Remark. Since the co-latitudes of the sun and moon and the comprehended angle (expressed by their difference of longitude,) form a quadrantal spherical triangle; therefore the true central distance between these particular objects may be more readily determined by the following concise method than by the above general Rule, viz. To the logarithmic co-sine of the difference of longitude, add the logarithmic co-sine of the moon's latitude; the sum of these two logarithms, abating 10 in the index, will be the logarithmic co-sine of the true central distance between the sun and moon. Example 1. August 6th, 1825, the moon's longitude, at noon, was 1:S:0:34", and her latitude 3:23:20% north; at the same time, the sun's longitude was 4:13:38:46"; required the true central distance ? Difference of long. = 95:38:12" Log, co-sine = True central distance =95:37:36" Log. co-sine =. which is precisely the same as that given in the Nautical Almanac. Example 2. August 7th, 1825, the moon's longitude, at noon, was 1:20:4:42", and her latitude 2:30:42 north; at the same time the sun's longitude was 4:14:36:18; required the true central distance? 8.979051 True central dist. = 84:31:55 Log. co-sine = which exactly corresponds with the computed distance in the Nautical Almanac. Example 3. Required the true central distance between the moon and sun at noon, August 8th; at midnight, August 8th; at noon, August 9th, and at midnight, August 9th, 1825 ? 136. 2.38 67:24:18 Log. co-sine. 9.584572 0.57.33 Log. co-sine. 9.999929 9.584511 Difference of longitude = Distance at midnight Aug. 9th = 55:34:16" Log. co-sine 9.752344 Now, from these four consecutive lunar distances, the distances at the intermediate periods, or every third hour may be readily determined in the following manner, viz. Find the proportional parts of the difference at the middle interval between the four distances (that is, between the second and third distances,) answering to 3 hours, 6 hours, and 9 hours: correct these proportional parts by the equation of second differences agreeably to the rule given, for that purpose, in page 34 ;-then, these corrected proportional parts being applied to the second lunar distance by addition or subtraction, according as the distances are increasing or decreasing, the sum or difference will be the true distances at the given periods :-thus, The proportional parts of 5:52:20% (the middle first difference) answer. ing to the intermediate periods, viz. The distances for the intermediate periods corresponding to the first and to the last 12 hours; that is, for every third hour between the first and second distances, and between the third and fourth distances, may be also very readily determined by means of the Formula which are given in page 117 of the Nautical Almanac for 1825. APPENDIX. SHOWING the direct application of logarithms to the solution of problems connected with the doctrine of compound interest; which develops the extraordinary powers of logarithmical arithmetic more than any other department of science which has been touched upon in this work. Definition.-COMPOUND INTEREST is that which is deduced not only from the sum of money lent as the principal, but also from the interest arising thereon; which interest, as it becomes due at the stated times of payment, is added, or supposed to be added, to the principal. Although it is illegal to lend money at compound interest, yet in purchasing annuities, pensions, or leases in reversion, it is usual to allow the purchaser compound interest for the use of his ready money.-And, these points being premised, we will proceed to the solution of the most interesting problems relating to this department of science ;-for which purpose the following Tables have been computed. Note. The rates, or ratios, of £1 sterling contained in these tables were computed by the rule of proportion in the following manner, viz.— As £100 £3:: £1 to £. 0300 ;-hence, the amount of £1 for one year is £1.0300; which, therefore, is the ratio; and so on for the rest. The respective numbers annexed to these ratios are expressed by the common logarithms corresponding thereto :-thus, the logarithm of 1.0300 is 0.0128372; and the logarithm of .0300 is 8.4771213, and so on of others. In this part of the work it has been deemed advisable to take out the logarithms to seven places of decimals; though for ordinary purposes six places of decimals will be found amply sufficient. 688 Rates per Cent. Yearly Payments. Half Yearly Payments. £. Ratio. 3. Logarithm. 1.0300 = 0.0128372 Ratio. = Quarterly Payments. CONCISE TABLES For facilitating the various Logarithmical Calculations connected with the Doctrine of Compound Interest. Yearly Payments. Quarterly Payments. Dec. .0300 8.4771213 .007500 = 7.8750613 .0350 = .01625 |