the four points are collinear and D lies in the segment complementary
to the segment ABC. The property of the separation
of pairs of points by pairs of points is projective. Also it can be
proved that Harm. (ABCD) implies that B and D separate
A and C.
Definitions.—A series of entities arranged in a serial order, open or closed, is said to be compact, if the series contains no immediately consecutive entities, so that in traversing the series from any one entity to any other entity it is necessary to pass through entities distinct from either. It was the merit of R. Dedekind and of G. Cantor explicitly to formulate another fundamental property of series. The Dedekind property[1] as applied to an open series can be defined thus: An open series possesses the Dedekind property, if, however, it be divided into two mutually exclusive classes u and v, which (1) contain between them the whole series, and (2) are such that every member of u precedes in the serial order every member of v, there is always a member of the series, belonging to one of the two, u or v, which precedes every member of v (other than itself if it belong to v), and also succeeds every member of u (other than itself if it belong to u). Accordingly in an open series with the Dedekind property there is always a member of the series marking the junction of two classes such as u and v. An open series is continuous if it is compact and possesses the Dedekind property. A closed series can always be transformed into an open series by taking any arbitrary member as the first term and by taking one of the two ways round as the ascending order of the series. Thus the definitions of compactness and of the Dedekind property can be at once transferred to a closed series.
12. The last axiom of order is that there exists at least one straight line for which the point order possesses the Dedekind property.
It follows from axioms 1-12 by projection that the Dedekind property is true for all lines. Again the harmonic system ABC, where A, B, C are collinear points, is defined[2] thus: take the harmonic conjugates A′, B′, C′ of each point with respect to the other two, again take the harmonic conjugates of each of the six points A, B, C, A′, B′, C′ with respect to each pair of the remaining five, and proceed in this way by an unending series of steps. The set of points thus obtained is called the harmonic system ABC. It can be proved that a harmonic system is compact, and that every segment of the line containing it possesses members of it. Furthermore, it is easy to prove that the fundamental theorem holds for harmonic systems, in the sense that, if A, B, C are three points on a line l, and A′, B′, C′ are three points on a line l′, and if by any two distinct series of projections A, B, C are projected into A′, B′, C′, then any point of the harmonic system ABC corresponds to the same point of the harmonic system A′B′C′ according to both the projective relations which are thus established between l and l′. It now follows immediately that the fundamental theorem must hold for all the points on the lines l and l′, since (as has been pointed out) harmonic systems are “everywhere dense” on their containing lines. Thus the fundamental theorem follows from the axioms of order.
A system of numerical coordinates can now be introduced, possessing the property that linear equations represent planes and straight lines. The outline of the argument by which this remarkable problem (in that “distance” is as yet undefined) is solved, will now be given. It is first proved that the points on any line can in a certain way be definitely associated with all the positive and negative real numbers, so as to form with them a one-one correspondence. The arbitrary elements in the establishment of this relation are the points on the line associated with 0, 1 and ∞.
This association[3] is most easily effected by considering a class of projective relations of the line with itself, called by F. Schur (loc. cit.) prospectivities.
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| Fig. 69. |
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| Fig. 70. |
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| Fig. 71. |
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| Fig. 72. |
Let l (fig. 69) be the given line, m and n any two lines intersecting at U on l, S and S′ two points on n. Then a projective relation between l and itself is formed by projecting l from S on to m, and then by projecting m from S′ back on to l. All such projective relations, however m, n, S and S′ be varied, are called “prospectivities,” and U is the double point of the prospectivity. If a point O on l is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate O to A, give the same correspondent (Q, in figure) to any point P on the line l; in fact they are all the same prospectivity, however m, n, S, and S′ may have been varied subject to these conditions. Such a prospectivity will be denoted by (OAU2).
The sum of two prospectivities, written (OAU2) + (OBU2), is defined to be that transformation of the line l into itself which is obtained by first applying the prospectivity (OAU2) and then applying the prospectivity (OBU2). Such a transformation, when the two summands have the same double point, is itself a prospectivity with that double point.
With this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude. Accordingly they can be associated in a one-one correspondence with the positive and negative real numbers. Let E (fig. 70) be any point on l, distinct from O and U. Then the prospectivity (OEU2) is associated with unity, the prospectivity (OOU2) is associated with zero, and (OUU2) with ∞. The prospectivities of the type (OPU2), where P is any point on the segment OEU, correspond to the positive numbers; also if P′ is the harmonic conjugate of P with respect to O and U, the prospectivity (OP′U2) is associated with the corresponding negative number. (The subjoined figure explains this relation of the positive and negative prospectivities.) Then any point P on l is associated with the same number as is the prospectivity (OPU2).
It can be proved that the order of the numbers in algebraic order of magnitude agrees with the order on the line of the associated points. Let the numbers, assigned according to the preceding specification, be said to be associated with the points according to the “numeration-system (OEU).” The introduction of a coordinate system for a plane is now managed as follows: Take any triangle OUV in the plane, and on the lines OU and OV establish the numeration systems (OE1U) and (OE2V), where E1 and E2 are arbitrarily chosen. Then (cf. fig. 71) if M and N are associated with the numbers x and y according to these systems, the coordinates of P are x and y. It then follows that the equation of a straight line is of the form ax + by + c = 0. Both coordinates of any point on the line UV are infinite. This can be avoided by introducing homogeneous coordinates X, Y, Z, where x = X/Z, and y = Y/Z, and Z = 0 is the equation of UV.
The procedure for three dimensions is similar. Let OUVW (fig. 72) be any tetrahedron, and associate points on OU, OV, OW with numbers according to the numeration systems (OE1U), (OE2V), and (OE3W). Let the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and let x, y, z be the numbers associated with L, M, N respectively. Then P is the point (x, y, z). Also homogeneous coordinates can be introduced as before, thus avoiding the infinities on the plane UVW.
The cross ratio of a range of four collinear points can now be defined as a number characteristic of that range. Let the coordinates of any point Pr of the range P1 P2 P3 P4 be
| λra + μr + a′ | , | λrb + μrb′ | , | λrc + μrc′ | , (r = 1, 2, 3, 4) |
| λr + μr | λr + μr | λr + μr |
and let (λrμs) be written for λrμs −λsμr. Then the cross ratio {P1 P2 P3 P4} is defined to be the number (λ1μ2)(λ3μ4) / (λ1μ4)(λ3μ2). The equality of the cross ratios of the ranges (P1 P2 P3 P4) and (Q1 Q2 Q3 Q4) is proved to be the necessary and sufficient condition for their mutual projectivity. The cross ratios of all harmonic ranges are then easily seen to be all equal to −1, by comparing with the range (OE1UE′1) on the axis of x.
Thus all the ordinary propositions of geometry in which distance and angular measure do not enter otherwise than in cross ratios can now be enunciated and proved. Accordingly the greater part of
the analytical theory of conics and quadrics belongs to geometry- ↑ Cf. Dedekind, Stetigkeit und irrationale Zahlen (1872).
- ↑ Cf. v. Staudt, Geometrie der Lage (1847).
- ↑ Cf. Pasch, Vorlesungen über neuere Geometrie (Leipzig, 1882), a classic work; also Fiedler, Die darstellende Geometrie (1st ed., 1871, 3rd ed., 1888); Clebsch, Vorlesungen über Geometrie, vol. iii.; Hilbert, loc. cit.; F. Schur, Math. Ann. Bd. lv. (1902); Vahlen, loc. cit.; Whitehead, loc. cit.
