For instance, if in the first figure we take a plane and three points in it, we have to take in the second figure a point and three planes through it. The three points in the first, together with the three lines joining them two and two, form a triangle; the three planes in the second and their three lines of intersection form a trihedral angle. A triangle and a trihedral angle are therefore reciprocal figures.
Similarly, to any figure in a plane consisting of points and lines will correspond a figure consisting of planes and lines passing through a point S, and hence belonging to the pencil which has S as centre.
The figure reciprocal to four points in space which do not lie in a plane will consist of four planes which do not meet in a point. In this case each figure forms a tetrahedron.
§ 42. As other examples we have the following:—
| To a row | is reciprocal | an axial pencil, |
| to a flat pencil | ” | a flat pencil, |
| to a field of points and lines | ” | a pencil of planes and lines, |
| to the space of points | ” | the space of planes. |
For the row consists of a line and all the points in it, reciprocal to it therefore will be a line with all planes through it, that is, an axial pencil; and so for the other cases.
This correspondence of reciprocity breaks down, however, if we take figures which contain measurement in their construction. For instance, there is no figure reciprocal to two planes at right angles, because there is no segment in a row which has a magnitude as definite as a right angle.
We add a few examples of reciprocal propositions which are easily proved.
| Theorem.—If A, B, C, D are any four points in space, and if the lines AB and CD meet, then all four points lie in a plane, hence also AC and BD, as well as AD and BC, meet. | Theorem.—If α, β, γ, δ are four planes in space, and if the lines αβ and γδ meet, then all four planes lie in a point (pencil), hence also αγ and βδ, as well as αδ and βγ, meet. |
Theorem.—If of any number of lines every one meets every other, whilst all do not
| lie in a point, then all lie in a plane. | lie in a plane, then all lie in a point (pencil). |
§ 43. Reciprocal figures as explained lie both in space of three dimensions. If the one is confined to a plane (is formed of elements which lie in a plane), then the reciprocal figure is confined to a pencil (is formed of elements which pass through a point).
But there is also a more special principle of duality, according to which figures are reciprocal which lie both in a plane or both in a pencil. In the plane we take points and lines as reciprocal elements, for they have this fundamental property in common, that two elements of one kind determine one of the other. In the pencil, on the other hand, lines and planes have to be taken as reciprocal, and here it holds again that two lines or planes determine one plane or line.
Thus, to one plane figure we can construct one reciprocal figure in the plane, and to each one reciprocal figure in a pencil. We mention a few of these. At first we explain a few names:—
| A figure consisting of n points in a plane will be called an n-point. | A figure consisting of n lines in a plane will be called an n-side. |
| A figure consisting of n planes in a pencil will be called an n-flat. | A figure consisting of n lines in a pencil will be called an n-edge. |
It will be understood that an n-side is different from a polygon of n sides. The latter has sides of finite length and n vertices, the former has sides all of infinite extension, and every point where two of the sides meet will be a vertex. A similar difference exists between a solid angle and an n-edge or an n-flat. We notice particularly—
| A four-point has six sides, of which two and two are opposite, and three diagonal points, which are intersections of opposite sides. | A four-side has six vertices, of which two and two are opposite, and three diagonals, which join opposite vertices. |
| A four-flat has six edges, of which two and two are opposite, and three diagonal planes, which pass through opposite edges. | A four-edge has six faces, of which two and two are opposite, and three diagonal edges, which are intersections of opposite faces. |
A four-side is usually called a complete quadrilateral, and a four-point a complete quadrangle. The above notation, however, seems better adapted for the statement of reciprocal propositions.
§ 44.
| If a point moves in a plane it describes a plane curve. | If a line moves in a plane it envelopes a plane curve (fig. 15). |
| If a plane moves in a pencil it envelopes a cone. | If a line moves in a pencil it describes a cone. |
 |
| Fig. 15. |
A curve thus appears as generated either by points, and then we call it a “locus,” or by lines, and then we call it an “envelope.” In the same manner a cone, which means here a surface, appears either as the locus of lines passing through a fixed point, the “vertex” of the cone, or as the envelope of planes passing through the same point.
To a surface as locus of points corresponds, in the same manner, a surface as envelope of planes; and to a curve in space as locus of points corresponds a developable surface as envelope of planes.
It will be seen from the above that we may, by aid of the principle of duality, construct for every figure a reciprocal figure, and that to any property of the one a reciprocal property of the other will exist, as long as we consider only properties which depend upon nothing but the positions and intersections of the different elements and not upon measurement.
For such propositions it will therefore be unnecessary to prove more than one of two reciprocal theorems.
Generation of Curves and Cones of Second Order or Second Class
§ 45. 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:—
| Theorem.—If two projective flat pencils lie in a plane, but are neither in perspective nor concentric, then the locus of intersections of corresponding rays is a curve of the second order, that is, no line contains more than two points of the locus. | Theorem.—If two projective rows lie in a plane, but are neither in perspective nor on a common base, then the envelope of lines joining corresponding points is a curve of the second class, that is, through no point pass more than two of the enveloping lines. |
| Proof.—We draw any line t. This cuts each of the pencils in a row, so that we have on t two rows, and these are projective because the pencils are projective. If corresponding rays of the two pencils meet on the line t, their intersection will be a point in the one row which coincides with its corresponding point in the other. But two projective rows on the same base cannot have more than two points of one coincident with their corresponding points in the other (§ 34). | Proof.—We take any point T and join it to all points in each row. This gives two concentric pencils, which are projective because the rows are projective. If a line joining corresponding points in the two rows passes through T, it will be a line in the one pencil which coincides with its corresponding line in the other. But two projective concentric flat pencils in the same plane cannot have more than two lines of one coincident with their corresponding line in the other (§ 34). |
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. We shall therefore in future prove seldom more than one of two reciprocal theorems, and often state one theorem only, the reader being recommended to go through the reciprocal proof by himself, and to supply the reciprocal theorems when not given.
§ 46. We state the theorems in the pencil reciprocal to the last, without proving them:—
| Theorem.—If two projective flat pencils are concentric, but are neither perspective nor coplanar, then the envelope of the planes joining corresponding rays is a cone of the second class; that is, no line through the common centre contains more than two of the enveloping planes. | Theorem.—If two projective axial pencils lie in the same pencil (their axes meet in a point), but are neither perspective nor co-axial, then the locus of lines joining corresponding planes is a cone of the second order; that is, no plane in the pencil contains more than two of these lines. |
§ 47. Of theorems about cones of second order and cones of second class we shall state only very few. We point out, however, the following connexion between the curves and cones under consideration:
| The lines which join any point in space to the points on a curve of the second order form a cone of the second order. | Every plane section of a cone of the second order is a curve of the second order. |
| The planes which join any point in space to the lines enveloping a curve of the second class envelope themselves a cone of the second class. | Every plane section of a cone of the second class is a curve of the second class. |
By its aid, or by the principle of duality, it will be easy to obtain theorems about them from the theorems about the curves.
We prove the first. A curve of the second order is generated by two projective pencils. These pencils, when joined to the point in space, give rise to two projective axial pencils, which generate the cone in question as the locus of the lines where corresponding planes meet.
