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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 1 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..
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.
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 1l the first month, 21 the second, 4l 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,
To find One Geometrical Mean Proportional between any Two
MULTIPLY the two numbers, together, and extract the square root of the product, which will give the mean proportional sought.
To find a geometrical mean between the two numbers 3 and 12.
36 (6 the mean.
To find Two Geometrical Mean Proportionals between any
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.
To find two geometrical means between 3 and 24.
To find any Number of Geometrical Means between Two Num.
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, òr divide continually from the last or greatest term.
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. & 6x2= 12, & 12 x2 = 24, & 24 X 2=48. Or 96*2=48, & 48 -2 = 24, & 242 = 12, & 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.
As in these three, 6, 8, 12 ;
where 6 : 12 : ; 8 6 : 128,
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;
where 6 : 18 ::8-6: 1812.
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 4 ti' and ý tý= = 4; 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.
Fellowship is either Single or Double. It is Single, when the sharer 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,
Add together the numbers that denote the proportion of the shares.
To the corresponding share or part.
So is each man's particular stock,
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.
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.
and as 6 : 240 :: 2 : 80 the 2d 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 ;
4 : 1:: 110 : 27) tuns = A's loss ;
4: 1:: 97 : 241 tuns B's loss; also as
4 : 1:: 133 : 334 luns c's loss;
Sum 85 tuns, the proof.
3. Two merchants, c and d. made a stock of 1201, of which c contributed 75l, and the rest : by trading they gained 301 ; what must each have of it?
Ans. c 181 158, and D 112 58. 4. Three merchants, E, F, G, made a stock of 7001, of which e contributed 1231, F 3581, and G the rest : by trading they gain 1252 108 ; what must each have of it?
Ans. E must have 221 1s Od 29.
64 3 8 03.
39 5 3 135 5. A General imposing a contribution* of 7002 on four villages, to be paid in proportion to the number of inhabitanis 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 Ist to pay 1161 38 4d. the 2d
163 6 8 the 3d
186 13 40 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 x's 751; what quantity of land must each one have ?
Ans. I must have 20 ac 3 ro 3943ps
23171 7. A person is indebted to o 571 158, to p 1081 38 8d, to 4 221 10d, and to R 731 ; but at his decease, his effects
* 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