at this stage The theory of distance will be considered after the
principles of descriptive geometry have been developed.
Descriptive geometry is essentially the science of multiple order for open series. The first satisfactory system of axioms was given by M. Pasch.[1] An improved version is due to G. Peano.[2] Both these authors treat the idea of the class of points constituting the segment lying between two points as an undefined fundamental idea. Thus in fact there are in this system two fundamental ideas, namely, of points and of segments. It is then easy enough to define the prolongations of the segments, so as to form the complete straight lines. D. Hilbert’s[3] formulation of the axioms is in this respect practically based on the same fundamental ideas. His work is justly famous for some of the mathematical investigations contained in it, but his exposition of the axioms is distinctly inferior to that of Peano. Descriptive geometry can also be considered[4] as the science of a class of relations, each relation being a two-termed serial relation, as considered in the logic of relations, ranging the points between which it holds into a linear open order. Thus the relations are the straight lines, and the terms between which they hold are the points. But a combination of these two points of view yields[5] the simplest statement of all. Descriptive geometry is then conceived as the investigation of an undefined fundamental relation between three terms (points); and when the relation holds between three points A, B, C, the points are said to be “in the [linear] order ABC.”
O. Veblen’s axioms and definitions, slightly modified, are as follows:—
1. If the points A, B, C are in the order ABC, they are in the order CBA.
2. If the points A, B, C are in the order ABC, they are not in the order BCA.
3. If the points A, B, C are in the order ABC, A is distinct from C.
4. If A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC.
Definition.—The line (A ± B) consists of A and B, and of all points X in one of the possible orders, ABX, AXB, XAB. The points X in the order AXB constitute the segment AB.
5. If points C and D (C ± D) lie on the line AB, then A lies on the line CD.
6. There exist three distinct points A, B, C not in any of the orders ABC, BCA, CAB.
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| Fig. 73. |
7. If three distinct points A, B, C (fig. 73) do not lie on the same line, and D and E are two distinct points in the orders BCD and CEA, then a point F exists in the order AFB, and such that D, E, F are collinear.
Definition.—If A, B, C are three non-collinear points, the plane ABC is the class of points which lie on any one of the lines joining any two of the points belonging to the boundary of the triangle ABC, the boundary being formed by the segments BC, CA and AB. The interior of the triangle ABC is formed by the points in segments such as PQ, where P and Q are points respectively on two of the segments BC, CA, AB.
8. There exists a plane ABC, which does not contain all the points.
Definition.—If A, B, C, D are four non-coplanar points, the space ABCD is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ABCD, the surface being formed by the interiors of the triangles ABC, BCD, DCA, DAB.
9. There exists a space ABCD which contains all the points.
10. The Dedekind property holds for the order of the points on any straight line.
It follows from axioms 1-9 that the points on any straight line are arranged in an open serial order. Also all the ordinary theorems respecting a point dividing a straight line into two parts, a straight line dividing a plane into two parts, and a plane dividing space into two parts, follow.
Again, in any plane α consider a line l and a point A (fig. 74).
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| Fig. 74. |
Let any point B divide l into two half-lines l1 and l2. Then it can be proved that the set of half-lines, emanating from A and intersecting l1 (such as m), are bounded by two half-lines, of which ABC is one. Let r be the other. Then it can be proved that r does not intersect l1. Similarly for the half-line, such as n, intersecting l2. Let s be its bounding half-line. Then two cases are possible. (1) The half-lines r and s are collinear, and together form one complete line. In this case, there is one and only one line (viz. r + s) through A and lying in α which does not intersect l. This is the Euclidean case, and the assumption that this case holds is the Euclidean parallel axiom. But (2) the half-lines r and s may not be collinear. In this case there will be an infinite number of lines, such as k for instance, containing A and lying in α, which do not intersect l. Then the lines through A in α are divided into two classes by reference to l, namely, the secant lines which intersect l, and the non-secant lines which do not intersect l. The two boundary non-secant lines, of which r and s are respectively halves, may be called the two parallels to l through A.
The perception of the possibility of case 2 constituted the starting-point from which Lobatchewsky constructed the first explicit coherent theory of non-Euclidean geometry, and thus created a revolution in the philosophy of the subject. For many centuries the speculations of mathematicians on the foundations of geometry were almost confined to hopeless attempts to prove the “parallel axiom” without the introduction of some equivalent axiom.[6]
Associated Projective and Descriptive Spaces.—A region of a projective space, such that one, and only one, of the two supplementary segments between any pair of points within it lies entirely within it, satisfies the above axioms (1-10) of descriptive geometry, where the points of the region are the descriptive points, and the portions of straight lines within the region are the descriptive lines. If the excluded part of the original projective space is a single plane, the Euclidean parallel axiom also holds, otherwise it does not hold for the descriptive space of the limited region. Again, conversely, starting from an original descriptive space an associated projective space can be constructed by means of the concept of ideal points.[7] These are also called projective points, where it is understood that the simple points are the points of the original descriptive space. An ideal point is the class of straight lines which is composed of two coplanar lines a and b, together with the lines of intersection of all pairs of intersecting planes which respectively contain a and b, together with the lines of intersection with the plane ab of all planes containing any one of the lines (other than a or b) already specified as belonging to the ideal point. It is evident that, if the two original lines a and b intersect, the corresponding ideal point is nothing else than the whole class of lines which are concurrent at the point ab. But the essence of the definition is that an ideal point has an existence when the lines a and b do not intersect, so long as they are coplanar. An ideal point is termed proper, if the lines composing it intersect; otherwise it is improper.
A theorem essential to the whole theory is the following: if any two of the three lines a, b, c are coplanar, but the three lines are not all coplanar, and similarly for the lines a, b, d, then c and d are coplanar. It follows that any two lines belonging to an ideal point can be used as the pair of guiding lines in the definition. An ideal point is said to be coherent with a plane, if any of the lines composing it lie in the plane. An ideal line is the class of ideal points each of which is coherent with two given planes.
- ↑ Cf. loc. cit.
- ↑ Cf. I Principii di geometria (Turin, 1889) and “Sui fondamenti della geometria,” Rivista di mat. vol. iv. (1894).
- ↑ Cf. loc. cit.
- ↑ Cf. Vailati, Rivista di mat. vol. iv. and Russell, loc. cit. § 376.
- ↑ Cf. O. Veblen, “On the Projective Axioms of Geometry,” Trans. Amer. Math. Soc. vol. iii. (1902).
- ↑ Cf. P. Stäckel and F. Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss (Leipzig, 1895).
- ↑ Cf. Pasch, loc. cit., and R. Bonola, “Sulla introduzione degli enti improprii in geometria projettive,” Giorn. di mat. vol. xxxviii. (1900); and Whitehead, Axioms of Descriptive Geometry (Cambridge, 1907).
