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GEOMETRICAL PROPORTION AND PROGRESSION,

THE most useful part of Geometrical Proportion, is contained in the following theorems:

THEOREM I. If four quantities be in geometrical proportion, the product of the two extremes will be equal to the product of the two means. Thus, in the four 2, 4, 3, 6, it is 2 × 6 = 3 x 4 = 12.

And hence, if the product of the two means be divided by one of the extremes, the quotient will give the other extreme. So, of the above numbers, the product of the means 12 ÷ 2 = 6 the one extreme, and 12 ÷ 6 = 2 the other extreme; and this is the foundation and reason of the practice in the Rule of Three.

THEOREM 2. In any continued geometrical progression, the product of the two extremes is equal to the product of any two means that are equally distant from them, or equal to the square of the middle term when there is an uneven number of terms.

Thus, in the terms 2, 4, 8, it is 2 × 8 = 4 × 4 = 16.
And in the series 2, 4, 8, 16, 32, 64, 128,

it is 2 × 123 = 4 × 64 = 8 × 32 = 16 × 16 = 256.

THEOREM 3. The quotient of the extreme terms of a geometrical progression, is equal to the common ratio of the series raised to the power denoted by 1 less than the number of the terms.

So, of the ten terms 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, the common ratio is 2, one less than the number of terms 9; then the quotient of the ex

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Consequently the greatest term is equal to the least term multiplied by the said power of the ratio whose index is 1 less than the number of terms.

THEOREM 4. The sum of all the terins, of any geometrical progression, is found by adding the greatest term to the difference of the extremes divided by 1 less than the ratio.

So, the sum of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, (whose ratio is 2,) is 1024 -2 1024 + = 1024 + 1022 = 2046. 2-1

The foregoing, and several other properties of geometrical proportion, are demonstrated more at large in the Algebraic part of this work. A few examples may here be added of the theorems, just delivered, with some problems concerning mean proportionals.

EXAMPLES.

1. The least of ten terms, in geometrical progression, being 1, and the ratio 2; what is the greatest term, and the sum of all the terms ?

Ans. the greatest term is 512, and the sum 1023. 2. What debt may be discharged in a year, or 12 months, by paying 14. the first month, 2/. the second, 47. the third, and so on, each succeeding payment being doable the last; and what will the last payment be?

Ans. the debt 40957., and the last payment 20482

PROB. 1.

To find one geometrical mean proportional between any two numbers.

RULE. Multiply the two numbers together, and extract the square root of the product, which will give the mean proportional sought.

Or, divide the greater term by the less, and extract the square root of the quotient, which will give the common ratio of the three terms: then multiply the less term by the ratio, or divide the greater term by it, either of these will give the middle term required.

EXAMPLE.

To find a geometrical mean between the two numbers 3 and 12.

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To find two geometrical mean proportionals between any two numbers. RULE. Divide the greater number by the less, and extract the cube root of the quotient, which will give the common ratio of the terms. Then multiply the least given term by the ratio for the first mean, and this mean again by the ratio for the second mean: or, divide the greater of the two given terms by the ratio for the greater mean, and divide this again by the ratio for the less mean.

EXAMPLE.

To find two geometrical mean proportionals between 3 and 24.
Here, 3) 24 ( 8, its cube root, 2 is the ratio.
Then, 3 × 2 = 6, and 6 × 2 = 12, the two means.
Or,
24212, and 12 ÷ 2 = 6, the same.
That is, the two means between 3 and 24, are 6 and 12.

PROB, JII.

To find any number of geometrical mean proportionals between two numbers. RULE. Divide the greater number by the less, and extract such root of the quotient whose index is 1 more than the number of means required, that is, the 2d root for 1 mean, the 3d root for 2 means, the 4th root for 3 means, and so on; and that root will be the common ratio of all the terms. Then with the ratio multiply continually from the first term, or divide continually from the last or greatest term.

EXAMPLE.

To find four geometrical mean proportionals between 3 and 96.
Here, 3) 96 (32, the 5th root of which is 2, the ratio.

Then, 3 × 2 = 6, and 6 × 2 = 12, and 12 × 2 = 24, and 24 × 2 = 48.
От,
96248, and 48 ÷ 2 = 24 and 242 = 12 and 12 ÷ 2 = 6.
That is, 6, 12, 24, 48, are the four means between 3 and 96.

OF MUSICAL PROPORTION.

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.

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When four numbers are in Musical Proportion; then the first has the same Proportion 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;
where, 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 , : T1⁄2, 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 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 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 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,

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 proportional numbers.

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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; theref., 340

hence, as 340 :

207 = 133 tuns, loaded by C.
85 :: 110

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3. Two merchants, C and D, made a stock of 120l.; of which C contributed 75l., and D the rest; by trading they gained 307.; what must each have of it? Ans. C 187. 15s., and D 117. 58.

4. Three merchants, E, F, G, made a stock of 700%; of which E contributed 1231., F 358l., and G the rest; by trading they gain 1257. 10s.; what must each have of it? Ans. E must have 221. 1s. Od. 2559. 64 3 8 39 5 3 135

F
G

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 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 116%. 13s. 4d.

the 2d

163

the 3d

.........

68 186 13 4 68

the 4th ......... 233

* Contribution is a tax paid by provinces, towns, villages, &c., to excuse them from being plun. dered, and is paid iu 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 500%. a year, M's 320l., and N's 75%; what quantity of land must each one have?

Ans. L must have 20 ac. 3 ro. 391 ps.

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7. A person is indebted to O 571. 15s., to P 1081. 3s. 8d., to Q 221. 10d., and to R 73l.; but at his decease, his effects are found to be worth no more than 1702. 14s.: how must it be divided among his creditors?

Ans. O must have 371. 15s. 5d. 29.

sured?

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8. A ship worth 900l., being entirely lost, of which belonged to S, to T, and the rest to V; what loss will each sustain, supposing 540l. of her were inAns. S will lose 45l., T 90l., 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,,,, and }; what are their shares? Ans. W must pay 9s. 8d. 39.

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10. A detachment, consisting of five 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 men, and the 5th of 36 men? Ans. The Ist must furnish 18, the 2d 17, the 3d 16, the 4th 13, and the 5th 12 men.*

DOUBLE FELLOWSHIP.

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.

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