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 1 2 3 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
t 
 
 ./ •■ 
 
 ^1 
 
 
 IX o T k: s 
 
 r 
 
 -ON- 
 
 PRACTICAL ASTRONOMY 
 
 COMPILKD FOR THE UsE OF THE CaDETS 
 
 ^^ 
 
 /tPi 
 
 ji~ ;^ 
 
 —OF THE— 
 
 ROYAL MILITARY COLLEGE OF CANADA 
 
 -BV- 
 
 LiEUT.-CoLONEL J. R. OLIVER, R.A., 
 
 Professor of Surveying and Military Topography. 
 
 KINGSTON: 
 
 PRIKTED AT THfe DAILY NEWS STEAM PRINTINU HOUSE. 
 
 188<>. 
 
 w » I 
 
PR 
 
 ROY^ 
 
 Pn 
 
TVOTISS 
 
 — ON — 
 
 PRACTICAL ASTRONOMY 
 
 Compiled for the Use of the Cadets 
 
 — Of THB— 
 
 ROYAL MILITAEY COLLEGE OF CANADA, 
 
 —BY — 
 
 Lieut.-Colonel J. R. OLIVER, R.A., 
 
 Professor of Surveying and Military Topography. 
 
 KINGSTON: 
 
 PKHfTED AT THE DAILY NEWS STEAM PRINTING HOUSE. 
 
 1880. 
 
 J 
 
a 
 
 I'' 
 
 >\t 
 
V 
 
 fvl 
 
 I'' 
 
 CORRECTIONS AND ADDITIONS. 
 
 ^t 
 
 Page 5.— 2d line above Fig. 2 should be "In the case of the alti- 
 tude of the sun being measured. 
 
 Page 25— 2d line from top should be ''=8tar's declination ±Z6V' 
 
 Note to Page 7.— In the Nautical Almanac is given a method 
 of finding the latitude by an altitude of the pole star when not on 
 the meridian. The formula is : 
 
 Latitude=a— i? cos. h + ^p^ sin. 2 h tan. a. 
 When a is the star's altitude, h its hour angle, and ^ its north polar 
 distance in circular measure. 
 
 Note to Page 17.— Another way of finding the meridian very 
 accurately is by taking transits of two stars of nearly the same 
 Right Ascei'sion, one of which should be as near the pole, and the 
 otlfer as far from it, as possible. The difference of their declination 
 should not be less than 50°. The formula is : 
 /• , , ^) COS. d COS. d' 
 
 when d is the deviation (in horary units) Z the latitude, i5, «, and d 
 the sidereal times of transit, the R. A and the declination respec- 
 tively, of the star near the pole, and ^', o!, ^ those of the other star. 
 It is necessary to know the rate of the watch, but not its eiTor, as 
 the interval between the transits has simply to be corrected lor rate 
 and converted from mean into sidereal time. 
 
 lit-t-{a:-a) is positive the deviation will be east of north. If 
 negative the deviation will be west. 
 
NOTES ON PRACTICAL ASTRONOMY. 
 
 I 
 
 (Tlie following notcB arc intended in great measure to serve the 
 purpose of skeleton lectures, and presuppose a knowledge on 
 the part of the student of the elementary facts of Astronomy 
 and of the different methods of reckoning time. Blanks 
 li^ve been left for the necessary figures.) 
 
 In all extenoive surveying operations it is absolutely necessary 
 that the surveyor shoulcl know how to determme the variation ol 
 the compass; to lay down a meridian line correctly; to ascej-tain 
 the true local mean time ; and to find the latitude and longitude. 
 
 The instruments ordinarily used in the field for astronomical pur- 
 poses are the sextant (with artificial horizon), the tneodolite or 
 transit, the portable transit telescope, and two or three good 
 watches or chronometers. The pocket sextant is graduated to read 
 siiiffle minutes and the large sextant reads to ten seconds ol arc. 
 The artificial horizon may be of mercury or molasses In every 
 case an observed altitude, whether obtained by the theodolite or 
 sextant, should be corrected for index error before any further steps 
 are taken. A set of mathematical tables and the Nautical Almanac 
 for the year must always be at hand. 
 
 Latitude and Longitude are the co-ordinates by which the 
 position of a point on the surface of a sphere is determined. 
 
 Taking the earth as a sphere, the length of a degree of lorigitude 
 will be at the equator the same as that of a degree of latitude, but 
 will diminish from the equator towards the poles, where it becomes 
 nothing. The length of a degree of longitude at any place wi l be 
 equal to that of a degree of latitude multiplied by the cosine ol the 
 latitude of the place. 
 
 For practical purposes it is as well to imagine the earth as a 
 stationary sphere at the centre of the visible univrrse, and the 
 heavenly bodies as projected on the surface of another sphere, hav- 
 ing the same centre as the earth but at an infinite distance Irom it. 
 The apparent motions of the sun, moon, and planets are measured 
 and mapped on the surface of this great sphere, which latter appar- 
 ently makes a complete .evolution round the earth in a lew minutes 
 less than 24 hours. The fixed stars retain their relative places m 
 the sphere, but the other heavenly bodies appear to move on its 
 surface. 
 
» 
 
 • . 
 
 Tlio (!0-or<linjite8 by vvhicli tlio ixwitions of the lioftvonly l)<)dio8 
 on tlio sphcro am dotoriiiiiiccl uro called Declination and Riglit 
 Ascension : tlie first vtorrespondin^' to Latitude,- and tlie second to 
 Longitude. Tiiero is, iiowover, tins ditt'erenco— Longitude is reck- 
 oned eastwards 180° and westwards 180", starting from a given 
 meridian. Right Ascension is measured for the whole circle east- 
 wai'ds from 0" to 300", the zero point being one of the intersections 
 of the equator and ecliptic, technically known as the *' Ist point of 
 Ariei»." 
 
 THE LATITUDE. 
 
 The latitude of a place is the same as the altitude of the pole 
 above the horizon at that place. It is noHli if the north polo is in 
 sight, and vice versd It is usually found by the meridi .. altitude 
 
 of a heavenly body, iuch as the sun 
 or a star. If there is the slightest 
 
 Fig. L 
 
 / 
 
 3. 
 
 Q 
 
 doimt it is safest to draw a figure , the 
 
 plane of the paper representing the 
 
 plane of the meridian. Let A be the 
 
 place and S the heavenly body. We 
 
 will suppose the altitude of the latter 
 
 to be measured from the southern 
 
 horizon. Draw IIB tangential to ' 
 
 the earth's surface at A to represent 
 
 the horizon. Set off the angle BAS 
 
 for the measured altitude, and from 
 
 AS set off SAQ for the declination 
 
 of the heavenly body as obtained 
 
 from the Nautical Almanac. If the 
 
 latter is south ^^ will be above J./S' 
 
 and vice versd. The AQ will be the direction of the intersection 
 
 of the meridian and equinoctial, and if we set off AP at right 
 
 angles to AQ, P will be the position of the pole, and the angle 
 
 BAP will be the latitude. 
 
 Now ^J./'=90"-^^<S— >S'-4^=90°-altitude-declination; or 
 (if the declination is north)^90° + declination — altitude. 
 
 In the case of the altitude being measured above the horizon 
 below the visible pole, as in figure 2, we shall have- 
 Fig. 2. 
 
 Lat.=P^//. 
 
 = QAH-90°. 
 
 =HAS+SAQ-90°. 
 
 = Altitude + declination —90°. 
 This can only occur within the 
 tropics. 
 
1 
 
 i 
 
 -ff 
 
1 
 
 -ff 
 
 —1— 
 
 The altitude may be obtained either by the theodolite or sextant. 
 On land it is generally necessary to use an artificial horizon with 
 the sextant, if the meridian altitude of the sun is so great that its 
 double altitude is a large, angle than the sextant can measure a 
 star must be observed instead, and this is rather a difficult thing to 
 do. It is therefore, as a rule, better to employ the theodolite, it 
 must, of course, be carefully levelled and the index error ot the 
 vertical arc allowed for. T^he pole star is a convenient star to 
 employ on account of its slow motion, the radius of the circle it 
 describes in the 24 hours being only 1°— 20'. Fig. 3. 
 
 In figure 3 P is the £ole, iV the 
 north point of the horizon llNR^ S and 
 S' the positions of the star when on the 
 meridian at its upper and lower transits 
 respectively. Then SP=/S' P=Bta.r'6 
 north polar distance or co-declination— 
 S¥ or S'JV is the star's meridian alti- 
 tude, and JVF is the latitude. The 
 time at which the star will be on the 
 meridian must be found from the Nauti- 
 cal Almanac. The star's altitude is 
 taken at the right time, and corrected 
 for index error and refraction. Then 
 we shall have, Latitude = star's correct- 
 
 ed altitudeistar's J!^.F.D. according as the star is taken below or 
 above the pole. ^ ^S* *• 
 
 Should the star when at S be south 
 of the zenitli the case will be differ- 
 ent. This is shown in figure 4 
 in which the plane of the mper 
 represents the horizon. WI^SS is 
 the horizon— iV^ and S its north and 
 south points— Z the zenith— P the 
 pole— aS" and S" the star. If we ob- 
 serve the star at S" then its altitude 
 will be S"S, and the latitude (or JV^P) 
 will he I^S- PS" -S"S 
 or =180°— star's J!iPJ) -altitude, 
 = star's declination -f 90° —altitude. 
 
 It must be remembered that all quantities which change from 
 instant to instant, such as the equation of time and the sun s decli- 
 nation, and which have to be taken from the Nautical Almanac, 
 must be corrected for longitude. For instance, suppose that the 
 latitude of a place in the neighbourhood ot' Kingston had to be 
 obtained by a meridian altitude of the sun. The declination ot the 
 latter is given in the N. A. for apparent noon at Greenwich, and 
 when it is on the meridian of Kingston it is about 5h. 6in. p.m. 
 apparent time at Greenwich, and the sun's declination for that hour 
 must be obtained by a proportion. 
 
i 
 
—9— 
 
 Jr. 
 
 TO FIND TEE TIME AT WHICH A STAR WILL BE ON THE MERIDIAN. 
 
 The data required for this are the star's Right Ascension, and the 
 sidereal time of mean noon. The latter must be corrected as above 
 for longitude and is the hour angle through which the 1st point of 
 Aries has moved by 12 o'clock noon since it was last on the 
 meridian. 
 
 In figuie 5 let the plane of the paper represent the plane of the 
 equator, P the pole, PM the meridian and y the first point of 
 Aries, and S the star. Then yS is Fig. 5. 
 
 the star's Right Ascension and yM 
 the sidereal time of mean noon. Then 
 the star will be on the meridian at an 
 interval of sidereal time after mean 
 noon corresponding to the arc /6*J/, or 
 yS—yM. If the star was at S^ yM 
 being greater than yS\ it would have 
 passed the meridian by an interval 
 corresponding S'M^ or yM—yS'. 
 This interval must be reduced from 
 sidereal to mean time and then de- 
 ducted from 24. 
 
 Example.— To find at what time the pole star would be at its 
 upper transit on a day when the side- 
 real time of mean noon, corrected for 
 longitude, is 21h. 30m., the Right 
 Ascension of the star being taken as Ih. 
 15m. Here the star has passed the 
 meridian by an interval of 21h. 30ra. — 
 Ih. 15m., or 20h. 15m., and will there- 
 fore be on the meridian in 3h. 45m. 
 (sidereal) after noon. 
 
 The same result is also obtained thus: 
 
 From 24h. Ora. 
 
 Subtract 21 30 =arc ySM 
 
 Fig. 
 
 6. 
 
 2h. 30m. = arc My 
 Add 1 15 =arc yS 
 
 3h. 45m.=arc Jf/^ 
 As in the case of the latitude it is always safest to draw a figure. 
 
 The watch or chronometer used should be one whose rate of 
 gomg can be depended on— that is, it ought to gain or lose the 
 same amount in equal timesi 
 
 In fixed observatories where the clock is regulated to show sidereal 
 time, and the horiR are numbered from to 24, the clock's rate is 
 
A\ 
 
 continually tested by star transits. But in the field it is not so easy 
 either to check the rate or to keep the chronometer in good work- 
 ing order, since the mere moving abont tends to derange its rate. If 
 the watch gained or lost at a regular rate and its error at a certain 
 instant was ascertained, then the true time could be found at any 
 other instant by applying the rate. But no chronometer is perfect, 
 and the best is always subject to two errors — the " Rate" and the 
 " Error of Rate," the latter term meaning that the watch's rate is 
 liable to vary from time to time. If the surveyor stays for a few 
 days at any particular place he can always determine the rate at 
 that time, and if he has more than one watch he can soon find out 
 which has the least error of rate. 
 
 The chronometers used at sea are set to show Greenwich mean 
 time. On land, it the longitude of a station and the true local mean 
 time are known, Greenwich mean time can of course be at once 
 found. One watch may be set to show Greenwich and the other 
 local mean time. 
 
 OIVKN THE .SIDEREAL TIME AT ANY INSTANT TO FIND THE MEAN TIME. 
 
 The Nautical Almanac gives the sidereal time of mean noon at 
 Greenwich for every day in the year. That is, the sidereal time 
 that has elapsed at 12 o'clock since the first point of Aries passed 
 the meridian. To find the sidereal time of mean noon at any other 
 place we must add or subtract 9s. 8565 for each hour of longitude, 
 according as the latter is west or east. Let T be the local sidereal 
 time, t the sidereal time of the preceding mean noon, found as above. 
 Then T—t is the interval in sidereal time which has elapsed since 
 mean noon. This, converted by tables into mean time, gives the 
 hour. 
 
 For instance. In 75°, or 5h. west longitude on a certain day and 
 liour we might have — 
 
 Sidereal time 21h. 9m. 24s. 
 
 Sidereal time of preceding noon at Greenwich. .18 47 
 
 2 22 24 
 Subtract correct-cu for longitude 49*28 
 
 Interval in sidereal time from mean noon 2 21 34*72 
 
 which may be converted into mean time. 
 
 Conversely, when mean solar time is given and we want to find 
 the sidereal time, we take the interval from preceding mean noon 
 and convert it into sidereal time. Add to this the sidereal time of 
 mean noon (taken from the N. A. and corrected for longitude) and 
 we have the sidereal time. 
 
 Sidereal time is usually found by calculating the hour angle of a 
 star from its observed altitude. This, added to the star's Right 
 Ascension if the hour angle is west, or subtracted from it if east, 
 
c> 
 
—13— 
 
 TO FIND TIIE LOCAL MEAN TIME, AND THENCE THE LONGITUDE, BY AN 0U8ERVED 
 
 ALTITUDE OF A HEAVENLY BODY. 
 
 To do this the latitude of the place must be known. The altitude 
 should be taken at a time when the Fig. 7. 
 
 heavenly body is rapidly rising or 
 falling — that is, when it is at some 
 distance from the meridian. In figure 
 7 the circle is the horizon, P the 
 pole, Z the zenith, S the heavenly 
 body, Sll'ite observed altitude. NPZ 
 will be the meridian, ZPS will be a 
 spherical triangle, in which PZ is the 
 co-latitude of tlie place, PS the co- 
 declination of the heavenly body, and 
 ZS its co-altitude. The three sides of 
 this triangle being thus known we 
 can find the hour angle ZPS from 
 the formula. 
 
 Sm PS Sm PZ 
 when 8 is the semi-perimeter of the triangle. 
 
 Dividing the hour angle by 15 will give its value in time— say T. 
 If the body observed is the sun we shall now have the apparent 
 solar time, and by adding or subtracting the equation of time we 
 shall obtain mean time. 
 
 If the body is a fixed star or planet, from its known R. A., sub- 
 tract T if the hour angle is east, or add it if it is west. This gives 
 the sidereal time of the instant, from which the mean time can be 
 inferred. The watch or chronometer time is noted at the instant 
 the observation is taken. If the watch shows local mean time its 
 error is at once obtained. If it shows Greenwich mean time the 
 difference between this and the calculated local mean time gives the 
 longitude. 
 
 20th February, 1880. Long. 5 h. 30m. west. 
 
 Hour angle of a Tauri was 2h. 30m. 178. west, at 9 p.m. by watch. 
 Find watch error : — 
 Star's R. A. = 4h. 29m. 48. 
 Hour Angle =2 30 17 
 
 6 59 21 = Sidereal time of the instant. 
 Sidereal time of mean noon=21h. 59m. 14s. 
 
 Correction for longitude 
 
 55 
 
 22 
 
 
 
 9 
 
 24 
 
 
 
 1 
 
 59 
 
 51 
 
 6 
 
 59 
 
 21 
 
 (N.B. 
 
 8 59 12=Sid'l interval since m.n. 
 =8 57 44 in mean time. 
 Or the watch was 2m. 16s. fast. 
 
 -A sidereal hour = a mean hour- 10 seconds, nearly.) 
 

 -.L-„ 
 
—16— 
 
 If tho theodolite or transit is set iip so that when the horizontal 
 plate is clamped the telescope moves in the plane of the merid.aTi 
 Se true time^an be very easily found by noting the time of the 
 trans t across the central wire of the sun or a star In the former case 
 the mean of the transits of the east and west limbs is taken for that 
 of the centre. This gives apparent noon, and by applying the equa- 
 to of time mean n?on can L found. Ab an exauipie : The mean 
 of the times of transit of the two limbs was llh 45m. by the >va tch 
 on a day when the equation of time was 15m. 2s. to be Bub tract a 
 from apparent time. To f nd the watch error- At the ^"stan of 
 ransit it was apparent noon or 12h., and subtracting 15m. 2s. from 
 tWs we have llh Mm. 58b. as the correct mean time of the instant. 
 The watch was therefore 2 seconds fast. 
 
 If a star is used the case is different. The star's Right Ascension 
 (found from the N.A.) is the sidereal time ot its transit, and this has 
 to bo converted into mean time. For instance: On a certain day 
 the transit of a star was observed at 8 p^m. The star s RA was 
 13h 53m., and the mean time corresponding to sidereal time 16ii. 
 53m. on that day was 8h. 3m. Hence the watch was 8 minutes slow. 
 
 THE IKEBIDIAN. 
 
 The meridian can be found with tolerable accuracy by the method 
 of equal altitudes of a star. A still better way is by a theodolite 
 observation of a circumpolar star (say Polaris) 
 at its greatest elongation. Let P be the pole, 
 S the star, Z the zenith. Then FZS i^ a 
 spherical triangle, right-angled at 6. /"^ is 
 the star's co-declination, and PZ Jhe co-lati- 
 tude of the place. Sin FZS sin PZ=sin P6 
 or, if I be the observer's latitude and a the 
 star's declination, 
 
 SinPZfcS^ 
 
 cos 
 
 I need be only approximately known. 
 
 When the star is at its greatest elongation it 
 will move for some time along the vertical 
 wire of the theodolite. The time for this must 
 be found from the Nautical Almanac and the 
 equation (in the case of Polaris) is 
 
 Sidereal time of G.E.= — star's R. A. ±6 hours. 
 
 Turn the axis of the telescope on the star, the horizontal plate 
 being clamped at zero. Then make the vernier read the angle PZ6, 
 and, bv means of a lantern, fix a picket exactly in the centre ot the 
 cross wires. This picket will be due north of the theodolite. 
 
—17— 
 If tbo 8tar'8 N. P. D. bo conBiderablc equation (a) will become-. 
 Sidereal time of G. E. = star's R. k.±<ZPS (in time), and the 
 ancle ZPS must be found froni the equation 
 ^ Cos. ZPS=tm. N. P. D. xtan. latitude. 
 
 The star's altitude is given by the equation 
 
 sin. latitude. 
 Sin. altitude=^^g^ ^j. j). 
 
 The meridian may be still more aeeurately found by the superior 
 and inferior transits of so.ne star very near the ^ ig. »• 
 
 Dole, the clock rate having been tonnd by 
 transits of several stars on two saccessive 
 nTghts In the figure ZSS' is the vertical 
 ;iLe of the instrument, S and S t/^e pooi ions 
 
 ^ =15 (^'-<-12h) (in sidereal time). 
 
 Again, in triangle ^^'f^we ji»^^^^, 
 Sin. Wsin. FZS=^m. FS sin. ^^^_^^^^ 
 
 or cos. I sin. PZS=<iOB. d sin. (15 ^ 
 
 which gives PZS. Ift'-t be less than 12h. the 
 line if&' will be west of ZP. 
 
 Church bears 76° 41' west of true north. 
 
 THE DETERMINATION OF LONGITUDES BY SIGNALS. 
 
 T Pt ^ and B be two stations, and let the exact local time at each 
 be known Then if a phenomenon occurs of an instantaneous char- 
 acter which is visible at both places, the difference between the 
 'ocal thnesof its occurrence is the difference of the longitudes ol 
 the stations. 
 
 ia\ From a point between ^ and ^ a rocket may be thrown up, 
 and the instant of its explosion be noted at both places By means 
 o?l series ot'such observations the difference between the longitudes 
 of London and Paris was found. 
 
 (h) The eclipses of Jupiter's satellites also serve as signals. For 
 this purpose the times of their occurrence are registered in the 
 NantCT AUi anac. They give very good approximations to the 
 W^de^ when; both the disappearance and reappearance 
 
 of the same satellite are visible. 
 
(A Instead of Bijjnal. the etations may bo connoctod by telegraph 
 (o) A"8\«'^^'?;.^;^'"^^^ bv proper mechanical contrivancctt, to 
 wiroB. Then it is uobsidio, uy pruw«i •"« . mi :,.»,„., 
 
 • / . ..f « Mu^ titno of a traiiB t that takes place at 71. ino inici 
 
 viding the true local time at each is known. 
 
 TO FIND THE VARIATION OF THK COMPAH« BY AN AMPLITUDE. 
 
 The amplitude of a heavenly body is at its angular distance from 
 tl JeLror^ west points of the fiorizon at its rismg or setting. 
 
 Let X be the heavenly body in the Fig.lO. 
 
 horizon. Its compass bearing is 
 noted. Then in the qiiadrantal tri- 
 angle FZX are given P^^^oAati' 
 tude /'X=co-decrination or N.I .JJ. 
 ofliie object, and ^X=90; We 
 have to And the angle ^^ ^1?^ 
 thence its complement X^IL ine 
 
 amplitude. ^ 
 
 Cos PX=sin.PZco8.i'ZX 
 
 or Sin. decrn=cos. lat. sin. amplitude. 
 .-. Sin. amplitude=sin. decl'n. sec. lat. 
 ' Of course the difference between 
 this calculated value of the amplitude 
 and that given by the compass will 
 be the variation. 
 
 TO FIND THE VARIATION BY AN AZIMUTH. 
 
 To do this the altitude of a heavenly body is observed with the 
 sextant at the same instant that its compass JJig- li- 
 
 bearing is noted. 
 
 Let X be the heavenly body at the instant 
 of observation, P the pole, and ^ the zenith. 
 Then, in the triangle PZXZXv^^h^=^^__ 
 -altitude, PZ=tKe co-latitnde luid PX- 
 the co-declination. The angle f f\will bo 
 the azimuth angle required, and is obtained 
 from the equation-^ ^__^^^^.^^^_^^^ 
 
 Sin.3 h PZX= mrPZ^^^^^ZX 
 
 where s is the semi-perimeter. 
 
—21— 
 
 THE PORTABLE TRANSIT INSTRUMENT. 
 
 This instrnment is generally used Fig. 12. 
 
 on extensive surveys, as the local 
 time, latitude, and longitude can be 
 very accurately found by it. 
 
 It consists of a telescope mounted 
 0-1 Y's, as shewn in the figure, and 
 adjusted so that its axis ^"oves in a 
 vertical plane, generally *^*^f *„'^ 
 meridian. In t^e eye-piece are one, 
 three, or five vertical wires, illumi- 
 nated (at night) by a lantern on one 
 of the trunnions-on the other trun 
 Dion is a spirit-level and circular arc 
 for setting the telescope at any 
 required angle of elevation. There 
 is also a level for getting the two 
 Y's on the same level. 
 
 There are three adjustments tor 
 
 the instrument. 
 
 (1) To get the Y's on the same level. 
 
 m To get the central wire (or the 
 line of cfilimation) into its correct 
 position as regards the axis of rota- 
 tion- , . ^. Til of in to make the axis of the 
 
 (3) To correct for dev,aUo.. Th» , ^ ^ n,ake ^^.^ ^^ 
 
 {;l:rof%=/3xIctVe"t Id west when thU adi«st«.e«t ha, 
 
 ^rt adjustment i, ^^ ZV^^^^^'^'^ 
 moving tUo instrument m azimuth »'» "'«J^?^^^^ ,,„« having been 
 point placed due north o^^^^^.f ^ ^ ''^Xmarn tL teleselpe is 
 previously ascertamed Jj ^^^ on the meridian mark, f the 
 
 'trJaking a transit the time, of the object passing each wire is 
 
 „i"d::nd^the n.ea„ fall the t.mes take... ^^.^ 
 
 The transit of a star gjves .he ^'de'^^l mo » ^^^^^^^^ ,^._^^^ 
 
 being tl,e same a. the star « ^'f '* j^^^^d „ the usual way. The 
 
 'ir ittiiriit IlLSral^lotr r good wat^hf the rate 
 of which is known. 
 
—23— 
 
 TO FIND THE LATITUDE BY OBSERVATIONS OF STARS ON THE PRIME VERTICAL. 
 
 The prime vertical is the great circle passing through the zenith 
 at right angles to the meridian. By means of a transit mstrument 
 placed in this position (that is with Fig. 13. 
 
 the vertical arc lying due east and 
 west) the latitude can be very accu- 
 rately found. 
 
 Let AB be the meridian, O'D the 
 prime vertical, Z the zenith, P the 
 north pole, and /S or S' the star in the 
 prime vertical. Let the sidereal time 
 of the transit be t, and a the star's 
 R. A. Then the hour angle SPZ 
 will be a—t. Let d be the star's 
 declination, and I the latitude. 
 
 Now in the right-angled triangle 
 PZSwe have 
 
 Cos. P=tan. PZcotan. PS 
 or Cos. (a— ^)=cotan. Z tan. d 
 
 This equation gives I very accurately, even with a portable 
 instrument. 
 
 If both the east and west transits of a star are taken half the sum 
 of the titnes ought to be the same as the star's R. A., and the error 
 of deviation may be thus corrected. It should also be noted that it 
 we reverse the telescope on its supports any error of collimation or 
 inequality of pivots will produce exactly contrary effects on the 
 determination of the latitude. The star may be taken with the 
 telescope in reversed positions, either twice the same day, or once on 
 two successive days ; the mean of the two latitudes obtained bemg 
 taken. 
 
 If the watch used goes tolerably well the latitude can be found 
 by this method without any reference to sidereal time by taking^ the 
 time of the east and west transits and reducing the interval from 
 mean to sidereal time. This will give the angle SPS', half of which 
 is SPZ. A star should be chosen which culminates a little south 
 of the zenith, and the telescope should be reversed in its supports 
 between the two transits. 
 
 With the instrument called the Z'-nith Sector the latitude can be 
 found with extreme nicety. This instrument moves in the plane of 
 the meridian and measures the zenith distances of stars which cul- 
 minate near the zenith (and which are therefore unaffected by 
 refraction) with great accuracy. Certain stars whose declinations 
 are very exactly known are preferred for this. From the measured 
 zenith distance* of the star and its known declination we find the 
 zenith distance of the pole, which is equal to the co-latitude. 
 
—26— 
 
 Fig. 14. 
 
 Latitude=90°-PZ 
 
 —90°— (star's declination ±Z6) 
 
 INTEEPOLAXION. 
 
 In taking out variable quantities from the Nautical Almanac it is 
 necessary to interpolate for the.local time and longitude of the place 
 of observation, since the data given are for Greenwich time. 
 
 If the rate of clmnge of the variable quantities is itself variable 
 we must allow for this if we wish to obtain a very accurate result. 
 
 As an instance : we wish to iind the sun's declination at appar- 
 ent noon on the 2d January, 1880, at a place in longitude 60 or 
 4h. west. 
 
 For Greenwich mean noon we find in the N.A. 
 
 Date. 
 
 2nd January 
 3rd 
 
 (( 
 
 Sun-s declination. 
 
 22° 57' 16"-2 
 22° 51' 45' -4 
 
 Variation in 1 hour. 
 ... 13"-21 
 ... 14"-35 
 
 Now, at apparent noon at the place it will be 4 p.m. apparent 
 time at Greenwich, and we take the variation at 2 p.m. Green- 
 wich as the a/verage variation for those 4 hours. 
 
 This variation is 13"-305, which, multiplied by 14.35 
 
 4 gives 53"'22 to be subtracted from the declina- 13-21 
 
 tion of 2d January — 
 
 22° 57' 16"-2 
 
 53''-22 
 
 22^ 56 22 -98— required declination. 
 
 12) 1-14 
 •095 
 13-21 
 
 lt> OKfO — Bt a P.M. 
 
 53-22 
 
 INTERPOLATION BY SECOND DIFFERENCES. 
 
 The difference between two consecutive tabulated numbers is 
 called a "first difference," and between two consecutive first differ- 
 ences a "second difference." In many astronomical problems second 
 differences have to be taken into account in interpolating. The 
 method can be best shown by an example or two. The formula 
 employed is 
 
 Where A is half the sum of two consecutive Ist differences, and 
 B is half their difference. The signs of A and B must be noted. If 
 the numbers are decreasing the Ist differences are negative, and if 
 the let differences are decreasing the 2nd differences are negative. 
 
—21— 
 
 Ex. 1.— Given the logs, of 365, 366 and 367 to 7 places of deci- 
 mals to determine log. 366*4. 
 
 Numbers. 
 
 Log. 
 
 l8t Differ'ce. 
 
 3d Differ'ce. 
 
 365 
 366 
 367 
 
 5622929 
 5634811 
 5646661 
 
 11882 
 
 11850 
 
 —32 
 
 Here k is j%, ^=11866, and ^=-16. 
 
 5634811 
 4746 
 
 3639557 
 3 
 
 3639554 
 The tables give the log. as 3639555. 
 
 11866 x^ 
 4 
 
 47464 
 '¥X (A)' = -3, nearly. 
 
 If the second difference had been neglected— *'.e., if we had 
 worked by simple interpolation, the result would have been 5639551. 
 
 Ex. 2.— Given log. cos. of 89° 32', 89° 33', and 89° 34', to find 
 
 ono OO' 1 K^ 
 
 log. COS. 89° 33' 15-^ 
 
 Log. COS. 89 32=7-9108793 
 Log. COS. 89 33=7-8950854 
 Log. COS. 89 34=7-8786953 
 
 Ist Difference 
 
 -157939 
 -163901 
 
 Snd Difference 
 
 -5962 
 
 Here we have to subtract ^ x half the sum of the Ist differences, 
 and (f|)2 xhalf the second difference, or 40416 in all 
 
 .-.log. COS. 89° 33' 15"=7-8910438. 
 
 TO FIND THE LONGITUDE BY TRANSITS OP MOON-CULMINATINO STARS, 
 
 A surveyor provided with a portable transit telescope and a watch 
 that keeps good time can obtain his longitude wit' considerable 
 accuracy by taking transits of the bright limb of the moon and of 
 certain stars, the R. A. and declination of which are nearly the same 
 as that of the moon at the time. It is not necessarjr to know either 
 Greenwich or local time. All that has to be done is to set up the 
 telescope in the plane of the meridian, note the times of the transits, 
 and reduce the interval between them from mean to sidereal time. 
 
—29- 
 
 Fig. 16. 
 
 In the N. A. are given, for every 
 
 day in the year, the sidereal times 
 
 of transit at"Gieeuwi(h of the moon 
 
 and of certain suitable stars, called 
 
 <'moon-culminating" stars ; also the 
 
 rate of change per hour (at the time 
 
 of transit) of the moon's R. A. As 
 
 the moon moves rapidly through the 
 
 stars from west to east it is evident 
 
 that at a station not on the meri- 
 dian of Greenwich the interval be- 
 tween the two transits will be 
 different from that at Greenwich ; 
 and the nioon's rate of motion per 
 hour being known a simple propor- 
 tion will (if the station is near the 
 meridian of Greenwich) give the 
 
 difference of time between the station and Greenwich, and thence 
 the longitude. If the station is far from the meridian of Greenwich 
 a correction will have to be made for the change in the rate of change 
 of the moon's R. A. The rate of change at the time of transit at the 
 station is found from the N. A. by interpolation by 2nd differences, 
 and the means of the rates of change at Greenwich and at the sta- 
 tion is taken as the rate for the whole interval of time between the 
 transits. 
 
 An example will best illustrate the method : 
 At a certain station to the west of Greenwich on the 25th Oct. 
 the interval between the transits of a Tauri and of the moon's bright 
 limb, reduced to sidereal time, was found to be Oh. 4m. 28. 
 First, to find the approximate longitude— 
 
 Greenwich Transits. 
 
 a Tauri 4h. 28m. ISs. 
 
 Moon's 2d limb. 4 2 26 
 
 25 
 
 52 
 
 4 
 
 2 
 
 21 
 
 50 
 
 60 
 
 
 From the N.A. we find 
 that the moon's R.A. 
 changes 1618.17 per 
 hour. 
 
 iW.f 
 
 r=8-127. 
 
 13108. 
 The approximate longitude is 8h. 7m. 378. 
 
 To find the correct longitude we have to determine the variation 
 of the moon's R.A. in one hour at the station. From the N.A. we 
 find 
 
 On the 25th at lower transit it was 159s- 67 
 " " upper " " 161 -17 
 
 « 26th lower " " 161 -88 
 
 1*50 difference. 
 0-71 " 
 
—31— 
 
 By interpolation by 2d diiferencea wo find tlic rate of change at 
 the station to be about 1618-73. 
 
 And the corrected longitude is 8h. 6m. 50s., which is 47 seconds 
 less than the approximate longitude first found, or a diiference of 
 about oi miles in the latitude of Kingston. 
 
 TO FIND THE LONGITUDE BY LUNAR DISTANCES. 
 
 The moon moves amongst the stars from west to east at the rate 
 of about 12° a dav. Its angular distance from the sun or certain 
 stars may therefore be taken as an indication of Greenwich mean 
 time at any instant— the moon being in fact made use of as a clock 
 ill the sky to show Greenwicli mean time at the instant of observa- 
 tion. Tiie local apparent time being also supposed to be known, 
 and thence the local mean time, we have the requisite data tor 
 determining the longitude of a station. 
 
 In the Nautical Almanac are given for every 3d hour of G.M.T. 
 the angular distances of the apparent centre of the moon from the 
 sun, the larger planets, and certain stars, as they would appear froni 
 the centre of the earth. When a lunar distance has been observed 
 it has to be reduced to the centre of the earth by clearing it of the 
 efi'ects of parallax and refraction, and the numbers in the N.A. give 
 the exact G. M. T. at which the objects would have the same 
 
 distance. . . , «. i? n j 
 
 It is to be noted that though the combined eftect ot parallax and 
 refraction increases the apparent altitude of the sun or star, in the 
 case of the moon, owing to its nearness to the earth, the parallax is 
 greater than the refraction, and the altitude is lessened. 
 
 Three observations are required— one of the lunar distance, one of 
 the moon's altitude, and one of th« other object's altitude. The 
 clock time of the observations must also be noted. The sextant is 
 the instrument used. All the observations can be taken by one 
 observer, but it is better to have three or four. If one of the objects 
 is at a proper distance from the meridian the local mean time can 
 be inferred from its altitude. If it is too near the meridian the 
 watch error must be found by an altitude taken either before or 
 after the lunar observation. 
 
 Four or five sets of observations should be made and written down 
 in their proper order. 
 
 Time by watch. Alt. of star. Alt. of moon's lower limb. Dist. of moon's far limb 
 Ist obs'n .... • • • • • • • • • • • • 
 
 2nd II .... •••• ••"• .••• 
 
 3rd II •••' *••• 
 
 4th M • • • • • • • • • • • ' • • * • 
 
 4) 
 
 Totals. 
 
 Mean 
 
It 
 
 « 
 
Fig. 16. 
 
 —aa— 
 
 If tliere is only one observer it in best to take the observations in 
 the following order, noting the time by a wateh. lat, alt. of sun, 
 star or planet ; 2<1, alt. of moon ; 3(1, any odd number of distanccH ; 
 4tli, alt. ot moon ; ntli, alt. of sun, Htar, or planet. Take the moan 
 of the distances and of their times. Then reduce tiio altitudes to the 
 moan of the times; /«., form the pro[)ortion— ditference of times 
 of altitudes : dilK of alts. ::diti'. between time of 1st alt. and moan 
 of the times : a fourth number which is to be added to or subtracted 
 from 1st alt. according as it is increasing or diminishing. This will 
 give the altitudes reduced to the mean of the times, or correspond- 
 ing to that mean. 
 
 The altitudes of moon and star must be corrected as usual, and 
 the augmented semi-diameter of the moon added to the distance to 
 give the distance of its centre. The lunar distance has then to be 
 cleared of the effects of parallax and refraction. 
 
 TO DETERMINE THE LUNAR DISTANCE CLEARED (Hr PARALLAX AND REFRACTION. 
 
 Let Z bo the observer's zenith, Zm 
 and Zh the vertical circles in which 
 the nu)on and star are situated at the 
 instant of observation. Let m and h 
 be their observed jdaces, M and -6' 
 tiioir places after correction I'or par- 
 allax and retraction: then Zm, Z«and 
 ms are known by observation : 7M 
 and ZS are obtained by correcting 
 the observations. The object of the 
 calculation is to determine MS. 
 
 Now, as the angle Z is common to 
 the triangles wiZv and MZS, we can 
 tind ^ from the triangle m^« in which 
 all the sides are known. Next, in tri- 
 angle MZS there are known MZ^ ZS, 
 and the included angle Z, from which 
 MS can be found. MS is the cleared 
 Lunar Distance. The numerical work 
 of this process is tedious. 
 
 The cleared distance having been obtained we proceed in accord- 
 ance with the rules given in the N. A. 
 
 The G. M. T. corresponding to the cleared distance can be found 
 either by a simple proportion or by proportional logs. 
 
 It admits of proof that if Z> be the moon's semi-diameter as seen 
 from the centre of the earth (given in N. A.) D' its semi-diameter 
 as seen trom a spectator in whose zenith it is, D" its serai-diameter 
 as seen at a point where its altitude is A, then 
 
 D" — D=^{iy—B) sin. A. verv nearlv. 
 

 ' 
 

 
 —35— 
 
 Snpposinjij a traveller to have entirely lost his reckoning, but to 
 be furnished with the requisite instruments, viz : Sextant, Artificial 
 Horizon, two Chronometers — one to show CTreeiiwich, the other 
 loeal time. Nautical Almanac, and set of Nautical Tables. He can 
 proceed as follows : — 
 
 First, by means of a meridian altitude of the sun he could find 
 his latitude approximately (not exactly, for not knowing Greenwich 
 mean time he cannot be certain of the sun's declination.) 
 
 In the afternoon, by means of a measured altitude, he could deter- 
 mine the error of one of the chronometers which indicates local 
 time. This also would be only approximate, since he uses an 
 approxinuUe value <»f the latitude, and is ignoratit of Greenwich 
 mean time. 
 
 If for any reason he caimot take the meridian altitude of the sun 
 he can ascertain the error of the watch by means of equal altitudes. 
 This again would only be an approximate value, since he does not 
 know tlie rate of his watch. He nn'ght also use any two altitudes 
 for the watch error and latitude. 
 
 By means of a Lunar Distance he can then determine Greenwich 
 mean time. 
 
 Lastly, by repeating the observations, he can determine all the 
 above quantities correctly.