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? 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? 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. This Table contains the proportional log. corresponding to all portions of time under three hours, and to every second under three degrees. It was originally computed by Dr. Maskelyne, and particularly adapted to the operation for finding the apparent time at Greenwich answering to a given distance between the moon and sun, or a fixed star; but it is now applied to many other important purposes, as will be seen hereafter. Proportional Logarithms may be computed by the following RULE. 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 04026: ? 3 hours reduced to seconds 40:26: given time, in secs. 10800" Log. 4.033424 Log. = 3.384891 Proportional log, corresponding to the given time = 0.6485,33 As hours and degrees are similarly divided, therefore, in the general use of this Table, the hours and parts of an hour, may be considered as degrees and parts of a degree, and conversely. And to render the use of it more extensive, one minute may be esteemed as being either one degree, or one second, and vice versa. Since proportion is performed by adding together the arithmetical complement of the proportional logarithm of the first term, and the proportional logarithms of the second and third terms, rejecting 10 from the index, the present Table is of great use in reducing the altitudes of the moon and sun, or a fixed star, to the mean time and distance, when the observations are made by one person, as will appear evident by the following Example. Let the first altitude of the moon's lower limb be 27:25:20", and the corresponding time per watch 21:428:, and the last altitude 25:24:20", and its corresponding time 21:55"57; it is required to reduce the first altitude to what it should be at 21:4933, the time at which the mean lunar distance was taken? 1st time 21:42 8: 1st time 21:42 8: 1st alt. 27:25:20% 27:25:20% Last do. 25.24.20 Last do. 21.55.57 Mean do. 21.49.33 As 0.13.49 Diff. 0. 7.25 Diff. 2. 1. 0 13:49, arithmetical comp. prop. log. 8. 8851 Is to 725 proportional log. So is 2: 1 proportional log. = 1.3851 = 0.1725 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 minútes, 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 5'10", in longitude 64:45 west of the meridian of Greenwich? Diff. in 24 hours = = .+ 22:55 p. log. = 0.8952 = 1:15:51:13 = 1:16:14: 8? To find the Sun's Right Ascension. 3:52 prop. log. : Greenwich time 9:29" prop. log. = = 1.6679 = 1.2783 = 9.1249 + 1:32" p. log. 2.0711 Sun's right asc. at noon, May 6, 1824, = 2:53"31'.7 + The arithmetical complement of the proportional log. of 24 hours esteemed as minutes. Remark. Since the daily difference of the equation of time is expressed, in the Nautical Almanac, in seconds and tenths of a second; if, therefore, these tenths be multiplied by 6, the daily difference will be reduced to seconds and thirds :-Now, if those seconds and thirds be esteemed as minutes and seconds, the operation of reducing the equation of time will become infinitely more simple; because the necessity of making proportion for the tenths, as above, will then be done away with :-remembering, however, that the minutes and seconds corresponding to the sum of the three logs. are to be considered as seconds and thirds. Example 2. Required the sun's longitude, right ascension, and declination, and also the equation of time, August 2d, 1824, at 19:22", in longitude 98:45 east of the meridian of Greenwich? |