But the line AB bisects the line CG at right angles in E; and hence the angle DEA is a right angle. B In the same manner, every other line drawn correspondingly to ED is perpendicular to AB. They are all, therefore, in one plane perpendicular to AB. Also, in the same manner, all these lines are equal to EC; and hence their extremities are in the circumference of a circle, whose centre E is in AB, and whose plane is perpendicular to AB. PROPOSITION IV. Through any point without a sphere innumerable straight lines, and likewise innumerable planes, may be drawn to touch the sphere; the linear tangents will be all equal; and the points of contact of the lines and of the planes with the sphere will be a circle. 1. Let O be the centre of the sphere, P a point without it, PA, PB, etc., tangents to the circles CAH, CBH made by planes through PO, and A, B, etc. : then AP, BP, etc., will all be tangents to the sphere. 2. For PA, PB, etc., are perpendicular, to the radii AO, BO, etc.; and hence are tangents to the sphere itself; and as the planes that may be drawn through PO are innumerable, the points A, B, etc., in them are innumerable. 3. Through each of the points A, B, etc., a plane may be drawn perpendicular to AO, BO, etc. These will be tangent planes to the sphere. Moreover, the lines AP, BP, etc., being perpendicular to AO, BO, etc., they will respectively be contained in the tangent planes at those points. But P is in each of these lines, and hence in each of the planes which contain them: or all the tangent planes pass through P, and are innumerable. 4. Since the right-angled triangles PAO, PBO, etc., have two sides, PO, OA equal to the two PO, OB, the third sides, AP, BP are D equal. Whence all the linear tangents from P to the sphere are equal. Draw AD perpendicular to PO, and join BD. Then in the right-angled triangles PAO, PBO, the sides AO, OP are equal to BO, OP, the remaining angles are equal, viz., APO to BPO, and AOP to BOP. Whence, since AP, PD are equal to BP, PD, and the included angles are equal, the bases AD, BD are equal, and the remaining angles to the remaining are equal. The angle PDB is, therefore, equal to PDA; that is, PDB is a right angle. Wherefore, AD, BD, etc., are in one plane perpendicular to PD (Prop. 111. Chap. 11.), and being all equal, they are in the circumference of a circle whose centre is D in the line PO, and whose plane is perpendicular to PO. Cor. To a given sphere a tangent cone can be drawn having its vertex at any given point without the sphere; and such cone will always be a right cone. PROPOSITION V. If an oblique cone be described on a circular base, then there is another plane parallel to which all the sections are circles.* Let AGB be the base, and V the vertex of the cone; and let the plane through the axis perpendicular to the base cut the cone in AV, BV, and the base in AB. Draw in the plane AVB a line A'B', making the angle VA'B' equal to VAB, meeting AB in D; and through A'B' draw a plane perpendicular to the plane AVB. The section of the cone by this plane will be a circle. Now the angles VAB, VA'B' being equal, the points B', A, A', B, are in the same segment of a circle; and hence AD.DB A'D.DB'. Again, the planes through AB, A'B' being both perpendicular to the plane AVB, their intersection EG is also perpendicular to that plane, and hence to the lines AB, A'B' themselves. But by hypothesis and construction, AEB is a semicircle; and B A B hence DE AD.DB = A'D.DB'; whence DE being perpendicular to A'B', and having DES = A'D.DB', the curve A'EB' is also a semicircle. * This secondary series of circles are called sub-contrary or antiparallel to the former : the former name is most frequently used, --the latter the more convenient. CHAPTER VIII. GEOMETRY OF THE SPHERICAL SURFACE. DEFINITIONS. 1. A sphere is that figure every point of the surface of which is equally distant from a certain point within it, and that point is the centre of the sphere. 2. The distance of any point on the surface from the centre is called a radius of the sphere. 3. The diameter of the sphere is any line drawn through the centre, and terminated both ways by the surface. 4. A diametral plane is any plane drawn through the centre; and its intersection with the surface of the sphere is a great circle of the sphere. 5. An eccentric plane is any plane which cuts the sphere, but does not pass through its centre; and the intersection of the surface of the sphere by an eccentric plane is called a less circle of the sphere, 6. A tangent plane to a sphere is a plane which meets the surface of the sphere in one point, and which being produced in all directions, does not cut it. 7. If a perpendicular be drawn through the centre of the sphere to a diametral plane, it intersects the surface in two points, which are called the poles of the great circle made by the diametral plane. 8. If a perpendicular be drawn through the centre to an eccentric plane, it cuts the sphere in two points, which are called the poles of the less circle. 9. The poles of a less circle are called the contiguous and the remote poles respectively, according as the pole is in the less or in the greater segment of the spherical surface made by the plane. 10. When two great circles of the sphere intersect each other, they divide the surface into four portions, called spherical lunes ; and the enclosed volumes made by the cutting planes and the intercepted surfaces are called spherical wedges. 11. When three great circles intersect each other two and two, the portion of the surface enclosed by three contiguous segments of them is called a spherical triangle, as ABC in the figure. 12. As the three great circles intersect each other two and two a second time on the opposite sides of the sphere at A', B', C', there will be eight spherical triangles formed by the arcs of the three circles, viz., ABC, A'BC, AB'C, ABC', and A'B'C', AB'C', A'BC', A'B'C. The first four and the second four sets are called respectively the primary and the symmetrical associated systems of spherical triangles. The primary system is such as to have either AB, BC, or CA in each triangle; and the symmetrical as to have either A'B', B'C', or C'A' in each of the triangles which compose the system. VOL. II. B A P с 13. Whichever of these triangles be first traced, the entire and completed system will be the same; and hence, any one of them may be taken as the original triangle of the system. It has been usual to consider the central one as the original; but we may readily change the system, as may be easily seen, so that any other of the three shall be the original one. We shall in our investigations, however, consider, except otherwise specially stated, the central one ABC to be the fundamental triangle of the associated system. The sides of the other three are either equal to these or supplementary to them: and hence the remaining three triangles will be spoken of as the supplementary triangles of the associated system. If we wish to specify which of the supplemental triangles we speak of, it will be conveniently done by mentioning the side of the fundamental triangle which is common to the two: thus, BA'C is supplemental with respect to a, AB'C with respect to b, and BC'A with respect to c. 14. If with the angular points A, B, C, as poles, three other great circles be described, their intersections a, b, c, will be the angular points of another spherical triangle abc, which is called the polar triangle with respect to ABC, the original triangle being called with respect to the polar one, the primary triangle. 15. If both the primary and polar system of associated triangles be completed, the systems are called the primary associated and the polar asso- e ciated systems. 16. The arcs AB, BC, CA, are called the sides of the triangle ABC. 17. By a spherical angle is meant the dihedral angle contained by the planes of the circles whose arcs form the boundaries of the lunes on the surface of the sphere. 18. Great circles passing through the poles of a great circle are called secondaries to that circle, and the great circle itself is called the primary with respect to the secondaries. 19. The spherical excess of a triangle is the excess of the three angles of that triangle above two right angles. 20. When two sides or two angles of a spherical triangle are both greater than } , or both less than 17, they are said to be of the same affection ; and when one (either side or angle) is greater and the other less than 7, they are said to be of different affections, or of unlike affection. 21. A quadrantal triangle has one of its sides a quadrant. It is conveniently called a bi-quadrantal and a tri-quadrantal triangle, when two sides or three sides, respectively, are quadrants. PROPOSITION I. The plane of a great circle is cut by the planes of any two of its secondaries in lines which contain an angle equal to the spherical angle formed by those secondaries. Let CDE be the primary great circle, and ACB, ADB any two of its secondaries (Def. 18), cutting the plane CDE in CE, DF: then the angle DOC is equal to the spherical angle DAC. E For, since ACB, ADB are secondaries, their line of section AB is perpendicular to the plane CDE (Defs. 7, 18); and hence the planes AOD, AOC cut CDE in lines CO, DO, which are perpendicular to OA (Pls. and Sols., Chap. 11., Prop. xvi.). Wherefore the angle DOC is the measure of the inclination of the planes of the secondaries; and hence again, equal to the spherical angle CAD (Def. 17). Cor. 1. If two great semicircles ACB, ADB, be bisected in C and D, the arc CD of a great circle joining them, is the measure of the spherical angle CAD. Cor. 2. The pole of a great circle is at the distance of a quadrant from every point of that circle: for since AC, AD, subtend the right angles AOC, AOD, they are quadrants. Cor. 3. All the secondaries of a great circle are perpendicular to it, for their planes pass through OA, which is perpendicular to its plane; and hence the measure of the angle formed by the circle and its secondary is a quadrant. Cor. 4. Every great circle perpendicular to another is secondary to it. Cor. 5. The poles of all the secondaries are situated in the primary. PROPOSITION II. The distance of the poles of any two circies is the measure of the spherical angle contained by these. (See the last Fig.) Let ACB, ADB be any two circles intersecting in A, B; and let P, Q be their poles respectively: then the arc PQ is the measure of the angle CAD. For let the circle PQ be drawn to meet the circles ADB, ACB in D and C: then since P is the pole of ACB, PC is at right angles to ACB, and equal to a quadrant (Prop. I., Cor. 2, 3); and for the same reason QD is at right angles to ADB, and equal to a quadrant. Hence to the secondaries ACB, ADB, CDPQ is the primary; and since CP is equal to DQ, each being a quadrant, we have PQ equal to CD. But since AC, AD are secondaries to CD, the arc CD is the measure of the spherical angle CAD (Prop. I., Cor. 1); and hence, also, PQ is the measure of that angle. SCHOLIUM. It is essential to remark, that in this demonstration, the poles are taken in continuation of the circle CD in the same direction, viz., both in the direction from C towards or beyond D, as P and Q of the figure, or from D towards or beyond C, as Q' and P'. When the poles are taken contrarily, as P, Q', or P', Q, their distances measure the supplementary angles DAE, and FAC. The two cases may be distinguished by the terms same direction and cross direction. |