Divide the log of the given number by the index of the power, and the quotient will be the log. of the root. To the log. of the length of the keel, reduced to tonnage, add the log. of the breadth of the beam, the log. of half the breadth of the beam, and the constant log. 8.026872* ; the sum, rejecting 10 from the index, will be the log. of the required tonnage. Example. Let the length of a ship's keel, reduced to tonnage, be 120.5 feet, and the breadth of the beam 35. 75 feet; required the ship's tonnage ? * This is the arithmetical complement of the log, of 94; the common divisor for finding the fonuage of ships. Given the Measured Length of a Knot, the Number of Seconds run by the Glass, and the Distance sailed per Log, to find the true Distance by Logarithms. RULE. To the arithmetical complement of the log. of the number of seconds run by the glass, add the log. of the measured length of a knot, the log. of the distance sailed, and the constant log. 9.795880*; the sum of these four logs., rejecting 20 from the index, will be the log. of the true distance. Example 1. The distance sailed by the log is 180 miles; the measured length of a knot is 43 feet, and the time by the glass 32 seconds; required the true distance? 32 seconds, arith, comp. log. 8.494850 43 feet, log. 1. 633469 180 miles, log. = 2. 255273 Constant log. 9.795880 True distance = 151.2 miles. Log. = 2. 179472 Example 2. The distance sailed by the log is 210 miles; the measured length of a knot is 51 feet, and the time by the glass 27 seconds ; required the true distance? 27 seconds, arith. comp. log. 8.568636 51 feet, log 1.707570 210 miles, log. 2.322219 Constant log. 9.795880 . True distance = 247.9 miles. Log. = 2. 394305 This is the sum of the arithmetical complement of the log. of 48 (the general length of a knot) and the log of 30 seconds, the true measure of the half-minute glass. TABLE XXIX. Proportional Logarithms. From the common log. of 3 hours, reduced to seconds, subtract the common log. of the given time in seconds; and the remainder will be the proportional log. corresponding thereto. Example. Required the proportional log. corresponding to 0:40.26!? Proportional log. corresponding to the given time = 0.6485. 33 As hours and degrees are similarly divided, therefore, in the general use Since proportion is performed by adding together the arithmetical com- fixed star, to the mean time and distance, when the observations are made by one person, as will one person, as will appear evident by the Example and its corresponding time 21:55:57!; it is required to reduce the first altitude to what it should be at 21:49:33:, the time at which the mean lunar distance was taken ? 1st time 21:42 8! Ist time 21:42 8! Ist alt. 27:25:20" 27:25:20 Last do, 21.55.57 Mean do. 21.49.33 Last do. 25. 24. 20 As 13"49:, arithmetical comp. prop. log. = 8. 8851 = 1.3851 So is 2: 1! proportional log. = 0. 1725 To prop. log. of reduction of Moon's alt.. = 0.4427 = 1: 4.57" And in the same manner may the altitude of the sun, or a fixed star, be reduced to the time of taking the mean lunar distance. Remark.-Although this Table is only extended to 3 hours or 3 degrees, yet by taking such terms as exceed those quantities one grade lower, that is, the hours, or degrees, to be esteemed as minutes, and the minutes as seconds, the proportion may be worked as above: hence it is evident that the Table may be very conveniently applied to the reduction of the sun's, moon's, or a planet's right ascension and declination to any given time after noon or midnight ; and, also, to the equation of time ;-for the illustration of which the following Problems are given. PROBLEM I. To reduce the Sun's Longitude, Right Ascension and Declination; and, also, the Equation of Time, as given in the Nautical Almanac, to any given Meridian, and to any given time under that Meridian. RULE. To the apparent time at ship, or place, (to be always reckoned from the preceding noon *,) add the longitude, in time, if it be west, but subtract it if east; and the sum, or difference, will be the Greenwich time. From page II. of the month in the Nautical Almanac, take out the sun's * See precepts to Table XV.-page 25, longitude, right ascension, declination, or equation of time for the noons immediately preceding and following the Greenwich time, and find their difference; then, To the proportional log. of this difference, add the proportional log. of the Greenwich time (reckoning the hours as minutes, and the minutes as seconds,) and the constant log. 9. 1249*; the sum of these three logs., rejecting 10 from the index, will be the proportional log. of a correction which is always to be added to the sun's longitude and right ascension at the noon preceding the Greenwich time; but to be applied by addition or subtraction to the sun's declination and the equation of time, at that noon, according as they may be increasing or decreasing. Example 1. Required the sun's longitude, right ascension and declination, and also the equation of time, May 6th, 1824, at 5?109, in longitude 64:45: west of the meridian of Greenwich ? . = + Correction of sun's right asc. 1:32" p. log. = 2.0711 Sun's right asc. at noon, May 6, 1824, = 2:53:3127 Sun's right asc. as required . = 2:55" 3'.7 + The arithmetical complement of the proportional log. of 24 hours esteemed as minutes, |