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

In Fig. 278, let A BCD, a rectangle, have its sides A B and A C divided into equal parts. Then the area of the figure will be obtained by multiplying one side by the other, or putting a for the side A B, and b for the side A C, then the area will be a x b, or a b. This will be the correct area of the figure, whatever the length of the sides may be. If, as shown, the area be divided into 4 x 7 = 28 equal rectangles, then a would equal 7, and b equal 4, and a b = 7 x 4 = 28, the area. If A B equal 28 and A C equal 16, then will a - 28, and b = 16, and a b - 28 x 16 = 448, the area. .

Fig. 278.

4-09. - Graphical Multiplication: Three Factors. - Let

A B CD E FG(Fig. 279) represent a rectangular solid which may be supposed divided into numerous small cubes as shown. Now, if a be put for the edge A B, b for the edge A C, and c for the edge CD, then the cubical solidity of the whole figure will be represented by axbxc = abc. If the edge A B measures 6, the edge A C 3, and the edge CD 4, then abc = 6x3x4= 72 = the cubic contents of the figure, or the number of small cubes contained in it.

Fig. 279.

410. - Graphic Representation: Two and Three Factors. - Figs. 278 and 279 serve to illustrate the algebraic expressions a b and a b c. In the former it is shown that the multiplication of two lines produces a rectangular surface, or that if a and b represent lines, then a b may represent a rectangular surface (Fig. 278) having sides respectively equal to a and b. And so if a, b, and c represent three several lines, then ab c may represent a rectangular solid (Fig. 279) having edges respectively equal to a, b, and c.

Fig. 280.

Continue to: