S1S respectively. lines which pass through a u and
alit respectively.
As A is any given point on the curve, and ul any line through it, we have solved the problems:
Problem.—To find the second Problem.—To find the second point in which any line through a tangent which can be drawn known point on the curve cuts from any point in a given tangent
the curve. to the curve.
If we determine in SI (fig. 16) the ray corresponding to the ray S2SI in S2, we get the tangent at SI. Similarly, we can determine the point of contact of the tangents ul or u2 in fig. 17.
§ 51. If five points are given, of which not three are in a line, then we can, as has just been shown, always draw a curve of the
second order through them; we select two of the points as centres of projective pencils, and then one such curve is determined. It will be presently shown that we get always the same curve if two other points are taken as centres of pencils, that therefore five points determine one curve of the second order, and reciprocally, that five tangents determine one curve of the second class. Six points taken at random will therefore not lie on a curve of the second order. In order that this may be the case a certain condition has to be satisfied, and this condition is easily ob
tained from the construction in § 49, fig. 16. If we consider the conic determined by the five points A, SI, S2, K, L, then the point D will be on the curve if, s, and only if, the points on DI, S, D2 be in a line.
This may be stated differently if we take AKSIDS2L (figs. 16 and 18) as a hexagon inscribed in the conic, then AK and DS2 will be opposite sides, so will be KS' and S2L, as well as SID and LA. The first two meet in D2,
the others in S and DI respectively. We may therefore state the required condition, together with the reciprocal one, as follows:—
Theorem.—The envelope of second class which is generated by two projective rows contains the bases of these rows as enveloping lines or tangents.
Proof.—If s and s' are the two rows, then to the point ss' or P' as a point in s' corresponds in s a point P, which is not coincident with P', for the rows are not perspective. But P and P' are joined by s, so that s is one of the enveloping lines, and similarly s'.
us
PROJECTIVE]
Pascal's Theorem.—If a hexagon Brianchon's Theorem.—If a be inscribed in a curve of the hexagon be circumscribed about second order, then the intersec a curve of the second class, then tions of opposite sides are three the lines joining opposite vertices
points La a line. are three lines meeting in a point.
These celebrated theorems, which are known by the names of their discoverers, are perhaps the most fruitful in the whole theory of conics. Before we go over to their applications we have to show that we obtain the same curve if we take, instead of S1, S2, any two other points on the curve as centres of projective pencils.
§52. We know that the curve depends only upon the correspondence between the pencils S1 and S2, and not upon the special construction used for finding new points on the curve. The point A (fig. 16 or 18), through which the two auxiliary rows u1, u2 were drawn, may therefore be changed to any other point on the curve. Let us now suppose the curve drawn, and keep the points S1, S2, K, L and D, and hence also the point S fixed, whilst we move A along the curve. Then the line AL will describe a pencil about L as centre, and the point D1 a row on S1D perspective to the pencil L. At the same time AK describes a pencil about K and D2 a row perspective to it on S2D. But by Pascal's theorem D1 and D2 will always lie in a line with S, so that the rows described by D1 and D2 are perspective. It follows that the pencils K and L will themselves be projective, corresponding rays meeting on the curve. This proves that we get the same curve whatever pair of the five given points we take as centres of projective pencils. Hence
Only one curve of the second Only one curve of the second
order can be drawn which passes class can be drawn which touches
through five given points. five given lines.
We have seen that if on a curve of the second order two points coincide at A, the line joining them becomes the tangent at A. If, therefore, a point on the curve and its tangent are given, this will be equivalent to having given two points on the curve. Similarly, if on the curve of second class a tangent and its point of contact are given, this will be equivalent to two given tangents.
We may therefore extend the last theorem:
Only one curve of the second Only one curve of the second order can be drawn, of which class can be drawn, of which four four points and the tangent at one tangents and the point of contact of them, or three points and the at one of them, or three tangents tangents at two of them, are and the points of contact at two
given. of them, are given.
§ 53• At the same time it has been proved :
If all points on a curve of the All tangents to a curve of second second order be joined to any class are cut by any two of two of them, then the two pencils them in projective rows, those thus formed are projective, those being corresponding points which rays being corresponding which lie on the same tangent. Hence—meet on the curve. Hence
The crossratio of four rays The crossratio of the four joining a point S on a curve of points in which any tangent u is second order to four fixed points cut by four fixed tangents a, b, c, d A, B, C, D in the curve is in is independent of the position of dependent of the position of S, u, and is called the crossratio of and is called the crossratio of the the four tangents a, b, c, d.
four points A, B, C, D.
If this crossratio equals—1 If this crossratio equals—1
the four points are said to be the four tangents are said to be
four harmonic points. four harmonic tangents.
We have seen that a curve of second order, as generated by projective pencils, has at the centre of each pencil one tangent; and further, that any point on the curve may be taken as centre of such pencil. Hence
A curve of second order has at A curve of second class has on
every point one tangent. every tangent a point of contact.
§ 54. We return to Pascal's and Brianchon's theorems and their applications, and shall, as before, state the results both for curves of the second order and curves of the second class, but prove them only for the former.
Pascal's theorem may be used when five points are given to find more points on the curve, viz. it enables us to find the point where any line through one of the given points cuts the curve again. It is convenient, in making use of Pascal's theorem, to number the points, to indicate the order in which they are to be taken in forming a hexagon, which, by the way, may be done in 6o different ways. It will be seen that 12 (leaving out 3) 4 5 are opposite sides, so are 2 3 and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1.
If the points 12 3 4 5 are given, and we want a 6th point on a line drawn through 1, we know all the sides of the hexagon with the exception of 5 6, and this is found by Pascal's theorem.
If this line should happen to pass through r, then 6 and 1 coincide, or the line 6 is the tangent at i. And always if two consecutive vertices of the hexagon approach nearer and nearer, then the side joining them will ultimately become a tangent.
We may therefore consider a pentagon inscribed in a curve of second order and the tangent at one of its vertices as a hexagon, and thus get the theorem:
697
Every pentagon circumscribed about a curve of the second class has the property that the lines which join two pairs of nonconsecutive vertices meet on that line which joins the fifth vertex to the point of contact of the opposite side.
This enables us also to solve the following problems.
Given five points on a curve of Given five tangents to a curve
second order to construct the of second class to construct the
tangent at any one of them. point of contact of any one of
them.
If two pairs of adjacent vertices coincide, the hexagon becomes a quadrilateral, with tangents at two vertices. These we take to be opposite, and get the following theorems:
If a quadrilateral be circumscribed about a curve of second class, the lines joining opposite vertices, and also the lines joining points of contact of opposite sides, meet in a point.
If we consider the hexagon made up of a triangle and the tangents at its vertices, we get
If a triangle is inscribed in a If a triangle be circumscribed curve of the second order, the about a curve of second class, points in which the sides are cut the lines which join the vertices by the tangents at the opposite to the points of contact of the
vertices meet in a point. opposite sides meet in a point
(fig. 20).
§ 55. Of these theorems, those about the quadrilateral give rise to a number of others. Four points A, B, C, D may in three different ways be formed into a quadrilateral, for we may take them in the order ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be taken as the vertex opposite to A. Accordingly we may apply the theorem in three different ways.
Let A, B, C, D be four points on a curve of second order (fig. 21), and let us take them as forming a quadrilateral by taking the points in the order ABCD, so that A, C and also B, D are pairs of opposite vertices. Then P, Q will be the points where opposite sides meet,
Every pentagon inscribed in a curve of second order has the property that the intersections of two pairs of nonconsecutive sides lie in a line with the point where the fifth side cuts the tangent at the opposite vertex.
If a quadrilateral be inscribed in a curve of second order, the intersections of opposite sides, and also the intersections of the tangents at opposite vertices, lie in a line (fig. 19).
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 he expressed more clearly if we consider ABCD as a fourpoint inscribed in a conic, and the tangents at these points as a fourside circumscribed about it,—viz. it will be seen that P, Q, R are the diagonal points of the fourpoint ABCD, whilst the sides of the triangle PQR are the diagonals of the circumscribing fourside. Hence the theorem
Any fourpoint on a curve of the second order and the fourside formed by the tangents at these points stand in this relation that the diagonal points of the fourpoint lie in the diagonals of the fourside. And conversely,
If a fourpoint and a circumscribed fourside 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.
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. 21.
The theorem reciprocal to the last is
Any fourside circumscribed about a curve of second class and the fourpoint formed by the points of contact stand in this relation that the diagonals of the fourside pass through the diagonal points of the fourpoint. And conversely,
If a fourside and an inscribed fourpoint stand in the above relation, then a curve of the second class may be described which touches the sides of the fourside at the points of the fourpoint.
§ 56. The fourpoint and the fourside in the two reciprocal theorems are alike. Hence if we have a fourpoint ABCD and a fourside abed related in the manner described, then not only mdy 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 acurve 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 Every curve of second class is a
a curve of second class, 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 13 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 crossratio of the four rays which join A to any four points on the curve is equal to the crossratio 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.
§ 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.
§ 6o. That the circle belongs to the curves of the second order is seen
''N
P
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 Si, 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.
If we join a point in space to all points on a circle, we get a (circular) cone of the second order (§ 43). Every plane section of this cone is a conic. This conic will be an ellipse, a parabola, or an hyperbola, according as the line at infinity in the plane has no, one or two points in common with the conic in which the plane at infinity cuts the cone. It follows that our curves of second order may be obtained as sections of a circular cone, and that they are identical with the " Conic Sections " of the Greek mathematicians.
§ 6r. Any two tangents to a parabola are cut by all others in projective rows; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows therefore similar. Hence the theorem
The tangents to a parabola cut each other proportionally.
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