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).