# Geometry in Space; Containing Parts of Euclid's 11th and 12th Books, and Some Properties of Polyhedra and Solids of Revolution, with Exercises

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## Description

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1888 edition. Excerpt: ...7' 77 77 3: II = tripl' ratio of a pair of homol' edges. THEOREM (r7)--If four lines are proportional, and if similar polyhedra are similarly described on the first pairas edges, and similar polyhedra are similarly described on the second pair as edges, then as the polyhedron on the firrt is to the polyhedron on the second so is the polyhedron on the third to the polyhedron on the fourth; and, con-oersely, if two similar polyhedra have the same ratio as two other pob/hedra, similar to each other, then similar edges of the first pair have the same ratio as similar edges qf the second pair. LEMMA--If two ratios are equql, their triplicates are equal; and conversely. Let A, B, C, D, P, Q, X, Y be mag's such that, A: B =B: X=X: Y, C: D= D: P= P: Q. r, ifA: B = C: D, thenB: X = D: P, and X: Y = P: Q, whence, compounding the ratios, we get A: Y = C: Q, i. e. triplicate of A: B = triplicate of C: D. 2, ifA: Y=C: Q, assumethatA: B=C: a.=u.: B=B: (. A: Y=C: y...-y=Q...........(r) Now if a. D, then B P, and.'. yQ; which contradicts (1). And if u. D, then B P, and 7 O; which again contradicts (1). It remains that u. D... A: B = C: D. Now let AB, CD, LM, NO be four given lines. On AB, CD let sim'r polyhedra P, Q be sim'ly described, and on LM, NO let sim'r polyhedra X, Y be sim'ly described. 1, suppose AB: CD = LM: NO. Then P: Q = tripl' ratio of AB to CD = tripl' ratio LM to N0 = X: Y. _ 2, suppose P: Q = X: Y. Then tripl' ratio of AB to CD = P: Q X: Y-= tripl' ratio of LM to NO. LM: NO. AB: CD THE DEL1AN PROBLEM--To make a cube whose volume is ziouele that of a given cude. Let a. be an edge of the given cube. Find (E-R-p. 276) two mean prop'ls X, y between a. and 2 0., so that u.: x...show more

## Product details

• Paperback | 28 pages
• 189 x 246 x 2mm | 68g
• United States
• English
• black & white illustrations
• 1236833384
• 9781236833389