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

One advantage derived from algebra is that the rules made are general in their application, For example, the rule of Art. 397, bc/a = d, is applicable to all cases of homologous triangles, however they may differ in size or shape from those given in Fig. 269 - and not only this, but it is also applicable in all cases where four quantities are in proportion so as to constitute four proportionals. For example, the case of the four proportionals constituting the arms of a lever and the weights attached (Arts. 375-378). For, taking the relation as expressed in Art. 277 -

PxCF = RxEC,

we may substitute for C F the letter n, and for E C the letter m, then m will represent the arm of the lever E C (Fig. 262), and the arm of the lever F C. Then we have -

Pn = R m,

and from this, dividing by n (Art. 372), we have -

P = R m/n; (110.)

or, dividing by m, we have -

R= Pn/m; (111.)

which is a rule for computing the weight of R, when P and the two arms of leverage, m and n, are known. For example, let the weight represented by P be 1200 pounds, the length of the arm m be 4 feet, and that of n be 8 feet, then we have -

R = Pn/m = 1200x8/4 = 2400 pounds.

This rule, R = Pn/m, is precisely like that in Art. 397 -

bc/a = d - in which three quantities are given to find a fourth, the four constituting a set of four proportionals.

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