PROBLEM. To find in how many Years any Principal or Sum of Money will double itself, at compound Interest, by yearly, half-yearly, or quarterly Payments. RULE. Let the logarithm of the ratio (Table A) be considered as the decimal part of a natural number; find the logarithm corresponding thereto, and subtract it from the constant logarithm 9. 4786098: the remainder will be the logarithm of the time in which a given sum of money will double itself at any proposed rate of interest within the limits of Table A: if the payments of interest be annual, the time will be expressed in years; otherwise, in half years, or quarters of years, as the case may be. Note. The constant logarithm is thus determined:-Let the double of any given sum of money be represented by the number 2, the logarithm of which is 0.3010300; consider this as the decimal part of a natural number: then, the logarithm corresponding thereto is 9.4786098; which, therefore, becomes a constant expression for all modes of payment and rates of interest. Example. Required the number of years in which any given sum of money will double itself, at compound interest, by yearly, half-yearly, and quarterly payments; the rate being 5 per cent. per annum? Constant log. = First, For Yearly Payments : Rate, 5 per cent. ; log., Table A, = 0.0211893; consider this 9.4786098 as the decimal part of a natural number, the log. of which is 8.3261167 Number of years, as required, = 14.2067 Log. = 1. 1524931 Second,-For Half-yearly Payments : .. 9.4786098 Constant log. = the decimal part of a natural number, the log. of which is = 8.0303528 Time, in half years, = 28.07094 Log. = 1.4482570 = Number of years, as required, 14.03547; which, therefore, is the time in which a sum of money will double itself, at 5 per cent. compound interest, by half-yearly payments. Rate, 5 per cent.; log., Table A,=0.0053950; consider this as the decimal part of a natural number, the log. of which is = Time, in quarters of years, = 55.79797 9.4786098 7.7319914 Log. = .. 7.7466184 Number of years, as required, 13. 94949; which, therefore, is the time in which a sum of money will double itself, at 5 per cent. compound interest, by quarterly payments. It is after this manner that the following table has been computed; excepting, however, the last column, or that for simple interest, which is merely expressed by the quotient of £100, divided by the given rate per cent. thus, £100 £5 20 years; which, therefore, is the time in which a given sum of money will double itself, at 5 per cent. per annum, simple interest. A TABLE, Exhibiting the Time in which any Sum of Money will double itself, at several given Rates per Cent. per Annum, Compound Interest, and also at Simple Interest. By the above table it is evident, that if a sum of money be put out at compound interest, at the rate of 5 per cent. per annum, it will double itself, by yearly payments of the interest, in 14 years; by half-yearly payments, in 14 years; and by quarterly payments, in 13 years; whilst at simple interest, it will not double itself in less than 20 years. Hence, if the given sum be £1250, it will amount, in 13 years, by quarterly payments, to the sum of £2500; in 27 years, to the sum of £5000; in 41 years, to the sum of £10000; in 55 years, to the sum of £20000 ; and so on in geometrical progression: while at simple interest, the same sum would only amount, in the same space of time, viz., 55 years, to the sum of £4737. 7s. 6d. Remark. The preceding problems and examples contain all that is essentially necessary to be known, independently of theory, in the doctrine of compound interest. The author is not aware that the direct logarithmical solutions of the various complex cases connected with this subject have been given by any other writer: many, indeed, have published theorems for this purpose; but these theorems (such as those given by the late ingenious Dr. Maskelyne, in his very learned Introduction to Taylor's Logarithms, under the head Compound Interest,) are expressed in such a scientific manner as to be of little use, in a mere practical point of view; being much better adapted for employing the minds of the curious in mathematical researches and investigations, than for abridging the labour attendant on arithmetical computations. THE USE OF THE GENERAL VICTUALLING TABLE, Contained in Vol. II., Page 661. As this Table contains the exact daily proportion of sea provisions for any given number of men within the ordinary limits of victualling, it will be found of considerable utility to the Pursers of the Royal Navy, in closing their annual accounts, and in completing the ship's provisions to any specified time: it will also be of great use to officers serving as Commanders and Pursers ;-and, perhaps, to those gentlemen in the Victualling Department of His Majesty's service who are employed in the examining and auditing of the Naval Victualling accounts of the abovementioned officers. REMARKS. 1. As the size of the page would not admit of separate columns being employed for the salt beef and salt pork, and as the allowance of each of these species is precisely the same; one column only has been introduced into each page of the Table on account of those articles of victualling ;— which column contains the exact proportion of each.-Hence, in taking out the proportions from this column, corresponding to any given number of men, care must be taken to put down such proportions twice; that is, first for salt beef, and then for salt pork; or, otherwise, to double those proportions, at once, for salt meat generally. 2. In like manner, as the size of the page would not admit of separate columns being employed for bread and beer, and for oatmeal and vinegar ; it will be necessary, in taking out the proportions of those species, corresponding to any given number of men, to put down as many gallons of beer as the second column expresses pounds of bread; and as many gallons of vinegar as the last column expresses gallons of oatmeal. The following problems will illustrate the principal uses to which this Table may be applied. PROBLEM I. Given the Number of Men victualled for one Day, to find the corresponding Proportion of each Species of Provisions. RULE. If the given number can be found in the left-hand column of the table, the corresponding proportion of each species of provisions will be found abreast of it in the same horizontal line; but if it cannot be exactly found, which in general will be the case, write down any two or more of the tabular numbers that will make up the given one, opposite to which put down the corresponding quantities of provisions: then, the sums of these quantities will be the true proportion of each species of provisions. Example. Let the number of men victualled for one day be 45685; required the true proportion of each species of provisions corresponding thereto ? Men. Bread. Beer. Salt Meat. Flour. Pease. Sugar. Cocon. Tea. Oatmeal. Vinegar. 45000 give 45000 45000 33750.0 16875.0 1406.2 4218. 12 2812.8 703.2 401.6.4 401.6. 4 600 do. 600 600 450.0 225.0 18.6 56.4 37.8 9.6 5.2.12 5.2.12 85 do. 85 85 63.12 31.14 2.54 7.15 5.5 1.5 0.6.1 0.6. 1 45685 give 45685 45685 34263.12 17131.14 1427.54 4282.154 2855.5 713.13 407.7.3 407.7.3 Note.-In making out the Purser's annual victualling account, the proportions of salt beef and pork, as given in the table, are to be doubled, or thrown into one sum under the head of salt meat, as above. If there be any fresh meat issued during the period of the account, subtract the amount thereof from the number victualled, and then take out the proportion of salt meat, flour, and pease, corresponding to the remainder: thus, suppose the quantity of fresh meat issued to be 22238 pounds. No.vic. for one day is 45685 Salt meat. Flour. Pease. 23447. Now, 23000 gives 17250.0 8625. 0 718.6 400 do. 300.0 150. 0 12.4 47 do. 35.4 17.10 1.32 23447 gives 17585.4 8792. 10 732.5 See Pursers' Instructions (Appendix), No. 21, page 117. PROBLEM II. Given the Complement of Men, and the Number of Days for which they are to be victualled; to find the Proportion of each Species of Provisions. RULE. Multiply the complement of men by the given number of days for which they are to be provisioned, and the product will be the number to be |