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. EXAMPLE. To find two geometrical means between 3 and 24. Here 3) 24 (8; its cube root 2 is the ratio. Then 3 x 26, and 6 x 2 = 12, the two means. Or 242 12, and 12 ÷ 26, 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 3 x 2 = 6, & 6×2 = 12, & 12 × 2 = 24, & 24×2=48. Or 962=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. 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: 18-12. 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 †,†, 1⁄2, are in arithmetical progression; for and ¦ + ¦ = 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 de 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 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. 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, 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. 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 or as 4: 1:110: 271 tuns = a's loss ; and as also as Sum 85 tuns, the proof. 3. Two merchants, c and D, made a stock of 1201. of which c contributed 75l, and D the rest: by trading they gained 301; what must each have of it? Ans. c 18 15s, and D 117 5s. 4. Three merchants, E, F, G, made a stock of 7001, of which E contributed 1231, F 3581, and & the rest: by trading they gain 125l 10s; what must each have of it? Ans. E must have 221 18 Od 29. 64 3 8 03. 39 53 135. 5. A General imposing a contribution* of 700 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. 163 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 500 a year 's 3201, and N's 75l; what quantity of land must each one have? Ans. L must have 20 ac 3 ro 391ps. 13 1 30,45 3 0 23178. 7. A person is indebted to o 571 15s, to p 1081 38 8d, to a 221 10d, and to R 731; but at his decease, his effects are found 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. VOL. I. 17 are to be worth no more than 170l 14s; how must it be divided among his creditors? Ans. o must have 371 15s 5d 2.5303 104399 27498 10439 10439 Ex. 8. A ship worth 9001, 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 insured? Ans. s will lose 451, т 901, and v 2251. 9. Four persons, w, x, y, and z, spent among them 25s, and agree that their shares are to be in proportion as, ,, and what are their shares? : Ans. w must pay 9s 8d 3449. X 6 5 344. Y Ꮓ 410 144. 310 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 first consisting of 54 men, the 2d of 51 men, and 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. DOUBLE FELLOWSHIP, as has been said, is concerned in cases in which the stocks of partners are employed or continued for different times. *Questions of this nature frequently occurring in military service, General Ha. viland, 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. |