This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

The discussion of the subject of Ratios has thus far been confined to its relations with the mercantile problem of Art. 364. The rules of proportion or the equality of ratios apply equally to questions other than those of a mercantile character. They apply alike to all questions in which quantities of any kind are comparable. For example, in geometry, lines, surfaces, and solids bear a certain fixed relation to one another, and are, therefore, fit subjects for the rules of proportion. It is shown, in Art. 361, that the corresponding sides of homologous triangles are in proportion to one another. Hence, when, of two similar triangles, two corresponding sides and one other side are given, then by the equality of ratios the side corresponding to this other side may be computed. For example: in two triangles, such as ECD and EAB (Fig. 269), having their corresponding angles equal, let the side E C, in the triangle ECD, equal 12 feet, and the corresponding side E A, in the triangle EAB, equal 16 feet, and the side ED, of triangle E CD, equal 14 feet. Now, having these three sides given, how can we find the fourth?, Putting them in proportion, we have, as in Art. 361 -

Ce: Ae:: De: Be;

and, substituting for the known sides, their dimensions, we have -

12: 16:: 14: BE;

and, by Art. 373 -

12 x B E = 16 x 14.

Dividing each member by 12, gives -

BE = 16x14/12.

Performing the multiplication and division indicated, we have -

BE = 224/12 = 18 2/3 Thus we have the fourth side equal to 18 2/8 feet.

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