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OF MUSICAL PROPORTION.

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THERE is also a third kind of proportion, called Musical, which being but of little or no common use, a very

short account of it may here suffice.

Musical Proportion is when, of three numbers, the first has the same proportion to the third, as the difference between the first and second, has to the difference between the second and third.

As in these three, 6, 8, 12;

where 6 : 12 :: 8 6 : 12 – 8,

that is 6 : 12 :: 2 : 4. When four numbers are in musical proportion; then the first has the same ratio to the fourth, as the difference between the first and second has to the difference between the third and fourth,

As in these, 6, 8, 12, 18;

wliere 6 : 18 :: 8 6 : 18 12,

that is 6 : 18 :: 2 : 6. When numbers are in musical progression, their reciprocals are in arithmetical progression ; and the converse, that is, when numbers are in arithmetical progression, their reciprocals are in musical progression.

So in these musicals 6, 8, 12, their reciprocals ) , l'I, are in arithmetical progression; for + t = =; and [ + = = ; that is, the sum of the extremes is equal to double the mean, which is the property of arithmeticals.

The method of finding out numbers in musical proportion is best expressed by letters in Algebra.

FELLOWSHIP, OR PARTNERSHIP.

FELLOWSHIP is a rule, by which any sum or quantity may be divided into any number of parts, which shall be in any given proportion to one another. By this rule are adjusted the gains or loss or charges of partners in company; or the effects of bankrupts, or legacies in case of a deficiency of assets or effects; or the shares of prizes; or the numbers of men to form certain detachments; or the division of waste lands among a number of proprietors.

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Fellowship is either Single or Double. It is Single, when the shares or portions are to be proportional each to one single given number only; as when the stocks of partners are all employed for the same time : And Double, when each portion is to be proportional to two or more numbers ; as when the stocks of partners are employed for different times.

SINGLE FELLOWSHIP.

GENERAL RULE.

Add together the numbers that denote the proportion of the shares. Then say,

As the sum of the said proportional numbers,
Is to the whole sum to be parted or divided,
So is each several proportional number,

To the corresponding share or part.
Or, 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. TO PROVE THE WORK. Add all the shares or parts together, and the sum will be equal to the whole number to be shared, when the work is right.

EXAMPLES.

1. To divide the number 240 into three such parts, as shall be in proportion to each other as the three numbers 1, 2 and 3.

Here 1 + 2 + 3 = 6, the sum of the numbers.
Then, as 6 : 240 :: 1 : 40 the 1st part,

and as 6 : 240 :: 2 : 80 the 3d part,
also as 6 : 240 :: 3 : 120 the 3d part,

Sum of all 240, the proof.

Ex.2. Three persons, A, B, C, freighted a ship with 340 tuns of wine; of which, A loaded 110 tuns, B 97, and c the rest : in a storm the seamen were obliged to throw overboard 85 tuns; how much must each person sustain of the loss ?

Here 110 + 97 = 207 tuns, loaded by A and B;
theref. 340 207 = 133 tuns, loaded by c.
Hence, as 340 : 85 :: 110

4 : 1 :: 110 : 27 tuns = A's loss;
4 :

97 : 24 tuns = B's loss;
4 : 1 :: 133 : 33% tuns = c's loss;

Sum 85 tuns, the proof.

or as

1 ::

and as also as

F

G

3. Two merchants, c and D, made a stock of 1201; of which c contributed 751, and D the rest : by trading they gained 301; what must each have of it?

Ans. c 181 15s, and D 111 5s. 4. Three merchants, E, F, G, make a stock of 700), of which e contributed 1231, F 3581, and G the rest : by trading they gain 1251 10s; what must each have of it?

Ans. E must have 221 1s Od 2359.

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39 5 3 175 5. A General imposing a contribution * of 7001 on four villages, to be paid in proportion to the number of inhabitants contained in each; the 1st containing 250, the 2d 350, the 3d 400, and the 4th 500 persons; what part must each village pay?

Ans. the 1st to pay 1161 13s 4d. the 2d

163 6 8 the 3d

186 13 4 the 4th

233 6 8 6. A piece of ground, consisting of 37 ac 2 ro 14 ps, is to be divided among three persons, L, M, and n, in proportion to their estates: now if L's estate be worth 5001 a year, M's 3201, and n's 751; what quantity of land must each one have ? Ans. I must have 20 ac 3 ro 391*. ps.

13 1

3 O 7. A person is indebted to o 571 15s, to p 1081 3s 8d, to Q 221 10d, and to R 731; but at his decease, his effects

M

304

N

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* Contribution is a tax paid by provinces, towns, villages, &c. to excuse them from being plundered. It is paid in provisions or in money, and sometimes in both.

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are found to be worth no more than 170/ 145; how must it be divided among his creditors?

Ans, o must have 37/ 15s 5d 29.
P

10 15 2 2
14 8 4 Oranje

R

47 14 11 20

X

Ex. 8. A ship, worth 9001, being entirely lost, of which belonged to s, io T, and the rest to v; what loss will each sustain, supposing 540/ of her were insured?

Ans. $ will lose 45), 1 901, and v 225), 9. Four persons, w, x, y, and z, spent among them 25s, and agree that w shall pay { of it, x }, y, and z}; that is, their shares are to be in proportion as į }, }, and : what are their shares ? Ans. w must pay 9s 8d 349.

6

5 34: 4 10 174

3 10 3 10. A detachment, consisting of 5 companies, being sent into a garrison, in which the duty required 76 men a day; what number of men must be furnished by each company, in proportion to their strength; the 1st consisting of 54 men, the 2d of 51 men, the 3d of 48 men, the 4th of 39, and the 5th of 36 men ? Ans. The 1st must furnish 18, the 2d 17, the 3d 16, the

4th 13, and the 5th 12 men*,

Y

z

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DOUBLE FELLOWSHIP, as has been said, is concerned in cases in which th• stocks of partners are employed or continued for different times.

* Questions of this nature frequently occurring in military service, General Haviland, an officer of great merit, contrived an ingenious instrument, for more expeditiously resolving them; which is distinguished by the name of the inventor, being called a Haviland.

RULE.

Rule*. -Multiply each person's stock by the time of its continuance; then divide the quantity, as in Single Fellowship, into shares, in proportion to these products, by saying,

As the total sum of all the said products,
Is to the whole gain or loss, or quantity to be parted,
So is each particular product,
To the correspondent share of the gain or loss.

EXAMPLES.

1. A had in company 501 for 4 months, and B had 601 for 5 months; at the end of which time they find 241 gained ; how must it be divided between them?

Here 50

60 4 5

200 + 300 = 500

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Then, as 500 : 24 :: 200 : 9} = 91 12s = A's share.

and as 500 : 24 :: 300 : 14} = 14 8 = B's share. 2. c and d hold a piece of ground in common, for which they are to pay

541. c put in 23 horses for 27 days, and D 21 horses for 39 days; how much ought each man to pay

of the rent?

Ans. c must pay 23l 5s 9d.

D must pay 30 14 3 3. Three persons, E, F, G, hold a pasture in common, for which they are to pay 301 per annum ; into which e put 7 oxen for 3 months, F put 9 oxen for 5 months, and G put in 4 oxen for 12 months; how much must each person pay of the rent?

Ans. E must pay 5l 10$ 6d 1f-9.

11 16 10 0,45 G

12 12 7 20. 4. A ship's company take a prize of 10001, which they agree to divide among them according to their pay and the time they have been on board: now the officers and midshipmen have been on board 6 months, and the sailors 3 months ;

F

as

* The proof of this rule is as follows: When the times are equal, the shares of the gain or loss are evidently as the stocks, in Single Fellowship ; and when the stocks are equal, the shares are as the times; therefore, when neither are equal, the shares must Þe as their products.

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