(9) Through three given points on a sphere, one and only one circle can be drawn. § 591. (10) All points of any circle of a sphere are equidistant from either of its poles. § 594. (11) The polar distance of any point of a great circle is a quadrant. § 596. (12) If a point on a sphere is a quadrant's distance from each of two given points on the sphere, it is the pole of the great circle passing through these two points. § 597. (13) If one point on a sphere is a quadrant's distance from another, it is the pole of some great circle passing through the other. § 598. (14) The intersection of two spheres is a circle, the plane of which is perpendicular to the straight line joining the centres of the spheres, and the centre of which is on that line. § 604. 5. THEOREMS RELATING TO TANGENTS TO A SPHere. (1) A plane which is perpendicular to a radius of a sphere at its (3) The plane of two straight lines tangent to a sphere at the same 6. THEOREMS RELATING TO THE PROPERTIES OF SPHERICAL ANGLES, TRIANGLES, AND POLYGONS. (1) The angle formed by two intersecting arcs of great circles has the same measure as the dihedral angle formed by the planes of the circles. § 613. (2) The angle formed by two intersecting arcs of great circles has the (3) All arcs of great circles drawn through the pole of a given great (5) The sum of any two sides of a spherical triangle is greater than the third side. § 621. (6) The sum of the sides of a convex spherical polygon is less than a great circle. § 622. (7) An isosceles spherical triangle and its symmetrical spherical triangle are identically equal. § 626. .(8) If two sides of a spherical triangle are equal, the angles opposite those sides are equal. § 627. (9) If the first of two spherical triangles is the polar of the second, then the second is also the polar of the first. § 631. (10) In two polar triangles, the measure of any angle in one is equal to the measure of the supplement of that side in the other of which its vertex is the pole. § 633. (11) If A is any angle of a spherical triangle, and a' the corresponding side of the polar triangle, then the measure of a' is equal to the measure of the supplement of A. § 634. (12) If any spherical triangle is equiangular, its polar triangle is equilateral, and conversely. § 635. (13) If two spherical triangles on the same sphere, or on equal spheres, are mutually equiangular, their polar triangles are mutually equilateral, and conversely. § 636. (14) The sum of the angles of any spherical triangle is greater than two right angles and less than six right angles. § 637. (15) A spherical triangle may have one, two, or three right angles, or one, two, or three obtuse angles. § 639. 7. THEOREMS ON THE EQUALITY OF SPHERICAL TRIANGLES. Two spherical triangles lying on the same or equal spheres are identically equal, or symmetrical, if they have — (1) Three sides of the one equal, respectively, to the three sides of (2) Two sides and the included angle of the one equal, respectively, (4) Three angles of the one equal, respectively, to the three angles 8. THEOREMS RELATING TO AREAS. (1) Two symmetrical spherical triangles have equal areas. § 646. (3) The ratio of two lunes on the same sphere or on equal spheres is equal to the ratio of their angles. § 649. (4) The area of a lune is to the area of the whole sphere in the same ratio as the angle of the lune is to four right angles. § 650. (5) If two great circles intersect on a hemisphere, the two triangles formed by their arcs and arcs of the great circle bounding the hemisphere are together equal to a lune having the same angle as the angle between the great circles. § 651. (6) The area of any spherical triangle expressed in spherical units is equal to the spherical excess of the triangle. Area = A + B + C-180. § 653. (7) If S is the sum of the angles of a spherical polygon of n sides, the area of the polygon is [S(n − 2) 180] spherical units. § 654. (8) The area of the surface generated by a line-segment revolving about an axis in its plane is equal to the length of the projection of the line-segment on the axis multiplied by the circumference of the circle whose radius is equal to that segment of the perpendicular bisector of the revolving line-segment which is intercepted between it and the axis. § 655. (9) The area of a zone is equal to the product of its altitude and the circumference of a great circle. A = 2 πrh. § 657. (10) The area of a zone is equal to that portion of the lateral area of the enveloping cylinder which is intercepted between the same planes as the zone. § 660. (11) The area of a sphere is equal to the product of a diameter and the circumference of a great circle. A 2 πrd = 4 πr2. § 661. (12) The areas of two spheres are in the same ratio as the squares of their radii, or as the squares of their diameters. A: A' = r2: r/2, or d2: d'2. § 663. (13) The area of a sphere equals four times the area of one of its great circles. § 664. (14) The area of a sphere equals the lateral area of its enveloping cylinder. § 665. (15) The area of a spherical unit expressed in plane units equals 9. THEOREMS RELATING TO VOLUMES. (1) The volume of a sphere is equal to one-third the product of its area and its radius. V = π3. § 667. (2) The volumes of two spheres are in the same ratio as the cubes of their radii, or as the cubes of their diameters. V:V' = p3 : p13, or d3: d'3. § 669. (3) The volume of a spherical pyramid or a spherical sector is equal to one-third the product of the area of its base and the radius of the sphere. V = Ar. § 673. (4) The volume of a spherical segment is equal to h (wa2 + πb2) + Th3, where a and b are the radii of its bases and hits altitude. § 674. 10. MISCELLANEOUS THEOREMS. (1) Through any four points not lying in the same plane, one and only one sphere can be passed. § 605. (2) The perpendiculars to the four faces of a tetrahedron, erected at (3) The six planes perpendicular to the edges of a tetrahedron at (4) One and only one sphere can be inscribed in any given tetra- (5) The planes which bisect the six dihedral angles of any tetrahedron have one point in common. § 611. (6) The shortest line that can be drawn on a sphere between two points is the arc of a great circle, not greater than a semicircle, joining the two points. § 642. APPENDIX The following brief introduction to Trigonometry is designed to give to the high school or academy pupil as much of that subject as he may need for a course in Physics or Elementary Mechanics. It contains no solid geometry, and may be read as soon as the pupil has completed Chapter V of this text, or earlier if desired. the ratio is called the sine of the angle A, MP AP AM the ratio is called the cosine of the angle A, AP the ratio is called the tangent of the angle A. MP If the point P were chosen differently on the boundary AC, and the perpendicular were drawn to AB, would the ratios of the sides of this new triangle be equal to the corresponding ratios of the sides of the triangle PAM? Or, if the point P were chosen in the boundary AB and the perpendicular drawn to AC, would the ratios of the sides be altered? |