and E, F the intersections of tangents at opposite vertices. 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, H in a line, and the quadrilateral ABDC a line containing the four points R, P, I, K. These three lines form a triangle PQR.
The relation between the points and lines in this figure may be expressed more clearly if we consider ABCD as a four-point inscribed in a conic, and the tangents at these points as a four-side 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 circumscribing four-side. Hence the theorem—
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 circumscribed four-side stand in the above relation, then a curve of the second order may be described which passes through the four points and touches there the four sides of these figures.
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| Fig. 21. |
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, determine 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. 21.
The theorem reciprocal to the last is—
Any four-side circumscribed about a curve of second class and the four-point formed by the points of contact stand in this relation that the diagonals of the four-side pass through the diagonal points of the four-point. And conversely,
If a four-side and an inscribed four-point stand in the above relation, then a curve of the second class may be described which touches the sides of the four-side at the points of the four-point.
§ 56. The four-point and the four-side in the two reciprocal theorems are alike. Hence if we have a four-point ABCD and a four-side abcd related in the manner described, then not only may a curve of the second order be drawn, but also a curve of the second class, which both touch the lines a, b, c, d at the points A, B, C, D.
The curve of second order is already more than determined by the points A, B, C and the tangents a, b, c at A, B and C. The point D may therefore be any point on this curve, and d any tangent to the curve. On the other hand the curve of the second class is more than determined by the three tangents a, b, c and their points of contact A, B, C, so that d is any tangent to this curve. It follows that every tangent to the curve of second order is a tangent of a curve of the second class having the same point of contact. In other words, the curve of second order is a curve of second class, and vice versa. Hence the important theorems—
| Every curve of second order is a curve of second class. | Every curve of second class is a curve of second order. |
The curves of second order and of second class, having thus been proved to be identical, shall henceforth be called by the common name of Conics.
For these curves hold, therefore, all properties which have been proved for curves of second order or of second class. We may therefore now state Pascal’s and Brianchon’s theorem thus—
Pascal’s Theorem.—If a hexagon be inscribed in a conic, then the intersections of opposite sides lie in a line.
Brianchon’s Theorem.—If a hexagon be circumscribed about a conic, then the diagonals forming opposite centres meet in a point.
§ 57. If we suppose in fig. 21 that the point D together with the tangent d moves along the curve, whilst A, B, C and their tangents a, b, c remain fixed, then the ray DA will describe a pencil about A, the point Q a projective row on the fixed line BC, the point F the row b, and the ray EF a pencil about E. But EF passes always through Q. Hence the pencil described by AD is projective to the pencil described by EF, and therefore to the row described by F on b. At the same time the line BD describes a pencil about B projective to that described by AD (§ 53). Therefore the pencil BD and the row F on b are projective. Hence—
If 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 any four points on the curve is equal to the cross-ratio of the points in which 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. We mention only a few of the more important ones.
The locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone of the second order.
The envelope of planes which join corresponding lines in two projective flat pencils, not in the same plane, is a cone of the second class.
Cones of second order and cones of second class are identical.
Every plane cuts a cone of the second order in a conic.
A cone of second order is uniquely determined by five of its edges or by five of its tangent planes, or by four edges and the tangent plane at one of them, &c. &c.
Pascal’s Theorem.—If 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.
Brianchon’s Theorem.—If a solid angle of six edges be circumscribed 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.
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| Fig. 22. |
§ 59. Projective Definitions of the Conics.—We now consider the shape of the conics. We 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. If the line at infinity has no point on the curve the latter is altogether finite, and is called an Ellipse (fig. 21). If the line at infinity has only one point in common with the conic, the latter extends to infinity, and has the line at infinity a tangent. It is called a Parabola (fig. 22). 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 they meet. The curve is in this case called an Hyperbola (see fig. 20). The tangents at the two points at infinity are finite because the line at infinity is not a tangent. They are called Asymptotes. The branches of the hyperbola approach these lines indefinitely as a point on the curves moves to infinity.
§ 60. That the circle belongs to the curves of the second order is seen at once if we state in a slightly different form the theorem that in a circle all angles at the circumference standing upon the same arc are equal. If two points S1, S2 on a circle be joined to any other two points A and B on the circle, then the angle included by the rays S1A and S1B is equal to that between the rays S2A and S2B, so that as A moves along the circumference the rays S1A and S2A describe equal and therefore projective pencils. The circle can thus be generated by two projective pencils, and is a curve of the second order.
