Let PT reprefent the Earth, T the Centre, MD V a Part of the Moon's Orbit, DD Part of the Sun's Orbit, and the rational as well as A A is the fenfible Horizon. Now, an Obferver at T will fee the Moon in the Line DR, when another at P at the fame Time will see her in the Line DA in the fenfible Horizon, and the Angle P) TMT) ADR is the Horizontal Parallax fet down in the Nautical Almanack for every Noon and Midnight, and lies between 54' and 62'; this Angle diminishes as the Object approaches the Zenith; for fuppofe the Moon in the Line Pq the qDo PDT is less than the < ADR = <TDP, ftill diminishing until it comes to the Zenith Z, where it is nothing. To find the Diminution of Parallax in Altitude, fay, as Radius is to the Horizontal Parallax :: fo is the Co-fine of the Apparent Altitude: to the Parallax in Altitude. The Parallax of the Moon is greater than any of the rest of the Planets, owing to her being neareft the Earth, the vaft Distance of Sun and Stars rendering their Parallax fo fmall, that they are often neglected in Nautical Calculations; for fuppofe the Sun at F or S, then the < TFP=4 TF is the Sun's Parallax, about 8 Seconds. Having the Earth's Semi-diameter and the Parallax of any of the Planets, their Diftance may be found, by faying, as the Tangent of the Parallax is to the Earth's Semi-diameter in Miles :: fo is Radius to the Distance. Having the Distance the Parallax is found by faying, As the Diftance is to Radius: ; fo is the Earth's Semi-diameter: to the Tangent of the Parallax. The Earth's Semi-diameter is 1146 Nautical Leagues, the Sun's Distance from the Earth is 27,809,344, the Moon 69,059, Mercury 10,764,563, Venus 20,115,400, Mars 42,372,897, Jupiter 144,635,833, and Saturn 265,283,603 Leagues, To find the apparent Time, and thereby regulate the going of the Watch. A MONG the Methods propofed, that by equal Altitudes of the At the Time when the Watch ftands in need of being regulated, for the Obfervations intended, let the Sun's Altitude be taken at any convenient Time in the Forenoon, 2, 3, 4, or 5 Hours diftant from the Meridian. Set down the Altitude with the correfponding Time exactly (the Index being already fet to the Morning Altitude): Note down the Time of the fame Altitude in the Afternoon; half the Sum of these two Times, is the apparent Time fhewn by the Clock or Watch when the Sun was upon the Meridian of that Place. But it muft here be obferved, that if the Change of Declination be confiderable during the elapfed Time, it must be allowed for, by adding the Difference to, or fubtracting it from, the fecond Altitude, according as it is increafing or decreafing. Left that an Altitude taken in the Forenoon cannot, by the Interpofition of Clouds, have a correfponding one in the Afternoon; it is therefore proper to take several in the Forenoon, in order to fecure a correfponding one in the Afternoon. And, if feveral equal Altitudes can be taken on both Sides of the Meridian, it will be beft to find the Noons for each Pair, and the Means of all the Noons thus found for the true one. When there is Reafon to believe that the Watch gains or lofes confiderably, other Sets of Obfervations may be taken on fucceffive Days, whereby the Daily Variation may be found and allowed for; by which Means the Artift will have little more to do in finding his Longitude by Obfervation, than to reduce the obferved Diftance of the Objects to the true Distance of their Centers; the Ship's Timę being thewn by the Watch previously regulated. 3 EXAMPLE May 20, 1796, fuppose that at 8 H. 40 M. in the Forenoon, and H. 16 M. Afternoon by Watch, the Sun had equal Altitudes, and the going of the Watch be required? S. May 18, 1796, in Lat. 49°N. fuppofe at 8 H. 10 M. 58 S. Forenoon, and at 3 H. 58' 34" you have equal Altitudes of the Sun, Required the going of the Watch? The Distance of the Time froin Noon when the firft was taken is 3 3 H. 49 M. 2 S. doubled is 7 H. 38 M. 4 S. and the daily Decreafe of Declin. at this Time is 23′ 41′′. Now as 24 H.: 23'41":: 7 H. 38′ 4′′ :.7' • I Hence the Index of the Quadrant must be fet 7' forward on the Arch, to correfpond with the Morning Alt. whence the Watch 2 will be found 4' 46" too faft. Here it is fuppofed that the Ship is lying by, or makes no Way through the Water; but if fhe is failing to or from the Sun, proper Allowance must be made for her Run, during the elapfed Time, but the following Methods of finding the Time will answer every Pur pofe at Sea. To find the apparent Time by the Sun's Altitude. Find the Ship's Latitude and Longitude by Account, at the Time of Obfervation, by carrying the Reckoning forward to that Time. With a Quadrant well adjufted, take the Altitude of the Sun's lower Limb. Take the Difference between the Semi-diameter and Dip of the Horizon, and add it to the obferved Altitude, the Sum will be the Sun's apparent Altitude. Take the Difference between the Sun's Refraction and Parallax in Altitude, and fubtract it from the apparent Altitude; the Remainder will be the true Altitude of the Sun's Centre; hence the true Zenith Distance. Turn, the Ship's Longitude into Time, and either add to or fubtract from the Time per Watch, according as it is Eaft or Weft; the Sum, or Difference, will be the reduced or fupposed Time at the Place of Obfervation. Take the Sun's Declination out of the Nautical Almanack, and proportion it to the reduced Time: With the Sun's true Declina tion find the Polar Distance; then, Add together The Zenith Diftance, The Co-Latitude, and Polar Distance into one Sum. From half this Sum fubtract the Zenith Distance, noting the half Sum and Remainder, then add together Indexes. The Log. Co-Secant of the Comp. of the Lat. 1 Rejecting their The Log. Co-Secant of the Polar Distance, The Log. Sine of the half Sum, and The Log. Sine of Difference into one Sum. Half the Sum of these four Logarithms will give the Log. Co-fine of half the Hour Angle; which being doubled and turned into Time, by allowing 15 Degrees for every Hour, &c. or more briefly by the Table, will give the true Time, if the Altitude was taken in the Afternoon; but if in the Forenoon, its Complement to 24 Hours will be the true Time, reckoned from the preceding, or Noon be fore. NOTE. The Refraction is found in Table V. of this Book. The Dip The Sun's Semi-diameter in Page 3d of the Month in EXAMPLE I. M. Suppose a Ship at Sea in Lat. 39° 54′N. and Long. 35° 30′W. of Greenwich by Account, on the 7th of May, 1796, at 5 H. 30 P. M. by Watch, the Altitude of the Sun's lower Limb was obferved to be 15° 45', the Eye being 18 Feet above the Water. Required the true apparent Time when the Obfervation was made? Obf. Alt. Sun's lower Limb 15° 45' 0" Semi-diam. 15' 53" Lat. 39° 54' 74° 6' 19" 50 6 72 49 19 Now as 24 H.: 16' 10" :: 7H 52′ 5′ 17", which added to 17° 5' 24", the Sun's Declination the preceding Noon, gives 17° 10′ 41′′ the Sun's true Declination at the Time and Place of Obfervation, which being fubtracted from 90, leaves 72° 49′ 19′′N. the Polar Distance. Zen. Dift. Co. Lat. Polar Dift. Co. Secant 0,11511 Co. Secant 0,01983 Hour Angle 83 23 65H. 33' 32" true Time at Ship. 5 30 32 Time Watch. per 3 o Watch flow. NOTE. The Co. Secant (rejecting the Index) of any Angle, is equal to the Arithmetical Co-fine of that Angle. EXAMPLE II. Suppofe in the Forenoon, on the 10th of October, 1796, in Lat. 51° 30'N. and Long. 52° Eaft, the Altitude of the Sun's lower Limb fhould be found as under, the Eye being 22 Feet above the Sea. Required the true Time of the Day? H. M. 8 21 Times. Altitudes. H.M. Lat. 8 14 12° 28' 51° 30' Time 12 8 19 13 20 8 30 14 51 Co. Lat. 38 30 3) 25 3 40 39 App. Time Long. in Time 3 28 16 53 20 21 13° 33' 00" Sun's Decl. Oct. 9 6° 38' 48" 7 1 32 O II 37 Here 3 H. 50 M. 32 S. fubtracted from 24 Hours, leaves 20 H. M. 28 S. the true Time from Noon, Oct. 9, or 8 H. 9 M. 28 S. in the Forenoon, Oct. 10, by the common Way of reckoning Time. Watch faft 11 M. 32 S. NOTE. As the Obfervations were made in the Morning, 12 Hours must be added to it to, give the Time from last Nɔon. |