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

In previous articles the signs in multiplication have been given to products in accordance with this rule, namely: Like signs give plus; unlike signs, minus. This rule may be illustrated graphically, thus: In the rectangular Fig. 283, let it be required to show the area of the rectangle AG C H, in terms of the several parts of the whole figure. Thus the area of A GE J equal A BEF- GBJF and the area of EJCH equals EFCD - J F H D. And the areas of AGEJ + EJCH equals the area of A G C H. Therefore the sum of the two former expressions equals A G CH. Thus A BEF - GBJF+EFCD-JFHD = AGCH. Let the several lines now be represented by algebraic symbols; for example, let AB = EF=a; let GB = JF - b; let AE=GJ=c; and EC - JH = d, and let these symbols be substituted for the lines they represent, thus A B E F - GBJF+EFCD - JFHD = AGCH.

ac - bc + ad - bd=(a - b) x (c + d).

An inspection of the figure shows this to be a correct result. It will now be shown that an algebraical multiplication of. the two binomials, allotting the signs in accordance with the rule given, will produce a like result. For example -

a - b c + d

ac - bc + ad - bd.

Fig. 284.

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