ARITHMETIC.

241

2. Compound Arbitration.

(71.) Compound arbitration respects the cases in winch the ex-changes between three, four, or more places are concerned.

A person who knows at what rate he can draw or remit directly,and also hath advice of the course of exchange in foreign places,may trace out a path for circulating his money, through more orfewer of such places, and also in such order, as to make a benefitof his skill and credit: and herein lies the great art of such nego-tiations.

But to determine in what order, and through how many placesto circulate a bill, no general rule can be given, as it depends en-tirely upon a per-on’s judgment, and a close attention to the re-sults'of former cases of the like kind. The following rules, how-ever, may assist in determining, whether a direct or an assignedcircular draft be preferable.

Rule l. Distinguish the given rales or prices in the circularcourse, into antecedents and consequents; and place the antecedentsin one column, and the consequents in another, on the right,fronting one another, by way of equation. And in this distribu-tion into antecedents and consequents, each consequent must beof the same kind with the last consequent, which must be the sumwhose value in exchange is required.

2. Multiply the antecedents continually for a divisor, and theconsequents tor a dividend; .and the quotient of the productswill be the value of the sum required by such exchange.

3. Then compute its value by the direct exchange, or by anyother circular exchange, and by comparing the values togetheryou will perceive the most advantageous method.

Ex. If London would remit 1000/. sterling to Spain , the directexchange being 42^1. per piastre of 272 mervadies ; it is requiredwhether will he more profitable, the direct remittance', or by re-mitting first to Holland, at 35s. per /.. ; thence to France at 58d.per crown ; thence to Venice at 100 crowns per 60 ducats; andthence to Spain at 3G0 mervadies per ducal.

Antecedents.

1/. Sterling58</. Flemish100 crowns1 ducat272 mervadiesHow many piastres420 x 60 x 3(50 x 1000

Consequents.

= 358 or420ff. Flemish.

= 1 crown.

= 60 ducats.

= 360 mervadies.

= 1 piastre.

= 1000 /.

210x30x45x 10 2835000

Then

5SX 100X272 20X17 403

5750J)-§ piastres = the value of 1000/. by the circular ex-change.

But ■ 1 piastre : : 1000/. or 240000c/: 56474 7 piastres,

tlie value by the direct exchange.

Thus it is plain that the circular exchange is the most advan-tageous, as it produces 1034J piastres more than the other.

Alligation.

(72.) Alligation is the method of mixing together several sim-ples of different qualities, so that tlie composition may be of amiddle quality : and it is commonly distinguished into two prin-cipal cases, denominated alligation, medial, and alligation al-ternate.

Case 1. Alligation Medial.

(73.) Alligation Medial is the method of finding tlie rate of thecompound, trom having the rates and quantities of the severalsimples given.

Rule. Multiply each quantity by its rate; then divide the sumof the products hv the sum of the quantities, or the whole com-position ; and the quotient will be the rate of the compound re-quired.

Ex amp le. A composition being made of 51b. of tea at 7s. perlb. t)lb. at 8s. 6</. per lb. and 141b. at 5 s. lQit. per lb. what is alb. of it worth ?

.r. d. /,. s. d.

7 — x 5 — 1 15 — •

R 6x9=3 16 65 10 x14 = 4 1 8

28)9 13 2(—6s. 10(1. 3jqr. per lb; Alls.vol. i.— no. 16.

Case If. Alligation Alternate.

(74.) Alligation alternate is the method of finding what quan-tity of each ot the simples, whose rates are given, "ill composea mixture of a given rate; so that it is the reverse of alligationmedial, and may therefore be proved by it.

Rule 1. Write the rates of the simples in a column under eachother. 2. Connect, or link with a continued line, the rate of eachsimple which is less than that of the compound, with one or anynumber of those which are greater than the compound ; and eachgreater rate with one or any number of the less. 3. Write thedifference between the mixture rate and that of each of the sim-ples, opposite the rates with which these are linked. 4. Then ifonly one difference stand against any rate, it will be the quantitybelonging to that rate ; but if there be several, their sum will bethe quantity.

U appears from the above rule, that many of the questions ofthis case will admit of various answers each ; but from an alge-braic. process it appears that they will all have infinite varieties ofanswers; alter one or more answers are found by the rule, a greatnumber more are found by increasing or decreasing the quanti-ties in any proportion, or by only increasing or decreasing any.one or more single pairs of yoke-fellows, in any proportion, andleaving the other rates as they are ; but that answer is commonlydesired which gives the rates in the least integer numbers, andthose the nearest to each other.

Example, flow much corn at 2s. 6d. at 3s. 8d. at 4s. and at4s. 8d. per bushel,, must be mixed together, that the compoundmay be worth 3s. lOd. per bushel?

Rates.

2410=12 or 2 at 2 6-| 442410=12 or 2 at 3 8

!_L48( 1 164 2 = 18 or 3 at 4 O'

!-16 J 16-j- 2=18 or 3 at 4 8

Sometimes one or more of the ingredients, and sometimes the -whole composition is limited to a certain quantity ; which maybe divided into the three following cases or limitations.

Limitation 1. When the whole composition is limited to acertain quantity, and that is not found from the method of link-ing, and taking the differences; augment or diminish the quantityot each ingredient, in tlie same* proportion as the given quantityis greater or less than the total quantity found from the linking,by saying, As the total quantity so found, is to the given quan-tity, so is the quantity of each ingredient, found by linking, to-the required quantity of each.

Limitation H. Y\ hen one of the ingredients is limited to a ter-rain quantity, and that quantity is not found by tlie method oflinking; either augment, or diminish the quantities of all therest, in tlie same proportion as tlie given quantity is greater orless than tlie quantity of the limited simple found by hiikimr, In-stating as in the lirst limitation ; or, only augment, or diminish intlie above proportion, that purl of the quantity of the ingredientswitli which the limited one is linked, which is tlie difference oftlie mixture rate and the rate of the limited simple, ami add theresultingquantity to tlie other parts, instead of the said difference ;keeping tlie quantities ot the other simples unaltered.

Limitation 111. if more than one of tlie simples lie limited,find, by Case I, what will be the rate of a mixture made of thegiven quantities of the limited sinijiles only; then consider thisas tlie rate oi a limited simple, whose quantity is the sum of tliefirst given limited simples ; from which, and the rates of the li-mited simples, by the second limitation, calculate tlie quantity of.each.

Single Position.

(75.) This rule is also called false position, or false sup-position, because it makes a supposition of. false numbers, as ifthey were the true ones, and by their means, discovers tlie truenumbers sought. '1 he single rule uses only one supposition, buttlie double rule two; whence their names.

To the rule of position belong such questions as cannot besolved by the direct process of any of the former rules; ami inwhich the required number or numbers do not ascend above tlielust power; such, for example, as most ot the questions usuallybrought to exercise tlie reduction of simple equations in algebra.But it will not tiring out true answers, when the numbers sougiitascend above tlie first power; for then the results are not propor-

3 Q tional.