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

Let

A B C D (Fig. 280) be a rectangular surface, and B E D F another rectangular surface, adjoining the first. The area of the whole figure is evidently equal to -

(AB + BE)xAC.

The area is also equal to -

or, since A C = B D, the area equals -

or, if symbols be put to represent the lines; say a for A B, b for B E, and c for A C, then the two representatives of the area, as above shown, become: The first -

(a + b) x c = area; and the last -

(a x c) + (b xc) = area. Hence we have -

(a + b) c = a c + b c.

This result exemplifies the algebraic multiplication of a binomial, which is performed thus: Let a + b be multiplied bye.

The problem is stated thus:

(a+ b) c.

To perform the multiplication indicated we proceed thus:

a + b

c____

a c + b c

multiplying each of the factors of the multiplicand separately and annexing them by the sign for addition. Putting the two together, or showing the problem and its answer in an equation, we have -

(a + b) c = a c + b c,

producing the same result, above shown, as derived from the graphic representation.

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