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GEOMETRICAL PROPORTION and PROGRESSION.

In Geometrical Progression the numbers or terms have all the same multiplier or divisor. The most useful part of Geometrical Proportion, is contained in the following theorems.

THEOREM 1. , When four quantities are in geometrical proportion, the product of the two extremes is equal to the product of the two means.

Thus, in the four 2, 4, 3, 6, it is 2 x 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 = ? = 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 x 8 1 4 X 4 = 16.
And in the series 2, 4, 8, 16, 32, 64, 128,

it is 2 x 128 = 4 x 64 = 8 x 32 = 16 x 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. Consequently the greatest term is equal to the least term multiplied by the said quotient.

So, of the ten terms 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, the common ratio is 2, and one less than the number of terms is 9; then the quotient of the extremes is 1024 = 2 = 512, and 29 :512 also.

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THEOREM 4. The sum of all the terms, of

any geometrical progression, is found by adding the greatest term to the difference of the extremes divided by i less than the ratio.

So, the sum of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,

1024 - 2 (whose ratio is 2), is 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 17 the first month, 21 the second, 41 the third, and so on, each succeeding payment being double the last; and what will the last payment be?

Ans. The debt 40951, and the last payment 20481.

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PROBLEM I.

To find One Goometrical Mean Proportional between any True

Numbers.

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

EXAMPLE

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

12

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PROBLEM II.

To find Two Geometrical Mean Proportionals between any Two

Numbers.

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.

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EXAMPLE

To find two geometrical means between 3 and 24.
Here 3) 24 ( 8; its cube root 2 is the ratio.
Then 3 x 2 = 6, and 6 X 2 = 12, the two means.
Or 24 = 2 = 12, and 12 • 2 = 6, the same.
That is, the two means between 3 and 24, are 6 and 12,

PROBLEM III.

To find any Number of Geometrical Means between Two Numbers,

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 one mean, the 3d root for two means, the 4th root for three 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 means between 3 and 96. Here 3) 96 (32; the 5th root of which is 2, the ratio. Then 3x2=6, & 6x2=12, & 12 x 2 = 24, & 24 x 2 = 48. Or 96--2=48,& 48 -2=24, & 24-2=12, &12:2=6.

That is, 6,12, 24, 48, are the four means between 3 and 96.

OF

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.

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 ) , Pr, are in arithmetical progression ; for + t = pt =*; 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

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 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.

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