GEOM ccntrcs S and S’ of the two pencils. This circle cuts 12 in two points H and K. The two pairs of rays, I2, is and li’,k', joining these points to S and S’ will be pairs of corresponding rays at right angles. The construction gives in gen-'ral but one circle, but if the line 32 is the pcrpcndiciilar biscetor of 55', there exists an inliiiite ii1imbc.r, aml lo rz'r:'_I/ rig//it angle in the DU.tLITY.] E T R Y 395 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 deter- mine 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 n.I' prncil CII)'I'I'S]707I(l.s' Cl ri'_://i/. miglc [ii (In: other. l‘i:i.'ciri.i-; U1-‘ DL.’.Ll'l‘Y. § 41. It has been stated in 1 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 Fig. 13. one figure which possesses certain properties gives rise in many cases to the construction of another figure, by replacing, according to definite rules, cl»-inciits of one kind by those of another. The new figure thus obtained will then possess properties which may l)e stated as soon as those of the original figure are known. We obtain thus a principle, known as the 11:-iziciple of (Zualz'l_i/ or of 1'ccz'procz'(_i/, which enables us to eoiisti'iict to any figure not containing any measurement in its construction a reciprocal figiii'c, as it is called, and to deduce from any theorem a 7'cci'procal 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 ailoptctl. We begin by repeating in this form a few of our former state- inciits :— Two points dctcrniine a line. Three points which are not in a line «lcterniine a plane. A line and a point without it de- termine a plane. Two lines in a plane detci-n'.ine a point. Two planes determine a line. Three planes which do not pass through a line determine a point. Aline and a plane not through it determine a point. Two lines through a point deter- mine a plane. Tlicse 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 trilie-_li'al angle. A triangle and a trilicdral angle are therefore reciprocal figures. .'iniilarly, 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 tetraliedron. § 42. As other examples we have the following :— 'l'o a row is rcciprocal an axial pencil, ., a Hat pencil ,, 3. flat, pencil, ., a held of points and lines ,, a pencil of planes and lines, ,. the space of points ,, the space 01‘ 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 pcneil ; and so for the other cases. This correspondence of reciprocity breaks down, however, if we take figures which contain incasurcincnt in their construction. For instance, there is no figure reciprocal to two planes at r1'_r/Izl miglcs, because there is no segment in a row which has aiiiagnitude as definite as a right angle. lVe add a few examples of reciprocal propositions which are easily prove.d. Tlic-r;rcm.—If A, B, f‘, Dareany four points in space, and if the lines All and ('D_ineet, then all four points lie in a plane. hence also AC and BD, as well as AD and UL‘, meet. Them-cni.—If a, [3, -y, 6 are four planes in space, and if the lines (1/3 and 76 meet, then all four planes lie in a point (pencil), hence also afi and 3/6, as well as a6 and (3')/, ineet. TllEOI‘.lZ.‘[.—If of any mmzbcr of lines cz-cry one meets crcry otlicr, 7!:/ulsl all do not lie in a point, then all lie in a Z lie in (1 plane, than all lie in a 1: (me. point (pencil). _§ 43: Reciprocal figures as explained lie both in space of three dimensions. It the one is confined to a plane (is formed of cle- iiiciits which lie in a plane), then the reciprocal figure is confined to a pencil (is formed of elements which pass through a point). mention a few of these. A flgiire consist-ing of n points in a plane. will be called an 71-point. A llgurc consisting of n planes in a pencil will be called an n~llat. At first we explain a few nanics :— A ll‘.-‘nre consisting of 12 lines in ii. plane will be called an n-side. A figure consisting of 1: lines in a pencil will be called an n-edge. It will be understood that an 71-side is different from a polygon of IL sides. The latter has sides of finite length and 71. verticcs, the former li:is sides all of iiilinite extension, and every point where two of the sides incct will be a vertex: between a solid angle and an '11-edge or an 9!-flat. ticularly—- A four-point has six sides, of which two and two are opposite, and three diagonal points, which are intersec- tions: of opposite sides. A four-ll-at has six edges, of which two and two are opposite, and three diagonal planes, which pass through opposite edges. A foiir—side is usually called a complete quadrilateral. A similar difl'crence exists We notice par- A foiir-side has six verticcs, of which two and two are opposite, aml three diagonals, which join opposite vertices. A four-edge has six faces, of which two and two are opposite. and three diagonal edges, which are iiitersec- tions of opposite faces. The above notation, however, seems better adapted for the statement of reciprocal propositions. If a point moves in a plane it de- scribes a plane curve. _ If a plane moves in a pencil it en- velopes a cone. If a line moves in a plane it cii- vclopes a plane curve (fig. 14). If a line moves in a pencil it de- scribes 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 a 1 wears either 1 1 fl‘ ’ ' tl 1 as tie ociis o mes passing irougi a fixed point, the “vertex” of the cone or as the enveloie of )lanes ’ passing through the same point. To a surface as locus of points cor- responds, in the same manner, a sur- face as envelope of planes; and to a curve in space as locus of points corre- sponds a developable surface as enve- Of the latter we shall not say any more at present. lope of planes. It will be seen from the above that we may, by aid of the prin- ciplc 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 nicasurenient. For such propositions it will tliercforc be unnecessary to prove more than one of two reciprocal theorems. Cl'll'ES AND Coxizs or SECOND ORDER or. Sncoxn CLASS. § 45. If we have two projective pencils in_ a. plane, correspondiiig rays will mcet, and their point of intersection will constitute some locus which we have to investigate. Rcciprocally, if two pro- jective rows in a plane are given, then the lines which join corre- sponding points will envelope some curve. We prove first :— Thcorcm.—If two projective flat pencils lie in a plane, but are neither perspective nor concentric, then the locus of intersections of correspoiid- ing rays is a curve of the second order, that is, no line contains more than two points of the locus. Proof.—'e draw any line t. This cuts each of the pencils in a row, so that we have on ttwo 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). Tlicorem.—If two projective rows lie in a plane, but are neither per- spective nor on a common base, then the envelope of lines joining corre- spoii(ling points is a curve of the second class. that is, through no point pass more than two of the enveloping lines. 1’roof_—Ve 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 join- ing correspond'Zng points in the two rows passes through '1‘, it will be a line in the one pencil which coincides with its corresponding line in the other. But two projective coii- ceiitric flat pencils in the same plane cannot have more than two lnics of one coincident with their correspond- ing liiie in the other (§ 34). It will be seen that the proofs are reciprocal, so that the one m_ay be copied from the other by simply interchanging the words point and line, locus and ciivelnpe, row and pencil, and so on. l e shall
therefore in future prove seldom more than one of two reciprocal