430. Other examples are added to facilitate the practice of this calculation.

If ducats gain at Hamburg 1 per cent. on two dollars banco; that is to say, if 50 ducats are worth, not 100, but 101 dollars banco; and if the exchange between Hamburg and Konigsberg, is 119 drachms of Poland; that is, if 1 dollar banco gives 119 Polish drachms, how many Polish florins will 1000 ducats give?

30 Polish drachms make 1 Polish florin.

431. We may abridge a little further, by writing the number, which forms the third term, above the second row; for then the product of the second row, divided by the product of the first row, will give the answer sought.

to say,

Question. Ducats of Amsterdam are brought to Leipsic, having in the former city the value of 5 flor. 4 stivers current; that is 1 ducat is worth 104 stivers, and 5 ducats are worth 26 Dutch florins. If, therefore, the agio of the bank* at Amsterdam is 5 per cent. that is, if 105 currency are equal to 100 banco, and if the exchange from Leipsic to Amsterdam, in bank money, is 331 per cent. that is, if for 100 dollars we pay at Leipsic 1331 dollars; lastly, 2 Dutch dollars making 5 Dutch florins; it is required to find how many dollars we must pay at Leipsic, according to these exchanges, for 1000 ducats?

* The difference of value between bank money and current money.

Answer, 26391 dollars, or 2639 dollars and 15 drachms.

CHAPTER IX.

Of Compound Relations.

432. COMPOUND RELATIONS are obtained, by multiplying the terms of two or more relations, the antecedents by the antecedents, and the consequents by the consequents; we say then, that the relation between those two products is compounded of the relations given. Thus, the relations a : b, c :d, e:f, give the compound relation ace: bdf.*

433. A relation continuing always the same, when we divide both its terms by the same number, in order to abridge it, we may greatly facilitate the above composition by comparing the antecedents and the consequents, for the purpose of making such reductions as we performed in the last chapter.

For example, we find the compound relation of the following given relations, thus ;

*Each of these three ratios is said to be one of the roots of the compound ratio.

Eul. Alg.

18

So that 2: 5 is the compound relation required.

434. The same operation is to be performed, when it is required to calculate generally by letters; and the most remarkable case is that, in which each antecedent is equal to the consequent of the preceding relation. If the given relations are

a: b

b:c

c: d

d: e

e: a

the compound relation is 1 : 1.

435. The utility of these principles will be perceived, when it is observed, that the relation between two square fields is compounded of the relations of the lengths and the breadths.

Let the two fields, for example, be A and B; let A have 500 feet in length by 60 feet in breadth, and let the length of B be 360 feet, and its breadth 100 feet; the relation of the lengths will be 500 360, and that of the breadths 60: 100. So that we have

Wherefore the field A is to the field B, as 5 to 6.

436. Another Example. Let the field A be 721 feet long, 88 feet broad; and let the field B be 660 feet long, and 90 feet broad; the relations will be compounded in the following manner.

437. Further, if it be required to compare two chambers with respect to the space, or contents, we observe that that relation is compounded of three relations; namely, of that of the lengths, that of the breadths, and that of the heights. Let there be, for example, the chamber A, whose length = 36 feet, breadth 16 feet, and height 14 feet, and the chamber B, whose length = 42 feet, breadth 24 feet, and height = 10 feet; we shall have these three relations;

So that the contents of the chamber A : contents of the chamber B, as 4: 5.

438. When the relations which we compound in this manner are equal, there result multiplicate relations. Namely, two equal relations give a duplicate ratio or ratio of the squares; three equal relations produce the triplicate ratio or ratio of the cubes, and so on; for example, the relations ab and ab give the compound relation a abb; wherefore we say, that the squares are in the duplicate ratio of their roots. And the ratio a: b multiplied thrice, giving the ratio a3: b3, we say that the cubes are in the triplicate ratio of their roots.

439. Geometry teaches, that two circular spaces are in the duplicate relation of their diameters; this means, that they are to each other as the squares of their diameters.

Let A be a circular space having the diameter = 45 feet, and B another circular space, whose diameter 30 feet; the first space will be, to the second, as 45 x 45 to 30 X 30; or, compounding these two equal relations,

1

Wherefore the two areas are to each other as 9 to 4.

440. It is also demonstrated, that the solid contents of spheres are in the ratio of the cubes of the diameters. Thus, the diameter of a

globe A, being 1 foot, and the diameter of a globe B, being 2 feet, the solid contents of A will be to those of B, as 13: 23; or, as 1 to 8.

If, therefore, the spheres are formed of the same substance, the sphere B will weigh 8 times as much as the sphere A.

441. It is evident, that we may, in this manner, find the weight of cannon balls, their diameters and the weight of one, being given. For example, let there be the ball A, whose diameter = 2 inches, and weight 5 pounds; and, if the weight of another ball be required, whose diameter is 8 inches, we have this proportion, 23835 to the fourth term, 320 pounds, which gives the weight of the ball B. For another ball C, whose diameter inches, we should have,

23:1535:.... Answer, 2109 lb.

15

is required, we

may always express it in integer numbers; for we have only to multiply the fractions by b d, in order to obtain the ratio ad : bc, which is equal to the other; from which results the proportion

If, therefore, a d and b c have common divisors, the ratio may be reduced to less terms. Thus,

24

· ៖ថ៍ = 15 X 36:24 × 25 = 9:10.

443. If we wished to know the ratio of the fractions and, it is

1 1
Ъ

evident, that we should have -: =b: a; which is expressed by

a

saying, that two fractions, which have unity for their numerator, are in the reciprocal, or inverse ratio of their denominators. The same may be said of two fractions, which have any common numerator; for с =b: a. But if two fractions have their denominators equal,

с

as

a b

2:2, they are in the direct ratio of the numerators; namely, as

C C

3

a: b. Thus, ÷ ÷ = ƒ ÷ ÷ 6:32:1, and &:

:

1° : V = 10 : 15, or, = 2 : 3.