be presented[1] here as twelve in number, eight being “axioms
of classification,” and four being “axioms of order.”
Axioms of Classification.—The eight axioms of classification are as follows:
1. Points form a class of entities with at least two members.
2. Any straight line is a class of points containing at least three members.
3. Any two distinct points lie in one and only one straight line.
4. There is at least one straight line which does not contain all the points.
5. If A, B, C are non-collinear points, and A′ is on the straight line BC, and B′ is on the straight line CA, then the straight lines AA′ and BB′ possess a point in common.
Definition.—If A, B, C are any three non-collinear points, the plane ABC is the class of points lying on the straight lines joining A with the various points on the straight line BC.
6. There is at least one plane which does not contain all the points.
7. There exists a plane α, and a point A not incident in α, such that any point lies in some straight line which contains both A and a point in α.
Definition.—Harm. (ABCD) symbolizes the following conjoint statements: (1) that the points A, B, C, D are collinear, and (2) that a quadrilateral can be found with one pair of opposite sides intersecting at A, with the other pair intersecting at C, and with its diagonals passing through B and D respectively. Then B and D are said to be “harmonic conjugates” with respect to A and C.
8. Harm. (ABCD) implies that B and D are distinct points.
In the above axioms 4 secures at least two dimensions, axiom 5 is the fundamental axiom of the plane, axiom 6 secures at least three dimensions, and axiom 7 secures at most three dimensions. From axioms 1-5 it can be proved that any two distinct points in a straight line determine that line, that any three non-collinear points in a plane determine that plane, that the straight line containing any two points in a plane lies wholly in that plane, and that any two straight lines in a plane intersect. From axioms 1-6 Desargue’s well-known theorem on triangles in perspective can be proved.
The enunciation of this theorem is as follows: If ABC and A′B′C′ are two coplanar triangles such that the lines AA′, BB′, CC′ are concurrent, then the three points of intersection of BC and B′C′ of CA and C′A′, and of AB and A′B′ are collinear; and conversely if the three points of intersection are collinear, the three lines are concurrent. The proof which can be applied is the usual projective proof by which a third triangle A″B″C″ is constructed not coplanar with the other two, but in perspective with each of them.
It has been proved[2] that Desargues’s theorem cannot be deduced from axioms 1-5, that is, if the geometry be confined to two dimensions. All the proofs proceed by the method of producing a specification of “points” and “straight lines” which satisfies axioms 1-5, and such that Desargues’s theorem does not hold.
It follows from axioms 1-5 that Harm. (ABCD) implies Harm. (ADCB) and Harm. (CBAD), and that, if A, B, C be any three distinct collinear points, there exists at least one point D such that Harm. (ABCD). But it requires Desargues’s theorem, and hence axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD′) imply the identity of D and D′.
The necessity for axiom 8 has been proved by G. Fano,[3] who has produced a three dimensional geometry of fifteen points, i.e. a method of cross classification of fifteen entities, in which each straight line contains three points, and each plane contains seven straight lines. In this geometry axiom 8 does not hold. Also from axioms 1-6 and 8 it follows that Harm. (ABCD) implies Harm. (BCDA).
Definitions.—When two plane figures can be derived from one another by a single projection, they are said to be in perspective. When two plane figures can be derived one from the other by a finite series of perspective relations between intermediate figures, they are said to be projectively related. Any property of a plane figure which necessarily also belongs to any projectively related figure, is called a projective property.
The following theorem, known from its importance as “the fundamental theorem of projective geometry,” cannot be proved[4] from axioms 1-8. The enunciation is: “A projective correspondence between the points on two straight lines is completely determined when the correspondents of three distinct points on one line are determined on the other.” This theorem is equivalent[5] (assuming axioms 1-8) to another theorem, known as Pappus’s Theorem, namely: “If l and l ′ are two distinct coplanar lines, and A, B, C are three distinct points on l, and A′, B′, C′ are three distinct points on l ′, then the three points of intersection of AA′ and B′C, of A′B and CC′, of BB′ and C′A, are collinear.” This theorem is obviously Pascal’s well-known theorem respecting a hexagon inscribed in a conic, for the special case when the conic has degenerated into the two lines l and l ′. Another theorem also equivalent (assuming axioms 1-8) to the fundamental theorem is the following:[6] If the three collinear pairs of points, A and A′, B and B′, C and C′, are such that the three pairs of opposite sides of a complete quadrangle pass respectively through them, i.e. one pair through A and A′ respectively, and so on, and if also the three sides of the quadrangle which pass through A, B, and C, are concurrent in one of the corners of the quadrangle, then another quadrangle can be found with the same relation to the three pairs of points, except that its three sides which pass through A, B, and C, are not concurrent.
Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved. Also a conic can be defined as the locus of the points found by the usual construction, based upon Pascal’s theorem, for points on the conic through five given points. But it is unnecessary to assume here any one of the suggested axioms; for the fundamental theorem can be deduced from the axioms of order together with axioms 1-8.
Axioms of Order.—It is possible to define (cf. Pieri, loc. cit.) the property upon which the order of points on a straight line depends. But to secure that this property does in fact range the points in a serial order, some axioms are required. A straight line is to be a closed series; thus, when the points are in order, it requires two points on the line to divide it into two distinct complementary segments, which do not overlap, and together form the whole line. Accordingly the problem of the definition of order reduces itself to the definition of these two segments formed by any two points on the line; and the axioms are stated relatively to these segments.
Definition.—If A, B, C are three collinear points, the points on the segment ABC are defined to be those points such as X, for which there exist two points Y and Y′ with the property that Harm. (AYCY′) and Harm. (BYXY′) both hold. The supplementary segment ABC is defined to be the rest of the points on the line. This definition is elucidated by noticing that with our ordinary geometrical ideas, if B and X are any two points between A and C, then the two pairs of points, A and C, B and X, define an involution with real double points, namely, the Y and Y′ of the above definition. The property of belonging to a segment ABC is projective, since the harmonic relation is projective.
The first three axioms of order (cf. Pieri, loc. cit.) are:
9. If A, B, C are three distinct collinear points, the supplementary segment ABC is contained within the segment BCA.
10. If A, B, C are three distinct collinear points, the common part of the segments BCA and CAB is contained in the supplementary segment ABC.
11. If A, B, C are three distinct collinear points, and D lies In the segment ABC, then the segment ADC is contained within the segment ABC.
From these axioms all the usual properties of a closed order follow. It will be noticed that, if A, B, C are any three collinear points, C is necessarily traversed in passing from A to B by one route along the line, and is not traversed in passing from A to B along the other route. Thus there is no meaning, as referred to closed straight lines, in the simple statement that C lies between A and B. But there may be a relation of separation between two pairs of collinear points, such as A and C, and B and D. The couple B and D is said to separate A and C, if
- ↑ This formulation—though not in respect to number—is in all essentials that of M. Pieri, cf. “I principii della Geometria di Posizione,” Accad. R. di Torino (1898); also cf. Whitehead, loc. cit.
- ↑ Cf. G. Peano, “Sui fondamenti della Geometria,” p. 73, Rivista di matematica, vol. iv. (1894), and D. Hilbert, Grundlagen der Geometrie (Leipzig, 1899); and R. F. Moulton, “A Simple non-Desarguesian Plane Geometry,” Trans. Amer. Math. Soc., vol. iii. (1902).
- ↑ Cf. “Sui postulati fondamentali della geometria projettiva,” Giorn. di matematica, vol. xxx. (1891); also of Pieri, loc. cit., and Whitehead, loc. cit.
- ↑ Cf. Hilbert, loc. cit.; for a fuller exposition of Hilbert’s proof cf. K. T. Vahlen, Abstrakte Geometrie (Leipzig, 1905), also Whitehead, loc. cit.
- ↑ Cf. H. Wiener, Jahresber. der Deutsch. Math. Ver. vol. i. (1890); and F. Schur, “Über den Fundamentalsatz der projectiven Geometrie,” Math. Ann. vol. li. (1899).
- ↑ Cf. Hilbert, loc. cit., and Whitehead, loc. cit.
