three-space ABCD, then the three-space ABCD is identical with the three-space A'B'C'D'; that if two points of a line lie in a given three-space ABCD, then so do all points of the line; that if three points of a plane lie in a three-space, so do all points of the plane; and that if and only if two planes are in the same three-space they have a line in common. The notion of a three-dimensional region can then be defined and studied analogously to V. Congruent figures can be defined as in VI.

A

B

FIG. 52.

Assumption VI provided for the existence of a plane, but since nothing has as yet been said about the existence of points. which are not coplanar, we add the following:

Assumption XV. If A, B, C are three non-collinear points, there exists a point D not in the same plane with A, B and C. Assumption XVI. Two planes which have one point in common have two points in common.

Assumption XV provides for the existence of at least one three-space and from Assumption XVI it follows that all points are in the same three-space.

All the theorems of elementary three-dimensional geometry can be developed on the basis of these assumptions. But to do so would be to write a large book.*

* A book giving a complete and rigorous treatment of elementary geometry would be a most important influence in improving the teaching of the most ancient and perfect of sciences. Such a book could rarely, if ever, be used in the classroom, but if it were in the hands of the teachers it would serve to keep before them in something like its actual form the structure of which they are trying to give their students a first glimpse.

XII. CONCLUSION

The logically important questions as to the independence and categoricalness of our assumptions must be passed over with a reference to the two papers in the Transactions on which this essay is based. The ideas of consistency, independence and categoricalness (sufficiency) are explained in the essay by Professor Huntington in this book, and the independence of Assumption XIII is established in the essay by Professor Woods.

A reader who is sufficiently interested to pursue the subject further is strongly urged to go into the question of the independence of the assumptions and to try to discover for himself some of the examples which constitute the independence proofs. For convenience in this sort of study we have collected the assumptions in the following list.

I. If points A, B, C are in the order { ABC'} they are distinct. II. If points A, B, C are in the order ABC) they are not in the order {BCA}.

Definition. If A and B are distinct points the segment AB consists of all points, X, in the order AXB}; all points of the segment AB are said to be between A and B; the segment together with A and B is called the interval AB; the line AB consists of A and B and all points, X, in one of the orders {ABX}, {AXB,} {XAB}; and the ray AB consists of B and all points, X in one of the orders {AXB} and {ABX}; A is called the origin of the ray AB.

F

E

B

C

FIG. 53.

III. If points C and D (CD) are on the line AB, then A is on the line CD.

IV. If A and B are two distinct points, there exists a point C such that A, B and C are in the order {ABC}.

V. If three distinct points A, B and C do not lie on the same line and D and E are two points in the orders {BCD} and (CEA), then a point F exists in the order {AFB) and such that D, E and F lie on the same line.

VI. There exist three distinct points, A, B, C, not in any of the orders {ABC}, {BCA}, {CAB}.

Definition. If A, B, C are three non-collinear points, the set of all points collinear with pairs of points on the intervals AB, BC, CA is called the plane ABC. The points X of the plane such that the interval AX does not contain a point of the line BC constitute, together with A itself, one side of the line BC. The other points of the plane, not on the line BC, constitute the other side of the line BC.

The notation (A, B) denotes a pair of distinct points.

VII. If AB, then on any ray whose origin is C there exists one and only one point D such that (A, B) is congruent to (C, D).

VIII. If (A, B) is congruent to (C, D) and (C, D) is congruent to (E, F) then (A, B) is congruent to (E, F).

IX. If (A, B) is congruent to (A′, B′) and (B, C) is congruent to (B', C') and {ABC} and {A'B'C'}, then (A, C) is congruent to (A', C').

X. (A, B) is congruent to (B, A).

XI. If A, B, C are three non-collinear points and D is a point in the order (BCD), and if A'B'C' are three non-collinear points and D' is a point in the order {B'C'D'} such that the point-pairs (A, B), (B, C), (C, A), (B, D) are respectively congruent to (A′, B′), (B′, C′), (C', A′), (B′, D') then (A, D) is congruent to (A′, D′).

B

FIG. 54.

A

Definition. If O and Xo are two points of a plane a, then the set of points [X] of a such that (0, X) is congruent to (0, Xo) is called a circle. O is called its centre and any of the intervals OX is called a radius. The points, except the points [X], on radii of the circle are said to be interior to the circle. The points of a not on radii are said to be exterior to the circle.

XII. A circle passing through a point, A, interior and a point, B, exterior to another circle in the same plane has in common with the other circle at least one point on each side of the line AB.

XIII. If A is any point and a any line not passing through A, there is not more than one line through A coplanar with a and not meeting a.

XIV. If A, B, C are three points in the order {ABC} and B1, B2, B3, . . . are points in the order {ABB1}, {AB1B2}, such that (A, B) is congruent to each of the point-pairs (B, B1), (B1, B2), ..., then there are not more than a finite number of the points B1, B2, . . . between A and C.

XV. If A, B, C are three non-collinear points, there exists a point D not in the same plane with A, B and C.

XVI. Two planes which have one point in common have two points in common.