V<^ \%^^ IMAGE EVALUATION TEST TARGET (MT-S) u I I CIHM Microfiche Series ([\1onograplis) ICMH Collection de microfiches (monographies) Canadian Institute for Historical Microreproductions / Institut Canadian de microreproductions historiques I Technical anrt Bibliographic Notes / Notes techniques et bibliographiques The Institute has attempted to obtain the best original copy available for filming. Features of this copy which may be bibliographically unique, which may alter any of the images in the reproduction, or which may significantly change the usual method of filming, are checked below. Coloured covers/ Couverture de couleur I I Covers damaged/ D Couverture endommag^ Covers restored and/or laminated/ Couverture restauree et/ou pellicula I I Cover title missing/ n Le titre de couverture manque Coloured maps/ Caites gdographiques en couleur [~~^ Coloured ink (i.e. other than blue or black)/ D Encre de couleur (i.e. autre que bleue ou noire) Coloured plates and/or illustrations/ Planches et/ou illustrations en cnuleur Bound with other material/ Relie avec d'autres documents □ Tight bindi \q may cause shadows or distortion along interior margin/ La reliure serree peut causer de I'ombre ou de la distorsion le long de la marge interieure D n Blank leaves added during restoration may appear within the text. Whenever possible, these have been omitted from filming/ II se peut que certaines pages blanches ajouties lors d'une restauration apparaissent dans le texte, mais, lorsque cela etait possible, ces pages n'ont pas ete filmees. Additional comments:/ Commentaires supplementaires: L'Institut a microfilm^ le meilleur excmplaire qu'il lui a eti possible de se procurer. Les details de cet exemplaire qui sont peut-«tre uniques du point de vue bibliographique, qui peuvent modifier une i-nage reproduite. ou qui peuvent exiger une modification dans la methode normale de f ilmage sont indiques ci-dessous. □ Coloured pages/ Pages de couleur □ Pages damaged/ Pages endommagees □ Pages restored ard/or laminated/ Pages restaurees et/ou pelliculies Pages discoloured, stained or foxed/ Pages decolorees, tachetees ou piquees ^/ Pages detached/ Pages detachees Showthrough/ ansparence 0" □ Quality of print varies/ Qualite inegale de I'impression □ Continuous pagination/ Pagination continue □ Includes index(es)/ Comprend un (des) index Title on header taken from:/ Le titre de TentCte provient: □ Title page of issue Page de titre de la I I Caption of issue/ livraison itre de depart de la livraison ead/ ique (periodiques) de la livraison □ Masthead/ Gener This item is filmed at the reduction ratio checked below/ Ce document est filme au taux de reduction indique ci-dessous. 10X 14X ,8jj T 12X 16X 22X 26 X 30X J 20X 24 X 28X J 22 1 u'il :et de vue e tion es The copy lllmed here has heen rnproduceci thanks TO the generosity of: National Library of Canada The images appearing here era the best quality possible considering the condition and legibility of the original copy and in keeping with the filming contract specifications. Original copies in printed paper covers are filmed beginning with the front cover and ending on the last page with a printed or illustrated impres- sion, or the back cover when appropriate. All other original copies are filmed beginning on the first page with a printed or illustrated impres- sion, and ending on the last page with a printed or illustiated impression. The last recorded frame on each microfiche shall contain the symbol -^ (meaning "CON- TINUED"), or the symbol V (meaning "END"), whichever applies. Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in ons exposure are filmed beginning in the upper lef i hand corner, left to right and top to bottom, as many frames as required. The following diagrams uiustrate the method: L'exemplaire fllm6 fut reprodult grSce d la gAnirositd de: Bibliothdque nationale du Canada Les images suivantes ont 6x6 reproduites avec le plus grand soin, compte tenu de la condition et de la nettet6 de l'exemplaire film*, et en conformity avec les conditions du contrat de filmage. Les exemplaires originaux dont la coiiverture en papier est imprim^e sont film6s en commenpant par le premier plat et en termlnant soit par la dernidre page qui comporte une empreinte d'impression ou d'illustration, soit par le si^cond plat, selon i^^ cas. Tous les autres exemplaires origiriaux sont filmds en commengant par la premidre pege qui comporte une empreinte d'impression ou d'illustration et on termlnant par la dernidre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbols — ^ signifie "A SUIVRE" le symbole V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent dtre filmds A des taux de r6duction diffdrents. Lorsque lu document est trop grand pour dtre reproduit en un seul clich6. 11 est tilm6 d pastir de Tangle sup6rieur gauche, de gauche d droite. et de haut en bas. en prenant le nombre d'images ndceosaire. Les diagrammes suivants illustrent la mdthode. 32 X 1 2 3 4 S 6 PREFACE, This manual has been drawn up for the use of the Cadets of the Royal Military College of Canada. The first five chapters on Practical Astronomy embrace that portion of the subject with which all Land Surveyors in this country ought to be familiar. The remaining chap- ters, together with the part of the work which treats of Geodesy, touch on the more important parts of the ad- ditional course, as regards those subjects, laid down by Government for candidates for the degree of Dominion Topographical Surveyor. It has become absolutely necessary to draw up some compilation of this kir' be- cause, while many of the Cadets are anxious to v. ify themselves as far as possible in the above-mentio,.od course, the number of different books they would have had to refer to in order to obtain the requisite knowledge would have entailed on them a heavy expense. In order to make the work as cheap as possible the number of diagrams has been cut down to a minimum, it being in- tended to supply the place of expensive plates of instru- ments ct cetera by lecture illustrations. The author has also made the higher portion of the Astronomical course 03 lU'f IV Preface. as brief as possible. It will be found treated in the fullest manner in Chauvenet's Astronomy. Geodesy being both a difficult and a very extensive sub- ject no attempt has been made to write anythinj,' like a treatise on it. All that has been aimed at has been to give a sketchy account of its most salient points, adding a few details here and there. The student who wishes to pursue the subject further is referred to standard works, such as Clarke's Geodesy. The author has to acknowledge having made more or less use of the following: Chauvenet's Astronomy, Puissant's Gdodtook on Surveying, and Jeffers' Nautical Surveying. He has also to thank Lieut.- Colonel Kensington, R.A., for valuable assistance in in- vestigating some doubtful formulas. I Kingston, Cadada,) January, 1883. [ I I CONTENTS. PART I. PRACTICAL ASTRONOMY. CHAPTER I. Idea of the great sphere "declination circle,' tude." "declination "sensible horizon General view of the universe. The fixed stars. Their classification magnitudes, and distances, The sun. The planets. Their rela- tive sizes and distances from the sun. Apparent motions of the heavenly bodies. Their real motions. Motion of the earth with reference to the sun. The solar and sidereal dav. Mean and apparent solar time. The equation of time. Sidereal time. The sidereal clock CHAPTER n. Meaning of the terms "pole," "meridian " "hour circle," "zenith," "latitude," "longi- ' "right ascension," "altitude," "azimuth " , , ,. . •, "rational horizon," "parallels of latitude'" declination parallels," "circumpolar star," "transit," "paral- lax. Refraction The Nautical Almanac. Sidereal time. The celestial globe. Illustration of the different co-ordinates on the great sphere CHAPTER III. Uses of practical astronomy to the surveyor. Instruments emploved in the field^ Their particular uses. Corrections to be applied to an observed altitude. Cause of the equation of time Given the sidereal time at a certain instant to find the mean time To find the mean time at which a given star will he on the meridian Given the local mean time at any instant to find the sidere.il time Illustrations of sidereal time. To find the li<,iir angle of a "iven' star at a given meridian. To find the ine,-.n time by .•-niaralti- tudes of a fixed star, To find the local m..an tinu. by an !.bserved altitude of a heavenly body. To find the time by a meridian transit of a heavenly body _ , PAGE. II 22 VI Contents. CHAPTER IV. CHAPTER V Sun dials. Horizontal dials. Vertical dials CHAPTER VI. ''^ iS''TL'^lniH?P?' ?^ M'"onieter. The Reading Micro- scope. The Spirit Level. The Chronometer. The Electro Chronograph. The Sextant. The Simple Reflecting cTrcie" Jircle !r""^ '"^'^'"""S ^''■'='«- The^PrismaS R^flectilTg 36 66 77 Tt, . u, ^ CHAPTER VII. anerto,„fde,m,onon ,h= ialilude, V p=r™al e,;S. CHAPTEH VIII, f SI/ £te,. ^^.r o","," as'of °,L :sii; =;. '^is'.rc'o^r:'.'!'.''.";. . '!!■'. r *.'!r .'! '"•™; CHAPTER IX. '^'"''Jlrn' .•"^'^°'^^ °f finding the latitude.-By a single altitude h^ ^ "^ -H- """"''S "'"^' ^y observations of the pole star ou"of the meridian. By circum-meridian altitudes. °. g^ CHAPTER X. Interpolation by .second difterences. E.xamples. To find the Green- moon'!!n': *■'"" ^corresponding to a givL right ascension of the To finH .h^7'" ^,^y,- ^"t^rpolation by diflerences of any order IoLS. H /°"g""^« by moon-culminating stars. To find the longitude by lunar distances T fi j.u ,. , CHAPTER XI. on /h/if^P '"'^V^"^ ''?"'■ ^"^'^^ Of a given heavenly body when heav/nlv°hT- T° '?"d the equatorial horizontal parallax of a Tn finJ^.h ^^'i;" g'.^«" distance from the centre of the earth. 10 find the parallax in altitude, the earth being regarded as a (^nZ\ ^'f 'c^'^'op!'--^- Differential variationsof co-ordinates thP H^ K '"]^'\ '."«'j"^l't''^s in the altitudes when finding hP tT ^ ^'^."''J altitudes. Effect of errors in the data upon lutl computed from an altitude. Effect of errors of zenith r,i,l '•^^''''"f'°"i ^"d time upon the latitude found by cir- cum-mendian altitudes. The probable error joj 33 89 Contents. Vil PART II. GEODESY. _ » III _,. . ^ CHAPTER I. cfr^'rcs ,!? V""," °Wate spheroid-proved by measurements ,, , . CHAPTER H. aSiZ'''}-'- ^^'^'"J^'^dop.ed for mapping country Tri- „ . , CHAPTER HI. cal excess. Correcting the a ■• , ","' . V"' P'^ne. The spheri- sides of the triaSs The n- '; , .A !^"^i^- Calculating the CHAPTER IV. the earth. F^/rueH^seS «K'd"°ul oHhti^'"'^ °' sphere described with radiusequaltoX normal of *hV?rf»,^'"''-7 Reduction of a difference of lafit ,ri» ^^ .u ^ ■, spheroid. responding difference of Lf.H the spheroid to the cor- ili^on Tj^iS^i'^^^A ins .he la,ii„d= o onfj^'i'n,, S aziZh of'?;/,'"""' '"«'"'''« 133 -■l*fete«_„ vrii ^'ontents. 'n fho North A '"" "10 Uffariia .,. I , ['"'J Hie area ""sets to a parallel ''hodsofclelineatin ^"AI'TEK V N5 J'>nom,.(r,v,i . ... <^'"APTl<:i< VI »C0 ''>iK"nomeincal lev..ir *^'"'^'"I'KK VI '■■I"' «M of ,1,5 . CHAM-g,, v,V '.'"-• Pen,i„l,„„ ninsri ■■"'■ ''" '""y ascemin ^ ""'' "''-' «'me of 182 '75 Devilles "if area parallels parallel '45 actions, cator's, i66 above Ke- ; used di/ler- i. and d tile To erva- »7J the »rce idu- e of 1 of ven !0f ir's er- .. 182 NOTE TO PAGE 52. By drawing a figure it can be easily shown that, in the case of a horizontal dial, ii

y the sunlight reflected fiom then- surfaces, and when viewed through a good telescope, look like small moons, instead of m'ere points of hght, as m the case of the fixed stars, and may be noticed also to pass through phases like the moon especially in the case of the two that are inside the eTrth^ orb. The variability of their brightness is caused pr^y' by this, partly by change in their distances from the earth The well-known rings of Saturn are now supposed to con sistofa shower ofmeteorites revolving round him Supposing us to be situated in the northern hemisphere and not too far north, if we watch the apparent motions of he heave^y bodies in the sky we shall Zee tCfToX list ^f n u "''f '"*''' ""^ ^^^^ ^^^J'^^t about th? the ax t oTr ' n'"' *'^ ^^^P^^'*^ '^ *^^ -- about the 2ist of June. During the winter half of the year his nsing and setting is south of the east and west pdn s of the nonzon, and during the summer half they are north of It; while at two intermediate periods, known as the equinoxes he rises due east, remains in sight for la hours and se 3 due west. At midwinter the afc he descXe^ ZZt ''' '' ''' '°"^^^' ^"^ ^* -^^— er tt When the moon is first seen as a young moon she is a I'ttle to the east of the sun. She rapidly'move" th^.h i The Moon. the sky towards the east, so that about full moon she rises as the sun sets, and later on is seen as a crescent rising before the sun in the early morning. T4ie^ight ) L.U^ to^whichsh€nses^in4h""• "-'■! ""•""- mstant the sun ,s on the meridian is called ".ipparent 1 r The equation of time. ^ r noon. Noon as shown by a perfect clock is called mean noon." The interval between the two is called the "equation of time." Its greatest amount is about the ist of November, when the sun culminates about iih. 4jm. 41S. A.M. The equation then diminishes till about the 24th December, when mean and apparent noon coincide. After that the equation increases (the sun culminating after noon) till It attains a maximum of 14I. minutes about the nth February, and then continues to decrease, becoming ;^t.ro again about the 14th of April. It attains a maximum "t 3m. 50S. about 14th May, becomes zero about 14th June, 6i mmutes about 25th July and 2ero 31st August. The cause of the equation of time is as follows. If the earth moved round the sun in a circle and at a unifprm rate, and if the axis on which it itself turns were perpen- dicular to the plane of its orbit, the sun would culminate each dr.y at noon exactly. But the earth moves in an ellipse and at a variable rate, and its axis is inclined to the plane of the ecliptic at a considerable angle, the com- bined effect being that we have the equation of time. The great circle on the earth whose plane passes through the centre and is at right angles to the axis is called the "equator," and the projection of its plane in the heavens is also called the equator, and sometimes the equinoctial. If the sun, in its apparent annual path moved at a uniform rate and traversed the equinoctial in- stead of the ecliptic we should have no equation of time An imaginary sun moving in this way is called the "mean sun." In addition to the time kept by an ordinary clock and that kept by the sun— in other words "mean time" and apparent solar time"-we have a third kind called sidereal time," that is, the time kept by the stars. It has been already mentioned that the interval between two successive culminations of the same star is a little less than 24 hours ; the time it takes, in fact, for the earth lo Sidereal time. to make a single revolution on its axis. If we divide this interval into 24 equal parts we have 24 sidereal hours; and if vve construct a clock with its hours numbered up to 24 instead of 12, and rate it to keep time with the stars, it is easy to see that the hour it shows at any instant will give the exact position of the stars in their apparent diurnal revolution round the earth, ('locks and chronometers of thisdescription are used — the former in fixed observatories, the latter for surveying purposes. The subject of sidereal time will be referred to later on. Before proceeding further it will be necessary to explain the meaning of the various astronomical terms in ordinary use. de this rs; and I to 24 rs, it is 'ill give diurnal iters of itories, ter on. explain ■dinary I. CHAPTER II. EXPLANATION Ol' CERTAIN ASTRONOMICAL TERMS. NAUTICAL ALMANAC. THE For practical purposes the earth may be considered as a stationary globe situated at the centre of a vast transparent sphere at an infinite distance to which are attached the fixed stars, and which revolves round it in a little less than 24 hours. The sun, moon, and planets appear to move on the surface of this great sphere, the sun in the ecliptic, the rest in their respective orbits. The extremities of the earth's axis are called the poles ; and the poles of the great sphere are the points where the axis produced meets it. Great circles passing through the poles are called "meridians." This term applies both to the earth and the great sphere. In the case of the latter they are also called "declination circles." Meridians are also called "hour circles," and the angle contained between the planes of any two meridians is called an "hour angle," be- cause it is a measure of the time the sphere takes to revolve through that angle. It follows that the hour angle is the angle formed by two meridians at the poles. In speaking of the meridian of a place we mean the great circle pa ung through the place and the poles; and a great circle passing through the poles of the great sphere and the zenith (or point in the sky immediately 12 Latitude and longitude. ridian for the instant, ki^§t the observers head) is the as regai ds the great sphere. To fix the relative position of points on the earth's sur- face we employ certain co-ordinates, called "latitude" and "longitude." The former is the angular distance of any point from the equator, and is measured along a meridian north or south as the case may be. The latitude thus varies from zero at the equator to 90° at the poles. Longitude is the angular distance of the meridian of the place from a certain fixed initial meridian, and is measured either by the intercepted arc of the equator or by the angle contained by the two meridians. Longitude is measured east for 180° and west for 180°. Different countries reckon from different initial meridians. The English use that of Greenwich. The present system has many inconveniences, and it is to W. hoped that someday the world will unite in adopting some fixed meridian and will reckon longitude through the whole j6o degrees in- stead of as at present. The position of the heavenly bodies on the great sphere is determined by similar co-ordinates, but the latter are called "declination" and "right ascension," the former corresponding to latitude and the latter to longitude. Declination is measured from the equinoctial towards the poles, and right ascension eastward from a certain meridian. The latter is, however, reckoned through llir whole 36o",and is counted byhour^, minutes, and sec ids instead of by degrees, i hour corresponding to isdegiees. The point where the ;{ero or 24-hour meridian cuts the equator is called the "first point of Aries," and is desig- nated -V the symbol /". It is also one of the intersections of theeq. ^'o'- -ith theecliptic. On referringtothe Nautical Almarn. ^ viii hf ; een that the co-ordinates of the stars are co.uim;>' lly changing. The fact is that, owing to the slow cci.ical .; orion of the earth's axis known as the "pre- % Altitude and azimuth. n cession of the equinoxes," the planes of reference are changing,'. This, however, causes no practical iticonveni- ence. as the relative positions of the stars remain the same. It should be noticed here that the terms "latitude" and "longitude" are also used with reference to the heavenly budies, and are liable to cause confusion. These co- ordmates are measured from and along the ecliptic, and are not required for the problems here treated of. Hesides the above-mentioned co-ordinates which relate to the relative position of points on a sphere another set IS necessary to fix the position of a heavenly body with reference to the observer at any instant. They are called "altitude" and "a;jimuth." The first scarcely needs ex- planation. The second is the angle formed by the verti- ca' plane passing through the observer and the object with the plane of the observer's meridian. The altitude and azimuth of a star at any instant are, in fact, the angles read by the vertical and horizontal arcs of a theodolite respectively when the latter has been clamped with its Aovo due north, and the telescope has been directed on the star. Azimuth is generally reckoned from the north round by the east, south, and west; but it is sometimes reckoned from the south. The plane of the "sensible horizon" is the horizontal plane passing through the observer's position, and there- lore tangential to the earth's surface at that point. The "rational horizon" is a plane parallel to that of the sen- sible horizon and passing through the centre of the earth The projections of these two planes on the great sphere coincide, being at an infinite distance. It is easy to see that about half the great sphere is in sight at any instant. The portion that is visible depends generally on the latitude of the place and the sidereal time of the instant. At the north pole the whole north- ern hemisphere would be always in sight and no other % 14 Definitions. part At the south pole the view would be limited to the southern hemisphere. At the equator both poles would be on the hor,.on, and every point on the j^reat sphere would come m sight in succession. At intermediate places a certam portion round one pole would always be above the hon.on, while another portion round the other pole would never be visible. "Parallels of latitude" are small circles made by the in- tersection with the earth's surface of planes parallel to the equator. Similar circles on the great sphere are called declination parallels." A little consideration will show that w.thin a certain distance of the equator at each side at mid / '" t^k' /"''' '" '^' y'^'' P^'' °v-head at mid-day. The belt enclosed between the two parallels within which this takes place is known as the "tropics." sutlZT.^ ''<,I'"'''^ '''""' '"^"'^^ explanation. When speaking of the "hour angle" of a heavenly body at any instant we mean the angle formed at the pole by the meridian circle of the instant and the declination circle passing through the body. By the term "circumpolar star" is meant a star which never sets but appears to describe a complete circle round -he pole. These stars cross the merdian twice in the twenty- our hours. One crossing is called the "upper wTn 'th : '''"" ''' ''°"^'" *^^"^'^ •" A* ^he points be- tween the transits at which the stars have the greatest azimuth from the meridian they are said to be at the greatest elongation," either east or west. The words "transit" and "culminate" have the same meaning when used with reference to stars which rise and tio7Tv ''; '' '^' ''''"^' ^" '^' "PP^^^"* '•^'-tive posi- lon of objects owing to a change in the observer's posi- lon^ Astronomically it generally signifies the difference in the apparent position of a heavenly body as seen by an } Parallax. J5 observer from what it would be if viewed from the centre of the earth. Parallax is greatest when the object is on the horizon, and nothing when it is in the zenith. The moon, from being near the earth, has a considerable parallax. That of the sun does not exceed 9". The positions of the sun, moon, and planets given in the Nauti- cal Almanac are those which they would have as seen from the earth's centre, and it is therefore necessary to correct all observations on those bodies for parallax. Parallax causes the object to have less than its true altitude. Refraction has the opposite effect. The latter, like the former, diminishes with the altitude. Near the horizon— say within 10 degrees of it— its effect is very un- certain, and observations of objects in that position are therefore unreliable. At an altitude of 45° the refraction is about i'. As it varies with the temperature and atmos- pheric pressure the barometer and the thermometer must be read if very exact results are required. '"^ The corrections for refraction and parallax are not to be found in the Nautical Almanac, but are given in all sets of mathematical tables. The N. A., as a rule, gives only variable quantities — such as declination, right ascen- sion, equation of time, etc. It is rather a bulky volume, but the portions of it in general use by the practical surveyor could be comprised in a small pamphlet. The most use- ful are the sun's declination and right ascension, the equa- tion of time, the sun's semi-diameter, and the sidereal time of mean noon — all given for every day in the year ; the declinations and right ascensions of the principal fixed stars, taken in regular order according to their right ascensions ; and the tables for converting intervals of mean time into sidereal time and 7ricc versa. To these may be added tables of moon-culminating stars, and tables for finding the latitude from the altitude of the pole star when off the meridian. I 'i i6 The Nautical Almanac. riven r\ ^u""" ^"' P^"?" °f ">» ')'■='"'«« g van for each month ,„ the Nautical Almanac, and of the da a for fi.ed stars, are reprinted below. All he quanti- t on tT" °' "T "' ''■■"""'^h on the day in' ques- .on They must, therefore, be corrected by a propor- t.on for any other hour o, longitude. Thus.'^when ^ s noon at a place in 90' west longitude, or si.v h^urs we,, of Greenwrch, ,. ,s 6 p.m. at the latter. ThereforeTt an observafon were taken at the western station at noon the quan.,t,es requ.red would have ,0 be corrected for "heir change m six hours. Owing to the earth's uniform revolution round its axis Zfh7 a' , ""'^^ ''"^" "* Greenwich was 3 p.m and the sidereal time iih., they would be g a m and ^h' respectively at a place in longitude 90" west ^ In the Nautical Almanac the day is supposed to com mence at noon and to last for 34 hours. Thus ga mTn' 1st ot January. This astronomical method of reckoning mean time must not be confounded with sidereal t^. which is quite a different thing. *™'' fhl^^Tl""" *^' ^''' °^ ""^^ "^°"th are given also at the end of the preceding month. Thus, weLd sTlyl ZZVfFr'''''''Z'''''^'' ^^ ''^ 3ad being real^ at ng '' "^'^ " '°^ convenience in interpo- Nautical A Imanac. 17 JUNE, 1880. AT APPARENT NOON. THE SUNS Apparent Right Var. Apparent AscensionUour Declination. hoVr.MeSL, Sidereal Eflaation time of of Time the to be Semi- i *"*<./rom , diameter j «rf,/,,rf to Var Var. passmg U/„..„ J i^"^,- in I I, the II Time, jhour. :h m s s I i4 39 071 110241 10-258 j 2j'4 43 671 3 4 47 1310 I0'274 '[ 22 |4 51 i9'87 10-289 |4 55 26-gg 10-304 i4 59 34 44 10-317 N.22 8 59-6 19-79 22 16 42-9I18-82 ' 24 2-9|i7-85 3 4220 5 7 5024 5 II 58-54 10-329 10-340 ; 10351 5 16 708 10-360 5 20 15-82 10-368 '5 24 2474 10-375 1315 28 33-8r 10-381 '4115 32 4302 10-386 j i5||5 36 52-34110-3801 22 30 59-4116-86 22 Z7 32-3I15-87 22 43 41-3 14-88 22 49 26-5|i3-88l I 8-72 22 54 47-612-88 ;i 8-7S 22 59 44-6.-II-87 ,1 8-79 23 4 17-3 10-86 23 8 25-7, 9-84 23 12 9-6I 8-82 23 15 290) 7-79 23 18 23-71 6-77 23 20 53-8,1 5-74 m s j s 2 21-83 0.384 2 12-42 0-400 2 2 61 0416 I 52-43 0-431 I 41-90 1.1-446 I 31-03 0459 1 19-86 0-471 ' 8-41 0-483 o 56-7o;o.493 44 '75 (0-502 3260 0510 20-27 0-517 0523 ° 4-8310-528 ° 17-35 |o-532 i8 Nautical A Imanac. JUNE, 1880. AT MEAN NOON. I 43 o Apparent Right Sr-'fAscension Oil Tues. Wed. Thur. Frid. Sat. Sun. h m s 4 39 III 4 43 7-09 i4 47 13-45 I 14 51 20- 19 14 55 27-28 (4 59 3470 THE SUN'S Equation of Time, to be added to subt. from Mean Time. Apparent Declination. Semi- diameter. Sidereal rime. / // N.22 9 0-4 22 16 43-6 22 24 3-5 22 30 59-9 22 37 32-7' 22 43 41-7 15 48-1 : 15 47-9 ; 15 47-8 15 47-7 ^j 15 475 :; 15 474 .i m s ! 2 21-82 2 12-40 j 2 2-59 ! I 52-41 I 41-88 I 31-02 h m s 4 41 2293 4 45 19-49 4 49 iG-05 4 53 12-60 4 57 9-16 5 I 572 Mon. Tues. Wed. 7 , 8 j 9 ' 5 3 42-43 5 7 50-44 5 II 58-71 Thur. Frid. Sat. 10 II 12 5 16 721 5 20 15-91 5 24 24-80 Sun. Mon. Tues. 13 14 15 '■ 5 28 33-84 5 32 43-01 5 36 52-29 22 49 26-8, 15 47-3 ;; 22 54 47-91 15 47-2 Ij 22 59 44-8' 15 47-1 I 19-85 I 8-40 o 56-69 ~3 4 17-5 23 8 25-8 23 12 9-7 15 470 15 46-9 15 468 23 15 290 23 18 23-7 23 20 53-9 15 467 15 467 IS 46-6 o o o 44-75 3260 2027 4-83 17-55 5 5 2-28 5 8 58-83 5 12 55-39 5 16 51-95 5 20 48-51 5 24 45-07 5 28 41-62 5 32 38-18 5 36 34 74 APPARENT PLACES OF STARS. 1880. AT UPPER TRANSIT .\T GREENWICH. Month and « Andromedje Day. R.A. Dec.N. Jan. I II 21 31 h m 2 s 11-88 11-75 J3 11-62 '-' 11-51 ", ,j 28251 1 55^ 9i 54-4 51-8 'H y Pegasi. (AlgenibJ h O s 4-24 4-13 402 393 8-6 77 6-8 5-7 '9-^' 11^5 fJ;26-3o ^^162-7 19-50 19-39 19-30 ii-«''' 3i ^81-6 -I 93 6^5 '" ^4-51 ^^i59-8 II 25-37 124-51 '2373 12 The Celestial Globe. •• 19 To revert to the subject of sidereal time: Sincrthe sidereal clock stands at ^ero or 24!!. at the instant the ist point of Aries is on themeridian, and as the clock keeps time with the stars in ^apparent diurnal revolution round the earth, it follows tlmt when any particular star is on the meridian its right ascension is the sidereal time of the in- stant. Thus, if the stars R. A. were 6h. the clock should show thattime at the instant of the stars transit, and its error may be ascertained by mountim^ a telescope .0 as to move only in the plane of the meridian, and noting the instant of transit. If we want to find the mean time of a stars transit we have only to convert the star's R A into the corresponding mean time of the instant, in 'the manner to be presently explained. Conversely, a star's transit gives us the sidereal time of the instant, and hence the true mean time. The celestial globe is of great use in studying astronomy. It IS a model of the great sphere supposed to be viewed from outside. The positions of the stars on it are the points where straight lines, joining them with the earth would intersect it. The equator and ecliptic-the lattei' being the sun s annual path through the stars-are marked on It, as also the sun's place in the ecliptic for every five Jays. The axis on which it turns is that of the poles 1 he meval ring passing through the latter represents the meridian and the flat horizontal ring the plane of he rational horizon. ^ y ^i me t.-o?"'fl'^' '^''^"''' °^'^" ^^^°^^ '^ t° «how the posi- tZ. T T'\^' ^"^' '"''""' ^^'^^^ ^•^S^'-d to the spec- tatoi. To do this we raise the pole by means of the ToTrr- '''' ""7'^^" ^^ ^^ *« ^'- ^* - altitude br2 tt °"i ''"'^ '" '^'' ^"^^^"^^ °f *he place, and bung the sun's place in the ecliptic for the day to the meridian The half of the globe above the horizon wl now roughly represent the position of the visible hemis- phere at noon. To find the position of the sphere at any 20 The Great Sphere. other hour it is only necessary to turn the Tlobe through ITL stirs at I r ""''!'" f "' °"* ^'^^ ^'^'^^^ P-'tions ot tne stars at 8 p. m. we should have to revolve the dobe westwards through an angle of x.o^ Conversely we can find the name of any constellation or star by noting IccSgTy.'" ''' ''' -^ '- '-- -^ s.U..X:Zl easTtr'*.'^''^' P'f '"^ *^^°"^^ the zenith and the angles to the mend.an, ,s called the "prime vertical " Pig' 3. In Figure 3 the small circle at the centre represents the ^nelk- '1 ')' ''''' ^'"'^ *^^ ^^-t ^Ph-- Strictly to the Tal r: ''""'' '^ ^ "^^^ P«'"t ^" comparison to the latter, and the pomts on the great sphere would ap- f ough from tions flobe 2 can 3ting jlobe the ■ight Explanation of Terms. f 21 iionzon, and p pi the earth's polar axis meeting fh^ grea. sphere i„ .he points P l. z if/he .e"' N he nad,r or po,n. on .he sphere diame.ricaliy opposi.e "^ The plane of the paper represen.s .he plane of tl e po.n.s of he horizon, . j ,s .he equa.or, E r O the eq „,oc,.a r the firs. poin. of Aries, and P ,' P . .L .n .,al dechnat.on circle passing through i., fro™ wh ch '- Dorfiln f "" °" '"' "'"''"''''' ■""'dian. Z S B Ind ri r ?•'"■'? '"■'=''=• P''^^'"'! "'""gh the .enith and S and mee.mg the horizon a. B. Z B is of coursi .'re',. s„h Tf '■'P'''''"' "'^ ^PP""^"' '""'ion of the grea. sphere wi.h respec. to .he earth, Do.^n','?"'' ^'f'r?' ""' ""«'■= '' ° '• '^ "-^ '^"'"de of .he po n. A, and A O j=Z O Q, which is .he zeni.h distant Mepij, /t;?et^.°1'^ It should be noticed .ha. .he whole of .he hemisphei above the plane H B R is visible .0 the observer a A (nor m'I ",^«"™" "f '•>' ^'ar S is zero, its declina;ion (north) S hour angle S P Z, its al.itude S B, zeni.h d,s.a„ce S / and azin^uth S Z R. The star S- hLs R. A r «, decimation (south) Q s! hour angle nil, altitude S' R, zen.th distance, S. Z, and azimuth ze o. The dereal „me of .he instant is r P Q, or the arc r Q a triangle P Z S ,s called the "astronomical trianrie " It should be noted that in all calculations it north declina ..on ,s reckonedpositive,southdecli„atio„must becounted negative, and wee z;m«. "uuiea CHAPTER III. Uses of practical astronomy to the surveyor. Instruments employed in the field. Methods of using them. Taking altitudes. Problems RELATING to time. The principal uses of practical astronomv to the sur- veyor are that it enables him to ascertain his latitude, longitude, local mean time, and the azimuth of any given hne; the latter of course ^'iving him the true north and south line and the variation of the compass. In fact the only check he has on his work as regards direction when running a long straight line across country is by determin- ing its true azimuth from time to time, allowing (as will be explained hereafter) for the convergence of meridians. The mstruments usually employed are the transit theo- dolite, sextant or reflecting circle with artificial horizon, solar compass, portable transit telescope, and zenith tele- scope. To these must be added a watch or chronometer keeping mean time, a sidereal time chronometer (this is not, however, absolutely essential), the Nautical Almanac for the year, and a set of mathematical tables. With the sextant or" reflecting circle we can measure altitudes and work out all problems depending on them alone, and also lunar distances. The transit theodolite may be used for altitudes, and also gives azimuths. The solar compass IS a contrivance for finding, mechanically, the latitude. 1 Instrumenta. i J ^ meridian line, and sun's hour anj^le. The zenith tele^ scope gives the latitude with great exactness, and is par- ticularly suited to the work of laying down a parallel of at.tude The transit telescope enables us to determine the mean and sidereal time, latitude, and longitude. The transit theodolite answers the same purpose, but is not so delicate an instrument. It is, however, of almost uni- versal application, and nearly every problem of practical field astronomy may be worked out by its means alone if the observer has a fairly good ordinary watch. The sex tant has been called a portable observatory; but in the writer's opinion the term is more applicable to the last named instrument. The sextant is not so easy to manage and only measures angles up to about Ii6,° so that s8" is practically the greatest attitude that can be taken wJth it when the artificial horizm has to be used. The latter as generally made, is disturbed by the least wind, and then gives a blurred reflection. maHng the observation nearly worthless. There is little use in having the arc graduated to read to within a few seconds if the contact of the images cannot be made with certainty to within a minute or two. All observations taken with the transit theodolite should If the nature of the case admits of it, be repeated in re- versed positions of the telescope and horizontal plate and he naean of the readings taken, as we thereby get rid of r'of Thi '1"'"''^''^"i '"'^•^' '^"^^ "*''- '"^--"tll errors hus, for an altitude, the plate having been evened, the vertical arc set at .ero, and the bubble of Ihe telescope level brought to the middle by the twin screws the verticahty of the axis is tested by turning the upp plate in azimuth i8o°, and seeing if the bubble is still in the centre. If it is not it is corrected, half by the lower plate sere., half by the twin screws, and the operation repeated til the bubble remains in the centre in ever" position. Ihe altitude is then taken, the telescope H A Ititudes. turned over, the upper plate turned round, and the alti- tude again read. In each case both verniers should be read. The first step after taking an alticude with either sextant or theodolite is to correct it for index error, if there is any. The following lists give the corrections to be applied in each case to an altitude of the sun's upper or lower limb to obtain that of his centre : THEODOLITE. Index error. Refraction. Parallax. Semi-diameter. SEXTANT. Altitude above water horizon. Index Error. Dip of Horizon. Refraction. Parallax. Semi-diameter. Double Altitude with artificial horizon. Index Error. Divison by 2. Refraction. Parallax. Semi-diameter. The semi-diameter has to be added if the lev er limb IS observed, and vice versa. When taking an altitude for time with the artificial horizon the easiest way to get the correct instant of contact is to bring the two images into such a position that they overlap a little while recedinjr from each other. Ai^the instant they just touch the observer calls "stop," the assistant notes the exact watch time, and the vernier is then read. This plan necessitates observing the lower limb in the forenoon and the upper in the afternoon. The dip depends on the height of the instrument above the water, and, like the refraction and parallax, is to be found in the mathematical tables. In the case of a meridian altitude for latitude the sun or star, after rising to its greatest height, appears for a short time to move horizontally. When this is the case the altitude may be read off. Fixed stars require, of course, no correction for parallax or semi-diameter. As the refraction tables require a correction for temperature and atmospheric pressure the height of the thermometer and barometer should be noted t . Equation of Time. r 25 If an altitude has to be taken with the sextant and H.t.fic.al horizon, and the sun is too high in the heavens for^t^he ,nstrun.ent, a su.table star n^ust be observed Tn! In surveying operations the latitude is generally known aF^roxjn.ately. This gives the approximate altitVdeTo" a rner,d.an observation; for the altitude of the intersec- K>n of the mend.an and equator being 90' minus the lautude. we have only to add to or subtract from thi tTtud"^ '^'''' declination, and we have t^e al- \ T f . , , . y . f i ^ AVSE OF THE EQUATION OF TIME. In Figure 4 P is the pole, E C a portion of the ecliptic, and E Q a portion of the equr.tor; each being equal to go°. C and Q are on the same meridian, and P Q is also a quadrant. Now, let S be the sun, and suppose it to move at a uniform rate from E to C. Let ^■ S' be an imaginary sun (called the "mean" sun') moving m the equator at the same rate as the real sun. Now let the two suns start together from E, and after a cer ain interval let their position be as shown in the figure Since they move at the same rate. E S will be equal to t ir> , but as a consequence the meridians P S and PS' S p c?;°'"''^f ' ^; ^^"^"^ ^°t ^h^-d of S. The angle time aZ.' '^" '"° "^'"^'""^ ^^ *h« equation of IZn f ' •. ° '""' """"'^ ^"'^^ simultaneously at C and Q It IS evident that, though S^ gains on S at first, it will, after a certain point, cease to gain and lose insteld Since the equation of time-in other words the differ- ence between apparent and mean solar time-is con- the'mtn ."'"^' '' "' """* *° '"^ ^^°- ^^e Almanac the mean time corresponding to apparent time at any '/^ rn:^ c ^'-K^t^ £^c<.f„^-c_^ '1- ^ ' -ss im Azimuth by an Altitude. 41 Cos» 1^ Cos 5 Cos (s-S P) Sec; Sec a where a is the altitude of the object. S Pits polar dis- tance, X the latitude, and s= "+'* + ^± ve>^^ ^^^^^^.^.^^.r^ort^ce to sur- ^"^h as alSl^^e' "" ^*-^^'"^"th instrument, of any line Z A (Fie Vo : ^'^^^^^^""^'"'cal bearing directing the tel scope on th"e L°""' "J''^ "^^ ^^ horizontal plate reading and Then" ""'' *'''"^*'^ urmng it on the heavenly body and S^r'd-'"'"'; ^"' the'horilTal plate reading. It is better to repeat the observation in reversed positions of the instrument and take the mean eneorf°'"^^^^ ^^^ediffer- llne and 5h T'''"'^^ '""^^"^^ °" the hne and the heavenly body gives the angle A Z S, and the triangle P7^ u !^ "' the angle P Z S, whence vSh a "?"" '^'"^"^ ^'^«« bearing, and therefore Z P fh^".^''^ ^ -^ P the required In taking an alt-Xnth of .k ''''°" °^ *^^ "^^"'^ian. altitudeonf, we mu"" ^^'s fbUct' th "^ '^'^ ' ^'"^^^ to get the altitude of the centre rfn' f"^'"^^^-^*^- vertical and horizontal wire the sun! " '" ^ tangential to both. To ^et the Z%^^^^ '' '""^^ semi-diameterwemustmnf. 1 u ^""^^ correction for Jatter by the seTanT .f T ? ^ '^" ^^'"""^^ ^^^^e of the tions ar^ got ri • -7 Tj c ;;: 7-=sin Z P S tan A where A' is the star's azimuth, Z P S its hour angle at the time of observation, and A its azimuth at greatest elongation. ■>_ lu^ui^fm-miui. tn— r-in--Ti-iiTi-— r — r ntK tir'nninr-iin— ii-ni-i>wwTlr:itiiim i i t J^''^'^^"«« h Hifrh and Low Stars. 47 TO FIND THE MERIDIAN BY OBSERVATIONS OF HIGH AND LOW STARS. This is a very useful method, as it is independent of the pole star, and can therefore be employed in the southern hcnusphere where that star is not visible. Choose two stars differing but little in right ascension oneof wh.ch culminates near the .enith and the othe; near the south horizon (or the north horizon if in the o hern hemisphere.) Level the theodolite very care of the Vr ^"'' "^'^^ ^"^P^ °"^ ^^^ *^^ collimation Hne of he telescope W.11 coincide with the meridian at the ^enith, however far U may be from it at the hori.or • and the field T"*"^^ "'^'" -'^ ^^"^^^ -" -°^« the cent're of he field of view at nearly the same time as if the ele horizon Having^L^;,^^^^^^^^^^^ ^^^ stars will cross the meridian observe the tranlit of fZ upper star, noting the watch time. Th s wm give the watch error approximately, and we shall now knTw the transit. By keeping the telescope turned on that starti that mstant arrives we shall get it very nearly n th" plane of the meridian ; and by repeating the process with another pair of high and low stars we sh^all have the di^c tion of the meridian with great exactness. For this method we require a transit theodolite fitted with a diagonal eye piece. The nearer the upper stars are to the zenith the better. ^^ ^ The Canadian Government Manual of Survey recom mends for azimuth the formula : ^ tan. P Z s=^^LPA55?jii!nAP^S i-tanPStan/lcosZPS as applied to observations of the pole star- but it r^ quires special tables in order to work it out i WW,' ejl H P H I ^ ^i» B^.i ! . 'i i , - i"^ flB|BWWp W WWHBWllimwP 48 Meridian by Pole Star. The following is the proof of this formula : We have the fundamental formulce — 1 f sin fl sin C = sin c sin A I ^ cos c — cos a cos b cos C= ; ; — r sm a sin b COS. a — cos b cos c cos A: (I) (2) (3) sm b sm c From (3), cos a cos 6=cos^ b cos c-(-sin b sin c cos 6 cos A ^, cos c — cos a cos b From (i & 2), cot C = -; -. — j-—. — -^ — riuiii v* «. -a/, sm a sin 6 sm C cos c — cos' b cos c— sin 6 cos b sin c cos A •.tan C = sin b sin c sin A sin b sin c sin A sin' b cos c — sin b cos 6 sin c cos A cosec 6 tan c sin A iii I — cot b tan c cos A In the triangle P Z S let P S=c, Z S= a, aud P -Tlrrfc Z=CandP=A Then— cosecPZtanPSsin^PS tan l'^S-^_^^^p2tanPScosZPS tan P S sec ^ sin Z P S ■ I— tan P S tan ;i cos Z P S t t] w w eq ha sh tai OS A OS A CHAPTER V. SUM DIALS. thrown on anv olanp «nrf=„ ''«<=earm, its shadow of the sun alwaCn. ll """' '^" S'ven hour angle be the sun's d™Hn 1" ' t^T T '^'" ''"" "■'^'"" Poin, in .h. line Slve asVhetT"'.'"^ "^"'"'^ will always lie in .he same straitlt H„« f" ™™'' "" a"gk. On this principd al f„ ! ? '"'' «""" '"'" The position of th'e sh?dow 1:: h^^.s'^r'™^'^- at the instanf anri fi, r • "" ^ '^our angle •tae; so tha in^"!',"::^^;';:"*-;- ""e .>^.„„, ,„L >.ave to applv the equatL ofTme ""' """" "»' "' .he°,ltercas?r"LH'*".'"'r°"'^' " -'->• ^ thrown on a ho^'o'a, Ite ttheT^ " '= ""^''- « wall. P ^^ ' '" *^^ Matter on a vertical equator the d'al wonM evidertl ""^ "' ""^ P°''^- A' the having a hori.orta 'edg 'S'™^'!,' "' % -"'-' P'a.e shadow lines would be P^a fd fo he ,L ar/tt ' ^'^ 'ances apart for equal intervals of 'ti^ Cu d'"; ^^ mm 50 Sun Dials. I increase according to the sun's distance from the meridian, and would become indefinitely great when he was on the horizon. At the poles the stile would be a fine vertical rod, from the base of which 24 straight lines, radiating at mtervala of 15 degrees, would indicate the hours. The line on which the shadow was thrown at the time corresponding to Greenwich mean noon might be assumed as the zero or 24-hour line. At other places the stile must be set so that its angle of elevation above the horizontal plane is the same as the latitude of the place. HORIZONTAL DIALS. A horizontal dial generally consists of a triangular metal stile fixed on a horizontal plate on the top of a pillar. Fig. 13 is n elevation and Fig. 14 a plan. The angle of elevation of the stile is made equal to the latitude of the place, and if the variation of the compass is known, the latter may be used to get the dial with its stile in the plane of the meridian. The hour lines on the plate are marked out thus : let Fig. 13. A B (Fig 14) be the base of the stile. Fig. 14. and A its south end. Draw A C so th^t n x~r~ 7~ the ,ati.„de and at any point^C^irSt C B^r "ne E B F pTpta'Lfa,".; Isn'MtT^rfT' D », D .., D b. D 4., &c., n,ee,i„g E B F in . j'!"" and n,akh,g ,ha angles B D ., b'd a'. I S ^.^ a, b ,." ac, each equal to 1=; detrrees Fr^r^ a j y ^ u , lines .H.on,2 a. , „.,^, r%.::rj t^^e'^t i'nes : A . for 9 A. m., A t for lo a. m., A a for „ am and so on The proof of the correctnes of this con't'ruc" tion ,s easily seen by imagining the trianele A R r , I, turned round A B till i. is perp^endicil r tttha pLe" of he paper or dial plate, and the triangle c D c'T K turned up on . .. till it abuts on B C when D w n ° cde wuh C, and A C will be parallel lortepoIaT^isTd" perpendicular to the plane of D cc'. When the divisions on the line E B F run off the niaf . we contmue them thus: In A C (the , p m 1 n.l , \ any point », and through it draw a H e para ieHo^ S 9 AM. l,„e) meeting A 6., A .. &., fn A >' & and make », equal to .A o ,• to op^, &c.. and ttiugh'; ," onlhfothrsTdt"""" "™= '"'"= -orninghour lines VERTICAL DIALS. These have the advantage that they may ^e made nf n very large size and placed in conspic L ^osl dot There are vanous ways of constructing them As^Zle .^ trnro'f '^ ^°,f^ \''' ''-'' havi^gaLndh^t It, m front of a wall with a southerly aspect. (Fig. 15.) 1 i 52 Vertical Sun Dials. Fig. 15. The disk should be roughly per- pendicular to the sun's rays at noon about the equinoxes. The The bright spot in the middle of the shadow of the disk on the wall indicates the hour. Tb' hoi- lines are found thus: At the time the sun is on the meri- dian mark the position of the bright spot on the v/all. Let A be the hole in the disk and B the spot. Measure A B. Through B draw B C vertical, and draw a line B D so that B D is equal to A B, and the angle C B D to the sun's polar distance minus the co-latitude. Make the angle B D C equal to the supplement of the sun's polar distance. It fol- lows from this construction that if the triangle BCD were turned round B C till it touched A the points D and A would coincide, and C D (and therefore the imaginary line C A) v^ould be parallel to the polar axis. Now take a watch, set to noon at the time of the sun's transit, and mark the positions of the spot on the wall at the success- ive hours. Straight lines joining these points with C will be the hour lines. Of course a large triangular stile CAB might be substituted for the disk ; o- we might use a rod C A fixed in the plane of the meridian, and having the angle A C B (which it makes with the verticaf equal^ to the co-latitude. CHAPTER VI. THE REFRACTING TELESCOPE other. The former 1h ' '"'' " ^^= Pi"'^ =" ">e objec. a. i.s foZi:; ^31" ^.rjs' ™^^^ °' '"» tographic camera and th. ^- ''"' °' " P^o- "u. ifihe .eSpe -^ :ar:er.: s"h""fH'''^r '^"^^ natural position a 00™^™!"™ of fo 7 °''''" '" "^ by means of which tt^Tu^T^ "' '""""^ '= 'mployed, This has. however tl!e H? . ""^^^ " ^«='" '"^"'^d- much ligi, and L '1 '^"f."^"'^^'' "f <^utlin^ off too land objfct;. '' "'"' '" """" '^'«-°Pes and for "ne^tivZ-'in^LHrh tte" '^ ''-'7 '■''"<''•• «-' '»^ t»o lense; of the et pLe "^h" '"T^" '"'""" "'- "^ed in telescopes desi/ned for Th " "" "'"" ^=™""->' objects without makTnf „/ ■""" examination of "positive," in whkh h! Z T"'"' ^"""-"J"' 'be is outside the eye pLee rr rf'''^'™'="=^^ '---Wean?-teto;rrra;:i:rs 54 The Telescope. plane at the common focus of the object glass and eye piece. The position of the focus of the former depends on the distance of the object — that of the latter on the eye of the observer. The one is the same for every indi- vidual. The other has to be adjusted to suit the observer — short-sighted people having to push the eye piece in, while those who have long sight require a longer focus. The larger the object glass is the more rays from the object are collected on the image, and the brighter it is. The greater the magnifying power of the eye pic. e the more apparent are any defects of definition in the image. The magnifying power of the telescope is measured by the fraction focal length of object glass. Thus, if this focal length of eye piece, fraction were 4, the linear dimensions of the object seen through the telescope would be four times what they would be when vjewed with the naked eye. Therefore^, ^, for a given eye piece, the longer the telescope is.^he" smaller will be the field of view, or portion of the earth or sky visible. 1 he angular diameter of the field is, in fact, the angle subtended by the diameter of the eye piece at the centre of the object glass. In large telescopes the field of view is so small that it is necessary to use a "finder," which is simply a small telescope attached to it so that the axes of the two shall be parallel. A diagonal eye piece is one in which there is a mirror or prism between its two lenses by which the rays of light are turned at right angles and emerge from the side instead of the end of the eye piece. It is used for observing objects when the altitude is so great that it would be uncomfortable or impossible to look up through the tele- scope tube. Lenses have to be corrected for chromatic aberration and spherical aberration. Take the case of an object glass con- -7^ P it ar na gh m] pr: The Telescope. ' have different foci tL u °[ '^^'^"Si^ihty would other. Such a lens is calle/ ..■r;^hr:Sic!" """" '"" By "spherical aberration ' ;.. mpai-l th. Hi. rays caused by the central pc-, ^of let „ ^ T" " surfaces having a different focu. om its ou J Ir '' "'I St c^bitTor'^^ ^^ "■' -CeTtrr -r pro;tiy"r:tedr:ifri'st °ut^^ ^'-= ^- ■!- Spherical aberration is detprt^ri k, portion of the lens wi h a ctc'la^dTrT*'^""*^^^ focusing it on an obie.r .V . ^ °^ P^P^'" ^"^ 6 1 uu dn ODject, afterwards removmo- *i,^ ^ i and covering the oiitPrr^orf -^u . "^ ''-'"oving the disk .he focus o.'ZTZZT'ZT'''''''' "'-- -»- If one part of the object platal wire. If „o,v the screw A " .r„e7tnr ' '"'"• one star, and B till n n cut« th? .V , '" " '^'"^ •ween the two is measured J ,^ "■'. """ *^'''"« be. and fractional div" .^n^ofa t'„™ ItTakes'^t'-- "' ^ to » n. This is not theexact m.,f i / ""«^ "" ™ "P "„, it serves to illnstr^e the prtadple ''""''"" ''""'"^^' THE READING MICROSCOPE. for reading the facH ai part^ o^f hT^' °' "^™'" graduated circles ofl-,™.-. divisions of the is fi«d, the ciS VZ'IZZTTL t^ """°="°p^ ■nstrument and moving with it Th ''°P' °' '^^ one screw and moveabk f ame J*? ™"°'"^'" has only of cross-wires in the common foculahTr^' T'^ and eye piece of the microscone Tl? '^"^ «'"' used in exactly the same w!*^ 1^'^ cross-wires are 'elescope. only that thToh^ "• """'^ "^ " "-eodolite arc, on which fh micrt eoolm Tf '^ ""^ S^-''''"-f readings will give the slope. ^^^^ ""^ *^« tail^7o?d»i«t^l:",j2t^ "^'' ^--" "' -- wh];Hr^t?„Vh":Lott';;iiS'''' =-^-' <-> by means of the ad- -tin, '^'^''■"•'°>"'- Then, both its ends read ^tZ ' ""°'= '"= ^'■'^'>'' «" THE CHRONOMETER ourf^'nTa^^ranSSrh r ""'"' ^'^^ ""•" changes of ,emperatu™have. he?"'r° "'^''™'^''^ "■" the time of its osc llation ?» P"'"'"''"'''" "P°» structed to Iceep either "l^^'''""™'^'"^ ""^X be con- used on board BriSh shil "' ''"f^'' '™e. Those wich mean time The Jea'-rnf"''' 't^''°" <^'-»- that it should keep a regjlar rate th",' "^T"™"^"- '^ 0"ly gain or lose a certf.n 12 ', "' ""' " ^'""'W •his can be depe^ ed on ^ cfn T" "™- " the true time at any instant h, T ^^^ ascertain numberofdavsandhoaJst.lr''''?''"^''''' '*'= ^°' «he error of the chronom r ^L la" d^ "^'"^ *"""•"" by comparison with other cZnteterroXt """"^ mical observation Th« r, "'"eters or by an a rono- -re perfect ir.he c^ronrir ris^^, "'= "'P' '"« venient to have a small than^a^^rge rltrtfalTr^- eight daT'T^errr"^"^ ""^"^ '° -" ^""er tw^ or seventh day. IttiZlTrr"^'"'' ""^ '>«- e-ary "P at the rlgula: in e7:ra ' fie tr'™.'" '? " "' -ed part of the spring c^m^f t'to'X, Tnd' ir^ I'm t ll J I i i 1 62 The Elecho Chronograph. larity of rate may result. If a chronometer has run di-^vn it requires a quick rotatory movement to start it aftt' ,'t has been wound. Transporting — On board sbip chronoireters are allowed to swing freely in their gimba;-. so that t^.y may keep 3 horizontal position ; but en land they should be fastened with a clamp. Pocket chrononieters should 'ilways be kept in the same posifion, and if carried in the pocket in the day shou'd be huag up at night. Chronon;eiers h.ave usually a different rate when travelling from WiuU: they keep when stationary. The travelling rate r, ^y be found by comparing observr'tions for time taken ai the same place before and after a journey, or fro in observations at two places of which the difference of longitude is known. For mean time observations an ordinary watch may be used by comparing it with the chronometer, provided ihe rate of the watch is known. Chronometers are generally made to beat half seconds. THE ELECTRO CHRONOGRAPH. Under this head may be included all contrivances for registering small intervals of time by visible marks pro- duced by an electro magnet, and thus recording to a precise fraction of a second the actual instant of an occurrance. By this means an observer at a station A can record at a distant station B the exact instant at which a given star passes his meridian, and thus the difference of longitude of the two stations may be ascer- tained. REFLECTING INSTRUMENTS. THE SEXTANT. A person accustomed to work with the pocket si ^ nt will have little diffit in using the larger kint ■ the latter, with its adjai,.»i;ents, is so fully describe :^ most J - ' i IL J The Sextant. 63 OntnT .:"^"^>';"^;hat little need be said ^ho~:^^^r^, of.erc. ,,,,,t::r:2 or with a plate of glass floating on the mercurv Th;' oof. when one haif of a set of observa.io,. has been ,aken the roof should be reversed end for end R„r ,t I sun. double altitude the da. glasl ^the ^ pt^,^ e:c^tj;;:t::h^.f^-- noted when the circles just touch. As this requir s h" he images should be receding from each other he lui tude of the lower limb must be taken in the for noon and of the upper limb in the afternoon. For a lunar distance of the sun direct the telescope on the moon and use one or more of the hinged dark glasses for the sun The A common fault of the sextant is that the optical power of the telescope :s too small. There is httle use in beTng ab e to read the graduation to .0 seconds if the eye cTn not be sure of the contact of the images within 30^ THE SIMPLE REFLECTING CIRCLE This is Simply a sextant with its arc graduated for the Ic rt"""T"°''""'^'^"^^^'^^^h^-dexarmp o- at h enT Th'' ""'''''1 T'"^^ ^"^ carrying a vernier a each end. The mean of the two verniers can be taken g t dd T'tT ^"' ^"^ '-'- ''- ^° eccentricity^!;:: got nd of. This arrangement also tends to diminish the errors of graduation and observation of i^r reflecting circles have three verniers at intervals .jM 64 The Repeating Reflecting Circle. THE REPEATING REFLECTING CIRCLE. In the repeating reflecting circle the liori^on glass (m Fip, 17), instead of being immovable, is attached to an arm which revolves about the centre of the instrument and which also carries the telescope (0 and a vernier (v). The index glass (Af) is ^'^- ^7- carried on another revolving arm, which also has a vernier vK The arc is graduated from 0° to 720° in the direction of the hands of a clock. To use the instrument the index arm is clamped and its reading taken. The telescope is then directed on the right hand object (6), the circle revolved till the images coincide, and the telescope arm clamped. The index arm is then undamped, the telescope directed on the left hand object (a), and the index moved forward till the images again coincide, when its vernier is read. The difference between the two readings of the index vernier IS twice the angle between the objects. This repeating process may be carried on for any even number of times. The first and last readings only are taken, and their difference, divided by the number of J*p**i4i©»8, gives the angle. If the angle is changing, as in the case of an alti- tude, the result will be the mean of the angles observed, and the time of each observation having been noted the mean of the times is taken. This instrument will not measure a greater angle than the sextant. Its advantages over the latter are that there 15 no index error, and errors of reading, graduation, and eccentricity are all nearly eliminated by taking a sutKcient number of cross-observations. f, <■ ' /.i^. i/<-. ,' ■ 1 bi J about 4,' """f """""'lescope arm at an angle of ^ei.h. :/it;:,-,:':e<,';r„^t; LiT'^r ^^*» altitude, of Ob;:" ^^L'lhelrUh. """ '"' '"""^ ^°""« In ,t, ™-^ '""■SMATIC REFLECTING CIRCLE I fixed prismatic reflector (/) which halfcovers the object klass. The index mirror (w) IS carried on an arm which revolves round the centre of the circle and has a vernier (v v^) at both ends. This instrument will I measure anj-les of any di- 'mension, and has also the following; advantages: (i) Eccentricity is completely I eliminated by using both Fig. i8. verniers. (2) Thp rpfl«, bodies across that plane — generally either the meridian or the prime vertical. In the former case it enables us to find the true local time, either mean or sidereal, and also serves to determine the longitude by means of transits of moon-culminating stars. Ir. th.. latter case it gives us a very accurate method of ascertai'^ ing the latitude by transits of stars ac ss the rrime v tical. In the focus of the telescope are one or two horizontal wires, and an odd number of equi-distant vertical ones— generally five- jf wh.Ji .ne central rhould be in the opti- cal axis of the instrument, and at righ: anghs to the a'is of the trunnions or pivots; and if, in -' ,.uon, this axis is truly horizontal, the line of colli' \tion vill move in a ver- tical plane. The telescope is ovi 'ed with a ver cal graduated circle, with a level a cht which serves as a finder to set it at any required avigle of elevation. It v>hs also a diagonal eye piece for transits of objects of consid- erable elevation, and a very delicate striding level for getting the pivots perfectly horizontal. At night the light b g c fl P w ti re e I f "I The Transit TcUscopc. . 67 of a lantern is thrown into the interior ,r. ,11 • T w,res by „,eans of an opening v^,h"len° """""= "■' »..d to u.: r„;iVoTanVht"^:3i::f::-.rt -"-■ server calling ortoo" r,h T- ''"■°"°'"^'"-' ">= "b- In the case rf the un 1 'T T^'" '=^='> «'-• noted is when the %":» 7.^ k' "'""^'^ '"^ ■"''^■" Hther in connng :;t% tt^^T^tZ "^ "'!,■ for Its semi-diameter to na« ,1, _■ "* required w^ds added or sub^a ^ed, ."' ""=?'';'." ''> ?"- The first adjustment to be attende/^;= ,l\ r " n^ation. This may be effected bTttinVth . r"'" on some well-defined distant objecf or ^ ^^^ ""'" - - at its greatest elongat on 'xh; tel." ' "f^TP^^^^ verscH in its supports, e'nd Z Jt^^TiiT ''"' "■ bis.crs the object, the collimation is aH n^h V^f^^ one side of it must be moved tova dth-,/;^ '^"^ *° val by the limation screws The iL. ^' '"*"'■ moved laterally by means 7iLl '"^^'•""^^"t is then one of the V 'sup^ ^ilUhT^ire^^ J^^^^^^^ when the telescope is again reversed and th. °^J''*' peated till the collimation is perfect ^'°'"'' '"■ The horizontality of the axi? nf f h^ • by the striding level and fooTslw " 'If th" ^'"T'^ generally an error of it. „, T '. ^^ ""« '««■ has change (Ling to^Ltelt'ionsT tl^p til"" "1" '<- flexure, &c.) it will be fnnn^ '^'"Pe'^ature, accidental pivots by getting the!n in^ such a ^^1^? 'T' ''^ will have equal but opposite read^ntf *^" '^"^^ tions. Thus, if in one ^^si ion !hf ^ . '" J"'""'"'^^ P°^'- reads xo, while the west end t .Th „ " /^'^ ^f ^^« e .St end should read 12 and the west xo ''''"' *^' '^. r»m 68 The Transit Telescope. 9 1 1 If one of the pivots has a larger diameter than the other it is evident th;. when their upper surface is level their axis will not be so. This will entail a constant error which will be investigated presently. Thi verticality of the central wire must be tested by levelling the pivots and noticing whether the wire re- mains upon the same point throughout its whole length when the telescope is slowly moved in altitude. If the collimation is out of adjustment, but the level- ling correct, the line of collimation will sweep out a cone. If the collimation is correct but the levelling in- accurate, it will describe a great circle, but not a ver- tical one. If both are right it will move in a vertical plane. We have now to make this plane coincide with some given one— say that of the meridian. The north and south line may have been already approximately ob- tained by means of a theodolite, and we can now find it exactly by one of the following methods. (i) By transits of two stars differing little in right as- cension, one as near the pole, the other as far from it as possible. Let a be the right ascension, 8 the declination and t the observed clock time of transit of the star near the pole; «S ^S and t^ the same quantities for the other star, d the azimuth of the instrument — in other words, the error or deviation to be determined — and tp the lati- tude. Then d is found from the formula, , f , , . ,,, „ I cos ^cos ^' d= Ha> — «)- -it^-t) ) cos f sin (S — d^) The rate of the clock must be known, but not its error; the interval t^ — t must be corrected for error of rate; and, if a mean time watch is used, converted into sidereal time. d being in horary units must be multiplied by 15 to ob- tain the error in arc. If the declination of the southern star is sruth it will. J The Transit Telescope. g_ ton L? :r'v ''""""';' P^" °f '"'^ f°™"'> »' King. on (Lat. 44 ,4, .^ o.oj;, fo, .^^ .^^^^ ^^.^ ^^ ^^^^^ S= .he'd'e'*r^?eni;;r '7 '"-'" " too ^.V^ ^ safes. .0 draw aCelThtcfr "'""^ " ''^^'^^-^ the instrument is known. ^''*^"" ^"^^"^ To prove the formula: J cos tp snT?^^ cos ^p sin (^~5') ^}^zJ^!;' ^^ P°'^' ^ the zenith. ^ ^B the plane m which the teJe- scope moves, and A Z A- the tn.e meri- dian. Wr ' " - aza». have to find the angle Let S and S^ be the two stars at transit, . the unknown clock error, t the clock time when the star S was on i '*g- 19 M I 70 The Trantit Telescope. the meridian. The true time of the star being at S will be t-\-e. Let a be the R. A. of the star. Then a was the time when the star was on PB; .*. Z P S=^+^— a. Let f be the observer's latitude, d the star's declina- tion, d the deviation of the instrument. From the triangle P Z S we have; Sin S Z sin S Z B=Sin P S sin S P Z (i) And Z S=P S— P Z very nearly=y»-<5 .-. sin {

nstrnment at the Royal thickest m? i ""^^^ P'''°* ^^'^^ f°"nd to be the uiicKest, moving the bnhhlp a a- • ■ •■" uc lue Tllerefore. when th;!m„ %* ""°"' °" '"=>'"5»1- their lower surfaf. "^ "", ° "' "'' P'™*^ ™= '^vel theiraxTforo IVr,'"^ "=" ''"^ '- ^visions, and corrected fo eh ' value of on T- '"'' "'"'f'"^' '^ '"' Pi"0t highest bvthr^l ^ ""'"' '•-«•■• ^'h »"=« error wiU b fl t^t""™!.""" ""'"" •="'• '"e total pivot is highet^and': ■ :t:r "■' '"°-- '^''^" '"' «"^- -0 APP.V THE LEVEL COaRECT.ON TO AN OBSEEV„,ON The l,v ■ ™ "■= ™-*"=" ^'•'^SCOPE. " .■el!;:„:ai,t£:'t;fr^^^^^^^^^^^ Now^'iJ'--'^_''^'"SR Sin P Sltl. Sin R cos^decjination R and P =JLgi j^- altitud e COS. declinatron Jiltitude P li S' r \ I ( 72 The Level Correction. West 35 West 55 The correction for the transit in time will be-— - 15 Example— At Kingston, Canada, the transit of Arcturus was observed, the level readings being : ist position East 45 — 2nd " East 25— To find the correction in time. Here we have Latitude 44* 14' N Star's declination 19 47 N Star's altitude at transit 65 33 Level correction = 35±55_:^5zl25 ^^^ ^ ^.^.^j^^^^^ west end being highest, and the pivot correction altered this to 6 divisions. The value of one division of the level was 6".45, therefore the angle R was 38".7 east of the meridian, and the transit took place too soon. P = H X 38". 7 = A^y". 2, and the correction was 2.48s. to be added to the observed time of transit. When the instrument is in perfect adjustment the error of the watch or chronometer can be at once obtained by means of meridian transits, as described at page 34. FINDING THE LATITUDE BY TRANSITS OF STARS ACROSS THE PRIME VERTICAL. If S is a star on the prime vertical, P the pole, Z the zenith, and W Z E the prime vertical, S P Z is a right-angled triangle ; and if we know the angle S P Z and the side P S we can find the side P Z by the equation Fig. 22. Cos S P Z=tan P Z cot P S I Prime Vertical Transits. or. If a IS the star's right ascension, /the s^^^^^7^~^ Its crossing the prime vertical d ilT/ v ^ °^ i^ the latitude of the place ' ^^^^^'n^tion, and re.u Cos(«_0=cot^tan5 •It the star is in thp nncit,/^,, c / the equation becoLr '" *"' °' "*' »="<''-'» Cos (<— a)=cot ». tan '''^' p- observation may be m»T ^1 ™'" ''" ''°°™- The instrument oTwUh TtntZ ™h'V^ """^"^ "»=" other delicate ,«»!<,:' latrude ?' '" '''^' " '" Sin. altitude =^L sin ^ (Since cos P S = cos P Z cos Z S ) exc^rth^^flls'edecl-r^^rb t*"^ ''""' -"-' latitude of the place Th I ^ " ""'" ^"'^ *' zenith are to be preferred h "" "'"""""^ "=" ">= observed time of ^;S^:^//;C-:-;'-^ in the telescope clamped'at n^' L^es t? b"'''°'=' ^"? *^ horizontal arc If „„ , ' ""^ ""^ans of the transit telescope L nre Ld" d'"'. ^''^ "" """^"^ (from the approximate l,,rH,u°'""' '' '° '='''<="'="'= which culn^'nyersTver d ; : 3t„:r^f :l '^'"^'^ =" ='" cross the prime vertical, a.,d d ec'Tl \° '""'"'" that instant. It will now be „ea Iv if^r °" " " position. The emr m i ""any m the required -ethesiderea,reoTL::i.t;:X^:^-- -r*^- J I 1 74 Prime Vertical Transits. east and the west verticals. The mean of the two will be the time of the transit over the meridian of the instru- ment, and should be equal to the right' ascension of the star. If the two results are not equal their difference shows the angle which the plane of the instrument makes with the true prime vertical. In working these observations we may use either a sidereal or a mean time chronometer, in the latter case making the usual reductions, and always allowing for the rate. If two transit telescopes are available, one of them may be set up in the plane of the meridian for the purpose of ascertaining the exact chronometer or watch error by star transits. A large transit theodolite serves . instead of two transit instruments, and in this case an ordinary good mean time watch will suffice, the mean time of the observations being reduced to sidereal time. If both the east and the west transits tre observed the dif- ference of time in sidereal units is double the hour angle P, and the latter may therefore be obtained without any reference to the actual watch error, provided the rate is known. It should also be noted that if we reverse the telescope on its supports any error of collimation or ine- quality of pivots will produce exactly contrary effects on the determination of the latitude. Two stars may be observed with the telescope in reversed positions on the same day, or the same star on two successive days, and the mean of the two resulting latitudes taken. It will be found advisable to calculate beforehand the altitudes and times of transit (either mean or sidereal, as the case may be) of a number of suitable stars. If the plane of the telescope is not in the prime vertical the calculated latitude^ will be too great. Suppose the deviation to be to the ea^st of north and that the tele- Prime Vertical Transits cope describes a vertical circle passing through fb^E > ZWi. Then V Z\ which bisects S S', will be the calculated co-latitude. The correction for the deviation may be computed thus. The star's R. A., minus the mean of the p- - ^me^of transit corrected for clock error, wiH be thetngle JlL'e : ' "^ *'' "^ht-angled triangle Z P I, tan PZ cos ZPZ' = tan PZ' = tan FS cos SPZ' or tan- deep a.°d nvo feet in diameter, n,:, ,„,, t^, |^„ P f^ fZw T"'T^""""'-' *^""=>- b^'-^'^" 'he Lake of the Weeds and the Kocty Ilountains. The meridian mark snould, if possible, be at least half a mile distant. ... black or white vertical stripeTa ntet on a stone serves tor the day time. A, night a'buIlTe;t 76 The Personal Equation. lantern may bo used, the glass being covered by a piece of tin with a vertical slit cut in it. Or, as the lantern is liable to be blown out by the wind, it may be enclosed in a wooden box with a vertical slit. The larger transit theodolites may be used as transit instruments, and have the advantage over them that when the meridian line has been ascertained the prime vertical can be at once set off. THE PERSONAL EQUATION. It often happens that two persons, equally well trained in taking observations, will differ by a considerable and nearly constant quantity in estimating the precise instant of an event, such as the transit of a star across a wire. This difference is called their personal equaiion, and an allowance should always be made for it when observations made by two individuals have to be combined. In the case of the transit instrument this equation may be de- termined as follows: Let one observer note the passage of a star over the first three wires and the other observer note the transits over the remaining wires. If the two observers' estimation of the instant of transit differ, it is evident that (provided the wires are equidistant) the difference will appear on comparing the intervals of time. For instance, if A notes the transits across the first three wires at los., 20s., and 30s., and B notes the remaining two at 39S.5 and 49S.5, it is plain that A would consider the star to be on any wire half a second later than B would, and their personal equation is therefore os.5. By repeating the same process on other stars, and taking the mean of the result, a more accurate estimate is obtained. The personal equation has been found liable to vary with the state of health of the individual. The difference in the estimated instant of a transit is only a particular case of the personal equation. CHAPTER VIII. THE ZENITH TELESCOPE. The zenith telescope is a contrivance for the exact d. termination of the latitude by measurinc^wf/h7h minuteness the differences or^tr^-T ' ^'^^*^'* tancesoftwostars one of f ' ^ "^ "" ^'"^'^ ^^^^ the zenith distance of the equator, we have, and adding, 2^= J+i^IjI7Z/ thett;:::7fi::L^;^Lrra„T;''^r -. ^-^ ing thei.. actual values/ Mo Lt, if /anT::! '""T :nr;eir---7;i--— ^ r>f*u t .■ ^ '"^° account the difference of the refractions at the two altitudes. fnrln '"^[[""^^'^t is practically a telescope about 45 inches foca length attached to a vertical a^s round whi^h revolves, having been first clamped at a certain ang e o li' 1 ^«*te:*„j,*v,«L '■ i .1 i 78 The Zenith Telescope. elevation. The latitude must be known approximately, and a pair of stars selected which are of so nearly the same meridian zenith distance at that latitude that they will both pass within the field of view of the telescope without our having to alter its angle of elevation. As a rule, z and z must not differ by more than 50' at the most. If the axis is truly vertical and the telescope remains at the same vertical angle at the observation of both stars, then it is plain that the difference of z and z may be read by a micrometer in the eye piece. It is usual to observe only stars which pass within 25 degrees of the zenith. The telescope has a long diagonal eye piece with a micrometer in its focus, and the micro- meter wire is at right angles to the meridian, ^^here is a very delicate level attached to the telescope, and a vertical arc which serves as a finder. By reading this level at each observation we can detect and allow for any change in the angle of elevation of the telescope. The above is the merest outline of the principle of the instrument, and reference must be made to other works for the details of its construction. The method of using it is this: The latitude being already approximately known, a pair of stars is found from a star catalogue, both of which will pass within the field of view without altering the elevation, and which have nearly the same right ascension. The reason for this is that their transit may take place within so short an interval of time that the state of the instrument may remain unchanged ; but a sufficient interval must be allowed for reading the micrometer and level and reversing in azimuth ; say, not less than one minute or more than twenty. The meridian line must have been previously ascertained by transits of known stars, or otherwise, and the chronometer time calculated at which each of the stars will culminate. The telescope having been brought into the meridian, ready for the star which culminates first, and set for the mean The Zentt: Celescope 79 wire at the calculated instant ofTt.anl H ™"°'"^'f manner If'aftj^h , ' ''"=°"'' ^'^^ '" ">= ==>"<= must 1; r LI ed b/r: ""• "° '^^^' '' ■""=•■ °"'. " teleLope'^'' ^'^ ^""-""^^"^ ^^ --"^^^ '^^ the transit This method of finding the latitude is known as Tal cotts having been invented by Captain Takott oHhe' U. S Engineers. Its defects are that it is often difficuk to obta.n a sufficient number of suitable pairs o? stlVs o which the dechnations are accuratelv known As a -X we have to use the smaller stars, whose FJ;cef are not very well known, and must therefore observe alarge num her of pairs to eliminate errors. TO FIND THE CORRECTED LATITUDE ern^tr 'f *t ""'T''" ""''^^"^ ^'^ ^^^^ ^^ ^he south- ern star, m, the same for any point in the field assumed as he micrometer zero, and .„ the apparent zenith distance represented by m„ when the level reaSingis.ero. Suppose also, that the micrometer readings increase as the zen th distances decrease. Then, if the level reading were ze 1 the star's apparent zenith distance would be 8o The Zenith Telescope. Let / be the equivalent in arc of the level readinK, posi- tiwi when the reading of the north end of the level is the greater. Lot r be the refraction. Then the true zenith distance of the southern star, or z, is: The quantity r„ +w„ is constant so long as the relation of the level and telescope is not changed. We have, there- fore, for the northern star, Hence 3—z'—m'—m-\-l'-\-l-\^r—r' and the equation for the latitude previously given will become : X=^ (^+5') + i (m'-m) + i {l'tl) + ^ {r-r') TO FIND Ti'E CORRECTION FOR LEVEL. Calling the readings of the north and south ends of the bubble n ana s, and the inchnations at the observations of the north and so-th stars, expressed in divisions of the level, L' and L, we shall have V- n—s L= n- 2 2 and if D is the value of a division of the level in seconds of arc, we have l'=U D /=L D and the correction for the level will be i (/'+0=i (L'+L) D=''^-Jl±A D 4 TO FIND THE VALUE OF A DIVISION OF THE LEVEL. Turn the telescope on a well-defined distant mark. Set the level to an extreme reading L, bisect it by the micrometer wire, and let the micrometer reading be M Now move the telescope and level together by the tangent \ The^enith Telescope . g screw till the bubble mves s^^A\^^' T" " — treme, bisect the mark LJn k 1 "^•^'- the other ex- 'nicrometer reading bM?THf ^ ""^ "' ''' *^^ the level in turns of th! n!'" "^^"^ "^ ^ ^'^^^''^n of lurns ot the micrometer will be ^nd if R is the value in st^t of arc of . , r of the micrometer th*. vai» r. r , •*,^^ "^ '^ volution arc will be ' "'"' ^ °^ *^« ^^^^1 ^n seconds of TO FIND Till D=Rrf ^emth distance by the formulce ' ^"^''" ^"'^ cos ^-=cot } cos r=cosec «J sin /I Whence, knowin/' the star'Q P a „ ^ .l . error, .e find the chronome i ta. rf^h ^'■-"O'""- Ration. Set the telescope for he^nhh ir"''''/'""- it upon the star ,n „r ,„ • '""/'=""" distance », direct «. elcngatio; 7. . rcrr^Ith':^ "'^ ""'"f «-'• note the time if bisec^.^ "^'"""eter wire; .■n.s. A3 the :Lr » ^es ; ^Irrea^ -' -^' -ad- often as possible while it i"" LLTh "!'= ?™«= as tU\:;:.t:.t^\"--™"-'.wroTbist^^^^ 1 > 3> s> o , --- ""'"-o ui uisec- readin,s, „ "the micromete'r Te'^Lrf,! '"•'■"°'""'=' greatest elongation (,). and T 7 ^f Photographic Sciences Corporation 23 Wk:.>i MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 m \ f ^\ ^v V ? ,^ ^ (/. 82 The Zenith Telescope. screw to move the thread through the angular distance ij {m — ;»i) R=i\ Also (m — >Ha) R=«3 Therefore, subtracting (Wa — Wi) R = /i — jj or R:r= Wj -;m, To correct for any change in the level reading, let l^ and /a be the level readings corresponding to m^ and ;«,; then (/g — /j) D is the change required. The angular value of D is unknown ; but, since D = (^/K, the correction to be applied to (/j — i^) is {/g — /j) dR; and (Wa — Wi) R=ti — ij, ± (/^ — /,) dR or R=^ U-H A value of R is thus obtained for each of the observa- tions, and the mean of the results taken. This mean has then to be corrected for refraction, thus : From the tables find the change in refraction for i' at the i-enith distance z. Let this change be dr ; then R dr will be the correc- tion to be subtracted from R. REDUCTION TO THE MERIDIAN. If a star has not been observed exactly on the meri- dian it may be taken when off it, and the observation re- duced. The following is one method of doing this. Keep- ing the instrument clamped in the meridian, the star is observed at a certain distance from the middle vertical thread and the time noted. This will give its hour angle, and if we denote this by t (in seconds of time) the reduc- tion is obtained by the formula ^ (15 ty sin x"sin 2d This is to be added to the observed zenith distance of a southern star, or subtracted from that of a northern one, and, in either case, half of it is to be added to the latitude. The Zenith Telescope. REFRACTION. 83 When the .en'th distances are small the refraction vanes as the tangent of the zenith distance. Let r=a tan z r'=ci tan z' Then r~r'=a (tan s— tan z) __ sm (2 — z') cos z cos s' =(^-^y') " sin i' ^^ ''^ cos* 3' "^^'■^y a may be taken as 57".;, and the difference of th^ micrometer readings used for (z~z') ^ THE PORTABLE TRANSIT INSTRUMENT AS A 7ENITH TELESCOPE. adL'^'n T'^'l" uT' ''^'^^^"P^ ^'-^ - micromete^ added to It, and the level of the finder rim]. ;.- ^ sufficiently delicate, it may be used a^ a L " f te :scZe' reversing the mstrument in its Ys between the oS^: Th^'^!7'^^' ?^''^'"^"^^ give the mean places of the stars The «/^«..;., places are those which have to be used and must therefore be determined. ' ^ wliiilliHlili II CHAPTER IX. ADDITIONAL METHODS OF FINDING THE LATITUDE. TO FIND THE LATITUDE BY A SINGLE ALTITUDE TAKEN AT A KNOWN TIME. Here we have in the triangle P Z S the hour angle P, the side Z S (90°— the objects alti- tude), and P S the polar distance. From these data we have to find P Z. From S draw S M perpendicular to P Z produced. Let cJ be the declination, (p the latitude, and a the altitude. In the triangle P M S we have : cos P=tan P M cot P S=tan P M tan d M Z=P M— P Z=P M + y>~9o° Also 'A^- . cos PM : cosZ M::cos P S : cosZS or cos P M : sin (P M+^)::sin <5 : sin a Therefore sin (P M+^)= sin ojcos P M sin d (2) Equation (i) gives P M and (2) gives P M + ^ In this method, if the star is observed when far from the meridian a small error in the hour angle produces a large error in the computed value of the latitude. The altitude should therefore be taken when the object is near the meri- dian. TDE. "AKEN tl the large itude neri- Lalitude by Altitude of Pole Star. 85 TO FIND THE LATITUDE BV OBSERVATIONS OF ThT^ STAR OUT OF THE MERIDIAN. Up be the polar distance of the pole star m circular measure p'^ h a very small quantity. Let P be the pole, Z the zenith, and S the star at an hour anp^le h or SPZ. Draw ^ N at right angles to P Z and take ZU equal to Z S. Let P N be denoted by .r. MNby3..SPby;^, the star's altitude by a, and the latitude by /". Then P Z=Z M + M P=Z S + P N-N M '''' '' or go~l=go—a+x~y .'. l=a — x+y We have to find .v and y. (1) From the right-angled triangle S P N we have cos S P N=tan P N cot P S .'. tan .v-=tan p cos h or, approximately, .-v =p cos A (2) Denoting S N by , we have from the same triangle Sin S N=sin S P sin S P N or sin 7=sin p sin h .'. approximately, q=p sin h. (3) In the right-angled triangle S N Z we have cos Z S=cos S N cos Z N .'. sin fl=cos q sin (a+y) or sin (a+y)= ^^'^^ or approximately sin a+|^tcosa=- : COS^ sm a =sin a (i-l-^ qi) y cos a=^ q9 sin a orj'=J<7» tan a =i>» sin* /{tan a MMi ! i f 86 Circum-Meridian A ItiUides. Hence, in circular measnre l=-a—p cos h-^^ p^ sin'* h tan a or in sexag-esimal measure l=a~p cos h + ^ />» sin i " sin^ h tan a This is the method given in the explanations at the end of the Nautical Almanac. To find the latitude we have only to take an altitude of Polaris, note the time (which will give us the sidereal time), and apply certain correc- tions as directed in the Almanac. FINDING THE LATITUDE BY CIRCUM-MERIDIAN ALTITUDES. When the latitude has been found by a single meridian altitude the result is only approximately true. It may, however, be obtained with great exactness by taking a number of altitudes of the sun or a star when within about a quarter of an hour of the meridian on either side of it. The altitudes may be taken with the sextant, reflecting circle, or theodolite, and the observations should follow ea-h other quickly, and at about equal intervals of time. The watch error must be exactly known, and the time of each altitude noted. The mean of the altitudes is taken, but the hour angle for each must be obtained separately. In the case of the sun this is done by cor- recting the observed times for watch error and subtracting them from the mean time of apparent noon. If a star is used the mean time corresponding to its R. A. will, of course, give the hour angles— The formula is Latitude=go* — a±d — x' Where a is the mean of the altitudes, d the declination of the object (negative if south), and x" a quantity equal to • • ^» 2 sin' — 2 sin J. X cos. approx. lat. x cos. dec'n. x sec. alt. ; h being the hour angle. Circiim-Meyidian A ltitude%. ^_ 87 To prove the formula 2 sin» — and .r"= ^ v ^°^ ^ ^°s ^ • //A Sin I cos a Let P be the pole, Z the zenith, and S the sun or star near the meridian. Let a be the star's altitude, h its hour anirle and d its declination. Let a + x be the star's meridian altitude. Then a+x-=^d=Cio—l We have now to determine the small quantity .v ^^ Now, sin PZ sin PS cos ZPS^cos ZS-cos v'z< ^^■ ^^^^ or cos I cos d cos A=sin «-sin / sin d ' "' .: cos I cos d (i-cos h) -= -sin a + cos (l-d) = — sina-fsin(a + ;v) .*. 2 cos / cos d sin* - - 2 Therefore, approximately -2 sin -^ cos (a + ^) 2 smj — X"= ?_ V ^°S ^ cos i sin I cos a d is, of course, negative if south 2 sin«~ The value of the expression __^'" 2 .. sml"~ (^"own as the "reduction to the meridian") is found for each hour t7cire :.^:^ -' ''' -- -^" ^^e vai::^ tatn the mean of ten altitudes of the sun's lower l!mb oh' served w.th a powerful theodolite, was ,,■ 59 ao' Tht," :^- -^■^■nn 88 Circum-Meridian A Ititudes. \) t when corrected for refraction, parallax, and semi-diameter gave 40° 14' 3i".55 as the true mean altitude of the sun's centre. The sun's declination was 19° 53' 45".8 south The mean of the values of the reduction for the observed hour angles, as taken from the table, was i6".26, and the calculated value of x was I7".36. go o o ^•t'tude 40 14 31.5 5 -o ^ ,.■ . 49 45 28.45 Decimation... 19 53 45,80 ^_ 29 51 42.65 1^36 '^ Latitude=29 51 25.29 Strictly speaking, a further correction ought to be made for the change in the sun's declination during the obser- vations. In the case of a star we must add 0.0023715 to the log oix" to correct the hour angle for the difference between the sidereal and mean time intervals; for the star moves faster than the sun, and therefore gives a larger hour angle for the same time. Additional accuracy is obtained by taking half the ob- servations east of the meridian, and half west of it, the intervalsof time between the successive observations being made as nearly equal as possible. The hour angle changes its sign after the meridian passage of the object. I w -diameter, the sun's ".8 south, i observed 3, and the CHAPTER X. > be made le obser- » the log. between ir moves ?er hour the ob- ' it, the ns being r angle object. t^TERPOLATlON. ''^^^^ODS OF FINDim THE LONGITUDE. and longitude of tre place 'fo7'' '°'" '^^ ^°^^^ ^'-e given are for GreemvS"me. "''""' ^^"^^ *^^ ^^^- accurate result. "^^ "^"^ *° obtain a very su':s:s:::~:;::;^^v^thg.^^ ^ -ain .ear, at a ^t:\Z^!: ^ :fJr:::^ Of For Greenwich mean noon we find in the Almanac Date c > J '^ow, at apparent noon at the ohr^ :, -ii v apparent time at Greenw.VI, VPf^ " "'" be 4 p.m. ^;-ree„..a.t.:r!;:;--:--;-rr Th.s va„at,on is X3-.305, which n,„„ip„-ed by I X^ 90 Intcipohition, 53".22 to bo subtracted from the dc- dination of 2d January — 22' 57' lC".2 .^3".22 22 56 22,98=required dec'n. IJ.2I 12) 1. 14 •095 13.21 TO one =^ V«rl»tlnn ^J •305 It a I'M 53-22 il V -3 INTERPOLATION BY SECOND DIFFERENCES. The differences between the successive values of the quantities fjiven in the Nautical Ahnanac as functions of the time are called the first differences ; the differences be- tween these successive differences are called second differences; the differences of the second differences are third differences, and so on. In simple interpolation we assume the function to vary uniformly ; that is, that the first difference is constant, and therefore that there is no second difference. If this is not the case simple interpo- lation will give an incorrect result, and we must resort to interpolation by second differences, in which we take into account the variation in the first difference, but assume its variation to be constant and that there is no third difference. The formula employed is f{a + k)=f{a) + Mi + \5k''- where A is half the sum of two consecutive first differ- ences and B is half their difference. It is thus derived: We have by Taylor's Theorem / (.V + /;)=/ (.v) + Ml + hh 2 + &c . (A) and if /t is small compared with .v ilie successive terms of the series grow rapidly less. VarUtlnn at i r.M (A) Interpolation. m Suppose a— I, 11, and ~F=a, the 1st differences. " and &c., a' ~a=b, and so on. Nmv, if FCO is the function corresponding to the argu- nient I +n w we have F(«'^F + «a+ "P^II^b+'K»-j) (n~2) ^^ , , u -r \-8cc. (a) 1.2 1.2.3 • If « be taken successively equal to 0,1.2, &c., we shall obtam the functions F, F', F" &r nnH ;J a- . , , i , r , (xc, and mtermediate values are found by usmg fractional values of;/. To find the proper value of n in each case let T-f^ denote the va ue of the argument for which we wish to interpolate a value of the function ; then n w=t, and n= t ; that is, n is the value of / reduced to a fraction of the interval w. Ex.-Suppose the moon's R. A. had been given in the Almanac for every 12th hour, as follows : W ■ I 1 1 94 Mar. 5, oh Muoit's R. A. Interpolation . i^TDiff^^ 2nd Dijf. 3rf niff, ^thDiff\5thDif «»ai.i. on2in. 5Sni2Ss .30I „ " 5. 12h|22 27 15 .4i +28D1- 47S.O4 fM2h23 23 3 .39 27 37.89 32.18 6.53 + IS .74 " 7, oh 23 50 15 .03 27 12 .24 ^5 .05 ^ 53 j J ^g " 7, I2h o 17 9 .8J 26 54 .20 ' i« oa 7 — OS.66 required the moon's R. A. for March 5, 6h. Kere T=March 5, o^, t^C^., u>=x2\ n:=^\.=L. ^,,. .f we denote the co-efficients of ., ,, ., &e. in^« Lv A i5, L, &c., we have - ' „=+,8". 47..04, A^„ ,, F=^" 58" a8..39 b=~ 4.79, C=B ^ =+^.^, c'c^-.-+ ip.74> D^C ' M— 3 .r= 6 — tItt, D^=<=— 4^62 o«.3o o».o7 Og.02 Moon's R.A. on March 5, 6^ or F^^^ =22» 12-56.74 TO FIND THE LONGITUDE BY TRANSITS OF MOON- CULMINATING STARS. This is a simple and easy way of finding the lon^ritude tTLT/''^ ''" '^ '"°^^"' ''^^^^ -^ ^ -y at ZT2 ' r '" '''''' °^ °"^ ■'^^^^"^ '" an observed ^ans t may throw the longitude out as much as half a mmute m t.me, or 7-i n^inutes in arc. It is, howeve a ^^es 11 T' '^""'"' *° ^ ^"'•^^^-' -- ^"'h wa'c Of " ' ''■'"''* ^'^^^^^''^^ ^"d -" ordinary prefted'f rS,:.-^^^^^^ ^^^^ ^— is to d The instrument is set up in the plane of the meridian, ■'--r I^^^ii^^deby Moon-Culminations. gg knew either Greenwich or Ic'tim uHL'TT 1° watch should be taken into' accoT^ 't fn lal J transit are noted, and the interval of ti„,e between t .s reduced from mean to sidereal time! ''™ In the Nautical Almanac are given, for everv dav of .(,„ year, the sidereal times of transit a C^TZthllt^ moon and of certain suitable. •,s c-ill,. '•"'"^'^ °f "'<= in-r" ^, I . ''"^'J'e^ ^^s, callei. noon-culniinat- ■ng stars ; also the rate of change per hour fat f hT of transit) of the moon's R. A As the m„ rapid^- through the stars from west t Z. TL7ZZ tl.at,a sta.,o„ not en the meridian of Greenw.cl the .nterva, between the two transits will be diffe,e„Tto ,h, at Greenwich; and, the moon's rate of mo.irpeHou being known, a simple proportion will ,if the stat o nl longi^,dr"?f M '""°" """ ''"•'■"*''• ""'I 'hence the longitude If the station is far from the meridian of Greenwich a correction will have to be made for "he change i„ ,he rate of change of the moon's R A xt rate of change at the time of transit is fonnH f? .f Nautical Almanac by interpolation by ^ c™d iffrncer and the mean o the rates of change at Greenwich and at" he station is taken as the rate for the whole interval o time between the transits. mierval of An example will best illustrate the method :- At Kingston, Canada, on the 24th Februarv .sx, .1. transits of the star -. Tauri and of'the „ 00 's ti^ iimb were observed at 6h. om. ys., and 6h. rm. gs r sp" vely, mean time. Uifferenee, 46 seconds, „i 46 ,?,, Sidereal units. 4"s.i^ in 1^ I 1 i 96 Longitude by Moon-Culminations. Greenwich TransitslL^^"""-^^- ^9^. 163.62 Moon I... 4 7 57 .44 Difference in sidereal time= Add interval at Kingston^ iim. 19S.18 46 .12 Total change of moon's R.A= 12m 53.3=7253.3 By interpolation by second differences the variation of the moon's R.A. per hour at Kingston at the time of transit was found to be 142s .23 At Greenwich it was 1^.2 .68 J 2)284 -91 Mean rate of variation 142.455 1-2:^5 ^^h. = 5h.09i6 5h. 5m. 293.76 west longitude. It should be noted that in this case the moon was west of the star at transit at Greenwich and east of it at Kingston, having passed it in the interval. The following is a specimen of the part of the Nautical Almanac relating to moon-culminating stars. n Mom-Culminating Stars. !5s.3 tion of ime of [itude. s west it at utical o 00 00 U) O ^ O NH H h < :?; h V O o H ■V^s„jo-joo ■u a « o apnjiuSBi^ 2 N ix O CO ■*oo <> d "> 't- fO 5h to to i _oo'_ I-H l-H .^ ^ CXI >- o c rt "3 FINDING THE LONGITUDE BY^Ti^dISTANCES. This method is an important one to the travelh'n^ n r ::;r';; ^'^ -^^^^"^^ -^°^^ chJnTmit^r; other Ifl^;- " '"^trument used is the sextant or some other reflectmg one, and the observation is a very siZe eve; an"erTl'°''^T^''"^ ^^^ angle, causes how! ever, an en or m longitude of about a quarter of a degree 1 ■ I 98 Longitude by Lunar Distances. The moon moves amongst the stars from west to east at the rate of about 12° a day. Its angular distance from the sun or certain stars may therefore be taken as an in- dication of Greenwich mean time at any instant— the moon being in fact made use of as a clock in the sky to show Greenwich mean time at the instant of observation. The local mean time being also supposed to be known, we have the requisite data for determining the longitude of a station. In the Nautical Almanac are given for every 3d hour of G.M.T. the angular distances of the apparent centre of the moon from the sun, the larger planets, and certain stars, as they would appear from the centi of the earth. When a lunar distance has been observed it has to be reduced to the centre of the earth by clearing it of the effects of parallax and refraction, and the numbers in the Nautical Almanac give the exact Greenwich mean time at which the objects would have the same distance. It is to be noted that, though the combined effect of parallax and refraction increases the apparent altitude of the sun or a star, in the case of the moon, owing to its near- ness to the earth, the parallax is greater than the refrac- tion, and the altitude is lessened. Three observations are required— one of the lunar dis- tance, one of the moon's altitude, and one of the other object's altitude. The altitudes need not be observed with the same care as the distance. The clock time of the observations must also be noted. The sextant is the instrument generally used. All the observations can be taken by one observer, but it is better to have three or four. If one of the objects is at a proper distance from the meri- dian the local mean time can be inferred ."rom its altitude. If it is too near the meridian the \vatch error must be found by an altitude taken either before or after the lunar observation. I Longitude by Lunar Distances. 99 written down in their proper order: latob?;i>"^^*'.'.r'="- ^'^•.°^«'»'- -'t- Of moon'B lower limb. Dist.ofmoon'B f.rllmb ard " 4th " 4) Mean ToUli. If there is only one observeVit is best to take" the ob- servations in the following order, noting the time by a watch, ist. alt of sun, star or planet; ad, alt. of moon; 3c., any odd number of distances; 4th, alt. of moon; 5th at of sun star, or planet. Take the mean of the dis! ances and of their times. Then reduce the altitudes to the mean of the times; ,.c., form the proportion-differ- ence of times of altitudes : diff. of alts.::diff between t.ine of ist alt. and mean of the times : a fourth number vvaich IS to be added to or subtracted from ist alt ac cc rdmg as it is increasing or diminishing. This will give the altitudes reduced to the mean of tSe times, or Z- responding to that mean. The altitudes cf moon and star must be corrected as added t"?h.'!"'™"''^ semi-diameter of the moon added to the distance to give the distance of its centre. 1 iie lunar distance has then to be cleared of the effects of parallax and refraction. TO DETERMINE THE LUNAR DISTANCE CLEARED OF PARALLAX AND REFRACTION. Let Z be the observer's zenith, Zm and /,v the vertical circles in which the inocn and star are situated at .he instant of observation. Let m and s be their observed places, U and S their places after correction for parallax and refrac- tion : then Zm, Zs, and ms are found by observation, andZ iM and ZS are obtained by correcting the observations. The ob- Fi. 27 100 Longitude by Lunar Distan ces. ject of the calculation is to determine M S. Now, as the angle Z is common to the triangles mZs and M Z S, we can find Z from the triangle mZs in which all the sides are known. Next, in triangle MZS there are known M Z, Z S, and the included angle Z, from which M S can be found. M S is the cleared lunar dis- tance. The numerical work of this process is tedious. The cleared distance having been obtained we proceed in accordance with the rules given in the N.A. The Greenwich mean time corresponding to the cleared distance can be found either by a simple proportion or b}- proportional logs. It admits of proof that if D is the moon's semi-diameter as seen from the centre of the earth (given in N.A.), D' its semi-diameter as seen by a spectator in whose ;;enitli it is, D" its semi-diameter as seen at a point where its alti- tude is a, then D" — D=(D' — D) sin a, very nearly. For details of the methods of finding differences of longitude by the transportation of chronometers, and by the electric telegraph, vide Chauvenet or Loomis. CHAPTER XI. MISICELLANEOUS. TO KIND THE AMPLITUDK AND HOUR ANGLH OF A GIVEN HEAVENLV RODY WHEN ON THE HOK,;^ON. Tho ampUtHdc is the nn-^Ie that the plane of tl,e vertical c.rcle thron.h an ohject n,al, we have sin p = — , a TO FIND THE PARALLAX IN ALTITUDE, THE EARTH BEING REGARDEI'^ AS A SPHERE. In Fig. 2g A is the observ- er's position, Z the zenith, C H the rational horizon. A H' the sensible horizon, and S the heavenly body. Let p be the horizontal parallax (H'), p' the parallax in altitude (S), h the altitude (S A H'), and d the distance of the /,,„ .,, heavenly body (S C). From the trian ,de S A C we have !i" ^__ — _ s'" S _ A C sin Z A S ~ sin S A'"C~ S^ = —7- = sin 6 a ^ or sin/)' = cos // sin p The angles/) and p' being (except in the case of the moon) very small, we may substitute them for their sines, and the equation becomes p'=p cos h STAR CATALOGUES. If we want to find the position of a star not included amongst the small number (197) given in the Nautical Almanac we must refer to a star catalogue. In these J J Star Catalaf;ues. 103 catalogues the stars are arranj^^ed in the order of their right ascensions, with the e cha g„ 1 ' "IT "''"" .''"^ '"""<'"• To change A» in^hetl'tirude' trL^^titrd";^" '' " the equation. "'^ **^ differentiate Sin«^sin^sinoVcos^coso-cos^ Regardin^r

rime vertical. U d(^ — o and d(p=o we have *^ cos tan q Hance an error iu the star's djclination produces the l«;ast effect when the star is on the prime vertical (since tan (7 is a maximum when sin Z=i), and that, of different stars, those near the equator are the best to observe. In high latitudes it will often be necessary, in order to avoid low altitudes, to obsf.rve stars at a distance from the prime vertical. In this case small errors in the data v/ill affect the clock correction. But if the star is ob- served on successive days on the same side of the meri- dian at about the same azimuth, the clock's rate will be accurately obtained, though its actual error will be un- certain. If the same star is observed both east and west of the meridian, and at the same distance from it, constant errors d 7"V" ^^.*^™"-^--^ of this kind. In fi iding the la tudes of stations we are in general dependent on the direction of the plumb line; and should there, as often happens, be a local abnormal deviation of the later from the true perpendicular, the resulting latitude' wi 1 be erroneous. This was proved many years agX taking the latitudes of two stations on oppJte side^ o a mountain in Perthshire, and measuring Jhe true hoi 'l! tal distance between them, when it was found that the ^^^e of the Earth. Betweeen A iind B... 2=;' co •tJ and C 17" while .he actual differences, as found by triangula.ion. Between A and B 24" 2^7 B and C r4".ig As a rule, the deviation seldom exceeds a f.yv seconds except in the neighbourhood of great mount.i^ Z as at the foot of the Himalayas. tvher^eT -"rmXas' Where there is considerable deviation in level countries It .s no doubt caused by neighbouring portions orthe Z %TJ:7 '''-' ''''-'' - li.lfter than the avt! sy..rnetrica,ly g.ouped round ita.::Wn ^n tL t^l^^ and ong..uae of each obtained by astron^micll obta t ons. The actual distance and azimuth of the central station rem each of the others being known by tr'an 1 latitude and lon"itude of th. 7 1 ^'^ ""^^^ Dared with ,v\ *''^ ''^"''"^^ station being com- pared wuh the latuude and longitude as obtained by ill r-i i 'i 1^ V 114 Figure of the Earth. if astronomical observations will give the deviation of the plumb line. If a and b are the semi-major and semi-minor axes of an ellipse, the distance of the centre from either focus is |/rt»Z_^3, and this quantity divided by a is called the "eccentricity." This is generally written e. The quantity a -b. is called the "compression" or "ellipticity," and is denoted by c. The latest calculations make the com- pression of the earth about ^^j, the ratio of the semi- axes being believed to be 292 to 293. The true measure of the compression is the difference of the semi-axes divided by the mean radius of curvature of the spheroid. The equator has also been found to be elliptical, its major axis being about *oo yards longer than its rninor axis. It should be noted that the expression e has different meanings in different books. English writers occasion- ally employ it for the compression or ellipticity, while in American books it is used in the same sense as here, namely, for the eccentricity. Even in different chapters of the same work the letter e is often used both for the compression and the eccentricity. The accompanying | figure represents a section of the earth. PP' is the polar axis, | QE an equatorial di- ameter, C the centre, I F a focus of the ellipse, A a point on the surface, A T a tangent at A, and! Z A O perpendicular ^''s- 3° to A T. Z' is the geocentric zenith, and Z' C E' is its declination. The latter is called the geocentric or reduced lati- Figure of the Earth. "5 /«rf. of A. Z O' E' is thegeographkal ^r^^^tZ^^iiMA^^^ t •.°' ^^^'^ ^^"^d tJ^e reduction of the latitude. It IS evident that the geocentric is always less than the geographical latitude. LetCE=:«. CP=6. Let .- be the compression and . the eccentricrty. a—b a , CF _b_ a e= C F2 CE3' CF CE ~P F C E3 "■^~ (^£2 63 ^i-^^i_(i_,). That is, e2. or, c= i/gcH^ (I) TO FIND THE REDUCTION OF THE LATITUDE. Taking the centre of the ellipse as the origin of axes the equation of the ellipse will be «? ^ 62 ^ Let ^ be the geographical latitude 9* " geocentric " We have, tan a>=^— ^ dy and from the triangle ACB, tan ^'= X or. Differentiating the equation of the ellipse, we have _y_^ b^dx X a2 dy tan 5^= -^tan ^—(1-^2) tan ,p lo hnd the reduction, or^_^', we use the general development m series of an equation of the form tan A'=-/) tan y, which is x—y=^q sin 2y-fj ^2 sin 4v4-&c. 1: i 1^ ii6 Figure of the Earth. in which (7=^-— Applying this to the development of (2) we find, after dividing by sin i" to reduce the terms of the series to seconds, and putting x=^=- — tan

f "^ Figure of the Earth. u9 117 ^» + J'^ I— ^a -a^ -~ =• (i~e') tcixi

Vi—e^ ^nd hence, p=a{^IZ3jl^}l \ I — I Vi~e^ sin'

'y levelled would, wh directed on one station, intersect the other on the tele scope being turned over. CHAPTER II. OEODETICAL OPERATIONS. The methods adopted in the old world for mapping large tracts of country have been reversed in America. Instead of starting from carefully measured bases, and carrying out chains of triangulation connecting various principal points in such a manner that the relative positions of the latter with respect to each other may be ascertained within a few inches, though several hundred miles apart, the system pursued (if we except the U. S. Coast Survey and some other triangulations) has been to take certain meridians and parallels of latitude inter- secting each other ; to trace and mark out these meridians and parallels on the ground ; to divide the figures enclosed by them into blocks or "checks ;" and to further subdivide the latter into townships, sections, and quarter sections. Although the method of triangulation is incom- parably the most accurate, the American plan has the advantage of rapidity and cheapness. As the latter is very simple, and is fully explained in the Canadian Gov- ernment Manual of Survey, it will not be further touched upon here. At the commencement of a triangulation a piece of tolerably level ground having been selected, a base line, Tnauffulatiofi. . lai cliam of tna,n.|,.s ,, s,,^,t«l. I„ ,i,e f„r,„e,. case th^ nancies are expanded as rapidly as possible ,11 ,he„ Ire lame enourf, to cover the whole country with a netwo k "f pntnary tr,a,„des. This is done by taking ang losfrl d tan , f " "" "'°""""" '"""■ ■•""' calcnia ing thei I stances by ,ng„,„„ctry. The instru.nont is then placed enTt : in '":r"'r r'""': -■" -«'- -"- S ;.ei; Id';': rb :ere,''° ■r',:;,'':j"'-'--'^ »- and e.te„de• iiitir siaes die often from ^o to 60 nr The louges s.de ■„ the British triangulation was ,T The stdes of the secondary triangles are from abou , to 20 mtles, and those of the tertiary triangles fiveor"Ls The larger triangles should be as nearly equilateral as .rcmnstances admit of. The reason for having te„ so ■s ha, w„h ,h,sform small errors in ,he measuremen" of ,he,r angles wll have a minimum effec, on the cdcn .e, lengths of the sides. Such triangles are a ed "well-conditioned" ones. The original base has to be reduced to the level of the re^tTc fthri^'h :r f "-^^ '^^^^^^" ^'^ ^^^-^^ -^^r verticals through its ends intersect the sea level must be V-.** e 122 Tnangulation. II ascertained. The exact geographical position of one end, and the azimuth of the other with respect to it, must of course be known. The angles of all the principal triangles must be measured with the greatest exactness that the best instruments admit of, the lengths of the sides calcu- lated by trigonometry, and their azimuths worked out. The work (when carried on on a very large scale) is still further complicated by the earth's surface being not a sphere but a spheroid. The accuracy of the triangulation istestedby what is called a "base of verification." That is, a side of one of the small triangles is made to lie on suitaole ground, where it can be actually measured. Its length, as thus obtained, compared with that given by calculation through the chain of triangles, shows what reliance can be placed on the intermediate work. As instances: The triangulation commenced at the Lough Foyle base in the North of Ireland was carried through a long chain of triangles to a base of verification on Salisbury plain, and the actual measured length of the latter was found to differ only 5 inches from the length as calculated through several hundred miles of triangulation. An original base was measured at Fire Island, near New York, and afterwards connected with a base of verifica- tion on Kent Island in Chespeake Bay. The actual distance between them was 208 miles, and the distance through the 32 intervening triangles 320. The difference between the computed and measured lengths of the base of verification was only 4 inches. In Algiers, two bases about 10 kilometres long were connected by a chain of 88 triangles. Their calculated and measured distances agreed within 16 inches. If the country to be triangulated is very extensive— as, for instance, in the case of India— instead of covering it with a network of triangulation, it may^ntersected in the first place by chains of triangles, either single or double. Base Lines. 123 and bases measured at certain places/usually wher--^ these chains meet. In India the chains run generally either north and south or east and west, and form a great frame or lattice work on which to found the further survey of the country. A double chain of triangles forms, of course a series of quadrilateral figures, in each of which both the diagonals, as well as the sides, may be calculated. The following is a brief account of the measurements ot some celebrated base lines : In 1736 a base line had to be measured in Lapland for the purpose of finding the length of an arc of the meridian by triangulation. A distance of about 9 miles was mea- ured in mid winter on the frozen surface of the River Tornea. By means of a standard toise brought from France, a length of exactly 5 toises (about 32 feet) was marked on the inside wall of a hut, and eight rods of pine terminated with metal studs for contact, cut to this exact length. It had been previously ascertained that changes of temperature had no apparent effect on their length The surveying party was divided into two, each taking four rods, and two independent measurements of the base were made, the results agreeing within four inches. The time occupied was seven days. The rods were probably placed end to end on the surface of the snow. The same year a base 7.6 miles long was measured near yuito in Peru, at an altitude of nearly 8000 feet The work occupied 29 days. Rods 20 feet long, terminated at each end by copper plates for contact, were used The rods were laid horizontally, changes of level being effected by a plummet suspended by a fine hair. The rods were compared daily with a toise marked on an iron bar which had been laid off from a standard toise brought from Paris. This base was the commencement of a chain of triangles for the measurement of a meri- dianal arc. Three years later another base, 6.4 miles long 124 Base Lines. was nieasured near the south end of this chain and onlv occnp,ed ten days. The party was divided into'.wo c:™" pan,es wh.ch measured tne line in opposite directions melced'trtTr'™'' """"'■ "^ "'"'' ''"«^'" ™^ "- He^th I , "^^f "'■"•"ent of a base on Hounslow Heath, which was chosen from the great evenness and openness of the f;ro„nd. Three deal rods, t.pped „' h bell metal and .o feet long, were used at firs . But it was ™™i y'tf'th;-;""' r ^"1='^" ^^ ='"'"^- '" '^""■ Cth ^f t, fr^P'^"'' t''" Slass tubes of the same iscfrti: d" » rP'""°" '°' temperature had been ascertained, were substituted, the temperatures of the ubes betng obtained by attached thermometers The ^eng h of the base when reduced to the sea lev and 6^ Faht. was 9,134! yards. This distance was subsequentfv 40 I nks half an mch square in section. A second siniH.r chatn was used as a standard of comparison The chai,^ was laid in five deal coffers carried on trestles Id kept stretched by a weight of a8 pounds t'c act Z olT'' f "mf " ™= """''' '' =" ="''- ^^ •" steel TJL r° "?^''^'«-^'"ems (glass tubes and steel chains) agreed within two inches. Two bases, each about ji miles long, were subse- quently measured in France-one near Par s, the other at Carcassonne in the south. Four rods were ised The ' were composed of two strips of metal in contact (patilum and copper), forming a metallic thermometer carried o^ a stout beam of wood. Each rod was supported on two .ron tripods fitted with levelling screws, a^d there was In arrangement for measuring their inclination. The Lough Foyle base was measured with Colbv's compensation bars; an arrangement in which the ,,„e„,« expansions and contractions of two parallel bars o differ ant metals (brass and iron), ,0 feet long, are utili/ed o ktp Base Lines. bury Plain base was measured in the same way Colby's" bars were subsequently used for ten bases inlndiabu were not found togive very reliable results there ' sattn J7'°'?'"' ""^ ^°'^^'^ arrangement is the compen- atmg apparatus used in the United States coast smvev ^ It consists of a bar of brass and a bar of iron a I ttl.!^ than SIX metres long and parallel to each ohe"' Th bar'.' Sr";?^^"^'^^^"^"^^"^' but free to movl'^t th other Their cross-sections are so arranged that llll versejy as tHfespecific heats, allowance being made for hei difference of conducting power. The bras bar is tl r"; -^'--^f on rollers mounted in suspen ! to'itts ruJo: r brLvr °" ''''' -'-' '-^-' i- 'iM,Cl tS-. 11 II 126 Base Lines. fig. 31 The annexed figure shows the arrangement at the two ends the left hand part being the compensation end. It will be seen that the lever of compensation (/) is pivoted on he lower bar (a), a knife edge on its inner side abutting on the end of the iron bar (b.) This lever terminates at Its upper end in a knife edge (^ in such a position that whatever be the expansion or contraction of the bars it always retains an invariable distance from their other end. This knife edge presses against a collar in the shdmg rod (d), moving in a frame (/) fixed to the iron bar, and is kept back by the spiral spring (s). The rod IS tipped with an agate plane (p) for contact. The vernier {v) serves to read off the difference of lengths of the bars as a check. At the other end where the bars are united a sliding rod terminates in a bluftt horizontal knife edge (g) its inner edge abutting against a contact lever (A) pivoted at (»)• This lever, when pressed by the sliding rod, comes in contact with the short tail of the level (k), which is mounted on trunnions and not balanced. For a certain position of the sliding rod this bubble comes to the centre and this position gives the true length of the measuring bar. Another use of the level is to ensure a constant pressure at the points of contact, p and g. To the lever and level is attached the arm of a sector which gives the inchnation of the bar. J ^ J Base Lines. 127 level sector and vernier, are read through glass doors tTesVe"s T^'n'^' :"" ^"^ "-"«'<' on a ptof rSgned'^rAl*!''""""^^'''""'-"'"^-^''^^ paSr'the'°;r'T".'""'' '°"«' '"=^^"'-='' -i"- "•- ap. paratus, the greatest supposable error was commit.^ th 'onrtr^"'- ""^ '"^ *''^" -.::„thT^ an' men. On another base, six and three quarter miles ?„n„ annr"''' '"" *^^ '^^^ "-an one ten"h oTan °ch' and the greatest supposable error less than three-temhs' GellTrttaes".'"" '"•'' '■' -"^^="""8 a base in eorgia thre . t.mes, twice in winter and once in summer at temperatures ranging from i8- to 10/ Faht Th^ discrepancies of the three measures with their respecliv! -cans were, in n.illimetres,-8.ro,-o.3., and "sTi ' qul^trfec7ilr."''' "TT' "'^' "■= >PP-«- - n°t .r^^^ure is'r„";oX •''"''"^ ™ "-""" "■' a w^;^str:rt„^%rst:: ■" ^^r^s surveys ,t may be sufficient to nn.asure the b».. . F -V.C ui pjanx, which is made to adhpr#> ♦« fK^ ground by means of pointed spikes on its under L^ce! oarnrd'^^^'' ^"" ^^^«°"^d' baked, boiled in drying oil painted and varnished, may be u«?pd Th u f j"^ , ' be levelled or h=.v« fk T , ^"^^^ ^^°"^d either ieveued or have their angle of inclination read. If the li u »i'— t'-Cv_J_-, 56 miles out 170 machin- ;en was ad been t across ! on un- .'e been in the asuring lites of imuth" he sur- ts em- •dolites .verage "or the ites of In the 3 have ry and The of the rniers, le the which lue to jntire- irniers differ- Drtant 'l!<,fH-V.- Triangulation. 1 , ^Z7 Of the smaller theodolites there are two ki^^^dT^il ticukr reading, the telescCo can be dSed ,„ "' ''"" required I„ a ^eUe.atin/.heodomf le C:,X'^ fixed to the stand, and when the instrument Ts set 1 ' ,nr .he purpose of measuring a horizontal angk it s ' ui.e , verniers. If an angle is read off on each and th« * i Srhl""'"'^' °™' ^"'' •"= -rr;m a's ed.t tx:zT '"^ --- ^^ '^^ saUairr Ch! h-d.object.andclamped,!::fad"Vt r^t ;",r Th ', f °'" '^ "6"'" =" °" 'h^ l^ft hand obtct The lower plate ,s then clamped, the upper one set f^l ve"rn er'ma °'\'""''"' °" ""^ right^Ldob e The ^a-;::-r^;e:^:^tr:r-ii-- a=Xti"-o--Lt■" °' °pp-'- du'^ ±ad k (cot A + cot B^ and smce ' sin (A + B) cot A+cot B=ii5LiA + Bl sm A sin B f^osTA^=BP-JE5F(ATB) ', 144 Triangulation. it follows that da= ± ad A- 2 sin C cos (A — B) -(• cos C and da will be a minimum when A"=-B. In either case we have the result that the best con- ditioned triangle is the equilateral. 1 1 w i ■J m ;st con- CHAPTER IV. DETERMINATION OF THE GEODETIC LATITUDES LONOI TUDES, AND AZIMUTHS OF THE STATIONS OF A TRIANOULATION, TAKING INTO ACCOUNT THE ELLIPTICITY OF THE EARTH Where the lengths of all the sides of a triangulation have been computed it becomes necessary, in order to plot the positions of the' stations on the chart, to obtain their latitudes and longitudes. The first step to be taken is to determine by means of astronomical observations the true position of one of the stations, and also the azimuth of one of the sides leading from it. We can then, knowing the lengths of all the sides of the triangles and the angles they make with each other, deduce the azimuths of all the sides, and calculate the latitudes and longitudes of the other stations. Before geodetical operations had been carried to the perfection they have now attained it was considered suf- ficient to solve this problem by the ordinary formulae of spherical trigonometry, taking as the radius of the earth the radius at the mean latitude of the chain of triangles / L 146 Geodetic Latitudes, 6-c. Thus in the triangle PAA' (fig- 35) where P is the pole of the earth, and A, A', two stations, if the latitude and longitude of A were known, and also the ler.jth and azimuth of A A', we should havo tin- two sides A P, A A', and the included angle PAA', and could use Napier's .^j,,^ ^^ analogies to determine the remaining parts of the triangle and thus obtain the latitude and longitude of A', and the azimuth of A at A'. But this method is deficient in exactness, especially as regards the latitude, and the fol- lowing has been adopted as giving better results. Let A N be the normal at A, and suppose a sphere to be described with centre N and radius N A meeting the polar axis at^. Also let p A, p A' be meridians on this sphere. We then calculate the geographicr.l position of A , not by the ordinary formulas of spherical trigonometry (since the side A A' is very small relatively)" but by the series I. a-^b—c cos A+J c2 cot b sin»*'A +J c3 cos A sin2 A (J+cot2 b)+... II. i8o°— B—A + c sin A cot b 4- ^ c2 sin A cos A (i + 2 cotj" b) + i c« sin A cos2 A cot 6 (3 + 4 cot^ b) — J-c' sin A cot b (1 + 2 cot2 b)... III. C= 'sif6'^"^+srrh'^"-^^°^Acot6 '^i^b ^^" ^ ^°^' ^ (I + 4 cot2 /;)_i _li sin A cot^ b... sin V C^^ n . ■ y. iif- f 6t. ^-r ■Vij Lt-, / f -1l^ / f Geodetic Latitudes, S-c. 149 The imaginary sphere used in the above investigation will, of course, coincide with the spheroid for the parallel of latitude through the point A. Any plane passing through the normal will cut the surface of the sphere in the arc of a great circle, and the spheroid in a line, which, for about three degrees, will be practically a geodesic line. The following is another way oi treating the sub- ject. Instead of taking the n jruial at ne of the points A A' as the radius of the imagi-iurv sph.re let us take the normal at the point B, mid-way v^■ee.n them, as in I-i.r. 36, and for the sake of simplicity let these points be on the same meridian. Let A N, A' N' be the normals at A A', produce them to Z and Z' respec- tively, and draw A c, A' e parallel to the major axisOE. The astronomical lati" tudesof the two points are Z A e, Z' A' e. If now we draw B C the normal at B, C will fall between N and N'. The curve given in the figure is the elliptical meridian. The circular curve drawn with radius C B is not shown ; but it would pass a little outside of A and A'. For practical purposes we may suppose it to pass through those points. Join C A, C A', and produce them to s and z respectively. 2Ae,zA'e' will be the latitudes of A and A' on the imagmary sphere, one being less and the other greater than the latitudes on the spheroid. The differences ZAs!,Z.Az may be considered the same. Let each be c I , ^-n^-u^^j^ ^ f^-V^ / i-6 J ^srsSBESwa i 150 Geodetic Latitudes, 6-c. 3 designated--. Let L and L' be the astronomical lati- tudes of A and A', /. and /' their latitudes on the sphere, and X the latitude of B. Then already investigated. In what next follows K is the distance A A' in yards of any two stations A, A', u the same distance in seconds ot arc K the radms of curvature of the meridian, N the normal (both in yards), e the eccentricity (=0.0817), and a the equatorial radius. Equation («') gives us the values of wand V, (V) ^ives us M . and (c) gives Z'. If we neglect the denominato of the fraction in (c) we have 2'=i8o- + Z— (M'— M)sini(L + L') ' or Z'^i%d' + z- u sin Z . , -^^^,smnL + L') The last term of this equation, which is the difference t^::::^:''''' ^^^ ""^^"^' '- ^'^ —genceof. ■ ■ . cU,.- ^•f-'^i- ■tr: / :^a > / / i v . - ^yr ,. r.^.^. I/H-Al' ••■. ■^v<.yu^ - ( M ' - /ii/ -:...vx C : 7.. ., .- J I srence nee of r-^'"^" ^^"^ ■ ■ ■ d J J _^Geodetic^atitudes, &c. '■ If the triangnlation is limited^hT^^^i^i^Tr^T convenient to express L' Af n ^^. '^ '"^y ^^ "^ore lar co-ordinates re eld' to axes ha " " 'r^ °' "^^^"^"■ station A. the axis of/b h.rthe '?• '' °"^'" ''^'^^ axis of .the freodesic-'line iourr'^'" '' ^' ^"^ ^^^ to the meridian.The equations are ' P-P^-^icuIar L'= '^Rsinr-^-^iNsHTr" M'=M± ^- N sin I" tan fL± — ^_ ) ^ ^Rsini'V X cosL' tan L'* N sin i" "=edi„ the next three probkms! ' '"" ''^ "»^.*« THE LINE JOINING THEM DIRECTION OF Here we have eiven T r ' »* Lan W« have then to fl„d^/ and /'from the equations ^"~L and /'=-L'+~^ 2 2 ^icuIartothe^eHdiathLn^h^olh^tin-''^-''^-'- Let y be the number of secondQ in f k ".eridian het.een L and CZlu^^Z^::"' pointy ' from 1. ^ ' ^ ^^*^^ ^^''""th of the .:: .1 ^--o <^' I-IATH^ ,/^ //'; _;,2i-^«='5— i-Stess;:^:' 1 152 Geodetic Latittides, &c. Then we shall have .V'-=-(M'— M)cos/' j"— =/ — /' — \ sin i" x-i" tan / x=~x N sin I" ^_jy" N sin I" tan Z= — M = sin Z cos Z K=-w* N sin I" The signs of (L — L') and of (/ — /') must be carefully at- tended to. EXAMPLE. ' \t Given ' L=49« 4' 25" L'=49 22 33 M' — M, or difference of longitude— 38' 47"=2327* to find Z and K Here L + L'-=98° 26' 58" A ^ ^ (L+L)=49 13 29 L'— L-» o 18 8 i(L'— L)=o 9 4=544' » To find the value of — 2 log £2—7.81085 log \ (L— L)=2.73549 2 log cos HL + LV 0.62994 Jog — —0.17628 2 ^ Z— L- =49° 4' 26".5 (5 being negative) 5 /'=L'+ ^-=4q°22'3i".5 1 irefully at- =2327* ;ive) Geodetic Latitudes, &c. To find x" Log (M'-M) =3.3668785 log cos //= 9.8136470 Jo&^:=3.i8o5255 X =1515" To find the value of the 2nd term oiy" log i sin 1 "=4.38454 2 log a;' =6.36105 log tan /=o.o6ig7 log 2nd term=o.8o756 2nd term»=o° o' 6" /'-/ -=0 18 5 :v'=o 18 IlcsrIOgi" To find the azimuth Z Log a;"=3. 1805255 logy'=3.o378887 Jog" -V =0.1426368 ^=125° 45' 21" To find log N sin 1" Log N (in yards)=6.8443224 log sin i"=4.6855749 Log N sin I "=1.5298973 To find log u" Logj/';=3.o378887 iog cos ^=9.7666596 To find k'"^'''^^-'''"'' log «"=3.27I229I logNsini"=i.5298973 4.8011264 J^= 63226 yards. I •if \t 154 Geodetic Latitudes, &c. ' To find the co-ordinates. Value of X. Log a'=3. 1805255 log N sin i"= 1.5298973. log x=\'yio^?'i% ^=51336 yards Value of y. Logy;=3.o378887 log N sin i"=i. 5298973 log ^=-4.5677860 J' =36965 yards TO COMPUTE THE DISTANCE BETWEEN TWO POINTS, KNOWING THEIR LATITUDES AND THE A2IMUTH OF ONE FROM THE OTHER. Let L and L' be the latitudes, Z the azimuth, and let^ ■^'^ <, >->- Then we sh.j.\\ have, as before J- = g ^ (L— L ') col'; 2 2 N= ,0 2 (i— e2 sin3 /l)J /'=L'H- «? Assume tan / ^ cos ,, . , „, sm / . -^ then, sm (

. log sin L=9.8782652 log tan Z=o.i426358 log tan ^=0.0209020 f =— (46° 22' 42") ^— 38 47 To find L" log tan L=o.o6i9663 log sin (f—;ji) =9.8643024 co-log sin 0-0.1463154 iog tan L' -0.0665841 - ig" 2 ^ 30 yj — w— 47* I' 29" To find t? L 49 4 25 49 2-z 30 L-L"=— 18' 5 "=1085" L+L"=98 2655 log c» ^.7.81085 log(L— L")=3.0353i 2 log cos \ (L+L'0=9.62994 log ^=0.47610 8= -3" L'_L"-^=49°22'33" =49° 4' 26/'.5 /'=L'- d =49 22 31 .5 ;■ ! i^ To find u" and K — log ^=3.3668785 log cos /--g.8136471 co-log sin Z—0.0907036 log m"'=3.27I2292 log N sin i'/==-i.5298973 log K- , >ii265 K^^L.3^26 yards. • G<^odetic latitudes, &c. On the North American boui^d^i^rVev "irTTs^Tr following method was omploved in\lT2 '^^ ^^^ muths of two distant noinV I , ^ *^^ """^"^^ ^zi- of which were known.' '' ^'^'^"'^^ ^"^ ^-^'^"^es noJjLlrndPthUl/^^Tr'-r' °^ -^-^ B is the sphere, we have in Ihe s^h T' '''"'^"^ *^^ ^^'"^h as a sides PA, PBfanVtLTnrA pTef "^^J^^ ^^ find the angles A B Th; J^ ^'''^"' ''"^ have to g'es A, B. This IS done by the usual for^I^, tan A (A+B)-^ cos AP-BP cos . AP-BP sm — tif tan \ (B-A)-z. ___^ v o . P . AP+BP "^^^^a sm ^— - 2 2 which give -^ilB and fc^ 2 2 Then,A=^B_B-A 2 2 B=A±B B-A > «, /?, from the formul ' ' "' ''^'"'^^^ *^^ ^"^^^ sin a= sin AP sin /?= sin BP ^75 -'- 4/--^ '' tln'/^o"-;' ^' ^^^ *^^ *-^ spheroidal azimuths, igo —A )-=cos a tan (go'-A) tan(B'-go")=cos^tan(B-9o°) cut'^from'oXfntT ^'T t^" ^ ^°"^ ''^ ^^ ^« ^^ / one point to another through forests. ^ L I p 158 Geodetic Latitudes, &c. To find the accurate length of the arc on the surface of the earth between two very distant points of known lati- tude and longitude is a very difficult and not very useful problem. It is, however, often advisable to calculate the distances between stations that are within the limits of triangulation, as a check upon the geodesical operations ; and in the case of an extended line of coast, or in a wild and difficult country where triangulation is impossible, this problem is most useful for the purpose of laying down upon paper a number of fixed points from which to carry on a survey. In the triangle PAB mentioned in the last article we have, as before, the sides PA, PB, and the angle P, as data. By solving the triangle we obtain the length of the arc AB. If the azimuths can be observed at the two stations the accuracy of the result will be greatly increas- ed, and we can obtain the difference of longitude of the two stations as follows : — It may be proved that the sphe- rical excess in a spheroidal triangle is equal to that in a spherical triangle whose vertices have the same astrono- mical latitude and the same differences of longitude : from whence results the rule PA-PB A+B , P tan — = 2 cos- cos- PA + PB X cot cos ^ diff. lat. xcot A+B sin J sum of lat. which gives P, or the difference of longitude. As a rule, a small error in the latitudes is of no import- ance unless the latitudes are small : but the azimuths must be observed with the greatest accuracy. The angle P being known we can get the length of the arc AB, and must then convert it into distance on the earth's surface, using the radius of curvature of the arc for the mean lati- tude. surface of lown lati- ery useful culate the ! limits of lerations ; in a wild npossible, ^ing down ii to carry irticle we igle P, as gth of the : the two y increas- de of the the sphe- that in a astrono- )ngitude : Devillc's Methods. o import- azimuths rhe angle AB, and ; surface, aean lati- , to only a few inches in loo miles ^' ''™°""" of solving certatan„M °'"';™"'^ ^'"P'" "^'hods tables of Lrrhmsofl^."' '" "^'"""^ ""^ "'^''"^ °f the distance ot oi„f I. J ^0?:;:: ^Tr 'ot°/ "T" oT.ts.'.rt- i^ r T 4-™^^^^^^^ A and B wil, be. pra^iXls^^i^TSr^r" .a"n7fre:s I'e^xi^i':-' rr='^- <'■^- less than 90' by ha f thecnnv " '^ """^ '^ "'" ^^ constant the conve I "ce will f "" " "" *='^"« ^ the equator (where ttk„rh- ? "' "" ''"'^^ f"™ problems involv S^ Vo station^' ' fT^^ "", ""'^^^ '" convergencensed ,! .. f/==-^ cot/) 2 K Next join AB by a great Circle arc. The angle BAC will be half the convergence, and AB=AC, approximately. Draw PD bisectmg P, and therefore at right angles to AB. In the triangle APD we have D— 90* converg ence 2 and rAD'='9o' i ' Offsetts to a Parallel. Therefore, cos PAD=tan AD cotT or sin^^^— I^"^^ _ a t 2 -^AD cot /, approximately =J X cot / „ Therefore, j;=^,y2 cot /— .v sin ^5BX51?2£5 2 This is equally true if .r and ^ are measured lengths. 165 i CHAPTER V. METHODS OF DELINEATING A SPHERICAL SURFACE ON A PLANE. Since the surface of the globe is spherical, and as the surface of a sphere cannot be rolled out flat, like that of a cone, it is evident that maps of any large tract of coun- try drawn on a flat sheet of paper cannot be made to ex- actly represent the relative position of the various points. It is necessary, therefore, to resort to some device in order that the'points on the map may have as nearly as possible the same relative position to each other as the corres- ponding points on the earth's surface. One method is to represent the points and lines of the sphere according to the rules of perspective, or as they would appear to the eye at some particular position with reference to the sphere and the plane of projection. Such a method is called z. projection. The principal pro- jections of the sphere are the "orthographic," "stereo- graphic," "central or gnomonic" and "globular." A second method is to lay down the points on the map according to some assumed mathematical law, the con- dition to be fulfilled being that the parts of the spherical surface to be represented, and their representations on the map, shall be similar in their small elements. To this ^Wk- Projections. 167 class belongs Mercator^s Projection, in which the meridians are represented by equi-distant parallel straight line "d the paralle s of latitude by parallel straight lines a d^t ang es to the mendians, but of which the distances from each other mcrease in going north or south from the equator m such a proportion as always to give the t ue hearings of places from one another. The third method is to suppose a portion of the earth's surface to be a portion of the surface of a cone whose axis coincides with that of the earth, and whose ve ex somewhere beyond the pole, while its surface cuts or ouches the sphere at certain points. The conical sur ace as then supposed to be developed as a plane. whTh It of course admits of being. The only conical develop as tfte ordinary polyconic." in l\L°''"'Tf"' '''°'""°'' '^ ''"P'y ""= °"= employed in plans and elevations. When used for the delineat on of a spheneal surface the eye is supposed ,0 be a an "n fin, e distance, so that the rays of ifgh. are para 1 1 the P^ane of projection being perpendicular ,„ their direc ion eitner that of the equator or ol a meridian. When a hemi sphere ,s projected on its base in this manner the rela.™ positions of points near the centre are given w'h o r able accuracy, but those near the circumference are com dtdutd ^'T'^'- ^"l'^"^ °f "- P.ojection arTs": deduced. Amongst others it is evident that in the case of a hemisphere projected on its base ail circles pL,n. through the pole of the hemisphere are p o ected »« straight lines intersecting at the centre. CircTes hti I'g heir plane, parallel to that of ,he base are projected I*" equal circles. All other circles are projected as e fin of wh,cl, the greater axis is equal to the' dia„,e r o T r':;3:,":'or:b^u!.7 ^Xi !l i68 Projections. Stereographic Projedion.—ln this projection the eye is supposed to be situated at the surface of the sphere, and the plane of the projection is that of the great circle which is every where 90 degrees from the position of the eye. It derives its name from the fact that it results from the intersection of two solids, the cone and the sphere. Its principal properties are the following: i. The pro- jection of any circle on the sphere which does not pass through the eye is a circle; and circles whose planes pass through the eye are straight lines. 2. The angle made on the surface of the sphere by two circles which cut each other and the angle made by their projections are equal. 3. If C is the pole of the point of sight and c its projection; then any point A is projected into a point a such that c a is equal to tan (arc CA rx ..„ — I — -S where r is the radius of the sphere. From the second property it follows that any very small portion of the spherical surface and its projection are similar figures ; a property of great importance in the construction of maps, and one which is also shared by Mercator's projection. The astronomical triangle FZS can evidently be easily drawn on the stereographic projection. Z will be the pole of the point of sight. The lengths of ZP and ZS are straight lines found by the rule given above, and the angle Z being known the points P and S are known. The angles P and S being also known we can draw the circular curve PS by a simple construction. The orthographic and stereographic projections were both employed by the ancient Greek astronomers for the purpose of representing the celestial sphere, with its circles, on a plane. Gnomonic or Central Projection. — In this case the eye is at the centre of the sphere, and the plane of projection is Projections. i6g a-plane touching the sphere at any assumed point. The projection of any point is the extremity of the tangent of the arc intercepted between that point and the point of contact. As the tangent increases very rapidly when the arc IS more than 45°, and becomes infinite at go*, it is evi- dent that this projection cannot be adopted for a whole hemisphere. Globular Projection.— This is a device to avoid the dis tort.on which occurs in the above projections as we approach the circumference of the hemisphere. In the accompanying figure let A C B be the hemisphere to be repre- sented on the plane A B, E the position of the eye, O the centre of the sphere, and EDOC perpendicular to the plane A B. M and F are points on the sphere, and their projections are N and G. Now the representa- tion would be perfect if A N : N G : G O were as A M : M F : F C. This cannot be obtained Fig. 39, .^'^^^y^;.^"* ''' ^^'''1 ^« approximately so if the point E IS so diown that G is t!.e middle point of A O and F the middle point of A C. In this case, by joining F O and drawing F L perpendicular to O C, it may easily be shown that L D ,s equal to O L, which is O Fxcos 45" or rxo.71 nearly.-M«.««^ G O is half the radius and F L halt the inscribed square— tfeererfase FL:GO::OC:OL but F L: GO::LE ; O E .-.LE : O E::OC : OL consequently, LO : O ■• : . C L : O L, or O L^.=0 E C L but O L^=.F L»=D .. L C, .-. O E. C L^D L. C L or O E-»D L that is, E D—O L I 170 Projections. The above projections are seldom used for delineating the features of a single country or a small portion of the earth's surface. For this purpose it is more convenient to employ one of the methods of development. Mercator's Projection is the method employed in the construction of nautical charts. The meridians are repre- sented by equi-distant parallel straight lines, and the parallels of latitude by straight lines perpendicular to ^he meridians. As we recede from the equator towards the poles the distances between the parallels of latitude on the map are made to increase at the same rate that the scale of the distance between points east and west of each other increases on the map, owing to the meridians being drawn parallel instead of converging. If we take / as the length of a degree of longitude at the equator (which would be the same as a degree of latitude supposing the earth a sphere), and /' that of a degree of longitude at latitude ;, then /'-/ cos >^, or /'.•/:: i : sec ;. Now /.' : / is the proportion in which the length of a given dis- tance in longitude has been increased on the map by making the meridians parallel, and is therefore the pro- portion in which the distance between the parallels of latitude must be increased. It is evident that the poles can never be shown on this projection, as they would be at an infinite distance from the equator. If a ship steers a fixed course by the compass this course is always a straight line on a Mercator's chart. Great circles on the globe are projected as curves, except in the case of meridians and the equator. In this projection, though the scale increases as we approach the poles, the map of a limited tract of country gives places in their correct relative positions. The Ordinary Polyconic Projection.—In conical develop- ments of the sphere a polygon is supposed to be inscribed in a meridian. By revolution about the polar axis the vxmf-sss.^i^^ifi.fi^,^, Projections. 171 polygon will describe a series of frustums of cones, li the arc of the curve equals its chord the two surfaces wilF be equal. In this manner the spherical surface may be looked upon as formed by the intersection of an infinite number of cones tangential to the surface along succes- sive parallels of latitude. These conical surfaces may be developed on a plane, and the properties of the resulting chart will depend on the law of the development. The Ordinary Polyconic is a projection much used in the United States Coast Survey. It is peculiarly appli- cable to the case where the chart embraces considerable difference in latitude with only a moderate amplitude of longitude, as it is independent of change of latitude. Before describing it it must be noted that whatever projection is used the spheroidal figure of the earth must be taken into account, its surface being that which would be formed by the revolution of a nearly circular ellipse round the polar axis as a minor axis. In the Ordinary Polyconic each parallel of latitude is represented on a plane by the development of a cone haying the parallel for its base, and its vertex at the point where a tangent to a meridian at the parallel in- tersects the earth's axis, the degrees on the parallel pre- serving their true length. A straight line running north and south represents the middle meridian on the chart, and is made equal to its rectified arc according to scale.' The conical elements are developed equally on each side of this meridian, and are disposed in arcs of circles described (in the case of the sphere) with radii equal to the radius of the sphere multiplied by the cotangent of the latitude. The centres of these arcs lie in the middle meridian produced, each arc cutting it at its proper latitude. These elements evidently touch each other only at the middle meridian, diverging as they leave it. The curva- / . -SCTssssTjiBaBSsM:, I.l 172 Projections. tureofthe parallels decreases as the distance from the poles increases, till at the equator the parallel becomes a straight line. To trace the meridians we set off on the different parallels (accordinfj to the usual law for the length of an arc of longitude) the true points where each meridian cuts them, and draw curves connecting those points. To allow for the ellipticity of the earth we must use for the radius of the developed parallel N cot /, where N= a (i—e^ sin 3 /)J^ a being the equatorial radius, e the eccentricity, N the normal terminating in the minor axis, and / the ar >!« it makes with the major axis. It is evident that the slant height of the cone— say y— is N cot /, and that the radius of the parallel on the spheroid is N cos /. The length of an arc of M° of a parallel will be »°-^o N cos /. In practice, instead Fig. 40 of describing the arcs of the parallels with radii, it is more IZrV"- T''''''' *'^" ^^°- ^^^'^ ^ 43- the earth ; both in feet. BB' is about 0.16 CC ACB may be taken as a right angle; and AC*, the arc AC, and the straight line AC, ar e"alt equal?' We shall have then, if - th e dint finr" K ii mi l ii 111 {j i'- U t. BC=K tan B'AC + CC-BB' ^K tan B 'AC + 0.00000002 K^ where B'AC is the observed angle of elevation of B. This formula supposes that AC'B is practically o^' If the dis- ^.O-A.'*'*/ \ IMAGE EVALUATION TEST TARGET (MT-S) /. .? C^ ^ / A 1.0 I.I |50 "^B ^ b£ 12.0 2.5 11-25 II 1.4 A-KX^l-V^ rtograpnic Sdences Corporation 1.6 23 WEST MAi.: STREET WEBSTER, NY. 14580 (7)6) 872-4503 // ^A, .v^^ 4 ^, V ^0 K<^^ ^1^ i \ 176 Trigonometrical Levelling. tance is so great that this is not the case we shall have in the triangle ACB ,sin BAG BC=K- sin B To find the angle B, we have in the triangle AOB, B=i8o=-(0+BAO) =180— (0 + 90° + BAG) =- 90— (O + BAG) Hence, sin B=-cos (0+BAG) sin BAG BG=Kx cos (b + BAG) And BG'-=BG + GG'-BG + 2K (rY" =^'^3252 Log K =4.76220 2 N sin i' K I — 2 r 2.89472=log 784''7=o° 13' 4".7 2 N sin I " ^-^^° ^'' 53".i I i8o Trigonometrical Levelling. Thirdly, value of the difference of level. Log K =4.7621984 Log tan 88" ir 53".! =1.5022427 Log ist term=3.2599557 2nd terni= 691 3rd term= +627 4th term= 9 Second Term. Log^= 28393 Log a =3 Log 2nd term = 5-8393 2nd term = 0.000069 1 Log. diff. Ievel=3-26oo884=log 1820.07 yards. Third Term. Log ^^3.538 2 N log I St term = 3 2599 log3rdterm=57982 3rd term =00000627 REFRACTION, die. Fourth Term. 34387 Log _M I 2 N'J log Ks =95244 log 4th term = 2 9631 4th term =0-0000009 TO FIND THE CO-EFFICIENT OF TERRESTRIAL REFRAC- TION BY RECIPROCAL OBSERVATIONS OF 2EN1TH DISTANCES. Let A and B be two sta- tions, and let their heights (ascertained by levelling) be h and h'. Consider the earth as a sphere, and take O' its centre. Call the radius r and the angle AOB v. Let Z be the true zenith distance of B at A, viz., ZAB, and Z' that of A at B or Z'BA. The dotted curve shows the path of the ray of light. A' and B' are the apparent positions of the sta- tions. P'g- 46. / The co-efficient of refraction is the ratio of the differ- U ence between the observed and real zenith distance at Ml 1 07 yards. Term. =34387 =95244 1 = 29631 = O'000OOO9 REFRAC- 2ENITH differ- nce at Trigonometrical Levelling. 181 either station to the angle v. Thus, if A is the co-effici.^ and z 2 the observed zenith distances, we have k equal to L — Z Z' — zi But these are not always the same. -or V V In the triangle AOB we have 2 tan —tan ^ (Z— Z)= h' + 2 r+h These equations give 2 and Z. If we substitute for tan^ the first two terms of its ex- pansion in w«^s; the second equation may be put in the form h'—h=s tan i (Z'—Z) 2 r — [ izr*) where s is the length of AB projected on the sea level. The co-efficient of refraction may also be obtained from the si.nultaneous reciprocally-observed zenith distances ot A and B without knowing their heights. Thus : Z=z + k V, and Z'—.z' + k v •: e+z'+2 k i'^i8o°-f y or 1-2 k='±'-J^^l V The mean co-efficient is .0771. For rays crossing the sea It 13 .o8og, and for rays not crossing it .0750. The amount of terrestrial refraction is verv variable and not to be expressed by any single law. In flat, hot countries where the rays of light have to pass near the ground and through masses of atmosphere of different densities the irregularity of the refraction is very great- so much so that the path of the rays is sometimes <-onvex to the surface of the earth instead of concave In Great Britain the refraction is, as a rule, greatest in the early mornings ; towards the middle of the day it de- creases and remains nearly constant for some hours, in- creasing again towards evening. : CHAPTER VII. THE USE OF THE PENDULUM IN DETERMININO THE COMPIiEtiSION OF THE EARTH. The spheroidal form of the earth causes the force of gravity to increase from the equator towards the poles, and this force may be measured at any place by means of the oscillations of a pendulum. If we had a heavy particle suspended from a fixed point by a fine inextensible thread without weight we should have what is called a simple pendulum. If this pendulum were allowed to make small oscillations (of not more than a degree in amplitude) in vacuo, and in a ver- tical plane, the time of oscillation would be given by the formula Iff ) Where / is tho number of seconds, / the length of the pendulum in feet, and g the force of gravity. Therefore, taking^ as constant, if there were another pendulum /' feet long and vibrating in t' seconds, we should have t:t".:\/l: 4//' or, if the time were constant and g changed to g', Pendulum Observations. or HI were constant and ^ variable ft'" -. ' If « and n are the number of osci i^mti, then, »': M::^.^' From (2) we have, ir'=-^/r _n2 „8^ 183 nations in tfee time y^ (3) rnelLremluhetlhX"^' T'/ ^'T^ *'^^^^^^^^" ^^ tain number o os^ £^^^ P'"'"'""! '^^' ^^^^^^ ^er- what ar/called "c;rou^d>f^° H 7' """"'^ ''^'^'' "= sible .0 calculate theleZh I ^ """f' """ " '^ P"'" would oscillate in .he same it a-".'lT ""'"'"T "■=" by finding the position ^0^ "cLt ! 0?™''°^''- ™'' wou',^.h:;:::d'^rr 7? ^^^^^—^^ and if a pent Z t '''r?" "'= '"'"^''angeable, '--eco^™r.trri::r:z,^'----e is "er-r„:r.^t^,rdr^^^^^^^^^^^^^ -j ^ - have, by the formula ^n^olt ctva^t^T^rm " "' S~g [i+(|w-0sin2yj and, since ;^' "' n" fi. if « is the number of oscillations in i ^ I i 184 Pendulum Observations. a given time at the equator and »' the numbert at the station. n r 'a—ns [iMi '"— ^) sin" ] (5) I being the length of the pendulum at the equator, m being known, and » n', or / /', being found by experiment, we at once get the value of c from equation (4) or (5). Borda's pendulum, which was used by the French astronomers to find the length of the second's pendulum (that is, a pendulum oscillating in a single second) at dif- ferent stations, consisted of a sphere of platinum sus- pended by a fine wire, attached to the upper end of which was a knife edge of steel resting on a level agate plane. The length of the simple pendulum corresponding to Borda's was obtained by measurement and calculation. In 1818 Captain Kater determined the length of the seconds pendulum in London (39-13929 inches) by means of a pendulum which had two knife edges facing each other— one for the centre of suspension, the other at the centre of oscillation— so that, provided the two knife edges were at the correct distance apart, they could be used indifferently as points of suspension ; the pendulum being, of course, inverted in the two positions. The pendulum was made to swing equally from either point of suspension by adjusting a sliding weight. The distance between the two edges gave the length of the simple pen- dulum. The advantage of such a pendulum is that it contains two in one, and that any injury to the instrument is de- tected by its giving different results when swung in the two positions. This pendulum was afterwards super- seded by another of similar principle, in which, instead of }er» at the (4) dulums in- ave (5) qaator. m ixperiment, ) or (5). he French 5 pendulum ;oiid) at dif- itinum sus- per end of level agate rresponding calculation, igth of the s) by means facing each 5ther at the ! two knife 2y could be e pendulum tions. The her point of 'he distance simple pen- it contains ment is de- ATung in the ards super- I, instead of Pendulum Observations. 185 using a "^liding weight, one end of the bar of which it consisted was filed away until the vibrations in the two positions were synchronous. In using the pendulum it is swung in front of the pendulum of an astronomical clock, the exact rate of which is known. By means of certain contrivances the number of vibrations made by the two pendulums in a given time can be compared exactly, and the number made by the clock being known that of the experimental pendulum is obtained. Certain corrections have to be applied. One for changes in the thermometer, which lengthen or shorten the pendulum : a second for changes in barometric pressure, which by altering the floatation effect of the atmosphere on the instrument, affect the action of gravity on it ; a third for height of station above the sea level, which also affects the force of gravity, the latter diminishing with the square of the distance from the centre of the earth ; and a fourth for the amplitude of the arc through which the pendulum swings, which, in theory, should be indefinitely small. The number of pendulum oscillations in a given time has been observed at a vast number of stations in various parts of the world, and in latitudes from the equator to nearly 80". The most extensive series of observations was one lately brought to a close in India, the pendulums used in which had been previor; ;, tested at Kew. The general results of all the pen iiilum experiments gives about 292 : 293 as the ratio of the earth's axes, which is the same as that deduced from measurements of meri- dianal arcs. L L I