Sinopsis
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. 1861 Excerpt: ...Geometrical Method. The formulas for these operations may be deduced geometrically, by a process very similar to that required for the corresponding rule for cube root. For this purpose, let the following problem be proposed:--Required the dimensions of a rectangular parallelopipcdon whose breadth and length exceed its height by d ami d' respectively, and whose solidity is g. Let y = the height; then y--d= the breadth, and y--? = the length. Multiplying together the three dimensions, we have (1) f + (d+d')f + d?y = q. Put s = d--ct, and p = dct; then (2) y + sy+py = rj, the general form employed in the foregoing pages. We will suppose the required root of this equation to be irrational, and let m = the initial figure. Since the entire root is to be the height of a right prism whose several dimensions are as given in the problem, we will form a right prism (Fig. 1), having AB = m, the height; A C = m--d, the breadth; and AD = )i--d', the length. The solidity of the prism is m(m--d)(m--?) = ntz--(d--d')m2--da m= mz--sm2--pm = 8. Subtracting the solid contents of the prism from that of the required prism, we have, for a remainder, q--S. If we now enlarge our prism (Fig. 1) by the addition of this remainder, its solidity will be that of the required prism. To preserve the given relations, the addition must be made upon three adjacent sides or faces, in the form of a covering of uniform thickness. The addition will therefore be composed of three flat prisms to cover the three faces (Fig. 2), and three oblong prisms and one cube to fill the vacancies at the edges and corner (Fig. 3). Now the thickness of this enlargement will be additional height of the prism, and consequently the remaining part of the root. Hence, to obtain a second figure of the root, we may di...
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Reseña del editor
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. 1861 Excerpt: ...Geometrical Method. The formulas for these operations may be deduced geometrically, by a process very similar to that required for the corresponding rule for cube root. For this purpose, let the following problem be proposed:--Required the dimensions of a rectangular parallelopipcdon whose breadth and length exceed its height by d ami d' respectively, and whose solidity is g. Let y = the height; then y--d= the breadth, and y--? = the length. Multiplying together the three dimensions, we have (1) f + (d+d')f + d?y = q. Put s = d--ct, and p = dct; then (2) y + sy+py = rj, the general form employed in the foregoing pages. We will suppose the required root of this equation to be irrational, and let m = the initial figure. Since the entire root is to be the height of a right prism whose several dimensions are as given in the problem, we will form a right prism (Fig. 1), having AB = m, the height; A C = m--d, the breadth; and AD = )i--d', the length. The solidity of the prism is m(m--d)(m--?) = ntz--(d--d')m2--da m= mz--sm2--pm = 8. Subtracting the solid contents of the prism from that of the required prism, we have, for a remainder, q--S. If we now enlarge our prism (Fig. 1) by the addition of this remainder, its solidity will be that of the required prism. To preserve the given relations, the addition must be made upon three adjacent sides or faces, in the form of a covering of uniform thickness. The addition will therefore be composed of three flat prisms to cover the three faces (Fig. 2), and three oblong prisms and one cube to fill the vacancies at the edges and corner (Fig. 3). Now the thickness of this enlargement will be additional height of the prism, and consequently the remaining part of the root. Hence, to obtain a second figure of the root, we may di...
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