Elements of Algebra

CHAP. X. OF ALGEBRA. 163 what height, a stone will pass by descending for the space of an hour, or 3600 seconds; we must say, As $1^2 : 3600^2 :: 16 : 207360000$ feet, the height required. Which being reduced is found equal to 39272 miles; and consequently nearly five times greater than the diameter of the earth. 502. It is the same with regard to the price of precious stones, which are not sold in the proportion of their weight; every body knows that their prices follow a much greater ratio. The rule for diamonds is, that the price is in the duplicate ratio of the weight; that is to say, the ratio of the prices is equal to the square of the ratio of the weights. The weight of diamonds is expressed in carats, and a carat is equivalent to 4 grains; if, therefore, a diamond of one carat is worth 10 livres, a diamond of 100 carats will be worth as many times 10 livres as the square of 100 contains 1; so that we shall have, according to the Rule of Three, As $1 : 10000 :: 10 : 100000$ liv. Ans. There is a diamond in Portugal which weighs 1680 carats; its price will be found, therefore, by making $1^2 : 1680^2 :: 10 : 28224000$ livres. 503. The posts, or mode of travelling, in France, furnish sufficient examples of compound ratios; because the price is regulated by the compound ratio of the number of horses, and the number of leagues, or posts. Thus, for example, if one horse cost 20 sous per post, it is required to find how much must be paid for 28 horses for $4\frac{1}{2}$ posts. We write first the ratio of the horses - $1 : 28$ Under this ratio we put that of the stages - $2 : 9$ And, compounding the two ratios, we have - $2 : 252$ Or, by abridging the two terms, $1 : 126 :: 1$ liv. to $126$ fr. or 42 crowns. Again, If I pay a ducat for eight horses for 3 miles, how much must I pay for thirty horses for four miles? The calculation is as follows: $8 : 30$ $3 : 4$ By compounding these two ratios, and abridging, $1 : 5 :: 1$ duc. : $5$ ducats; the sum required. 504. The same composition occurs when workmen are to be paid, since those payments generally follow the ratio M 2