By Cremona L.

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A BC df, five straight lines (Fig. 7). two complete If quadrilaterals abed, a b c d are such that five pairs of vertices ab and a b be and 6V, ca and i 8. triangles are b c a ) Or polygram. (tri- by THE PEINCIPLE OF DUALITY. 37] 17, AA CC one point S. So too the triangles ABD, AB D fore DD f in are in perspective also will pass triangles f follows It . BCD, B C D 18), ; BB , there through common point BB and meet will the S, to AA that the are also in therefore CD and meet in a point on the straight line s, which is deter mined by the point of intersec tion of BC and B G and by that perspective : CD D of !

STAUDT, correlative propositions Geom. der Laye, Art. 66. THE PRINCIPLE OF DUALITY. 34] 27 Geometry of space. Two correlative propositions are deduced one from the other by interchanging the elements point &nd plane. in the B 1 Two points A determine a straight line (viz. the straight line which passes through the . , AB given points) which contains an infinite number of other points. A 2. B (not lying on the line) deter planes a , /3 determine a straight line (viz. the straight line the intersection of the given a/3, planes), infinite straight line a and a point mine a plane, viz.

The data of the problem may also be the centre 0, the axis s, and the x t vanishing line j of the first figure (Fig. 7). In this case, if a straight line a of the first figure cuts j -7 / in J and P, the point be collinear with in s J corresponding to J will J and and at an infinite And as the straight line a distance from 0. corresponding to a must pass both through J and through P, it is the parallel drawn through P to OJ. To find the point A corresponding to a given point A, we must draw the straight which corresponds to a straight line a drawn arbitrarily ^x^ line a A f OA is the required point A a the constructions just given, letof (4) Assuming knowledge be the the of s centre, axis, again homology, and j the vanishing line of the first figure.