Elements of Algebra

CHAP. IX. OF ALGEBRA. 155 475. Let the two proportions be $6 : 4 :: 15 : 10$, and $9 : 12 :: 15 : 20$, their combination will give the proportion $6 \times 9 : 4 \times 12 :: 15 \times 15 : 10 \times 20$, or $54 : 48 :: 225 : 200$, or $9 : 8 :: 9 : 8$. 476. We shall observe, lastly, that if two products are equal, $ad = bc$, we may reciprocally convert this equality into a geometrical proportion; for we shall always have one of the factors of the first product in the same proportion to one of the factors of the second product, as the other factor of the second product is to the other factor of the first product: that is, in the present case, $a : c :: b : d$, or $a : b :: c : d$. Let $3 \times 8 = 4 \times 6$, and we may form from it this proportion, $8 : 4 :: 6 : 3$, or this, $3 : 4 :: 6 : 8$. Likewise, if $3 \times 5 = 1 \times 15$, we shall have $3 : 15 :: 1 : 5$, or $5 : 1 :: 15 : 3$, or $3 : 1 :: 15 : 5$. CHAP. IX. *Observations on the Rules of Proportion and their Utility.* 477. This theory is so useful in the common occurrences of life, that scarcely any person can do without it. There is always a proportion between prices and commodities; and when different kinds of money are the subject of exchange, the whole consists in determining their mutual relations. The examples furnished by these reflections will be very proper for illustrating the principles of proportion, and shewing their utility by the application of them. 478. If we wished to know, for example, the relation between two kinds of money; suppose an old louis d'or and a ducat: we must first know the value of those pieces when compared with others of the same kind. Thus, an old louis being, at Berlin, worth 5 rixdollars and 8 drachms, and a ducat being worth 3 rixdollars, we may reduce these two values to one denomination; either to rixdollars, which gives the proportion $1\mathrm{L} : 1\mathrm{D} :: 5\frac{1}{3}\mathrm{R} : 3\mathrm{R}$, or $:: 16 : 9$; or to drachms, in which case we have $1\mathrm{L} : 1\mathrm{D} :: 128 : 72 :: 16 : 9$; which proportions evidently give the true relation of the old louis to the ducat; for the equality of the products of the extremes and the means gives, in both cases, 9 louis *24 drachms = 1 Rixdollar* *$5\frac{1}{3}$ Rixdollars = old Louis d'or*