Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/412

Rh 398 GEOM § 55. Of these theorems, those about the quadrilateral give rise to a number of others. Four points A, B, C‘, 1) may in three ditfcrent ways be formed into a quadrilateral, for we may take them in the order ABC D, or ACBD, or ACDB, so that either of the points B, C, D may be tiken as the vertex opposite to A. Accordingly we may apply the theorem in three dilfcrent ways. Let A, B, C, 1) be four points on acurve of second order (ﬁg. 20), and let us take them as tormiug a quadrilateral by taking the Fig. 20. points in the order ABCD, so that A, C and also B, D are pairs of opposite vertiees. Then P, Q will be the points where opposite sides meet, and E, F the intersections of tangents at opposite vcrtiees; The four points P, Q, E, F lie therefore in a line. The quadrilateral ACBD gives us in the same way the four points Q, R, G, II in a line, and the quadrilateral ABDC a line containing the four points ll, 1’, I, K. These three lines form a triangle PQR. The relation between the points and lilies in this ﬁgure may be expressed more clearly if we consider ABCD as a four-point in- scribed in a conic, and the tangent at these points as a four-sidc circumscribed about it,—viz., it will be seen that P, Q, R are the diagonal points of the four—point ABCD, whilst the sides of the triangle PQR are the diagonals of the cireumscribing four-side. Hence the theorem—- TIIEOREZI.—-Any four-point on a curve of the second order and the four-side formed by the tangents at these points stand in this relation that the diagonal points of the four-point lie in the diagonals of the four-side. And conversely, If a four—point and a eercmnscribed four-side stand in the abore relation, then a curve of the second order may he described erhich passes through the four points and touches there the four sides of these ﬁgures. That the last part of the theorem is true follows from the fact that the four points A, B, C, D and the line a, as tangent at A, deter- mine a curve of the second order, and the tangents to this curve at the other points B, C, D are given by the construction which leads to ﬁg. 20. The theorem reciprocal to the last is- TnEor.r:..—Any four-side circumscribed about a ewree of second class and the fowr-point formed by the points of contact stand in this relation that the diagonals If the four-side pass through the diagonal points of the four-point. And conversely, E T It Y If a four-side and an inscribed four-point stand in the aborc relation, then a curve of the second class may be described u-hich touches the sides of the four-side at the points of the four-points. § 56. The four-point and the four—sidc in the two l'c('ip1‘0c:1l theorems are alike. llcnce if we have a four-point ABCl) and a four—side abcd related in the n1:mner described, then not only may a cur'e of the second order be drawn, but also a curve of the second class, which both to11ch the lines a, b, c, it at the points A, B, C, D. The curve of second order is already mo1'e than determined by the points A, B, C and the tangents (1, I», c at A, B, and C. The point 1) may therefore be any point on this curve, and (l any tangent to the curve. On the other hand the curve of the second class is more than determined by the three tangents (t, b, c and their points of contact A, B, C, so that d is any tangent to this curve. It t'«_.llovs that every tangent to the eurve of second order is a tangent of a curve of the second class having the same point of contact. In other words, the cu1've of second order is a curve of second class, and vice rcrsa. Hence the important theorems :— 'l‘nEORE.l.—Every curve of second 'l'lIEORE.I.—Et'P:‘_II curre of s'.:-nu-i’ order is a curve of second class. class is a <:lu1.'e of scculld order. The curves of second order and of second class having thus be--n proved to be identical shall henceforth be called by the. eonnnon name of Con ics. For these curves hold, therefore, all properties which have been proved for curves of second order or of second class. 'e may therefore now state l’aseal’s and Brianehon’s theorem thus— I’ascal’s T/Lcorem.——If a hexagon be inscribed in a conic, then the intersections of opposite sides lie iii a line. J}:-ianehon’s Thcurem.—If a hexagon be circumscribed about a conie, then the diagonals forming opposite centres meet in a point. § 57. If we suppose in ﬁg. :20 that the point 1) together with the tangent it moves along the curve, whilst A, B, C and their tangents a, b, e remain ﬁxed, then the ray l)A will describe a pencil about A, the point Q a projective row on the ﬁxed line BC, the point 1'‘ the row I), and the ray EF :1 pencil about But El’ passes always through Q. Hence the pencil described by AD is projective to the pencil described by El’, and therefore to the row described by F on b. ' At the same time the line BD describes a pencil about 1} pro- jective to that_deseribed by AD (§ 53). Therefore the pencil 13D and the row F on b are projective. Iicnce— TllEOP.E.I.—I/' on a conic a. point A be taken and the tangent a at this point then, the cross—ratio of the four rays which join A to an 3/ four points on the curve is equal to the cross—9'atio of the points in tcltich the tangents at these points cut the tangent at A. § 58. There are theorems about cones of second order and second class in a pencil which are reciprocal to the above, according to § 43. 'e mention only a few of the more important ones. _ Thco7'em.—Tl1e locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone "of the second order. Theorem.—The envelope of planes which join corresponding lines in two projective ﬂat pencils, not in the same plane, is a. co11e of the secoud class. Tlworcm.—Cones of second order and cones of second class are identical. Theorem.—Evcry plane cuts a cone of the second order in a conic. T111-:0RE.I.—A cone ef second order is Iuziguely rletcrminal I23/ﬁre‘ of its edges or by ﬁre of its tangent planes, or by four edges and the tangent plane at one of them, it-e., it-e. Theorem. (I’aseal’s).——lf a solid angle of six faces be inscribed in a cone of the second order, then the intersections of opposite faces are three lines in a plane. Theorem (Brianehon’s).—If a solid angle of six edges be circ1nn- scribed about a cone of the second order, then the planes through opposite edges meet in a line. Each of the other theorems about conics may be stated for cones of the second order. § 59. We have not yet considered the shape of the conies. 'c know that any line in the plane of the conic, and hence that the line at inﬁnity, either has no point in common with the curve, or one (counting for two coincident points), or two distinct points. lf the line at inﬁnity l1as no point on the curve the latter is altogctlu:r ﬁnite, and is called an Ellipse (ﬁg. 20). If the line at inﬁnity has only one point in common with the conic, the lttlltf extends to inﬁnity, and has the line at inﬁnity a_tangcnt. It is called a I’ara- bola (ﬁg. 21). If, lastly, the line at inﬁnity cuts the curve in two points, it consists of two separate parts which each extend in two branches to the points at inﬁnity where theymcct. The curve is in this case called an Ilyperbola (see ﬁg. 19, 24, or 2.3). The tangents at the two points at inﬁnity are ﬁnite because the line at inﬁnity is not a tangent. They are called /l.s‘_1/mploics. The branches of the hyperbola approach these lines indeﬁnitely as a point on the curves moves to inﬁnity. § 60. That the eircle belongs to the curves of the second order is seen at once if we state in a slightly different form the theorem [r1:oJEe'r1'r:.