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To find four geometrical means between 3 and 96.

Here 3) 96 (32; the 5th root of which is 2, the ratio.

Then 3 x 2 = 6, and 6 × 2 = 12, and 12 x 2 = 24, and 24 x 2 = 48. Or 962 48, and 48 ÷ 2 = 24, and 24 ÷ 2 = = 12, and 12 ÷ 2 = 6. That is, 6, 12, 24, 48, are the four means between 3 and 96.


THERE is also a third kind of proportion, called Harmonical or Musical, which being but of rare occurrence in questions purely arithmetical, a very short account of it may here suffice. It will however be again noticed both in algebra and in geometry, but especially in the latter.

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.

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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.

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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,,, are in arithmetical progression; for + & = = ; and += }: ; that is, the sum of the extremes is equal to double the mean, which is the property of arithmeticals. The method of finding a series of numbers in musical proportion and progression is best expressed by algebraic methods and symbols.


FELLOWSHIP is the 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 losses 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.

Fellowship is either Single or Double. It is single, when the shares or portions are to be proportioned each to one 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.




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 of the several proportional numbers,
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 gain or loss.

TO PROVE THE WORK. Add all the shares or parts together, and the sun will be equal to the whole number to be shared, when the work is right.


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.

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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 ;

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3. Two merchants, C and D, made a stock of 1207; of which C contributed 757, and D the rest: by trading they gained 301; what must each have of it? Ans. C 187 15s, and D 117 5s. 4. Three merchants, E, F, G, make a stock of 7001; of which E contributed 1231, F 3581, and G the rest: by trading they gain 125/ 10s; what must each have of it? Ans. E must have 221 1s Od 23.q.

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5. A General imposing a contribution * of 700l on four villages, to be paid in proportion to the number of inhabitants contained in each; the first containing 250, the 2d 350, the 3d 400, and the 4th 500 persons; what part must each village pay? Ans. the first to pay 1167 13s 4d.

the 2d ......

163 6 8.

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

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 3207, and N's 751; what quantity of land must each one have? Ans. L must have 20 ac 3 ro 3915 pls.

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7. A person is indebted to O 577 15s, to P 1081 3s 8d, to Q 227 10d, and to R 737; but at his decease, his effects are found to be worth no more than 1707 14s; how must it be divided among his creditors?

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to T,

8. A ship, worth 9007, being entirely lost, of which belonged to S, and the rest to V; what loss will each sustain, supposing 540l of her were insured? Ans. S will lose 451, T 90l, and V 2257.

9. Four persons, W, X, Y, and Z, spend 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,,, and what are their shares? Ans. W must pay 9s 8d 3449. X..... 6 5 3599.


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10. A detachment, consisting of 5 companies, being sent into 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


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

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 corresponding share of the gain or loss.

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.

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, as in Single Fellowship; and when the stocks are equal, the shares are as the times; therefore, when neither are equal, the shares must be as their products.


1. A had in company 50l for 4 months, and B had 607 for 5 months; at the end of which time they find 247 gained: how must it be divided between them?

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and as 500 24 :: 300: 143 = 14 8 = B's share.

2. C and D hold a piece of ground in common, for which they are to pay 547. C put in 23 horses for 27 days, and D 21 horses for 39 days; how much ought each man to pay for the rent? Ans. C must pay 231 5s 9d. D...... 30 14 3.


3. Three persons, E, F, G, hold a pasture in common, for which they are to pay 30% 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 51 10s 6d 159. 11 16 10 0. G........ 12 12 7 29.



4. A ship's company take a prize of 1000l, 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; the officers have 40s a month, the midshipmen 30s, and the sailors 22s a month; moreover, there are 4 officers, 12 midshipmen, and 110 sailors: what will each man's share be?


Ans. each officer must have 231 2s 5d 09. each midshipman.. 17 6 9 39993. each seaman ........ 672 0193.


5. H, with a capital of 10007, began trade the first of January, and, meeting with success in business, took in I as a partner, with a capital of 1500l, on the first of March following. Three months after that they admit K as a third partner, who brought into stock 2800l. After trading together till the end of the year, they find there has been gained 17767 10s; how must this be divided among the partners? Ans. H must have 4571 9s 44d 39.

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6. X, Y, and Z made a joint-stock for 12 months; X at first put in 201, and 4 months after 201 more; Y put in at first 301, at the end of 3 months he put in 207 more, and 2 months after he put in 407 more; Z put in at first 60l, and 5 months after he put in 107 more, 1 month after which he took out 301; during the 12 months they gained 507; how much of it must each have?

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INTEREST is the premium or sum allowed for the loan or forbearance of money. The money lent or forborn is called the Principal; and the sum of the principal

and its interest is called the Amount. Interest is allowed at so much per cent. per annum, or interest of 100l for a year, is called the rate of interest. Thus : when interest is at 3 per cent. the rate is


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But, by law, interest ought not to be taken higher than at the rate of 5 per cent.

Interest is of two sorts; Simple and Compound.

Simple Interest is that which is allowed for the principal lent or forborn only, for the whole time of forbearance. As the interest of any sum, for any time, is directly proportional to the principal sum, and also to the time of continuance ; hence arises the following general rule of calculation.

As 1007 is to the rate of interest, so is any given principal to its interest for one year. And again,

As 1 year is to any given time, so is the interest for a year, just found, to the interest of the given sum for that time.

OTHERWISE. Take the interest of 1 pound for a year, which multiply by the given principal, and this product again by the time of loan or forbearance, in years and parts, for the interest of the proposed sum for that time.

Note. When there are certain parts of years in the time, as quarters, or months, or days; they may be worked for, either by taking the aliquot or like parts of the interest of a year, or by the Rule of Three in the usual way. Also, the division by 100 is done by pointing off two figures for decimals.


1. To find the interest of 230/ 10s, for 1 year, at the rate of 4 per cent.

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2. To find the interest of 5471 15s, for 3 years, at 5 per cent. per annum.

As 100

Or 20

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