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L'exemplaire film6 fut reproduit grice d la g6n6rosit6 de I'dtablissement prSteur suivant : Bibliothdque nationale du Canada Les cartes ou les planches trop grandes pour §tre reproduites en un c«n| clich6 sont film6es d partir de I'angle sup^rieure gauche, de gauche d droite et de haut en bas, en prenant le nombre d'images n6cessaire. Le diagramme suivant illustrs la mdthode : 1 2 3 4 5 6 ■■<— ■■ — I.*, ■aw If n mSSMBm EXAMPLES OF ASTRONOMIC AND GEODETIC CALCULATIONS y *. 1 EXAMPLES OP ASTRONOMIC AiND GEODETIC CALCULATIONS FOR THE USE OF LAND SURVEYORS V By Capt. E. DEVILLE, P. R. A. S. Late officer of the French Navv Dominion Geographical Surveyor Provincial and Donninion Land Surveyor &c / QUEBEC PRINTED BY P. G. DELIbLE 18V8 4 T UOVKRNMENT OF CANADA DKPAinWIl^NT OF THE INTERIOR DOMINION LANDS OFFK'K liOAIM) OF KXAMINKK's FOK l)(rMlM().N i,a.\1) SUI.' VKVolJr Moved hy Mr. Andrew Russell, s.vonded l)y Miijcr Web!) iilld )'es()l\'(>(l : •• That the Secretary Ix' directed lo <()nuiiuiiicate to •• Capt. DeviijJ': the appreciation oT the value ol" his • work by the l)oard, us an aide nienu)ire to Surveyor^; •• in the Held." -The IJoard is particularly pleased with the solution • g-ivon by him, ibr practical pur])oses, ol" certain (reo- • detic i>roblems by ihe method set Ibrth, dejiendini.- on • the convero-ence ol" Meridians; and also ol" (^ipt ])]■:- ville's labor in calculating- the orio-inal and very ' useiiil ta})les appended to the work."" AHBREAIATIONS. M. T.— Mean time. A. T. — Apparent time. S. T.— Sidereal time. Crr. — Grreenw^ieh . H. C. R. — Horizontal circle readinu'. ch, chs, — Gunter's chain or chains. I ^ TABLE OF CONTENTS. CHArTER I. DEFINITIONS AND PRELIMINAKY OPEHATIONS. EPHEMEI5IS. Art. 1. 2. 3. 4. 5. 0. 7. 8. I>. 10. 11. 12. 13. 14. l>»j: Division of the circle j^ To convert an arc into time 12 To convert time into arc j;; T™^ '^Z'Z'Z'. 13 To convert astronomical into civil time 14 To convert civil into astronomical time 1 r, To find the Greenwich time corresponding to the time of another place | r, G-iven the time of a place, to find the time of another place 2 ^^ Interjiolation -t^ Conversion of the simultaneous times of a place.. 21 To convert apparent into mean time and con- versely 21 To convert a sidereal interval into a mean one and conversely o^, To convert mean into the corresponding- sidereal time o» • «i') To convert sidereal into the corresponding mean time " .,_, Alt. riix«». |.>. To liiul tin' mean linic ol' ilif sun's or a istar's t raiisi t i'/) 1(1. liovvcr truiisils 2(1 17. Solar and stellar ()l)S(M'vations '21 ]X. Corrcclioii ol' altitudes •'.() ('IIAITEK' II, TIME. 1!>. Dellnitious ni ilO. To liiul the time )>y the sun's meridian l.iansit... .'!2 '2\. To liud the time by a star's meridian transit -I'l '2'2. To liiid the name ol'an unknown st:ir passin«»' the meridian 08 2-\. To lind the time by the su)i"s altitu ('iLArTr:K iv. AZIMI'TII. 2i». Delinitions.... 47 •')(). To di'duee, from an observation, the astronomical azimuth of a line or object 4S 81. A/imuth ])y the meridian transit of the ])olestar. 4!< I Alt. I'nRi'. '■\-2. A/iiiiulhUy t lie <'l(»ni>'ati()ii ol' tlic pole st;>r •')! M.'). Axiuiutli 1)\ *>l)s(i\ .itioii orilic ]>()1(' star ill :ui\ time ')'■'> o4. Cliiii)ljic solution ol ihc imMcdino- ])i()l)l('in '>' H.'). Azimuth by oljscrvntion ol' PoliniN when vertical with other .stars oS :M'). A/imuth hy altitudes of the s\ni <»<» 87. Aximuth l>y altitudes of stars (14 HS. (Jraphic solution oi'the two i)i'ec(Mlinii' ]>rol)leins. ()4 iV.l To liud the variation oTthe eompass i!i! CHArTEli V. ('();VVE1!(J?:NC'K ok .mki.mdi.ws. 40. J)eIinitions (Ill 41. To liud the converaenee ol' meridians ])etween two i)oints '<• 4"J. To rel'er to the meridian of a point 1>, an azimuth reckoned from the nu-ridian of another point A . Ti 4o. To lind the differences of latitude and longitude between two points when their distance and azimuth are known 7- 44. (riven the azimuths of a line at l)oth ends, and the mean latitude, to lind the length of the line 74 45. Greodetic surveying* 7'') 46. Plotting the survey SI 47. To run a line Sl> 48. To lay out, on the ground, a ligure of which the plan is given correcting the course by astro- nomical observations ,S;) 40. Griven the azimuths of sides, to lind the angles of a trianule S4 All I'liKP. .')(). To produi'f^ a p;i,rallt'l o{' laUtudc by layiiii^ oui (•hords of a u;'iv4. V.'Hioal dial ;>4 TA1!1J<:S. TAHi.r, I. Mean n»iVa(;tion !♦? II. Sun's parallax in alliiudc "••7 III. Times oi' elongation of the pole star U^ IV. A/imuth of th<' pol9 V. For lindini»th<' a/imuth ol' Polaris \vh«^n vertital with other stars 100 VI. Quantities havint^ relation to the Ijo-ure and dimcnisions of the Earth 107 VII. Correction to apply to the timtssht^wn by a sun dial 109 i \ PREFACE. In almost all countries, the duties of a Land Surveyor consist only oi'Iaying- out and dividing land; geodetic surveying belongs to the business of the civil and mili- tary engineer corps, composed of men who have mastered the highest branches of science. We have no such bodies here, and the Land Surveyor is often called upon to un- dertake extensive topographic surv^^s and those of geo- graphic exploration. But, in what books is he to hnd the knowledge necessary to fuliil those duties ? I w^as often asked the question and could not answer ; the works on the subject, intended for eminent mathemati- cians, are too abstruse ; w^hat is needed, and w^hich I have endeavoured to prepare, is a treatise pre.s- nting under a practical and elementary form, the solution of the problems most frequently met with in practice. I had to alter some of the usual methods, and in some instances to devise entirely new ones, in order to main- tain the same simplicity through out. Having only my own experience to direct me in this work, I know it presents many imperfections, I hope nevertheless, that as a whole, it will prove useful and acceptable to sur- veyors generally. 2 li: 10 PREFACE. Great extension is li'ivcn to the chapter on Azimuth. By the methods shewn, iind with the aid of the tabh\s annexed, it will l)e possible to lind the direction of the meridian at uhriist any time of the day or night. Instead of the long and tedious formulae of geodesy, I have given somi^ simple and expeditious ones, depending on the convergence of meridians. The results are nearly the same as with the series deduced from the precise formulae, when such terms are neglected as contain powers of the distance above the second. This is sviffi- ciently accurate for all practical purposes. At places without telegraphic connection, surveyors are sometimes requested to establish sun dials. Chapter VI explains the most convenient processes to construct them. The tables have been carefully computed with the latest numerical values ; they haA'e been, like the work generally, specially devised for Canada. This work must be used only in the Northern hemis- phere, as ma}.y of the methods w^ould be wa-ong tSouth of the Equator. CHAPTER I. DEFINITION AND PRELIMINARY UPERATIONS- EPHEMERIS. (1) I. Division of the circle. I The circle is divided into 360 degrees or 24 hours The degree or hour is divided into 60 minutes, the mi- nute into seconds and the second into 60 thirds. This last division is seldom used, the decimals of the second beino* taken instead. ° An angle is composed of a number of degrees and fractions of a degree or hours and fractions of an horn- Thus we say : an angle of 60 degrees 30 minutes and 45 seconds or an angle of 4 hours 2 minutes and 3 seconds, and they are written thus : 60° 30' 46" 4h Qom Q3S (1) The reader will find, at the end of Nautical Almanac some explana- tions on the elements of astronomer and the me of the Ephemeris and tables, h ft : ■! h ^ ' 12 PRELIMINARY OPERATIONS. We have, according to definition : 360° — l; 24'^ 90° 6'^ 15° — z 1^ 1° ^m 15' : 1" 1' = 49 15" =: V It will be observed that the minutes and seconds of arc have not the same value as the same parts of the hour, although having the same denomination. 2. To convert an arc into time. Multiply by 4, the product of the degrees in the arc will give minutes of time, that of the minutes, seconds, and that of the seconds, thirds. Divide, then, the number of thirds by 60, to reduce them to decimals of a second. Example 1. — To convert into time 5° 12' 06" 20"' 5° X 4 12' X 4 6" X 4 gives u u 48-^ 24^ 5° 12' 06" = 20'" 48" 24' or = 20-" 48«, 4 Example 2.— To convert into time 76° 43' 28" 304'" 76° X 4 gives 43' X 4 28" X 4 172' 112 ot 76° 43 gS'^ = 304"^ 172^ 112' OP . = S"" 06"> 53« 87 'i i\ TIME. 1^ 3, To convert time into arc. Reduce the hours to minutes, add the given number of Ininutes and divide by 4, the result will be degrees. Reduce the remainder to seconds, add the given number of seconds and divide by 4, the quotient will give mi* nutes, and generally a decimal fraction of a minute, which may be reduced to seconds by multiplying by CO.' Most of the tables of logarithms contain tables for con- verting time into arc and conversely, they require no explanation. Example 1.— To convert into arc 10'' 29™ 58' 629"" : 4 118' : 4 gives 157' remainder 1"' 29'5 lO*" 29™ 58^ = 157° 29'.5 or = 1570 29' 30'' Example 2.— To convert into arc 7*» 51" 27' or 471'" : 4 207' : 4 gives 117*^ remainder 3' 51'.75 (Jh 52m 279 _ 117° 51'.75 117° 51' 45" 4. Time. The transit, meridian passage or culmination of a heavenly body is the instant when that body is on the \n^^ 14 PRELIMINARY OPERATIONS. moridian. Transits are diA'ided into upper and lower, according as they take place above or below the pole. A day is the interval of time between two successive upper transits of a heavenly body. It is a solar, lunar or sidereal day, according as the body is the sun, the moon or a star. (1) Sidereal days are nearly equal, but true solar days are not, as the motion of the sun is not uniform. To obtain a uniform measure of time, astronomers have devised a mean sun, moving uniformly. The interval between two transits is a mean solar day for the mean sun, and an apparent or true solar day for the true sun. Time is known as mean, apparent, or sidereal, accord- ing as it is reckoned in mean, apparent, or sidereal days. The astronomical day begins at noon and is divided into 24 hours, numbered from 1 to 24. The civil day begins at midnight, twelve hours before the astronomical. It is divided into two periods of 12 hours each, marked A.M. from midnight tc noon and P.M. from noon to midnight. 0'', astronomical time, cor- responds to noon the same day of the month and 12'', astronomical time, to midnight at the beginning of the following day. Civil time is, usually, mean solar time. 5. To convert astronomical Into civil time. "When the number of hours is less than 12, add the designation P. M. (I) More accurately speaking, a sidereal day is the interval between two upper transits of the vernal equinox. TIME. 15 When it is more than 12, subtract 12'', make its the desi- gnation A.M. and add one to the day of the month. Example 1.— What is the civil time on Sept. 9 at 18''. 20'". 23», astronomical time ? Ans-Sept 10, at G''. 20"'. 23^ A.M. Example 2.— What is the civil time on Jnly 4 at 7'". 45'". oO^ astronomical time ? Anx.—.lnly 4, at 7''. 45'". 30^ T.M. 6. To convert civil into astronomical time. For r.M. times, make no change, but for A.M. ones, add 12'' and subtract one from the day of the month. Example 1.— What is the astronomical time on Auo- 4 at 9" 35'" 43'' A.M. ? "^ Ans.—kng. 3, at 21" 35"' 43\ astronomical time. Example 2. — What is the astronomical time on Jan. 24 at 2" 07"> 31«. r.M. Ans.—J'dii. 24, at 2'' 07*" 3 P. astronomical time. 7. To find the Greenwich time corresponding to the time of another place. When the place is West of G-reenwich, add the longi- tude to the given time, or subtract it when it is East. If the subtraction cannot be made add 24'' or if the result is more than 24'' subtract 24", subtracting or addin"-, at M |i: 16 PRELIMINARY OPERATIONS. the same time, one to the day of the month. When the longitude is given in arc, it is converted into time as in Art. 2. Example 1. — What is the Greenwich time on Oct 24, at 9" 28"' 42« M. T. of Quebec ? Long, of Quebec 71° 12' 51" W. 71° 12' 52" W. = 4" 44'" 51^ W. (Art. 2) Quebec M. T. = Oct. 24, at 9 28 42 Greenwich M. T. = Oct. 24, at 14'' 13"' 38^ Example 2.- -What is the Greenwich time on Sept. 29, at 21" 13"' 41« M. T. of Toronto. Longitude of Toronto 5'' 17"' 33^ W. Loner, of Toronto = 5" 17"^ 33' W. Toronto M. T. = Sept. 29 at 21 13 41 Greenwich M. T. = Sept. 29 at 26 31 14 or = Sept. 30 at 2" 31"' 14« 8. Given the time of a place, to find the time of another place. If I Take the sum of the longitudes, if they are of different denominations, or their difference if of the same. Add it to the given time, if the second place is East of the first, or subtract if West. When the substraction is not possible or the result is more than 24", 24" must be added to or taken from it, subtracting or adding, at the same time, one to the day of the month. Example 1. — What is the Ottawa time on Oct. 27 at 3" 27"' 30" M. T. of Moisy P. Q. TIME. 17 Long, of Ottaw.. 5^ 02"' 54" W. Lciig. of Moisv 4" 24'" 12" W. o y Long, of Ottawa Long, of Moisy DilFerenco Moisy M. T., Oct. 27 at 5'' 02'° 54«. W. 4 24 12 AV. 38 42 3 27 30 Ottawa M. T., Oct. 27 at 2'' 48™ 48" Rmmp/e 2— What is the time at Quebec on Ann- ^^ -a 23" 47'" 25" M. T. of Winnipeg. Long, of Quebec 4" 44"' 51" W. Long of Winnipco- ,].. 28"'30^W. uuii^u Long, of Winnipeg Long, of Quebec (3'' 28™ 30". W. 4 44 51 . W. or Difference i 43 39 Winnipeg M. T. Aug. 21. 23 47 25 Quebec M. T. Aug. 21. 25 31 oT Aug. 22. V 31™ 04" Rvample 3.-What is the time at Sydney (Australia) on June 6 at 15" 23'" 30^ Montreal M. T. 4" 54'"^13^''w ^"^'"'^ ^^" ^''" ^^' ^' ■^^''^- ""^ ^^^^'*'''"^ Long, of Sydney Long, of Montreal Sum Montreal M. T. June 6 10" 05'" 00". E. 4 54 13. W. 14 59 13 15 23 30 or Sydney M. T. June 6 30 22 43 June 7 6" 22"^ 43* If Ig MIELIMINATIY orERATlOjfS. 9. Interpolation. The quant itios in the Nantiral Almanac aro given for ci'rtain Greenwich times. As they are continually vary- ing, corrections are necessary to ascertain their values at any intermediate instant, and these are to be projjortional to the intervals elapsed since the Nautical Almanac times. Consequently, we must first obtain the Greenwic^h time at the instant for which the value is required, and then, iind the correction for the difference between it and the Nautical Almanac time. Supposing- this quan- tity to be given for every Greenwich mean noon, the variation in 24 hours will be known and a simple pro- portion will give the correction for the elapsed time since Greenwich mean noon. It is to be added or sub- stracted according as the quantity given in the Nautical Almanac is increasing or decreasing. AVlicn the variation in 1 hour is given, the correction will be found })y multiplying it by the number of hours and decimal of an hour elapsed since Greenwich mean noon. Otherwise the best method is to proceed by aliquot parts and say : if for 24'' the variation is so much : for so many hours it will be so much ; and so on. AVhen any of the aliquot parts are not to be used, they must be crossed out before adding. E.vamph 1. — To find the Sun's declination on Nov 21, 1877 at 7" 24'" M. T. Greenwich. EPIIEMERIS. 19 Wo find ill the Nautical Almanac, page II of the month (Right hand side) : (1) Smi'.sdocliiiationouNov. 21 at 0''M.T.Gr. 20° 01' 10". 3 S 20 14 11 . " oo Declination on Nov. 21, jVar in atO''M. T. Gr. 20°-0r.1()". S. I 24'' Prop, part for 7'' 24'" 3 [>[) ' Declination on Nov. 21, _ at 7" 24"' M. T. Gr. 20° O.V l.V. S. I 1 i] i -< Oi) 3 14 20" 4 32 11 o 17" 24"'| 3' ;-)!)" Examph 2.— To find the equation of time on Dec 14 1877 at 0'' A. T. Sherbroke, V. Q. ' ' Sherbrooke's Longitude, 4'' 48"\ W. The Nautical Almanac gives for apparent noon, page I of the month ( left hand side) : ' -^ Equation of time on Dec. 14 4"* 58^ 89 " 15 4 20 89 Var. in l** — \\ 2 Sherbrookc Longitude A. T. Dec. 14 0'* 00™ 00« 4 48 00 W. Gr. A. T. at 0'' A. T. Sherbrooke Dec. 14. 4 48 00 or 4", S (I) Page 1 of the month gives the elements for apparent noon and ra-e II tor mean noon. When apparent time is known, page 1 is used and when' mean time, page II. i! ! 26 PiifiLIMINAtlY OPERATIONS. Equation of time on Dec. 14 at 0" A.T. Or. 4'" r,H' 80 Var. in 4'' 48'" ;> 70 Equation of time on Dec. 14 at 0'' A. T. Sherbrooko 4'" 5:'/ VI Var. in 1" 1*. 2 Multiplied by 4.8 9 6 48 Var. in 4'' 48'" = 5' 70 Example 3. — To find the sun's declination on Sept. 8, 1877, at T)" 24™ M. T. of Montreal. Longitude of Montreal 4'' 54"» W. We find in the N. A. ibr Oreen\vivi(di appjirent i)Onu, TIME. lent intervals of mean solar time " They require planaiion. (1) 23 no <>x- 13. To convert mean into the corresponding sidereal time. Take out of the Nautical Almanac the sidereal time at mean noon, then add the x>roi>er correction ior the time elai)sed since Greenwich mean noon, Avhich is equal to 9" 8505 for every hour of longitude or 3'" 5»J^ G for 24 hours. Add the given m(?an time converted into a sidereal interval by means of the tables, the result is the reqr-=red sidereal time. Wh(!n greater than 24'', sub- tract this quantity without altering the date, which is the same as for mean time, sid«3real days having no parti- cular date. This operation and the next one miglit also be performed with the time of transit of the iirst point of Aries, given in the Nautical Almanac. ^.tw/zyVt'.— Find the sidereal time on June 11, 1«77 ut n" 25"' 00'' M. T. of Chicoutimi P. Q. Long, of Chicoutimi 4'' 44'" 20 \ "W. By table " For converting mean into sidereal inter- vals " 17 Til 9.V" M. T.^17" 02" 41" 50. S. T. = 25 04 ]1. 17'' 25"' M. T.=17'> 27"' 51« 07. S. T. (1) Dr. PettTH Astronomiche tdfcln und forinelii, WillK-lm Muiiko, ILiinlMirK, contain, among many UHeAil tables, Home tr)<)n< convenient than lli()se of tlij ^fautical Almanac for thone converHion«, i-: y 24 PRELIMINARY OPERATIONS. S. T. on June 11 at 0" M. T. Gr. Var. in 4'> 42"^ 20" S. T. on June 11 at 0" M. T. Chicoutimi M. T. interval converted into S. T. t:5. T. on June 11 at 17'' 25"* M. T. Chicoutimi 5" 19™ 03« 22 46 71 Var. in 24" 3'" 56" 6 4 40"' 4 20^ 9* 17 19 49 93 27 51 67 39 43 6 57 m 05 47'" 41" 60 Var. in 4" 42"* 20" 46" 71 III 14, To convert sidereal into the corresponding mean time. Find as above, the sidereal time at mean noon of the place ; subtrai^t it from the given time, and reduce the re- mainder to a mean time interval by the Nautical Almanac tal)le. When the subtraction is not possible, 24 hours are added to the given sidereal time. Example. — Required the mean time on Nov. 6. 1877 at 9" 13"* 53" S. T. of Rouse's Toint. Longitude 4" 53"* 28" W. S. T. on Nov. 6 at 0" M. T. ar. Correction for 4'' 53"* 28" S. T. on Nov. 6 at 0'* M. T. Rouse's Point Rouse's Point S. T. (1) S. T. from mean noon Var. in 15" 03"* 13" 48 48 21 24" 3™ 56" 6 4 40m 39 43 6 57 15 04 01 69 10 1 64 33 13 53 00 3 20" 8 65 02 18" 09"* 51" 31 Var. in 4" 53"' 28", 48" 21 (1) 24'> are added lo make the subtraction possible. TIMES OF TKANSIT. o- inK:'^ "^"' ^°"^-^''«"«- ^''•"-' ""0 -eau time IS-^ S. T. =- 17" 57'" 03«.07 M T '^'" 8 58.53 f' 50.86 Kouse's Toiiit M. T. Nov. 18'' 06'" 52^78 15. To find the mean time of the sun's or a star's transit. The time ofthe sun's transit is apparent noon (see Art. 4) The corresponding mean time will be found as in Art 11 The sidereal time of a star's transit is equal to its rioht ascension. It is converted into mean time as in Art. U ( Au'-T^tZ^^s^'"'^ the mean time of transit of a Aquilae (Altair) at St. Hyacinthe. T. Q., on Sept. 18, 1877. Longitude 4'' 52'". "VV. (i) IVar. in S. T. on Sept. 11 at 0'' M. T. Gr. ll'' 50"^ O'-^^ Correction for 4'' o2"' 43 3'" 56\6 ^\J-.^,*0;:.^".St-IIyacinthell 50 50 Altair sKight Ascension 19 44 50 feidereal interval from mean lioon Y 54 00 By table " For converting sidereal into mean time intervals 7" S.T.= 6 58 51.2 M.T 54'" = Bn ^1 o M.T. of transit at St-Hya- -___ cinthe, Sept. 18 71. 52'" 42« (1 ) In (I.iH e.xn.ni.le, fractions of a second are not t.aken into account. 26 PRELIMINAUY OPERATIONS. 16. Lower Transits. Stars near the pole have, at least, two meridian transits above the horizon every day, one above and the other below the pole. The time of upper transit is found as explained; the lower transit takes place 12 sideral hours or ll'' 58'" 02' M. T. after or before the upper transit. The calculation may be performed as above, by adding 12 hours to the star's right ascension. The transits occuring at intervals of 24'' S. T. or 23"' 5(jm Q48 ]vi/p there may be tw^o upper or two lower transits in the same mean solar day. Example. — Required the mean time of lower transit of Polaris at Quebec, on Oct. 13, 1877. Longitude 4'' 45'" W. Yar. in 8.T. on Oct. 13 at 0" M.T. Or. 13" 28'" 3G^2 24" 3'" 5G\G Correction for 4" 45'" 46 .8 S. T. at 0" M. T. Quebec 13 20 23.0 12" H- Tolaris' liight As- cension (1) "^ 37 14 32 .7 Sidereal interval from mean noon 23 45 09 .7 By table for converting si- dereal into mean time intervals 23"8.T = 22 56 13.9 39 .4 0.8 46^8 45'n 9\7 Mean time of lower transit Oct. 13 44 52.6 9.7 23" 41'" 16'.2 (1) 'J4'' are added to that niunher, in tlun case, to make tlie Biibtractiou possible. OIJ.SEIIVATIOA^M. 27 17. Solar and Stellar observations, Methods of observing. We will suppose the instrument used to be a transit theodolite (1). This being' properly adjusted, if a star is to be observed, the teleseope is directed to it, and the intersection of the wires fixed precisely upon it. The time of the clock or watch is noted, if required, at the instant when the star appears to be exactly covered by the intersection of the wires. Having read the verniers, the telescope will be turned over, and directed again to the star by turning the vernier plate through 180°, when the same operation as above is repeated. The field of the telescope must be illuminated to render the wires visible at night. Transit theodolites are gene- rally provided with means for illuminating. If not, a light may be held in front of the object glass, but this method is very imperfect. For solar observations, a colored glass of a suitable tint is adjusted to the eye piece and the intersection of the wires directed to the centre of the sun. This is only an approximate method ; if more precision is required, the limbs, instead of the centre, must be placed on the wires. In this case, the limbs to bi ■ observed with the telescope reversed, are the opposite to those in the first position, so that the mean of the results is equivalent to an observation of the sun's centre. AVhen both altitude and azimuth observations are being made, and the limbs sighted, the image of the sun is placed so that it leaves one of the wires, while it is ad- vancing towards the other. Suppose, in Fig 1, the appa- rent direction of the sun's motion to be represented by (1) \ simple transit may be uaeil, however, when altitmles are not reqtiired. ■^^»" »»Ui« 28 niELIMINAllY OPEKATlONS. the arrow, and the wires by AD and BC, Clami^ the telescope so that the image of the sun is in ss', and keep A]) tangent to the disc by moving the tangent screw of D 'I C B J'ig' 1. the horizontal vernier plate, that of the vertical circle remaining clamped. The sun will appear to move only vertically, and at a certain time, it will be tangent to both wires, at which time the observation is taken. In the reversed position, Fig 2, the image is placed in ,S5'' in the angle DOC opposite to AOB, where it was in the iirst or direct position. The horizontal wire is kept tangent to the limb by means of the tangent screw of the vertical circle, till the disc is tangent to the vertical wire. The object of this method is to have to move only one tangent screw at one time, as it would be diihcult to make a good observation while moving two screws. If the sun's motion be different from that shewn in the figure, the disc's image should always be so placed, in relation to the wires, that it is advancing toward one wire while leaving the other; the wire towards which it is adA'ancing is the one to be moved by the tangent screw, because, the only visi])le part of the wire being that pro^ OBSERVATIONS. 20 t Joiied oil tlur disc, it is nioro oasily soon whon it leaves than when it comes ui^on the limb. When no colored glasses are provided, the telescope is adjusted I'or ibcus on a distant object, and the eye piect* drawn ont a little. Directing it to the sun, the images oi' the disc and wires should be made fall on a white screen, such as a sheet of paper, held behind the eye piece. The observation is then made as before, when the sun is seen tangent to the wires, projected on the sheet of paper. The distance of the screen is adjusted by moving it for- ward and backward until the image is sharply delined. This distance, and the size of the image are decreased when the eye piece is drawn out and increased when pushed in. The telescope is readily directed to the sun by revol- ving the instrument till the shadow of the vertical circle is in its own plane, and turning the telescope up or down till its shadow^ forms a circle. The sun is then in the field. The vertical circle reading is not always the altitude ; sometimes it is the zenith distance, sometimes that dis- tance or the altitude plus 90°. The altitude is the diffe- rence between the reading when the telescope is hori- zontal and the reading when directed to the sun or star. Should the number, used as reading with the telescope ho- rizontal, not be accurate there will be an error in the al- titude arising from it ; but when the observation is made in both positions of the telescope, this error affects both altitudes precisely by the same quantity, but with diff- rent signs. It will, therefore, disappear in the mean result. The altitudes, horizontal circle readings and times employed in the subsequent calculations are svipposed to jaju_ *.m I" ao niELIMINARY OPERATIONS. be always the moan of two observations, made as deseri- l)ed above, in both positions of the t(^h\scope. lU-sides the advantages already alluded to, the results are inde- pendent of certain instrumental errors, such as collimr.- tion and inclination of horizontal axis. f 18. Correction of altitudes, :n For altitudes of the sun, su]>tract the retraction (Table I) and add the parallax in altitude (Table II). If the observation was not taken as directed in Art. 17, there might be a correction for semidiameter Avhich we a\ ill not consider it here. When great accuracy is not required, the correction for parallax may be dispensed with. The altitude's cor- rection is then the same as for a star : refraction only. Example 1. — The apparent altitude of the sun ])eing 21° 07' 15", find the true altitude. Apparent altitude 21° 07' 15" Refraction (Table!) 2 81 Difference rarallax( Table II) True altitude Example 2. — The apparent altitude of Polaris being 47° 46' 22" find the true altitude. 21 04 44 8 21° 04' 52" Apparent altitude Refraction ( Table I ) True altitude 47° 40' 22" 53 47° 45' 29" r .31 I (JH APT Ell II. TIME 19. Definitions. The zoiiith di«tance is the complomcut of tlie allitiKh. thorelbre it is found by subtracting- the altitude from 90°.' The pohir distance is the complement to 90° of the de- clination when the heavenly body is north of the equa- tor, or 90° plus the declination, when south. In the sou- thern hemisphere, the contrary would be the case At the head of the Nautical Almanac columns are the letters N or S or the words North or South, which denote on what side of the equator the heavenly body is. The colatitude is the complement to 90° of the latitude A cologarithm is a logarithm subtracted from 10. It is found by subtracting each figure from 9 except the last one which is subtracted from 10. With a little practice they can be read from the ordinary logarithmic tables almost as easily as the logarithm itself. By their use subtraction is performed by addition, as adding the colo- garithm or subtracting the logarithm will prodLe results difiering exactly by 10 in the characteristic. 32 Tl.MK. \Vc may roiuark thai : Cologarithm sine a is the same as Lc gaiilhni tosooaiil u " cosine a " siu-aiit a Tancrent rt " colanureiir^/ :iiul ihoy may bo usod iiididi'ivully (1) 20. To find the time by the sun's meridian transit. A littlo before noon, the telescope oi' the insh'unicnt should be directed to astronomical south, and placed at the proper altitude. When the sun's centre is Ix^hiiul the vertical wire, it is apparent noon (Art. 4.). The cor- responding mean time will be found as in Art. 11 and Ik It w more accuriite to note the times of transit of the East and West limbs and take the mean, which will cor- respond to the sun's centre, or to apparent noon. When there is no colored glass, the observation is made on a screen, as explained in Art. lY. 21. To find the time by a star's meridian transit. M il Having fixed the telescope to astronomical south, when the star passes the vertical wire, the time of the clock or watch is noted. The mean time of transit is deduced as in Art. 15 and 16. The stars south of the zenith will fifive more reliable results than the stars to the north of it, because they move quicker. (I) Some tables, such as Caillet'.s, Gautliier Villars, Paris, contain tlie lo- garithms and cologarithnis. Tliey also give the logarithms of trigonometric lines for angles expressed in arc and in time, which is often very convenient. ]{V MHKIDIAN TKANSIT. no 22. To find the name of an unknown star passing the meridian. AVhcn iionooi'tho stars noar tho moridiaii arc known, ilio briqhtoHt will bo observed, and the time ol' transit and altiiude noted. If the clock or watch is not far astray, the name of the star will l)e found ihus : For stars south of the zenith, — Find the dillerence bcl- ween tlie star's zenith distance and the latitude, it is the approximate declination, South, if the zenith distanc(^ is greater than the latitude and North if it is less. To ihe sidcral time at Greenwhich mean noon (from llie Nauti<'al Almanac) add the mean time of observation, the result is the approximate right ascension. For stars between the pole and zenith. — Add the zenilh distance to the latitude, it will give the approximate nor- thern declination. The approximate rir^ht ascension is found as above. For stars below the pole. — Add the altitude to the co- latitude to obtain tho approximate Northern declination. Find the approximate right ascension in the same ma- nner as above, adding 12 hours to the result. See then in the Nautical Almanac which star has the corresponding declination and right ascension; it is tho star observed. If there be no star in that position, search should be made amongst the planets. Example.— kt St. Hj^acinthe, V. Q., on Sept. 18, 1877, the meridian transit of a first magnitude star was observed at 52° 50' altitude South ; the time shewn by the watch's face was 7'' 52"' P. M., required its error. Latitude 45^ 39' N. Longitude 4'" 52'" W.. I'*lk, tn TIME. Sl-IIyaciiilhc M. T. Sept. IH S. T. oil Sept. iHal, 0''M. T. Or. 7'' ")J"' 11 ;')() Approximate riglit ascension 10'' 42" Altitude of Star 52° 50' Zenith distance 37 10 Latitude St-Hyacinthe 45 3t> Approximate declinalion 8° 2!)' N. ill tlie Nautical Almanac lisl, we iiiid that the above rii;lit ascension and declination correspond to a Aquilae (Altair). The calculation being- carried on as in the example of Art. 15, we lind: M. T. cf Altair\s transit 7'' 52'" 42^4 AVatch time 7 52 00 AVatch slow 42\4 23. To find the time by the sun's altitude. To obtain accurate result, the observations should be made not less than three hours before or after noon. Observe the sun's altitude (Art. 17), correct it (Art. 18) and find the polar distance of the sun (Art. 19). Add together the true altitude, the latitude and the polar distance, take half the sum, and subtract the altitude from it. Find the cologarithm (Art. 19) cosine latitude, the cologarithm sine polar distance, the logarithm cosine half sum, and the logarithm sine half sum minus the altitude. Add and take half the total, you will have the logarithm sine of half the sun's hour angle. This beinir taken out of the table, and converted to time ( Art. 2 ) nv TiiK sun's altittm)!:. ^ivos the uppuivnt iimo irth,. «,„. wns \V,.,s( „r,h,. „,,,• ;;•;;»; il Ka,sl, tJio hour nno.lo must I.. sul,lmr|..„„.„.,• ..x,,lainc.cl in I'ositioii Ol' t('l0SO()])l' Direct lie versed Sun's allilude AVatch liim. I'' IS'" ()(;■< J'. M JietjuinHl the wiiteh error on mean iimc !.atitud(i 45" 21)' N. Loiii-itude V .',4'" AV. (.|^ Montreal approximaie M. T. Auj.-. 1, 4'' [jim Longitude ^ t r . n- 4 04 \\ GreenAvirh approximate M. T. Au- 1, <) 4:> ^^^j,!. ^ (1) Tlu- formulji is: cos L .sin A where S = ! ' ^ 2 A = True altitude. L = Latitude. A = Polar distance. P = Sun's hour angle. (2) In tliLs example, seconds of arc and fractions of a second of time ^ not taken into account. are 36 TIME. Poldf Didame. Sun's declination on Ani?. 1 at 0" M. T. G-r ir 57' N Correction for 9" 43" Sun's declination on Var. in 1" Multiplied by Auff. 1 at 4" 49"' Montreal M.T. H 51 N Polar distance 72 09 38".02 9 . 7 2r3614 34218 Var. in 9" 4:3'" 3G8".794 Eqnalion of Time. Equation of time on Auir. 1 at 0" M. T. Gr." ( To be added to A. T.) Correction for 9" 48"' Equation of time on Aug. 1 at 4'' 49"' Montreal M. T. Var. inl^ 0M5 Multiplied by 9. 7 {')"' 02'' 1 105 135 6'" or Var. inO*' 43m r.455 ll ''<. Mean of the sun's apparent altitudes 25° 44' Refraction minus parallax 2 True altitude Latitude Polar distance Sum Half sum Half sum minus altitude Sum 25 42 45 29 colog. cos 0.15421 72 09 colog. sin. 0.02143 143 20 71 40 log. cos. 9.49768 45 58 log. sin. 9.85669 19.53001 J BY THE sun's ALTITUDE. h1 Half sum or log. sin. half the hour angle 9.7G500 Half the hour angle 350 gg/ in time 2'' 22"" 24'' Hour angle or apparent time 4 44 43 Equation of time (to be added to A. T.) 6 01 M. T. of observation Mean of the watch times Watch slow 4 50 49 4 48 34 2"" 15« Example 2.-The following observations were taken at Toronto, on Dec. 18, 1877. Position of telescope Direct Reversed Sun's altitude 10° 22' 10" 10 40 15 "VVatch time 8" 57'" 11" A.M. 8 59 47 Required the watch error on mean time Latitude 43° 39' 26" N. Longitude 5" 18"' W. Toronto Approximate M. T. Dec. 17 oQh 53,,, Longitude ' "^ ^^ ^^ Greenwich approximate M. T. Doc. 18, 2"~1G"' = 2" 3 Pofar Disla/irc. Sun's declination on Dec. 18 at 0'' M. T. Gr. 23° 25' 00" S Correction for 2'' 10'" 9 Sun's declination on Dec. 17 at 20'' ~ 58™ M.T.Toronto 23 25 09 S Polar distance J ] 3 25 09 Var. in 1'' Multiplied by 3".8 2 .3 11.4 70 Var. in 2'' IG'" = 8".74 88 TIME. I Equation of Time. Var. in 1'' Eq. of time on Dec. 18 atO^M.T. Gr. (tobe subtracted from A.T.) 3'' 01'" G Correction for 2" 16'" 2 Eq. of time on Dec. 17 at 20"' 58'" M.T. Toronto 2'" o8^ 7 V.'li Multiplied by 2.3 372 248 Var. in 2'' l(l"'=2\8r)2 M. Q ihc ollowino- altitudes of p Orionis (Ki,,]), were i.lj, |h,. star being East of the meridian. Position of telescope Star's altitude WaUh lime Direct lieversed 21° IMY 21 44 7'' ;:]0"' 0;y V. M 7 SI ;]:j Required the watch error on mean time. Latitude 4(j^ 20 N. Longitude 1'' oO- W, (]) Sidereal. Time. S^T. on Sept. 28 , y.,, j-^^ o.i. . o,„ ,.^, ;::;at O" M. T. Or. 18" 28- 14^ 5 ' ' " ''^' "'^ (V)r. for4''r)0"' 47 n S. T. at 0'" M. T. Three Iiivers 18'' 29'" 02' 1 Var. in 4"' oO'" | 47«.(; (1) In ti.is example, <,„a,.tilios loss ,1.,,. I nnn.".^7;;;^or 1 time, are not faktn inio acroiiiit- second of 40 TIME. Mean ol' the star's altitudos 21° Si' lieiraction 2 01 O't 46 20 colog. COS. 0.1G08G 98 21 colon-, sin. 0.00403 True altitude Latitude Polar distance of the star Sum Half sum Half sum minus altitude 61 33 log. sin. 166 16 83 08 loff. COS. 9.07758 9.94410 Sum 19.18717 Half sum or log-, sine half the hour angle 9.59358 Half the hour\ingle 23° 5' 45" in time l** 32'" 23" Hour anale or 24'' minus the star's time 3 04 46 Star's time Star's right ascension S. T. of observation S. T. at 0" M. T. Throe liivers 20 55 14 5 08 42 26 03 56 18 29 02 Sidereal interval from mean-noon 7 34 54 l>y table " for converting sidereal into mean time iiit(n-vals '' 7'' S.T. : 34'" 54" M. T. of observation Mean of the watch times 6 58 51.2 33 54.4 53.9 7 33 40 7 30 49 Watch slow 2'" 5r C'fRAPHIO METHOT. 41 25. Graphic method tor finding the time by altitudes of the sun or a star. Describe a circle. Through the centre O (Fi- 3 and 4) draw two lines forming an angle EOH equal to the anfEOA I') ''!' r'''^'" ^'^^ ^^^^^^ *^ '^^ ^^^^^ude and EGA to the declination, A being to the left of E if the declination is North, and to the right if South. From A, let tail a perpendicular A6- on EO and with O as centre and Oc as radius, describe a circle. Throu-h A and B draw two lines As and B. parallel to Ho" and LG. At . erect a perpendicular ss' to A., on the right hand side, if the sun or star be in the West or on the left Jug. 3. if ill the East. Join G to the intersection point of .sV wiUi the inner circle, and the angle EG.', reckoned in the direction of the arrow, will be the apparent or star's time according as the sun or a star was observed. The mean time is deduced as in Art. 23 and 24. 4 42 TIME. i Fii«-. ;{ istho .soluliou ol" Example 1, Aii. 2:], JIOK is ihv cohiliiiulc, 44" W, AOE i.s the sunVs docliiiatioii, 17" At' Nortli, liOlI the true altitude, 25° 42' and EO.s', wliich i,s eciual to 71° 12' or 4'' 44'" 48^ i.s the apparent time. Fin-. 4 is the solution of Example 2, Art. 2-3, HOE is Toronto colatitude, 46° 21', AOE the sun's declination 23° 25' South, BOH the true altitude 10° 26' and JLOs, which is equal to 313° 20' or 20'' 53'" 20' is the apparent time. These figures should be made on a much larger scale, and the angles platted by means of a scale or table of chords. r is OHAPTER III. LATITUDE 1 IS on Os, ?nt lie, 5 of 26. Latitude by the Meridian altitude of the Sun. When the sun is near the meridian, its motion in alti- tude is very small. If one of the limbs be observed a little before apparent noon and the other a little after, the mean of the altitudes may be assumed to be, within the limits of precision of this work, the meridian altitude of the sun's centre. The observation should be made with the telescope in both positions, direct and reversed. Find, then, the true altitude (Art. 18 ). Add the sun's polar distance, and subtract from 180^ The result is the latitude. Example.— On May 3, 1878, at Chicoutimi, P. Q., the altitudes of the sun on the meridian were : with telescope direct, lower limb reversed, upper limb Required the latitude of the place. Longitude of Chicoutimi, 4'' 42"' W 5r 09' 39" 57 41 25 41 LATITUDE. Sun's (leilinalioii on May 3 at 0"' A. T. CIr 15° 47' 21" N Correct, for 4'' 42'" 3 26 Sun's declination on May 3 at 0" A. T. Chicout. 15 50 47 N Polar distance 74° 00' 13" Var. in 1 hour 4."." H5 Multiplied by 4.7 30005 17540 V.in4"42'"=20G".005 1 f J Mean of the sun's apparent altitudes Kefractio 1 Difference rarallax True altitude Polar distance Sum Latitude 57° 25' 32" 37 57 24 55 4 57 24 50 74 00 13 131 34 12 48° 25' 48" N 27. Latitude by the meridian altitude of a star. For stars south of zenith, make the same calculation as for the sun, For stars between the zenith and pole, subtract the polar distance from the true altitude. For stars below the pole, add the polar distance to the true altitude. (1) (1) The formulae are : For stars passing the meridian south of zenitli L=1SU° — (A -|- ^) " between zenith and pole L=A — ^ " below the pole L=A-f- A Where L is the latitude, A the altitude and ^ the polar distance. BY THE ALTITUDE OF rOLARIS. 45 If the star bo unknown, its name would bo found in the mannor indicated at Art. 22. Example. — On Jan. (I, 1877, at Motabotchonan, (Lako vSt. .Tolin) V. Q. tho aliitudo of tho polo star at upi)or transit was 40^ 4(1' W lioquirod Iho lalitudo. Doclination on Jan. »», 1877 Polar distance Apparent alliludi' lie fraction True alliiude I'olar dislantc Lalitudo 88^ no' ;37" 1 20 2'} 40'' 4(r 50" 40 Ml 40 01 1 20 2:} 48^^ 25' 38" N 28. Latitude by the altitude of the pole star at any time. It is rocjuisito to have a watch, whose error on mean or sidereal time has been determined (Chapter II), and to note the time of observation. The latitude will bo calculated by means of the tables at the end of the Nau- tical Almanac " used in determining the latitude by ob- " servations of the pole star out of the meridian." The method of using them is as follows : Find the true altitude and take 1' from it. (1) From the watch time, deduce the sidereal time (Art. 13) With that time, take out the " first correction." It must be added to the true altitude if the sign be + and subtracted if—, (1) This correction might possibly be dispensed with in subsequent edition-? of tlie Nautical Almanac. The explanation ,u;iven at the end of it, should be read before using the tables. I 46 LATITUDE. |i L To the result, add the " second " and " third correc- tions " ; the sum is the latitude. The error in the latitude for one minute oi' time never exceeds 21". Example. — On July 25, 1877, at Cape liosier Light, Gaspt', r. Q., the altitude of the pole star was 47° 48' 53 ' at 7" 33'" 12* r. M. Required the latitude of the liht, II. (\ I{. on the sun. IMrect j 84° 40' 10" Ke versed (1) | ;U 80 50 844° 28' 00" 844 8o 01) The calculated azimuth of the sun was 101° 12' 00". Required the azimuth of Lark' Island linhl. Azimuth of the sun 101° 12' 00" Mean of the II. (1. I{. on sun 844 2!> 00 Difference m; 48 OO Mean of the 11. C. II. on Lark Id. li^•ht 84 40 00 Azimuth of Lark Id. li^-ht 1'>1° 28' 00 31. Azimuth by the meridian transit of the pole star, The upper transit of Polaris takes place 24 miirites after its passage in the same vertical plane with e Ui-st^e majoris (Alioth). This interval, 24 minutes, increases yearly by 17 seconds. The lower transit occurs 20 minutes after the passage of (1) In reality, these readings were 214° 39' 50'^ and 1G4° 23' 00'^ bnt for simplicity, it iH convenient to add 180° to the H. C. R. in tlie pos'iiion reversed. >0 AZIMUTH. Polaris in the same vcr(ical plane with y Cassiopeiae. This interval, 2G minutes, increases yearly by 19 seconds. (1) * Ursa Maj(tr *> Tol iiri^ Pole * o A 1 w^* 1. Ursa* Minor * * * Cass opoia * * ^4'"- G. i; Both stars are al)out on the same line with the pole star (Fig. 0.) They allbrd a very sim^de method of de- termining an azimuth. The instrument being adjusted, and the horizontal circle clamped, direct the telescope to the pole star, clamp the vernier plate and lower the telescope to the altitude of " Alioth " or " -y Cassiopeiae." Ivepeat the same operation till the lower star crosses the vertical wire, note the time, and 24 or 20 minutes after, the pole star will be on the meridian. It is more accurate to find the time of transit, (Art. !;> and IG) and observe it with a watch whose error has ])een determined by one of the methods of Chapter II. A very reliable result will be obtained by observing the pole star at the same interval of time after as before tha transit. The process is as follows, the 11. C. It. being noted in each case. 1*^. Direct the telescope to the object. 2^. A certain time before transit, say 2 minutes, l)ring the A'ertical wire on Polaris. (1) CalpiilHled for 187S niul Lnlitiide 47" ELONGATION OF POLARIS. 51 3°. Reverse, and 2 minutes after transit, repeat the observation on Polaris. 4°. Sight to the object. The mean of the readings on Pohiris corresponds to the meridian. In Canada, an error of one minute in the time afl'ects the azimuth by about 30". Example. — At Tadousac, V. Q., the following oliserva- tions were made, the pole star being obser^-ed 2 minutes before and 2 minutes after meridian transit : Position of telescope. Direct Ke versed (1) n. C. Pv. on r>lack Pt. light. n. C. Pv. on Polaries. 101° 14' 05" 101 13 5o 3^ 3G' 00" 3 88 00 Required the azimuth of Black Pt. light. Azimuth of Polaris 300^ 00' 00' Mean of the H. C. R. on Polaris 3 37 00 Difference 3oG 23 00 Mean of the XL C. R. on Pdack Pt. liuht 101 14 00 Azimuth of Black Pt. light or ol7 37 00 187° 37' 00" 31, Azimuth by the elongation of the pole star. This method is very i jcurate, and is always employed when great precision is aimed at. Table III gives the approximate times of elongation. A little before that time, the distance? of the star from (I) See note ol" Art. SO. H 52 AZIMUTH. h the meridian is increasing, then it moves only in alti- tude, and after elongation, it begins to move in the oppo- site direction in regard with the vertical wire. Table IV gives the azimuth at Eastern elongation, from which the other is readily deduced. It is calculated with the mean positions of Polaris for the latitudes 42^ to 54° and the years 1878 to 1890. When more accuracy is required, add the logarithm cosine declination to the cologarithm cosine latitude, the result is the . j^arithm sine of the azimuth. AVhen the star is at Eastern elongation, the azimuth is the smallest angh* found in the table, corresponding to the logarithm sine ; at "We.scern elongation, it is 8G0° minus that angle. (1) Example.— On :iime 20, 1878, at Trois-Pistoles, P. Q., the II. C. R, on the pole star at Western elongation was 33!l° 41', and on Black Point light (t^aguenay) 251° 18'. Find the azimuth of light. Latitude of Trois-Pistoles 48° 08' N. V. Bij Table IV. Azimuth of Polaris at Eastern elongation 2° GO' 30" Western " 357 59 30 II. C. E. on Polaris 339 41 00 Difference II. C. n. on Black Pt. light Azimuth of Bla-k Pt. light 18 18 30 251 18 00 209° 36' 30" ct)s D (1) The formula is : sin Z = . -, where Z is the n7.iiniith, D (he (h'cli- n:iti()n of Polaris ami L ihe lalitiidc. OBSERVATION AT ANY TIME. 63 I >-. 2°. By Logarithms. Declination of Polaris on Jan. 20, 1878, 88° 30' W N. Log. COS. declination Colog. COS. latitude Log. sire azimuth 3G0° minus azimuth Azimuth H. C. K. on Tolaris Ditlerenco H. C. K. on light Azimuth of Black rt. liuht 269° 37' 04" 33. Azimuth by observation of the pole star at any time. 8.3GG9G 0.175G1 8.54257 1^ 59' 50" 358 00 04 339 41 00 18 19 04 251 18 00 '/ It is requisite to have a watch whose error has been determined ( see Chapter II.) The observation is made in both positions of the telescope, and the altitudes, II. C. 1\., and times of observation noted. The sidereal time of observation is deduced from the time of the watch by the method of Art. 13, and the right ascension of Po- laris subtracted from it. When necessary, 24 hours are added to make the subtraction possible. Convert, then, the remainder into arc ( Art. 3,) add its log. sine to the log. cosine of the declination and the colog. cosine of the altitude (1). The sum is the sine of the azimuth. The star is East of the meridian when the difference between I'ch- (1) The apparent altitude may be used. o4 AZIMUTH. the (sidoival time and right ascension is greater than 12 hours, and AVest when less. Therefore, in the iirst case the azimuth is the smallest angle corresponding, in the talde to the log. sine azimuth, and in the other case it is 300° minus that angle (1). Example 1. — The following observations w^ere taken on May 2(j, 1877, on the exterior line of township Ashuamp- mouchouan. ( Lake St. John ) V. Q. Position of Telescope. Direct Kevers. (2) Time of 1 he watch. 7'' or 15^ r.M. 7 00 09 H. C. R on a light placed in the line. 233°41'40" 233 41 20 II.C.R.on Polaris, 359°46'30" 359 47 30 Alti- tude of Pola- ris. 47° 18' 47 20 i) The watch was 54" slow, and the longitude of the place 4'' 50'" W. Required the azimuth of the line. Mean of the watch times AYatch slow M. T. of observation May 20 7" 05"' 12^ P. M 54 7 06 OG 1 1) This is only an approximate method. The formulae are : sin T cos D sin Z = 1 cos A and T = S — R wliere Z is the azimuth, D the declination, A the altitude, S, the sidereal time of observation and R the right ascension. (2) See note of Art. 30. OBSERVATION AT ANY TIME. 00 S. T oji May '2i\, at 0'' M. T. Gr. 4h !,;,„ j,^, Corroction for 4'' 50"' 48 S. T. on May 20, at 0" M. T. Ashuamp. 4 17 2(i l>y table for ('Oiivertiiig ineaii into sidereal in- tervals, :Var. in 7'' M. T. = 7 01 OS S. T. «'" (I 01 0" «; Sidereal time of o]),serv. 11 24 41 Right ascension of rolariH 1 12 5-; ereneo I'j" i] L'" 48^ : = 15: '" 57' Log-, sine 152° 57 0.6570 Log. cosine declination (88° 30' 00 ") 8,3716 Colog. cosine altitude (47° 10') 0.1688 Log. sine azimuth of Polaris 18.1983 360° minus azimuth of Tolaris 0^ 54' 17" Azimuth of Polaris 350 05 43 Mean IL C. K. on Polaris 359 47 00 Difference 359 18 43 Mean H. C. R. on light 233 41 30 Azimuth of the line 503 00 13 or 233 00 13 Example 2.— The follow^ing observations were taken at Halifiix, N. S., on November 20, 1877 : I 5G Position oi" telescope. Direct lie vers. (1) The watch was 3"" 15' slow and the longitude of the place 4'' 14'" W. Iveqnired the azimuth of the reference light. . AZIMUTH. 11. C. Pv. Watch time. on Polaris. n. C. R. on a Reference light. Altit. of Po- laris. 8" 25-" 03« P.M. 0° 22' 30" 8 27 23 21 40 359° 14' 40" 359 14 30 45° 52' 45 53 vUi Mean of the watch times AVatch slow gh 20"' 13" P. M. 3 15 IlaliliixM.T. of observation Nov. 20, 8 29 28 S. T. on Nov. 20 at 0'' M.T.ar. Var. in 4'' 14'" «. T. on Nov. 20 at 0" M.T.Halifax By table for converting- mean into sidereal in- tervals 8'' M. T = 8 01 19 S. T. 29'" 29 05 28" 28 15'' 58'" 25« 41 Var. in 24" 3"' 57* 4 14 39 9 15 59 OG Ji V in 4'' 14'" 41" 24 29 58 S. T. observation Polaris Riuht ascension 25 14 27 Diflerence 44'"29«=11° 07' 15 ' 1 r^'f (1) See note of Art. 30. <}UAP1II(.' METlIOll, Log. sine 11'^ 07' 1-5" Log. cosine declination (88° 30' 48") ('o]og. cosine allilnde (4o° 52' HO") Log. sine AzimiUli oi' Polaris Azimnth of Polaris Mean II. V. U. on Polaris Difference Mean II. C. K. on light 0.28o3 8.3670 0.1572 17.8104 0° 22' 13" 22 05 00 08 350 14 35 Azimuth of reference light 350° 14' 4.3 -^7 34. Graphic solution of the preceding problem. Draw OC (Fig. 7) equal to the polar distance of the star upon a scale of a convenient number of minutes to the inch. Make the angle AOC equal to the difference between the sidereal time and right ascension. Throush C, draw CE parallel to AO, and make AOD equal to the zenith distance of the star. OD is, by the scale of the plan, the azimuth or 360° minus it, according as the star is East or West of the meridian. Fig. 7 is the solution of Example 1, Art. 33. OC is equal to the polar distance 80' 12" at the scale of 80' to the inch, AOC is equal to lO** IP* T f .00 M. Tr.""' ^^^^ ^^'' ^^^ '^ ^^^ zenith distance 42° 41' and OD, measured with the scale of 80 to the inch gives 54' for the complement to 360° of the azimuth. Scale : 80' to the incli. y7>. 7 )K AznruTii. l! I 35. Azimuth by observation of Polaris when vertical with other stars. Selection of s/arx. — To the sidoroal tiino at (h•oon^^'i(•h motiii noon, add tho moan tiino l)e.st suited lor your obsor- valion. Tho rosult is tho approximato oorrosponding" sido- rt'al timo. In Table V, tho star will bo ohoson, whoso sido- r(nil tiino of transit ovor tho vortical oi" Polaris, civon at lht» hoad oi" tho column, is nearest to tho sidoroal timo Ibund. To J'ukI /lie vwan lime k'/ich Ihe sr/a/cd shir is rer- linil irilh Polarh. — It is necessary to know it approxi- mately, in order to be ready I'or observation. From tho sidoroal time w^hon vortical with Polaris, (Table A') su1)stract the sidereal time at Greenwich moan noon, tho dilloronco is, within a low minutes, the required mean timo. 'lA hours are added, when nec^essary, to make tho sul)traction posible. Ohsrrni/ion. — About 10 minutes bolbre tho lime found, direct the telescope oi' the instrument to Polaris and ('lamp the horizontal circle and vernier plate. Fix th(^ telescope to the altitude given in Table V I'or the star and tho latitude of the place ; its direction will then bo so nearly tow^ards the star as to (^ause )io ditliculty in identi lying- it, those given in Table V being the brightest oi' that part of the firmament in which they are situated. AVait till the star is vertical with Polaris, which will be seen by moving the telescope round tho horizontal axis, from one star to the other. Fix it, then, on the polo star, whose azimuth for that instant is given in Table V. This table is calculated with the nu>an positions of stars for 18T8. I'or those North of the zenith, the transit below tlio polo is tho only one taken into consideration. POLAIMS VKirriCAIi WITH OTllEll STAK 69 A rt\sull iiiort' jucuralo ran Ix' ol)laiiU'cl ])y calciilatiiig the a/imuth with the apparent positions of stars I'or the day of observation. Tho loi^-. sino of the azimuth is found l)y adding' tog-ether the h)g". sine of the dilferenee of the right ascensions, the log-, cosines of tlu> declinations, the colog. cosine of the hititude and th(^ colog-. sine of the dis- tance from Tohiris, which hitter is given in Table V. It will be seen, ])y the table, wether the azimuth is to be taken between 270^ and :',()0^ or l)etween 0^ and 00°. (1) Examjt/e. — On Dec. (>, 1877 the pole stnr was observed when vertical with a Ursae majoris. The horizontal circle reading on the pole star was 0° CO' 00" and on a reference light '2oV PA)', required the azimuth of tho light. Latitude 48" 02' 1\ Jh/ Tahic V. he tar be in test Ited. iich tal ole •V. Azimuth of Polaris (Table V) 11. V. ]l. on Polaris Difference II. C. Iv. on reference light Azimuth of reference liuht 1° 0(5' 30 27 283 S() ;i) Tlie furmulae are : sin Z - 234° 03' sin T cos co^ D' Hill d los L and T = II — K' wliere Z is the aziiniitli, D and D' tlie (Kclination.<', il and U' the ri^ht ascensions of Polaris aii'l llie other star, Jtlieir dist.nnoe, and 1/ tlic latiliuK'. (W) AZniT'TIl. lI^. /)// lj()<»nrif/nns. ]{i,nht asct'iisioii ol'a I I'siic innjoris lo'' .■)(>■" l;}- J)o2 ])i(i;'iviii(' 1)'' 41'" f)V or 14')°2n'l.V' Log-, sine 145'^ 20' 1,5" !t.7')n27 Log-, cosine doclinatioii Polaris 8.3(174") a Ursao !».()(;.)H() Colog-. sine distamo IVom Polaris (Table V) ().-'nH4S Colog'. cosine lalilude 0.17477 m ]jOg'. sine azimuth Azimulh H. C. l^ on star Diircronce 11. C. ]{. on light Azimuth ol'roi'crcncc light 28.27077 r 05' 2!>" 80 00 20 20 2;5n JJO 00 234*= 02' 20" 36. Azimuth by altitudes of the sun. r i • Observe as explained at Art. 17, noting in each posi- tion oi' the telescope, the altitude of the sun and the horizontal circle readings on the sun and object M'hose azimuth is required. Find the true altitude (Art. 18) and the sun's polar distance. Add together the altitude, latitude and polar distance; take half the sum and subtract from it the polar distance. Ai/rrrrDKs or tiik stn. 61 •Sl- lar the Ad«l l.»ui'lli('r llu' colou'. loisiiic ol" tlu> :il(iiucK', ilir coloo'. cosine ol' the hititude, the log-., cosiiio luili' the .sum of the iillitud<\ hititude lUid pohir di.staiiee, iind the lun:. rosiiie ol* hull' the siniie siuii iniiius tlie polar distaiu «>. Hair the total is the log-, cosine ol' hall' the sun's uziimith il'it was East or ol' the su])plemeiit ol' the same angle, if West of the meridian. This method is one of the simplest and most conve- nient for surveyors. The observation does not require more than o or 4 minutes and can be made without any loss of time, as for instance, when the llagman is going- forward and the surveyor waiting for him. The observation, to be reliable, should not be taken less than 3 hours before or after noon, and the sun's alti- tude not less than "/'. (1) Kctniiple. 1.— On July 15, 1877, at 7'' A. M., Tointe For- tune, P. Q , the following observations were taken to Iind the azimuth of a signal placed at Carillon : Position of telescope. II. C. Pt. on ! II. C. \l on «un. Direct Ileversed (2) 71° 4r 71 51) Signal. 345° 3G' 345 30 Sun's altitude. 12° 31)' 12 53 Ilequired the azimuth of the signal. L-atitude 45° 33' N. Longitude 4" 58'" W (3) (1) The formulae are : Cos .', Z = i./^g^(«-A) ^„j s _ i±ii±_A cos L CO.S A 2 where Z Is the azimuth, A the true altitude, ^ the polar (llstaiite, and L the latitude. (2) See note of Art. ^0. (o) Arcs less than l' are not taken intu accouiit in tljis cj^auipU*, I' 62 AAi.Mr'rii. rt. Fortune M.T,.luly II. Lono-itudt> GvooinvicliM. T., .luly 14, 1!!'' ()()'" 4 n.s W. )."J. 5.S'" ^'> Sun's (locliiiatioii on July 1.'), ni d'' ]\I. T. (ir. L'r n;>' N Correction I'or 2"' <) Sun's docl. on July 14, nt l!i'' M. T. Pi. Forlunv 21 JJJI N Polar distanri} " (18° 21' I i I 4 :1 ii Moan ol' the ai)paront altit. 12^ 40' Itel'raction minus i)arallax 4 True altitude Latitude Polar diistauK' 12 42 eoloi^'. co.sine O.OlOTi; 4.") J]:] colog-. cosine 0.10472 US 21 Sum 120 30 Half sum 0-] 18 log. cosine 1).6.')2.)0 Half sum minus polar dist. ■") 03 log. cosine 0.008-M Sum Half sum or log. cos. Indf the azimuth Half the azimuih Azimuth of the sun Mean II. C. R. on sun Diilerence Mean II. C II. on signal Azimuth of signal 10.8103:) 9.00817 71 ').") 71 oO Of) V") 1 - O / • 345° 44' Example 2.— On Oct. 2, 1877, at 3" 20"' P. M. Esquimalt, B. C, the following observations were taken ; A[/rri'ri)i;s ok Tiir, sr\. 6r> rosilidii (if l('lt'8('()l)»', Dii't'ct licVtM'st'd (1) II. <*. U. II. ('. K...II oil sun. u liiii^-. SuiTs Jilliliule, 172°::I7' m" 17-"5 02 oil i:!4° 4.V i'0"i -21" (i.V :]0" l:{4 4.") 10 i L>(> ,■).■) iV, lu'(]uiri'(l tile ;i/iiimtli (»riho lijiu'. Liilihulr AS^ I'd' :);;" N Loiin-iiud,. s l |'" W. {2) Ivsquiiiijilt M. T. Oct, 1', I-.oiig'itu(l(.' (hvi'iiwicli M. T. ()(•(. 1'. Sun's doclinatiou on Oct. 2, at 0'' i\r. T. Gr. Correction lor 11'' JU'" Sun's doclination on Oct. 2, at ;}'' 20'" M.T. l<]squ. Polar distanct' Moan of the api>arent altitudes 20^' 51)' 27" Ite fraction 2 81 :!'' 20'" H 14 \\ 11" 34'" 11»', <; ' VV U" S 11 l.> 1> • > r>4 2!) S l);j^ .54' 20" Var. in V' .")S".2 Multiidicd by 11 .0 i!40.2 .IS 2 ,"iS2 V. in 11'';U"'=(;7.VM 2 or 11' I.V DiH'erence Parallax True altitude 20 50 5(; 8 20 57 04 (1) See note of Art. 30. (2) In tJiis example, the logariiliuia are taken out to the nearest 15'' onlv. 64 AZIMUTH. Ill True altitude Latitud«; Polar distance 20° 50' 04" colog. co.siiie 0.02!>TO 48 20 8:J colog-. co.siiic 0.17S24 1>;3 54 2!l {Sum 10:3 18 00 Half sum 81 39 03 log. co.sine 1». 10203 Half sum minus polar distance 12 15 20 log. cosine 0.98'J'J8 10.35995 ^um Half sum or log. cosine 180^ minus half the azimuth 9.07997 180° minus half the azimuth 01° 24' 22".5 Half the azimuth 118 35 37 .5 Azimuth of the sun 237 11 15 Mean of the H. C. II. on sun 172 49 45 Difference Mean of the H. C. K. on ihig. Azimuth of the llau. 04 21 30 134 45 15 199° 00' 45" II I ii: 37. Azimuth by altitudes of stars. The method is exactly the same as for the sun, and the same rules apply. The stars should be chosen as much East or "West as possible, ])ecause they -will, then, give more reliable results. For the same reason, the altitude should be small, but, on account of the uncertainty of re- fraction for small altitudes, it should not be less than 5°. 38. Graphic solution of the two preceding problems, Describe a circle (Fig. 8 and 9) and draw two diame- ters, NS and QQ', forming an angle ecjual to the colati- nuAPiiu; METHOD. h.') tiulo. Make ACS equal to the altitude and QCJJ, to [\w sun's or star's declination, ]] being to the lei't of Q lor Southern declinations and to the right for Northern one.s. From A, let lall a perpendicular Aa on NS, and from the centre C ^vith the radius (Ja, describe a circle. Througli A and 13, draw An and Us parallel to NS and QQ'. At''.v, N E S A ;ie •h re ie •e- le- ti- /7>. 8. erect ss' perpendicular to As, on the left side if the star is West and on the right if East of the meridian. Join C to the point s\ where the perpendicular meets thi- in- ner circle, and the angle NCs', reckoned in the direction of the arrow, is the azimuth. (1) Yig. 8 is the solution of Example 1, Art. 30. SCQ is the colatitude of Vt Fortune 44" 27', SCA the true alti- (l) See note of Art. 25, 66 AZIMUTH. I n\ iudc 12'' 4-2', J]CQ Iho doclinatioii North 21" .TJ' and NCs' tlie sun's a/iniuth, 71° oo'. Fig. is the sohiti(.n of Kxaniplf 2, .\rt. :;(;. rSCQ is X the colatitude ol' Esquimalt 41'' ;3:]', SCA the tmo alti- tude 20^ 57', BCQ the declination South 3^ rA' and NCs' the sun's azimuth 2;)7° 11'. 39. To find the variation of the compass. AVhon you observe, take the magnetic azimuth oi" the sun or star ; the variation is the ditFerence between the calculated and magnetic azimuths. It is East, when the calculated is greater than the magnetic and West, when it is less. It' this difference was greater than 180°, 300° should be added to the smallest azimuth before sub- tracting. 1 7 10 le 0° VAKIATlON OF Tin: COMPASS. 07 Example 1.— The sun was obsorvod lo the N. ■1%'' -JO' AV ol' tlu* compass, its c'likulatcd axiinulh wns .''2')'^ 4o'. w quired the variation. Mao-netie azimuth Ul"" 40' Astronomical '• ^Vlb 4o Variation /)° 57' W. Exainjik '1. — A star was observed lo the N. 8° 15' E. oi' the compass, the cab uhited azimuth was 355° 4-3, re- (juired the variation. Mag-netio azimuth, 8° 15' or 868° 15' Astronomical Variation 355 43 12° 32' W. Example 3.— A star was observed to the N. 10° 30' "\V. of the compass, the calculated azimuth was 7° 10', re- quired the variation. Astronomical azimuth, 7° 10' or 307^ 10' Magnetic '^ 349 30 Variation 17° 40' E. The magnetic needle and variation must be used only in compass surveying* ; in angular surveying they must be absolutely discarded. AVe know that the magnetic needle is not motionless ; indeed, it may be said to be always moving. It is all'ected ])y diurnal, annual, secu- lar and abnormal changes. At Toronto the dillerence caused by the former between 8 A.M. and 2 P.M. amounts, during some oi' the summer months, to 10', and the ab- normal changes have sometimes been greater than 2 degrees in 8 hours. (1) These numbers would be still (1) See " Abstracts of MiiKiietical observations made at tlie Magnetical ob,> i.e nn lire of ii gro.'it circle. ?' ■(> (•ONVKR(}EX('K OK MERIDIANS. r irom the iiKnidian ofC. By subslractiiii^- or iul(lin,c>- the convergonte to one of thoin, we ol)tiiiii the other, thiit is, wo refer it to the meridiiui oi" the otlier point. Suppose that, having surveyed a line AGIIK , wo deduce the azimuths of the lines AG, Gil, UK, from the nnules of the survev and the astronomical ohscr- />;-. 10. vations taken at A ; any of those azimuths will then })e reckoned from the meridian of A, that is to say, will ))e the same as would l)e found by direct observation on that meridian, if these lines were prolonged till they inter- sected the meridian. 41. To find the convergence of meridians between two points. Find, by a traverse tal)le or by logarithms the depar- ture in Gunter's chains, add its logarithm to that of Table VI, col. 11, corresponding to the mean of the latitudes of the points. The sum is the logarithm of the convergence in seconds of arc. Kranip/e.—^Tlm departure ))etween two points is 47-*» chs.. the mean latitude 4S°, required the converuen<'«v J CoUHRf'TlOiN OK AZIMUTH. 71 ].on-. conver. ibr 1 eh. dopait. (Tal)le VI, col. 11) O.808IO Log-. 47-^ . 2.r,748>; Log". ('011V(M'i^(?n('e 42. To refer to the meridian of a point B an azimutli reckoned from the meridian of another point A, •ar- il )lo ol' iioo 47:'. ) ( sul)trtici ) irr> is ' ; of A, 'i ; thocoiivcriiviicc i East ^ "' "• ^ .^^1^1 ^ ])(»t\veon the points to tho a/imuth givon Examp/c. — Boing- in tho Church stooplo of Caughnaua- ga, noar Montreal, the azimuth of a signal at I'to. Claire was observed and found equal to 270° f)(V, the distance ])et\veen the points was 581, ;>o chs. and the mean lati- tude 4.')° 2«)'. What would be the azimuth of the steephi if observed at the Pte. Claire signal V (1) I3y the traverse talkie, we have Pte. Claire .')77.'iO chs. AV. of Caughnawaga. Log. convergence for 1 ch. di>]iarlur(* (Ta])]e Vf. col. 11.) 0.8211(1 Log. 577.30 2.70140 Log', convergence Convergence Azimuth given 1 180" Azimuth at Pte. Claire 2.58250 882" = 0' 22" 00° 50' 00" 00° 49' 38" (1) Tlie azimuth given, being ilLiduced from astronomical observation at ('aughnawaga, is reckoned from (lie meridian M' that place, and it is easilv «een that the problem requires, in referring it to the meridian of Pte. Clairf, the correction for convergonco of meridians, besides adding 180°. 1-2 ('(INVr.nCFA'CE OI' MKIMDIAX. 43. To find the differences of latitude and longitude between two points, when their distance and azimuth are known* If lie for the a/iinuih to the moridiuns of both points and take the moan. Find, from it and the. distance, the dis- tance in hititude and departure. To the logarithm of the former, add that of Table VI. col, 2, corresponding" to the mean latitude ; the sum is the lognrilhm of the difference of latitude in seconds of arc. Add together the log. of Ta])le VI, col. 5, corresponding to the mean latitude and the logarithm of the departure; the sum is the logarithm of the difference of longitude in seconds of ar<'. The mean latitude is necessary to take out the loga- rithms from Table VI. If that of one point only was known, the other would be found by an approximate calculation of the difference of latitude, using the given azimuth and latitude by the former method. This may be done by construction. Example. — The azimuth of the Mission Church at Pte. Pileue, Lake St. John, P. Q., observed from a station on the East shore of the lake, is 208° 28' and the distance I^IO;") chs. The East shore statioji is in latitude 42° 28' 18" N. and longitude 71° 58' 31" "W. llequired the latitude and longitude of the Church. Mean latiliide. With 1305 chs. and 208" 28' as azimuth, the traverse ta]»le gives : Distance in latitudt l)i' tarture Wi'y chs. K, 1 220 A\' LVTITUDICS AND Fi<)N(}ITUI)i:s. L»)!>". Of>r> <•> 2.S2:j Log. 1 ch. in seconds of latitude (Tii})le VI. co\. 2). 0.814 Approximate latitude of Church Mean latitude Log. approximate dillerence of latitude 2.G37 Approximate difference of latitude 434"=7' 14" Latitude of the East shore station 48° 28' 18'' 48 35 32 48° 31' 5r>" Mean azimvth. Log. 122G 3.08849 Log. converg. for 1 ch. departure (Tab VI, col. 11). 1).8(JG27 J^og. convergence Convergence 2.95476 901"=]5' 01" Azim. reckoned from East shore meridian 298° 28' 00'' Azim. referred to Mission church meridian 298 12 59 Mean azimuth 298° 20' 29".5 Latitude. Log. 1395 3 144-Y Log. cosine 298° 20' 29" 9.G76U Log. 1 ch. in seconds of latitude (Table VI, col 2) 9.81380 Log. diiference of latitude in seconds 22.63481 Difference of latitude 431".3=7' 11".3 N Latitude of East shore station 48° 28' 18".0 Latitude of Mission church G 48° 35' 29".3 r 355 7» <'(>.\vi:i;) !i.!)01 ')."> Lo[]^. diUbronce of loiiL-itudt' in .-^croiuls ^-''.OHOOT Diircroiico ol' loiigitiulo 1-J01"=20' 04" AV Loiiji-iliult' of East shoiv station 11" 58' 81" W I^oiigitudo of Mission chuT* li 72" 18' 8.V' J; ii 44. Given the azimuths of a line at both ends, and the minn latitude, to find the length of the lino, Take \ho diHi'i'onct' of the aziuiutlis, it is jho conv.'r- U'eiico. ItodiHH! it to seconds, take its logarithm, add tlie cologarithm ol' convergence lor 1 ch. dei>arture, onding to the mean latitude, deduced I'rom Table VI, col, 11, and the cologaritlnn sine of the mean azimuth The sum is the logarithm of the distanee in chains. (1) This method gives only iin approximate result, and is of no use when the azimuths are near 0° or 180°. The azimuths ought to he very accurate nnd the points far apart. Exaniple. — From the to}) ol' Montreal mountuin, the azimuth of Iligaud mountain was ol)served and found equal to 203° 53' 33" From the latter, the azimuth of the former was o])served and found 83° 22' 2(V'. Re- quired the distance. IMean latitude 45° 29'. (I) The formula i^ I) // — Z ', — : , vvliiM't' Z iidd Z' are tli'.' :i/.i- . /Z' - z^' ninths, c the fonvorgcnoe I'or 1 cli. (T. VI, (oI. U ) and U iho distan^Mn Gunter'H chains. Isl, Azimuth 2tl Coiivorg-tuice Moan azimuth J.orr. 18C7 3.27114 Col. couvoiovnco Ibr 1 , h. doparturo (Tal)lo VI, col. 11) igQ^j, Col. sine moan azimuth 0.002»il) 203° 53' 203 22 70 33" 20 31 203 38 07 = 1S07" 00 JjO (''>NVKi!(ir:,N('i; ok MKiniM \n.< round llic Vi'itical axis; tlio 11. C. R. is the aiiulc ol' iult r- s<»(tion. Tho samo opcralioii is thou rojH'atod ut C and J). It is to 1)0 obsorvod that, with an inslruniont p:raduatod us in FijT. T), tlio amnios aro rockonod IVom tho station ])ohind, throiig'h tho right, to tlio station forward ; thoy can, thoreibro, assumo all tho valuosl)ot\V(MMi 0' and :;t;o . Lot tho astronomical courses ])o : Sid(\ C^ourso A P, N. 7»r' E. nc s. 82 ]<:. C]) s. 2'; AV. 1)E s. SI E. Thon tho ang'los road on Ihc cinlo a . . v<>cordod in Iho Hold ])ook, Avill bo : M i II Station. H. C. U. B 252° C 238 ]) TO 2d Method. — Tho angle of dolloction is Tormod by ono side produced and tho next one. It is rockonod either from 0= to 3G0° or only from 0"^ to 180^, adding thon tho indication of right or left. To measure it, wo procetnl as in the first method, tho only diflerence being that tho telescope is turned over after the station behind has been sighted, and before directing it to the station for- ward. When the angles are reckoned from 0"^ to 3G0°, the angle of deflection is the reading of the circle ; the field book would then show^ for the survey of AIjCDJ*] : (fOKDKTir SUUVEYI\(}. 77 Station. H. c. n. JJ 72^ V .•)S ]) '2.j() ()thor\vis(\ lh(« aii'^Io Ix-inir r.'.koiKHl only from 0° to 180°, the dcsiii nation "riirht" is n to the midini^whon it is l(>:s.> tlian 180= ; whon it is more, the complement to JJiJO^ is taken, adding' to it the designation "h'ft." Tlie survey of AliCDE, recorded l)y this mellKxl would be : Station. I Anule. n C 1) 12'' liiM-ht. r)8 Kio-ht. 110 Left. M Mel hod. — A traverse anule is formed by one of the sides and a parallel to a certain liiu' adopt(>d as the meri- dian of the survey. This line is not necessarily an astro- nomical or a mag-netic meridian. To proceed by this me- thod, set the instrument at !>, fix the vernier at the assu- iiu>d azimuth of AB, and unclamp the horizontal cinle. Direct the telescope to A, fasten the horizontal circle, and, having- loosened the vernier plate and ti rned over the telescope, sight to C. The II. C. II. is t:.e traverse angle. The same operation is, then, repeated at C, U..., using at each station, the last II. C. II. recorded, in the same manner and instead of the assumed azimuth of AB. If we take 7(3° for the azimuth of AB, the meridian of the survey is the astronomical meridian. At B, we set the vernier :;l 7 with a prolnutor, and lot any iiumhor, .V Tor iufstaiico, rt^- proseiit the possible error in dra\viii2," a line. The di- rection of a traverse course, on the plan, depends only on two lin(\s : a parallel to the meridian and the course itsell'; the possihh> error, then, is only twice 5'. By other methods, it is as many tiini's .V as there arc courses before the one considered ; and, in that case, the best way to make a correct plan woidd be to compute the angles which should have l)e,>n f)l)S(n-vtMl i)i traversing, and plot th(^ survey as a traverse. Traversing- has other advantages, v. hich it would lie too long to enumerate ; we shall, therefore, in the follow- ing articles, suppose the surveys made by that method, the astronomical meridian of one of the stations being the meridian of the survey and th(» instrument so placed, that the horizontal circle readings at that station l)e 0° to North, 1)0° to East, 180° to South and 270° to West. The II. 0. 11. at any station will, thus, be an azimuth reckoned from the meridian of the survey. The station, at this meridian, we shall call astronomical (1). When the survey is extensive, the orientation of the instrument must be rectilied from time to time. The sun or a star will be observed as explained in chapter IV and the II. C. \l. on sun noted at this second astronomic 1 station. The difference between it and the calculated azimuth should ])e equal to the convergem-e of meridians l»etween t' e astronomical stations, l)ecause the II. C. K. is the sun's azimuiik, reckoned from the meridian of the lirst astronomical station, whilst the calculated azimuth is reckoned from the second. The error of orientation will consequently l)e found ])y referring one of the azimuths, the II. C K. for instance, to the meridian of (1) l'cc:iihu» it H wlicrt' llit> i ..mnlli was (i)m'iv<'(l. P' KO CONVERGENCE OF MERIDIANS. the otlior The dilFerence bctwc«'ii them, il any, is the error. It is equally distributed among the courses by dividing it by the number of stations. Multiplying the result by the number of any course gives the correc- tion for it. Example. — The following traverse was made, the astro- nomical station beins; the first one : fi Stations. 11. C. K. Distances. 1 309° 39' 15" 378.0 chs. 2 317 2G 15 489.5 3 324 55 30 1154.6 4 290 37 15 702.2 5 272 17 30 642.0 G 256 31 45 588.4 1 259 42 15 846.0 At station 8, the sun was observed and its azimuth found by calculation 267° 11' 50", the H. C. K. on sun being 267° 59' 10". llequired the error of orientation and the corrected traverse. Mean latitude 48° 39'. N. Departure between St. 1 and St. 8, 3988.9 chs. W. 3.60085 Log. 3988.9 Log. convergence for 1 ch. departure (Table VI, col 11) 9.86830 Log. convergence Convergence II. C. Iv. on sun 11. C. li. on sun, referred to the meridian of St. 8 Azimuth of the sun at St. 8 3.46915 2945" = 49' 05" 267° 59' 10" 267 10 05 267 11 50 Va'yoy of orientation r 45' ."»'' 1.-'/ I'LOTTINCr THE 8URVEV. Corrected /rmrrse. 81 Stations. 11. C. R. observed. Cor- rection. H. C. R. corrected. 1 2 3 4 5 6 7 300° 30' 15" 317 26 15 324 55 30 200 37 15 272 17 30 256 31 45 259 42 15 -1- 15" 30 45 1' 00 1 15 1 30 1 45 300° 30' 30" 317 26 45 324 56 15 200 38 15 272 18 45 256 33 15 250 44 00 46. Plotting the survey. From the last station, the survey might be carried on with the same meridian as before, and plotted with it, the only correction to the course being the error of orien- tation ; but it is more simple to use the meridian of the last astronomical station and lay it down on the plan, making with the first one an angle equal to the conver- gence between them. Each course, then, is plotted with the meridian it is reckoned from. According to it, in the example of Art. 45, the last course 259° 42' 15", should be corrected by the difference between the calculated azimuth and the II. C. R. on sun, 47' 20". At St. 8, the instrument should be set up, and the survey carried onward with that corrected course, 258° 54' 55". Courses from 1 to 8 should be plotted as is usually done, and at St. 8, a meridian drawn making wuth the first one an angle equal to the conver- gence, 49' 05", the following courses being plotted with it as a meridian. It is not necessary to correct the orientation immedia- tely after the observation is taken ; the survey may be continiied, and the correction applied only when th<. S2 (H)Xvkii(;en'c"c of mkuidians. (lay's work is ovservations to rectify its course ; on what azimuth must ho produce tlie lino as a straight (1) lino ? {'!) Miutn lle of Art. 4.'{, and that of llie tlOili mile and mean latUude deduced. Willi tilt' l:Ult'r,llu' iniiia! a/.imiilli is icfired (o the nuiiilian of the COtli tuile. TO RUN A LINK. 83 Aziuiul/i. Departure for the azimuth 285° and the distance 4800 chs Log. 4G36.5 Log. convergence for 1 ch. departure (Table VI, col. 11) 403G.5 chs E 3.Gt)G19 Log. convergence Convergence Initial azimuth Azimuth at the GOth mile it.830o3 13.40G72 3138"=52' 18" 285° 00' 00" 284^ 07' 42" J* 43. To lay out on the ground, a figure of which the plan is given, correcting the course by astronomical observations. Each side must be run with the azimuth reckoned from the meridian where the course M^as corrected. {Sup- pose, for instance, the figure to be a square township and the course to be corrected by observations at the four corners. The azimuth of each side, deduced from the plan, must be re- ferred to the meridian of the corner which IS its initial point. Examp/e.-^UQCimred the azimuths to be used in running the exterior lines of a square township, ABC]), (Fig. 12) whose sides are 8 miles and initial azimuth NAB, 45°, cor- recting the course by observations at Mean latitude 47° N. //^•. 12. the four corners. Azimuts reckoiu'd from the nu'ridian of A BC 135 VI) DA 315'^ 84 CONVERGENCK OF MERIDIANS. a ])op;irlnvo bctwoon A and B A and C B and C 4')2.o5 OOo.lO 452.55 chs. Log. departure 2.05507 2.05070 Log. convergence for 1 'h. departure 0.84200 0.84200 5' 15" Log. convergence 2.40857 2.70000 Convergence 315"=5' 15" 0:50"= 10' 30" Azimuth reckoned from the meri- dian of A i;]5^ 00' 00" 225^ 00' 00" 315° 00' 00 Azim. to be used 135^ 05' 15" 225° 10' 30" 315^ 05' 15' 49. Given the azimuths of sides, to find the angles of a triangle, |l I Sometimes the angles of a triangh> cannot be mea- sured directl)^ either because one of the anguhir points of the triangle cannot bo seen from another or for any other reason. But they may be found if the azimuths of the sides have been observed, by leferring them to the same meridian and taking the ditference. Example. — From a station on the Eastern Shore of Lake St. John, the o})served azimuth of the Mission Church at rte. Bleue was 208° 28' ; a traverse between that station and the mouth of the Peribonka river, being worked out gave for the distance 1827 chs. and for azimuth 323° 28', At the mouth of the reri})onka river, the o])served azi- muth of the Mission Church was 180° 35'. Kequired the angles of the triangle East Shore, Peribonka, Mission Church. Mean latitude 48° 37' N. (I) (1) III this example, tlie aziiniitlH of the traverse and of tlio Church from llie I'iasl Shore Station, hi'iiii,' reekono.l from tlie meridiati of tliat sl;itioii. the tliinl aziimilli is referred to it and the diilereiu'es taken. A\(iIiKS (»K A TKIANdlii:. S') »' (\t\'f JJi'piuiure lor iiio azimntli :j2;r 28' and llu' distance 1827 chs. lOHT.-) vlis. Loo-. 1087.') :i.0n04:] Log. coiivcrg'onco ibr 1 di. (Tahh' VI, col, 11) !».8G7r)4 Log. convorgoiico 2.00807 Convorgonco 802"==13' 22" Azimuth reckoned from the meridian ol' Teribonka 180^ 35' 00' A/imuth referred to tlie meridian of E. fc5hore Station 180^ 48' 22" mea- ^^C • R. Peribonka. Mission Churcl , ;E. Shore St. Fig. i:j. Amrlf. at Peribonka. Azimuth of Church East Shore Anorle at Peribonka 189^ 48' 22" 143 28 00 40° 20' 22" 8« CONVlillOKNCK OF MICUIDIANS. yl/zi'/^ (d 31ission Church. Aziinuth ol' lilast kShorc " reribonka Aiiulo at Misfsiou Chunli 118° 28' 00'' U 48 22 108° 39' 38" Aiiii'/e at East ^horc. Azimuth oi" Mission Church " reribonka Angle at East Shore 323° 28' 00" 2U8 28 00 25° 00' 00" 50. To produce a parallel of latitude by layinjs; out chords of a given length. The ani-le of doflection between two chords, BAD, Fi£^. 14, is equal to the convergence of meridians for the //V. 14. length of a chord, and the azimuths NAB and NAC, are equal to 90° minus half the convergence and 270° plus the same. i'f tht i'ai;amj:ls ok i.ArrrrnK. S7 Knniiff/r. — To lay oul, l>y rhords lsi» chs. Utwj; Ww l)aralli'l of 40^ :>l' N. Iwojj^. 480) 2.(58! i:Jl Log. convorgviico lor 1 ( li. dcpmlun' (TaMr VI, t'ol. 11). 1».87871 Log. convorgonco l*2.')G8()iJ Convergence, or (It'llcct ion angle .'570"=(;' lo'' i Convergence ;}' O.V Azimuths ol' chords 80^ 5G' 5;V'- 270° 00' 05'' This parallel is the second base lino in Ma^'itoba and the North West Territory. The chords are the bases ol' the townships, the East and "West sides being meridians. Since they converge, the North base is smaller than the South one. The dilFerence is equal to the length of side^ 489 chs, multiplied by the sine oi' the convergence. This angle ])eing small, its sine is equal to the num})er ol" secon Log. sint ofl second sot — 4.384o4 Log. Isl. 18.0(5407 1st. oll'se t = 0. ch., OlloO — 1 links 2d " ss 0. 11">0 X 4 = 5 " 8d " = 0. llfiO X 9 — 10 " 4th " := 0. 1159 Xl6== 18 " r)th " = 0. 1159 X25 — 20 " Gth " = 0. 1159 X3C — 42 " 52. Projections. • ;i i m The preceding method may be used for making pro- jections. Th(^ offsets may be calculated for each of the parallels of the map, adopting, for their interval, a number of miles suited to th(^ scale. Having drawn two perpendicular lines, N^S and WE, (Fig. 15), these offsets will be platted at 1, 2, and the points obtained joined by straight lines. In table VI, col 7, the distance (1) This parallel is part of the boundary between Canada and the United States. It was laid out, by the International Knundary Connnirsion, as des- cribed here. Pll<».FK< THINS. 80 ill (liiMiis, iM'twtM'ii lli(. |)!inill.'l WJ'^ ;iii«l 111,, iioxt (.iM«, w ill Ix" Ijiki'ii oiil, Net oil" IVom C to 1) and I'd i)lo(loa as /y^ 15. it l( KVV. All tho parallels being piojected, the ilislance on each ol' them, between two meridians will be taken out of Table VI col. 9, set oil' on the map, and the points ". ". ". 1 f>, ffJ^, ,'/, '/, I! if 00 (.'ONVEK(iEN('E OF MEUlDIANf^. On the Knh paniUcl, we liiul in the «aino iiuiiincr : Distance between the meridians 38o0, 2 ehs. 12 3 4 OlFsets 24.2 !H).7 217.") -J^''-^ rhs. and so on lor ollu-r i>aval!el.^^. (lis. CHAPTER VI SUN ])IALS. 53. Horizontal dial. ^ Fix on a plate NS (fig-, ifj) a triaiin-ular sheet of metal BAG, whoso angle CIJA is equal to the latitude, the plane of the triang-le being- perpendicular to the plate. This last condition is attained by presenting- a square on North Soiitl IX vin iv>. 10. VJI VI both sides of ABC. With a spirit level, make the plate BO horizontal, the line BA 1)eing- nearly in the meridian and the hypothenuse BC facing to South. Observe the sun's meridian transit and turn the plate BO, keepino- it horizontal, till the shadow of the triangle is exactly^n the prolongation AO of BA, when the sun is on the me- no StJN DIALS. ridian. Fix the dial firmly in that position and, with a j'-ood watch set to noon w^hen the shadow is on AG, mark the points o, n, n, a where the shadow is at 1, 2, 3 r. M., 5, G, 7 A. M. Join Ba, Ba, Ba, which are the hour lines. These lines may be drawn directly as follows. Let BO, lic^. 17, be the dial's meridian or noon line. Draw BC, making with BO an angle OBC equal to the latitude. At any point of BC, erect a perpendicular CE, measure VII VII VIII •m Fi^^. 17. EO equal to EC, and at E, erect the perpendicular FG- to OB. Through 0, draw the lines os, os', os" inclined of 15°, 30°, 45° on OB. Join Bs, Bs', B.s" which are the hour lines. For some of them, the construction \vould extend outside the plate, ])ut they may be found 7 1 HORIZONTAL DIAL. 93 IX by drawing parallels to the 9 A. M. and 3 P. M. lines, measuring ah' equal to ah, nr' to ni\ ae,' to ne, and joining l\b\ Bf.-', W. We supposed that the sun was observed with a tran- sit. When none is at hand, choose a level surface or piece of ground, and plant on it a staff vertically by means of a phimb line. Describe several circles from the foot of the staff as a centre, and note the time of a clock or watch when the extremity of the shadow crosses each of them. The mean (1) will be the time shewn by the clock or watch at the sun's meridian transit, and the dial will be in the right position if the shadow is on the line BO at that time. A more satisfactory method by means of Alioth or y Cassiopeiae and Polaris, is given at Art. 31. To apply it, suspend a plumb line to a tree or an elevated point SmiQ .^ Polo Star Fig. 18. 'Gto ined :hich ction 'ound and let the bob dip in a pail of water, to lessen the oscil- lations. Place a light behind you, to illuminate the plumb line and render it visible. The time when both (1) Noticing tliat tlie meiin of .\. M. and I*. M. litnts, siu-li as 9 A. M. anj 1 I'. M. in 1 1 A. M. and not 5 o'clo(;iv. 04 SI)N DlAIiS. stars arc covered l)y it is noted, and 24: or 20 minutes after when Polaris is on the meridian, a stake A is placed so that, looking from its top, the star is covered by the plumb line. Another stake will, then, be planted at I>, so that, looking from it, the other A is covered by the plumb line whose shadow, cast by the sun, will fall on 13, at the time of meridian transit or apparent noon. 54. Vertical dial. This dial is formed by a style, fixed on the side of a wall, and bearing at the end a disc nearly perpendicular to the sun's rays at noon, in March or September. At its centre, a hole with sharp edges, projects a bright spot on the wall, in the middle of the disc's shadow, cast by the VII ■t.m'M VII vni sun. The position of that spot among the hours lines indicates the time. This dial has some advantages ; it can be construe/. ;d of large dimension.: and placed on a high wall, so as to be seen from a great distance*. It is, also, more easilv made accurate. VKUTK'AL DIAL. ft;-) T(» -raduali' it, procood as lor the oilier dial, at tlio .sun's meridian transit, and mark the place ( Fi'- 19) at that time, of the bright spot. By moans of a plnmh line, draw a vertical line through that point ; it is the noon line. Measure the distance between and the middle of the hole B, and describe the triangle AOC, where CO is equal to the measured distance OB, AOC to the sun's polar distance minus the colatitude and OCA to Uie supplement of the polar distance. With a good watch, set to noon at the time of the sun's transit, mark for each successive hour, the positions s, s, s, of the bright spot on the wall and join As, A.i, An,'.'.'.... which are the hour lines. There are several methods of drawing those lines di- rectly ; but, although not very complicated, they w^ould be out of place in this work. Sun dials show the apparent time, and we know (Art. 4) that civil is mean time. The clocks, therefore, must not be set to the time shewn by the dial, it must be pre- viously corrected by Table VII. ■■H HMH TABLE I. MEAN REFRACTION. (1) 97 Apparent altitude. .2 p .in O Apj»arent altitude. .2 ci o Apparent o altitude. .2 o Apparent 1 o altitude. | .•1 2^ o / / // o / / // o / / /' / // / // 00 3«) 00 14 00 3 50 29 30 1 43 45 58 76 15 20 31 00 30 3 42 30 00 1 41 46 56 77 13 40 28 00 15 00 3 34 30 1 39 47 54 78 12 1 00 2') 00 30 3 27 31 00 1 37 48 52 i ''9 11 . 20 22 00 10 00 3 21 30 1 35 49 51 i 80 10 40 20 00 30 3 14 32 00 1 33 50 49 : 81 09 2 00 18 20 17 00 3 08 30 1 31 51 47 ] 82 08 20 10 55 30 3 03 |33 00 1 30 52 46 ! 83 07 40 15 35 18 00 2 58 1 30 1 28 53 44 84 06 3 00 14 25 30 2 52 34 00 1 26 54 42 85 05 30 12 57 19 00 2 48 30 1 25 55 41 ' 86 04 4 00 11 44 30 2 43 135 00 1 23 56 3i» 87 03 30 10 44 20 00 2 39 ! 30 1 22 57 38 88 02 5 00 9 52 30 2 35 36 00 1 20 58 36 89 01 30 9 07 21 00 2 31 30 1 19 59 35 90 00 6 00 30 8 28 7 54 30 22 00 2 27 2 23 37 00 ! 30 1 17 1 16 60 61 34 32 TABLE II. 7 00 30 8 00 30 9 00 7 24 6 57 6 33 6 12 5 53 30 23 00 30 24 00 30 2 20 2 10 2 13 2 10 38 00 1 30 139 00 30 1 15 1 13 1 12 1 11 62 63 64 65 31 30 28 27 26 sun's jPiiral inaltimrte. .Utit. I'aralhix. 2 07 40 00 1 09 GG 9 30 5 35 25 00 2 04 30 1 08 67 25 10 9 10 00 5 19 30 2 02 41 00 1 07 68 24 20 8 30 5 05 26 00 1 59 30 1 00 69 22 30 8 11 00 4 51 30 1 56 42 00 1 05 70 21 40 7 00 4 :^"9 27 00 1 54 30 1 04 71 20 50 6 12 00 4 28 30 1 52 43 00 1 02 72 19 60 5 30 4 17 28 00 1 49 30 1 01 73 18 70 3 13 00 4 07 30 1 47 44 00 1 00 74 17 80 2 30 3 58 29 00 1 45 30 59 75 16 90 (1) Barometer 30 inches: Fahrenheit Thermometer -|- 50°. I 08 TABLE III Times of elongation of the pole star, on the 1st, 11th and 21.sl of each month for latitude 47° N. Jjong-itude o'' W. and the year 1878, (1) "■^ ^ ^^^m f^ IHHM ^^■" I O 1 o o i *i ■*j Month. ! i 1 Ist • S3 ' w lllh. S3 C o 2 1 W 21s t. o January... 0*^ 25'" A.xM. W 42'" P. M. 11'^ 03'" P M. w February . 10 20 P. M. 1( 9 40 " (( 9 01 H l( March .... 8 29 i( 11 1 j 7 50 " a i 11 U u April G 28 (( u 6 02 A.M. E 5 A.M. E May 4 43 A.M. E 4 04 1 3 24 u .( June 41 u u 1 2 02 <( 1 23 (( (1 July 4H ti it 04 (( 11 21 p. M. .'i August.... 10 37 P.M. n 9 58 P. M. (( 9 19 n (( September 8 34 u f( 7 56 <( 7 17 n u October... 6 38 u u 5 58 <( 5 09 A.M. W November 4 20 A.M. w 3 47 A.M. w 3 08 (( (( December .J 28 <( a ; 1 i 49 (( 1 10 (( (i (1) These times increase by 21" yearly, about. 99 ,nd 21st ,1 I u A.M.' K 1 1 to o -M O CM CO a o CO .2 oo -M r-l § ^ 0) "tS u SO Pi o a "«1 •opmipj^ ° 71 ro -f o -r I- x ci c — • -M M -t< "f -t f -^ rf -f 't 't »ft o »o •.": o o 00 1— 1 ri «f5 t- !!C ~ ri m i- ci 'M i« i- o 'f -ti -f -f ifti •« in iC5 ift o o =■ T-H 51 CO ^ "f in t- ci -H ro ift !-• — irj lO CO -h -t -t -f -f »f5 «f5 »« o o o o O — 1 ° ^ ^ rt r-^ ^ ,H rH ,-( C-l M (M C^l 5^1 : 1888. -t -o t- r. -- ro '>r CO o ro '-0 CO M -fi -t -f f »n m lO lO o o o o '— • o _ -H ^ r- ^ r-i -H -^ M !M 1 • 00 -t< '^ CO c: *>! -f to OD — ' CO :o CJ IM -t -fi 't lO ift lO O i« o o o o »-• ° rH ^ »-i T-H rH ,-( ^ .-( (>1 'M 0.1 (M M C/J CO lO t-- CO O 71 -t t- Oi — 1 rf^ t- O CO -f- -f -f itj in lO i« ift O O O rH ^ ° 1— lTH»-ii-Ht— If— i'^rH(M?.lC<|(MCq 1885. «ft t- 3i — CO »n b- CI -iC0»0I^O(MirtX'-<-i< 't-t't'OiOOinOOO OrHrl ° ^ ^ ,_,,_( ^ rH rH (M (M 5^1 (M (M (>1 S2 00 I— 1 O00O(M-+i-jDC0OC0iO00rH^ ° r-l ,-> ,-t T-i r-i r-> r-( l^l 01 '>t 0-1 0--* 0^ CO CO t-H . — _ — . , .- t-COO(M'fCOOO'HCO«DOi'MiO Ttt-^iftiOiOiOiOOOOOi— ir-l ° r-lr-i-Hr-i.-ir-i»-Co I— c; r-t 't '-0 crj (M »ft rt (1> .A -a o O i 100 I I N J3 1- X .c j5 ^ o " o P CO "^ I— I •rH I— « o SJ (2 •0|)ll)ip!rf T 'M ro -+ in "^ I- o) ci o '— n ^-j -f -f -f -t -t -f -f -t -t lO lO >n in i.-s c GO 2 M -* 2 I CO - -t .s <1 lO c o o o o i« o lO o :r o o <-i o ci CO o ift :o «-! o cr. t^ lo c ,d 03 (M * S I' < .5 CO - C5 - CI " CI CI 6 <1 o •- inoioinooooooooo CO* CO t- CO *^ lO -*• so ci .-• o a; CO I— crH^^l— (r— Sj ® fi g 5 2"=^ ".AS I CqcC'*»Ci»t-00OO>-iClC0'»f< is . 101 M T-i -f n in m M (M o o o • ■ • r- in CO ro CO CO r-< — »— t .-1 -H 1— 1 -f CO C^l M ?1 C-l 36 40 1 " 10.0 37 40 i " 9.0 38 40 i » 8.0 :: :: s z> lO la -t CO CO **< rf Tt< 12 59 13 59 t- :: 5 in m H T— < 1— t 19 40 20 40 i CO "HH ■ O lO •opinijirx ° 'M CO -f in tr I'- -/•^ r^i o "H ci CO -f t -t -r 't •»< -r -t -t o in in in in ■B 1 1 ^ r* :: x :: r*- J o ^h :: *>! co s -t o II 1 TI 'M (M CO CO CO CO CO 1 = a 2 1 • 1 in »n o r m »n o o in o m in in "^ T. x-^ -o -f '^5 c oo 'o CO <-< «:' in rl C CO '>> o 1 a — — — 1 ^ — 1 »-H in in o QO ^ Asi ^ (M S 1 ,^ Q o o ^3 <5 (/J I— ins::~:::3U3::sinc::: M CO 1 h- c: — ri r/5 1- •- -^ m in -f ** -t< CO I. S •+ -t< CO CO ro CO CO CO CO CO CO CI :o >^ •^ CO ^ in 'o t- :/: ci o '- c-i CO 't' m ^ _ 4* •*a s i V 1^ ^ I- : CO :: ci - o ■: — < 'm .: co -f< c§ < M 01 iM -M CO CO CO CO CO J2 ■/• CO m in o in in o o o o in o m c o C9 "i ! CO CO CO CO ;> Oi CO .1^ re tt a CO X -- « o 1 ^" ^" O in o in o o m m in in o in o o h:r>-fcO'-ic b-inco^xoco ^3 M (M M !M M 1— ( 1— t .-H 1— 1 T-H CO m H < CO C3 o V X "3 OOOOCrOOr-^r-— i^rH 5 CQ. -t-t-ti"fi'i<-t-f'*''f-f't-+<'* Ph ° C5 CO t- *-0 in -t CO "M --1 O O 00 t- ^ «^ -3 CO CO CO CO CO CO CO CO CO CO (M 'M M _. i" ^ "S 5 s 3 i) (M -M C^l » ft rH O N in::;:-::::--::::^:::: bo P J? <3 CO .£! ^ —IrHOOOOOO CI CSS^ClCi t: 3 in o m in m m in m ^ -f -* Tt< -^ o .ii ° CO -t m ^ t- 00 C5 o rH (>-| M -*< in T-( ^ ^ ,-H rH rH r- O-I •>! (M (M O-l C^l g .2 rb i Ph o ;. -r "O III o tar. len ve Polur .sine from aris. (MCO-^^inOt-COOSOi-ftMcO"* •pfiTP-^-rrjt'^'-fiT^iniomiOin OJ «t-J3 M)« — « IS ♦» C o c3 ^H^§| h:i m I I I I I :,l ' Jl I ■ 1 rh! ,1 1 I' I : 102 CO 1— QO r£5 M fi CS -M CO f-t o l^c M 15; O ;zi s: r£l o "^ ■4— -t^ o o ^ O r— 1 -^j (S o C) • • •^ Ol ;> t: -^ Oj •f) w k O) H^ M < •i-H 1^ H O O y:5 o • opniijiirj (M ro -f K-5 to I- cc r; o p-> ri ro "^ 2 8 Ci a 00 ^1 |j ^ .r5 ? ^ o o C W T-i 1-1 (M n IM £ V o V o c: in — 1 r-i 1— ( lO CO f irs i« ro fO - ^ cocot-t-i-t-t-t-otoo-oo -t o VD I- CO C5 c: — 1 ri ^•; -t* m -o ^ _ ^ „ ^ ^ J, ^1 5.| >j 5., ;.j J.) 5 >-» o o o c O in 00 S ^ M § V ;M ■: ro CO ro CO CO CO CO CO CO CO CO CO CO CO CO :o CO CO o-; CO -M — ' O Cri C/'- I- 'O lO -t< CO (M ^ »n m ic: irt T -f -t r -i^ -t^ -^ -* ■^ X o o ft ^5© '.'^ ^ ^ a o o V 1 V 5^1 ■>! ^1 'r< CO CO CO • .a <5 1 V o 1 OOiOiOinOOiOOOiOinO 5-1 C: CO » Tf ^J O I- lO 'M Ci O -co -— ir-1 inOinrti-rJiTfi 00 1- CO CO 1.1 V CO C^ ri (M M (M IM M l CO CO rHOClCOt— COiO-fCOC-l'— "©0:1 CO CO C-1 C-J C^l CI C-l C-l CM CM C^l CM rH Magnitude. S.T. when vertic. with Polaris Colog sine dis- tance fi om Po- laris o C^l CO rJH irt cr t^ 00 n T-l C-1 CO ■* m^ 1 ■5 ^0 -t 1 lO lO 1 ^1 - s« a o o rf !M o CO o o I— 1 lO in >o c: O "0 -t -f rp 70 70 70 70 M CI CO M ^ ■^ -t ■* JO CO - lO lO o • • • C5 O '^0 29 1:9 30 29 31 29 - CO TtH CO CO ifl lO o 1 o t- ■* lO Tf rf^ C5 o o M CO CO l-H O CI M 'M rH M CO ■* n U5 o ^ 'O a» ^ -M 'd p) Ctf GO 1:^ QO r-\ m ei 0) t>> o ^ -M ^1 o 6*H ^ yj M a -M OQ ^H 0) ^ -t.J o iz; r^ -4-) u *^^ -+ t? lO (— H O c3 -M w» o -t-J CM H ^ a> ill c a ;h 'o o rd a •IH cc •pH o Ph •opu)!p!T^ c-1 ^0 -t »o "o t- 00 c^ — -M CO -t< -t -t Tfi -r -t -f -f -t^ 10 »n in »n "3 B "x M .d 1 1 O •J '■£■ o \ ^, ^ 1 ^ 1 a i •s 1 < 1 " t c: - - :: - :: .0 ^ :: :: :: ^ inomirsoiooooifsoiAo "" — — J r:> c^ cc CO 1-- t- ->:J ci iti lO •^ -f CO CO CO CO CO .*0 CO CO CO CO CO in::-:::::::::: J ::::::::: CO 6 -S ^^ 1 ci 00 t>» t« t- -.r ".T m in m m in »n in in «n in in in >n un in ° t- CO Cl C: — 1 5>-. CO -* in -0 I- CO Ci •-3 a ^ CO ^ GO o . . ,, Annual Azimuth. . increase V oo^oiniooooinminm o5 X t- in 'it CO M ci 00' "—1 in m m Cl GO ivn-::s::-;::;::io:::: CO CO _ 41 OCiCl ri5ii»CO00QOCO00C0CO in in in in in in in in in in in in in ° ccasc;--H(Mco-t^in--ct-a)C5o ■M v> T'-i CO :o .-o CO CO ct: CO CO CO '^ • 8 1—1 m o • o Annual increase V ^ L- : :: :: co : cr. .: : 02 :: r-c rH „ ,_ 5.^ Jv, • "5 <5 oinooininioininooinin CO V5 in CO rH c; I- in :o r-^ ci CO CO M ri 3-1 (M 3^1 — 1 r-H rH rH ri ° ^ in::::::::::: :::::::::: CO cocococo(-ococo-f-t-tHTf<-t-+ ininininminmininmininin cscoi-oin-tco'MrHocioot- CO CO CO CO cr: CO CO CO CO CO c^j 7i m • pfi (M C5 P^ • o CJ Ci CO. l-H 00 • o Annual increase t-: - ::ooiCi: : 03 ::rH rH rH rH (M (M • s oinominoooinmoc in O-*iC0i-iCi00«iTtrHOt-Tj1rH IM -> -£3 o •■^ t ^ o (—1 -u -S -* '-< "^ si 03 £ S • 1— t N bo f-H •1— ( •OpTl'Jl^Tjrj C'l rc -^ -t< -t -t 'i* »« cr t- Gc C-. c; — I 'M ro -f -f -^ --f -t" «ft in iSi >c C'13t-C0C5O.-i(MC0 2 S CO J* -t I- £ 5^1 "" O CI CI CI :: ci ci CI CI CI »o QC5. -t-> -1 O ift I* 3 I -I .2 (M CS o ^ o . (V C "-I t. a o *-« »o irt in lO lO ic m m in ift lO o c CI CI CI CI Ci CI CI CI CI C^l CI CO CO o o cococococococococooocococo ° I— crm-tcoc^i'— oc^oot-:cio COCOCOCOCOCOCOCOC5C1C7MCJ V ^in2co^^::co::::::t-:::: o in lO CO t— t- -* rt* ■* in :: :: CO I- t— t— t^ CO CO CO o o ixi m m m ooooooooooooo ° ci CO "t >n CO t- 00 c: o '-I ci CO -t< ^ ^ rH rH ,-H ,-1 rH i-H (M Cl CI C-l M s <1 o m in o o o »n in o in in o in ^ cot— ir-t-t^t-cococoinin>n-H in::::s-:::i:::i-:;;i^ CO CS 03 « u -J: "o ft* ■" c P •= o .2 « tf ^ o o to ci CO ■--*< »n CO t- CO C5 o i-i c^i CO -^ -^•^•^■'^■^Ttt'^Ttimioioom ^1 T^ o o o 1 • • • 1 '« Ci 1—1 1—1 , ^. >— 1 ^^ O ^^ ro CO p— ' 5^1 CO 1 1— 1 ^" f— 1 ^ lO • ^ O If? lO lO ^ irt --o 00 O' CI -t -.o CO '-^ -i^ o J-. CI ■-t't-t'>n»^iO o E ?— 1 'J3 CO o o 0) aJ ^t-. X. O. C. ,-lC^i. co-f CI c^i CI CO CO CO ro CO s < »o O' lO o o o o o o m o lo i« ^ ■<+ CO* I- c. i-< CO Id 1— cr. -H -t* -o o OOCOCOCO^-f'^'t'^iCilCiQO si -1 C5C;a:OC:OOCOi-H.-irH,-HC^l C0C1-H,— ClXt-COiOrtiCOCli-i CI CI CI CI ^ 1— .— i-H f-H 1-1 rH T-( r-l • CO ■*— ' c p o o S . C-! r-H Ci CO o o • o 4 i <1 =^ V ^Tt*ift. CO. i-^. x. o. o--- cO»ft »oiaif5>o»om»oinin»oio>nin ° '-'OCiOOt-COlC'^OOC-lrHOO C o i CO I— GO -t-i CQ o rC o "^ l-H 72 o Ph O fee o •Opn)l)Br£ 1 ° 5-1 CO -t* m •-? I- 'r; r: o — -M r: -+ v tu V '^ S >— i:i;::jj^'!M^^^^^-i • c i 73 "1 =^ o • c in C' o »c in o o o o o o o ' — ^ S .9 ^ V'»» •••••••• >w ~, "^1 ^ — c. c- --I r- -M ^o -+ ir^ -.2 t— 00 Ci ?=5 ^^ O ""^ p-* coro-fi'^-^'+'-t't-T't'ti-r-!' iTJ c o \ iM !?1 C^I o L- CO Oi c: -- 1^1 ro -t lO - t- CO i aJ V -o S ^ t- :; :: -J :i :: ;: 00 ^ c: :i :: J • c 2 ■/I <; o o S CO ■ ooooctnirsoiooioo'C cs -4— » Cl O 1-1 (M'CO ^•nt-OOO^fO'+l (^ O 00 s »0 O rH ,-( 1— 1 r-t •M — c o D _. *" V « •5 'O OCSCOCOOCCOXt— t-«:-t~-t-t- •C a (M 'M C-l '>^ (M -M ri !M 3-1 (M CI (M (M ■* O *0 I- CO Ci C: r-i Ol CO -t >0 ^2 ^ ^ ,_! „ ^ rH (M 7) (M (M (M ri (M V v V '« s Qo::-'jCi::::::c:o^^s r/3 = fe 1— 1 r— ■ •^ " 5-1 ^-> ■J. 3 s cs S ro • m «o m o o — lo c lo o in o o -^ "" (M CO "t CO t- CO Ci i-H (M -f «0 t-' C5 -:; :iCo:i ::ci:; ^ot: rH i-H 1-H rH M • a c w. i -< " u p '5^ fl oc • ico»oir50»cm«oioomo»« 1^ ^ ^ o -*-» Ci'--(MTt--s-^- es rH::^^-::---.^^^-^ - Cfi N 0 t- o H — t— ( 1—1 ^. ^ ^ 5 — (M CO : »o o «o 1 ; i- -o K-5 ^ o >« (M m o »rt 1 • ■ 1 30 rH CO 1 -t to in 1 ^ - ,^ CO CO t- »-H 1— ( 1—1 -t ift o 1 1-1 tH ^ (M CO -^ u:5 O >A TABLE VI. Quantitios havin£>' relation to the fig'uiv and dimensions of the Earth. '♦J a o / 42 00 30 43 00 30 44 00 30 45 00 30 46 00 30 47 00 30 48 00 30 49 00 30 50 00 30 51 00 30 52 00 30 53 00 30 54 00 One (rnn- ter's olinin in seconds of Lati- tude. (1) 0.G521 .G520 .6519 .6519 .6518 .6518 .6517 .6517 .6516 .6516 .6515 .6514 .6514 .6513 .6513 .6512 .6512 .6511 .6510 .6510 .6509 .6509 .6508 .6507 .5597 Logarithm. (2) 9.81429 25 21 18 14 10 06 02 9.81399 95 91 87 84 80 76 72 68 64 61 57 54 50 46 42 39 .0) S (3) o o o o One Gun- 1 tor's chain n seconds i of longi-ji tude. (4) 0.8742 0.8811 1 0.8882 0.8955 0.9030 0.9107 0.9186 0.9267 0.9350 0.9435 0.9523 0.9613 0.9706 0.9801 : 0.9899 i 0.9999 I 1.0102 \ 1.0208 ' 1.0318 > 1.0430 j 1.0546 ' 1.0665 I 1.0788 ! 1.0915 i 1.0045 I Logarithm. (5) 9.94161 .94504 .94853 .95208 .95570 .95938 .96312 .96694 .97082 .97476 .97878 .98286 .98702 .99126 .99557 .99995 1.00441 .00896 .01359 .01830 .02309 .02797 .03294 .03801 .04316 = >-> (6) 114 116 118 120 123 125 127 129 131 134 136 139 141 144 146 149 152 154 157 160 163 166 169 172 V 42 00 30 43 00 30 44 00 30 45 00 30 46 00 30 47 00 30 48 00 30 49 00 30 50 00 30 51 00 30 52 00 30 53 00 30 54 00 •ij m 108 TA15LE Yl.— Qmtuiifed. Quantities haviiit*- relation to the figure and diinensions of the Earth. i ' II \ ' n 'Ml -3 H-1 Onefk'fjree of Latitude in chiiins (7) 42 Oi) 80 43 00 80 44 00 30 45 00 30 46 00 80 47 00 80 48 00 30 49 00 80 50 00 30 51 00 80 52 00 80 53 00 30 54 00 5520.0 5521.4 5521.9 5522.4 5522 5528 5528 5524 5524 5525.8 5525.8 552G.2 5520 5527 5527 5528 5528 5529 . 1 5529. G 5530.1 5530.5 5581.0 5531.5 5532.0 5582.4 .9 .4 ,8 .8 .8 4) .V 03 (8) c 0.2 One decfree of lionj^i- tiide in chains. (9) 4118.1 4085.7 4058.0 4020 . 898(;.(i 8958.0 8919.0 3 884. 8 8850.2 8815.4 8780.8 8744.9 37(19.2 8G78.2 3037.0 8G00.4 8568 . 6 852G.5 3489 1 3451.5 3413.6 3875,5 3887.0 8298.4 3259.4 ,0; 5C .10) Lo^aritlini ol' tlie conver- gence of me- ridians for one cliain de- parture. (11; 10.8 10.9 11.0 11.1 11.2 11.8 11.4 11.5 11. G 11.7 11.8 11.9 1 .0 12.1 12.2 12.3 12.4 12.5 12.6 12. G 12.7 12.8 12.9 13.0 I.7G712 .77490 .78255 .79007 .79747 . 80508 .81261 .8201G ,82775 .83580 .84290 .85045 .85810 .8G576 . 87885 .88101 .88867 .89G87 .90409 .91184 .91962 .92744 .98529 .94318 .95112 253 258 258 258 252 252 252 253 258 253 258 254 254 254 254 255 256 257 258 259 261 262 263 265 -a 42 00 80 43 00 30 44 00 80 45 00 80 4G 00 30 47 00 30 48 00 80 49 00 30 50 00 80 51 00 30 52 00 30 .53 00 80 54 00 mt piisions :( Latitude. o / 42 00 30 43 00 30 44 00 30 45 00 30 40 00 30 47 00 30 48 00 30 49 00 30 50 00 30 51 00 30 52 00 30 53 00 30 54 00 109 TABLE VII CORRECTION TO APPLY TO THE TIME 8HEWN BY A S'JN DIAL. When marked | , the correction is to be added to the dial's time ; when marked — it is to be subtracted from it. Day of the month. January. Feb'ty. Marcli . April. May. June. 1 + 4» 4- 14"> 4- IB'" + 4"° — 3=« — 2°* 11 " 8 " 15 " 10 " 1 '^ 4 " 1 21 " 12 " 14 " 7 — 1 " 4 1 July. August. Sept. October. Nov. Dec. I -.. 4tn + 6" Qm r — lO"* — 16"° — 11"» 11 " 5 " 5 — 4 '• 13 1 " 16 " 6 2] " 6 " 3 " 7 " 15 " 14 « 2