PRINCIPLE OF DUALITY
§ 41. It has been stated in § I that not only points, but also planes and lines, are taken as elements out of which figures are built up. We shall now see that the construction of one figure which possesses certain properties gives rise in many cases to the construction of another figure, by replacing, according to definite rules, elements of one kind by those of another. The new figure thus obtained will then possess properties which may be stated as soon as those of the original figure are known.
We obtain thus a principle, known as the principle of duality or of reciprocity, which enables us to construct to any figure not containing any measurement in its construction a reciprocal figure, as it is called, and to deduce from any theorem a reciprocal theorem, for which no further proof is needed.
It is convenient to print reciprocal propositions on opposite sides of a page broken into two columns, and this plan will occasionally be adopted.
We begin by repeating in this form a few of our former statements :
Two points determine a line. Two planes determine a line.
Three points which are not in a Three planes which do not pass
line determine a plane. through a line determine a point.
A line and a point without it A line and a plane not through
determine a plane. it determine a point.
Two lines in a plane determine Two lines through a point
a point. determine a plane.
These propositions show that it will be possible, when any figure is given, to construct a second figure by taking planes instead of points, and points instead of planes, but lines where we had lines.
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,
„ a flat pencil a flat pencil,
„ a field of points and lines „ a pencil of planes and lines,
„ 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 Theorem.—If a, /3, y, b are any four points in space, and if four planes in space, and if the the lines AB and CD meet, then lines a0' and yS meet, then all all four points lie in a plane, four planes lie in a point (pencil), hence also AC and BD, as well hence also ay and 133, as well as
as AD and BC, meet. aS and i3y, 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 lie in a plane, then all lie in a
plane. 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 A figure consisting of n lines in a plane will be called an in a plane will be called an nside. npoint.
A figure consisting of n planes A figure consisting of n lines in a pencil will be called an in a pencil will be called an
nHat. nedge.
It will be understood that an nside 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 nedge or an nflat. We notice particularly
A fourpoint has six sides, of A fourside has six vertices, of which two and two are opposite, which two and two are opposite, and three diagonal points, which and three diagonals, which join are intersections of opposite opposite vertices.
sides.
A fourflat has six edges, of A fouredge has six faces, of which two and two are opposite, which two and two are opposite, and three diagonal planes, which and three diagonal edges, which
pass through opposite edges. are intersections of opposite faces.
A fourside is usually called a complete quadrilateral, and a fourpoint a complete quadrangle. The above notation, however, seems better adapted for the statement of reciprocal propositions.
§ l4•
If a point moves in a plane it If a line moves in a plane it
describes a plane curve. envelopes a plane curve (fig. 15).
If a plane moves in a pencil it If a line moves in a pencil it
envelopes a cone. describes a cone.
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 pro
perty of the other will exist, as long FIG. 15. 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.
End of Article: PRINCIPLE OF 

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