EXAMPLE. To find a geometrical mean between the two numbers 3 and 12. 12 36 (6 the mean. 36 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. Or 24 2 Then 3 x 26, and 6 × 2 = 12, the two means. 12, and 1226, the same. That is, the two means between 3 and 24, are 6 and 12. PROBLEM IIF. 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. OF HARMONICAL PROPORTION. THERE is also a third kind of proportion, called Harmonical or 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 be tween 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 be-, tween 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,,,, are in arithmetical progression; for += = }; 1/2 1; 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 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, 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 Then, as 6 240 :: 1 : and as 6 : 240: : 2: also as 6 240 sum of the numbers. 40 the 1st part, 80 the 2d part, 3: 120 the 3d part. Sum of all 240, the proof. 2. Three persons, A, B, C, freighted a ship with 340 tuns of wine, of which a loaded 100 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. 3. Two merchants, c and D, made a stock of 1201; of which c contributed 751, and D the rest by trading they gained 307; what must each have of it? Ans. c 187 15s, and D 111 5s. 4. Three merchants, E, F, G, make a stock of 7001, of which E contributed 1281, r 3581, and & the rest: by trading they gain 1257 10s; what must each have of it? Ans. E must have 221 1s Od 23¶• 64 3 8 03. 3953 1. 5. A General imposing a contribution* of 700 on four * 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. 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 1st to pay 1167 13s 4d 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 5007 a year, M's 3201, and N's 751; what quantity of land must each one have ? Ans. L must have 20 ac 3 ro 391ps. 7. A person is indebted to o 571 15s, to P 1087 3s 8d, to a 221 10d, and to R 731; but at his decease, his effects are found to be worth no more than 1701 14s; how must it be divided among his creditors? 5302 Ans. o must have 371 15s 5d 2+3+39. 70 15 2 27430. 8. A ship, worth 9007, being entirely lost, of which be. longed to s, to T, and the rest to v; what loss will each sustain, supposing 5401 of her were insured? Ans. s will lose 451, т 907, and v 2251. 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, 1, 1, and 1 : what are their shares ? Ans. w must pay Is 8d 344q. 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 1st consisting of 54 men, |