This rule may generally be so abridged by cancelling equal quantities on both sides, and abbreviating commensurables, that the whole operation may be performed with very little trouble, and it may be proved by as many statings in the Single Rule of Three, as the nature of the question may require.

CASE I.

When it is required to find how many of the first sort of coin, weight, or measure, mentioned in the question, are equal to a given quantity of the last.

RULE.

Place the numbers alternately, that is, the antecedents at the left hand, and the consequents at the right, and let the last number stand on the left hand; then multiply the, left hand column continually for a dividend, and the right hand for a divisor, and the quotient will be the answer.

EXAMPLES.

1. Suppose 100 yards of America=100 yards of England, and 100 yards of England=50 canes of Thoulouse, and 100 canes of Thoulouse 160 ells of Geneva, and 100 ells of Geneva=200 ells of Hamburgh: How many yards of America are equal to 379 ells of Hamburgh? Antecedents. 100 of America

100 of England

Consequents.

= 100 of England.

- 50 of Thoulouse.

100 of Thoulouse = 160 of Geneva. 100 of Geneva =200 of Hamburgh. 379 of Hamburgh?

379X5

Abriged.
Ant. Con.

5 8

379

Therefore, 8236 yds. of America=379 ells of Hamburgh.

ILLUSTRATION.

The two 100s of both sides cancel each other. Let the last cyphers of the next three antecedents and consequents be cancelled, which is dividing by 10. Then divide the second antecedent and consequent by 5, and the quotients will be 2 on the side of the antecedents, and 1 on the side of the consequents; then 2 will measure the third antecedent and consequent, and the quotients will be 5 and S. 10 will measure the 4th antecedent and consequent, and the quotients will be 1 and 2. Now, there being 2 left on each side, they cancel each other, and as there is no farther room for abridging by reason of the odd number 379, the operation is finished, and the answer found, as before.

2. If 20 at Boston make 23 at Antwerp, and 155 at Antwerp make 180 at Leghorn: How many at Boston are equal to 144 at Legborn? Ans. 1071ib.

3. If 12 at Boston make 101b at Amsterdam, 10 at Amsterdam 12b at Paris: How many pounds at Boston are equal to 8015 at Paris? Ans. 801.

4. If 140 braces at Venice be equal to 150 braces at Leghorn, and 7 braces at Leghorn be equal to 4 American yards: How many Venetian braces are equal to 32 American yards?

Ans. 52.

5. If 40 at Newburyport make 36 at Amsterdam, and 901b at Amsterdam make 116 at Dantzick: How many pounds at Newbu ryport are equal to 260 at Dantzick? Ans. 2245.

CASE II.

When it is required to find how many of the last sort of coin, weight or measure, mentioned in the question, are equal to a given quantity of the first.

RULE.

Place the numbers alternately, beginning at the left hand, and let the last number stand on the right hand; then multiply the first row for a divisor, and the second for a dividend.

EXAMPLES.

1. Suppose 100 yards of America=100 yards of England, and 100 yards of England 50 canes of Thoulouse, and 100 canes of Thoulouse 160 ells of Geneva, and 100 ells of Geneva 200 ells of Hamburgh: How many ells of Hamburgh are equal to 2367 yards of America?

The learner will readily

100 Gen.

This needs no further illustration.

see, that this case being the reverse of the former, they are proofs to each other.

2. If 20 at Boston make 231b at Antwerp, and 155 at Antwerp make 180 at Leghorn: How many at Leghorn are equal to 144 at Boston ? Ans, 1445. 3. If 12 at Boston make 10 at Amsterdam, and 100 at Amsterdam 120 at Paris: How many at Paris are equal to 80 at Boston? Ans. 80.

4. If 140 braces at Venice be equal to 150 braces at Leghorn, and 7 braces at Leghorn be equal to 4 American yards: How many American yards are equal to 52,4 Venetian braces?

Ans. 32 yards.

5. If 40% at Newburyport make 36 at Amsterdam, and 90 at Amsterdam make 116 at Dantzick: How many pounds at Dantzick are equal to 244 at Newburyport? Ans 22313.

ARBITRATION OF EXCHANGES.

By this term is understood how to choose, or determine the best way of remitting money from abroad with advantage; which is performed by conjoined proportion: Thus,

1. Suppose à merchant has effects at Amsterdam to the amount of $3530, which he can remit by way of Lisbon at 840 rees per dollar, and thence to Boston, at 8s. 1d. per milree (or 1000 rees :) Or, by way of Nantz, at 53 livres per dollar, and thence to Boston at 6s. 8d. per crown; It is required to arbitrate these exchanges, that is, to choose that which is most advantageous?

1 dollar at Amsterdam = 840 rees at Lisbon.

1000 rees at Lisbon =

840X97X3530

1000X1

97d. at Boston.

3530 dollars at Amsterdam.

= £1198 8s. 84d. by way of Lisbon.

i dollar at Amsterdam = 53 livres at Nantz.

livres at Nantz

= 80 pence at Boston.

5 × 80 × 3530

1X6

3530 dollars at Amsterdam.

£1059 by way of Nantz.

Here it may be observed that the difference is £139 83. 8d. in favour of remitting by way of Lisbon rather than by Nantz, which depends on the course of exchange, at that time; but the course may vary so, that, in a short time by way of Nantz may be better; hence appears the necessity and advantage of an extensive correspondence, to acquire a thorough knowledge in the courses of exchange, to make this kind of remittance.

2. A merchant in England can draw directly for 1000 piastres in Leghorn at 50d. sterling per piastre ; but he chooses to remit the sum to Cadiz at 19 piastres for 7000 maravedies; thence to Amsterdam at 189d. Flemish for 680 maravedies; and thence to Liverpool at 9d. Flemish for 5d. sterling: what is gained by this circular remittance, and what is the value of a piastre to him?

Ans. Gain £28 14s. sterling nearly. Value of a piastre 56d. 3.55qr. sterling. 3. A merchant in New York orders £500 sterling, due him at London at 54d. sterling per dollar, to be sent by the following circuit; to Hamburgh at 15 marks banco per pound sterling; thence to Copenhagen at 100 marks banco for 33 rix dollars; thence to Bourdeaux at one rix dollar for 6 francs; thence to Lisbon at 125 francs for 18 milrees; and thence to New York at $14 per milree : did he gain or lose by this circular remittance, and what was the arbitrated value of a dollar by this remittance?

Ans. He gained.

Value of a dollar was 69d. sterling nearly.

FELLOWSHIP.

THE Rules of Fellowship are those by which the accompts of several merchants or other persons, trading in partnership, are so adjusted, that each may have his share of the gain, or sustain his

share of the loss, in proportion to his share of the joint stock, together with the time of its continuance in trade.

SINGLE FELLOWSHIP

Is, when the stocks are employed for any certain equal time.

RULE.*

As the whole stock is to the whole gain or loss, so is each man's particular stock to his particular share of the gain, or loss.

PROOF. Add all the particular shares of the gain or loss together, and, if it be right, the sum will be equal to the whole gain or loss.

EXAMPLES.

1. Divide the number 360 into four parts, which shall be to each other, as 3, 4, 5 and 6.

2. A, B, C, and D companied; A put in £145; B, £219; C, £378, and D, £417, with which they gained £569: What was the share of each ?

£ s. d.

159 1958 A's share.

145: 71 3 8

375 B's ditto.

219: 107 10 311

Whole stock. Gain. As 145+-219+378+417: 569 ::

552 C's ditto.

1159

3 D's ditto.

3. A, B, C, and D are concerned in a joint stock of $1000; of which A's part is $150; B's $250; C's $275, and D's $325. Upon the adjustment of their accompts, they have lost $337 50c. What is the loss of each? Ans. A's loss 850 624c. B's $84 374c. C's $92 814c. and D's $109 68fc.

4. A and B companied; A put in £15, and took of the gain; What did B put in? 5-3-2. Then, As 3: 45: 2: 30 Ans. 5. A, B and C freighted a ship with 68900 feet of boards: A put in 16520 feet; B 28750; and C the rest; but in a storm, the captain threw overboard 26450 feet: How much must each sustain of the loss? Ans. A, 63413 feet. B, 110363 and C, 90711 do. 6. A gentleman died, leaving three sons and a daughter, to whom he bequeathed his estate in the following manner : To the eldest son, he gave 312 moidores, to the second, 312 guineas, to the third,

*That their gain or loss, in this rule, is in proportion to their stocks is evident: For, as the times, in which the stocks are in trade, are equal, if I put in of the whole stock, I ought to have of the gain: If my part of the stock be 4, my share of the gain or loss ought to be 4 also. And generally the same rafio that the whole stock has to the whole gain or loss, must each person's particular stock have to his respective gain or loss.

312 pistoles, and to the daughter, 312 dollars; but when his debts were paid, there were but 312 half joes left: What must each have in proportion to the legacies which had been bequeathed them? Ans. 1st son £293 0s. 3d.-2d. son £227 17s. 103d.-3d. son £ 179 1s. 2 d. and the daughter £48 16s. 81d.

7. A ship, worth $3000, being lost at sea, of which belonged to A, to B, and the rest to C: What loss will each sustain, supposing $450 to have been insured upon her?

Ans. A's loss $312 50c.

8. A and B venturing equal sums of money, cleared by joint trade $140: By agreement, as A executed the business, he was to have 8 per cent. and B was to have 5 per cent. : What was A allowed for his trouble?

As 8+5 140: 8: 86,3 And, as 8+5: 140 :: 5 : 5311.
Ans. $32 30c. 7.

9. A bankrupt is indebted to A £120, to B £230, to C £340. and to D £450, and his whole estate amounts only to £560: How must it be divided among the creditors?

Ans. A, £58 183. 114d. B, £112 19s. 7fd. C, £167 Os. 4d. and D. £221 1s. Old.

10. A, B, and C put their money into a joint stock; A put in $40 ; B and C together $170: They gained $126, of which B took $42; What did A and C gain, and B and C put in respectively?

Ans. $24 A's gain, $70 B's stock, $100 C's stock, $60 C's gain. 11. A, B, and C companied; A put in £40; B 60, and C a sum unknown: They gained £72; of which C took £32 for his share ; What did A and B gain, and C put in ?

Ans. £16 A's gain, £24 B's gain, and £80 C's stock. 12. A, B, and C put in $720, and gained $540, of which, so oftten as A took up $3, B took 5, and C 7: What did each put in and gain?

Instead of the above rule, you may find a common multiplier to multiply the proportions by, or multiplicand to be multiplied by the given proportions, thus, 15)720(43 multiplicand to find the stocks. And 15)540(36 multiplicand to find the gains.

$

48x3=144 A's stock. 48X5 240 B's ditto. 48x7=336 C's ditto.

And

$

36x3=108 A's gain.

36x5=180 B's ditto.

36×7 252 C's ditto. as before. 13. A, B, C, and D companied; and gained a sum of money of which A, B and C took £120, B, C and D, £180. C, D and A, £160, and D, A and B, £140: What distinct gain had each ?

The sum of these 4 numbers is £600, and as each man's money is named 3 times, therefore, viz. £200 is the whole gain.Therefore £200-£ 120 A's B's and C's gain £ 80 D's gain ;And £200-£130 B's, C's and D's gain= £20 A's gain.-£ 200— £160 C's, D's, and A's gair €40 B's gain.-And £200-£140 D's, A's and B's gain £ 60 C's gain.