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

-Each of the four quantities in the aforesaid equation is termed a factor. Comparing the equation of the last article with that of Art. 43, it appears that the two are alike excepting that the factor 87 has been transferred from one member of the equation to the other, and that, whereas it was before a divisor, it has now become a multiplier. From this we learn that a factor may be transferred from one member of an equation to the other, provided that in the transfer its relative position to the horizontal line above or below it be also changed; that is, if, before the transfer, it be below the line, it must be put above the line in the other member; or, if above the line, it must be put below, in the other member. For example: in the equation of the last article let the factor 9 be removed to the second member of the equation. It stands as a divisor in the first member; therefore, by the rule, it must appear as a multiplier after the transfer; or -

45 x 87 = 9 x 435;

which is read, 45 times 87 equals 9 times 435. By actually performing the operations here indicated, we find that each member gives the same product, 3915; thus proving that the equality of the two members was not interfered with by the transfer.

373. - Equality of Products: Means and Extremes. - In

Art. 366, the four factors are put in the usual form of four proportionals. A comparison of these with the four factors as they appear in the equation in the last article, shows that the first member contains the second and third of the four proportionals, and the second member contains the first and the fourth; or, the first contains what are termed the means, and the second, the extremes'. From this we learn that in any set of four proportionals, the product of the means equals the product of the extremes. As for example,

3/2= 1 1/2; so, also, 6/4 = 1 1/2, an equality of ratios: hence the four factors, 2, 3, 4, 6, are four proportionals, and may be put thus:

Extreme, mean, mean, extreme. 2 : 3::4: 6

and, as above stated, the product of the means (3x4=) 12, equals the product of the extremes (2x6=) 12.

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