The sum of the angles of a spherical triangle is greater than two, and less than six right angles. Let A, B, and C be the angles of a spherical triangle. The arcs which measure the angles A, B, and C, together with the three sides of the polar triangle, are equal to three semi-circumferences (Pr. 9). But the three sides of the polar triangle are less than two semi-circumferences (Pr. 4); hence the arcs which measure the angles A, B, and C are greater than one semi-circumference, and, therefore, the angles A, B, and C are greater than two right angles. Also, because each angle of a spherical triangle is less than two right angles, the sum of the three angles must be less than six right angles. Cor. A spherical triangle may have two, or even three right angles; also two, or even three obtuse angles. If a triangle have three right angles, each of its sides will be a quadrant, and the triangle is called a tri-rectangular triangle. The tri-rectangular triangle is contained eight times in the surface of the sphere.* B * In all the preceding propositions, it has been supposed, in conformity with Def. 6, that spherical triangles always have each of their sides less than a semicircumference, in which case their angles are always less than two right angles. It should, however, be remarked, that there are spherical triangles of which certain sides are greater than a semi-circumference, and certain angles greater than two right angles. For if we produce the side AC so as to form an entire cir- A cumference, ACDE, the part which remains, after taking from the surface of the hemisphere the triangle ABC, is a new triangle, which may also be designated by ABC, and the sides of which are AB, BC, CDEA. Here we see that the side CDEA is greater than the semi-circumference DEA, and, at the same time, the opposite angle ABC exceeds two right angles by the quantity CBD. E Triangles whose sides and angles are so large have been excluded by the definition, because their solution always reduces itself to that of triangles embraced in the definition. Thus, if we know the sides and angles of the triangle ABC, we shall know immediately the sides and angles of the triangle of the same name, which is the remainder of the surface of the hemisphere. The area of a lune is to the surface of the sphere as the angle of the lune is to four right angles. A B G Let ADBE be a lune, upon a sphere whose centre is C, and the diameter AB; then will the area of the lune be to the surface of the sphere as the angle DCE to F four right angles, or as the arc DE to the circumference of a great circle. · First. When the ratio of the arc to the circumference can be expressed in whole numbers. Suppose the ratio of DE to DEFG to be as 4 to 25. Now, if if we divide the circumference DEFG in 25 equal parts, DE will contain 4 of those parts. If we join the pole A and the several points of division by arcs of great circles, there will be formed on the hemisphere ADEFG 25 triangles, all equal to each other, being mutually equilateral. The entire sphere will contain 50 of these small triangles, and the lune ADBE 8 of them. Hence the area of the lune is to the surface of the sphere as 8 to 50, or as 4 to 25; that is, as the arc DE to the circumference. Secondly. When the ratio of the arc to the circumference can not be expressed in whole numbers, it may be proved, as in B. III., Pr. 14, that the lune is still to the surface of the sphere as the angle of the lune to four right angles. Cor. 1. On equal spheres, two lunes are to each other as the angles included between their planes. Cor. 2. We have seen that the entire surface of the sphere is equal to eight tri-rectangular triangles (Pr. 17, Cor.). If the area of the tri-rectangular triangle be represented by T, the surface of the sphere will be represented by 8T. Also, if we take the right angle for unity, and represent the angle of the lune by A, we shall have the proportion, area of the lune: 8T:: A; 4. Hence the area of the lune is equal to 8AXT . or 2AXT. Cor. 3. The spherical ungula, comprehended by the planes ADB, AEB, is to the entire sphere as the angle DCE is to four right angles. For, the lunes being equal, the spherical ungulas will also be equal; hence, in equal spheres, two ungulas are to each other as the angles included between their planes. PROPOSITION XIX. THEOREM. If two great circles intersect each other on the surface of a hemisphere, the sum of the opposite triangles thus formed is equivalent to a lune whose angle is equal to the inclination of the two circles. Let the great circles ABC, DBE intersect each other on the surface of the hemisphere BADCE; then will the sum of the opposite triangles ABD, CBE be equivalent to a lune whose angle is CBE. For, produce the arcs BC, BE till they meet in F; then will BCF be a semi-circumference, as also ABC. Subtracting A F B E C BC from each, we shall have CF equal to AB. For the same reason, EF is equal to DB, and CE is equal to AD. Hence the two triangles ABD, CFE are mutually equilateral; they are, therefore, equivalent (Pr. 15). But the two triangles CBE, CFE compose the lune BCFE, whose angle is CBE; hence the sum of the triangles ABD, CBE is equivalent to the lune whose angle is CBE. Therefore, if two great. circles, etc. The area of a spherical triangle is measured by the excess of the sum of its angles above two right angles multiplied by the tri-rectangular triangle. Let ABC be any spherical triangle; its surface is measured by the sum of its angles A, B, C diminished by two right angles, and multiplied by the tri-rectangular triangle. Produce the sides of the triangle ABC until they meet the great circle DEG drawn without the triangle. The two triangles ADE, AGH H G A are together equal to the lune whose angle is A (Pr. 19); and this lune is measured by 2AXT (Pr. 18, Cor. 2). ADE+AGH=2AXT. For the same reason, BFG+BDI=2B×T; also, CHI+CEF=2C×T. Hence we have But the sum of these six triangles exceeds the surface of the hemisphere by twice the triangle ABC, and the hemisphere is represented by 4T; hence we have 4T+2ABC-2A×T+2B×T+2C×T; or, dividing by 2, and then subtracting 2T from each of these equals, we have or ABC=AXT+BxT+C×T-2T, Hence every spherical triangle is measured by the sum of its angles diminished by two right angles, and multiplied by the trirectangular triangle. Cor. If the sum of the three angles of a triangle is equal to three right angles, its surface will be equal to the tri-rectangular triangle; if the sum is equal to four right angles, the surface of the triangle will be equal to two tri-rectangular triangles; if the sum is equal to five right angles, the surface will be equal to three tri-rectangular triangles, etc. PROPOSITION XXI. THEOREM. The area of a spherical polygon is measured by the sum of its angles, diminished by as many times two right angles as it has sides less two, multiplied by the tri-rectangular triangle. E A D B Let ABCDE be any spherical polygon. From the vertex B draw the arcs BD, BE to the opposite angles; the polygon will be divided into as many triangles as it has sides C minus two. But the surface of each triangle is meas ured by the sum of its angles minus two right angles, multiplied by the tri-rectangular triangle. Also, the sum of all the angles of the triangles is equal to the sum of all the angles of the polygon; hence the surface of the polygon is measured by the sum of its angles, diminished by as many times two right angles as it has sides less two, multiplied by the tri-rectangular triangle. Cor. If the polygon has five sides, and the sum of its angles is equal to seven right angles, its surface will be equal to the trirectangular triangle; if the sum is equal to eight right angles, its surface will be equal to two tri-rectangular triangles; if the sum is equal to nine right angles, the surface will be equal to three tri-rectangular triangles, etc. BOOK X. MEASUREMENT OF THE THREE ROUND BODIES. Definitions. 1. A cylinder is a solid described by the revolution of a rectangle about one of its sides, which remains. fixed. The bases of the cylinder are the circles described by the two revolving opposite sides of the rectangle. 2. The axis of a cylinder is the fixed straight line about which the rectangle revolves. The opposite side of the rectangle describes the lateral or convex surface. 3. A cone is a solid described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed. The base of the cone is the circle described by that side containing the right angle which revolves. 4. The axis of a cone is the fixed straight line about which the triangle revolves. The hypothe nuse of the triangle describes the lateral or convex surface. The side of the cone is the distance from the vertex to the circumfer ence of the base. 5. A frustum of a cone is the part of a cone next the base, cut off by a plane parallel to the base. 6. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals. The convex surface of a cylinder is equal to the product of its altitude by the circumference of its base. Let ACE-G be a cylinder whose base is the circle ACE, and altitude AG; then will its convex surface be equal to the product of AG by the circumference ACE. In the circle ACE inscribe the regular polygon ABCDEF, and upon this polygon let a right prism be constructed of the same altitude with the cylinder. |