Elements of Algebra

152 ELEMENTS SECT. III. CHAP. VIII. Of Geometrical Proportions. 461. Two geometrical relations are equal when their ratios are equal; and this equality of two relations is called a geometrical proportion. Thus, for example, we write $a:b=c:d$, or $a:b::c:d$, to indicate that the relation $a:b$ is equal to the relation $c:d$; but this is more simply expressed by saying $a$ is to $b$ as $c$ to $d$. The following is such a proportion, $8:4::12:6$; for the ratio of the relation $8:4$ is $\frac{8}{4}$, or $2$, and this is also the ratio of the relation $12:6$. 462. So that $a:b::c:d$ being a geometrical proportion, the ratio must be the same on both sides, consequently $\frac{a}{b} = \frac{c}{d}$; and, reciprocally, if the fractions $\frac{a}{b} = \frac{c}{d}$, we have $a:b::c:d$. 463. A geometrical proportion consists therefore of four terms, such, that the first divided by the second gives the same quotient as the third divided by the fourth; and hence we deduce an important property, common to all geometrical proportions, which is, that the product of the first and the last term is always equal to the product of the second and third; or, more simply, that the product of the extremes is equal to the product of the means. 464. In order to demonstrate this property, let us take the geometrical proportion $a:b::c:d$, so that $\frac{a}{b} = \frac{c}{d}$. Now, if we multiply both these fractions by $b$, we obtain $a = \frac{bc}{d}$, and multiplying both sides farther by $d$, we have $ad = bc$; but $ad$ is the product of the extreme terms, and $bc$ is that of the means, which two products are found to be equal. 465. Reciprocally, if the four numbers, $a, b, c, d$, are such, that the product of the two extremes, $a$ and $d$, is equal to the product of the two means, $b$ and $c$, we are certain that they form a geometrical proportion: for, since $ad = bc$, we