§ 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—
| In the row, length of segment; In the flat pencil, angles; In the axial pencil, dihedral angles between two planes; In the plane, areas; In the pencil, solid angles; In the space of points or planes, volumes. |
Segments of a Line