Charles Everitt, Olaus Henrici, Edwin B. Elliott, John H. Grace, Bertrand A. W. Russell and Alfred N. Whitehead

“ If we next take two intersecting planes, then the point at infinity in their line of intersection lies in both planes, so that their lines at infinity meet. Hence every line at infinity meets every other line at infinity, and they are therefore all in one plane.
All points at infinity in space may be considered as lying in one ideal plane, which is called the plane at infinity. ”

“ Any science is called a “geometry” if it investigates the theory of the classification of a set of entities (the points) into classes (the straight lines) , such that (1) there is one and only one class which contains any given pair of the entities, and (2) every such class contains more than two members. In the two geometries, important from their relevance to existent space, axioms which secure an order of the points on any line also occur. These geometries will be called “Projective Geometry” and “Descriptive Geometry.” ”

“ The principle of simplicity would therefore lead us, in general, where an observation conflicted with one or more of those laws, to ascribe this anomaly, not to the falsity of Euclidean geometry, but to the falsity of the laws in question. This applies especially to astronomy. On the earth our means of measurement are many and direct, and so long as no great accuracy is sought they involve few scientific laws. Thus we acquire, from such direct measurements, a very high degree of probability that the space-constant, if not infinite, is yet large as compared with terrestrial distances. ”

“ The coordinates of a point taken at random in the plane will, as a rule, not satisfy the equation, but infinitely many points, and in most cases infinitely many real ones, have coordinates which do satisfy it, and these points are exactly those which lie upon some locus of one dimension, a straight line or more frequently a curve, which is said to be represented by the equation. ”

“ These are the principal properties of lines and planes in space, but before we go on to their applications it will be well to define the word distance. In geometry distance means always “shortest distance”; viz. the distance of a point from a straight line, or from a plane, is the length of the perpendicular from the point to the line or plane. The distance between two non-intersecting lines is the length of their common perpendicular, there being but one. ”

“ Parallelism.—We have now the following definitions:—
Parallel lines are lines which meet at infinity;
Parallel planes are planes which meet at infinity;
A line is parallel to a plane if it meets it at infinity.
Theorems like this—Lines (or planes) which are parallel to a third are parallel to each other—follow at once.
This view of parallels leads therefore to no contradiction of Euclid’s Elements.
As immediate consequences we get the propositions:—
Every line meets a plane in one point, or it lies in it;
Every plane meets every other plane in a line ”

“ In the general case of a singly infinite system of lines, the locus is a ruled surface (or regulus) . Now, when a line is changing its position in space, it may be looked upon as in a state of turning about some point in itself, while that point is, as a rule, in a state of moving out of the plane in which the turning takes place. If instantaneously it is only in a state of turning, it is usual, though not strictly accurate, to say that it intersects its consecutive position. A regulus such that consecutive lines on it do not intersect, in this sense, is called a skew surface, or scroll ”

“ But a plane is only one among an infinite number of conceivable surfaces. We may draw figures on any one of them and study their properties. We may, for instance, take a sphere instead of the plane, and obtain “spherical” in the place of “plane” geometry. If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last all hold, then all propositions up to the 28th will be true for these figures. ”

“ Projective Definitions of the Conics.—We now consider the shape of the conics. We know that any line in the plane of the conic, and hence that the line at infinity, either has no point in common with the curve, or one (counting for two coincident points) or two distinct points. If the line at infinity has no point on the curve the latter is altogether finite, and is called an Ellipse (fig. 21) . If the line at infinity has only one point in common with the conic, the latter extends to infinity, and has the line at infinity a tangent. It is called a Parabola (fig. 22) . ”

“ Foci.—The centre of a pencil in which the conic determines a circular involution is called a “focus” of the conic.
In other words, a focus is such a point that every line through it is perpendicular to its conjugate line. The polar to a focus is called a directrix of the conic.
From the definition it follows that every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it. ”

“ For the sake of brevity we shall presuppose a knowledge of Euclid’s Elements, although we shall use only a few of his propositions.
§ 1. Geometrical Elements. We consider space as filled with points, lines and planes, and these we call the elements out of which our figures are to be formed, calling any combination of these elements a “figure.”
By a line we mean a straight line in its entirety, extending both ways to infinity; and by a plane, a plane surface, extending in all directions to infinity. ”

“ Here again, as x takes all values, the point or points describe a path or paths, which constitute a locus represented by the equation. Except when y enters to the first degree only in ƒ (x, y) , it is not to be expected that all the values of y, determined as going with a chosen value of x, will be necessarily real; indeed it is not uncommon for all to be imaginary for some ranges of values of x. The locus may largely consist of continua of imaginary points; but the real parts of it constitute a real curve or real curves. ”

“ But any plane α cuts each of the planes π1, π2 in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane α.
A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines which meet on the axis determine a plane.
If the plane is parallel to the axis its traces are parallel to the axis. Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. ”

“ On the other hand, the fact that a straight line does or does not cut a circle is independent of measurement, it being dependent only upon the mutual “position” of the line and the circle. This difference becomes clearer if we project any figure from one plane to another (see Projection) . By this the length of lines, the magnitude of angles and areas, is altered, so that the projection, or shadow, of a square on a plane will not be a square; it will, however, be some quadrilateral. Again, the projection of a circle will not be a circle, but some other curve more or less resembling a circle. ”

“ The sum of the three angles of a triangle is always less than two right angles. The area of the triangle ABC is λ2 (π − A − B − C) . If the base BC of a triangle is kept fixed and the vertex A moves in the fixed plane ABC so that the area ABC is constant, then the locus of A is a line of equal distance from BC. This locus is not a straight line. The whole theory of similarity is inapplicable; two triangles are either congruent, or their angles are not equal two by two. Thus the elements of a triangle are determined when its three angles are given. ”

“ From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve, surface and solid, appropriately defined, are the premises from which the geometer draws conclusions. The geometrical axioms are merely conventions; on the one hand, the system may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of physical science ”

“ Every line through S which joins it to any point at an infinite distance in p is parallel to p. But by Euclid’s 12th axiom there is but one line parallel to p through S. The difficulty in which we are thus involved is due to the fact that we try to reason about infinity as if we, with our finite capabilities, could comprehend the infinite. To overcome this difficulty, we may say that all points at infinity in a line appear to us as one, and may be replaced by a single “ideal” point. ”

“ In two dimensions there is no necessity for a special line geometry, inasmuch as the straight line and the point are interchangeable by the principle of duality; but in three dimensions the straight line is its own reciprocal, and for the better discussion of systems of lines we require some new apparatus, e.g., a system of coordinates applicable to straight lines rather than to points. ”

“ Euclid’s definitions mentioned above are attempts to describe, in a few words, notions which we have obtained by inspection of and abstraction from solids. A few more notions have to be added to these, principally those of the simplest line—the straight line, and of the simplest surface—the flat surface or plane. These notions we possess, but to define them accurately is difficult. Euclid’s Definition 4, I., “A straight line is that which lies evenly between its extreme points,” must be meaningless to any one who has not the notion of straightness in his mind. ”

“ The axis which contains the foci is called the principal axis; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic.
In case of the parabola there is but one axis. The involution on this axis has its centre at infinity. One focus is therefore at infinity, the one focus only is finite. A parabola has only one focus. ”

“ The point conjugate to the point at infinity is called the “centre” of the involution. Every involution has a centre, unless the point at infinity be a focus, in which case we may say that the centre is at infinity. ”

“ Definitions.—The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,—hence from experience. The same source gives us the notions of the geometrical entities to which the axioms relate, viz. solids, surfaces, lines or curves, and points. A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid. ”

“ 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. ”

“ Conics.—If we have two projective pencils in a plane, corresponding rays will meet, and their point of intersection will constitute some locus which we have to investigate. Reciprocally, if two projective rows in a plane are given, then the lines which join corresponding points will envelope some curve. We prove first:—
It will be seen that the proofs are reciprocal, so that the one may be copied from the other by simply interchanging the words point and line, locus and envelope, row and pencil, and so on. ”

“ 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′. ”

“ Let P be a point in space not on the surface, which we suppose not to be a cone. On every line through P which cuts the surface in two points we determine the harmonic conjugate Q of P with regard to the points of intersection. Through one of these lines we draw two planes α and β. The locus of the points Q in α is a line a, the polar of P with regard to the conic in which α cuts the surface. Similarly the locus of points Q in β is a line b. This cuts a, because the line of intersection of α and β contains but one point Q. The locus of all points Q therefore is a plane. ”

“ If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre. Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them. ”

“ Representation of Points.—We have thus the following method of representing in a single plane the position of points in space:—we take in the plane a line xy as the axis, and then any pair of points A1, A2 in the plane on a line perpendicular to the axis represent a point A in space. If the line A1A2 cuts the axis at A0, and if at A1 a perpendicular be erected to the plane, then the point A will be in it at a height A1A = A0A2 above the plane. ”