It will be observed that not only are planes determined by points, but also points by planes; that therefore the planes may be considered as elements, like points; and also that in any one of the above statements we may interchange the words point and plane, and we obtain again a correct statement, provided that these statements themselves are true. As they stand, we ought, in several cases, to add “if they are not parallel,” or some such words, parallel lines and planes being evidently left altogether out of consideration. To correct this we have to reconsider the theory of parallels.
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| Fig. 1. |
§ 2. Parallels. Point at Infinity.—Let us take in a plane a line p (fig. 1), a point S not in this line, and a line q drawn through S. Then this line q will meet the line p in a point A. If we turn the line q about S towards q′, its point of intersection with p will move along p towards B, passing, on continued turning, to a greater and greater distance, until it is moved out of our reach. If we turn q still farther, its continuation will meet p, but now at the other side of A. The point of intersection has disappeared to the right and reappeared to the left. There is one intermediate position where q is parallel to p—that is where it does not cut p. In every other position it cuts p in some finite point. If, on the other hand, we move the point A to an infinite distance in p, then the line q which passes through A will be a line which does not cut p at any finite point. Thus we are led to say: 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.
We may therefore now give the following definitions and axiom:—
Definition.—Lines which meet at infinity are called parallel.
Axiom.—All points at an infinite distance in a line may be considered as one single point.
Definition.—This ideal point is called the point at infinity in the line.
The axiom is equivalent to Euclid’s Axiom 12, for it follows from either that through any point only one line may be drawn parallel to a given line.
This point at infinity in a line is reached whether we move a point in the one or in the opposite direction of a line to infinity. A line thus appears closed by this point, and we speak as if we could move a point along the line from one position A to another B in two ways, either through the point at infinity or through finite points only.
It must never be forgotten that this point at infinity is ideal; in fact, the whole notion of “infinity” is only a mathematical conception, and owes its introduction (as a method of research) to the working generalizations which it permits.
§ 3. Line and Plane at Infinity.—Having arrived at the notion of replacing all points at infinity in a line by one ideal point, there is no difficulty in replacing all points at infinity in a plane by one ideal line.
To make this clear, let us suppose that a line p, which cuts two fixed lines a and b in the points A and B, moves parallel to itself to a greater and greater distance. It will at last cut both a and b at their points at infinity, so that a line which joins the two points at infinity in two intersecting lines lies altogether at infinity. Every other line in the plane will meet it therefore at infinity, and thus it contains all points at infinity in the plane.
All points at infinity in a plane lie in a line, which is called the line at infinity in the plane.
It follows that parallel planes must be considered as planes having a common line at infinity, for any other plane cuts them in parallel lines which have a point at infinity in common.
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.
§ 4. 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;
Any two lines in the same plane meet.
§ 5. Aggregates of Geometrical Elements.—We have called points, lines and planes the elements of geometrical figures. We also say that an element of one kind contains one of the other if it lies in it or passes through it.
All the elements of one kind which are contained in one or two elements of a different kind form aggregates which have to be enumerated. They are the following:—
I. Of one dimension.
1. The row, or range, of points formed by all points in a line, which is called its base.
2. The flat pencil formed by all the lines through a point in a plane. Its base is the point in the plane.
3. The axial pencil formed by all planes through a line which is called its base or axis.
II. Of two dimensions.
1. The field of points and lines—that is, a plane with all its points and all its lines.
2. The pencil of lines and planes—that is, a point in space with all lines and all planes through it.
III. Of three dimensions.
The space of points—that is, all points in space.
The space of planes—that is, all planes in space.
IV. Of four dimensions.
The space of lines, or all lines in space.
§ 6. Meaning of “Dimensions.”—The word dimension in the above needs explanation. If in a plane we take a row p and a pencil with centre Q, then through every point in p one line in the pencil will pass, and every ray in Q will cut p in one point, so that we are entitled to say a row contains as many points as a flat pencil lines, and, we may add, as an axial pencil planes, because an axial pencil is cut by a plane in a flat pencil.
The number of elements in the row, in the flat pencil, and in the axial pencil is, of course, infinite and indefinite too, but the same in all. This number may be denoted by ∞. Then a plane contains ∞2 points and as many lines. To see this, take a flat pencil in a plane. It contains ∞ lines, and each line contains ∞ points, whilst each point in the plane lies on one of these lines. Similarly, in a plane each line cuts a fixed line in a point. But this line is cut at each point by ∞ lines and contains ∞ points; hence there are ∞2 lines in a plane.
A pencil in space contains as many lines as a plane contains points and as many planes as a plane contains lines, for any plane cuts the pencil in a field of points and lines. Hence a pencil contains ∞2 lines and ∞2 planes. The field and the pencil are of two dimensions.
To count the number of points in space we observe that each point lies on some line in a pencil. But the pencil contains ∞2 lines, and each line ∞ points; hence space contains ∞3 points. Each plane cuts any fixed plane in a line. But a plane contains ∞2 lines, and through each pass ∞ planes; therefore space contains ∞3 planes.
Hence space contains as many planes as points, but it contains an infinite number of times more lines than points or planes. To count them, notice that every line cuts a fixed plane in one point. But ∞2 lines pass through each point, and there are ∞2 points in the plane. Hence there are ∞4 lines in space. The space of points and planes is of three dimensions, but the space of lines is of four dimensions.
A field of points or lines contains an infinite number of rows and flat pencils; a pencil contains an infinite number of flat pencils and of axial pencils; space contains a triple infinite number of pencils and of fields, ∞4 rows and axial pencils and ∞5 flat pencils—or, in other words, each point is a centre of ∞2 flat pencils.
§ 7. The above enumeration allows a classification of figures. Figures in a row consist of groups of points only, and figures in the flat or axial pencil consist of groups of lines or planes. In the plane we may draw polygons; and in the pencil or in the point, solid angles, and so on.
We may also distinguish the different measurements. We have—
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Segments of a Line
§ 8. Any two points A and B in space determine on the line through them a finite part, which may be considered as being described by a point moving from A to B. This we shall denote by AB, and distinguish it from BA, which is supposed as being described by a point moving from B to A, and hence in a direction or in a “sense” opposite to AB. Such a finite line, which has a definite sense, we shall call a “segment,” so that AB and BA denote different segments, which are said to be equal in length but of opposite sense. The one sense is often called positive and the other negative.
