equation thus found is to be applied by addition to the computed time at Greenwich, when the proportional logarithms are decreasing; but by subtraction when they are increasing; as will be shown hereafter. 81. The above-mentioned equations may be computed in the following manner, viz. :-Let the difference of the proportional logarithms, answering to the lunar distances in the Ephemeris, be esteemed as a whole number: then, to the logarithm of this, increasing the index by unity or 1, add the logarithm of the interval in time; the logarithm of its complement to 3 hours, and the constant logarithm 8. 142667* ;the sum, abating 10 in the index, will be the logarithm of the equation in seconds. Example. Let the difference between the proportional logarithms answering to two lunar distances in the Ephemeris, be 80, and the interval or portion of time 110"; required the corresponding equation? = Difference of proportional Logarithms 80. Logarithm 2.903090 Equation, as required = 8. 142667 23:77 Logarithm. 1.375949 Hence, the correction answering to 80 and 1:10" is 24 seconds in whole numbers. 82.-Table B, Volume II., page 611.* Reduction of Latitude on account of the Oblate Spheroidal Figure of the Earth; or, to reduce the Geographical to the Geocentrical Latitude. In the explanation of Table XLI., which is given in page 105, the ratio of the equatorial semidiameter of the earth to its polar semiaxis, has been assumed at 230 to 229, agreeably to the Newtonian hypothesis; and therefore the excess of the spherical above the elliptic arch, in the latitude of 45 degrees, north or south of the equator, has been estimated at 11:53" :-this, however, does not exactly correspond with modern calculations; for, since the ratio of the earth's equatorial radius to its polar semi-axis is now admitted to be as 305 to 304, as established by the French philosophers; therefore, the excess of the spherical above the elliptic arch in the mean parallel of latitude is no more than 11:17. The spherical excess signifies the angular distance * The log. arithmetical complement of 72, viz. 24 × 3,-three hours being the common interval between the lunar distances in the Náutical Almanac. Z between the central and the apparent zeniths of an observer in the parallel of 45 degrees from the equator.-This excess, which is 36% less than the above, is determined in the following manner, viz.: To the logarithm of the equatorial semidiameter of the earth, add the logarithm of its polar semi-axis; the log. tangent of 1", and the constant logarithm 0.301030 :-from the sum of these four terms subtract the logarithm of the sum of the two semidiameters; then, the arithmetical complement of the remainder will be the logarithm of the spherical excess in seconds; as thus: 677"-Logarithm Sum of the semidiameters = 609, Logarithm . Spherical excess = 2.830838, or, 11'17"; which, therefore, is the correct angular distance between the central and the apparent zeniths of an observer at the mean parallel; or, the true reduction of latitude on account of the oblate spheroidal figure of the earth, in the parallel of 45 degrees north or south of the equator. Now this being known, the reduction corresponding to any given latitude betwixt the equator and the Poles may be determined in the following manner, viz. : The geographical latitude at the mean parallel being diminished by the spherical excess, the result will be the geocentrical latitude: hence, 45-11:17" = 44:48:43"; the log. co-tangent of which, rejecting radius, is 0. 002850.-Now, this (taken as a constant quantity) being subtracted from the log. tangent of any given latitude, the remainder will be the log. tangent of such latitude reduced to the oblate spheroidal figure of the earth: the difference between which and the given latitude will be the tabular reduction of latitude; as thus :To reduce the Geographical Latitude 50 degrees to the Geocentrical Latitude. Given geographical latitude 50: Constant quantity, as above Geocentrical latitude Difference. Log.tangent 10.076186 0.002850 49:48:53 Log. tangent 10. 073336 .117; which, therefore, is the reduction of latitude in the parallel of 50: north or south of the equator; and in the same manner were all the equations in Table B determined. Note. In taking out the equations from the present Table, proportion must be made for the excess of the minutes above the given degree: this may be readily done; viz. :-As 60 minutes are to the difference of correction between the two degrees that are next greater and next less than the given degree; so are the minutes of latitude to the required equation:-but, in general, the equation may be taken out at sight, without the trouble of making any proportion; and it is always to be applied by subtraction to the geographical latitude. 83.-Table C, Volume II., page 611.* Logarithmical Radius of the Earth, for reducing the Moon's Horizontal Parallax to the Earth's Oblate Spheroidal Figure. In treating of the reduction of the moon's horizontal parallax at pages 104 and 105, the ratio of the polar axis to the equatorial diameter of the earth was assumed at 229 to 230, as established by Newton, and the compression of the Poles as 3-Agreeably to that ratio, taking the equatorial diameter of the earth at 7917.5 English miles, its polar axis would be 7883.7 miles; which is 33. 8, or, in round numbers, 34 miles less than the diameter at the equator.-Modern philosophers however, have given up Newton's ratio, and substituted in its stead the mean of the two ratios that were determined by Le Lande and Delambre. This mean ratio makes the polar axis be to the equatorial diameter as 304 to 305 :-Therefore, taking the equatorial diameter of the earth at 7917.5 English miles; its polar diameter will be 7891.5. miles; which is but 26 miles less than the diameter at the equator; hence, the polar radius is only 13 miles less than the equatorial radius: and this is found to correspond with the phenomena which should arise from the precession of the equinoxes, and the nutation of the earth's axis. And since it thus appears that the equatorial radius of the earth is to the polar semi-axis in the ratio of 305 to 304; therefore, the compression of the Poles is 305-304 -—-—- 304, instead of ; as in 105. page 304 Now, since the greater the diameter of the earth, the greater must be the value of the moon's horizontal parallax; and, vice versa, the less the diameter of the earth, the less will be the value of the moon's horizontal parallax ;-therefore, since the polar semi-axis of the earth is 13 miles less than the equatorial semidiameter; and, consequently, since the radius of the earth must diminish at every point, from the equator to the Poles, so must the value of the moon's horizontal parallax.—The diminution of the moon's horizontal parallax may be computed agreeably to the formula in page 105, by substituting the constant logarithm 7.517126 (the log. arith. comp. of 304), for that given in the rule. But, since it is more convenient, in general practice, to deduce the value of the moon's horizontal parallax from the earth's radius, therefore, assuming the equatorial semidiameter of the earth as unity or 1, the relative value thereof at any point of the meridian, betwixt the equator and the Poles of the world, may be readily determined in the following manner, viz. : From the logarithm of the polar radius, subtract half the logarithm of the sum of the polar and equatorial radii :-Then, let the remainder be subtracted from the log. sine of the latitude; and the result will be the log. tangent of an arc: the log. co-sine of which will be the logarithm of the radius, or of the semidiameter of the earth in the given parallel of latitude. Example. Let the equatorial radius of the earth be 1; the ratio of the equatorial to the polar semidiameter as 305 to 304, and the latitude of the place 50, north or south; to find the logarithm of the earth's radius in that latitude: Polar radius 304-Logarithm . . 2.482874 Polar radius + equatorial radius = 609 half its logarithm. 1.392308 Remainder Given parallel of latitude = 50: Log. sine Now, the log. co-sine of this arc is 9.999162; which, therefore, is the logarithm of the earth's radius in the given parallel of latitude; and in this manner were all the logarithms in Table C computed. Now, to the log. sine of the moon's horizontal parallax (reduced to the meridian of Greenwich), add the tabular log. radius of the earth corresponding to the given degree of latitude (reduced to the oblate figure); the sum, abating 10 in the index, will be the log. sine of the moon's diminished equatorial horizontal parallax.—Or, to the common logarithm of the moon's horizontal parallax in seconds, add the tabular log. radius of the earth; the sum, abating 10 in the index, will be the logarithm of the moon's horizontal parallax reduced to the oblate figure of the earth. Note.--The elementary principles upon which the logarithmical rules in Articles 82 and 83 are founded, may be seen by referring to Mr. W. S. B. Woolhouse's admirable Paper on Eclipses; which is given as an Appendix to the Nautical Almanac for 1836, between pages 55 and 58. 84. Having thus explained, in the preceding Articles and Definitions, every point of importance in the new Nautical Almanac, we shall now proceed to the consideration of such problems as may be deemed introductory to the Science of Nautical Astronomy. INTRODUCTORY PROBLEMS TO THE SCIENCE OF It may be necessary to premise, that throughout the astronomical part of this work the time will be reckoned agreeably to the mean solar - day, viz. from mean noon to mean noon, or from 0 to 24 hours, the same as in the Nautical Almanac,* without paying any regard to the nautical or civil division of time at midnight :—this will conduce much to simplicity, as well as to uniformity; and do away with that confusion which frequently arises from the nautical distinctions of A. M. and P. M., when the time at a given place is to be reduced to the meridian of the Royal Observatory at Greenwich. PROBLEM I. To convert Longitude or Parts of the Equator into Time. RULE. Multiply the given degrees by 4, and the product will be the corresponding time :-observing that seconds multiplied by 4 produce thirds; minutes multiplied by 4 produce seconds, and degrees multiplied by 4 produce minutes, which, divided by 60, give hours, &c. * See the fourth paragraph, page 498 of the Nautical Almanac for 1836. |