create a new geometry that obeys the aforementioned axioms. | |||
| Segment Addition Axiom: If B is between A and C, then AB + BC = AC. | |||
| Angle Addition Axiom: If S is in the interior of angle PQR, then the measure of angle PQR equals the sum of the measures of PQS and SQR. | |||
| Area Addition Axiom: the area of a region is equal to the sum of the areas of its non-overlapping parts | |||
| Arc Addition Axiom: the measure of an arc formed by two adjacent arcs is equal to the measure of the two arcs. | |||
| Side-Side-Side Congruence Axiom: If three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent. | |||
| Side-Angle-Side Congruence Axiom: If two sides and the included angle of one triangle are congruent to two sides and the includeded angle of another triangle, then the triangles are congruent. | |||
| Angle-Side-Angle Congruence Axiom: If two angles and the included side of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. | |||
| Angle-Angle Similarity Axiom: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. |
