Ex. 3. In a hexagon ABCDEF, are known all the sides except AF, and all the angles except B and E; to find the rest. Suppose the diagonal BE drawn, dividing the figure into two trapeziums. Then, in the trapezium BCDE, the sides except BE, and the angles except B and E, will be known; and these may be determined as in example 1. Again, in the trapezium ABEF, there will be known the sides except AF, and the angles except the adjacent ones B and E. Hence, first for BCDE: (cor. 3 theor. 2). CD sin. C + DE sin. (C + D) tan. CBE = BC + CD cos. C + DE cos (C + D) CD sin. 72° + DE sin. 147° BC + CD cos. 72°+ DE cos. 1470 BC+ CD cos. 72° DE cos. 33° Whence CBE = 79°21"; and therefore DEB = 67°57′59". = BC cos. 79° 2 1 + CD cos. 7° 2 1" Secondly, in the trapezium ABEF, = CD sin. 72° + DE sin. 330 2548 581. = EF sin. F: whence Taking the lower of these, to avoid re-entering angles, we have B (exterior ang.) = 95° 4 6" ; ABE = 84° 55 54"; FEB = 63° 4 6" : therefore ABC 163° 57′ 55′′; and FED 131° 2 5" : and consequently the exterior angles at B and E are 16° 2′ 5′′ and 48° 57′ 55′′ respectively. Lastly, AF = AB cos. A - BE cos. (A + B) - EF cos. F = cos. 64° + BE cos. 20° 55′ 54′′ — EF cos. 84° — 1645°292. [Note. The preceding three examples comprehend all the varieties which can occur in Polygonometry, when all the sides except one, and all the angles but two, are known. The unknown angles may be about the unknown side; or they may be adjacent to each other, though distant from the unknown side; and they may be remote from each other, as well as from the unknown side.] Ex. 4. In a hexagon ABCDEF, are known all the angles, and all the sides except AF and CD: to find those sides. Here, reasoning from the principle of cor. 2, theor. 2, we have Ex. 5. In the nonagon ABCDEFGHI, all the sides are known, and all the angles except A, D, G: it is required to find those angles. Given AB = 2400 FG 3800 Ext. Ang. B = 40° Suppose diagonals drawn to join the unknown angles, and dividing the polygon into three trapeziums and a triangle; as in the marginal figure. Then, 1st. In the trapezium ABCD, where AD and the angles about it are unknown; we have (cor. 3, theor. 2). tan.BAD= = BC sin. 40°+CD sin. 72° BC sin. B+CD sin. (B+C) AB+BC cos. B+CD cos. (B+C) AB+BC cos.40°+CD cos. 72°* Whence BAD = 39° 30′ 42", CDA 32° 29′ 18′′. 2dly. In the quadrilateral DEFG, where DG and the angles about it are unknown; we have = EF sin. E+FG sin. (E+F) EF sin. 36°+FG sin 81° tan.EDG=. DE+EF cos. E+FG cos. (E+F) DE+EF cos. 36°+FG cos. 81°* Whence EDG = 41° 14′ 53′′, FGD = 39° 45' 7". 3dly. In the trapezium GHIA, an exactly similar process gives HGA = 50° 46′ 53′′, IAG = 47° 13′ 7′′, and AG = 9780.591. 4thly. In the triangle ADG, the three sides are now known, to find the angles: viz. DAG = 60° 53′ 26′′, AGD = 43° 15′ 54′′, ADG = 75° 50′ 40′′. Hence there results, lastly, IAB = 47° 13′ 7′′ +60° 53′ 26" + 39° 30′ 42′′ = 147° 37′ 15′′, Consequently, the required exterior angles are A = 32° 22′ 45′′, D= 30° 25′ 9′′, Ex. 6. Required the area of the hexagon in ex. 1. Ans. 16530191. Ex. 7. In a quadrilateral ABCD, are given AB = 24, BC = 30, CD = 34 ; angle ABC = 92° 18', BCD = 97° 23'. Required the side AD, and the area. Ex. 8. In prob. 1, suppose PQ =2538 links, and the angles as below; what is the area of the field ABCDQP? APQ = 89° 14', BPQ = 68° 11, CPQ = 36° 24', DPQ = 19° 57'; Ex. 9. It is required to inscribe a polygon in a given polygon, so that its sides shall be parallel to the same number of given lines. Ex. 10. It is required to inscribe a polygon in a given circle to fulfil the same conditions. Ex. 11. It is required to describe a polygon through any number of given points, so that its angles shall be situated in the same number of given lines. Ex. 12. It is required to inscribe a polygon in a given circle, so that its sides shall pass through the same number of given points. THE CONIC SECTIONS. DEFINITIONS. 1. THE Conic Sections are the figures made by a plane cutting a cone, either right or oblique. [Whether the cone be right or oblique, the sections made in it by a plane have precisely the same properties, but as the investigation is more simple when they are considered in the right cone, it is usual to employ this instead of the oblique cone.] 2. According to the different positions of the cutting plane there arise five different species of figures or sections; namely, two right lines intersecting in the vertex of the cone, the circle, the ellipsis, the hyperbola, and the parabola. The three last are peculiarly called conic sections, the properties of the right line and the circle having been previously laid down in Plane Geometry. 3. If the cutting plane pass through the vertex of the cone, and also cut the circular base, its section with the conical surface will be two straight lines, as VA, VB. This the student will readily prove. [Since the line which generates the conical superficies (Geom. def. 105) may be of indefinite length, the cone itself is of indefinite length, the figure delineated being only part of it. Hence the lines in which the plane cuts the surface are of indefinite length. Also, as by producing the line indefinitely on the other side of the vertex, an equal and opposite cone will be produced, the same plane section, or the lines VA, VB are also indefinitely continued beyond the vertex.] 4. If the plane cut the cone parallel to the base, or make no angle with it, the section will be a circle, as ABD. See theor. 117. Geom. 5. The section will be an ellipse when the cone is cut obliquely through both sides, or when the plane of section is inclined in a less angle to the plane of the base than the tangent plane to the cone is. [The student is required to prove that the section is comprised within finite limits, and that the curve is one continuous line.] 6. The section is a parabola when the cutting plane is parallel to a tangent plane of the cone, or when it makes the same angle with the plane of the base that the tangent plane parallel to it does. [It is to be proved that this curve is of infinite extent on the side BE estimated from A, the figure delineated being only a part of it.] 7. The section is an hyperbola, when the cutting plane makes a greater angle with the base than the tangent plane to the cone makes with it. 8. And if all the sides of the cone be continued through the vertex, forming the opposite cone, and the plane be also continued to cut the opposite cone, this latter section will be the opposite hyperbola to the former; as dBe. The two opposite hyperbolas are also called, and more properly, the two opposite branches of the hyperbola, considered as one curve. [It will appear hereafter, cor. 4, pr. II. Hyperbola, that whatever be the position of the plane of the hyperbolic section, the two figures are in all respects equal to one another. B It is required to prove that the opposite hyperbolic branches cut from the two opposite cones are of infinite extent from A and B in the directions of DE and de respectively. All sections made by planes parallel to the axis are hyperbolas.] 9. The transverse plane is that which is perpendicular both to the plane of the conic section and to the plane of the cone's base. 10. The transverse axis of any conic section is the portion of the line of intersection of the transverse plane with the plane of the conic section, and indefinitely produced. 11. The transverse diameter of the section is the portion of the transverse axis which lies between its two points of intersection with the curve; and is represented in the foregoing figures by AB. [The axis of a parabola is infinite in length, AB being only a part of it.] 12. The points in which the transverse axis cuts the curve, are called the transverse vertices of the curve; and often simply the vertices. [The ellipse and hyperbola have two vertices, but the parabola only one.] 13. The middle of the transverse diameter is called the centre of the conic section. [The centre of a parabola is infinitely distant from the vertex. The centre of the ellipse lies within, or on the concave side of the curve: and the centre of the hyperbola without, or on the convex side of the curve.] 14. The vertical angle of a right cone is the angle included by the two lines which constitute the section (def. 2.) made by a plane passing through the axis of the cone and the angle contained between the axis and one of these lines is called the generating angle of the right cone, it being the constant angle made by the generating line with the fixed line or axis. 15. Two right cones, whose vertical angles are the supplements of each other, are called supplementary or conjugate cones. [If two pairs of such cones be so placed as to have their vertices coincident, their axes will be at right angles to one another, and in the same plane with their lines of contact: and the axes will bisect the angles made by those lines of contact. The common tangent plane to the conjugate cones at their lines of contact will be perpendicular to the plane in which the axes and lines of contact are situated.] 16. Let LCI and HCK be the lines of contact of the conjugate cones, and AB, ab, their axes; take any point I in one of the lines of contact, and draw IH, IK cutting the axes in a and B, and being themselves consequently bisected in those points. Then if the cones whose axial sections are HCI and ICB, A E with their opposite sheets, be cut by planes parallel to the plane of the axes, and at distances from it equal to Ia and IB respectively, two pairs of opposite hyperbolic branches will be produced in the cones by these planes. If now these planes, with their sections, be moved down till they coincide with the plane of the axes, and their intersections with the common tangent planes coincide with HR and LI respectively, they will form the annexed figure, in which the pairs of branches are called the conjugate hyperbolas. That is, the branches FBG, DAE are conjugate to the branches fbg, dae; and mutually fbg, dae are called conjugate to FBG, DAE. G [These four branches of the curve, though not cut from the cone by the same plane, are yet to be viewed as forming part of the same system. Indeed, though for convenience of verbal description they have been cut by means of two planes, it is easy to see that by a different method of placing the axes of the conjugate cones the conjugate sections might have been made by a single plane-uamely, by depressing the axis of the obtuse angled cones below the present plane of the axis (the cone still having the same tangent planes perpendicular to the plane of the paper) by a distance equal to the excess of IA above IB.] 17. The lines AB, ab, will be the conjugate rectangular diameters, (see prop. 2, Hyperbola): and the rectangle HIKL is called the conjugate rectangle, or the rectangle of the axes. These are sometimes called the principal diameters. 18. The lines HK and LI are called the asymptotes of the hyperbola. 19. The perpendicular (to the transverse axis of the ellipse), from the centre C, and terminated both ways by the curve, the conjugate rectangular diameter (Ellipse, pr. 2.) of the curve, and the rectangle formed of AB and ab, viz. HIKL is the conjugate rectangle, or rectangles of the axes. a B 20. A tangent to a conic section is a straight line which meets the curve, but being produced both ways does not cut it. 21. A diameter of a conic section is a line drawn through the centre and terminated both ways by the curve, and the extremities of such a diameter are called its vertices. : 22. If a tangent and a diameter of a conic section be drawn through the same point in the curve, they are said to be conjugate to each other and any lines parallel to these are said also to be conjugate to each other. If they be lines terminated by the curve, and meet at a point within it, they are called conjugate chords; if they meet the curve and intersect at a point without it, they are conjugate secants; if they both touch it, they are conjugate tangents; if they both pass through the centre, they are conjugate diameters; and if one be a |