Items related to Elements of Geometry Volume 2; Geometry of space
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1898 Excerpt: ...are together equal to a lune whose angle is C. § 888 Hence area ABC+ area CA'B'--iC. Now the sum of the areas of ABC, B'CA, A'CB, and CA'B' is the area of the surface of a hemisphere, which with the adopted unit is 4. Hence, adding the three equations above, we have, 2 area ABC+4-2A +2B+ 2C. Therefore area ABC =A+B+C-2. Q.e.d. PROPOSITION XXXVII. THEOREM 896. If the unit angle is the right angle, and the unit surface the tri rectangular triangle, the area of a spherical polygon is measured by the sum of its angles minus twice the number of its sides less two. Given--the spherical polygon ABCDE. Denote its area measured in tri-rectangular triangles by K; the sum of its angles measured in right angles by S; and the number of its sides by n. To Prove K=S--2(«--2). Divide the polygon into triangles by diagonals drawn from any vertex A. The area of each triangle is measured by the sum of its angles less two. § 895 The number of triangles is n--2, there being one for every side except the sides intersecting in A. Hence the area of the polygon is measured by the sum of the angles of all the triangles minus 2(n--2). But the sum of the angles of all the triangles is equal to the sum of the angles of the polygon. Therefore K=S--2(n--2). Q. E. D. 897. Defs.--A spherical pyramid is a solid bounded by a spherical polygon and the planes of its sides; as O-ABCD. The centre of the sphere is called the vertex of the spherical pyramid, and the spherical polygon its base. 898. Defs.--A spherical ungula, or wedge, is a solid bounded by a lune and the planes of its bounding arcs. The lune is called the base of the ungula; the diameter in which the bounding planes meet is its edge. The angle of the bounding lune is also called the angle of the ungula. 899. The pro...
"synopsis" may belong to another edition of this title.
(No Available Copies)
Search Books: Create a WantCan't find the book you're looking for? We'll keep searching for you. If one of our booksellers adds it to AbeBooks, we'll let you know!
Create a Want
