If the error was in mean time, this must be when on the meridian, which being 14 point reduced to apparent time by applying the equa- to the left of N., the variation is 11 point east. tion of time with a contrary sign. The variation of the compass may also be To find the variation of the compass.-—If the found by the amplitudes of celestial objects. bearing of any object by the compass be com But, as from the effect of refraction they appear pared with its known bearing, the variation or in the horizon when they are about 33' below it, deviation of the points of the compass from their the centre of the sun, or a star, ought to be corresponding points in the horizon becomes of about 33' + W. dip above the horizon, when course immediately known; the difference of the their amplitude is observed to compare wit true and observed bearing being the variation. their true computed amplitude to find the varia Now when an object is on the meridian its tion. Or the lower limb of the sun ought to be true bearing is either due north or due south; about 17' + the dip above the horizon. hence the deviation of an object from the meri To compute the true amplitude, add together dian as observed by the compass, when the ob- the secant of the latitude, and the sine of the ject is known to be on the meridian, is the vari- object's declination, and the sum rejecting ten ation of the compass ; west when the object from the index will be the sum of the true ambears to the right; and east when it bears to plitude, east when the object is rising, and west the left of the meridian. when setting ; north when the declination is Again, the middle time between equal alti- north, and south when it is south. Then if the tudes of a celestial object being the time of its true and observed amplitude, be both north or being on the meridian, the middle point between both south, their difference, otherwise their sum, those on which it bears when it has equal alti- is the variation; westerly when the true amplitudes is its bearing when on the meridian; hence tude is to the left of the observed, and easterly if this middle point be to the right of the meri- when the true altitude is to the right of the obdian the variation is west ; if to the left the va served. riation is east. Let P, fig. 16, plate II., be the pole, A the Example 1. The sun at noon was observed east or west points of the horizon, C B or C'B, to bear S. by W. } W., what was the variation ? the declination of the object at rising or setting, S. by W.; W.is 14 point to the right of S. then A C or A C' is the amplitude, B C or BẮC therefore the variation is 13 point W. the declination, and BAC or B’AC' the colatiErample 2.—The sun, when he had equal al- tude; and rad. sin. BC=sin. A C.sin. BAC, titudes on the same day, bore N.N. E. and N.W. rsin. BC r.sin. BC by W.; what was the variation ? whence sin. ACE sin. BAC cos. lat. The middle point between N. N. E. and N.W. sect. lat. sin. declin by W. is N. by W.; W., the bearing of the sun rad. E.rample. If on Oct. 11th, 1828, in lat. 50° 46' N., long. 17° W., the sun rise E. 20° S. by compass at 6h. 32m. A. M., required the variation ? h. Time Oct. 11th 18 32 O's declination at this time 7° 3' S. sin. 9.088970 Long. 1 8 Lat. 50 46 sect. 10.198953 E. 11 11 S. Greenw. time 1940 Obs. do. E. 20 0 S. sin. 9.287923 m. Fine amp. Variation 8 49, W. the true bearing being to the left of the observed. To compute the true azimuth o any celestial rithms will be the sine of half the azimuth, to object from its altitude, polar distance, and the be reckoned from the north in south latitude, latitude of the pluce of observation. and from the south in north latitude, eastward Add together the altitude, latitude, and polar when the latitude is increasing, and westward distance, and take the difference between half when it is decreasing. the sum and the polar distance. Then add to Then, when the true and observed azimuths gether the secani of the altitude, the secant of are both east or both west, their difference is the the latitude, rejecting ten from the index of each, variation, otherwise their sum is the variation, the cosine of the half sum, and the cosine of the westward when the true is to the left, and east. remainder, and half the sum of these four loga- ward when it is to the right of the observed. For, adopting the notation employed in investigating the rule for computing the meridian distance of a celestial object, we have (fig. 10, plate II.) a+1+p. a +1+p -0. sect. I. sect. a A 2 P 2 2 AZC rad. 2 AZC V cos. a +1+P a.+1+p 2 -p. sect. l . sect. a. rad. 2 Example. On Feb. 16th, 1928, in lat. 30° 14' N., long. by account 31° W. at 4h. 30m. P. M. the alt. of Ó was 12° 36' -, bearing S. W. } W. per compass, height of the eye fifteen feet, required the variation ? COS COS. 2 sin. or sin. 2 2 . COS. 2 2 From the observed altitudes and the distance of are given to find the angles PMS, Z MS, and the sun and moon, and the compass bearing of ei- the difference of those angles is the angle ther object, to find the latitude, longitude, and Z MP; and in the triangle Z M P are given ZM, variation of the compass. MP, and the included angle Z MP, to find ZP, With the observed distance enter the Nautical the co-latitude. Hence the time at the place of Almanac for the time of the month in which the observation, and the true azimuth of either observation is made, and take by inspection the object, may be found ; and the time compared day and hour of Greenwich time, corresponding with the Greenwich time, previously found from most nearly with that distance. To that time the distance, will give the longitude; and the take the moon's semi-diameter and horizontal true azimuth compared with the observed one parallax, and, clearing the distance, find the will give the variation of the compass. Greenwich time from it. If this time differ much Erample.-On September 2d, in the morning, from that before taken out by inspection, take the altitude of 0 was 15° 45' + of ) 58° 40 out the moon's semi-diameter and parallax again, distance of their nearest limbs 77o O 40', lat. and, again clearing the distance, find from it the by account 48° N., height of the eye twelve fert, true Greenwich time. For this time take the bearing of the sun S. E. f E., required the latipolar distance of both objects, and proceed with tude, longitude, and variation of the compass ? ihe computation thus : By inspection in the Nautical Almanac it is Let M (fig. 17, plate II) be the true place of readily seen that the Greenwich time of this obthe moon; S that of the sun; MP,SP, their po- servation must have been about 19h. of Septemlar distances; and MZ, SZ, their true zenith ber 1st. To this time the moon's semidiameter is distances; and MS their true distance. Then in 15'Oʻ, and hor. par. 55' 2" the triangles PMS, Z MS, all the sides in each . h. m. s. h. m. S. Obs. alt. ) 58 40 0 15 45 .2865 Aug. semid. 15 12 Semid. 15 53 Hor. par. prop. log. •5146 58 55 12 16 0 53 Prop. log. •8011 Dip 3 3 25 28' 27" 34 refr. )'s app. alt. 58 51 47 O app. alt. 15 57 28 27 53 3 10 27 53 corr. h. S. )'s true alt. 19 40 O true alt. 15 54 18 Obs. dist. n. Is. 77 0 40 12 $15 Semidiameters Zenith dist. MZ 30 40 20 Zenith dist. SZ 74 5 42 ( 15 53 . 59 Then in the triangle ZMP we have ZM = 30° 40' 20ʻ, MP 71° 28' 31', and angle ZMP 10.000000 cos. MX 30° 32' 21" 9.935145 30° 40' 20 9.773127 : cos. PX 40 56 10 9.878201 9.997701 :: cos. MZ 30 40 20 9.934549 Variation 30 18 19 W., or nearly 23 points W. ON NAUTICAL INSTRUMENTS FOR CELESTIAL and consequently by geometry the angle I is OBSERVATIONS. half the angle E. The instruments used by seamen for celestial It evidently follows, from what has just been observations are, the quadrant, the sectant, and demonstrated, that the plane of a distant object, the reflecting circle, which are all essentially the as seen by reflection from two parallel mirrors, same instrument, and depend on the following will be the same as that of the object itself; and general principles, viz. if an object be seen by consequently, if the image of a distant object as reflection from two mirrors, the angular distance seen by reflection from two mirrors coincide with of the object from its reflected image is double the object, we are certain that the mirrors are the inclination of the mirrors. parallel to each other. For let A B, C D (fig. 18, plate II.), be two If C D' be a mirror perpendicular to CD, mirrors, whose planes produced meet in I. Let then an eye at S' would see the image of S in the SF be a ray of light from an object S, reflected direction of S'G E, in which case the supplefrom the mirror A B, in F G, to the mirror CD, ment of the distance of the object from its image and again from C D, in D Е, meeting S F pro would be double the inclination of the mirrors; duced in E. Then to an eye at E the angle and consequently a distant object and its image as SEG, or S ES', would be the angular distance seen by reflection from two mirrors perpendicuof S from its image s', as seen in the direction lar to each other would appear 180° apart, or EGS', after reflection from the two mirrors. diametrically opposite to each other. Now from the principles of optics, the angles Fig. 1, plate III., is a representation of the SF A and G F B are equal ; and by geometry quadrant as it is commonly fitted up. PO is SF A and B F E are equal; hence FI bisects the graduated arc or limb of the instrument, the angle G F E. Again F G being produced to A B a mirror perpendicular to the plane of the II, by the principles of optics, the angles FGC instrument, attached to the fat bar K, which reand EG I are equal, and by geometry F G C volves with it round the centre, and carries at its and HG I are equal; hence ÉG E, the outward extremity a vernier scale for subdividing the de angle of the triangle G F E, is bisected by G I, visions on the limb. E is a mirror also perpet |