419. Let the two proportions be, for example, 6: 415: 10 and 9: 12 = 15: 20, their combination will give the proportion 6 x 9: 4 x 12 = 15 × 15: 10 x 20,

420. 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:cb: d, or a : bc:d. Let 3 × 8 = 4 × 6, and we may form from it this proportion, 8: 46: 3, or this, 3: 4= 68. Likewise, if 3 x 51 x 15, we shall have

3:15=1:5, or 5: 1= 15: 3, or 3 : 1 = 15; 5.

CHAPTER VIII.

Observations on the Rules of Proportion and their utility.

421. THIS theory is so useful in the occurrences of common 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.

422. 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 to others of the same kind. Thus, an old louis being, at Berlin, worth 5 rix dollars* and 8 drachms, and a ducat being worth 3 rix dollars, we may reduce these two values to one denomination; either to rix dollars, which gives the proportion 1 L: 1 D

* The rix dollar of Germany is valued at 92 cents 6 mills, and a drachm is one twenty-fourth part of a rix dollar.

=5R: 3 R, or=16:9; or to drachms, in which case we have 1 L: 1 D=128: 72 16: 9. These 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, 9 louis 16 ducats; and, by means of this comparison, we may change any sum of old louis into ducats, and vice versa. Suppose it were required to tell how many ducats there are in 1000 old louis, we have this rule of three. If 9 louis are equal to 16 ducats, what are 1000 louis equal to? The answer will be 17777 ducats.

If, on the contrary, it were required to find how many old louis d'or there are in 1000 ducats, we bave the following proportion. If 16 ducats are equal to 9 louis; what are 1000 ducats equal to? Answer, 5621 old louis d'or.

423. Here, (at Petersburg,) the value of the ducat varies, and depends on the course of exchange. This course determines the value of the ruble in stivers, or Dutch half-pence, 105 of which make a ducat.

So that when the exchange is at 45 stivers, we have this proportion, 1 ruble : 1 ducat=45: 105 = 3:7; and hence this equality, 7 rubles 3 ducats.

By this we shall find the value of a ducat in rubles; for 3 ducats: 7 rubles 1 ducat: ..... Answer, 24 rubles.

=

If the exchange were at 50 stivers, we should have this proportion, 1 ruble 1 ducat 50: 105 10: 21, which would give 21 rubles 10 ducats; and we should have 1 ducat = =276 rubles. Lastly, when the exchange is at 44 stivers, we have 1 ruble 1 ducat 44 : 105, and consequently 1 ducat = 212 rubles= 2 rubles 387 copecks.*

424. It follows from this, that we may also compare different kinds of money, which we have frequently occasion to do in bills of exchange. Suppose, for example, that a person of this place has 1000 rubles to be paid to him at Berlin, and that he wishes to known the value of this sum in ducats at Berlin.

The exchange is here at 471, that is to say, one ruble makes 471 stivers. In Holland, 20 stivers make a florin; 21 Dutch florins make a Dutch dollar. Further, the exchange of Holland part of a ruble, as is easily deduced from the above.

A copeck is

with Berlin is at 142, that is to say, for 100 Dutch dollars, 142 dollars are paid at Berlin. Lastly, the ducat is worth 3 dollars at Berlin.

425. To resolve the questions proposed, let us proceed step by step. Beginning therefore with the stivers, since 1 ruble = 471 stivers, or 2 rubles = 95 stivers, we shall have 2 rubles : 95 stivers = 1000:.... Answer, 47500 stivers. If we go further and say 20 stivers: 1 florin = 47500 stivers: .... we shall have 2375 florins. Further, 2 florins 1 Dutch dollar, or 5 florins = 2 Dutch dollars; we shall therefore have 5 florins: 2 Dutch dollars 2375 florins:.... Answer, 950 Ditch dollars.

=

Then taking the dollars of Berlin, according to the exchange at 142, we shall have 100 Dutch dollars: 142 dollars = 950 : the fourth term, 1349 dollars of Berlin. Let us, lastly, pass to the ducats, and say 5 dollars: 1 ducat = 1349 dollars: ... Answer, 449 ducats.

426. In order to render these calculations still more complete, let us suppose that the Berlin banker refuses, under some pretext or other, to pay this sum, and to accept the bill of exchange without five per cent. discount; that is, paying only 100 instead of 105. In that case, we must make use of the following proportion; 105: 100 = 4493: a fourth term, which is 4281 ducats.

427. We have shewn that six operations are necessary, in making use of the Rule of Three; but we can greatly abridge those calculations, by a rule, which is called the Rule of Reduction. To explain this rule, we shall first consider the two antecedents of each of the six operations.

I. 2 rubles

II. 20 stivers

III. 5 Dutch flor.
IV. 100 Dutch doll.

V. 3 dollars

VI. 105 ducats

95 stivers.

: 1 Dutch flor.
: 2 Dutch doll.

: 142 dollars.

: 1 Ducat.

: 100 ducats.

If we now look over the preceding calculations, we shall observe, that we have always multiplied the given sum by the second terms, and that we have divided the products by the first; it is evident therefore, that we shall arrive at the same Eul. Alg.

18

results, by multiplying, at once, the sum proposed by the product of all the second terms, and dividing by the product of all the first terms. Or, which amounts to the same thing, that we have only to make the following proportion; as the product of all the first terms is to the product of all the second terms, so is the given number of rubles to the number of ducats payable at Berlin.

428. This calculation is abridged still more, when amongst the first terms some are found that have common divisors with some of the second terms; for, in this case, we destroy those terms, and substitute the quotient arising from the division by that common divisor. The preceding example will, in this manner, assume the following form.*

429. The method, which must be observed, in using the rule of reduction, is this; we begin with the kind of money in question, and compare it with another, which is to begin the next relation, in which we compare this second kind with a third, and so on. Each relation, therefore, begins with the same kind, as the preceding relation ended with. This operation is continued, till we arrive at the kind of money which the answer requires; and, at the end, we reckon the fractional remainders.

* Divide the 1st and 9th by 2, the and 12th by 20, the 5th and 12th (which is now 5) by 5, also the ad and 11th by 5.

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.
doll. B°. 1000 duc.

Ducat 1 ::

XXX,50 : 101 doll. B°.

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.

Question. Ducats of Amsterdam are brought to Leipsick, having in the former city the value of 5 flor. 4 stivers current; that is to say, 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 Leipsick to Amsterdam, in bank money, is 33 per cent. that is, if for 100 dollars we pay at Leipsick 1334 dollars; lastly, 2 Dutch dollars making 5 Dutch florins; it is required to find how many dollars we must pay at Leipsick, according to these exchanges, for 1000 ducats?

The difference of value between bank money and current money.