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Tous les autres exemplaires originaux sont film^s en commenpant par la premidre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernidre image de chaque mici Dfiche, selon le cas: le symbole —► signif ie "A SUIVRE", le symbols V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent 6tre filmds d des taux de reduction diff^rents. Lorsque le document est trop grand pour dtre reproduit en un seul cliche, 11 est film6 d partir de Tangle sup6rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images n6cessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 1 2 3 4 5 6 la^sv 1 c Tsssrr fsx /^/ Practical Astronomy FOUR n SURVEY PURPOSES n I'oK 11 Hi rsK u(- THK — CADETS AT THI- ROYAL MIUrAKY COLLEGE, I^I3MC3^ST03Sr, 0-A.]Sr-A.I5>V ,^^- ^ Compiled by MAJOR C. B. MAYNE, R.E., Professor of Military To f'u<:ni /•/!)>, Siinryiiig and Astronomy. () T T A \V A : Printed by Brown Chambeki.in. Qukkn's I'rintkr and Cijntroller of Static. :ery. 1890. 43 45 1 tot are inl \% I the hin are the Ed Th gn PREFACE. These papers form the basiB of the lectureB on Astronomy ^iren to the Cadets at the Boyal Military College, Kingston, ( Canada. They are parposely unaccompanied by diagrams, which have to be billed in by the Cadets themtrelves. The introductory chapter, giving a general view of the HeavAns, is necessary to enable he beginner to understand the mechanism of the visible Heavens, whilo at the same time it cannot fail to impress him with wonder and awe at the Infinity aud Power of the Almighty Creator as displayed in His works. The authorities, on which the following notes are based, are the works of Chauvent, Loomis, and Godfray, and also the astionomy course of the School of Military Engineering, England. Those notes do not in any way pretend to be complete. They are accompanied by explanatory lectures illustrated by dia- grams. They are only intended to tave time by doing away with the necessity of many written notes and for purposes of reference. R ROYAL MILITARY COLLEGE, KINGSTON. Georrrapliical Latituac, 44° Li' 50" N. Colalitmlc, 45° 4(1' 10" Loii<>-iUule, 5" 5'" 50' W. or 5-Oi)T2 hours W. Gooc'cMitric latitiulc == 44° 2' 20" N. Anj?lc of the vertical =--^ 0^^ 11' 2!>"-80. Co)is((intti for li. M. CoJleje. Corrc'i'tiou for S. T. at M. N. -}- 50-2()5S soc! " M. T. at S. N. 50-l.'50() sees. \o\x sill lat. =: 9 •8435737 = log cos c-< lat. log cos lat. — - 9'H552:5i>7 = log sin i-olat. log tan lat. = 0-0883340 = log cot. colat. log cot lat. = ]0-01UU)()0 = log tan colat. log sec lat. = 10-1447(>03 = log cosec colat. kx' cosec lat. = 10-15042()3 == log sec colat. sin 1" = -0000048481 los; sin 1' = (•)-()855740 Grcilc Alp/iabef. Ali>lia. . . .a Iota t Keta ,3" Gainnia. . .y Delta.... ^ <^ ZC'ta. on. Eta Theta ....(> Kaj.pa . Lambda Mu. Nu. Xi . Omicron . . " Pi - Kl lO . Sigma . . . . - rx s- Tau r Upsll Phi Chi on. Psi. . i\ , T> i + when above Pole, rolardistanee, >s r Dor ±: P ■< ' Dei'lliiation rh o North Hour angle ± t or Altitude II. A. - ]^ — " under when west of Meridian. " east " . ( -{- wiien S. of Zenith. "'{ - " X. V -ii w i , \ -\- when S. of Zenith. Zenith Distance ±: ;; - ' 44 v a A/iinuth ...... Parallaetif angle ..... llight asc'i'iision ..... First point of Aries ..... Equatorial horizontal ])arallax IIoriz(»ntal parallax for latitude of observer Parallax in altitude ..... Local sidereal tinie .... " apparent time ..... moan time ..... sidereal noon ..... apparent noon .... mean noon ..... Greenwich sidereal time apparent time .... mean time .... sidereal noon .... apparent noon mean noon .... Equation of time ..... Nauti<'al Almanac ..... Chronometer or clock .... Clock mean time ..... Clock sidereal time .... Prime vertical Radius of earth at equator «« " " " latitude <- ... '♦ " curvature of meridian at latitude

mall moons, instead of mere points of light, as in the case of the fixed stars, and may bo noticed also to pass through phases like the moon, especially in the ease of t le two that are inside the earth's oi bit. The variability of their br ghtness is caused partly by this, partly by change in their distances from the earth. The planets and their Halellitcs are practically opaque non- luminous bodies and can only be seen either when the light of the Bun falling on them is reflected to the observer, or when they come between the observer and the sun. When a planet is furthest from the sun it is said to bo in aphelion and in perihelion yrhen it is nearest to the sun The line joining these two points of a planet's orbit is the major axih of the orbit and is known as the apse line or tine of apsides. Similar points with reference to the earth are known as the apogee and porigeee points. th th All tho planets revolve in one direction round the sun, vis., oon« trary to the motion of the hands of a watch when viewed from the north side of the plane of the earth's orbit, or ecliptic ; and all the planets rotate on their own axes in the same direction ; and the satellites revolve round their planets in the same manner, except those of Uranus and Neptune which move in the opposite direction. The orbits of the planets are not in the same plane, but those of the principal planots lie in planes inclined not more than 7^ to the plane of the earih'n orbit on cither side of it. The nodes of a heavoily body arc the points of the orbit where the latter intersects \.be eclipiio. They are distinguished as the ascending and descending nodes according as the body is passing to the north or south side of tho ecliptic, respectively. Each planet revolves on an axis, which axis is inclined to the plane of the orbit and remains for a considerable time practically parallel to itself during its motion through space, though there is really a small continuous change. Two heavenly bodies are said to be m conjunction when they have tho same celestial lori»^fitude (see dotinitions, p. 17), and in opposition when their celestial longitudes differ by 180^. An inferior planet is in inferior co'^junction when it comes between the earth and the sun, and in superior conjunction when the sun is between it and earth . The apse lines of the orbits of the various planets do not lie in the same direction,but are continually altering their directions in space. The earth's apeo lino has an annual progressive movement of 11.78" i e. in the direction of the earth's movement. The general laws (f planetary motion already given are on?y ap- proximately true, and are tho resultant of Newton's laws of motion, and of his law of universal gravitation. The former are as follows : — 1. A body once set in motion and acted on by no force will move forward in a straight lino and with uniform velocity for ever. 2. If a moving body bo acted on by any force, its deviation from the motion stated in the fir^o law will be in the direction of the action of the force and pioportional to it. 3. Action and reaction are equal and opposite. And of the law of universal gravitation is as follows :— Every particle of matter in the universe attracts every other par- ticle with a force directly as the product of their masses and in- vorfely as the square of the distarce between them ; hence all the planets and satellites attract each other, and consequently their orbits and their planes are always slightly shifting. Tho direction of the axes of the planets, and their inclinations to their respective orbits, the directions of tho apse lines of these orbits, their ecceutrioties, and other data relative to them, have all either continuous or periodic secular variations owing to the attractions that the sun, the planets, and their satollites exercise one on another. Take %he earth for instance ; the centrifugal force of the earth's rotation on it8 own axis is greatest at the equator and nil at the poles, and hence the centrepetal force, due to attraction, is greatest at the poles, which are consequently drawn in more towards the centre of tho earth than is the equator ; and for this reason the polar diameter (7899.1 miles) is less than the equatorial diameter (792&.6 miles). Thus the figure of the earth is that of an oblate Hpheroid, and the attraction of the sun and moon on that portion of the earth's mass, which, owing to its spheroidal shape bugles out beyond the insciibed sphere, causes tho axis of the earth to move in space and give rice to the phenommou known as the preeestion of the equinoxes. STATISTICS OF THE SOLAR SYSTEM, TH£ EABTH. The Earth is an oblate spheroid. Greatest or equatorial diameter 7925*6 miles. Smallest or polar or axial diameter 7BM9'1 mile. Mean distance from Sun 91,4 0,000,000 miles. Surface 197,309,000 square miles. Volume 26u. 6 13,000,000 cubic miles. Specific Gravity (Water = 1) 5 b7. Weight 6,069,000,000,000,000,000,000 tons. Accelerating force of gravity 32.16 feet per second at the surface of the earth at the eqnator. Orbital velocity 18*2 miles a second. Equatorial velocity of rotation 0-29 miles a second. Sidereal revolution 365-256 mean days Axial rotation 23 hrs. 56 min. 4 s. mean time. tnclination of axis to elliptic 66°.32\ Solar equatorial horizontal parallax at the earth's mean distance 8" -849. TBS MOON, The Moon is a spherical globe with a diameter of 2160 m.les. Mearj distance from earth 238,833 miles. THJE SUN. Diameter 852,584 mileei. That is to i^ay, if the earth was at the centre of the sun, the moon would reach a little more than half-way out to its circum OTcnci'. The following table uhows the enormous size and weight of the sun as compared with tho-e of the whole of the other planets put tog'^lher and bow woii it is adapted to form tho centre of the solur hiystom. In the coturano in which tho uarih's data are thken as unity, the absolute data for the tun and planets are found from the lelalive ones by multiplying these latter by the absolute duta given above for the earth. ft Table I. 05 05 00 1 r- 05 00 i; I a ^3 g -I) O :^ H CO ■aoissujduioa Jwioj »• ^ ■> n >. u it 'K IN CI e- fr. ^• •jo^'Biiba t4 ^^ ? h. « &. g 3 35 s? ^4 o ■jB aon«'»iu JO .CiiooiOA ■ CJ II ~ - ■"■ ~ S s o o 1 -2 ^ CI ^^ oc or CO CI CI X, 1 ^ pi4 ce ■*: » ■:£> o o •pojjort IBIJOpiW ! »■ 1 a II 1 s; t-H r^ 1-H ?S CO o ^ t* 53 S J; * CI ^ UOt)Oni 0|J)U33O||9I1 '([Itid 1 To i s CO ^ CI ••* o 2 a o^ Is Sj ^ o % '6 a •H -^ fr o «-^ ■»>< CI 15 QO CO •.fiToopA |tnin»o t^ « » II 1 to t* ^H »-t ^H *"■ o o o c o o ^ s ro r^ s o CI av CO 35 U5 •«j —4 T ar CO ■nt o r- I'- 1—4 •iC^t.VBja JOODJO,.! II ?5 -3 CI --^ o "o 3 £ 2 g ?^ g C-l =? 2? s S2 s •a.i'ii'Bni '^ ? o o — o 9 $ CJ CO o JO ^)naBnb JO sbbk -a II CO S^ V M CO ec h- CO CO or g CO r* ■«f« o irl «c 05 CI cc U5 ^ II •-H r* \r Ifl ri _ 3 O =1 CO •^JIAtUS aijpads .lo A'^suaQ ■S rH a r, g g 9a ?3 23 OC t- « II O -■ o »-H o o c o « c 93 J? '5 S s S CO a ?l s 8 •oiuiiioA II o c w^ — g J? CI ^ o H ?. A 1^ »-4 B 8 §s ^ go' s§ ^ o •oaiijjns 1 II Is O O r^ o CI -*. "^ CI o ^ 1^ •M CO »^ Hi o ce J; •qiiBO uio.ir cc 1 ^ joio(ini||> wu Jill ^iuuuddv ?=' 00 i? 1 1~ ? - CO CI ■^ _^ rS M s s s ira 5? I- CI I-* •B.M13llIB!fI c3 i| t O o o — f ■11 H — ^ •sixw jofuni-iiiio-* .!() mis - — 1 K 1- = ? O — 1-H 115 » cs g •[oqiiiAs i>, • ; ; , « : Si; a 5 i a ID > 5 09 c3 O a 3 03 «1 s g 13 a a i 13 11 ,1; 'I Table h-^Concluded. s « t - 3 « u ./> rt.ii S 5 '^ ><2 5, 9) 00 CO -3 00 H is M n H h C K H S H •qooda jts apn)i3aoi usaK 1 2 -J. .• .• .- fc «• ?j »• ie .• o" o* o* o* o* >>' > o' o' W CO O 01 -H « T- 11} « CO. OS ^* 52 Si- 00 T I % 9 ^ 'JS X I I I + o o « 2 o V 4-1 "S c o £ •janoniv 59 cl o =• o o V e i< d 1: o a; 5 'JO CO ii I I I I •jnnouiv S « I'J 3! ^ •^uiiouiv + 1 + + + 1 1 '2 t 1- M S 5 s s c5 = yi s 1 = 047 0.009 c •ondnoa JO ju«ldo;s|XBjo ao,jBU|iour 'UOHBIOJ iBixn 10 9uin aiieiv o 2 IS s ts O C9 g^"?c3 S5 ^ gS S in ;=^g ■M l75 IS s E5 "c'^-a^^ ?1 ?1 S -■ ^ °5 M ' -3 •aS ^ u a a S s * 02 S if s a eg Q. a 3 fl 2 4> a s iZi a o o 91 .a b O « a S o. a; a c o ■3 e s K e -+ i « fl o .1 eg F*rom the table we Beo that the apparent mean diametefd of the Bua and moon are nearly the same This is dne to the fact that the ratios of their diameters to their mean distances from the earth are about the same. But tho lar^e eccentricity of the moon's orbit make'' its appaient ciiamr .vary between 29'.2l" at apogee and 'S'6\'iV' at perigee, while the small eccentricity of the earth's orbit cuuHus the earth's distance from tho sun to vary so slightly that the gun's apparent diameter varies only between 3r.3i" at aphelion and 32'.33" at perihelion. I'll .'.I ■ STAR DISTANCES. X a x> s ■+ a o > Tho cl.btan;tB of tho tixfJ tlaio Iioiu tho viiilit and fiurn otich other are so great as to he silmot^t beyond human conception. It wjtH for long believed that they could not be measured. It was, however, eventually iound that in the case of a fow of them, by tHking tho iipse line of tho earth's orbit as a base (i 83,000,000 miles) and tho star ns tho apex of a very actlto-angied triangle, the diiferenee of the angles adjacent to the b< ko could be measured and the acute angle thus detei mined. The length of the base being known the star's distance can then be approximately calculated. To give an idea how far off the nearest star is, it may be mentioned that a ray of light, travoUiLg at the rate of 18c),600 miles a second, would pass round tho earth (about 24,900 miles) in leds than one eeve^'th o/ a second ; it takes 8^ minutes to traverse the 91^ raiilionH of miles from thf^ sun to the earth ; and 3^ years to reach us irom the nearest etar. The following are some of the nearest stars and their approxim- ate distances, in terms of the number of years it takes their light to reach us travelling at a rate of 1S6,600 miles a second. a Centauri 3316 years. 61-Cygni 6.029 •• aLyroB 12.618 " Sirius 21.'i06 " Arcturus 25.636 " Polaris , 30.720 " Tho nearest star is more than 206,265 times the mean distance of the sun from the earth . Thus tho distance of the fixed stars is so immense, that in calou- lutioDs based on observations of these stars we may take the whole orbit of the earth as a dot in space ; the radius of the celestial Hphore as infinity ; and the centre of this sphere as the whole orbit of ^he earth or any part of it. Thus the dimensions of this orbit and of the sun and earth completely disappear in calculations with the stars, and the centre of the sun, or of the earth, or any other point of or inside the earth's orbit may be taken without error as %\ IP?" 8 the centre of the celestial sphere at the same time, and all lines drawn from any number of points on or within the earth's orbit to the same star can be considered as the name line. Our Run is only one of the stars, and the latter, though called " fixed," are in reality all moving according to the laws of dyna- mics. What these motions are we cannot tell, as we do not yet know tbe manrorin which the masses are distributed through space, [t hm, however, beon ascertained, not only tbut they are slowly changing thoir position with regaid to each other, but that in one part ol the heavens they a e getting farther apart, thui indicaiing that the moLion of our sun with his attendant planets is in the direction of that part, which motion is further corroborated by the fao.t that the btars in the i pposito part are getting closer to- gether. It may be inforre 1 that the stars are, as a rule, the centres ot plunetaiy hyotoms liise our own, and that posaibly some systems have planets in a stuto of development permitting of their habita- tion by living creatures. THB SUN AND JICN's REAL AND APPARENT MOTION AMONO THE STABS. Supposing us to be situated in the northern hemisphere, and not too far north, if we watch the apparent motions of the heavenly bodies in the sky wo shall notice the following facts. The sun rises latest and sets earliest about the 21st of December, while the opposite is the case about the 21st of June. During the winter half of the year his rising ana setting is south of the eaut and west points of the horizon, and during the summer half they are north of it; while at two intermediate periods, known as the equinoxes, he ri^es due east, remains in sight for 12 hours, and sets due west. At midwinter the arc he desc jibes through the sky is the lowest, and at midsummer the highest. When the moon is lirsl seen as a young moon sho is a little to the east of th( sun. She rapidly moves through the sky day after day towards the east, so that about full moon shu rises as the sun sets, and later on is seen as a orescent rising before the sun in the early morning. The intorval between the two moons is about 29^ days, and she rises each day on an average about 49 minutes later than the day before. It may be noticed that in summer ih? full moon describes a low aio in the sky and in winter a high one, thus giving the most moonlight when the days are shortest and tbe nights longest. The stars, it carefully observed, will be noiioed to rise each nignt a little less than four minutes earlier than they did the nit^ht before, so that at any given hour a certain portion of the sky which was visible at the same hour the night before will have disappeared in the west, and a similar portion will have come into view in the eaHt. In laot the whole mas^) of the stars appears to be overtaking the Buu (or rather the san to bo moving through the stars); and, aa a oonscquonoe, if the Htars were visible lo the day time this motion ouuUi bo plainly hoou. Tho points of rising and setting of the atari arc always the same. Cbo sun and all tho lura reach iheir greatest height ii) liie t:ky— or culroinalo, hh it is termed -at a point where they are due oorLh or 60Jth of tho Rpcctntor. The path of the sun* lies in the plane of the earth's orbit, i.e., in the ecliptic, and it makes a complete circuit with reference to a fixed star in about 3t)5j^ days, and therefore it appears to move at a mean rate of j||°jTr or of 59' 8" -33 a day. Tho puth ot the moon lies in a plane inclined at an angle of 6° &' 40" to the plane of the earth's orbit ie., to v>bc ecliptic, and it mukes a complete circuit with respect to a fixed star in 27d. 7h. 43m., and thus moves in its orbit at a mean rate of 13*^ 10' 3&" a day, or .>2' 56" (t.e. about its own diameter) per hour. Supposing lor oonvonionce th:it tho sun is moving round the earth, then the real motion of both the sun and the mojn among the start?) it< from west to oast, bui their appai'ent motion is from oast to west, on account of tho oarlh's diurnal rotation from west to east being more rapid thuu their own motions in tho same direction. As^tho sun and moon appear to move in the same direction (W. to E.) among tho stars, the moon will after inferior conjunction, from its greater rate of movement, separate from tho sun at a rate of about la** 10' 35"— 0" 69' «", or 12° 11 27" a day to tho eastward. Thus, if tho moon and sun aro in the same line with the earth on any given day, the moon will bo 12^ l\j^' e.stof tho sun at the same time next day ; 24® 23' oast of tho sun at tho same time two days after, and so on. The period of revolution of the moon with respect to a fixed star is called a sidereal ot periodical month, and its length is 27d. 7h. 43m. ll>46ib. BuL during this time tho sun has advanced among the stars about 27'^ in tho same direction »s too moon's motion, and it takes the laoon about 2d, 5h. to uutch it up and hence tho length oi a lunar month or lunation or synod>.cai poriod (as the period of revolu- tion with respect to tho sun is called) in 29d. 12h. 44m. 2.87s. The commencemeui of each mouth is when the sun and moon are in the same plane at right angles to tho ecliptic and passing through the oeutte of tho tarth. Tho moon rotates ou ltd axis from wost to ou^^t but once in the time that it takes to make a revolution of its orbits lound the earth, and consequently the same portion (or vory nearly so) of its sur- face is always prcsonttd to tho eurih. ;i • It is often convenient to suppose the sun to move round the earth which is then supposed to be stationary though spinning on its axis. 10 The line of inlerscctioD botwoen the pianos of tha earth's and moon's orbits, i. e iho line of nodes, is found to be retrograding along the eolipUc (or plane ot the earth's orbit) at an average rate e line has a progressive movement ot about 39. "45' 40" a year. All the disturbing causes which effeot the mioon's raoiion aire not fully known, and hence the data given in the 2fautical Almanao for the moon are approximate only, but they are sufficiently accurate or navigation purposes and reconnaissance surveys. SCLIPSES, OCOULTATION AND TRANSITS. When the moon hides a star horn our view, the star is said to be occulted, but when the moun comes between us and the sun, the sun is said to bo eclipsed. Also wiie:i the earth comes between the sun and moon, the moou is hUid to bo eclipsed; and when a planet comes between as and the ^un, it is said to transit across the sun. A solar eclipse may bo partial, total, or annular; a solar eclipse can only occur at or near a conjunction and a lunar eclipse at or near an opposition. PHASSS OF THE MOON . (Fig 2.) The moon is an opaque body nud tho light we receive from it is reflected from the sun. When ihe moon is between the earth and the sun we cannot see it, as the bright or lit up side is away irota as, unless it is actually in the plane of the ecliptic when it crosses the sun's iace and appears as a black dies and causes an ociipse. The moon is then new, 24 hours after it will be 12^ 1 V east of the sun and were it not for the overpowering brightness of the sun, i t would appear as a thin crescent at sunset at a small altiiuUo. 24 hours after this ugtxin it will be 24" 2b' oast of tho sun and at a higher altitude, and a broader orescent can only be seen, and so ou. But tho crescent can only be seen when tho moon is some 30^ to 40*^ from the sun, i.e. when it is between 2 and 3 days old. After 7 days the moon is nearly 90' from the sun, /. e. near the meridian at sunset and will appear as a half moon. This is the first quarter. Alter about 15 days, it will be 180" from the sun and will be seen as a juU moon which rises as the sun sets, and which is visible the «Qiirti night. 11 After abont 22 days the distanoe from the snn measured eastwards ia about 270°, and only half the moon is ngain 8eon,formini; the last quarter. This half moon doew not reach tho meridian until nearly gnn rise, and is only vimible durin*; the laHt H tiours of the night. From the last quarter i>i *ho next full moon, the moon a^jpoars again as a oroso-nt (dimini^hins^ daily In breadth) and U only Been in the morninsf shortly boforo sun rise. During tho fir^t hiili of the month tho moon is less than 180** east of the sun, and during this period tho western ed>»e or limb (denot- ed by ( I. L.) is the lit up or bright imb. In tho 2nd half of the month, the eastern limb (deaotcd by ( II. L.) bacomea the bright limb. PHASES OF THE PLANETS. Like the moon the planets are opaque bodies and only shine by tho reflected light of the san and thoy will tlioroforo present phases analogous to those of tho m' on. With two the inferior pl:incts Mercury and Venus, the phases will range from a full ronnl di'^o to a thin orescent which vanishes like that of tho moon to rouppear again. The full discs, however, are never ween owing to the Inii^htnoss of the sun. In the case of tho sapor or planets which never come between the eanh and the sun, more than half the disc will always be illu- minated. THE SEASONS. (Fig 3). The earth describes an elliptic orbit round the snn in about C65]^ days. It also revolves on its own axis in about a day. This axis remains practically parallel to itself for a considerable period of time, and is inclined to the plane of the orbit at an angle of about 66' 32'. Hence the phenomena of the seasons, and of the varying positions of the sun from day to day. Figure 3 shows the position of the earth with reference to the sun at the different soasonw. N is the north pole, S tlie south polo, and A a point in tho northern hemisphere. The left hand sphere shows the earth's position when it is midwinter at A, and the r'ght hand sphere when it is midsummer. The sun occupies the focus of tho earth's orbit nearest to the peri- helion point. The eccentricity of the earth's orbit is so small that it does not aflfect the temperature on tho earth's surface nearly so much as the inclination of the equator to the ecliptic. It is the obli- quity of the sun's rays and tho incnased lonath of their passage through tho atmosphore that principally cause the decrease of tem- perature towards the poles. I 12 It therefore repalts that on aooount of the present inclination of the equator to the eoliptio of 23^ 28', it is winter in the northern hemisphere when we are nearest the nun, and the sun'ti altitade is 46° fi6' loss than in summer. The reverse is tho case for tho sonthorn hemisphere, but the hot- ter summer is some 8 day^ nhor^or. It is summer in the southern hemisphere when it is winter in tho oorthern hemisphere and vice verta. In A. D. 1800 the «>ccontricity of tho earth'rt orbit was 0*0168 and is at present slowly decreasing. If this change went on continu- ously, the earth's orbit would bocomo circular and all tho Heasond would be of the same length. But it ran bo proved from tlio law of gravitation that this eccentricity only decreases for 24,000 years to a minimm value of 0'0033, after which it increa(rm of tho earth's orbit changes, yet the apse line always remains tho same length, and hence an increaf^e or decrease of eccentricity can only be effected by the perihelion distance being diminished or inoi eas- ed respeotivoly*. The glacial periods are ascribed by some authori- ties to these times of greatest eccentricity. • See Green's Geology, p. 532. tt CH:i?TER II. SPHERICAL COORDINATES. II a ^rc&t circle of a sphertj bo fixed in position and be aflsam d m a primitive circle of reference ho t 'oat all point-^ on the sphere can be loforred to this circle by a systora ot secondaries or ^roat drcUs perpnndiciilur lo the primitive, and which conseqaenly pass fh'-ou^h itH jj'les, then the ponitionof apoint on thesurface can be expressed by I'vo spherical co crdinates, viz. : — (a) Tho dirttanco of the point from the primitive circle measured on a H'U'Ohdary . (/>) The dirttan'^e int»;rc.ptod on the primi'ive between this Hcooiid ry circle and some given point on the primitive aH-;nmed as the origin of the co-ordinntes. Hefiro pn.«fling to the cole.'^tial nphero we will first cons d-^r hoT ihoHO i*phen. nl co-oidinates are applied to fix points on the oavih'n Huriaue, the earth lein^ suppi>HO I spherical. The polar axis of the earth in mat aiameteriound which the earth porformH its diurniil icvohition-i. and its extremities are cxliud re- spectively tliH north and 8'>uth po'.ts. The equator is the great circle on the earth's surface, who o piano pabHe8 tiwouj^b the centre of tho earth perpeuditjularly i> liie uxitj. It U a definitely fixed primitive circle of reference, aiil tveiy poirt on it is 90^ from either pole. The secondary circle* 0/ reference aro called meridians. Thoy are grout circles jfasftiv.g tlirou^h tho poles, and whose plants aw. conse- quently peipendiculai- to iho plane of the equator. Evojy p'uco on tho earth's turtaco has a meridian passing through it, anii ovcry mi'ridian is divided into tvvo nalves by the polar axis Then the pphorical co ordinates are : — (a) The terrestial latitude ot a place on the earth's suiia 'o is its angular distance from the equator raeiisured on its lue.idai , i.orth or south, up to 9i»°. (6) The terrestial lohgitude of a place on the eavth't* ^mf-xcd is measured by th' are of the equatoi' inteicepted bet\ve>.'!; vho nicii- diau of the place and some other arbitrarily fixed moi idian choheu as origif, such as that of (rreenwich, from which all lorr^itudcs are measured east and west up to 180°. Coming now to the celestial sphere, * the celestial axis, po'es equator, and meridians are finip'j those of the earth produced to meet tho celcHtial sph' re • Note — The celestial sphere is the projection of the heavenly bodies on a suppositious sphere of arbitrary radius, and whose centre is the observer's eye, or the axis of revolution of the instrument. ^\ ifiii 14 ( •■ Primitive, Azimut.h. Hour a ogle. R'iijht ascension. Celestial lo;i;j;ilU'ie. In dealing with the celeBtial sphere we can obtain four systems of spherical co ordinates, as follows : — t^.eaondariea, 1. Altitude, and 2. DjeUnation, and 3. Declination, and 4. (Jolostia,! latitude, tmd The ^ocon 1 syatoai corresponds moat nearly with lerreetrial lati- tu ie and loi gitude. FIRST STSTKM OP OO-ORDINATFS : ALTITUDE AND AZIMUTH. (fIG 4). The sevs'ib'.e horizon at any point on tho earth's surfaoc is tlie plane, tangetitial 1o -he earth's surface at t u»t point. It is independent of tho ht'ight of the observor'a eye /.*'., the observor's eye is suppo-od to be on i lie c^urth'^ gnrfrt^ie. Whon the ob-ior/ei-'.^ . yo is aboro the earth's fliirfica, the iHsfhle horizon who) looking over the eoa U wn'i to dip. ard tl'.e anglo bntwcen the horizontal line ])a8sing Inrngh th'' obnoi'var'n eyo .and tho line j).i-(hiMg to tho observed oi* visible ^ica-hofiz in iso, >llol the dtp f,f the horizon. (Sao p. .) Ihe ratioTi'il horizon is the gre;xt oirc'o who've piano pa*'^(?8 throiii^h tho centre of i\w earth parallel to the son.sible horiz'.Mi, Obs.^" vntinns Rro mudo on iho ejirth's surface with tho sonsibli h iriz m and are aiierw»rds reduced by a 8im))!e correction fcr parallax (>eti p. ) to wh it they would have baon, hid they beun taken at tho cont.ro of the earth With the raHonal horizon. In observations of stars, wb'.se d'stauoos re irtTiniie, the sensible pnd rational horizons mcy be taken as tl e mme. The ration d b u'izii' is Ine priruitivo circle of th:-i systeoi of co- ordinates. The vertical line (ZON) at any place is tho porpi ulicnla" to the pi 'in e of the horizon at ihatp>int; this lino proiuc.ad -.noot^ tae celestial sy h -ro fiboVi) ihe horiznn at the zenith (Z i and bilow it at the nidir (N). Tho t^ecoD Jaries to tho rational hoviz)naro vertical circlet lying in Vvrtioil p'anes passing through tho vertical lino !;0"peL'dicu'ar to the horiz)r:. I'he colesual meii'li.tn ^»as^e8 through the zenith and nadir, and ie tho> efora one of the-io vertica' circles. 1 hi? intcrsociion of the plane of t'-.o colostinl miU'jMi in with tho ! ine of tb'i horizon is willed tho north anit>uth (> e '.-r the. meridian i;"e (n 'J «), vnd the points in wh:ch this lino mo ;lh th > colchticil Bphe-e are c. !k«i the north and south pointH of tho hovizcin. The prtme Derf-cal i.j tho vortical circle pcrpeiid oular to the inei'i- diu», uiid "-' p!;ino intersoc.t>» that vi tlio h." to 9 i°. (b) The ftour ang e (t) ol the point is the arc of the tquator (are QD or angle QPD) intorcopiod botwcoti t lis hour oirclo and the meridian at the apper transit, and is reckoned eatiL and Wtist fiom the meridian up to 180° or 13 hours, allowing J 5** for eaob hour. At the upper transit of a star its hour angle in hrs. and at the lower transit 12 hrs The north polar distance (P) of a poi)!t on the celestial t phore is ita angular distance (PS) from the north pole. Henoe L^ = 90 ^ §, according as the point has a north or south declination reiipjctivoly. Tne parallactic angle (q) of a point in the (leleatial sphere in t^e angle at the point between the two great circles — one the hour cirulo, pasMJog through the point and the pole, and the other the vertical oiroio passing t' rough this point and the observer or his zenith. L'ue greutest elongation of a cirourapolar star is the greatest anglo in azimuth of the star from the meridian and is alwayu greater than the polar distance czoept when the observer in on the equator when the greatest elongu >n is equal to the polar distance. In this system we see thai the origin is dependent on the obeer- ver's position, i.e., on his meridian. THIRD BTSTBM OT OO-ORDINATBS ; DBOLINATION AND RiaQT A80EN8ION. (Fig. 5.) In this system the primitive circle is still the equator, and henoe the first co-ordinate or the declination, is tht same a^ before. The second co ordinatd is al^o meosuied on the equator, but from another origin which is not affeotod by tho observer's position. The point on the equator taken us origin >s the first point of the Aries. (Sym- bol Y. ) The earth's axis is inclinei abjut 88-'32' to the plane of the earth's orbit or the ecliptic and conboquently the plane of the equator is inclined about 2->*^ 23* to the plane of the ecliptic. Thit; ktter angle is called the obliquty of the ecliptic, &< d varies within small limits. The two plane-^ intersect in the ane of equinoxes, and the points in which it meets the oelesual sphere are culled ti e equinoctial points or the equinoxe&*. Uao of ihoso points is calici the fi''st point of Anet (Tf) and tha other the first point of Libra ( A- ). * From the days and nights being equal when the earth is at these points. If the earth's axis was not inclined to the eoliptio the days and nightn would always be equal. It The vernal equinox is the point at which the sun's oeotre appears to coincide with the firnt point of Aries, and at which the sun appears to ascend, about the 2l8t March, from the southern to the northern Bide of the equntor, that is, the vernal equinox is the first point of Libra. The autumnal equinox is the point at which the aun's centre ap- pears to coincide with the 1st point of Libra, and at which the sun appeals to descond, about the 2.jrd Sept., from the northern to the southern side of the equator, that is, the autumnal equinox is the first point of Arioe. The solstices ure the points of the earth's orbit 90 "^ from the eqninozcH, at which the sun attains its greatest northern and southern declination, amounting to about 23° 28'. The axis of tho earth desoribes a circle round tho polos, changing direction in spaoo about 50" 22 per tiiinum in a, direction contrary to the earth's motion in its orbit, which causes a corresponding retrograde change in the position of the equinoxes and solstices, known as the precess'on of the equinoxes. The apse Hiic of tho earth's orbit, on the oihor»hand, has a progressive motion of H".77 a year. TborC two motions cause the line of solstices and the apse line of tho earth's orbit to separate at a rate of about 62" a year. The co-ordinates of this system are as follows : — (a.) Declination (as before, see p. 16). (b. » The right ascension (R A.) of a point on the celestial sphere is the arc of the equator (arc T ^) intercepted between the two hour circles patrsing through the point and the Ist point of Aries respeo- tively, and is reckoned from the first point of Aries eastwards up to 360« or 24 hours. FOURTH SYSTEM OF C0-ORDINA.TES; 0BLE8TIAL LATITUDE AND CELESTIAL LONGITUDE. As we do not use these co-ordmutes in our calculation, we shall not touch on this system more than to say that the ecliptic is the pri- mitive from which the celestial latitudes are measured above and below it, and tho first point of Avios is the origin for tho celestial longitudes, which are mcasined eastwards from it up to 360°. As tho plane of the ecliptic is veiy nearly a fixed plane, the celestial latitudes of the different stars are praotically constant from year to year, while their longitudes alter about 50'22" to the west yearly on account of the precossii n of the equinoxes. The secondaries in this system are called circles of latitude. The right ascensions, declinations, celestial latitudes and celestial longitudes of all the principal heavenly bodies ure tubulated in the Nautical Almanac, which gives their values at particular instants of time for the Greenwich meridian, as if thoy were seen from the centre of the earth. 3 ■ %^ 18 ! ♦■ Since the data of the sun, moon, and planets are alw?»yB varying, and are only given in the Nautical Almnnao for certain instants of Greenwich time, they must be oorreoted for use at any other instant oftime(Seep. ). Let figure 7 represent the orthograpVio projoction of the celestial sphere (given in fig. 6) upon the plane of the horiRon. Now fitting together the first three systems of oo-ordiuaties we obtain the spherical triangle FZS, which is known as the aatronomical triangle, and nearly all problems of time, latitude and azimuths are simply the solving of this triangle by the aid of spherioal trigono- metry. nZisi* the meridian ; tcTi e, the prime vertical ; u> Q «, the equa- tor ; Z S H, the vertical circle drawn through a star S; P S D, the hour circle passing tl>rou mentally, P S is given by the Nautical Almanac, and hence the choice lies between one of the 3 quantities P Z, t, and A to fiid the other 2. But practically A is never known, ant henco tho choice lays between either P Z and t, being known to find tho other,, or to find A. But PZ is not as a rnle found directly by solving the triangle PZS. It is generally found indirectly by other moani* and can be easily obtained with suflScient accuracy for the solution of the astro- nomical triangle to find t or A. T^ 19 « TRANSFOEMATION Ol*' SPHERICAL COORDINATES. OIVBN THE ALTITtTDI (A) AND AZIMUTH (a) OF HBAVINLT BODT AND LATITQDB (v>) OF OBSERVER, TO FIND THE DECLINATION (P) AND HOUR ANQLE (t) OF THE BODT. That IB to transform the so-ordinates of the first sjstem into those of the second. From spherical trigonometry we obtain the following formaloe:— > COS a = ooB c cos 6 -\- sin c sin b cos A sin a 008 B = sin c cos b — cos c sin b cos A sin a sin B = sin b sin A In applying these to the astronomical triangle PZ9, let A = PZ9 = A a = PS = 90 — d* B=ZPS= < 6 = ZS = 90 — * = jf C = PSZ = } c = PZ = 90 - ^ Substitating these values in the above equations we get sin d* = sin ^ cos z -\- cos ^ sin z cos A cos d'cos t = oo6^ cos z — sin 4> sin z oos A cos ^ sin if = sin z sin A To adapt these to logarithmic computation Let m sin M = sin z oos A and m cos M = cos z Then the 3 equations can be written as follows: — sin (f = m sin (0 + M) cos (f cos t = m eos (^ -f ^) cos (j sin f = sin ^ sin A From which we get tan M = tan z cos A tan A sin M tan t = — rx-rnciT cos (9 -|- M) tan cf = tan (0 -f" ^) oo» ' When A is greater than 90^ from the north puint we must ase (<^ — M) for (4) + M). GIVEN THE DECLINATION Q) AND HOUR ANQLE (t) OF A HEAVENLT BODT AND THE LATITUDE OF THE OBSERYER () TO FIND THE ZENITH DI8TAN0B (z) AND AZIMUTH (A) OF THE BODT. That is to transform the coordinates of the second system into those of the first. 80 II n i.*: From whioh wo got tan M = taD z In applying the general Bpherioal equations given in the last prob- lem to the astronomical triangle, LetA = ZP3=< a = ZS = z B = PZS == A ft = PS = 90 — ^ C =« PSZ = q (j=PZ = 90 — y* Then we have COB « = sin y> sin ^ -\- cos y cos ^ co8 t Bin z COS A = COB y> sin ^ — sin f cos ^ cos t sin r sin A = oos ^ sin t To adapt these to logarithmic computation Let m sin M = sin ^ m 008 M = cos ^ sin t Then the 3 equations cati be written as foilowd : — cos 2 = m COB (M — y) sin 2 oos A = m sin (M — y) sin ^ sin A = cos ^ sin t cos < tan t cos M Bin (M. — y) tan (M — ^ y) cos A A minuB Bign for oos A or tan A indicates that the true asimath measured from the north point, = 180° — A. »0 FIND THE ZENITH DIHTANOK AND AZIMUTH OF A HIAVINLT BuDY WHEN ON THK U HOUR OiaOLE. That is when t = *i brs. = 90*^. Then putting cos t = o und sin t = 1 in the equations in last problem we get cos z = b\x\ *P sin ^ stn z COB A = cos y sin ^ sin * sin A = cos ^ or cot A = cos y tan ^ The same result and those of the two next problems, are more readily obtained from the forraulro lor right angled triangles. Taking Napier's rules for circular parts, the complements of thu two sides containing the right angle and the other side and angles being written down in the order they occur. Then from these we get cos any part = product of cots of adjacent ])art8. = proluct of sines of opposite parts Hence cos SZ = Bin (^90— PZ) sin (90 —PS) cos (90— PZ) = cot. (90— PS) cot A or oos z = sin y sin ^ cot A = cos y tan ^ 21 TO fIND THB HOUR ANGLK, AZIMUTH, AND ZKITITH DISTAMOK Of A OIVEN HEAVENLY BODT AT ITS OREATKST ELONOATIiN. Ill this case the parallaotic angle (q) at S is the right angle. Re- rranging the circulai parts and applying Napier's rules we get tan A COB < = , — ^ sin A = tan ^ COB ^ COS Z = COS ^ Hin f Bin 5 TO FIND THB HOUR ANGLK, ZENITH DIbTANOB, AND PARALL ACTIO ANGLE OF A GIVEN HEAVENLT BODT ON THE PRIMK VKRIIOAL OF A GIVEN PLAOl. In this case the angle A or PZ S is a right angle, and then by Napier's rales tan 5 COS t = tan 7 Bin ^ sin 9> OOB ^ Bin q ^ cos ^ where q is the parallactic angle PSZ. GIVEN THE ZENITH DISTANCE OF A KNOWN HBAVENLT BODT AT A GIVEN PLACE TO FIND THB HOUR ANGLE, AZIMUTH, AND PARALLACTIC ANGLE OF THB BODT. Id this caRC we ui^e the spherical formnla for solving a triarglo wh 80 three sides are known, namely Bin Y = \— (s — a) HID (5 — e) sin a sin c Whore s = half the sum of the sides = a + b -{■ c 'A It f-liould be noted that a and c aro the sides which form the angle B. To find the other anglen we have gin B nin A sin C sm h fin a sin c If wopi't A= rZS = A a = PS = 90^5 B = Z\S = t 6=ZS = z C = PS^ = q fl = PZ = 90 — V we can solve the problem. Bat this sabstitntion will be made hereafter when the problem is practically applied. ;i 22 GIVEN THE nO(JR AJJOLB AND THE DECLINATION OF A GIVEN HEAVENLY BODY, AND THE LATITUDE OB^ THE OBSERVER, TO FIND THE AZIMUTH AND ALTITUDE OP THE BODY. In this case we use thefipherical formula for solying a triangle of which two sides a and c and the included angle B are known, namelj : tan J (A + C) = '-2iil^=4 «oH B COS i (a 4- c) ^ i 1 • » f-t^ sin i (a ' c) . . ^ tan J (A — C) = ^— 1-^ — ; — ( cot * B '^ sin ^ (a -f- c) ^ From this we find A and C, and then to find b we huve . , f'in a sin B sin c sin B sin = = sin A sin C In the Canadian Dominion Land Survey, however, the following metnod is adopted in preference, when applied to observations of the polo star. We have . sin a sin B fiin A = sin 6 cos a = cos tor, the variation in the value of the equator of time is due to the two following causes : — 1. I he inclination of the equator to the ecliptic. 2. The eliptical form of the earth's orbit. Firstly. Suppose the mean and apparent suns to move with the same velocity, each in their own pianos. (See Fig. 10 a.) Then they will be in the same hour circle only at the equinoxes and sol> stices. At other times the hour circle paHHirg through the centre of the mean sun will be ahead of or behind that parsing through the centre of the apparent sun. The maximum difierence between the hour circles of the mean and apparent Huns from this cause is about 10 minutes, and occurs about the loUowing dates : — 5th Feb -f- 10 min. 2l8t Maroti 7th May — 10 min. 2l8t June 8th Aug + 10 min. 23rd Sept 8th Nov — 10 min. 2l8t Deo -\- means that the true sun is ahead. 2» Secondly. Neglecting theobliqaityof the eoliptio, suppose the two suns to move in the same eliptical orbit and in the plane of the eqaator. (See Fig. 10 b.) Then as the apparent son moves at a varying rate, after it leaves the perihelion point, at which it ooin* oided with the mean san, it goes ahead, and the two sang are farthest apart at a point nearly half way between the perihelion and aphelion points of the apaeline. They ooinoide again at the aphelion point after whioh the moan san goes ahead and the 2 suns are f\irtheflt apart at a point nearly half way between the aphelion and perihelion points of the apseline, coinciding again at the perihelion point. The greatest difference between the hoar circles passing through the two suns from this caube is about 7 minutes and occurs about the following dates : — let Jan min/ 4th April -j- 7 min. 28th June min. 28th Sept — 7 min.. The combination of these two causes of variation (see Fig, 10 c.) give the following results : — The equation of time is equal to soro (i.e., the hour circles passing through the apparent and mean suns coineide) four times a year and have maximum, positive and negative values, four times a year about the following dates : -|- 14'5 minutes'^ 00 -{■ means that true sun is ahead. the nth Feb 16th April.. 14th May.... 15th June... 26th July... Slst August 3rd Nov.... 24th Dec... + 3'9 00 6-2 0-0 163 00 « -f- means that true sun is ahead. the If M = mean time at any given moment A = corresponding apparent time at the same moment. E = equation of time at same moment. Then M = A ± E. The value of E is given in the Nautical Almanac for ;G-reenwich noon for every day of the year. This value must consequently be corrected for use at any other instant of time as it is always chang* ing. (See p. ). HOUR ANGLES. The following orthographic projection (see Fig. 11) of the celestial sphere and earth, on the plane of the equator, is a very useful and simple figure in solving all problems of time and saves much effort of memory. Looking down on the North Pole the earth rotates in the opposite direction to the hands of a watcht * i 28 Let P be the North Pole; PM the moriJian of tho observer and PO the meridian of Greenwich, rotating tis shown, w rule the celestial sphere remains fixed — thesso two meridians form a constant angle, called the longitude of M nioanured from Gr. y is the Ist point of Aries; Vy ihe hour cirolc paw^iing through it; PS the hour oirclo passiing thro.ig'i a heivoniy body S. As y and S are fixed points on tho (jolosual ("phoro, P/ and P3 do not revolve like PM and PG. As PM rotates, the line Pm tAiern its poMtion and at any instant the angle 7:\Fy or tho urc my me'U'urtid wontwardfi irom m is tho L.S.T. Tho angle yPS or tho arc yS moarfured oaHtwardrf from y is the K A. of the star, and so wo koo that whori m reachoH S, i.e.., when a heavenly body crosses tho mcrioiaii at ii« uppor tri-Uj^ic the L.S.T. = K.A. of tho liouvor.iy body. If S represents the 7/imn 3 K7i tho right ascon-ion arc ^S is uni- formly increasing 59' 8 "33 in arii, or Hm. 5f> fi')5 (rec. sidereal units, or 3m. 65.1109 sec. meiin iinit.-i, daily in tho tiirictiou of the earth's rotation, and iho L.Vl.T. at any nionicnt in tl)o arc ?nQS measured westwards from m. Also the &vi ^S voing tiio K, A. of tho mean sun at any given instant, thon when wi is nt >, tho are S^, measured westwards from S, lis tho L.S.T. of L.M.N., and when ^nisafy* the arc )^QS measure westwards fiom y, is the L.M.T. of L.S.N., or the L.M.T. of the tranmt of tho first point of Aries. If S represents tho mean sun, thon wo have L.M.T. = arcmQS G.M. r. = avcgH L.S.T. = mT G.S.T. r= gr^ and arc 'yS rr= R.A. of moan tmn. From the figure wo kco also that for a wfsf haur angle, i.e., when the heavenly body i.H west of ihe ni'.'ridian i'.n 1l.\. ut any given instant = L S.T. — K.A. of the heavenly Ivdy . For an east HA., tho K.A is gujutor than the L.S.T and will give negative value in tiie equation 11. A. = i-.. 5.T. — ii A., in which II. A. moana an oast a a -f* H.A. means a west hour !;nglo niid a hour angle. Also we see from the dotiiiiti.:n .;r cio;;k tirao on p. 25, that in observations of the sun a west II A. fr the sun trivcs local clock time directly, while an east H.A for the svti lia-' to b.; di^ducted from 24 hours to give local clock tin;c; or gonoroUv (see Fig. 12), the local olock time of obsorvaiion of a ho.avvi ly body + for W. =. tho local clock time of il.s upper nuit-it If the sun is obnervcd, tho loca transit = o or 24 hrs. 11. aj)piiront solar time of upper 29 If a star is observed, the local sidereal limoot upper transit =R. A. of star. From what han been said we see that longitude in arc and hour angles can be exprespcd equally well in sidereal or mean units ot time by dividiog the arc meat c Greenwich) or west {i.e., after Greenwich), But also for time oi" oi: servation {i.e., hour angle from local noon) if the time of observa- tion is not exactly at local noon. The exact number of hours, for which the correction for longitude and time of observation must be made, is determined by finding out, by the rule given on p. 29, the Greenwich time when the observa- tion took place. From this we obtain the total number of hours for which Greenwich data has to be corrected. The correction is usually found by multiplying the number of hours from the Greenwich time at which the data are recorded by the hourly change of the quantity under consideration, which change is usually given in the N. A. But this correction is not so easily found in all cases. (See Interpolation by successive differences, on p. ). Now to determine when those conections have to be added or subtracted we must inspect the quantities tabulated in the Nauti- cal Almanac. Then, If the quantities are decreasing then for a west longitude subtract correction, and for an east longitude add correction. ,i 31 If the quantities are increasing then for a west longitude add correction, and for an east longitude Bubtraot correction. OONTBPSION OF APPARENT INTO HEANTIMS AND YIOX TKR8A. 1 On page I of each month in the N. A. we find the equation of time at 6. A. N. that has to be applied to the G. A. T. at that in- istiint to find the coriespondingG. M. T. Od page II of each month we find the equation of time at G. M. N. for converting the G. M. T. at that instant into G. A, T. These data in the N. A. have to be corrected for other longitudes and times as above. Example. — At Kingston in 6*1 hours, W. long, the L. A.T. was 3h. 46ni. 6'&8. on the 16th December, 1888. Bequired the correspond- ing L. M. T. This L. A. T. is a west H. A. of 3-735 hours. Hence from p. 29 the Greenwich equation oi time for G. A. N. on the Ifith Decem- ber has to be corrected for 8 835 hours, and as the tabulated equa* tioDB of time are decreasing the correction is a minus one. As the equation of time is varying 1-2 lu sees, an hour, the total correc- tion is 10-73 sec. Eq. of time at G. A. N. 15th December, 4m. 21'648. Correction for long, and time of observation.... 10.738. Eq. of time at moment of observation 4m. lO'Sls. From the N. A. we see that this bus to be liedaoted from L, A. T. Hence irivon L, A. T. = 3h. 46m. 5*508. Eq. of time = 4 10 81 Koquired L. M. T 3 41 64-69 UNITS or TIME. We are now going to consider the relative values of the two stand- ard units of time for measuring intervals of time, quite irrespective of clock times. iUean solar time and sidereal time boirg both uniform, it ia quite ca^y to compare these two units of measure with one another. 366*24222 mean days = 366-24222 sidereal days, or 1 mean day == 1*00273791 sid. day or 24 hours of mean units. = 24hs. 3m. 668*656 sid. units of time. Th« eicess of 3m. 66s'556 sid. units ia called iLc acceleration of $idereal on 24 hrs. meantime and is the gain of a sidereal timekeeper in 24 hoara mean time. 32 J Similarly. 1 sid. day = 0«99'72695'7 mean day or 24 hours of Bid. units. = 23h. 56m. 4*091 mean units of time. The defect of 3m. 56'909b. mean units is culled the retardation of mean on 2i hours of sidereal time and is the loss of a mean timo- keeper in 24 hours sidereal time. S = a given interval of time expressed in sidereal units. M = the same interval expressed in mean unitP, and ft^ = - 36CK TIMEa TO CONVERT TJE LOCAL MEAN TIME AT ANT INSTANT TO THE OOBr BBSPONDTNG LOCAL SIDEREAL TIME AT THE SAME INSTANT. Now olook time is simply interval from noon measured west- wards. Let the line A (Fig. 13) represent the preceding mean noon and the line B the givon mean time, then the line AB is the interval of time elapsed since noon expressed in the number of mean units stated in the given mean time* Now if we know by any means the S T. at M.N. i.e., the S.T. at A, then we get the required S.T. at B, by converting the interval AB (expressed in mean units) into sidereal units and adding it to the S.T. at A. Consequently the rule for the conversion is L.S.T. required = L.S.T. of preceding L.M.N. -)- the sid- ereal equivalent of the given L.M.T. The N. A. gives us the G. S. T. at G. M. N. for each day, and from it we can find the L. 8 T. of L. M. N. as follows : — The Nautical Almanac gives yS when PG crosses S. ( Pig. 14). Now S is moving forward 3m. 36'556s. in 24 hours M.l . Hence by this time PK crosses S, S will have moved forward to S' or yS has increased to Y^S'. 3m. 56-6558; SS' = X longitude of K from Greenwich in hrs. _ 24 — 9.8566 sees. X long, of K from Greenwich in houis. This is a + correction for a W, longitude and a — for an east longitude. The correction for E. M. College, Kingston, in 5b. 5m. 508. W. longitude is = + 50 2658 see's. = + 50-27 see's, nearly. This constant correction we shall in future suppose to be applied to every G. S. T. of G.M.N, given on p. II of each month. Example.— Convert llh. 7m. 9.84s., a.m., standard mean time at Kingston, on the 17th of March, li-88, to L.S.T. We must remember that the mean time kept at Kingston is the 5 hour W. longitude time, (see p. 30.) 5h. W. long. M.T. Ifith March, 1888 23 7 9-84 Difference from Kingston, longitude W 6 60.00 L.M.T 23 1 19.84 G S.T. of p. G.M.N 23 38 Oi-lO Correction for long. 5*1 hrs. W. -|-.... 50*27 L.S.T. of p. L.M.N 23 38 61*37 23 hrs 23 3 46-6989 1 m 1 0*1643 19 sec 190520 0.84 sec '8423 Sid. Equivalents to L.S.T. required 22 43 58-1276 ■1 84 TO OONTIBT LOCAL SIDEREAL ItlME AT ANT INSTANT TO THE COaREB- PONDINQ LOCAL MEAN TIME AT TUB SAME INSTANT. The rule for the conversion is just the converse of the la?t :— L.M.T. required = L.M.T. of precoeding L.S.N, -f the mean equivalent of the given L.S.T. The Nautical Almanac gives yQS when P3 crosses T- Cig- IS ) Now S is moving forwa^^s Hra. 55^.909 mean units in 24 hourn. Hence by the time PK crosses X> S will have moved forward SS', or yQS has decreased to ^QS'. SS' = 3m. 558 909 24' X longitude of K from Greenwich in hours. = 9*8296 sees, x Jong, of K from Greenwich in hours. This is a — coneclion for a west longitude and a -|- correction for an east longitude. The correction for the E. M. College, Kingston, in 6h. 5m. 50p. W. longitude is = — 50* 1806 secp., which correc- tion we shall in futnre suppose apjlicd to every G.M.T. of G.S.N, given on p. Ill of each month. As there is nothing to mark the successive occurrence of sidereal noon, we cannot determine at first sight which is the preceding sid. noon to be used, because the sid. days are shorter than the mean days and as each sid. day begins directly the last one has Hnished, the M.T. of S.N. is constantly altering within the limits of eoch mean day. A little consideration will show that by consulting the S.T. of M.N. given on p. II of each montb (after applying the eorrection given on p. 33), we can easily determine which is the preceding S.N. to be used by the following rule : If the given L.S.T. is less than the L.S.T. at the succeeding or next coming L.M.N, then use the L.S.N, falling in the limits of the given or current day ; if it is greater, then use the L.S.IS . fall- ing in the limits of the past or preceding day. I'roof. Suppose that the current day is 4th March, 1888. TSeo Fig. 16). From the N.A. we find that, The S.T. of M.N. for the 4th March, is 22h. 50m 42«. " " " •' 5th " " -rih. 54m. 39.-. Now as sidereal noon has occurred eoraowbero bolweeti iheso two times, any L.S.T. greater than the S.T. of M.N. of he Bih (or succeeding day) can only have occurred between S.N. of the 3rd and S.N. of the 4th, and hence the preceding S.N. is that of the 3rd. or past day. If the L.S.T. is less than the S.T. of the M.N. of the 6th (or succeeding day), then it can only occur between SN. of the 4th and S.N. of the 5tli, and hence tho prt ceding S.N. is that of the 4th or current day. li 35 Often at the equinoxes we get two sidereal noons falling within the limits of the same mean day. A little consideration will show which one of these two is to be used as the preceding sid. noon. As the sid. day is shorter than the mean day by 3m. 3()'5559. sid. uuits, in each mean day we find certain sidereal times occurring twice, viz., the sidereal times lying within the limits of 3m. 66'&558. Bid. units after the preceding mean noon and before the nozt coming mean noon. Uhere can, however, be no mistake as to thene times as they are very nearly 24 hours apart. Example. Convert 22h. 43m. 53'li75 L.S.T. at Kingston on the 16th March, 1888, to mean standard clock time. As the given L.S.T. is lees than the S.T. of M.N. (corrected for long., see p. 33) for the 17th March, we must use the G.M.T. for G.S.N, for the 16th March. h. m. s. G.M.T. for G.S.N. 16th March 21 56-30 Correction for 5*1 hrs, W. longitude — . 50*13 L.M.T. for L.S.N 21 6-17 r I Moan equivalents for-{ I 22hrs 21 43m., 588... •128.. 00758.... 56 42 23 '7497 52-9555 57-8417 •1197 •0075 L.M.T Add difference of longitude. 23 1 5 19-8441 60-00 9'84 Standard mean clock time Ib'th March.. 23 or Uh. 7m. 9-848. a.m., 17th March. Let us take another example. Correct 22h. 43. 58-13h. L.S.T. at Kingston, on the IHth August, 1888, to local mean clock time. As the given L.S.T* is greater than the S.T. of M.N. (corrected for longitud j) for the 17th August, we mast use the G.M.T. for G.S.N, for the 15th August. h. G.M.T. for G.S.N. 15ih Aug 14 Correction for 5*1 hrs. W. long. — L.M.T. for L.S.N 22hr8. = m. 20 s. 21-17 50-13 Mean equivalents for 43m. = 588. =. . • J 38. =:.. 14 13 31-04 21 56 23 75 42 52-90 57-84 •13 L.M.T. 16th Aug 12 59 46-72 or Oh. 39ra. 45-72s. a.m. local mean time on the 17th Aug. I 1) n i.> 36 TO FIND THE HOUR ANGLE OF A f^TAR AT A OrVEV TIME AT A QITEN MERIDIAN. Find the L.S.T. of the given instant and take the star's R.A. from the Nautical Almanac. Then the H. A. is found according to following formula. HA=L.S.T.— R.A.,a+ result meaning a west H. A. and a — result an east H. A. To find the hour angle of the sun, reduce the L.M.T. of obser- vation toL.A.T. If the L.A.T. is less than 12 hrs., it gives the H. A. west directly; but if the L.A.T. is greater than 12 hrs. then (24 hrs. — L.A.T.) gives the H.A. east. Example. On the afternoon of the 29th Feb., 1888. at Kingston in long. 5-1 hrs. W., required the sun's hour angle at L.S.T. 23h. 10m. 46s. ^ „ , , As the given L.S. I", is greater than the S.T. of M.N. for the 1st March, we must use the S.N. falling in the limits of the 28th Feb. h , m. B. G.M.T. of S.N. 28th Feb 1 28 45-71 Correction for long . — 50'13 L.M.T. for L.S.N. 28th ^eb... 1 21 56-58 i 23h 22 56 13-92 Mean equivalents for-* 10m 9 58-36 (46s 45-87 L.M.T. 29th Fee, roq 34 53-74 = . 58 hours W. hour angle. Kingston longitude 5 . 10 " " Observation W. of Greenwich 29th Feb. 5 . 68 hours. h. m. 8. G. Bq. of T. 2yth Feb 12 37*08 Correction for 5-68 hours — 2 79 Fq. of time at observation — 12 34-29 L.M.T 34 53-74 L.A.T. or H.A. west =.... 22 19-45 15 West hour angle 5° 34'51"-75 aiVElf THE HOUR ANGLE OP A GIVEN STAR AT A GIVEN MERIDIAN TO FIND THE LOCAL MEAN TIME. In the case of a star find the L.S.T. from the formula (see pi 28) L.S.T. =R.A. ±:H.A. Then convert the L.S.T, into L.M.T. 9i In the case of the son, a west H.A. gives the L.A.T. directly, bat an east H.A. has to be deducted from 24 hoars to give the L.A.T. Then convert this L.A.T. into L.M.T. by means of the equation of time corrected for the Greenwich time of the observation. Example. On the morning of the 29th Feb., 1888, at Kingston, in long. 51 hrs. W., the eun'e hour angle cast of meridian was 30'' 2C' 42". Eeqnired the M.T. that should be shown at that instant by the clocks at Kingston keeping 5h. W. long. time. 15 1 30° 25' 42" Hour angle in time eaet. 2* 1" 42**8 L.A.T.28thFeb 21 58 27-2 = 21'97 hours. Kingston long west 5*1 (( Moment of observation distant from Green- } „^ ntr k«n«. wioh meridian of 28th Feb. J ^^ "^ ^°°"- Feb. 3'( Or distant west from G. meridian of 29th h. m. s. G. Sq. of time 29th Feb. + 12 36-98 Correction for 3*07 hours — 1*51 0*7 hours. Eqi of T. at moment of observation +0 12 35 •4*7 LAT 28th Feb. 21 58 27*20 LMT 26th Feb. Add diff. of iongtitude 22 10 5 52-67 60 00 Standard olook time req'd 28th Feb. 22 16 42*67 or lOh. 16m. 42 678 a.m. 29th Feb. TO FIND THl LOOAL l^IAN TIME AT WHICH A UIYKN STAB WILL BK ON THE MERIDIAN. On page 28 we have seen that at the upper transit of a star, the L.S.T. = E.A. of star. At the lower transit, the L.S.T. = E.A. of star + 12 hours. Then correct this L.S.T. into L.M.T. ill ii TIMEKEEPERS. There are two kinds of time-keepers in general use, viz., clocks and watches or rather chronometera, which are only very perfect watches. nwi 88 1 -. i !'■ I.-. J' } '-.■ 1 ' I Time-keeperB should be as perfect as possible for astronomical work and can be rated to keep S.T. or M.T. as both of these kinds of time are uniform standards of measurement differing only in rates. The mean time Ueper shows the interval of time in mean units between the successive upper transits of the mean sun's centre across the meridian ; and the sidereal time keeper shows the interval of time in sidereal units between the successive upper transits of the first point of Aries across the meridian. The interior arraogements and mechanism of M. and S. clocks and of M. and S. chronometers are exactly the same, but the S. time- keeper is regulated so as to show 24h. 3m. 56'555s. sid. units in the time that a M. time-keeper shows 24h. mean units, ho that the S. time-keeper is always gaining 3in. 56'555s. sid. units in eveiy mean day. As chronometers are more in use than clocks for general work, we will include all time-keepers under the nameof " chronometers " in the following remarks. THE XRROB AND RATE OF A OHRONOMETJBR. The error of a chronometer is the difference between the time it shows and the time it should show, and it is called a fast or -^oi'for when the C.T. is ahead of the proper time, and a slow or —error when it \a behind it. A -|- error requires a — correction and vice versa. The error of a chronometer is usually determined for noon on a given day. The chronometer correction at any instant is the quantity to be applied to the time shown by the chronometer at that instant to obtain the correct time. The correction and error are equal but of opposite signs. The chronometer rate is the daily alteration (increase or decrease) of the chronometer correction. No chronometer can be made to keep time truly, though they can be made to always gain or lose at a nearly uniform rate, i.e., gain or lose nearly the same quantity in equal intervals of time, it is sufficient that the rate should be constant and for convenience it should be made less than 10 sees, a day by regulating the balance lever or the weights on the balance wheel. Knowing the actual error at noon on any given day and the rate of the chronometer then the error for any other day and time is found by adding or subtracting the accumulated correction for error and rate for the given number of days and hours from the given day when the error was last found. Such a calculation, however, can only be made so long as we regard the rate as constant, but such uniformity of rate cannot be assumed for any length of time even with the *best time-keepers. 39 GoDHoquontly the intervals between taking astronomical observa- tioDB for clock correction should be so small that the rate may be taken as constant for the intervals. The length of these intervals will depend on the character of the time-keeper and the accuracy required. If we alter oar longitude between two observations, we must deduct the difference of longitude when we go east, and add it if we go west, to the supposed time elapsed between the two observa- tions before finding the error and rate at the place of the second observation. The rate of a chronometer 1*1 affected by any mass of iron near it, and also by its being moved as in travelling. Changes in tempera- ture affect it as well, bat the corrections for this cause of error are too complicated and minu'e for practical work. DETERMINATION OP ERROR AND RATE, The error of a chronometer at any given moment is found by the methods described under the heading of finding " Time." The daily rate of a chronometer at a given place is found by ascertaining its error on different days sufficiently distant from one another, and then deducing the rate on the assumption that it is uniform during the interval, by dividing the total gain or loss by the number of days and parts of a day of which the interval between the observations consists. The errors found by observation at any time of day are usually reduced to noon by allowing for rate when it is known. If the rate is not known, then the observations for error should be taken at about the same time each day, until it is ascertained. Even after the rate has been established it must be continually tested to prove its retrularity, and if the rates so found vary about a mean, as thoy piobably will do, the mean rate should be used lor the interval of time. The following is an example of how to find the rate of a time- keeper the errors being those for nearly the same time each day : as we DATE CLOCK SLOW DIFFERENCE LOSING R4TB REMARKS. DAILY m. 8. fi. S. 22nd Dec. 2 1-695 1855 9-28 Interval of 2 days. 24th " 2 32-60 *-i7th " 2 59 50 2700 2) 900 " 3 days. 1828 Mean daily losing rate 9 -148. I he rate of a M.T. chronometer, without reference to its error, can be found by observing at the same place, the same star when at ij II I H JM 40 eqaal altitade on the Bame side of the meridian on different nights, thas— From the time of the Ist observation deduct 3m. 65'909b. mQlti- plied hj the nnmber of days between the observations. From this remainder, which is the time of the 2nd observation if there had been no rate, deduct the time of the 2nd observation ; and divide the remainder with its proper sign, by the interval in days. A -\- sign means a losing rate, and a — sign a gaining rate. ExampU.—On the 4th Feb. at lOh. 66m. 26s., and on the 12th Feb. at 9h. 43m. 388., a certain star had equal altitudes. h. m. Bi Time of first observation 10 56 26*00 3m. 65-9098. X 8 days 31 2'' ^7 Time of second observation U 43 9 43 8) -f 20'73 Daily losing rate 2-B91 But if the barometer and thermometer readings are very different at the two observations then the result will not be quite correct (see p.p. ) To find the rate of a S. chronometer, in the above way, we need only take the difference between the two times and divide by the interval in days. The rate can be similarly found by timing the disappearance of a star behind a sharply defined vertical edge, oare being taken that the position of the observer's eye may be the same in both observa- tions. Let Tj (corrected as stated above) and Tj be the two times of observation ; D the nnmber of days between them ; and r the daily rate thenr= ^ p ' If the sun is thus used for finding the rate, then its alterations /\a and /^8 ill ^.A.. and deelination respectively must be considered. Lot ^t be the correction in time required for a small change of /\^, then, __ A ^ ^° g ^here q is the " 16. eoB (f ' ^ {)arallactic angle and can be found with sufficient accuracy from the atitude and an approximate value of t by means of the equation tan N = cot ^ cost . , tan * sin N and tan q = ^ cos (5 4- N) /\t is -{- when ^(f is a decrease and — when it is an increase. Th.B, r = T.-T. + Aa+A* In this formula the mean equivalent of l\ a must be used. ■t 41 COMPARISON or CHRONOMETERS. When there is more than one chronometer, one is chosen ah a standard and its ttror and rule are found by astronomical obaerva- tiona, and the errors and rates of the others are found by comparison with the standard one. If these comparisons are made some time after the observations the interval must be corrected for rate and consequently, to avoid the chance of an erroneous reduction, especi- ally if the rate is not well known, it is best to make the observations and comparisons about the same time each day. Chronometers are compared by noting coincident beats, and oonso- quontly we must consider the value of each beat. Some chrono- meters boat to 1 second, some to ^ second, and others beat 5 times in 2 seconds, or 8 times in 3 seconds or 13 times in H t^eoonds, etc. We will first consider chronometers beating to 1 and J seconds only. If they are both M.T. or both S T. chronometers, the boats will rarely coincide (as the ra'^s will rarely be the same) but will differ by a fraction of a second, vhich may bo taken as constant, as the difference in the rates of the two chronometers will not be percepti- ble during the few minutes in which the comparison is made. This fraction can be estimated to within a ^ in a second. In making the comparison between 2 chronometers showing tho same time, draw out the following form, calling the 2 ohronotneters A and B. Time by Chronometer A. m. B. Time by Chronometer B. h. m. 8. Number of beats. Corrected time of Chronometer B. h. m. B. To make the comparison look at A and listen to the beats of Bi When the seconds' hand of A is at a main division (60 or is the most convenient), the time of which 18 written down beforehand, begin to count 0, 1, 2, 3, etc., with the beats of B until the latter can be looked at and continue counting until the second hand of B is on some main division ; this time is recorded and the number of beats. In writing down the time of B, write down the seconds first, and then the minutes. The hours can be filled in afterwards. Take 6 of such comparisons so as to use the mean of the results. Then deduct the number of beats (reduced to seconds) and the estimated fraction from the 2nd column of the above form, to find the true time of comparison of B. 6 Mi fMJJ' \ •! 42 ill To test th: aoonraoy of the comparison the difference in tims between any two oomparisons in the firet and last columns should be the same. A less error than J second cannot be ensured by this method. Gompariaons to within ^ second can be made by two persons, eaoh taking the time of one chronometer One cries out "Are you ready ?" just before and then " Now," when the comparison is to be made. In comparing chronometers showing different times, we proceed dif- ferently. With 2 such chronometers, whose rates are fairly small (i.e. under 10 seconds a day), the interval between the beats will not be cocstant but will coincide at certain intervals. A sideieai chrocometor (if it has no rate) gains 3m. 66 555s. in 24 hours mean time or about 1 second in little over 6 minutes. Thus if both chronometers beat seconds, the sidereal chronometer will gain I boat in 6 minutes ; and if one beats ^ seconds and the other seconds or ^ seconds, then it will gain 1 beat in about every three minutes. Thus every few minutes the beats of an M.T. and S.T. chiono- meters coincide theoretically, but in reality several beats seem to do BO, becBuse in 6 minutes there are 360 second beats, and the differ- ence between the boats of a M.T. and S.T. chronometer is only ^J^^th of a second, which difference the ear cannot detect. Find out by two or three trials the number of beats which seem to coincide, and make the comparison rs before from the centre one of these beati<, but in this case the timo of A cannot be written down beforehand . But we can write down the hours and minutes of A befoehand, leaving only the seconds to be filled in at the moment of comparibon. In this method the error of comparison depends on estimating the centre coincident beat correctly, but any largo oi ror even in thin estimation only produces a email error of comparison. Thus if we are 10 beats out, the resulting error is ^^^ or ^ sec. which is too small to be appreciated. To test the accuracy of the comparisons, take the interval of timo between two observations of A, reduce its equivalent in the other units of timo, and add it to the time of the first comparison of B ; the result should be the time of the second comparison, if the com- parisons have been properly made. From the foregoing pages we seo that from the greater accuracy of the second method it we want to compare two chronometers showing the same kind of time, it is best to take a third chrono- meter showing a different kind of time and compare the other two with it in succession. If there is more tnan one kind of chrono- meter it is best to use a sidereal chronometer as the standard one, as its errors are more easily and accurately found by observation . If there are more than 3 they should bo compared thus, suppot^ing S to be the standard sidereal chronometer ; — Ml with S, ; Sj with M, ; M, with S^ ; S, with M, ; etc. 43 m The compatations are now best oarried out by means of the fol- lowing form :— rr^yr-^^:^ 00 S I 3 aj H H < O H -< Oh M H .3 fi.Q< o. 5 f ES »a «" s- o s «o >"" o ■* »- w 'JO 55 •^ O tM c- o S2 eta S3 St ,; §2 .rror o np. . to no e( gg OS o 03/3 a o ' e> £S X 3 ■3 V ■3 (S H ** • ^■3 * "If u O u "3. CO «-• • O « S u He, 5- C o § la ft<-' O O <-? .S a « «a sa a% — a °M • • (SCS 'O w «d a b >-J3 22 "1 o-j § Ho (U a d 2d2 o c M 1 , d V a H g: • rt o b»« o — u u •4 u « s «j . •-* a «J =■1 » i 2d2 8S 3 u H « > a 5 - Hw CO •«! A < o-'so2 a --3 3 >- ; K oj o '^ — >- M « C2 o.Sc a*' - oJ£ Oi,0 t- o « *2 cc'C a o u s o w Od V Q, O a o o CO u o o a o o a S o 03 O o o a O c o o ' 4 44 < 1 'J 1 ■ 1^1 t In the above form, any error made in one oomparison is not carried on. When there are only 3 chronometers the following form is a convenient one : — SlONATURl, f Sj Plaob, Bates for 24 hours -J M^ Dat«. ( S, Standard 8i. Ml. 83. p*! oi Istcomp. correction (from observa- tion i>r irom last day cor- rected for rate). Chron. Mi of Ist comp. • Ml— an doom p. Error of Ml at comp. 1 Chron. Sj nri;ii.| c )mp "True Ka " Truo Si of Istcomp. dtMluot LS.T. oJM S ■ True Ml of 2nd comp. deduct U\f.T. of M.N. Error <>l 82 al Si.^l comp. cor. for rule to noou. bid intervnl Irom M N. (Jeduci returdiition. Mean Interval iioni S.N. addacceleralioa. Krror Of Sa at noon. )nie Ml of 1st comp. Chron. Ml of Ut comp. True 62 of 2nd comp. Error of Ml ntlatcomp. cor, lor rate to noon . ConRKOTIONS AT NoOK. Si. Ml. Sj. Error of Mt at noon. + h + To compare pocket chronometers beating 5 times in 2 seconds, etc., with each other, r.a .£ with ink on the face ofthe dial, the seconds at which the beats coincide and then only compare the marked peoonds. If we are comparing them with a chronometer beatin^j to ^ Hoconds or seconds, then count the beats of the latter giving them their real names (as 43, 44, 45, etc.) and compare with the marked divisionw allowing for any portion of a second between the be.its if necuHHary, or we can look at and tap on the table the second boats of the pocket chronomctor and comparo hh before. This can bo done several times rapidly and the mean of the rcBults taken. Nl CHAPTER IV. TOE NAUTIOL ALMAKAC— INTERPOLATION BY 8U0018BIVB DIFFEREN0K8. The Naatical Almanac has already been referred to on many oscasioQa, but its arrangement and the nature of its data have not as yet been specially treated on. The N. A. gives certain data concerning certain heavenly bodies and the spherical co-ordinates defining their positions on the celes- tial sphere at given instants of time for a particular year. The spherical co-ordinates choHcn are thoee \Thich are independent of the observer's position and of the diurnal motion of the earth, viz. : declination and right ascension, and celestial latitude and longitude* It also gives the data by which the efFects upon the co-ordinates of an obiserver's change of position can be readily computed such as the parallax, the apparent angular magnitude of the sun, moon and planets, etc., and in general all those phenomena which change with time and which may therefore be regarded ab functions of time. All the quantities and co-ordinates given in the N.A. are the geocentric quantities and co-ordinates (i. e. as seen from the earth's centre) at the stated instant of Ortenwich local time, and as they are geocentric, they are, with the exception of the data for the moon, unaffected by the latitude of the observer. The moon is eo close to the earth at the spheroidal shape of the latter is fully felt in lunar observations. The data in the N.A. computed for Greenwich meridian are given for equidistant instances of Greenwich time. The data for any other instant of time must be found by interpolati0n, and to facilitate this interpolation the N.A. gives the rate of change of the quantities in some unit of time. DBSORII'TION OF THE CONTENTS OP THE NAUTICAL ALMANAC. The arrangement of the contents of the N.A. is not alwavs the same from year to year, but any particular data can be readily ibund from the 1 able of Contents attached to the book. At the end of the N.A. is an explanation of the various tables contained therein. Each month of the year has 18 pages specially devoted to it, and these pages are numbered consecutively in Roman numerals, J., II., &o., to distinguish them from the ordinary numbering of the pages of the book. Mm 46 I A caution is here given that throughout the N.A, the Astronomical, and not the Civil, Day it used, (see page 26) . The elements of the ephemeriB of the N.A. are computed for Greenwich time. The day of the month (p. I. of each month for apparent noon e> Aopted) is assumed to begin at mean noon of the corresponding civil day, and to end at mean noon next day. The principles of taking out the elements are given in the " Explanations " at the end of the N.A. ELEMENTS AND DATA FOR THE SUN. I.— 'The apparent right ascension aad declination of the Sun at apparent noon are given for every 24 hours apparent time on p. I. of each month, together with their variation in one hour. 2. — The apparent right ascension and declination of the Sic at mean noon are given for every 24 hoars mean time on p. II. of each month, and their respective variations in one hour are the same as those given at apparent noon on p. I, of the month. 3. — The equation of time is given for every 24 hours on pp. I. and II. of each month, — that on page I. is the correction to be applied to apparent time (found by observations of the sun) to find the corresponding mean time; and that on p. II. is the correction to be applied to mean time to find the corresponding apparent time. The variation in one hour is given on p. I. and applies equally to both. 4. — The Sidereal time of mean noon is given for each day on p. II. of each month, under the heading of "Sidereal Time." To find the corrections for other times or longitudes, use simple interpola- tion, or method given on p. 33. 6.~ The mean time of sidereal noon is given for each day on p. III. each month, under the heading of '* Transit of Firtt Point of Aries." To find the correotionH for other tlmpj or longitudes, use simple interpolation, or method given on p. o4. ti. — The semi-diameter of the sun is given at every mean noon on p. II, of each month, and is practically the name as that at apparent noon. This semi-diamoter is the geocentric semi-diameter (i.e., as seen from the centre of the earth), but it is, sensibly, tbe same as that seen from any other position on the earth. Intermediate quantities can be got by simple interpolation. 7. — The sidereal time of the sun's semi-diameter passing the meridian is required for transit observations, and is given on p. I. of each month for eveay day. The mean time that the semi-diameter takes to pass is found by subtracting 0'18s. from the above sidereal times. Simple interpolation is UHcd for finding intermediate quantities. 8. — The horiMontal parallax of th". sun is given for every 10 days only, on the first page of the book. It is practically the same for all latitudes, and simple interp )lation is used to find intermediate quantities. It has, however, to. be corrected for altitude (see page ). 47 ELSMINTS AND DATA FOR Tl.E STABS. For theso see pages headed Mean Place of Stars and Apparent placcB of Stars, in N.A. 1. — The right ascension and declination of five close circumpolar Htars are given at their upper transit acroes the Greenwich meridian for every day of the year, and of 192 other standard stars at the namo transit for every 10 days in the year. Simple interpolation is need for intermediate quantities. 2. — Ihe mean places of the tibovo stars, together with the ar.nnal variations of these co-ordinates, are also given for the date at which the sun's mean longitude is 280'*, which in 1889 was 0'351 days (see N.A.) before tlio Ist .January began. 3. — Bessel's formulae for reduction of mean places to apparent places, and also the method of doing sc by iudependent quantities, are given in the N.A. 4. — BesseCs day numbers for use with the above formula} are given for mean midnight lor every day ot the year. B. — Airy'a day numbers are sim'larily given for use with the formula) used in connection with th'> Greenwich catalogues. 6, — Under the heading of Quantities for correcting the places of Stars similar data are given for the snme purpose when using the method of independent quantities. m ELEMENTS AND DATA FOR THE MOON, 1. — The moon's right ascension and declination are given for every hour of each day in the year on pp. V. to XII. of each month, together with thoir rc8])0ctive variations in 10 minutes. 2.— TAe moon's right ascension and declination, its semi diameter and equatorial horizontal parallax, are given for every tianbit over the Greenwich meridian for every day under the head of Moon, cul- minating Stars or Moon, Ephemtris of, at Transit (see Table of Con- tents in N.A. ). The similar co-ordinates of certain stars having approximately the same right ascension and declination and also the sidereal time of the moon's semi-diameter passing the meridian, are given in the same place. The symbol I. against the word " Moon " means the moon's eastern limb, and II., the western; U means the upper, and L the lower, transit. 3. — Besides the values of the semi-diameter and equatorial horizontal parallax given above, thee© are also given on p. III. of each month at noon and midnight for every day of the month. The semrdiamei er must be corrected for altitude and the parallax must first be corrected for the latitude of the observer and then for altitude before they are used in calculations. 48 4. — Lunar distances, i.e., the distance of the centre of the moon from the Ban and certain conveniently sitaated planets and starg, are given for every three hoara of every day on pp. XIII. to XVIir. of each month, together with the proportional logarithms of the differences. The planets employed are four in number, Saturn, Jupiter, Mare, and Venus. The stars employed ai'e nine, and are called lunar distance stars ; they are a Arietis, a Tauri (^Aldebaran) JS Gemi- norum (PoWux), a Leonis (Regulus), a Virginis (Spira), a Scorpii (Antares), a Aquilaj {Affair), a Piscis Aastralis (Fomalhaut), and a Pegasi (Markab) . 5. The different Ptars up to the sixth muguitude that are occult- ated by the mooti iu its orbit and the times of these ocoultations at Green wioh, are given for every day ot the year, together with the limits of latitude within which they can be seen under the heading of E ement of Occuliaiions of Stars by the Moon. (see Table of CouteiilB in N.A.). DLXUENTS AND DATA FOR THE PLANETS. 1. — The apparent right ascension and declination of the planets at Greenwich mean noon are given for the planet concerned for every 1 to 4 days (depending on the distance of the planet) and also mean time of the meridian passage at Greenwich, under the heading of * , Ephemeris of at Mean Noon (see Table of Contents in N.A.). 2. — The apparent right ascension and declination of the planets at the instant of transit over the Greenwich meridian, their respective horiMontal parallaxes, and semi-diameters at that time, and the sidereal time of their semi-diameters passing the meridian, are given, for the planet concerned for every day, under the head of * , Ephe- meris of, at Transit (see Table of Contents in N.A.) The semi- diameter and horizontal parallax of a planet are treated like thoee of the sun . 3. — Tables are also given in the N.A. of the Greenwich mean times of the Eclipses of Jupiter's satellites, which give an approximate method of finding longitude. There are also throe other useful sets of Tables given in the N. A., namely: — 1. — ^Tables to find latitude by observations of the pole star out of meridian. 2 — Tables of time equivalents. (Vide pp. 21, 32.) 3. — Table showing what fraction of the year from mean noon of 1st Jan. any particular day of the year is. * Name of Planet. 49 INTERPOLATION OP SERIES BY SUCCESSIVE DIFPBRBN0E8. In an astronomical series (such as any of those given in the N.A.) the differences between the various quantities for each hour of longi- tude vary inconstantly throughout the day, i.e., they vary differ- ently during each hour of the aay, although they so vary according to a law. In the X.A. wo have given for the different heavenly bodies, series of values of various data and coordinates, each following a definite law, corresponding to given equidistant intervals of time, and we may want to find the value of a particular series at a time inter- mediate to any two of thofio given. This is effected by interpolation^ by which tho intermodijito pot^ition is determined from the known inoremonts of tho data corrotfponding to equal intervals of time. The equidistant quantities on which the values of the series depend are called the argmntnts*^ of tho series. In astronomical tables, as the data aro simply functions of ; ;ne, the argumor.t for working on is lime ; and, in oalcalating the vari- ous data employed, it would be far too laborious to repeat the pro- cess of computation for each particular moment of time. This com- putation, however, can be effected with sufficient accuracy for all practical purposes by using tho method of difference*, by which, when the law of the series is known, or when several terms of the series are given, other terms in accordance with the law may be introduced between them, or tho scries may be continued for some distance without breaking tho law. In the dilforont series given in the N A. the law of the respective nerios is not given, but only tho numerical values of certain terms of tho series at stated ei|ui-di8tant intervals of time are given, and with these values wo can, by tho methodof differences, approximate sufficiently to the law of the series for practical purposes and so iihtain the value of imy term of tho series at a time not given in the tables. The differences between tho sucoo'^div^o equi-distant values of the (luantilioH of any serioK are culled W\o lirst differences; tho differ- ences between the succoHsive first differences are called the second differences, and so on. if a series varies constantly, tho differences hoiween the sucjcessivo oqui-dihtaiit quantities aro equal, i.e., the lirst differences aro equal, so that there can bo no second ditforonces. If all the second ditVerencoa aro equal, then there can bo no third iiifforenco«. and so on. The more variable a series is, the greater will be that order of differences, as it is called, whose successive quantities aro equal. M •Si (,' ''to ' •The " ArKumont " of a table 1h tho known quantity which serves to UetermlQe the value ofaume dupeuUotil tuuulUiu. i ■p* 50 /.• m In interpolation by first differences we assume that the variation of the quantities of the series (i.e., the first differences) is oonstant; in interpolation by second differences we assume the variation of the first differences (i.e., the second differences) to be constant, and 80 on ; and in any given variable scries of quantities we should campute any required quantity by interpolation by that order of differences which is practically constant. Let a, b, c, d, e, tfec, reprouent a series of equi-distant numbers following a given law ; subtract each number from the number which follows it in series, putting the proper ahjebraic sign bejore it, then we Lave — Valucp. 1st dill'. a b b — a c — b c d d — c e — d e &o. 2nd diir. c—2b -{-a d _ 2c + b e—2d-}-c aid diff. d — 2c + 3b — a e — 3d-\-3c^b 4th diff. e — id-\- 6c — ib -f- a i^ i I 'I ll IK I * Let di = b — a then b d2 = c — 2b -{- a c a-\- di a + 2^, 4- d. di=d — 3c + 3d—a d = a-\-3d,-f- Sd^ + d^ d^ = e — 4d-{- 6c—4h-\-a e = a -{• 4d^ -\- dd^ -j- 'id^ -f d^ Thus wo see that the co-efficients arc the same as those of the binomial theorem, and bonce if 5|„^. ,; represents the (n + 1) th term of the eorica , , , n(n — 1) , , n(n — 1) (n — 2) , . s (n + 1)= a + nd, 4--^— -.^-^J^ J^^ -L^ Id.^ 4- ... When n is < 1, s (« -\. i) stands for a term between the first and second of the given HcrioH. When n > 1 and < 2. the intermediate term lies between the Htcond and third term } and so on. Now let the above series be written in another form as follows, where T — 3/», T — 2A, T — A, T, T + /i, T 4- 2/i, T -f 3A, &c., express equi-distant values of the argument ; 5,, corresponding values of the series ; then «ii. «ij 5, 5% «",a"', Ac, 51 ion of tant ; ion of t, and hoald der of I Arguments. Series. Orilers of Dlfforonces. Ist 2nd 3ra 4th; 5th 6th T— 3/1 T—2h St flu 6n ^x ^ T \ s \ la] \ b \ [c] \ d \ [e] \ f T -\- h T-\-2h T+ 3/1 gin a' b' 6" d^ e' Then if < <"> be the value of tlie aeries corrospondini;^ to argument T + nh, we have — ^ n-(«-i):(n-2).(,.-3) ;;.^___^^_ ^^^ If n be taken successively equal to 0, 1, 2, 3, ... &c., we obtain the values «, 5*, «^^, s' ^*, ... &., and the intermediate values are found by using fractional values of n. Now we have Substituting these in the above we get — ,«. I , I n (n-^l) , , (^n + l)n (n—l) . + (n + 1) n (n — 1) (n — 2) 1.2.^.4 rf. + ... &c. (B) 52 To facilitate taking out the quantities roquirod draw a lino under the term of the series from which we sot out, as the line under the term T in the above table. The quantities to bo used then lie above and below this lino. Again a^=a^-\-b o and substituting in (B) wo lind — , (n-^l).n J , (n+l)n(n—l) + 1.2 ' i.^.y (n+2 ) («+l )n(n— 1) 1.2.3.4 , d ^ ... &o . (C) To faoilitato taking out the reciuired quantities draw a linooyer the term of the series from which wo sot out, an the lino above the term T in the above table. Equations (H) and (C) arc each of them only approximations, but when the orrv)r of ono is positive the error of the other will generally bo negalivo, and so we shall obtiiin a more accurate expression if wo take the mean of both, llonco, combin- ing (B) and (C), and putting — a = ^ (os that fall next above and next below this line. Thus, put — bo=H^ + b'), d„ = h(d -v rf'), Ac. and we kaow that- ch = 6' — b, e' -— a' —d, &o. Therefore — b=b,-h which substituted in (B) give d= d,, — i e', &c,. 1) n(/j' l)(n~k) 1.2 • » "^ 1.2 3 (n + 1) n (n— 1) (n — 2) 1.2.3.4 7. (n + l)n(n~.l) (n— 2) (n — h) ^ ■^ 1.3.5,4.5 "* + ... &c. (E) 53 To fuoilitato taking out the quantities required, draw two linoB, one imdtr iho terra wo sot out from and the other over the term of the HoricH next after thin one, as in Exam}>lo II., j.. 55, and between these two lines tho moans b^, d^, ) we have (n -t^l) n (n - 1 ) 1.2.3. m. , E .-. 0-41CC ; a =- + 3 4270 ; Y^ =r •1730 ; 6 - +- 66 ; — -0573 ; c = +0 0-02 ; ll. m. S - 13 2U 8. 34-64 ^'+ ' liU'-VH (n4- 1) n (n- 1,2.3. 0-10 0-00 R.A. reciuireU - 13 22 7-52 V,x. II. Find the distance of the moon from Saturn at 5hrs. (rroonwich time, on the 13th October, 1883. Lunar Dis. n l8t Ditr. 2na DitU 3rd Dlff 4th I'lff. if o » n >t At XVIII. " XXI. " Noou. " III. hrs. Mlh Oct. II II 15th Oct. 11 II 02 51 CO 57 5!) « 57 '.0 37 58 6 2 — I 63 39 — 1 5.3 52 — 1 54 4 + 13 + 12 + 10 [+ a-5] - 1 o — 1 + 1 — 1 [-05] — I 5.1 14 ' VI. «» *. 55 ]5 48 + » — 1 " IX. ii <> m 21 25 - 1 51 '.'3 + 8 Midnight •1 II 51 26 54 - 1 51 31 Ufciing formula (^B) wo have 9 I II n ■- T- - -6666 ; «i -^ — 1 51 1 lo ; S = 57 10 2 nci^ = — 1 16 !»-338 n (n — 1) 1.2. rt(airs or in threes, and revolve round a cora- mou centre. It has been ascortainoi by the spectroscope that tho chemical elements present in the sun a-.u! sl:irs are identical with those composing our earth ; ut least, no new elements havo as yet been discovered. Tho stars of tho first six magnitudes number about six thousand in all. Thoso are all that can be soon by tho naked eye. There are upwards of 2(),0(>0,000 st.ars in tho first fourtoon m.ignitndes. If the brilliancy of a star of tho 6th magnitude be taken as unity, then tho relative briliianoios are as follows, according to Sir W. Horsohol : — Ist magnitude.. 100 2nd " 26 3rd " 12 4th " 6 5th •* '. 2 flth " 1 61 Rnt Professor Newcomb says that astronc mora are not agreed on this point, and that, roujifhly speaking, the brightness of a star of any magnitude is about |thH or J^rd of that of the next greater magnitude; more accurate results on this point, he adds, are much needed. If a star occupies an intermediate place between two magnitudes, suppose between the second and third, it is marked 2*1, 2-2, 2*3, and so on up to 2*9, according as its brilliancy. To distinguish the stars they were grouped by the ancients into constellations, which have been given the Latin names of mythologi- cal heroes, animals, oto- To distinguish the stars in each constel- lation they have each been given a letter in the Oreek alphabet, and when that is exhausted the Eoman alphabet is used. The brightest star is called a* the next ft, and so on, the name of the constellation in the genetive case being pat after it. But since then some of the stars have altered in brillianoy, and consequently the order of the letters does not invariably mean the relative order of brilliancy of the stars of a constellation. If two stars have nearly the same brilliancy they both receive the same letter, and have the numerals 1 and 2 affixed to them, in the order of their right ascensions, as x"^ and x"^ Tauri. J'lamstocd assigned numbers to the stars of each constellation in the order of their R. A.'s, i.e., in the order they cross the meridian, only whatever their brilliancy. Some of the principal stars of the first two magnitudes have receiv- ed particular names, thus : — a Canis Minoris Prooyon, iL Lyriv! Voga. (L l^obtes Arcturus. fx Canis Majoris Sirius. a ArgCis Canopus. // Orionis Rigel. fx Auriga' , Capolla. (i Oriotiis Hotelgoux, a Eridani Achernar. a Tauri Aldebaran. a Scorpii Antaros. a Aquild' Altair. a Virginis Spica. a Piscis Australl8..romRlhaut. ^ Geminorum Pollux. a Leonis Regulus. y Pcgasi Algenib. a XJrsa^ Minoris Polaris. a"(ieminorum Castor. a Pegas: Markab. The star nearest the North Pole of the heavens is known as the iAorf/i Polar Star, or the Pole Star, or Polaris. This star is at prehont a Urs:o Minoris. and it is rather less than li* from the Pole, which distance, from the movement of the axis of the earth (causing the phenomenon of the procession of the equinoxes), will decrease in about 150 years to ^°, after which it will iucreaso again. Thus, the position of the Pole siar varies, owingto the movementof the polo. ; 1,600 years ago (i.e. in BC. 2700) a l3raconi9 fulfilled this olHce ; in 12,000 years, a Lyrw, now 61°20' from the pole, will bo 14 I m m( 58 within 50ofit,and will be the polo star. (Seo fig. 18). In 26,800 years the pole and the first point of Aries will have returned to their present position, supposing the conditions controlling the motion remain the same. Those stars which appear to revolve round the pole, and do not set below the horizon are callo i circumpotar stars. Every star whose distance from the elevated pole is less than the latitude of the observer is a circumpolar star. The following is the generally acknowledged list of constellations.* The co-ordinates refer to the central pan of the constellation. The constellations north of the ecliptic, nro called the northern, those south of the ecliptic, the southern, and iliof^e cut by the ecliptic, the zodiacal constellations . II I 5 (i 7 s II 10 11 I Vl \-\ 11 IJ I't IT IS i:) •J) !?«: Taken from Chamber's Asironoviy , Z) 69 Northern Constellations. No. Name. AiiilroinLHlii .,...,, Aiiiiilii.Tlio Kanlc Auilf;n, Tlio Cliiirioteor Bootes, Tho Uuar Keeper . . ■;;iii»oIopariluH, The Caineolcopiinl ... Canes Venatlci, Tlio Huiitins; Dogs .. C'asKiopola feplii'iiH Co-onUr.atoH U. A. i bee. Stars of .Magnitude. h. m. 1 1!) ;iii t! u 1» 35 t') 11! 1 10 .! 'Jl HI [iif Sdliii'ski Clypensor Sartiini Si>l)l(..ski,Tlic SliicUl IS lo i Coma llorcnice.s, Tlic Ilalr i)f Ilcrenioc. . | ij in Conma lioicalis, 'llio Norllieiii Crown ■ 1.') lo Cygnus, Tlir Swan ' uo 2') I Delphliuis.Tbii Dolplihi J) 10 Draco, Tlio Dra^jon Iviuiileus, Thr l,iUlL' Unrsc Hercules I.acerta, The l.i/.anl Leo .Minor, rii<; Litth' Mo'i I.ynx.llio Lynx Lyr.i,Tlie Harp.... 17 L'O ■:i l(i I") 120 iM 10 .j IS 10 IV'gasus, Tiio Wingoil Ilor-o (till- .Mi'diivii's IlcatI IVrsi'usot Caput Meilus.i', I'ersiMis ami haf,'ltta,The Arrow Serpens, The Serpent Ildwskl Taurus I'oiilatMnski, Tlu' I'.ull ol Toiiia- 'rriangnlr.iii, 'riieTriaM;,'le Ur»a M.ijor, Tlio (ireal lijar.,,. Ursa Minor, tlio Little Dear.. Vuli;ecuU e! Anser, Tlio l'\).\ ami (ioost, (irand Total. 10 42 M 0<( (0 (10 (i'l i:w liii :m IJ I.". ilii II II ;>ii .j) '■]'t ;! .■;o 17 Ill ■10 IS I'l 10 10 17 .')0 ■ ) • t ■' ) 10 ■10 .OH 1") ".s I'D 1) 2", ~ I ill. 1 ! - 1 ' 1 IV. S 2i 27 19 31 13 ;i7 10 I I ;i7 - 4 •J 1,0 ir, 51 4 (iS .3 1(1 11 10 •Jl 14 32 132 \r, ■ioi 18 .33 ai 8.5 30 15 48 44 4 20 19 ti7 10 80 5 13 15 28 18 43 40 5 23 23 2.3 827 ¥ m m m 1 m '11 >A: mi 1 60 Zodiacal Constellations. Co-ordinates Star.s of Magnituilo "3 No. Name. R. A. Dec. o 18 N i. Ji. 1 ill. Iv. V. 11 '^ 1 AriPH. The 21am h. m. 2 30 17 •?. Taurus, The Bui 1 i 18 N 1 1 1 s 41 oS 3 ripmiiil TliH Twins 7 8 40 25 N 20 N 1 1 o 7 5 17 Iw :'s 1 Cancer, The Crab lo 5 liOO The Lion 10 20 l;j 20 15 N o N 1 1 1 fi 11 111 21! 21 17 Virgo, The Virgin :;;) 7 Libra. The Balance .... 15 15 s — •' 1 11 1) ";i a Hcorii'o. The Scorniou 16 ir, 2(is 1 1 lu 1 15 'U 9 SaizitlariuH The Arclior 18 55 32 s - 1 ;j 10 25 :m 10 Panripornufi The Go:it 21 20 h •1 I(i ','11 11 A'juarius, The Watei bearer 22 O.s — — ;i 1 IS 25 12 Places, The Fiiihei 20 10 N — — 1 ■A U IS 12 (irand Total 5 [o o7 s;} 22:1 ::,;i SoDTHEUN Constellations. No .'1 A 5 U 7 8 y 10 11 12 13 14 Name. Amelia Piieutnatlca, Tlie Air-pump .... |Mi<) .Sculptor AiipiUMlus Scul|)t()ris, The Appurulus ol' A pis. The I?L'o ... , Ara, The Altar .\ rtjo Navls, The Ship " Aryo " ('tela .Sculplorls, The Sculptur'STixils. Cauls .Mii.jor, The Great Dog CanU Mluor,The Little nog •. tx'utuurus, The Centaur C6tu8,Tho Wliale f.'hauiuleou, The Cliameleon Clrclnus, Th>; Compasses Coluiuba Noaehl, The Dove of Noah Corona Australia, The Southern Crowu. Co- orill natos lt. A. Itec. 35 h. 10 m. 20 ;i2 15 20 70 17 51 i 1 •iO 40 50 42 45 24 1 25 r.:. i;{ 48 •> 12 lu 50 78 15 04 6 25 .•(5 18 M 40 <(.r.\!au i. ii. iii.| 1 1 ~ 1 — • » -> !) 1 • ) 1 1 — 1 • 1 1 5 ~ :i 1 — 1 1 uitmlo. iv. V- 1 ti — 1;! 1 4 ■■■ !• 101 7 l.i 1 1 Ki — 17 1 1 ;i 10 1 1.1 l.'l Ii 27 54 32 17 •2 16 7 61 Southern Constellations. — Continued litlldu ^ No. Name, Co- ordiiiatcs. Star ot Ma;; nltude. "5 V. V. iTi R. A. Dec. 1 ii. iii. Iv. V. h. m. y II 17 •■^ <1 OS lo 10 Corvus, The Crow ii; u 20 2U 18 15 1 1 2 5 K C'raler, Tho Cup •J 7 17 2i 17 Crux Ausiralis, The Southern Cross.. , . 1-2 1,5 00 1 2 1 .> 4 10 ■' 1' 15 IS Itonulo, The .Sword Fish 1 40 (!2 — — 1 3 13 17 11 i;.i ,7 1 i: Eiiuuleus Plctorls, The Painter's Easel. 5 25 'A — — _ ;i 14 17 li» l-'l ;;:) 1 1 [) u,j •21 I'.rldaiuis, The Rivor Erldaiuis 3 40 20 .•JO I 1 1 s t; (14 fiiiwo ForuaxChemlca.ThoClieinical I'ur- ^ 1 1". 01 '" -•"• :i,s oo Grus, The Crane 22 3 20 15 47 57 — 1 1 : 7 a 11 lloroloyiunj, The Clock n ■J Ki IM IM Hydra, The .Snake 10 II 10 — 1 — t 41 40 i IS -, a n i IS •Jo :;ii Hydras, Thy \V'atersnak« t) 21 4U U '.'.5 70 55 20 — — 3 1 1 1 1 It 21 13 '25 TnduR, The Indian 15 Lcpus, Tlic Hare.. 1H \i'i \ L'.'ii I ;;i;4 I'S LnniiH. The Wolf. 1.') "5 45 •) s •','{ 84 J!f Monocero.s, The Unieorn ■ . ,. 7 >) — — 3 u I'J — III) Mons Meusie, Tlio Table Mouutai n 5 20 7.") — - — 1 s itudo. ^ ;;i Micio.seoi)luin,The Miero.-eniie 20 40 )n, Orion 5 30 2 1 ■ > 7 ft 37 1 '• IJ ;'7 I'avo, 'ilie Pi.'ucuek I'J 20 liS - 1 •» 5 1!) 27 « 101 i.;,i 1 i) ti ;!s riKenix. The I'lmiilx 1 21 40 50 ;J2 I o 1 4 o 25 12 3" risfls Australia, Tiie Houtliera Fish.... Jli ■ I't '.7 1 (i '11 riscis V'olans.Tliu Flying FIhIi Iboldal Xol KetleuluH Uliomlioidalls, The Khwni- 7 1 40 08 02 — — 1 1 1 i) i) II ;;!i r,i ■I'J Se.\tan8, Tliu WexlanI 10 — "■ — — 3 ;? Ki 32 i:! TeleKcoiilnni. Tliu Toluscone l.s 41) .'l.'i t ■i H '17 17 II Toucan, The American Goohc 2;5 45 0(i 1 .H 17 21 1 a lU 15 ol 7 15 ITriangle Trlanunhun Auislrulo, The Houtliern ir, 40 115 — 1 .) 1 i 11 1 ■'^' (;rimr Tho Groat Cloiil. Nubes .Minor Tho Little Cloud. Fleur do Lvs.... Tho Lilv. Eobur Cacoli OharUw'' OaU-. Mons.Mcijriolaus Tho Mountain \l(OMa!ti.s. Cor Curoii Charles' Heart. Pixis Nautica Tho Mariner's Cumpassi. Sohtarius Tho rfolitairo. Paaltorium (reor<^ianum tioori^o's Luto. Honorios Krodorici I'ho Honors of Frederick. Sceptum B('an'1onbui\'iciuin....Tho Sceptre of Brandoribnrgh. Toloscopiura Iferschelli Ilerscholl's Toloscopo. Globus Aorostaticus Tho BhUooji. Quadrans Muralis Tho ilural Quadrant. Lochium Funis The iiog Lino. Machina Kloctrica (he Hlcctrical Machine. Officina Typographicu Tho Printini' Prors. Foils ,.. Tho Cat. ' Tho zodiacal const ellnlio/m (12 in number) aro thoso throuj^h which the prolongation of tho piano of tho earth's orbit pass. They each extend for about 30° of a:c, and tho earth in its motion round tho sun in one direction m:ikos the sun appear to move through these constellations in the samo direction. 63 Total. 827 911 2102 illations, ns to by gh. which y each nU tbo I th080 In about B.C. 300 the precession of the equinoxes was not under. stood, and as at that time the line of the intersection of the planes of the ecliptic and cq'ictor lay just within the constellation ot Aries and Libra, the vernal equinox was called the first point of Aries, and tiio autumnal equinox the first point of Libra. But, owing to the retroj^r.ide motion of the equinoxes, discovered in about B.C. 120 by Hipparchus, the first point of Aries now lies in the constellation of Pisces. The relative positions apparently occupied by the stars are affect- ed by different disturbing causes, as follows : — 1. Annual parallax. 2. Refraction. '6. Aberration. 4. Precession. 5. Nutation. 6. Proper motion. 1. Annual parallax is the apparent alteration in the position of a few near stars, due to the relative alteration of the earth's position in its annual movement round tho sun. Its value at any moment is the angle subtended at the star by a line drawn from tho centre of tho earth perpendicular to the line joining the sun and tho star. 2. The efi'ect of refraction (see p. ) is to make a heavenly body appear to be higher in tho heavens than it is. Its effect is greatest when the heavenly body is on the horizon ; it does not affect a heavenly body in tho zenith. 3. The difference of the rates and directions of the earth's move- ment and of a ray of light from a heavenly body causes the latter to appear in a different pcint of tho heavens to where it really is. This apparent displacement is called aberration. Aberration is partly due to the annual and partly to the dicurnal motion of the ourlh. These may bo considered separately. Light travels at a rate of 186,660 miles a eec, and the earth moves in its orbit IS'2 milee a sec; or as the radius to the tangent of 20".4451 (Fig. 19), which is v nlled the eomiant of annual aberration because it IS constant for all stajs. The diurnal aberration varies with the position of the observe.', being greatest at the equator and nonexistant at the poles. The maximum velocity of rotation is at the equator, wheto it is about 0'2!> miles a sec, or . , ,.^ of the earth's orbital at velocity, and hence tho maximum b5h2 coefficient of diurnal aberration for an observer and a star on tho equa- tor is -TjTTiTT or 0'"309 in arc or 0'0206 sceond of time. d5'b2 For an observer in latitude 9', and for a star with declination cfi this diurnal aberration in sidereal seconds of time is for an upper transit. a lower " Ml ]m ml ± 0-0206 cos sec. c- }± G4 ThiH diurnal aberration applied to the R. A. foand in the K. A. gives the apparent K.A. Tbe diurnal aberration affects the R. A. of heavenly bodies by only the above amount ; but, when observing with a transit instrument, its value is so small that it may bo neglected except for a star very near the polo. It may be entirely neglected for sextant and theodolite observations. Planetary aberration differs from Stellar aberration, inasmuch as a planet sensibly changes its place with regard to the earth during the time its light takes to reach the earth. 4. The effect of precession is partly duo to the attraction of tho planets and partly to the attractions of the sun and moon. The attraction of the planets causes an alteration in space of the plane of tho ecliptic, but has no effect on the plane of tho equator. Tho effect of this attraction is to alter the amount of tho obliquity of the ecliptic and to cause a slow progressive motion of the lino of equinoxes along the equator. This movement is called the planetarii precession, and causes a common annual decrease in all right ascon- bions. Tho combined attraction of the sun and moon (that of the planets being imperceptible) upon the protuberance of tho figure of tho earth beyond the inscribed sphere causes the plane of the equator to shift its position and to give the line of intersecton between the planes of the ecliptic and equator a retrograde motion along the ec\\i^t\c, cMod iho luH'-solnr prer^ession. Its effect is to increase the celestial longitudes and to cause a change in the declinations of all heavenly bodies. The exact value of this precession varies accord- ing to the time of the year ; it depends on the varying distances of the earth from tho sun, and the moon from the earth, and tho effect of tho sun and moon is nil when they are in the plane of the equator and it is greatest when they are at their greatest declinations above or below tho equator. The planetary precession is the effect of a motion of the ecliptic upon the equator, which is supposed to be fixed, and causes changes in the right ascensions, tho celestial longitudes, and tho celestial lati- tudes of stars. The luni-solar precession is the effect of a motion of the equator upon the ecliptic, which is supposed to bo fixed, and causes changes in the right ascensions, tho celestial longitudes, and declination of stars. Tho luni-solar and planetary precessions combined give the gen- eral retrograde motion, along the plane of the ecliptic, of the inter- section of the planes of the equator and ecliptic, known aa tho precession of the equinoxes. (See p. 17.) 5. The attraction of tho sun and moon on the earth has another effect called nutation. This nutation is a small oscillating motion of the earth's axis, and therefore affects tho exact value of tho preces 65 of the >quator. lino of lanetan/ gion: It carries the poles Bometimes before and sometimes behind the mean place to which a uniform motion would have brought it, and also sometimes nearer to and sometimes farther from the pole of the ecliptic. Thus, the true path of the polos of the equator is of a wavy form round the pole ot the ecliptic, 0. After every allowance has been made for in the above causes in tho apparent positions of stars, each star is yet found to have an additional real motion of its own, which is called its proper motion, and which carrioHi it among the other stars at a certain rate in a dufiuite direction: Thus, 61 Cygni has an annual proper motion of 6."12 of arc. Aberration, precession and nutation cause an apparent change of position of the stars ; their proper motion produces 9 real one; STAB OATALOaUES. Tho Nautical Almanac gives the apparent places of nearly 200 standard stars for every ten days ot the year, and the apparent position of five close circumpolar ptars for every day, at mean mid- night. The positions of the former ior intermediate days ate found by simple interpolation. It often happens that there are two transits of a star in one day, since the sidereal day is shorter than the solar day. When this occnrs on any day on which the co-ordinates are given in the Nautical Almanac, both are given in small letters, and when the double transit occurs on an intermediate day, the date on which it takes place alone is given in small figures. In interpolating in such a case tho extra transit must never be overlooked. Thc^e standard stars serve all general purposes, but do not include certain stars, which may be observed with advantage in particular cases.* The positions of other stars are to be found in Catalogues of Stars, which are lists of stars arranged in the order of their right ascen* sions, with the data from which their apparent right ascensions and declinations may be obtained for any given date. Several catalogues of the fixed stars have been published by variouH authorities, giving the positions of the stars named at a given date or epoch by means of co-ordinates. Tho most valuable catalogue for general nse is the British Asso- ciation Catalogue (the £. A. C), which contains the places of 8,377 slai-B reduced to 1st January, 1850, and furnishes the constants for deducing the apparent from the mean places with the greatest facility. The Grreenwich five, seven and twelve year catalogues published from time to time are more reliable than this, though less uonaprehensive, and also their formula) for reduction are different. In those catalogues, instead of writing + or — against the declino- * The English Commiflslon who laid out the boundary between the United .states and Canada, had to work out, for use with the zenltb telescope, tlie positions of iniiiiy stars whioh suited their purposes better thao those given In the yautical Almanac, 9 1 . ' f , r •11 m 66 IP tlons of the stars according as they aro N. or S. of the equator, their north polar distances (N.P.D) are given. The N.P.D of stain with a N. declination is loss than 90°, and for stars with a S. declination it is over4>0°. The right a>'consion8 and declinations, i.e., the places in tae celestial sphere, of the so called fixed Htars are in fact over ohangiog, Ist. — By precession, nutation, and annual aberration, which are not changes in the absolute positions of the stars, but are either changes in the reference circles of the celestial sphere (^us proccsRioo and nutation) or apparent changes arising from the observer's motion (as annual parallax and aberration). 2nd. — By the proper motion of the stars themselves, which is a real change of the star's 21 )luto position . In the catalogues *he places of the stars are referred to a moan equator and a moan jquinox at a given ejioch. The place of the star so referred at any time is called its mean place at that time; that of a star referred to the true equator and the true equinox, its true place; and that in which the star appears to the observer in motioD, its apparent place. The mean place at any time will be found from that of the catalogue simply by applying the precession and the proper motion for the interval of time from the epoch of the catalogue. The true jilace will then be found by correcting the mean place for natation ; aud finally the apparent place is found l»y correcting the true place for annual aberration. The reduction of the apparent places of the stars from the mean places given in a star catalogue is effoolod by one of three different methods, as follows: — I. By B.A.C. constants. II. By independent quantities. III. By the constants in the Greenwich caialognes* (I.) When the B. A. Catalogue is used the reduction is as follows: Let a* = Mean right ascension (in arc) of star at^ epoch of catalogue. 5^ = Mean declination (in arc) of the star at epoch of catalogue. p = Annual precession (in arc) in right ascension. These s = Secular variation (in arc) of p, or amount quantities by which p changes in lOO years. f-areall given pi = Annual precession (in arc) in declination in the cata- or N.P.D. logne for fii == Secular variation in p^ , the given q zzz Star's annual proper motion (in arc) in epoch. right ascension, ji = Star's annual proper notion (in aro) in declination or N.P.D. 67 ■>x, it8 fm n = Number of years from epoch of oatalogae to boginniog of given year. Oq = Mean right aBoeneion (in arc) at beginning of given year (epoch 4- n). ^o = Mean declination (in arc) at beginning of given year (epoch -f n). A + sign to p, q, s, Ac., means a progressive motion, and a — sign a retrograde motion. Ttie mean right ascension and declination for the epoch of the catalogue are reduced to that of tbo current year, by adding as many times the annua! precession and proper motion in right ascen- gioD and north polar distance (or in diolination) as the number of years elapsed since the given epoch ; and n :^ number of years from lut Jannary, 1850, or more correctly, thus— flo = <* 100 n' So = h' + (j>' + 'nn + ■9" Then if t is the time from the beginning of the given year, reck- oned in fractional parts of a tropical year (soo table in Nautical Almanac) from the moment when the sun's moan longitude is 280'^, which moment is given in the Nautical Almanac; and a = Apparent right ascension (in arc) of the star at the required time of the year. ^ = Apparent declination (in arc) of the star at the required tine of the year. then the following forrauln- have been proved : — a = a„ 4- ^!Z + Aa + B6 4 Cc -f Di + B. S=ho + ^/' + Aa' + Bb' + Cc^ + Dd\ in which a,/>,c, d, «*, 6S c', d' are certain functions of the starh' right ascension and declination, and must therefore bo computed for each star ; but Uh they are sensibly constant for a long period of years their logarithms are given with each star in the eatalogue. A, B, C, D and E are ! he functions given in ihe Nautical Almanac, of the sun's true longitude, the moon's true longitude, the mean longitude of the moon's ascending nodi , uud the obliquity of the ecliptic, all of which depend on time, and may therefore be regarded as functions of time; the logarithms of these are given in the Nautical Almanac under the heading of " Apparent Places of Stars; Bessel's Day Numbers," for mean midnight of everyday of the year ; the correction E is usually neglected, as it is very small, being less than0"'06. In order to obtain the correction of the mean place of a star we have only to take from the catalogue opposite to the given star the logarithms of a, b, c, d and of a*, ft*, c*, d^, with their proper signs; and to write down under them respectively, from the Nautical Almanac, opposite the given day, the logarithms of A, B, G, D, with ^'\ ft HSi 'I' If* their proper signs, romerabering that the signs prefixed to the loj. arithvis affect only the natural numbers. Then add each pair together and tiud the uaturul numbor ourrenponding to their sam. ThoHnm ot the four natural numbers thuH obtained (regard being had to their tignti) will be the total correction required in right aHcension anil north polar distance on the given day. This correction applied to the moan place of the star at the beginnin^r of the year will gm apparent place of the star on the day required. (II.) Now a, b, c, d, a*, b^, c\ >coh of catalo^uo. Pq = mean N.P.D. (in arc) of stsir i; . beginning ot given year (epoch + n). P = apparent N.P.D. (in arc) of star at the required time of the year. Py is found from P^ just as 5„ was' found from 5i on page. Then— a (in time) = a^ + tq + Ee -^^ Ff + Gg + Hh + L + ^—30000. P (in are) =P, + tq' + Ee^ + Ff ' + Qg^ + Uh^ -|- L-\- n —300-00. The numbers E, F, G, H, L, are always positive, and their log- arithms are given in the JSautical Almanac for means of midnight of every day of the year, under the head of *' Apparent Places of Stars; Airy's Day Mumbers;" e, f, g, h, I are positive (except lor a very few stars within 3* 10' of the pole), and e^ ^f^ , g^ , h^ ^ li are alno always positive, and as their values remain sensibly constant for a long period of years, their logarithms are given with each star in the Greenwich catalogues. The places of all stars given in the Nautical Almanac are the cor- rected apparent places given by the above formuho, i. «., in the co- ordinates given, the effects of precession, nutation, proper motion and annual aberration have been allowed for. 69 Dinrnal aberration only afitcts the apparent right asoension of Htars, and the correction for it in time in latitude <^, and for a star with deolination ^ is ± 0-020()S. cos; (^ sec. 5> + for an upper transit and — for a lower one. This correction applied to the R.A. given in the N. A. gives the apparent B.A. EXAMPLS OF RKDUCTIOir OF THE APPARENT PLACES Or STARS BT THE ABUVE TUBEE METHODS. Required the reduction {/\a) of the right aBcension and (^8) of the declination of y Orionis (No. 1687 in the fi.A.G.) for precession, proper motion, aberration and nutation, at Greenwich mean mid- night, on 5th December, 1882. h. m. 8. Moan a, Ist Jan., 1850 = 5 17 5 33 32 years' precosy conccn*"ic circles, with the polo as the centre, and of radius, ..:radiiis ol equator X cos. declination. On this map the positions of tho stars required for observation are plotted, the right ascension ciiclos being numbered in the opposite clirection to the direction of tho hands of a watch, as that is the way tho stars appear to move when wo look towards tho North Polo. Now, on tracing paper draw a storoographio projection from the Humo point of projection, showing the meridian lino, with tho ))()sitioiis of tho polo and zonith on it, and the prime vertical (all of which aro easily found), and, if necessary, the other azimuth circles (which aro rather hard to draw, from tho long radii of some of thorn) ur.d tho parallels of altitado (which are easily drawn). To UKo ttie instrument always keep the polo of tho tracing on the pole of tho map. Then, if tho meridian line of tho tracing paper is ])lacod on any star we soe at once the local sidereal time whon that star will bo on the meridian, what stars will be in the prime vortical Hi tho same time, and their approximate altitudes in these positions, I tho meridian lino of the tracing paper is put on any sidereal hour lino (m;., right ascension circle) we see at once what stars aro on tho meridian, which have passed, which are coming on, which are in the prime vortical, and all their approximate altitudes at that inutant. 10 CHAPTER VI. THE USE OF THE SEXTANT, THE THEODOLITE, AND THE CliaONO.AIETER IN ASTUONOMIOAL OliSKRVATIONS FOR SURVEY PCRrOSES. — TUB HOLAR COMl'ASH.''' Tho principal uboh of practical astronomy to the Fiirvoyor are that it onablos him to aHCortain hin latitude, lorgitudo, local clock time, and the azimuth of any given lino; tho hittor of course giving him tho true north and eouth lino and tho variation of the comp.ihH. In fact, tho only check ho haw on his work an regards direction when running a long straight lino across country is by determining its true azimuth from time to time, allowing (as will bo explained under tho head of (Jeodosy) for tho convergence of meridianR. Tho instruments usually employed are tho transit theodolite, sextant or reflecting circle with artificial horizon, solar compass, portable transit telegcopo, and zenith toloHCopo. To these must bo added a watch or chronometer keeping mean timo, a sidereal time chrono- meter (this is not, however, absolutely essential), tho Nuuticul Almanac for the year, and a sot of mathematical tables. With the stxtant or reflecting circle we can measure altitudes and work out all problems depending on them alone, and also lunar distances. Tho transit theodolite may bo used for measuring altitudes, and also givoH azimuths. Tho solar ccmjiass is a contrivance for finding, mechanically, tho latitude, meridian lino, and sun's hour angle. The zenith tetcsrope gives the latitude with great exactness, and is particularly suited to the work of laying down a parallel of latitude. Tho transit telesctj't enables us to determine tho moan and sidereal timo, latitude uiid longitude. The transit theodolite aaswors tho same purpose, but Ih not BO delicate an instrument. It is, however, of almost univori-al application, and nearly every problem of practical field astronomy may be worked out by its means alone if tho observer has a fairly good ordinary watch. The sextant is not so easy to mnnngo, and only measured angles up to about IK)"^, so that &8^' is pnictically the greatest altitude that can bo taken with it ^rh^n tho artificial horizon has to bo used. In this chapter we will not consider the frar-S't Instrumcn;*' and tho zenith telescope, but will deal with thum sepUiur ou. • For permanent ohgervntory work, the HtudeiitshouM stu I] kucIi works M Cbauyenet'H Astronomy, etc. 15 THB SEXTANT. For the constructioD and adjaBtmenta of tho sextant and other reflecting instruments (see Survey course), wo will here only oongider how it is to bo used for taking astionomical observations. Tho sextant is a doable reflecting instramont, and is well adapted to tho purposes of the explorer, for it is at once portable and extremely simple of manipulation, requires no fixed support, and furnishes its data with the least expenditure of time to the observer. Boing hold in the hand, the ext'.eme accuracy of the larger fixed jDHtrumonts is not to be oxpeo..ed from it, but, when ased by a praolisod observer the precision of the results obtained with it is often surprising. For accuracy and portability combined, an 8-inch sextant, reading to 10", is probably tho best size. It should be by one of the best maivors; the handle should be roomy, and if the frame be bronzed ijo much the bettor. The 6-inch sextant, also reading to 10", ia preferred by some on account of its greater portability. Tho prismatic sextants of Pistor and Martins, of Berlin, are some- what larger than the 8- inch ; they offer some advantages of con- Mtruction, and measure angles up to 180''. They would be speoially useful in low latitudes, whore the sun passes near the zenith. Ecflocting iiistrumonts muHt be used at sea, but on land they would not be u^cd whon suitable iion-reilocting instruments are available, for they are not so reliable in principle or, in fact, of construction. TUB DETEUMINATION OF THE INDEX EftROU. The index error may bo found as follows : — Iht. liij a star. Bring the direci and reflected images of a star into coincidonco and road ort" tho arc. The index error is numeri- cally equal to this reading, and ia positive or negative accordingly UH ilio rouaing is to tho loft or tho right of tho zero of tho are, i.e., UH ii iH on or oil' the arc. Tliis method may bo usod with tho sea horizon or other distant objcH't in:;cuaii of a star, but not with so groat precision. 'Jnd. By the &un. Measure the apparent diameter of tho sun by lirNl bringing tho u])por limb of the reflected image to touch the lower limb of the direct image, and u^ain by bringing the lower limb of tho rolioctcd imago to toach the upper limb ot the direct imii^o. Doiiolo the readings in the two cases by r on tho arc, and r' oil the arc; then if 1) = apparent diameter of the sun and x is the reading uf the Hcxtant when the two imagus are in coincidence, i.e., tho index error, then whoa r > D wo have — r := D + x r'=D — x T "^ I*' T '\^ r* or x = -| r, — and ^ D ;= — ~— • M i Tho practical rulo dorivod from thiH is as follows : If tho reading in eithor case i» on tho arc mark it with a -f- Hign ; if it is otT tho ftro (i c, on tho arc, of excess) mark it with a — sign ; then tho index correction is one-half of tho algebraic sum of tho two readings with the oppoHito Hign.* For example — Reading on tho arc + 30' 40" oir " —30' 12" u Index correction 2)34" — 17" or HuppoHO again Heading oir tho arc — 28' 22" a a — 38' 3r." "'2)6f)' r>8" Index correction -f" 3ii' 29" To tost if thoHO angles have boon taken with toIor»»Mo accunicy, add tho readings together without regard t'» ul In ol uii i>|<|iiih!Ii> sIkii. t The »'rrorri ro«nItiiig from lli»> oitiinl coiiHtriictioM of tli«< s^'xlaiit far exotMMJ thorti' arlHiii^' lr<)m " (kthoduI wrov " ami IrDmcxpanHiou ot'tiie metal during the oltsiuvutiun. w reading off tho ho iiulox ngs with K'cunicy, i(i (livido (lay, an can rcnil 1 may Ito varying' ion may iitision of tlnir tho 1)0 ahrcr- vaiiio of u.iEOTs is l>oiii^' lui'oinciit I l>y tho ) iH, that on ^lass . ThilH. iitiy two ll " I'Olll !■- Ilsll.' SiKI, tlniit ffir lieinctul 11 objects in the same vertical plane, or the left hand object of any two in any other plane, muHt be the one directly observed; and when hold in the left hand, the highest of any two objects in the same vortical plane, and the right hand object of any two in any other piano, must be the one directly observed. In the last case the instrument is inverted, so that the left hand is uppermoht. Tho diHongaged hand is used in moving the index arm, and working tho tan>^ont screw on the vernier. The ^hole secret of "observing " well with a sextant consists in being able quickly to place tho instrument in tho plane passing .through the two objects and the observer's eye. In measuring angular distances between objects, \7e want, as a rule, tho angular distances between thoir centres. To do this, if the budios have a sensible diameter, like the sun, moon and planets, tho angular distance botwoon their edges, or limbs, as tlioy are called, is observed, and tho value of the t>omi- JWiutictil Almanac ; tho distance tlms found is, in tho case of tho sun and moon, to be diminished by iho sum of tho semi- diatnetors of the two bodies (say 3L5'), and in the case of tho moon and ustar or planet it is diminished or increased by the moon's semi- diameter (say Iti'), acconlingly as the briglu limb is nearer to or farther from the star than iho moon's cenire. This nroci also a check against the mistake of employing the wrong star ing • Kxtroiiio brlHlituusa of uii ulijool ulfucU lU miiiarout kIzu to Iho eju. Hec pajfo 82 for thon oloHCope, o that of Ji'votl the oiisity of )d bhudoH bo found tolONtJopo ?o of tho on error la^OH too imu^os 10 toloH- ) futijruo hould Ijo CO of tlio A\0 ^MOUt that thiH iho HOX- ^lit lino to Kwoop tfio 8tur ontro of tho star ih of Lho Lap, ovoii n", boforo Dtl iina^o > ncaronl tod, and 10 index iing tho may ho H, ill tho ho Homi- 10 moon I'MHOini- or to or oding Ih ur. i}pB|;u H'2. w If thcTO is any uncertainty nbont which limb of tho sun is being taken, wo may tost by trial and examination of tho graduated arc, or by knowing tho direction of working the tangent ficrow to increase or (locreaso tho angle^ The larger angle means tho sun'n outer limh. J)iHtanre« may bo measured with an ordinary sextant up to 120°, and with a Pistor and Martins' sextant up to 180**. About full moon, and when tho moon appears quite round to tho oyo, a mistake may readily bo made, and distances ho observed from tho wrong limb of the moon, but as tho Xaufiral Almanar in page XII. of ouch month, gives tho Greenwich time of full moon to tho tenth of a minute, it is easy to avoid such an error. We can always tell, up to within a fow hoars of full moon, which is its bright lin b. If thnro is any doubt wait for a more favorable oppor- tunity for making tho observation. OhHorvations for time and azimuth i hould, if possible, ho made when tho colcstial ohjoc.t is nearly, K. or W., and for latitude, N. or S. In ohnorving lunar distances it will bo found easiest to bring on tho (iistanco of the moon, from whatever object may ho ohsorvod with it, hy looking through tho ring of tho toloscopo support, and bringing the olijoct as near as possible by tho naked oyo, thon screw in tho tcloHCopo, and looking through it, make contact porfoot hy tho tan- gent screw. Tho toloncopo should, of coureo, bo set to the proper focus, which will ho tho case when on looking at a fixed star it will apjioar sviallt !Ht makor cannot ovorcomo, but wlUch oan bo les- sened by two obHerTatlona with ruverHed pidten (nee p. S2). i When lllterluii morcury, one or two pln-liole« must bo made lu the bottom of the paper. planes.* Id Dot be or of tho n is fore* acid ; or i to a fine h a piece iiaed, and , into a lerod loaf )uro, mor- a untrae. ry should ;rumonts, nos used, vol led, a horizon- ink, and tbo an^lo iry of tho I altitude, id instru- by 2, and parallax b of one tho 070 ICO of tho 7 ex\nt in v'\-/ : — not tho rays il horizon, 1 can SCO t you aro - zero, HO >acopo on htly, and oan bo leN- * bottom of 81 800 wliich fi:in moves ; and now, while bringini^ tho head slowly down towards the hori/. )n, Uoop thirf irnai^o in tho toh)Hcopo by push ini<,f()r ward ^^radually llio iiidox ;irni, until you havo tho tolo- Hcopo diroctt'ti ti-w.irds Iho artificial horiz )n. It the head and tho HcxiiMil aro ris^hlly Hiiiiatcd tho m«M'('ury aha will bo at once soon in tho tiolii of tho loIoHoopo, us well as iho rcilootod sun (rom tho index glaHS. It iIjo iMorcury huh is not Hoon without altorint^ the position of the indox arm, romoro tho toio^copo from the eyo, place the head so thai tho moroury hun i>4 |ihiinly utioti with tho observing eyo, and then, icHhout rnorimj tlw he(h/, bring tho soxtuni up to tho eye, when tho mercury sun is at onco soon ; and now, while slowly revolving tho Hoxtant for a Hlmrt dir'tanco ri^ht ami loft ab'vut thotelcscopo as an axis (dono by the wiist only), tho tolo^('apo being held steadily dirortod on tho morcury Hun alt tho time, slowly raovo tho index arm bacUwurds or fVirwurds, until tho ictloction sun is soon to shoot into tho field. When thoy neaily touch, clamp the indox arm and UHo tho tangent slow motion sitow tu got an accurate contact. Tho rcH-Ciod sun from tho index arm is easily known by its Iironi|)tly vanishing if the soxt::nt is rovolvod round tho telescope as an axis, as above dosoiibi-d. Tho beginner often fails to fidd tho sun in tho mercury with tho toloHCopo, because ho forgets that ho cannot see the sun in tho mor- cury with both eyes nt tho same moment, oq account of tho narrowness of tho aititi iai hori/.jn. When obsor\iiig ono of the limbs of tho sun or moon care must ho taken as to whifh limb is ol)sorvod. If the eyo-piece is an inverting ono (as is iMually the c:ise) iho limbs, as soon in tho telescope through the hori/- iii-gla^s direetly from the surface of tbo mercury, aro tlio siuno as Uio ti uo upper and lower limbs (seo Fi(j. I'L'), while the reflected image from tho index-glass has its limbs inverted. 'I hirofbre, iho upper and lower limbs aro respectively oliHorvcd (with an iuvertwig eyo-j>iocc) when tho movable or ledoctod object is ubo'o or below the liicd object seen in the mercury. Tho revci.so in the case of tbo oyopioco is not an inveiling on-*. If there is any doubt as to which limb is being observed it may bo tested by trial. Tho larger a'lglo moans the u|)per limbs have been observed. We can also tell by iho direction of tho motion of the tangent scrows as U) whether tho angle is getting larger or smaller by working it. Wiieii a scries of alUtu ies aro to bo taken, cju'ckly ono after another, wo havo only to louk at tho image in the mercury after reading the a;iglo in onlor to find both images on rotating tho telesc(.»po; failing this, ji very slight movement of the indox arm away Irom or tovvards tiie ohsorv*;!', according as tho bo'.'.y is rising; or fulling, will find the reflected image. 11 I M IMAGE EVALUATION TEST TARGET (MT-3) 1.0 _ _ mio I.I : lis Ifi 12.2 M 1 ''^^ ill >•'< 1-6 c Hiotographi! Sciences Corporation 23 WIST MAIN STREET WEBSTER, .4.Y. MSSO (716)172-4903 \ iV 4 '^^ o .'**» ' ■^ ^ <«^\ ;\ ^ o^ 82 i ! Altitudes of the stars, planets and moon are found in exactly the same way ; but in obsetving the altitude of the moon great care must be taken to observe the altitude of the brightest limb. Fig. 20 ihows that an upper or lower limb is always available. In practice it is inadvisable to observe the altitude of the moon for calculations for time, latitude or azimuth, on account of the rapid and variable change of her co-ordinates, requiring interpola- tion to second or third differences, and because the moon's coordin- ates given in the ^.A. are only approximate, as compared with those given of the other heavenly bodies. The moon is generally used only for finding longitude. If the glasses of the artificial horizon are at all prismatic the observed altitude will be erroneous. The error may be eliminated by observing a second altitude with the root in reversed positiot, and, in general, by taking one half of a set of altitudes with the rod in one position and the other half with the roof in reverse position, It is easily proved that the error in the altitude produced by the glass will have different signs for the two different positions ; so that the mean of all the altitudes will be free trom this error. This reversal is unnecessary in taking pairs of, or balancing, observations, tor errors arising from tho horizon roof are eliminated by using tho mean of the observations, but the same face of the cover should be used next the observer in the two observations. On account of the feeble power of the sextant telescope and consequent imperfect definition of the sun's limb, the apparent diameter of the sun is somewhat increased. This error may be removed by taking the mean of two sots of altitudes, one of the lower limb and one of the upper limb, except in combined or balancing observations, when tho same limb need only be observed for reasons given above. ObservatJars may be rendered more accurate by tho use of a stand to which the sextant can bo attached, and which is so arranged that the sextant can be placed in any required plane and there firmly held. The manipulation must bo learned from the examin- ation of tie stands themselves, which are made in various forms. If a celestial body is less than 10*^ above the horizon the mechanical construction of the mercurial horizon will not allow of its double altitude being taken, but from what has been said under the head of Hefr.iction, it has been pointed out that altitudes under this amount should not bo taken. With an ordinary sextant and artificial horizon, approximate altitudes of from 20° lo 60° may be measured, or up to 90° with a Pistor and Martins' sextant. In the observation of tho altitude of a star with the artificial horizon, it requires some practice to find the image of the star re- flected from the sextant mirrors ; and sometimes, when two bright Btars stand near each other,there is danger of employing the reflected 83 xactly the reat care Fig, 20 >. the moon |\t of the interpola- 3 co-ordin- vith those rally used imatic the iliminated position, 1 tho roct ) position. 3d by the itioDS : so -or. This lorvations, using tho should be scope and apparent V may be >ne of the obined or J observed use of a > arranged and there e examin- forms. rizon the t allow of aid under idcR under xtant and j° may be artificial le star re- vvo bright ) reflected image of one of them for that of the other. A very simple method of avoiding this danger, by which the observation is also facilitated, has been suggested by Professor Knorre, of Eassia. From very simple geometrical consideratious it is readily shown that at the instant when the two images of the same star— one reflected from the artificial horizon, the other from the sextant mirrors — are in co- incidence, the inclination of the index glass to the horizon is equal to the inclination of tho sight-line of the telescope to the horizon glass,, and is, therefore, a constant angle, and hence is the same for all stars. If, therefore, we attach a small spirit level to the index arm, 60 as to make with the indox glaHs nn a^gle equal to this constant angle, the bubble of this level will be in the centre of the tube when the two images of the star are in coincidence with the middle of the field of view. With a sextant thus furnished we begin by dire^ .ng the sight lino towards the image in the mercury ; we then move the index until the bubble plays, taking care cot to lose the image in the meicury ; the reilccted image from the sextant mirrors will then be found in the field, or will be brought there by a slight vibrating motion of the instrument about the 8ight line. It is found most convenient to attach the level to the stem which carries the microscope or reading-glass, as it can then be arranged so as to revolve about an axis which stands at right angles to the plane of the sextant, and thus bo easily adjusted. This adjustment IS eftected by bringing tho two images of a known star, or of the eun, into coincidence ; then, without changing the position of the instrument, revolving the level until the bubble plays. When contact has boon made with the lower limbs the images will begin to separate if the altitude is increasing, but if the alti- tude be decreasing they will begin to overlap ; in observations made with the upper limb tho reverse is th« case. Further, it will bo found much easier to wait for contact, in order to note the moment of it, when the images of the body observed are separated and approaching one another, than when they are over- lapping and separating from one another. The reason of this is that when two bright bodies are separating from one another the impression on the eye is, as a rule, that they still overlap after actual separation has taken place; therefore, that limb should, when possible, be observed which will give the required result. Evidently, with tho same kind of instrument the opposite limb will have to be observed in the morning and afternoon observations, i.e., when the object is rising or falling, to secure this condition. Waiting for contact, however, is only practicable when the celestial body is alter- ing its altitude rapidly; when it is near tho meridian the motion 18 too slow. From this fact we get another way of observing altitudes and distances, which is only successful v/ith quick-moving heavenly bodies on cloudless nights. It is to set the index to certoin anglea 84 in BnccesBion, according to ihe rate of movement of the object, and to wait nntil contact is made. Thus, there are two methods of taking altitudes: — (1). By bringing the images into contact, as already described. (2). By allowing them to make their own contact. Each method has its own advantages. In the latter case there will always be a pretty conetan*^ + or — personal error, and which can be eliminated in the former case by bringing the reflected image alternately into conttict from above and from below the direct imago ; but the contact as made by the hand with the tangent screw ie not so sure as when made by the heavenly body itself in its progroes. From this we see that the latter method can only be employed in combined or balanced observations, which are more particularly referred to below. The method of taking an altitude of the sun with a sea horizon is as follows: — Direct the telescope towards that part of the horizou which is beneath the ob'ect. Move the index until the image of the object reflected in the sextant mirrors is brought to touch the horizon at the point immediately under it. To determine this point, the observer should move the instrument round to the right and left (by a swinging motion of thfl body, as if t, .ning on his heel), and at the pame time vibrate it about the ^ighl-lino, taking care to keep the object in the middle of the field of view; the object will appear to sweep in an arc, the lowest };oint of which will bo raado to touch the horizon by a suitable motion of iho tangent screw. The lower limb of the sun and the bright limb of the moon aro the limbs usually made to touch the hori/on. The apparent altitude of the point observed is fonnd by correcting the sextant roading for the index and instrumental oriors, and sub- tracting the dip of the horizon (see p. ). To obtain the apparent altitude of the sun's or moon's centre wo must also add or subtract the apparent semi-diameter. QiNEBAL REMARKS ON OBSERVING WITH TUB SEXTANT. Endeavour with much forethought to balance the observations, uf>ing the same telescope, horizon roof, &c., both in taking equal altitudes east or west of meridian, or north or south of zenith, and also in taking lunar distances east and west of the moon. In this way the observations will be balanced or in pairs, and the mean of each pair will tend to be independent of all constant instrumental and refraction errors, and by comparing the means of these pairs with one another you know your skill as an observer, and can esti- mate with groiit certainty the accuracy that 'ho results have reached. Never rest satisfied with observations unless you feel sure tbat you know the limits of accuracy of your work. When observa- tions are reoorded mark them "good," "very good," "doubtful," &c,, by their sides. 85 ABRANQEMENT OF 0BSKBYATI0N8. The surveyor who remains only one part of a day and one night at the same spot mast make the best of his opportunities, and take observations for time and latitude as they ofifer themselves. For time the heavenly bodies sboald be near the prime vertical east and west of meridian ; for latitude, near the meridian, north and south of zenith. In observing, it is desirable to settle beforehand what stars, &o., are available^ All work should be laid out and arranged before hand, as also the order of the observations. Consult the Nautical Almanac to see at what times standard stars of the requisite magni- tade will come to the meridian on both sides of the zenith. A rough list, corrected from day to day, will be found very useful, in which can be inserted the times at which a few east and west stars are on the prime vertical, as well as the hours of calmination of those that can oe used on the meridian, A " star-finder " (see p. 13) is useful for obtaining this information. For latitude, two of these stars, at least, must be observed, one on each side of the zonith, according to the method of circum-meridian altitudes ; or, failing this, by single altitudes within one hour of the meridian. When Polaris is visible it can be observed for latitude at any time, the result being combined with those obtnined from stars Bouth of the zonith. During the intervals of observations of the stars near or on the meridian for latitude stars in the east and west are taken for time, the results being combined ; six observations being taken of each Stan INSTRUMENTS AND MATERIALS BEQUIBO FOR OBSERVATIOMa. Sextant complete. Artificial horizon complete. Memorandum or observation book and pencil. Thermometer i ^* ^"^oroid barometer with attached thermometer. Pocket chronometer (beating 5 times at the most in tw« seconds). Dark lantern for night work. POINTS TO BE NOTED IN OBSERVATION BOOK IN TAKING SiiXTANT OBSlQRyATIONS. 1. The nature and purpose of the observations. 2. Date and place of observations. 3. Approzijiate latitude and longitude. 4. Number of sextant used. PS 86 '■A -'^ I 6. 7. 8. 9. 10. 11. 12. 13. 14. in. 16. 17. Indez, and iDstramental errors of Beztant, and observations in detail by whicb the former has been found. No. of chronometer used, and whether adjusted to sidereal or mean time ; rate at which it beats. Chronometer correction at observation. Barometer and thermometer readings. Name of object and the limb observed. Its position, whether east or west, or if passing the meridian, north or south of zenith. Observed double altitudes. \ In a form (see below). Beats elapsed. Observed times. Corresponding times to observed double altitudes./ State of weather and sky. Statement whether the observations are good or not. Comparisons of watch or chronometer with standard, both be- fore and after observation. It may bo laid down as a standard rule that no astronomical ob- servation is ever taken without the chronometer times of the obser- vation being also noted. Everythin)? should be noted which affects an observation, so that it can be worked out by an independent person if necessary. of St METHOD or TASING SEXTANT OBSXRYATIONS. We will now consider in full the case where there is no assistant. A pocket chronometer should be used, on account of its measured and loud beat. It should not beat more than five times in two seconds, for convenience of oonnting the beats. In night-work a lantern is required to read and note the times and angles, but during the observation its light must not show, or else staru cannot be seen in the mercury. The observation book is prepared thus : — Double Altitudes Beats. Observed Times. Corrected Times. O 1 II h. m. B. h. m. B. and all the information possible already given above should be entered previous to beginning work. ktions in ereal or a form •below). 8T Select a conTenient spot near camp, where there is no danger of the observer being disturbed daring bis observations by an inroad of strange animals ; it should be well trampled down, smooth, and should not be overshadowed by trees or rockp, and tall brashwood near should be broken down. A carpet 4 feet square is now spread out, and about the centre is placed the artifical horizon, the quick- silver having been strained into a clean trough through a piece of chamois leather. If the night is calm the glasd cover may be re- moved ; but, as a breath of aii disturbs the surface, it seldom can be dispensed with, except in the tropics. Boggy ground will keep the mercury in a tremor ; so will also the stamping of horses or dancing of natives, if very near. The surveyor may choose his atti- tude ; for a young man tlie sitting cross-legged, tailor-fasbion, is recommended, as it secures very steady and rapid results ; but it 's cruel work for the spine and thighs if the observations extend over two hours, as they sometimes may. The observer sits down opposite the mercury, and arranges himself so that when his head is bent down he can see the reflected image ; on his right is the sextant-box, and on it the box of the artificial horizon ; on this is placed the chronometer, propped up towards the observer, so that its face can be distinctly seen and the beats heard. If the work is at night the dark lantern is placed so as to face the chronometer when its light is turned on. The observation book is ready, ruled, opened at the proper page, and placed in a convenient position ; if an assistant can be impressed into the work, who can jot down the observations correctly, so much the betten The observer first focusses bis telescope, puts down the suitable shades if required before the index and horizon glasses, and tries for index correction. He then, with the index arm set approximately to zero, directs the telescope of the sextant upon the body to be observed in the heavens, and swiftly, by turning the index arm and moving the instrument at the same time, brings down the reflected image of the body until it coincides with the image in the mercury, seen directly through the unsilvered portion of the horizon glass; this becomes in practice quite a mechanical process, if the body observed is near the prime vertical he can let it make its own contact, this being the more certain method ; if it is near its culmination he must maka contact with the reflected and mercury image alternately uppermost, by this latter method doing away with all personal error. From the moment of contact, the observer who has been listening to the beats of the chronometer (making 5 beats in 2 sees, suppose) will, while bringing down his sextant, and bringing his eyo on the chronometer, count 0, 1, 2, &c., up to 10 or 20 beaLs, and note the minutes and seconds of the last beat in the column of " observed times," the hours may be entered afterwards. The times of observation are thus all noted 10 or 20 beats (t.e., 4 or 8 seconds) '^U 88 beyond the chronometer time, and no notice of this is taken until after the mean of all the observations is worked out, when the 4 or 8 seconds are subtracted. Another way of recording the instants of observations is for the observer at the moment of contact to take up the count 0, 1, 2, 3, &o., with the beats of the chronometer, while bringing down hia sextant, and bringing his eye on the chronometer (turning the light on it if, necessary), and to note in the observation book, in the column of " beats," the number he has counted when the second hand of the chronometer roaches a convenient figure on the dial, which time is also noted in the column of "observed times." Eead, and note down the sextant arc in the column of « double altitudes" and proceed with the next observation. The observations are taken and noted as fast as is consistent with accuracy ; the rapidity will differ considerably, not only on account of clouds, mist, &c., but aUo with the health of the observer. Hand and eye cannot always act closely in concert if the mind is strained, or the body wearied with undue labor, and the observer must learn to see with equanimity that his best efforts cannot always bring about good results. After the last observation has been taken, deduct the number of beats reduced to seconds from the " observed times," and enter the results in the column of " corrected times," which are the actual times of the observations. With an assistant an observation can be taken, even at night, every half minute, but they should never be taken at longer inter- vals, when observing objects well away from the meridian, than one a minute, even if there is no assistant. When observing objects on the meridian for latitude, the observations need not be closer than one every two minutes on account of the slow change in altitude of the object observed. With an assistant to book the observations, and if great accuracy is not required, just before contact is made the observer says, " are you ready," to warn him, and then, when contact is made, he says, " now," tor the assistant to note and write down the time. Then the angle is iread and written down, and the next observation pro- ceeded with. All observations should be worked out on set forms, and in all cases the mean of a set of, if possible, 6 observations at least (if taken within & or 6 minntes of one another) should be taken as a single observation. Observations taken over a greater interval than 6 or 6 minutes should not be grouped together to form a single observation, as the arithmetical mean ot them ail will not correspond to the arithmetical mean of the times, owing to the fact that the rate of motion of a celestial body in altitude varies with its altitude, and so does not remain constant. The disadvantage of calculating from only one observation is, that slight errors of observation, inaccuracy in reading, or in noting the 89 exact instant of contact, may accumulate to vitiate the result ; whereas, the chances of accuracy are ic creased by numerouH observations, and a more accurate mean is probably arrived at. If the method of taking the altitudes at equal intervals of arc bo adopted, the symmetry of the observations prevents the probability of any grave error, such as may sometimes occur with only oue obdcrvation. It will be found of the greatest practical advantage to observe the same stars over and over again, night after night. Many of the logarithms will be constant, and the star will become so I'amiliar as to bo caught with the greatest ease. INSTRUCTIONS FOR USE OF, AND FOR OBSy.UVINO WITH A BEXTANT. 1. A good observation depends on well defined images being obtained by careful focussing, on their being so faint as not to fatigue the eye, and on their being of as nearly equal intensity as possible. 2. Focus the telescope before screwing it in. 3. The intensities of the two images should bo reduced and equal- ized by the use of the colored shades, and by moving the telescope by the mill-headed screw for the purpose, until the sight line is directed through the unsilvered portion of the horizon.glass, so that a portion of the reflected rays are lost. The latter method is generally necessary in measuring the distance between a moon and a star or sun ; for the latter body the index dark glasses are used as well. 4. When observing the angle between two objects always direct the telescope on the fainter one, so that the brighter object suffers by double reflection, 5. Take care there is plenty of play in the tangent screw in the direction of the measurement. 6. Never forget to use the dark glasses or colored eye-pieces in solar observations, or loss of sight may occur. 7. When observing altitudesof the sun use the coloured eye-pieces, to avoid er-ors arising from any non-parallelism of tho surfaces of the dark glasses and from their not working at right angles to the rays passing through them. 8. Find the index error before and after each set of observations. 9. When the artificial horizon is used the cover should be reversed for each half of the observations, to remove errors arising from any defects in the glasses, except in balancing observations ; then you must be sure to have the same face of the cover towards you on both occasions. One of the faces should be marked for this purpose. 10. If a high wind is blowing the mercury is apt to be ruflied by it, if the cover does not accurately fit the ground. Loose earth or sand heaped round the cover makes a wind-tight Joint. 12 ■;ri \ r- 1 z' 90 11. Opinions differ as to whether tho separation or first contact of images is most accurately observed. Perhaps alternate methocis are the best. 12. When observing altitudes of the snn, half tho observations shonld be taken to the upper and half to the lower limb. 13. Sets of at least 6 observations should bo taken within 5 or 6 minutes, and the mean of the observed altitudes and time used. No observation shoul i be eliminated afterwards, but only at tho time of observation, when the observer feels that an error has boon mtide ; otherwise, the law of probabilities, by which a correct result is approximated to, is nullified. 14. All observations should be booked exactly as read on the sextant and chronometer. 15. Errors arising from personal error, incorrect estimate of refraction and index error, are best estimated by combining obser- vations, i.e., taking pair§ on each side of zenfth and south, or of meri- dian, east and west. 16. When put down, the sextant should always be left standing on the legs provided for the purpose. 17. When travelling, clamp the arc lightly. THE THEODOLITE OE ALT.-AZIMUTH INSTRUMENT. For the construction and adjustments of the theodolite, see Survey Course. We will here only consider how it is used for taking abtrono- mical observations. METHOD OP TAKING THEODOLITE 0BKERVATI0N8, In observing with a theodolito it is most eesential that it should be supported on a very solid foundation. The nature of tho ground affects the method of doing this. Generally, it is sufiicient to drive stout stakes as far as possible into the eanb, and then to cut them off level with the surface, to form an immediate support for tho stand of the instrument. If the theodolite is sboltered by an observatory the floor of the lattei- must have no contact with the tbeodolito stand. Many consider that the method of getting one set of observations, by taking obscrva- tio'is of the same heavenly body, with a rapid reversal of the teles- cope in the intervals, as is sometimes done in azimuth observations, in order to eliminate colliraation errors, is to be avoidecf Irom tho groal possibility of shaking or moving the instrument in so doing, and that it is better to ascertain the collimation errors and apply them to the observed angles. This remark does not apply to two distinct sets of observations, each having the telescope in one direction throughout. •vationa 5 or 6 frd. Ko time of mnde; eBult is on the 91 The general principles and arrangementH for observing with a thoodoiito are the same as for a sextant, and also with regard to the care required in balancing observations (see pages 84 seq.) The instraments and materials repaired for theodolite observa- tions are :— A theodolite complete, with stand . Mcmorandam book and pencil. JJarometer.... I ^^ android barometer with attached thermometer. Fockotchronometer (beating not more than S times in 2 sees). Lanlorn for night work. The memorandum-book is prepared thus :— ' Level Readings Fai-o at Horizontal Angles. Altiludes. Observ- Correct- Kltiht Eft(!h ed Beats. ed or Knd. Times. Times. Loft. liight. I.ul'l. Verniers, VernliTB. > II 1 II 1 II / II 1 II li. m. s. The points to be noted in the memorandum book are : — 1. Nature and purpose of observations. 2. Date and place of observations. Approximate latitude and longitude. Number of theodolite used. Vertical and horizontal collimation errors, and position of ver- tical circle, for which the plus sign holds. Zero of striding level, whether at centre of level tube or at one end. Chronometer used, and whether adjusted to |sideral or mean time ; rate at which it beats. Chronometer rate and error at observation. Barometer and thermometer readings. Name of object and limb observed. Its position, east or west; or, if passing meridian, north or soath of zonilh, and if above or below the pole. 12. Beferring object used, its horizontal reading, and a freehand sketch ot it if thought necessary. 3. 4. 5. 6. 7. U. 10. 11 92 In form given above. one, 13. Horizontal angles of object observed. 14. Observed altitudes. 15. Beats coanted. 16. Observed times. 17. Corresponding times to observed altitudes. 18. State of weather and sky. 19. Statement whether observations are good or not. 20. Comparisons of watoh or chronometer, with standard both before and after observations. Everything should be noted which effects an observation, so that it can be worked out by an independent person if necessary. In taking observations of a star its position is taken with refer- ence to the point of intersection of the centre vortical wire with the centre horizontal wire. This is not so easy to do accurately as may be at first supposed, because the wires have a sensible diameter, which obliterates the star, and we cannot tell if the bisection is accurate or not. Several observations should be made of a star, and the moan used to reduce this chance of error to a minimum. For the same reason, instruments for observing stars often have the central " wires " double, the two single wires forming each of them being very close together, and the star is observed between them. In taking observations of the sun we cannot take the position of its centre directly ; the way in which it is done is to observe the edges alternately. In observing for azimuth we observe two edges ot the sun at the same time, one the upper or lower, and the other the right or left edge, in one of the four angles formed by the inter- section of the central intersecting wires, by putting one of those wires in advance of the sun's movement and keeping the other wire in contact with the sun's edge, with the tangent screw, until the sun's edge also touches the first wire. If only one limb of the sun has been observed — the upper or the lower — then the semi-diameter of the sun applied to the altitude of the obiierved limb gives the altitude of the sun's centre. And if only the right or the left limb of the sun has been observed for finding szimuth, then the sun's semi-diameter multiplied by the secant of the true geometric altitude applied to the horizontal reading ol the observed limb will give the horizontal reading of the sun's centre. For the algebraic signs to be given to these corrections tee p.p. If two obHervations of the sun's right and left edges are made at nearly the isame elevation, and the mean used, then no correction is required ior the sun's semi-diameter to the horizontal angles to get the direction of the sun's centre when it is at the mean of the angles of elevation. It )rin ren »ve. ird one, \, BO that h refer* with the y an may liametor, eotion is Btar, and im. For lave the of them en them. )sition of lerve the wo edges the other the intor- of those he other ew, until 3r or the Iticude of And if Tvod for 1 by the orieontal jading of to these made at eotion is 98 to get I of the n Similarly, if two observations of tho eun's upper and lower edges are made within a short interval of time and the mean used, then no correction is required for the enn's semi-diameter to the mean of tho vertical angles to got the altitude of the Hun's centre. It is easier to make such observations when the wires ore on tho Ban's surface and the sun is passing off them than when the san is coraing on to thorn, for with the dark glass on the eye-piece the wires cannot bo p.ien until the sun is behind them. The timing of an observation is done in exactly the same manner for a theodolite as for a sextant. When two sets of observations are made on the same side of the meridian for tho same purpose, and within a short time of one another, Iho tcleecope should be reversed in its bearings so as to reverse the etfocts of any existing uncorreeted collimation errors. In an ordinary inverting eye-piece the motion of a heavenly body is in exactly tho opposite direction to that in which it really appears to bo moving to the naked eye. Thus, if a heavenly body appears to the naked oyo to bo rising from left to right, in the telesoope it will appear to fall from right to loft. For heavenly bodies at a considerable altitude the diagonal eye- piece must bo used. The effect of this eye-pioco is to invert the tine horizontal motion, but not the vertical motion, of a heavenly body. Thus, if a star appears to the naked eye to be rising from left to right, through tho diagonal e^ >-piece it will appear tome from rigJit to left. For night work two dark lanterns are required — one for booking the observations, and the other for illuminating the wires. As a rule a spocial lamp is provided for this purpose. It is so placed that its light can bo directed down a hollow cross axis on to a small mirror in tho telescope tube, which reflects the light on to the wires and thus illuminates them. But as a rule this spocial lamp is too small to burn well and is a constant source of trouble. Consequently it is iliscardod by many surveyors and other arrangements made. Thus, in Afghanistan, the wires were lit up by a second dark lantern hold by an assistant who directed the light on to a narrow piece of stiff white paper, secured by a piece of thread or an india- rabbor band in front of tho object glass, and inclined to it at an angle of about 45*^. A small eham pearl, about the size of a pea, has been successfully used in the same way — the pearl being held in position, close in front of tho object glass, by means of a wire fastened to tho telescope tube. These arrangements have been found to be much more satisfactory than the special small axis lamp, which often refuses to burn if there is any wind, and never gives H good light except under the most favorable circumstances, while it also gets into the observer's way when he wants to read the horizontal vernier situated underneath it. M' m ji." 1 m 94 MBTHOD or TAKING EQUAL ALTITUDES WITQ THE THEODOLITE. Taking eqaal altitudes of a star with the theodolite requires no special explanation. In taking equal altitudes with the sun, when observing for azimuth, suppose we are using a diagonal eye-pieoe, then in the morning the sun appears in the telescope to rise from right to left, and in the afternoon to fall from right to left. In the morning set the vertical circle (u «., the central horizontal wire) at a given angle in advance of the sun's motion, and at the instant when the sun touches it make contact with the vertical wire, by means of a tanget screw, as in 1, Fiy. 23. Note the time, and the vertical and the horizontal readings, l^ien make contact with the wires, as in 2, Fig. 23, and again note the time, and the vertioal nnd horizontal angles. Now again alter the wires ahead of the san, uud make similar contacts, as in 3 and 4, Fig. 23, noting their res- pective times, and the vertical and horizontal acglest Then if theae four observations are taken at about equal intervals of time, and within 2 minutes of each other, as they should be, the mean of the horizontal and vertical angles respectively gives the position of the sun's centre at the mean o^ the times. In the afternoon similar obnervations are taken with the same altitudes, but in exactly the reverse order, as in Fig. 24, the times, and the vertical and horizontal angles being also noted. Some observers prefer the order 1, 3, 4, 2, Fig. 23, and vice versd in the afternoon. If the above observations extend over 6 minutes, then the mean of the horizontal and vertical angles will not give the position of the sun's centre at the mean of the times. If the wires are inclined to one another, then only 2 observations cuQ be made, when the sun is tangential to both the horizontal wire uud one of the inclined ones, as in 1, 1 or 2, 2. Fig. 25. 5>in. 6-in. COMPARATIVE VALUE OF THEODOLITE AND SEXTANT IN THE FIELD * An altazimuth instrument — or a theodolite possessing a complete vertical circle as well an a horizontal circle— is in many respects superior to a sextant. 1st, it measures horizontal angles directly, thus avoiding the labor of reducing oblique angles to the horizon ; and a round of several angles can be measured with far less trouble than with the sextant. 2ndly, it measures small vertioal angles of elevation or depression of objects, which frequently could not be seen by reflection from a mercurial horizon for the measurement of " Taken from Hints to Travellers, published by the Royal Geographical Society. OLITI. jquires no rving for ea in the ht to left, lorizoQtal Dd at the tioal wire, I, and the t with the vertioal if the san, their ros- n if these time, and laD of the ositioD of the same the times, and vice 16 mean of on of the lervatioDs tntal wire FIELD.* complete respects directly, horizon ; IS trouble angles of 1 not he ement of •graphical 95 the doable angle by a sextant. 3rdly, its telescopic power is nsaally far higher than that of a sextant. 4thly, it may be so raanipalated aa to eliminate the effects — without, in the first instance, ascertaining the magnituaes — of certain constant instrumental errors, such as eccentricity, oollimation and index errors^ And 5thly, the influence of graduation errors may — when groat accuracy is required — be reduced to a very considerable extent by systematic changes of the zero settings of the horizontal circle. The disadvantages of the altazimuth instrument as compared with the sextant are its greater cofct, and bulk and weight ; but in many instances these disadvantages will be more than counterbalanced by its superior capabilities. Messrs. Tronghton and Simms have given the following details regarding the cost, weight, and telescopic powers of these instru- ments, as constructed by themselves : — Initrument. Weight with Box. Weight of Stand. Price. Telescopic Pewers. Read- ings of Vernlors Details. 7-ln. fradlus) sex- lbs. 7 6 to 1« ISi 23 31 lbs. £ 8. d. 12 28 32 10 40 6 to 10 10" Artitiolal horizon 4-ln. diameter i transit iheo- doiito 9 1« 10 Otol2 12 to 15 12 to 18 1' 30" 20" C Without tran- sit axis, lerel ( and lamp. (With transit axis, level ( and lamp. DO. Wn. " " .. 6-ln. " " .. The Messrs. Casella oonstruot certain very light and cheap alta- zimuth instruments, with 3-inch circles ; power 5, weight with box 4lbs„ weight of stand 3^ lbs. divided to 1', price under £20. For astronomical observations the sextant is decidedly preferable to very small altazimuth instruments, but the latter are to be pre- ferred for the measurement of horizontal angles and small elevations or depressions. The traveller must necessarily adapt his equipment to his require- ments and the facilities he will possess for carrying his instrnments about. He may find it convenient to employ a sextant for astrono- mical, and a very small light altazimuth for terrestrial observations. Bat, whenever practicable, an altazimuth of moderate size, which may be used as a universal instrument, would undoubtedly be the most convenient and satisfactory. The instrument which is recommended by Lient-General J. T. Walker, C.B., F.R.S,, the Superintendent ot the Trigonometrical Survey of India for geographical explorations, as being well adapted for astronomical and for terrestrial observatiocj, and I ■f I Q6 not very balky, is the *'-inch transit theodolite by Messrs. Troughton and Simms. Several of these have been UHod in exjilora-, tions connected with the operations of the Grroat Trigonometrical Survey of India, and have given great satisfaction, being sufticionily accurate for the pnrpoeo, and not too heavy to be easily carried. These instruments are adapted for determinations of time, and heiico longitude, by the method of zenith distances, and also by that of meridional transits; the former being best suited lor the traveller when he can only devote a few hours to the operations, tho latter when ho is halting for a long time at one place. The two methods lead to strictly independent results, so that when both are employed they serve to check each other. The instrument is also well suited for latitude and azimuth observations; in fact, it can be employed in any of the investigations which an explorer may have to uado"- take by means of astronomical observations. On tho other hand, ag an instrument for the measurement '^f torrostrinl anr;jlns, whether horizontal or vertical, it is very valrible, and far superior to auy sextant, not only being more conveniently mauipulatcJ, but posHOSM- ing telescopic powers, which permit of tho detection and identifica- tion of objects that would often be sought for in vain with a sextant. Trigonometrical operations are, as a rule, far simpler and more easily reduced, and lead to more accurtjte results th.nn astronomical observations. A continuous triangulation, or a tra- verse with measured angles and distances, is necessarily impossible when the explorer has to pass through a country very rapidly; but he may frequently remain for several days at one place, and may then have opportuniticp. of greatly extending the scope of his operations by executing a triangulation. THE CnRONOMETER. The use of the chronometer has been more or less rufcrrod to under tho head of " Time-keepers" (p. 37 seq.), and in tho remarks on taking observations with the sextant and theodolite. The chronometer is simply a very perfect watch,with the balance so constructed that changes of temperature have the least ponsible effect upon the time of its oscillation. Tho rate is altered by moans of screws on the balance wheel, or by tightening or loosening tho spiral spring with a regulating lever. The great point in a chronometer is tbat it should keep a regular rate — that is, that it should gain or lose a certain constant amount in a given time. Tho more legular the rate kept the more perfect is the chronometer. It is also more convenient to have to allow for a small rate than for a large one. Care of Chronometer. — Tho seconds hand of a chronometer must never be touched, as all the important parts of the works connect 97 Messrs. oxi)lora. pmotrical ifficiomly ' carried, ■nd heiice >y that of tiavollor ho Jattor methods lomplojed roll suited employed to uncle'- r hand, as whothor or to any It pOdH08t(. identifies- a sextant. iplor and mlta than or a tra- mpoesible ' rapidly; place, and o[)0 of his ! for rod to I remarks 10 balance t poHsible by moans loOBening loint in a in, that it irae. The )nomotor. ) than for ter mast i connect directly with it. The minute and hour hands may be moved with- out iojary, thongh this shoald be done as rarely as possible. Bat the minute hand may sometimes be found a little in advance of or behind the second hand, i.*., it does not point to a minute division when the second hand points to 60 or 0. In this case the nearest minute division to the minute hand must be read. Consequently, to avoid any chance of error, the minute hand should be moved for- ward, not backward, to correct it, if it ever gets 15 seconds out. Tc enable a chronometer to maintain an uniform rate it should bo kept in as uniform a temperature as possible, and away from ex- tremes of heat and cold, and also from masses of iron, which may act as magnets, and magnetize the balance wheel of the chrono- meters. Chronometers should be moved as little as possible, and when being transported they should be guarded against all sudden mo- tion. At sea, where the motion of a ship is not sudden, but progres- sive, box chronometers are used, swung on gimbals so as to-be kept horizontal, and are placed as near the centre of motion as possible. But on shore, box chronometers fail, especially when allowed to swing on gimbals. Consequently, pocket chronometers are best for field use, and if a box chronometer is taken, it should be stopped, when being transported, by wedging up its balance wheel with a piece of cork^, and the chronometer itself should be packed round with cotton, hay, grass or bits of paper. It is set going again at the halting places by the time shown by the pocket chronometers. The advantage of a box chronometer over a pocket one is that the beats are louder and more easily heard, and that their value is uBoally a half-second, instead of some inconvenient portion of a Beoond,like those of a pocket chronometer. If possible, a chronometer should always be kept in the same posi- tion. Hence, if a pocket chronometer is actually carried in the pocket by day it should be suspended vertically at night. Winding Chronometers. — Chronometers arc generally made to run either two or eight days. The former are wound daily, the latter every seventh day. It is important that they should be wound up at the regular intervals, as if let to go too long, an unused part of the spring comes into play, and irregularity of rato may result. The number of half turns of the key required to wind the chrono- meter should be known so as to avoid a sudden jerk at the last turn. The hand holding the chonometer should be held quite still and not rotated in the opposite direction to the key. If a chronometer is stopped, and does not start again on being wound up, give it a quick twist round in the plane ot the face, so as to set the balance wheel in motion. Do not shake it to do this. *How to do this should b« learned from the manufacturer. '^^ 18 98 THE ELEOXEO-CHEONOaRAPH. Under this head may be included all. contrivances for registering small intervals of time by visible marks produced by an electro magnet, and thus recording to a precise fraction of a second the actual instant of an occurrence. 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 ascertained. Such instru- ments, however, are not suited to field work. For permanent observatory work, the works of Chauvenet and Loomis may be consulted. THE SOLAR COMPASS. tr. The solar compass differs from an ordinary compass in having a solar apparatus for determining the true meridian by the sun, in- stead of a magnetic needle for determining the magnetic meridian. This apparatus can also be attached to a theodolite or an ordinary magnetic compass. The solar apparatus (Fig. 26) consists mainly of 3 arcs of circles, on which can be set off the latitude of the place, the declination of the sun and the local apparent hour of the day respectively. These arcs are respectively called^the latitude, declination, and hour arcs or circles. The movable arm (a) of the latitude arc, when set for the latitude of the place, represents the plane of the equator. The axis at right angles to it, on which the fixed arm (b) of the declination arc pivots, represents the polar axis. The movable arm (c) of the declination circle carries 2 projecting brass blocks, one at each end ', each block has a small convex lens on one- half of it and a silver plate with some intersecting lines on the other half— the intersection of the lines on one block h^'ng opposite to the lens on the other, and the distance apart of the blocKS being equal to the focal length of the lenses. The declination arc pivots round the polar axis in a direction par- allel to the plane of the equator. This enables it to be turned end lor end, its ultimate direction depending whether the sun has a north or south declination, and also enables it to be set at an hour angle depending on the time of the day, corrected for the equation of time. If, now, the instrument, with the latitude, and sun's declination,* and the apparent hour angle set oS on the proper arcs, is turned round horizontally, until the sun's rays passing through one of the lenses • In RettloK the declinntlon arc a correction has to be made for the atmoBpherlo refraction. (See p. ). electro cond the rverat a Dstant at erence of oh instrn- ermaoeut may be having a e Bun, in- meridian. ordinary w of circles, ination of 7. These ur arcs or le latitude (b) of the enable arm ka, one at of it and half— the LO lens on aal to the )tion par- rned end lun has a ) an hour equation ination,* lert round tie lenses mospLerio 99 on the movable arm (c) of the declination arc falls on the intersec- tion of the lines on the opposite plate, then the plane of the latitude arc is in the plane of the true meridian, and gives the direction of the true north. To this apparatus is attached a horizontal graduated arc, with a pair of movable sight vanea with verniers, which enables the azimuths of various objects to be observed from the true north. The horizontal plate is provided with levels for levelling it. On the top of each of the blocks carrying the lenses and inter- secting lines is a small sight . These sights are called the equatorial sights. Their use is explained on p. 102. In the construction of this instrument the radius of the earth is ignored, but so long as the graduations of the vernier do not read to less than 10 sees, of arc the error caused by this assumption is in- appreciable, for the angle subtended at the sun by the earth's radius is less than 9 sees, of arc. Usually the instrument does not read to less than 1 minute of arc. A magnetic needle, and graduated arc is generally attached to the solar compass by which the magnetic variation is easily and accurately determined. When the instrument is in accurate adjustment, its horizontal plate perfectly level, and the latitude of the place, the declina- tion of the sun for the given day, and the local apparent time set off on their respective arcs, the imago of the sun through one of the lenses cannot be made to fall ia the intersection of the lines on the plate opposite it, unless the lin 3 joining the inter- section and the lens is in the plane of the vertical circle passing through the sun's centre. Further, the image of the sun cannot be brought on the intersection of the linos until the polar axis is placed in the plane of the meridian of the place, or in a position parallel to the axis of the earth. The slightest deviation from this position will cause the image to pass above or below the inter- section, and thus discover the error. We thus, from the position of the sun in the solar system, obtain a certain direction.absolutely unchangeable, from which to measure the horizontal angles required^ This simple principle is not only the basis of the construction of the solar compass, but the sole cau^e of its superiority to the ordi- nary or magnetic instrument. For, in a needle instrument the accuracy of the horizontal angles indicated, and therefore of all tho observations made, depends upon the delicacy of the needle and the constancy with which it assumes a certain direction, termed tho magnetic meridian. The principal causes of error in the needle, briefly stated, are the dulling of the pivot, the loss of magnetism in the necdle,the influence of local attraction, and the 08*001 of the sun's rays, producing the diurnal variation. 'li 100 ■I i From all these imperfections the eolar instrament is free. The sights and the graduated limb being adjusted to the solar apparatus (see p. 101 Seq.), and the latitude of the place and the decli- nation ot the sun and the local apparent time of day also set off upon their respective arcs, we are able not only to run the true meridian, or a due east and west course, but also to measure the horizontal angles, with minuteness and accuracy, from a direction which never changes and which is unaffected by attraction of any kind. Retraction, — Befraotion makes the sun appear higher than it is, hence,in setting the declination arc we must add the refraction to a N. declination and subtract it from a S. declination, to find the value of the observed declination. Now, the refraction given in tables for different b.titudes cannot be used directly for the purpose, for it is evident that in revolving the declination arc around the polar axis the declination arc will not lie in the plane of a vertical circle, except when it is placed in the plane of the meridian. The proper correction for refraction to be set off on the decllDa tion arc is found from the following formula) taken from pp. 34 and 171, vol. 1, Chauvenet's Astronomy : — Eefraotion in declination = K* tan z cos q» = K^ cot (5 + N). Where q = the parallactic angle. z =z zenith distance. ^ = the declination; N = cot cos t. ^ = the latitude. t =■ the hour angle of the son. K^ = a number whose mean value is 57" and which may be employed when a very precise result is not required. The accurate value of K* is as follows : — Altitude. Zen. Dl8t. Ki. Altitude. Zen. DIst. Ki. 11 M 90" 0° 57-73 9° 81 5477 50° 40° 57-67 8<» 82 54-10 30° 60« 57-48 7'' 83 5317 20° 70° 57-^1 6° 84 51-88 15« 75° 56-60 b'' 85 60-11 10° 80° 65-29 The altitude can bo approximately foutid by moans of a clinometer. A table of corrections Jor refraction in declination should be work- ed out for the latitude of the surveyors. The vertical column of the table would give the hour angles, and the horizontal lines the correc- tions for each 5 degrees of declination between 28 J" N. and 23 J° S. declination. A separate table is required for about every 2 degrees of latitude. It will be seen that only a mean refraction can be allowed for. e. the Bolar 1 the decli- )t off upon meridian, borizontal bich never than it is, ftction to a the value tables for for it is the polar ical circle, le declina )p. 34 and eh may be Ki. n 54t7 54'10 5317 61-88 6011 inometor. i be work- tnn of the be correc- id 2^° S. legrees of e allowed 101 Caution as to the False Image.'^A false image is Bometimes caused by the refleotion of the trae image from the surface of the arm of the declination arc, and is often a source of error. This must be guard- ed against. The false image is readily distinguished from the real image by being much less bright and not so dearly defined. J^justments of Solar Compass. — 1. To adjust the levels: Bring each bubble in turn to the centre of its tube by the foot screv^s, turn the instrument through 180^, and if the bubble does noc remain at the centre of the tube, correct half the error by the foot screwr and half by the level screws. 2. To adjust the intersecting lines and the solar lenses:— Detach the movable arm of the declination arc by removing the necessary screws, and attach in its place the adjuster (which is an arm furnished with the instrument for the pur- pose), replacing the screws of the pivot and also of the clamp. Place the arm removed on the adjuster with the same side against the declination arc as before it was detached. Then, by revolving the instrument round its vertical axis, and by suitably adjusting the foot screws and the latitude and declination arcs, turn the arm in the direction of the sun, and bring the image of the sun on the intersection of the lines. Tarn the arm half over and again observe the sun's image. If it remains on the intersection of the lines the plate is in adjustment; if not, loosen the screws which hold the plate, and move the latter so as to correct hair the apparent error.. Again, bring the image on the intersection of the lines, and repeat the operation until perfect. Now reverse the arm end for end, and adjust th ottier plate in the same way. When both plates are adjusted remove the adjuster and replace the declination arm and its attachments. In tightening the screws holding the plates, care must bo taken not to move the plates. 3. To adjust the vernier of the declination arc : Level the horizontal plate, set the vernier of the declination arc at zero, and turn the declination arm towards the sun and bring its image on the intersection of the lines on one plate by moving the declination arc up or down the latitLue arc, and by the foot screws Then revolve the arm round the polar axis so as to bring the opposite solar lens towards the sun. If tbe sun's image falls on the inter- section of the lines on the other plate, no adjustment is necessary ; if not, count half the error by the slow motion screw of the declina- tion arc. Bepeat the operation until perfect. The zero of the vernier will not now coincide with the zero of the arc. Make it do so, by loosening the screws which hold the vernier and by moving the vernier. 4. To adjust the solar apparatus to the compass sights: Level the horizontal plate, and set the vernier of the borizontal limb at zero. I- I :'!■ I 102 Baise the latitude aro until the polar axis is horizontal, as eBtimated by the eye, and set the vernier of the declination arc at zero, Direct the equatorial sights on some distant point, and observe the same through the compass sights. If the same object is seen through both, no adjustment is necessary. If not, the correction must be made, by moving the sights or changing the position of the vernier, which is best done by the instrument maker. The use of the Solar Compass. — With the solar compass we can ran a north and south, or east and west line, when the latitude and declination are known, and find the latitude of the observer when the declination is known, or the time of day when the latitude \b known. 1. To determine the latitude : Set off on the declination arc the declination of the sun (corrected for refraction) for apparent noon on the given day. A few minutes before this noon set up and level the instrument and set the declination arc at 12 noon on the hoar circle, and turn tho instrument until the declination arm is turned towards the sun. Move the latitude arc vertically, so as to bring the sun's image on the intersection of the lines, and keep the image in this position by altering the horizontal and latitude arcs until the sun's image no longer moves downwards on the plate. Then the reading on the latitude arc will give the latitude of the place. 2. The time of day is best determined by the solar compass at apparent noon as above. Set off on the latitude aro the latitude of the place, and on the declination arc tha declination of the sun for the day corrected for refraction. Follow the sun on the arm of the declination arc and note the watch time when the sun's image ceases to fall and begins to rise on the plate. This is the mean watch time of local apparent noon ; and when the equation of time and the difference of longitude for standard time is applied to the local apparent noon, we get the true mean time which the watch should have shown. Then we can get the error of the watch. The same thing can be done at any time of day using an approxi- mate hour angle to find the declination. After getting the sun's image on the intersection of the lines note the watch time and read the hour circle. Then, allowing 4 minutes ot time for each degree, we get the hour angle, from which we find the L.A.T. of the observation. 3. To run a north and south line after levelling the instramout and clamping the sight at z::ro : Set off the latitude, the declination for the time of day and the hour angle in their proper arcs, and revolve the instrument until the sun's image falls on the intersec- tion of the lines. The line of the sights will nortained at the time, like the index error of a sextant, or their effects eliminated by reversed or balanced observations. centric ^r takon in only ipposed se muHt earth's to suit would bserved 2. — COBRIOTION TO SENSIBLE HOBIZON. When altitudes are taken they have to be measured from some horizon. With the theodolite the horizon tangential to the earth's surface is used mechanically, and hence no correction is needed. With the sextant, either an artificial horizon or the sea horizon is nsed. The altitudes taken with an artificial horizon are double the true ones, being the angles between the true and reflected images, an^ after the instrumental error has by 2 to obtain the instrumental hence the altitudes taken with it, been applied, must be divided altitude. When the true sensible horizon, or an artificial horizon, cannot be used, then the sea horizon is employed. It is below the true horizon, being the tangent to the earth's surface passing through the ob- server's eye. Thus, if O represents the observer's eye, (Fig. 27) the altitude of a star S is S T, when taken with the sea horizon. To find the altitude SOT with the senBible horizon we must deduct the angle HOT from the angle SOT. The angle H O T is called the dip of the horizon, which may be defined as the angle of depres- sion of the visible sea horizon below the true horizon, arising from the elevation of the observer's eye above the level of the sea. The dip ot the horizon, neglecting atmospheric refraction (see p. ) is found thus : Let C A or G T = r, the radius of the earth ; O A = A, the height of the eye above sea level ; and the angle T O H or O T = D, the dip of the horizon. cosJ0 = cosOCT = 0T= ^^ ^ Then 00 r + h and sin D = / 1 — cos^ D = >/>- CH-a;. =V I h J>" very nearly, as Ms small compared with r. or, as D is small _ 1 / 2A ~ sin 1" \—f i i 14 or = u 106 From this value of D a doductioD of about ^^tb should be made as an allowanco for terrestrial refraction ^, the uncertainty of which renders it most desirublo that tiio sea horizon should not be resorted to, when possible, for obaorvationH on shore. Ghauvenet gives the correction for the dip, allowing for a moan atmospheric roiraetion, as D' = D — 078 1 D 092U i I 2/t Bin 1" \ r where h and r are expressed in the samu units. Taking the mean value of r = 20.888,6 25 feet, we have D' = 58"-82 |/ h in loot. If d^= distance of sea horizon in statute milcs,f then allowing for a mean atmospberie refraction d= I'SII i/ h Hi Icoi. To enable 2 observers ut heights h and h^ feet to just see each other, above the rise caused by the curvature of the earth alone, their distance apart in statute miles will bo = 1-317 (y~K + iTTi ). To find the dip of the sea at a given distance from the observer, less than the distance of the sea horizon, D in Bee's of arc = 22'-14 d + 39"-07 \ a where A is in feet, and d is the given distance in statute miles. 3. — ATMOSrnERIC REFRACTION. p. A ray of light from a heavenly body on passing through the earth's atmosphere, the density ot which becomes greater as the earth is approached, is so bent that its path becomes a curve which is concave towards the earth. The heavenly body S is seen by an obeerver at O in the last direction O S' of the ray. The angular difference SOS' between the true and apparent directions is called refraction^ and we see that the eii'ect of refraction is to make all heavenly bodies appear higher than they really are. Thus : True altitude = apparent altitude — refraction, or True zen. dist. = apparent zen. dist. + refraction. An inspection of the figure shows that tliie refraction is greatest at the horizon where it is about 35' or greater than the sun's dia- meter, and is nil at the poles ; also that there would be no refrao • Terrestrial refraction is principally due to tho heated condition of the earth's surface and is described in " Geodesy." t A statute mile = 6280 feet ; a nautical or geographical mile = 6086 feet ; 1 statute mile = U'tiG7665 uautical mile, and 1 nautical mile = l*lo2(i5 statute mile. W be mtde tj of which be resorted for a moan Bt see eaoh arth alone, observer. ailes. rough the tor &a the rve which Ben by an e angalar IS is called make ait 18 : greatest sun's dia- 10 refrao- ion of the t ; 1 statute lot tion for a spectatOi' sitaatod at the centre of the earth. When the son just appears above the horizon it is really just under it ; hence refraction increases the length of our days. The oval shape of the suo and moon, when near the horizon, is explained by the same phenomenon, the horizontal diamotor is unatfouted, while the vertical diameter is shortened by the unequal refraction at its upper and lower limbs, for the refraction alters very rapidly near the horizon. Tables of refraction. The density of the atmosphere, on which the amount of refraction depends, varies with the pressure and tern* perature, consequently tables of refraction are drawn up for different altitudes for a mean barometer prosaure and a mean temperature, and to this table a second table of corrections for other pressures and temperatures is attached. The effects of humidity are insen- sible because watery vapour diminishes the donuity of the air in the same ratio as it incroaHos its refractive power. Young's refraction Tables (see Appendix) are conveniently sim- ple and accurate enough for general purposes. But BoHsel's tables given in Chauvenet's Astronomy, though more troublesome to use, are the best when groat accuracy is required ; they are based on the formula : ^Refraction = a P^ \K tan. si or log. Eetraciion=log. a-\-A log. >S + / log. K. Where log. (%, A, and ;( are taken direct from the tables with argu- ment apparent zen. dist. z^ ; log /i = log B -|- log. T, both these last two logs, being taken from supplementary tables with arguments " height of barometer " and " attached thermometer " respectively ; and log, I' is given in a supplementary table 'with argument " ex- ternal thermometer." There are two columns given in Bussel's table, — column A contains data for apparent Z. D.; and column B the data for true Z. B. The above refraction tables do not hold good for altitudes under 10^, because near the earth the density of the atmosphere is affected by many causes for which allowance cannot be made, such as un- equally heated surfaces, vapour, etc. A good aneroid whose corx-eotions are known, with an attached thermometer, is sufficiently good for obtaining the pressure and tem- perature to ascertain the refraction. The refraction correction can bo a^^certained approximately in the following manner. Suppose, the earth's atmosphere to be oondensed into such a space th^t it has an uniform density throughout as at the earth's surface. The error caused by this assumption will be but small for altitudes over 10° and will be less as the altitude increases. Let \JL = the refraction index of vac\io into air. r ^ the refraction in minutes of arc. . I ih 108 Then = sin (fi -{■ r) = ^ sin , „^, ^ or as r is a small angle, being less than 30 of arc sin + r COS 6 sin l^ = ^f sin B Whence r = ^^^- . tan $ sm l^ Bat in the triangle AOE, CE = radius of earth + = radius (1 + n) 1 where n = ^^^ and itz^ be the apparent zenith distance sin ARC _ Bin g 1 sin OAR sin^i sin z^ or Bin Q = (1) Radius 800 Now tan $ = 1+n sin S 1 +w sin cos d l/l-sin 20 substituting the value of sin Q in this and eliminating — from the numerator and denominator we get 1 + ** tan0 :^ sin z 1 + sin z^ i/n: sin 2 2i -\-'2n -\- n^ \/qob^ z^ -f (2n -f n^) sin z^ ooaz^ Vi + (ii n + n2) Beu2 ^i -i = tan si j I -)- (2 n + n2) s'Jc2 z^ I = tan 2 1 I 1 — H2 n + n2) 8ec2 z^ -\- . . . I = tan z^ ( 1 — n 8cc2 ^i ) nearly. Now substituting* this value of tan g in equation (1) we get << 1 o^n^ ~1 r = ^^ . tanci( 1 sin 1^ ^ sec "sUo" Now if z^ = 80° , then soo''* z'^ = 33 and hence as z^ should never exceed 80°, the value of see 800 = . 04 as a maximum, which may therefore be neglected, and thus we have . tan z^ itt-1 sm 1^ i>. i WT iins W + n from 109 Now let (i^ be the refractive index for 32® F. and 30 in. barometer — 1'000294 ; and let ^ be the refractive index for t^F, and b ins. barometer. Then b 460 + 32 , , 30 =16-4048 460+ t b 460 +< Now the value of sin V is '0002909, and that of ^i is always u-1 greater than (1 -f sin 10 ; and by trial we find that ^.^ ^^ = ^ very nearly. Hence r In minutes of aro-= ^t&m^ = 16-4 ^ , ^ • tan «* or r in seconds of are = 60 ft tan z^ = 984 . ... . . tan z^ 1000 6 460+ t . tan 2* nearly! 460 + * Since ^ is so nearly equal to unity, we may, for rough work, at as with a box sextant, take r in minutes of arc = tan z^ \ n 2» 2n + n2) 9 get uld never 'hioh may 4. Semi-diameter. In order to have a common basis to work on, angular measure- ments should be made to the centres of heavenly bodies. But as the observations to any body, having a measurable diameter, must be made to one of its edges, or limbs, as they are called, we must add or subtract the semi-diameter of the heavenly body, as the case may be, to reduce the measurement to its centre. The Hemi-diameter of the sun is given in the N.A. for each day at G.MN., and owing to the sun's great distance from the earth as compared with the earth's radius this value is practically the same for the earth's centre or surface. But not so for the moon, whose mean distance is only 60 times the earth's radius. Hence when she is in the aenith she is nearer to an observer by -^^th of her mean distance, or the earth's radius, than when she is in the horizon, and consequently her semi- diameter appears sensibly greater when she is overhead than when she is on the horizon. This increase is known as the augmentation of the moon's semi-diameter . Lot X -- required augmentation in seconds of arc. s ■=^ moon s horizontal semi-diameter in seconds of arc. s -\-x=. moon's semi-diameter when above horizon in seconds of arc. 110 D = moon's distanco wher on horizon = Om, rf := " " above horizon = OM. h^ = apparent alt. of moon = MOm. R = radius of earth. Now the angalar value of the Bemi-diameter is inversely as the distance, hence s : s-\- X = d : J) D s — ds ora;= j— (1) Now when D = 60 R it has been found that s = 930", and henca for other values of D we have D : 60 E = 9i0" : s" T, 55800 R ,-. or D = (2) s ' Draw OP perpendicular to OM; then we haveOM = MP nearly; and angle COP = h^ nearly; and CM = Om nearly; and D = Om — CM = CP + PM = R sin Ai + d 7 n T? • 7.1 55800 R — Rs sin h^ ... or J = 1) — R sm Ai = —(3) Substituting these values of D and d in o(][uation (1) we get X" = s^ sin h^ 55800 — s sin A^ = c a2 Bin Ai + J c2 s^ sinS A^ + i c^ ^a _|. ... Whenc= 'OOOOnO _ (Log c ="5 -2495, and log J c2 =Tu'1979) As a rule the last 2 terms may be omitLoJ, and hence x= -OOOOlty s2 Bin A^ The moon's horizontal sami-diaraoter (s) is given on p. Ill of each month in the N.A. for (xreeuwioh noon and midnight for each day of the yoar, and hence must be corrected for longitude and hour angle in the usual way Stars have no sensible semi-diameter. 6. Parallax in Altitude or Correction to Rational Horizon. Just as obsorvalioDs must be reduced to the centre of the heavenly bodies observed, so also must they be reduced to the centre of the earth from its wurfaco. This reduction is known as parallax, which may be defined as the apparent change of place which fixed bodies undergo when viewed from two different point^, i.e., it is the apparent change in the position of a fixed body due to a real change in the position of the observer. T )Iy as the (1) ad honco (2) nearly; -(3) got p. Ill of t for each itudo and Horizon, hoavenly tre of the ax, which cod bodies it is the al change 111 As parallax may be applied to any of the spherical co-ordinates, it must be qualified as parallax in altitude, parallax in declination, parallax in right ascension, etc , to define what kind of parallax is meant. But for the present we are only going to deal vrith parallax in altitude. As altitudes are measured from the sensible horizon, they must be reduced to the rational horizon (see p. ) . Hence the parallax in altitude is the ditferenoe of the altitudes or zen^ dists. of a heavenly body, as Been from the surface and centre of the earth, respectively. Thk Earth as a Sphere. As far as the son is concerned, wo may consider the earth as a sphere. Let 2^ = apparent zon. dist. ZOS., z =True " " ZCS Then ZOS = ZCS + OSC, or OSC = ZOS — ZCS = z^ —s. This angle OSC is the parallaz in altitude, and is the angle subtended at the heavenly body by that radius of the earth which joins the observer's position with the centre of the earth. Let p = parallax in altitude = OSC* Then true zen. dist. = observed zen. dist. — par. in alt., or 2 = 2* — p or A = A* + p, where h and h^ are the true and observed altitudes respectively. From the figure we see that the higher the object the lees is the parallax in altitude. It is nil when the object is over head and greatest when the object is in the horizon. The value of OSjC in this latter case is called the horizontal parallax, while OSC in any other position above the horizon is called the parallax in altitude. Let 7t denote the horizontal parallax. OC _ 00 _ ^ _ sin p Then sin 7t KJH, cs sm 5' where R = radius oi earth at observer's latitude, and D = the dis- tance of the centre of the heavenly body from the centre of the earth. Hence sin » = sin 7t sin z^ R When p and n are small, and j: can bo taken as constant, as for the 8un and planets, then j? = n sin z^, or j? := 7t cos h^ In the case affixed stars we have no appreciable parallax in alti- tude, because R D is infinite and — or sin, 7t = 0. I i 112 , ! l^he horizontal parallax (n) of the sun, moon and planets varies from day to day with their distances from the earth, and are given in the N. A. Now the formula for parallax in altitude stated above gives the value of the parallax when the apparent Z.D. is known, but when the true Z,I). is given we require a different formula. Now z^ = 2 -\-p. Hence, sin ^ = sin 71 sin (z -\- p) Prom which wo get, sin 7t sin z tan p = T : ■^ 1 — sin 7t cos z Which can bo expressed in the following form which rapidly converges as p is small. sin 7t sin 2 , sin ^n sin 2z 1>= „;^i.> + sin 1" + sin 2" sin 'Tt sin Bz sin 3 •j#i The parallax of the sun and planets is so small that we may use pz=7te\n z without sensible error. But for the moon we must use the full formula given above. The Earth as a Sfheroid.* R In the case of the moon, j: cannot be taken as constant, because she is so close that the variation of the earth's radius for different latitudes (due to the earth's spheroidal shape) can be appreciated. In this case the horizontal parallax is greater at the equator and least at the poles, and we must conseqaently find the horizontal parallax for the latitude ^ of the observer before we can correct it for altitude. The maximum horizontal parallax is called the equatorial horizontal parallax. Let Tte = equatorial horizontal parallax. n = horizontal parallax for lat. ^. B = radius of earth at equator, r = " " lat. . i> = distance of moon's centre from earth's centre. Then, R r sin Ttf = ^ and sin 7t = g and sin Ttf : sin 7t = R : r r . or Bin 7t := |:c sm jte /I * For a more accurate computation, see next obapter. 113 8 varies e given And as n aud Tte are small we have n= r^Tte b»tg=i pin 2^ 300 Hence ^j = % | very nearly (see Geodesy) . mn_^\ aou ) The eqaatorial horizontal paralUx is the horizontal parallax given in the N.A. for the moon and all heavenly bodies; and from it the horizontal parallax far latitude ^ is deduced ; after which it must be corrected for the altitude of the observed heavenly body above the horizon. The variation of the parallax for latitude is not appreciable for the sun and planets on account of their distance irom the earth. SUMMARY. We see from the foregoing that to reduce the obderved altitudes of heavenly bodies, taken at the surface of th^/ earth with the sen- sible horiz m, to what thoy would have been if they had been taken at the centre of the earth with the rational horizon we proceed as follows: — 1. Apply any instrumental index or coUimation errors to the observed altitude. 2. To the result apply the correction for the kind of horizon used. 3. To the altitude thus obtained subtract ihe refraction for the given barometer and thermometer conditions. 4. To this altitude apply the semi-diameter, corrected for aug- mentation if the moon is observed ; — for upper limb, -f for lower lower limb. This correction is not necessary if an equal number of observations have been made to the upper and lower limbs of the heavenly body observed and the mean of them used. 5. To the altitude with the sens! I lie horizon thus found, add the horizontal pa^-allax for the observer's latitude corrected for altitude; the result is the true altitude with the rational horizon. The algebraic signs given above as those to be applied to altitudes. Jhese signs mvst be reversed for application to zenith distances. In problems on the stars, stops 4 and 5 are omitted, and step 2 is not necessary for theodolite observations. The above corrections must be made in the order stated, because if the corre^'iion for refraction be made after the others we get a wrong apparent altitude, and if the correction for parallax is made first we get a wrong altitude for the heavenly body's centre to make the correction by. 16 CHAPTER nir. PARALLAX IN ALTITUDE, AZIMUTH, DECLINATION, AND RIQUT ASCENSION DUE TO THE SPHEROIDAL SHAPE OP THE EARTH. M In this chapter we propose to deal with the change made in the spherical co-ordinates by a change of position of the observer from the earth's surface to the earth's centre or vice versa, taking into consideration the spheroidal shape of the earth. The effect of parallax in altitude is confided entirely to a vertical plane. But in consequence of the spheroidal shape of the earth the vertical or plumb line at the observer's place does not coincide with the line drawn from the observer's place to the centre of tbo earth; that is, the vertical line docs not pass through the earth't* centre unless the observer be at the equator or at either pole. Hence the zenith to which the plumb line points Ih not the same as the zenith to which the earth's radius points, and therefore the vertical plane at any place does not pass through the earth's centre (the point at which altitudes, azimuths, declinations, and right ascensions are measured) unless the vertical j)lane coincides with the meridian, i.e., when the heavonlj' body is on tbe meridian. Consequently wo see that, when the observer is not at the equator or either polo, we hf),ve to consider the effect of his position ou the parallax in altitude, in azimuth, in declination, and in ris^ht ascension. Now, as we can consider the earth as a sphere for all heavenly bodies except the moon, the following remarks only apply to this last ramed body. Since the effect of panillax is confined to a vertical plane, when the moon is on the meridian there is noparalhix in R.A. or azimuth, but its effect is wholly on the declination and altitude. In every other position of the moon, the vertical circle passing through it is inclined to the circle of R.A. or equator, and the parallax due to spheroidal shape of the earth affects the R. A. and azimuth as well as the altitude and declination of the tnoon. The geocentric zenith distance that we are going to consider is the angle which the straight line drawn from the centre of the earth to the centre of the moon makes with the straight lino drawn through the centre of the earth parallel to the vertical lino of the observer. These two vertical lines are conceived to meet the celestial sphere in the same point, namely, the geographical zenith, which is the common vanishing point of all lines perpendicular to the horizon. Thus both the true and apparent zenith distances will be measured upon the celestial sphere from tbe pole of the horizon. naiiT ITII. ide in the rver from king into a vertical earth the cide with ho earth; h's centre lence the he zenith plane at point ai sious are idian, i.e.^ Y wo gee , wo b.'j^ve titudo, in heavenly y to this 0, when azimuth, In ©very 'Ugh it is ? duo to as woU or 18 the earth to throuajh >bserver. .1 sphere h is tho horizon, measured 115 Tlie azimuth of a heavenly body is, in general, the angle whioh a vertical plane passing through the ceniro of the heavenly body makes with tho meridian. When such a vertical plane is drawn through tho centre of the earth it does not coincide with that drawn at tho place of observation, as they are drawn through two different vertical lines. Hence we have to consider a parallax in azimuth as well as in zenith distance. In dealing with R. A. and declination we must suppose a line drawn through the intersection of the normal or vertical line of the observer with the equatorial axis of the earth, and parallel to the polar axic. The hour angle of a heavenly body is, in general, the angle which an hour circle passing through its centre makes with tho plane of the meridian. When such a plane is drawn through the secondary axis it does not coincide with that drawn through the centre of the earth, and hence wo have to consider a parallax in right ascension a^^ well as in declination. Tho effect of parallax in always to decrease tho true or geocentric altitude and azimuth. It lias the opposite etfoct on tho apparent data Parallax sometimes increases and sometimes deci'oasos the true hour angle, right ascension, and north polar distance. Its effect on thcso data is best seen by drawing a figure or plan of the problem under consideiation. To Compute the parallax in Altitude and Azimuth. To work out this pioblom we have to consider the rectangular coordinates of tho moon with roforonco to the pianos of tho rational and sensible horizons, and tho rectangular co-ordinates of tho observer's place with reference to tho rauonal horizon ; the centre of tho earth and the observer's place being taken as origins and tho north and south lice as tho principal axis of reference. Let O be I ho observer's place, tho centre of tho earth, N O S the north and south or inoridian lino on tho sonsiblo horizon, and R tho same lino on tho rational horizon ; O0 = radiusof oarth = r; and let V = angle of tho vortical = angle C F =: angle O C Z. d = distance of moon from centre of earth = C M. d^ = distance of moon from observer = O M. A and Ai=tho azimuths of M measured at Z and Z^ respectively, with tho north end of tho meridian as zoro. Lot a, b, c bo the co ordinates of O, and s, t, u those of the moon with rof'eronce to the rational horizon, origin at C; ands^, t^,u^ those of the moon with reference to the sensible horizon, origin at O. Then we have: Along meridian. At right angles to meridian. F = a = r sin v b z= o ZH = s=: — (I) f sin 2^ sin A^ =x sin 2 sin A (2) /cos^i = cc!^ 2 — sin Tt COS V (3) Mult. (1) by Bin A and C^) by coa A and subtract (2) from (U Mult. (1) by 008 A and (2) by sin A and add (2) and (1) / sin z^ sin (A — A^ ) = sin n sin v sin A (4) f sin z^ cos (A — Ai ) = h-in 2 -f sin n sin v cos A (6) ^-♦««/A Ai\ sin 7t sin u sin A cosec 2 or tan (A — AH =: - — ; — : ■. I + sin 7t sin V cos A cosoo z let m = sin ^ sin v coseo z, then .,..,. m sin A tan (A — Ai ) = As A then 1 + >» COB A Ai is small put (A •— A' ) sin I" for tan (A — A' ) Ai = m sin A m^ sin 2 A ^ 2 bin 1" + .1 sin 1" when z and A are known. But if «! and A^ are known, then in the same way we find that . , . ... 7ni sin Ai tan (A — - Ai ) = ; - ^ ^ 1 — ml cos Ai **' ^ ^ - sin i" + 2sinl" + ' ' ' where m} = sin 7t sin v cosec 2^ . Now maltiply (4) by sin J (A — A^ ) and (5) by cos J (A — A^ )• Then adding, we get /sin z^ cos i (A — Ai ) = sin 2 cos J (A — A» ) -f- sin 7t sin y cos |A — | (A — A^ ){ „. 1 . , . . cos ^ (A + Ai ) am z^ = Bin 2 4- sin n sin v — ^ ' ^ cos I (A — A' ) Let tan r = tan v f ,, ^ ., ^ cos A (A — Ai ) or ora (l> (1) (2) (3) f4) (5) i — A' ) d that h . . .' — Ai ). -Ai){ 117 Then/ Bin ^l = Bin « + Bin it cos « tan y, and from equation (3) we have 1 COB 2^ = COB Z — Bin 7t 008 W. Then by the same prooesB as that by which we found (4) and (5) ^««"'' Bin(z-i-y) / Bin {Z^ -'Z)--= Bin 7t COB .' coBy COB (g + y ) COB y Then or / COB («^ — 2) = 1— sin It COB » Let w = sin 7t cos t; sec y. N wBin(g + y) tan («i — «) — i_^co8(2+y) u;8in(« + y) . u?2Hin3 (g+y) . ^ - ^ = ^mT "^ --i Bin 1" ^ when z is known If «Mb known then we have u; sin (g^ -f y) tan (gi — 2) — r^-MJC08(2i+y) ^aain iiz^-\-y) , 2 sin i M;sin(giH-y) To Compute the Parallax in B.A, This can be found by means of rectangular co-ordinates, but it is Bimplsf to find it directly as follows ;— Let Z be the geocentric zen. dist. and P the pole o/ equator M the true pllce of the moon as seen f. om the centre of the earth. m the apparent place as seen from the surface. Then M ffi iB the parallax in altitude and the true tour angle Z P M is changed by parallax to Z P Jiii , and therefore M P BH is the parallax in R. A. Let 7t =1 horizDutal parallax ot place. p = parallax in altitude = M. Mi a = moon's geocentric B A. a\ = •« observed R A. d = '* geocentric declination, ji __ i< observed declination. 2 _. «i geocentric zenith distance. z^ = " apparent zenith distance. t = " geocentric H A. (log 0-9933254= 1'9970916). 5J h m 118 Then angle Z P M^ = < + a^ — a = f* In spherical triangle M P M^ we have . , , n\n p ftin M^ Bin (ai — a) = — rrr; — '^ ^ sin M r From the triangle M^ P Z we get . -,, siu P Z sin t^ sin M^ = Honce, Hence, sin M^ Z sin p cos <^i sin t^ sin (a* — a) = — — -—ri • ^. r, ^ ' Bin M, P bin Ml Z but sin p = sin 7t sin M^ Z. sin (ai — a) = Put m = sin 7t 009 <^i sin ~h sin s cos ^ cos A we get T» n »r t*in ^ — COS z sin A^ ein J* — cos 2' sin d)' cosPZ M= ^ ^,-^" = : — r ^7—^ sin z cos .'.sin z^ sin J — sin (^^ (sin 2^ cos z — cos z^ sins) =8insein^i or sin z^ sin 5 — sin 4>^ sin (2^ — ■2;) = sin z sin ^i 119 (1) (2) (ai — a) 0) (4) lown, and jvn. This sn is west But Bin («' — 2) = Bin p = sin 7« sin 2 and eince Bin z^ (Bin ^ — sin « Bin 4)^ = "n z sin §» 008 «J Bin t COB 5^ Bin t^ (6) Bin PZM = Bin z Bin 2' then Bin z^ COB 5 sin < Hin V = sin « cos 5' Dividing (5) by this, we get tan 81 = sin 3 — sin 7* bid ^ cos 5 B'D ^ — • sin t^ sin < tan ^i or 8in<^isin7« . s_ Bin t tan ^^ — .-;;71 tand— j,in' ^i'^ ^ cos 8' — sin Ji . sin 7« cos ^' OOB- fti — a cos (' + "-^') (9) 3. l*- • k 120 Let oot h Then = cos ^ ' + loot ip^ eeo a\ — a Bin (Ji — S) = "in .^^ 8'° '* (^°^ i^ "~ ^^^ 3^ ^°' *) sin n(a^-A)} or tan {^^ —^) = c sin (6 — a) + c cos (6 — a) tan c sin (b — X) or tan (ai - a) =f:::r- -^ or in series as before. c cos (6 — a) «^-«= c sin (ft—a) I <^' sin 2 (6—3) ' + 2sm 1" (12) (13). sin 1" Equations (9) and (10) are nsed for finding the parallax in declination when we know the apparent declination (as affected by EarsUaz) and equations (6), (10), (12), or (13) are used when we now the true declination. Formula (6) furnishes the apparent declination in terms of the true declination, and of the true and apparent hour angles, as found by the preceding problem. It is the simplest formula known for tbe parallax in declination, but as it requires, lor accuruto results, a table of sines and tangents to seven places of decimals for every second of the quadrant, it is sometimes more convenient to be able to ascertain this parallax in declination more direotly, and consequently the other formula) given above have been deduced. Except for the moon, the first terms of series ^4) and (13) will BufBce, and we may also use a* for ^\t^foTt\ and it sin 1" for sin n, without any sensible error, and hence: ft cos * sin /* -or = — ;- cos a' a' — a = tan 6 = tan —a) — - cos 5 dt The differential of t must be taken in parts of radius. If the variation is required for one hour, dt will represent the arc of 16°, which is 0*2G17994, the radius being unity. (log 0-2617994 = 1-417969.) To find the hourly variation of the parallax in altitude. We have from equation (9) sin (5* — S) = «in ^ ^in 9^ coc-^ ' — sin 7t cos — 8 and N M^ = P M^ — P N = 6 — 8S and sin Z Q M = sin Z N M\ Therefore, sin Z M sin Z Q M sin Z M^ sinQ M or - ' sin Q Z M sin z sin N M^ sin 2^ or sin ('. - 8) Hin (b — 5») Bin (b — S') _ sin^^ _ sin (S + x) _ 8 -{- x Bin (6 d') sin z sin S ' S where 8 is the moon's semi-diameter when she is on the horizon and X is the required augmentation. Hence, a;= S sin (6 -.3^) - sin (6 -^) sin (6 — 3) 2 sin b'-8 cos = S (c^-5)-^V) = S 2 sin b'-S 2 sin {b — 8) cot {I) — 5) cos 5 = S I Bin (^ " ^] cot (6 — g) + 2 sin in which the last term may be neglected. 2 2 Hence, ic = S (^1 — 5) sin 1" cot (b — ^) 'hen in the 3 and draw n Q and R. N and the R, having — a\ = *-« X le horizon 123 Indirect Method op computinq Parvllax in Declination and IN Altitude. The foregoing methods of computing the parallax enable us to pflps directly from the geocentric to the apparent co-ordinates. The following" indirect molhed is eomeiimes more convenient. This consists in reducing both the geocentric and apparent co- ordinates to tho normal centre i. e. the point in which the vertical lino of the observor intorsectw tho axis of the earth. Wo may suppose this point both as the centre of the celestial sphere and us the centre of tin imaginary terrestrial sphere whose radius is equal to the normal. Since the normal centre is in the vertical lin(3 of the observer, the azimuth at this point is the same as Iho apparent azimuth. If, thoroforo, the geocentric co-ordinates are reduced to this point, wo shall then avoid tho parallax in azimuth and tho i)arailax in altitude w:ll bo found by the simple formula for the earth regarded as a sphere (see p. Ill) taking the normal as radius, Again, since tho normal centre is in the axis of the celestial sphere, the straight lino drawn 'rom it to the centre of the moon lies in the d«.'clinat'on circle of that body ; the place of the moon, thoroforo, asseonfrom the normal centre, differs from its geocentric place only in declination and not in R. A. Consequently, when using the following indirect method, we have only to find tho reduction of tho declination and the altitude (or zerith distarco) to the normal centre. To reduce the Declination to the normal centre. Let PPi bo tho earth's axis ; C the centre j O tho normal centre ; M tho centre of the moon ; Appendix V. wo have CO — - R tho equatorial radius. Then from R e^ Hin

. COS z^ = COS z. — sin rti d^ d. bin 2;* = sin z^ from which we deduce — . sin (2^— ^i) = sin 7t 1 sin «, . cos (2^ — 2,) = 1 — sin 7ti cos 2, or in series ^ p in 71, sin g^ tan (2^ - 2,) = I _ fein 7t, cos ^, I Beriers . sin 7ti si ng, sm Tt, sm Z ^^ ^' - ^^ = sin 1" "^ 2 sin 1" To find 7t we obtain rigorously from equation (14) and (17) ^_ B sin 7te ^^" ^1 ~ r (1 + n sin Tte h^ V b>° 5) 126 I But this exact exproBsion of 7t^ is seldom reqaired, and it will generally suffice to take sin 7t, = — sin Tte * r which will give the value of Ttj to within 0'2", even for the moon, in every case. To compute the Parallax in Altitude for the normal centre when the apparent zenith distance is given. We have d^ cos z^ = di cos 2i — m d^ sin 2^ = di ein Zi Multiply the first equation by sin z^ and the second by cos z^ and subii'ac'ing we get 81 n (2* — 21 ) = ;,- sin z^ di = sin Tti sin z^ To employ the foregoing indirect reductions. (18) I In finding the hour angle of the moon and its azimuth from its apparent zenith distance, first reduce its geocentric declination and apparent zenith distance to the normal centre, and then tolvo the artfronomical triangle in the usual way. If the apparent altitude of the moon is not observed its apparent zenith distance at the time of observation can be computed, as follows : (1) With the approximate longitude find the G M T of obser- vation (2) With this approximate GMT find, from the N A, the geocentric R A, do-ilination, and equatorial horizontal parallax of the moon. ( 3) Then the hour angle f = L S T — R A. (4) By the formula ^i — 5 = ^ ^te sin ^ cos J, compute the reduction of ^ to normal centre, and thus find ^i (6) With ^i, t, and ^ find the zenith distance (zi ) at the nor- mal centre, by solving the triangle with the formula when two sides and the included angle are given. R (6) From the formula Tti ^= — Ttc find the value of the reduced r parallax in altitude (yti ). v7) Then find 2^ — 2i from equation (18); and knowing zi we can find z^ or the true apparent zenith distance. (8) From this z^ we must deduct the refraction (seep. 106) tofird the observed apparent zenith distance. md it will 9 moon, iD when the by cos «i (18) fiom its jclination lon tolvo apparent sated, as of obser- T A, the rallax of pute the the nor- hen two reduced ig si we 6) to find II APPENDIX I. c? TO CONVERT L.A.T. AND L.M.T. INTO THE CORRESPONDING L.S.T. To convert L.A.T. into the corresponding L.S.T. we may either convert the L.A.T. into L.M.T. (see page 31) and then convert the L.M.T into L.S.T. (see page 33) ; or we may proceed more directly as follows : — From Fig. 11 we see that if S represents the apparent sun, then The apparent sun's R.A. = tS The L.A.T. = mQS And the L.S.T. = mT But mT = TS — mS = cy5S_ (24— mQS) = wiQS + tS — 24. Or generally L.S.T. required = given L.A.T. + true sun's R.A. The R.A. of the true or apparent sun here used is the quantity found on i>age I. for each month in the Nautical Almanac, cor- rected for the Greenwich apparent time of the observation. ^ As an example, let us take the reverse of the example given on page 30, viz. : Given the L.A.T., Oh. 22m. 19-5s. on the 29th February, 1888, required the corresponding L.S.T. h. m. s. L.A.T 22 19-5 Dijff. of longitude from Greenwich. 5 5 50-0 G.A.T 5 28 9-5 = 5-47 hrs. R.A. at G.A.N. 29th February... 22 4'7 35-3 Correction for 5 •4'7 hrs 51-2 R.A. at given time 22 48 26-5 L.A.T 22 19-5 L.S.T. required 23 10 46 In the same M'ay, we have, if, in Fig. 11, S represents the mean sun, then L.S.T. required = given L.M.T. + R.A. of mean sun. f^S I » ■■»■ It 2 The R A. of the mean sun at Grecnwicli noon is called "the sidereal iime at mean noon " in the mnUkal Almanac, and is found on pai^e II. of each month. Before l.emg used as above u LTt bo cir^.cted for the G.M.T. of tl- observation Take the examideonpage n:^ : Given theL.M. 1 . 2.)h. Im l.» Ms. on the 16th March, 1888, required the corresponding L.fe. 1. h. m. s. L.M.T. IGth March ,.• 2^ 1 IJ^^ Diti. of longitude from Greenwich, o o oO uu G.M.T. 17th March 4^_J7_^^ = 4-12 hrs. L.S.T. at G.M.N, l^h March. . . . 2:3 41 n7-(i5 Correction for 4-12 hrs ^^^ R. A. at given mean time 23 42 38 -20 L.M.T 2'1 L.S.T. required 22_43_J^8;lo APPENDIX II. f I TO CONVERT K.S.T. INTO THE CORRESPONDIXO L.^t.T. (SEE PAGE 34). Another way of effecting the conversion of L.S.T. into the cor- rcsi.onding L.M.T. is to make use of the formnla on page 33. By so doing Ave are saved the trouble of ascertaining on v>'hat date is the preceding L.S.N. By forniida on page 33, we have: _ The Viven L.S.T. =^ L.S.T. of preceding L.M.N. + the sidereal c(piivalent of the required L.^NI.T., or, the sidereal equivalent of the required L.M.T. = the given j^.S.T. — the L.S.T. of the i)receding L.M.N. Then convert this sidereal eroduce(l in (.1 + dA) sin dA. 1 sin 1". of the error we may sup- s {A + da). small quan- ) or more of the small quantities are multiplied together, we get, after subtract- ing the result from equation (1) as before : Sin a da = (sin h cos c — cos b sin c cos A) db -j- (cos b sin c — sin b cos c cos A) dc -(- sin b sin c sin A dA. Second 3Iethod : This method involves the use of the differential calculus, and is by far the quickest method for those who know how to make use of it. _ Thus differentiating equation (1) with a and A as the variables we get — sin a da or, as before, — sin b sin c sin -.1 dA, dA sm a da sin b sin c sin .1 And again, supposing all the elements to be variable, we have sin a da = sin b cos o db -\- cos b sin c dc — cos b sin c cos A db — sin b cos c cos A dc -f- sin b sin c sin ^1 dA = (sin b cos c — cos b sin c cos ^1) db -f- (cos b sin c — sin ^ cos c cos yl) f7c -j- sin b sin c sin A d A ( 2 ) = sin a cos C <7^> + sin a cos ^ dc -f- sin i sin c sin ^1 f^yl (3) or da = cos (7 db -\- cos ^ (/c + sin b sin C dA ( 4 ) Now the general formulre used are as follows : cos a = cos b cos c + sin ^ si'i c cos yl (5) sin a cos 7i = cos b sin c — sin b cos c cos A ( 6 ) sin rt sin JJ = sin J sin ^1 ( "7 ) The first of this is the one usually en; loyed for finding the effect of small errors, but the others may be more convenient at times. Treating equations (6) and (7) as we have already done to equa- tion (1) or (5), and assuming all the elements to be variable, we get : cos a cos Ji da — sin a sin H dJS = cos a cos 7? da — sin b sin A dB (8) T=z — (sin b sin c -j- cos b cos c cos ^1) db -f (cos b cos c + sin b sin c cos A) dc -\- sin b cos c sin A dA (9) and = — (sin b sin c -f cos b cos c cos A) db + cos rt dc -\- sin i cos c sin yl f?^ (10) sin a cos 7? f?J5 -j- cos a sin i? da = sin ^ cos A dA + cos ft sin A db (11) 6 \ ' Of the tliflPcrcntial equations (i>), (:{), (4), (8), (9), (10) and (ii) showing tlu' effects of small variations in the co-ordinates, (2) is the most useful. For example, in solving the astronomical triangle T'ZS to find time, we desire to know the effect of small errors on the hour angle t in observing z and in taking ^ out of the Xautiad Ahnxnac. Put A = t ii — z Ji = A h =^ 90° — 3 C =7 c = 90 — (p In this case we assume that f lias been known correctly and that therefore (I = d^ = and da = dz = 0. Hence fr om equation (4) = cos A d

''"ivon Clements of the triangle are A + error means that the g ven ^^^ reverse. Thus an The algol.rai. ^ip.s o <''« 'T , . Tcreasea to that amount, ■!;:;i'L+,f..;r^.>ar«'i:a«tir.;;Xt a.o„„t f^. the ,ff„.t of the K''"'" <■';■■»>■»■ , ,, u ,vo »ee that the errors have Fro,n equatu,,., (12), (U) » ' „ ^;',,^<,„ observing for time the l;:::erUoaier:,:^ur, Z ., "-near to the prime vertical. 1: I- i .if- APPENDIX IV. I i THE COXTRACTIOX OF SEMI-DIAMETER OP THE SUN OR MOON WIIEX NEAR THE HORIZON. When the sun or moon is observed near the liorizon a correction h required to be made in tlie value of the semi-diameter deduced from the data in the JVciuHccd Almanac, because tlie effect of refraction is to elevate the upper limb less than the lower one, and thus to give the disc an ellii)tical form. The vertical semi-diameter will therefore differ from the horizontal one, and all inclined senu- diameters will have different values dependent on their degree of inclination. For if h and A' rei)resent the true altitudes of the upper and lower limb, and r and r* the corresj)onding refractions, then (A + r) and (A^ + **0 ^^'^^^ ^^*-' ^''^ apparent altitudes, and their difference (A — h^) + (r — 'z'^) = the vertical diameter — 2 (.s — ?/.') J" y»i Where s = the horizontal semi-diameter and w = — ; The disc will then be an ellipse whose semi-axes are s and (s — w). Then the compression c as defined in Aj)pendix V is iS — w w and therefore any radius s' which makes an angle y with the horizon = s (1 — c sin2 v) r — r .1 -sin- y. This correction is very small, and hence it is not necessary to know V very precisely. ■:M APPENDIX V. THE FIGURE AND DIMENSIONS OF THE EARTH. „ = the xemi-major axis or equatorial vadm, = C li- ft = the semi-minor axis or polar rad.us = C P c _ the compression of the eartli. c _ the eccentricity of the meridian. a a CF ^ — CE CE^ — CE- that is e- = 1 — /> (/ CE- - = 1 - (1 - ^•) = ;. 298-1528 _ = 1 — c 299-1528 1 1 . _= nearlv, 299-1528 300 It or c = Whence e == -0810907 'log rl"^-»122045 and log «= = 3-8244000). ss 3: • ■if. . 10 Tlie absolute length of the semi-axes at the mean sea level accord- ing to Bessel, are : (I = 0074532 -lU yds, = 3002 -802 miles, />= 0951218-00 ' " =3949-555 " Owing to the ellii>tical shape of the meridians of the earth, the direction of the vertical or plumb line ZAO at any i)oint A on the earth's surface does not coincide with the line AC drawn to the centre of the earth, unless the point be on the equator or at either pole. The zenith Z to which the ]»lumb line i>oints is therefore not the same as that, Z^, to which the radius of the earth i)oints. The angle ZOE which the vertical line ZAX makes Avith the major axis of the earth is called the inotirujthifiil* latitude, and is always the " latitude " meant, excejtt where otherwise mentioned. The angle Z'CE, which the radius of the earth at the poiut A makes with the majtjr axis, is called the tjinrentrir latitude. It is evident that this !>eocentric latitiule is alwavs less than the hical latitude. This difference, that is angle ZAZ' or CAN, is called tht rednction of the latitude or the aiajh' of the vertieui. Z is called the (ieo'- that IS tan c' = ~. tan point A de. It is I tlie o-oo- or CAN, 'trticuL D=y. To find the radius of the earth for a given latitude. Let r = the radius for the Latitude ^ = AC Then r = ] ^3"_|_ ys Now the equation for the ellipse is t u 'l_ = 1 and -^ = 1 — e- Ilcnce .•0-^ + f .'/* (1-e^) - = a"' and -^^- = tan ^i = (1 — e"-) tan X From which by a sinijde elimination we lincl a cos

/ we g et r = <( I — •> t" sin" y -^ <;* sin'^ y 1 — e- sin- cr If in this formula we substitute t^ sin* y^ for c* sin^ f , we get the following near approximation : r == (/ y 1 _ t^s'sin""^ nearly. 7m /,•;«? the length of the normal terminatorti in the polar or minor axis for a given latitude. j^^.t X = the normal = AN (tig. 39). Then in the triangle ACN Ave have 111 III*. II lllll!J,>V A-m.-^^- .. - N AN sin iVCN __ sjn_(n0_+jr^) ^ , expressed in seconds of arc, then the length of the arc of parallel between them is X /\Jj sin 1" = a cos ^ (1 + —- sin^ f) ^L sin 1" 13 To find the raaius of curmture of a ^nerUlUm at a 'jl»en latUurU. radius by P we have a (1 — e2) P = = a (1 ^H' + '^O™'"'^ riocr a — e"-^ = 1.9970911] place, cl..e ..og.,h<.r o-> J^e ^^ "J ^ia Kw-n them k APPENDIX VI. I- I L srx DIAKinXO. Sni»iiosc tlio cartli to be transparent with an oi»a(|ue axis, Tlioii this axis will throw a s]ia(h)w, and this sliadow will always Vw in the same direction for the same hour ani«le whatever the deelinalinu of the sun may he. But the shadow of any partieular /h it will always lie in the shadow of the axis for any uiven Ikhu' angle. On this principle all sun dials are constructed, the shadow of the axis I)eing made to fall on some convenient circle of the earth on which the directions of the hour circles are marked. Then the jwsition of the shadow on this circle, with regard to the meridian line on the same circle, shows the sun's hour angle at the instant and therefore indicates t/ie hwal appurent solar tf/nt : so that in order to obtain ordinary mean time we have to apply both the e(|uation of time and the reduction to standard time to the local api)arent time given by a sun dial. In all sun dials the earth's axis is represented by a thin rod or the sharj) edge of a metal or wooden ]>late, called the .s'///A or f/HonioiK This style or gnomon is placed parallel to the earth's axis, and on account of the great distance of the sun (as com))ared with the earth's radius), it may considered jis actually coinciding with it, without causing any appreciable error. In like manner the didl plate is really ])arallel to, but is supposcil to actually coincide with some circle of the earth, and the hour lines on it may be conceived to be traced out by the shadows of the axis of the earth falling upon its surface. lleuce there may be an infinite number of different kinds of sun dials, and the i»ractical value of each dejx'nds on the ])osition of the i)lane of its dial plate (on which the sluulows of the earth's axis falls) with respect to the meridian and the plane of the rational horizon. If the shadows of the axis are received on the great circle of the equator, the dial is called an etjuatorial dial ,' if ui>on a vertical plane, a vertical dial, and a north, or south, or east or >rest rertirat dial if it faces in any of these given directions respectively : and if the siiadows of the axis fall upon the liori/.on the dial is called a horizontal dial. 15 These are the most usual forms of dials and the earth's axis is represented on them either by a thin rod of metal parallel to the earth's axis, or by a triangular metal or wooden plate, one edge of which is parallel to the earth's axis. l\ Tlioi, ■•lys lie i„ |ecliiiatii)ii 'f'f in tlic S, tll(H|o]| iveii lioiir le sliadow v\n-\v (if marked. ii"d to the i'h' at till' f'fni ; so IM'ly botii •U' to the in rod or L' .s7//A or lie earth's ^•onijtaivd '>iiieidin't rfic,// ly ; and L-alled a EQUATOUIAL DIAL. In this case the axis of the earth is represented by a tine I'od per- pendicular to the j)late of the dial and projecting on both sides of it. Since there are l."j° in an hour, the angles at the poles, and there- fore the arcs of the ecpiator which measure them from the meridian, increase iinifonnly 1.")° for each hour of clock time. Hence after setting oif half the thickness of the style on each side of the meri- dian line on the dial i)late, we have oidy to set if, from these lines, angles of 1.")°, :Kt°, 45°, etc., to represent res- c^tively the I and XI, the II and X, the III and IX, etc., (ipjKirent clock times. (See Fig. 40.) Such a dial must be graduated on both sides, because during half tlie year the sun is north of the equator or dial plate, and during tlie other half, it is south of the ecpiator or dial plate. An e(pia- torial dial will not show time at the eipiinoxes, for then the sun is in tiie plane of the dial plate itself, /.t. on the ecpiator. Another sim])lo form of an ele method of construction can be made as follows : — 'ue sun is '» looking when tbe styles are to about J mean sun Imiddlc of |e style is |lt' also has then the it't by the icatc'd by It 0(1 oqua- N". and S. t;, and by > suit any e IS a tri- ', and the iitrle equal be pole is f the style le style is ev surface ^bove the (or rather half the 1 are not flcs PKi, = 15°, Thus we struction 17 l.cl All (Fii;'. i'2) be the borizoiital i»r(>jec(i«ni of Ihe style on the lioii/.oiital dial plate, and let it have a certain Ihiekness == Ac« = l!/y == \h?. Draw i>AC- = latitude, and C' perpendieidar to AC. I'roduci- AIJ to I), niakiiiL!; I'.l) = IJC and set oft" at I) the anules 151)1 = 1.")°, IJDl' = :;()°, 'i;i).{ = 4.-)°, ete., to meet a line liE, |ias>iiij4 tbrouu'li IJ and |ieriteiidieular to Al>, at tbe points 1, 'i, .'5, etc. Join Al, Al', A"), ete. Then these form tlu' hour lines, A siiiiihir constriietion is earried out for tbe hour lines on the other side (if tlu' styli'. 'I'hf principle involved in this eoiistruction is tbe prolonu;ation of the lidiir lines niarki'd on tlii' eipiatorial dial to meet tbe plane of till' liori/.ou l»ecause thesi- ecpiatorial hour lines mark tbe direeti<)ns (if the planes of tbe hour cireles passiny- throun'b the axis, and their imilonuations mark the jioints in wbieb they eut tbe horizontal |ilane. When tbe divisions on the line perpi'udieular to W) runoff the phite, we can eontinue drawing' tbe hour lines thus: — Take any pdint in tbe :!-bour line and tbrounb it draw a line })arallel to the !l-Ii(iur line, 'i'ben from tbe y'iven point set off alouL^ this line the distances of tbe 4, ."i ." '.d (» hour lines e(pial to tbe distances on the line of tbe -J, 1, and <> hour lines respectively from the given point. A similar construction is carried out on tbe other side of the style I'di' layini;- soitii dials. These dials are very similar to hoiizontal dials, except that tbe dial plate is placed vertical. ^Vs a rule they sboidd he larife, and placed in a conspicuous position. In vertical dials, facini-' I'xactly north or south, we have to iind the arcs Nl, N-J, N:5, etc. (V"\ii. 4:3), from tbe right angled triangles, I'Nl, I'X-i, PX:i, etc, as follows :— tan X 1 = cos lat, tan 1.")° tan Nii = cos lat, tan :U)^' t;in N". ^= cos lat, tan 45° ("tc. == etc. If 1* bi' tbe noi'tb jiole, tbe figure ri'preseiits (/ soi'f/i t/it//, Ol" representing the edge of tbe style, .1 iiorlh iHnl only can be used in the soutlu-rn bemisplu're, south of latitudi' •2'-\\°, and a Konth iHiif is only of use noi-tb of latitudi' l':!.\", in tbe northern hemi- spliei-e, 'i\) Iind if a wall is I'xactly facing south (or north, as tbe case 2 18 may In-), ln-fori' placiiiLj a vertical dial (tii it cri-ct .a style iK-rpt-n. (licular to tlu' wall, and liaiiij; a plimili liiu' Croiii tlic oiitiT eiul of tliis stylo. Then at local a|i|iari'iit noon tlic shadows of tlic stvK' and ]»lninl> line will coincide, if tlic wall laces exactly south ( or nor th). VKKTIt'AI. KAsr AND WKsr OIAI.S. In this case the |>lane of the dial is in the meridian, and ihestvlo is a rectanjijnlar plate |ilace(l iierpcndiciilar to it, itrojectinLC on eacl) side of the dial plate, and paralhd to the axis of the earth. 'J'hen the shadows on the di.al plate for each apparent honr-aiii,'lc will l»e )>arallel to the styli' and to each other. The (i a.m. line (Fisjf. 44) is in tln' |>lane of the style, /.»., it is the projection of the style on the dial plati". The distance of the T a.m. line from the style = height of style < tan 1")° ; the (lis. lance of the S a.m. line from the styli' - height of style >', tan ;30°, etc. 'I'he afternoon hour lines are similarly marked on the other side of the dial plate. 'I'he east sidi- only shows the mornini;" hours, and the west side only the afternoon hours. i! orilKIl KINDS OF VKUTH'AI, IHAl.S. If the wall on which a vertical dial is to l)e fixed is not faciuLj exactly sontli, suppose, the vertical dial is lu'st niatle I'XpcriMuiil.illy thus : — Fix a l»oai', and draw \\C vertical. 'I'hen set olV the an^lc C'Hl) = sun's north polar distance - — co-latitude = the sun's meri- dian zenith distance. Make JJI) = AIJ, and make the anisic r>I)("= IS(t° —sun's north jiolar (', A and I) wouM coin»-iarent time*, and I *Tlie viirintioii in tlie ciniulioii ol'liiiie will not iiiivcimy pcrcoptililc clVi'cl on such ii roiicii time ini'trunient IIS ii .sun diiii. Ifeon.-idcred ncci'-^jury, tlie Wiilcli could l)u i-ut iigiiiu fur ^auli a|)|)iii-unt hour. I's iiorlli ► would and li II ioiikIi again lur 11) l'('r|icii. • t'lltl ol' lie styk' |>iitli (or Itlicstvlo till 'A oil K' cartli. liir-aiiLck' .'., 11 IS (' II Till tlu' (1 IS. :•: Ian • I on tlic I'sl side 1 I'a fill' iiiciilallv he wall. I Ih'11 the • lirrjicil- II t•' il APPENDIX VII. LOCAL ATTRAf'TION oR AISN'OllMAL DKVIATIoNS (>K IIIK I'LlMi; MM .x- Granting; tliat tlic y('(»mc'trical fitiiiiv of tlu' t'iirlli to Itc lliat of ellipsoid of revolution, it must not \>v ini'errccl that the (lircctioM of the ))Iuinl) line at any point always coincides jtrccisely with tlic normal of l!ie ellipsoid. It would do so if the eiirtli were an exact ellijtsoiil eomposetl of perfectly honi(»n;cneous matter, or if, oriuin- ally Iiomojienei-ns and plastic, it has assumed its present form solely vndc , iic influeni-e of <;iavitation coml>ine(l with the rota- tion on ii> j.xis, IJut experienci' has shown ilial the pliind) line mostly devial's Thmu the nornnd to the renular eHi|isoid, not only towards ihe n(>rth or south, hut also towards tiie east or west ; so that the zenith as indicated hy the jiliunl) line dill'ers from the true zenith eorrespondinii; to the n()rmal both in declination and rifjht ascension. 'J'hese deviations are due to local irienularilies or variations, hoth in the tiyure and density »d' the earth. Their amount is, however, very small, seldom amountin^• to more thai ■"," of arc in any direction. In order to elimiiuite the iidluence of these deviations at a jiiveii l)lace, observations are math' al > nundier of places as nearly as ]K)ssible symmetrically situated around it, and, assuming' thedimcM- sions of the u'cneial ellipsoid \n be a- \\v have uiven them, tlic direction of the jiliimb liui at the liiveii place is deduced from its diret tion at each of the assunu-d piaci's (i>y the aid of the n'codetic rneasuri's of its distance and liirection from each) ; or, which is the same thinu'. the latitude ;uul longitude «»f the place are deduced from those of each of the assumetl places; then the mean of the rosultinn; latitudes is the t/rm/ific latitude (which is identical with the (/eoi/)-(f/>/ii(if/ latitude), and ihe mean of all the residtinj;' lonyi- tudes is the >/in(7ifir loni;itnh Hhii^h'ilHiKI flu 'jftrf of' Tj'Xt/l ^{iti'Ui'tloii oil ^[xillUlt/l If an almoi'iual deviation of the plumh line occurs at a station, aniounlinix to '/', the plumh line detlectiny' t(» the west of the true vertical, what etVecl will it haxc on the a/.imuth of a line run on a hcariiin' of |)° west ol' north, the saitl line passing- over grounil rising' ,:° I'rom the ohscrver V Let P (I'ii;. 4(1) lie the pole of the licavens ; Z the lieodetic /ciiith ; Z' the the plumh line zenith, r IIII' o". COS I). COS i/j -| '/." sin D) 22 I Siiuilnrlv wo have XV : ZZ' = cos (ito — P\) : cos (!i(i — TZ) 1)1(1 PN = latitiiilf = V" aii( 1 I'Z <)() - v I [dice, NX' : ZZ' == sin c : c" /^ iicarly if we iiciilcct the small <|iiaiitity '/." sin I), Xow, ]ir(»j('ctiiiii' this allele on to the apparent horizon iiicrnieil a" to the n'eo'letic horizon, we yet, ohserve azininth = X'll' sec '/.". l>iit as a" is alwavs very small sec ik" = I nearly, and we n'et oltserveil iizimiith == '/." tan c \) a cos I) tan ,j ortriieaziiniitlir^ I)-=ol»serve«i aziiniitli '/" tan f | '/" cos 1) Ian ,;. appi-:ni)ix viir •KKSONAI, KKItnKS AM> I'KltsoN A I. Ki^tf AlKtN — I'KKSOXAL SCAl, Pn'So/lnl J'JijKilflnii. ill I)") fllllCll sec ,/", AC ii'ct 1)1 III III iiiakiiiii uhscrviitioiis. even l)y well traiiicil iiu'ii, i-cruiin iiii;i\iii(l.ilil(' |K'isnii!il errors always crcc]* in. This is diii' to tlio •Mvc aii personal errors are more or less eonstaiit lor tlie same class (if ohservatioiis, ami their value lor each class when ascertained is <;illcil tlu' /ii rKDiidl i', whose distance apart from each other is. however, the same as that, of II ami li. The ratio in which the distance'^' //' is divided he may still j-stimate correctly, the case ii; point lieiny that the iili^oliite positions of the star in space ami time are wrongly c-i una te.l 1. IV an eiiiial (iiiant it v The distance lietweeii " and iiiiili II of till' oliserver, and, if it idiild lie determiiieil, iniiiht he applied as a correction to all hi> iihservations. Hut, so loni;; as his oliservat ions are not comliineil with those of another the e.visteiicc of such ail error cannot he dis- I'liveretl, nor is it then of any consi'ipieiice in cases where only the illf. '•i K of clock-times is icipiired tti he known, as when ohserv- iiiLj the times of tic transit of two stars (correcteil for instnimoiital errors and rate), for this ditl\'reiice will evidently he the same as if the oliserver had no personal e'|iiatioii. ("oiisecpieiitly, we see that in order to comhine the ohsi-rvations • if two iiidi\ idiials it i necessarv to know the ditl'erence nliii^i< "/ I'lnnJilinn should lie a^ iieaiiy as ]M)ssilih' till same. Kveii lln imsliiri nj' l/i, l„i,lif h:is heeii round to ha\(' some etl'ect upon the ohservel''s estimate of the (do(d<-tiliies of contact ; and it can hardly lie doiihtcij that the pergonal ei|ii;i- lioii will tliictiiate more or less with the oliserver's health, or the condition of his nervous system. That the personal eipiation depends upon no organic dd'ect of either the eye or ear, luit npoii an ac~ or decreases diiriiiL;' a lap^e ol' years ; this proi;'ressi\c increa-e or decrease indi- cates tlu' Liradiial rormalion of iixed liahits of ohservation. So j'ar i'roiii iiivalidatiiiL;- the r(siilt>. of two smdi ohservers. this fact would intlicale that their ahsolule ]per-^oual eipiatiiui^ were in all pidhaliililN very constant foi- moderate iiiter\als oj' time, and there- fore had no ap)U'ccialtli' elVect upon their r<'sults, so loiin' as tlie^c re^nltsdid not di'pend upon a comhination of their oli^cr\ alioiiv with thosi' of other ohM'i\ers. 'I'lu- methods (d' lindim.;,' the relative pergonal ei|iiatlon of two (d>si'r\vrs in notiiiL;- transits of stars will he L;i\en when trt'atin^ on tlu' 'I'ransit Iiistninient. / ( /'S'l/ii II Si'iiii . I'rof. I'ierce has calle<| attention to the fact that experienced (dtservei's often acipiire a lixi'd erroneous haiiil of esiimatiiiLi; par ticiilar i'raction- of a second. 'I'liu^. eacdi oh^erver i- conceived to have his own /n r!<"ii- h\ t he ^ame ohscrvi'f, for in cr\ iT. (iltscrx ('!', IMC clianri' lor liic ofciirn-iuT of i-arli of tlii' j,'e iiuniher of ohservatioiis, and may Ik- regarded as rininu a part of it. Hence it follows that the a)»iilication , which involves tlu' errors of the pi'i'soiial Iv eliminate the ohserver's constant error ,f ilu' lu-rsonal eiiuation, which involves tlu" errors ol the perso (ics not necessari •A ohservation, hiit that it pndiahly does eliminati' it from a larye nunihcr of ohservatioiis. AIMM'INDIX I\, w iMi.uis AMI I'Koi'. Ai'.i.i: i;i;i;i;i;\ a iimns. A inmil»('r of olisn'v.'itidiis Ix'iiin' l:iliir|M)sf ul" dctn- iniiiiiiu" > an iiii|ioi'taiit iiroltlcm lo (Irli-riniiic the nin^f /imhahli values of tlic iiiikiiowii i|iiaiitili('s. Ill ii'liliji ( fli.'ti I'nit liiiis iir< liiilili K\ (TV oltsi'r\ atioii whifli is a iin ti.-iiltii'ct to error; I'or e\|ierieii('e teaches that i'e|ieatee.\lant, say. u'ladmite"! t" (h';^i'ees onlv. minht ii'ive tliesanh' resnit. lor it is e^|ieeially wiien we a|i|iroa(li fill limits i>f inir nil (isiiriiKj /inir,,-s that we iieeome sensihh- ol tlic (lis('i'e|ia)iei('s ol' oltserx at ion^. Hence, tin' acconhmce ol' (ilisei\a- tions witli small instiaiments is not to he taken as an int'allihic evidence of their aecni'acx . 'IMuTc :ivv two kinds ol' errors to he disiiniiiiished. viz.: ( I ) ■■"i,- sliiiit ^>\' I'l i/iiliir I I'i'oi's \ and (l') i(<-ii

  • iicii errors must he the suhject arel'iil |ireliininary si.;irch in |iliysical in(|uirles, so that their act imi may l»e altou'et her prex ciili or their ell'ect removed hv ealcnlat ion. !•' ))• evam|ile. amoiiu' tlic constant errors may he enumerated ret'r.iction, aheiralion, etc.: the I'lVcet of the t( iii|iel'alure of ro(|s ii^ed in nieasiirinu' a liasc line in a survey; the error of division of a LjrMdiiateil instrument when tin same division is iiseil in all the measures; any |ieculiarity of an instrimient which alTecls a |iarticnlar measurement hy the s.inic amount, as an ine(|indity of the ]ii\(its of a transit iiistiaimenl. defective adjustment of the coHimatioii, imperfection of lenses, dcfi'cts oi' micrometer screws, etc. ; to w liicli must lie ;id(!ed CiHI- stant peculiarities of the ohsi-rver, who, I or example. m'(y ;il\\ a\ > note the passaue of a •«tar over a tlirea(i of a transit instrument t( (if dctcr- »'l'\ ;itini|N • ti'i'iiiiiu' y It iiiiiy !■(' leaches ' tif< .^tan(H' ), assniiics for a coulai-t a jtosi- limi ill wliicli tlir iiiiam's arc really at some eonstant small distance a|iai'l, or a position in wliicli the images are really ovi'rlappiiig, etc. Tims \vc liavc tlin-c kinds ol" constant errors : 1. 'I'lii <>,■< Uriil^ sncli as refraction, alierration, etc., whose ell'ects, ulicn tlieir causes ai'e once llioioiigl.ly iindfrstood, may l»e calcn- lalcd '/ ///■/( our |iliysieal theories are im|)roved. ■_'. liistrmin iitiil, which are discovered hy an examination of onr iii^I riimeiits, (If from a discussion of the ohservalioiis made l>y them. Tlie-^e may also he remove(l when their causes arc fully understood, eiiher i>y a |iro|per mode of using the instrtuiu'nt or l>y sul)se(|uent eiiiri|iuialioii. :i. /*/•.-' changes (d' density of the strata of the atmos- phere : from uiia\oidalile (dianges in the several parts of an instru- iiiciit produced hy jiuoiiialous variations of tcmperaturi', or aiioMialoiiv <'oiii ractioii and expaii>ioii id' the parts of an instrum*'nt even at known temperatures; hut, more especially, errors arising from the imperfection (d the senses, as the imperfection of tlu' eye in nu'asiiriiiL;' xcrv small spaces, of the cai" in estimating small iiiter\als of time, (d' the toiudi in tlu' dtdicate handling of an instrii- liieiit, etc. This distiiictioii iietwceii coiisiaiit and irregular errors is, indeed, lo a ii'iMain extent, ralhcr relative than aiisidule, and ilcpemls upon 28 '. the sense, more or less reslrii'ted, in wliiel; we consider ohserviitioiis to l»e of the fiaiii) iiatiii'i' or luadi' under the saiiii' ririi'/iisfdnris. For exanipU', tlie errors of division of an instninient may lie regarded as constant erroi's wlien tlu' same division (•omes into all measures of the sauu' (|uaiitity, l»ut as irrenidar wiien in e\eiy measure a diflVreMt division is used, or when the same (|uanlity is measureil re|ieatedly witii diU'erent iuslnimeuts. Although a«'cile elVect U|ion the results. For this condiination we invoke the aid id' the method of least sipiares, which may he said to have for its (diject the resti'iction of tlie tdVect of irreuular errors within the narrowest limits, according; to the tlmory of prohahilities. .and, at the same time, t(» determine from the (dtser\ ations themselves tlu' errors to whitdi our results aie liahle. It is proper to ohserve lu're. however, fn f/iKii'tl iiijiniiHt JiilUii'iiiitu Hfiplliiitliiiis, that ///' tliiiirif of' tin nntlinii /.s" i/r<>itn irhldi tin oht^i rrntloiiK nrr liuhlr. 20 <(»ui!i:« rio\ OK TiiK or.sKKv.\rit>\s. When no more ol)S('rvatioiis art' taken than arc siitlicicnt )o f»' n iiiiii'i if. 'i'his is a lo^icul necessity, since we cannot accept for tnilli that which is contradictory or leads to conti'adictory i-esnlts. The coiiti'adiction is obviously to be removed by applyinij to llu* several oltservations ( or conceivinji' to i)e applied) prol)al»le nirm-- fio/is, which shall make tlu'ui auree with each other, and which we have I'cason to suppose to l)e eipiivaleiit in amount to tlie accidental errors severally. Iliit it must be iiere rcmarki-d that, in this stati'- iiiciit, it is by no means to be implied that an observer is to iirhiti'iirlh/ assunu' a systi iii of cori'«'ctious which will produce accordance; on the coiiirarv. the methoil of least sipiares. which we employ for the purpose, is ilesi^ned to remove, as far as pos- sible, every arbilraiy consideration, and to furnish a set (d' piinciples which shall always yuide us to the most |U'!I!I;< r o|;si:i!VArioNS. ( )bservaticr\ atioiis III a»tr( iioinv, ill u't'oiU'sy, ami in iiliysical sciriicc Lfciicrally Ix'lolm' to tl lb attt'T cla^ lint as !ar as iiractical astroiioii.y is conccnicd \vc only ('iii|iiiiv as a iiialtcr ol" I'aiM, dii'i'd ohsn \atioiis oi' alt it in If- ami ili^laucis cf wliifli tlir results ohtaiiii'il arc I'liiKiioiis. ('oiisf(|iiciitly, we will not luTf ili'al with tlif case ol' imlirt'ct ohscrvations. 'I'lif consid- t'ratioii of this siiltjcct will conic iimlcr the licail of •* (icoilcsy." riir: aimi iiMiriK .\i. .mi:\\ oi' iiii;i:ci or,si:i;\ a i io\s. Chanvciict in his Sjiln i-iful unil l'rrd, // till iiliSi I'i'itt II niS ili'i ii/ii(tlli/ i/'iiiil. tin Slim nf tin Sijiiil/'is III till /'I sill mil I /'/'Ill's IS il III I III III II III , ('liaiiveiiel. priiceediim uilli the iiiv fst iLZit ion, •>1im\\v that the |irilici|ile that //" iimsl ji/'uhiil/li riilms <;/' tin II nl// (H' iii;>i:i;\ a- 1 loNS ol' I 111. s.\Mi; V. INK. The deLi'rees (d" )ireci>iiiii dl' dilVereiit series of uh^erx at imi- of the same kind ma\ lie coiiinared loLidhci' l»v conipariii'_i' the erriir~> wliicdi are coininitted with e(|nal facility in the the e.\t reine errnrs, ami we may selecl for this |inr|)ose in each system tin i/'i'n/' n-liii-li mi-iif/ii s tin miililh jiliin I// tin si/'iisiif I /'/'Ill's II I'/'il iii/i il I II t III ii/'ili/'nf t In i /• iinli/n it mli , sn tinit tin lliiliilni' nf I I'i'i >/'s irlfnli il/'i li ss tlnl.'i this ilssii/in il i /'/'ni' IS tin siliin lis tin iiiln/l/i/' nf i/'/'n/'S ii'lin'li i.i'niil it. Tliat IS, We may select in each system tin /i/'nlinhli //•/•"/•, in order to coiii)i;ire 31 |«' M|M)|| tn lie |i .'iMru- t<> tlii> ■III piny, lllll'I'S (if |uc will *'n||si,|. ;iniiiii'<: ll'I'islic 'ri;ic|c II. '/, lh< ■III I III il S'l IKIi'l .< Ii.'ll the IllllHtll s I I I'i'iii'.-i iliNrr\ a i:>i:i;\ \- lOJlS (if CM-Ml'^ x'l'io. I'lifs a !l iii/i . (lill'ciciil scries of iiltscrvatioiis, for l»y ('<»iii|i;iriiiy llicir |>rol);iltl(> (Tioi's we ••niii|iarc tlicir ilcr llii iirnliiilili < i-nn' as tlie icnii of ccniipai' Isnli lici w fell flilTiTciit sei'ics (»r oli«.ei'vatiniis is purely arliilrarv. altlioiiuli it M'eiiis to lie naturally tlesii^nateil hy its miildle |Hisitiiiii in the x'i'ies ol' en'ors. 'I'liere are two otlier i'rrors wliicji have l»eeii ust'd I'ui' tlic same |iiirp(i>e. 'I'lic lirst is tlie nil iiii <>f' tli e(i^iti\ e ^11:'!!. // ( nor. these heiiiLi' all taken with the le (tt iii'i" is I lie ////'///' /•/•"/•, wliieli must not l)e conioiiiiileil with tlie alio\e mean ^A the errors. Its delinition is, t/i I iiini' l/il I I rmrs. si/i'tii'' III' ii'liii'ii 'N ///' niiiiii iif I III s'j iiih'i ■•< iif' iill III ('haiiveiiel nives the lollowiiin' relations between tlicsc ilirec kimls of errors ii'oliahle error n-si.".:; mean of the errIi-.i'rvation>- of the same kind are said to liaNc the saiiu' or dil- jcreiit ileurecs o|' preei-ion, according as they have the same or llVreiit mean (or proIiaMe) errors. We assume ^/ /i:^iiat ioii evcrythiiiLi- which can alVect the ohscrvatioiis ; hut wliether this condition has in every case lieeii realized can only he IcMined II jii'fili i-iiii'i, from the mean errors reali/.e(l l>y the ohserva- tion^ themx'h <•>. iiii;i;ci oi;--i:i;\ Ai loN-- iiwivi. iiii; sami: i»i:(ii!i:i': oi- imji;* is|on. In direct oli^crv atioii-N, which arc ('(pially u'ood, the arithmetical mean Liisc^ the iiio>t |irol>ahle value of the ohserved a> Toll ows ;'o -' p and I'l-om the constant relation hetweeii the mean and the pridtahle errors niveli aliove \\v have IS, we miitare Where i the mean error of aiiv single ol )servat!oii till' mean eiror of the ai'ithmclical mean. 32 This result fdlldws I'roiii ilic iinportant (lii-iin in lli;il tin i>rnlsl,,it of till' nil iiii of !' any '^ivcn series of dirti-tly oKscrved (|naHtitii- of e(|nal |«recision. T-et '/,, e till' oliserved \aliM' their iiritliinc from eaeh ohserveil value : so lliat eti'., the residual errors found \)\ suht ractiiii,' ) of a single ol tscr- \ atnui -t- - (..•-■) * ^m wheri' // the iiumher of olisei'vations, and - ('•'-') the mum of tlio (luanlities ''i '< ''■:'- ''.i'"' ^'^'■• 'I'his forninla yixcs a close a|i|ii"o\iinati(»n to the value cd' », hiit it is declucc'fl I'roni a considn-ratiiin (d" an inlinite serii's id' ernuN which would re(luce the mean error (d'.'',, to an inlinitesiinal, accord- iiiii' lo the ]trinci|des assumed. ,1 In (tir iijiin'oyhiiiU'otn to f/o ruhn <>/' i\ ir/nri' tin SI t'ii'.t is liniltiil. is to he oluained i»\ consideriiii^ the mean error of .c^^ itself, and. consei|ueiitly, also the mean errors of t lie residual errors /• •tc. If •tc., he the triK errors cd' the ohservcd values, and if tlie true value (d' ,/■ he (,/ii I o) we have the true errors ami rememhering that - (/•) (i /', - '), et( wo have (A/) // r 1- {>-) 1' (r) H '»- Thus the aiiproximate \aliie ) re(|uires t he ciiirccl loii II ')■', the value of wITudi de| tends ii| ion the \ aliie we may ascrilic li As the liest aiiproxiination, we may assume it to lie the mean error i „ so that // o- II <• wlii( h n'ives wlu'm-e and conse(|ueiitly II <■' :• (r-) - ['-') -I II — I -h u-Cm i: \ -('•"') ;{:J 'I'liiis I'lom t .r :i siiiiiU- nhorrvt 1„. :„tii:>l ivsi.liial errors tin- im-aii ainl prol.al.lf frn.r I valii.' arc I'nmi.l. Mimic*'. In-iu wlial has l.con aiil oil I'J'.ti' ihc mean aixl |in»l»al>lc errors .» I' I he aritlnnctical iiicaii w I 11 he loiiiul hv tlie !"ormiihe (fC.7 l."> I) \ ,1 {„ - I) |l,,ir (i-CTI,-) - 1 -S-JSUS ]. 'I'lie iirocediiiii nut h...l of tin.ru.s^ the prohahlc error iVoni tlio s(iiiares o r the resiiliial errors i> that wiiieh is most comiiion Iv ('III Illation eaii l„.r of o\»ser\ati»»iis is very S'"^'''^' '^ ,,l,,vc-.l ; hut when the iiimi l.iVahh. to al.r-.l-c tlie lahoiir, if i..,ss,hle. A snlheien a .p'' )X1- 1„. ..htaine.l hv the um- of the lirM I'owi rs of the resi- dual errors, a- follows ;- (live all the errors the i-oMlive si_ij;ii only 'hell the mean of the error> ('•) ami /' ano ' ((•St."):! l-:>a:!: '-(>') ('•) A looser a|)|'ro\iiiiatioii. for the reasons already given, is V (i-sl.-):; V „ (» ~ I) Kr.n, tliese forn.iihe we -et the folluwin- aiM>'oxiiiia 1 nrnhahh' errors of the aritlimetieal mean te values for the mean aiiositive aniis wi' have : 1 w ~ IC, ami n (u I) 2(/-) lor an iiiliniic series lor a Unite series. .) N C()nse(|nenlly, sMhstitntini; (liis value in tlie lonnnla lor /<,, we <,'ct — -1 (i-tl7l I t.) ->- o-i7(;'.t and e., — -I- • ••ToTl C'onsetjuently, I 0--l7 ] And lienee we see that wIkIi we have several series of ohserva- lions of the same kind, ami llif oltservations of each series heini; e(|tially j^ood, luit tlie different series themselves not beinix !^«S 'Ikmi f/n ifrfi/Zifs itt th> illjf'triiif SI ri> ■■< l'iilKlhli) i n'oi'K <>/' f/n ir /•< S/h rfirr (li'ilh- tru'ticdt NiriuiK. It is not possihle to find tin- wi'iii'ht of each individnal oliscrva- tion. Intleed, there is no reason why the indivirvation. SKKIKS OK hIKICCT OUSKKVAIIONS Nor II.W INH rilK SAMK l)i;(iUi:K Ol' I'KKl ISloN. Each series of observations is treated as a single observation hy series of '„ \\V''V[ »»hs('rv;i- ics hciii'j; f so, tllt'll Hi/ In llti in tin'f/i- I oltscrvii- scrvatiniis Auiv\ tiiiu- illy !i,()("l. lis of »lif- crics ; or. it'll muliT )K(ii{i:K oi' vutioji by M 'M\ TlicM (In- rnc.iii :iiiil |»rul»;ilil<' criurs of llir !4t'iit'i;il iiiimii (///„) .in as lulluw-s ; S/ 1{"') \,r .•-■) (/' I) - <"•) I (i-tiVi. I ^1 ') 'I'lii' wrii^lil ("•„) <>r ill)' ■r,.|icr;il iiioiii is (// I) - ("•) "', : ( //• /• ) 'I'l K'sC I'liriil lllr m'IMIlll rulitil||;i'. ;iln! llic ulirs !.M\('II oil |i:i'.;r cm Im' "Iriivi'ii rimii llir>.c li\ iiiiMiiii^ /'• I, u liti :■ ( //• /•-) :■(/••■) !{»') n. 'V\\v ^v\ i\«; I IN! <• i; III I l'>i; ill I, 1 1; \N--ri IN-^lltl Ml N I •'!' I II i.oiMii I ii: >n■ r.lllt-. Is It' l.'llsili ill I'lili I' nillirlil. r mil Til ITS uT :i nil M I. imiiiii;) I rl:iiii<'i| Altli"Hi;li, ill vviii K ill til.' li.'M, in^lniiii«'ii<'^ '•'■'> -■""" In III' pill U|l l'"l <'llf Ml iiii|Hirl;iiil I' • III uiilv in ■' timt'-^ liMVf I m'iicr;ill\ :il ,iiil. :i -l;iv "I --"iiif < |;,\s will l.f iiiii'lr III "I ,1 >■ to ililrl Mlllir III"' llic llirllll'^ ' l.,„i.,ti,.lV c.r iMlilii.l.', or iH.'ii. \Mll. :isimi.li:i.TiiiMr.v ,r ,1... nl,s..,v.r sMll |-i..iit. r...l;r mk'Si '■"■"•"'" siMiif"-' ;iii ol*"' i'\ III ill-.! nimt'iii . \\ sviiirr ; r:iii-|i<>i |i;|.,..;|M,' Im-I ;i«l\;iiit;i"'' w II ,_r I, 'Ml III' lull 1-- rii|IIM'l' I \\,r ihr prnlii'li I I i . iivMiliiMi'. :iMil i'c»ii-ri|iiii Miiii;i'liri;il, liu'ii |"'rl;il'l'' li">- '"•'> ( Mhiiw !-<■. \\ ■!"• i| Irlll ~, W il' I, ,,:,|,1.. ..„.U :inil H".!, I" l<"' ^M"''"' "" 1'';'"/^ illv iIh' wi'i'^li' "I" lii' iiscil u illi iIh- I, ill'-: mIm.iiI l^^ii ll'S., ill) iwii III IV|irllill ;iii' II' lti:iii :iri' liKi Iv li. I.I' ;ilVi'.'t«"l I'N I'"' :''■'■ , I,.,. I ^\ i,|r th,' i.l.MT\:ilii.lis ,1 ii,,. i.iil ..r liiil i-i';i|>iii'4 ll,,,|,. Ill,' ii|-r. III till' "I ii'lllll! AI'PI'NDIX M i.isr (»!• iNsi im\ii:ni s, m\, i;i:r(iM\ii. \iti:i> im; >i l;\l:^N m \ \w\<>\\ \ rui nii;ii:s. I ^i; IN iiAi'iii fe¥4 Till' Mi-liclc- uliicli -J,-." 1 :i mcirt' or !<•<< .•.im|.|('t.' uiiHii iri:i\' he clMssilicd iiinlcf llii' lul lii\\ iii^ lic:il niiiM'iil^. I. St.il in|i;ir\. Il"tr ImikUs, ImhiK- mihI ili;i|p-. "i. A|>|iliiiiii'i'-^ l<>r <'ull(T(iiiy-. 'riic Art ..r Travrl," aiilioiilil Itr uot lidin lir-t rate iiiaUi'i-. I ^'^ I i.M Mi:\ I ■>. .1 li i/i'/l Siilili li.|i| ilic in-trii iiK III u it 11 its iinlcv clani|M'il to aii\ part of lliraic, ami I lie rccc|i lical lor I lie in-crtiii'^- |i'lfsco|ic loii^ ciioiii^li to allow oj' it lniiiL;' jiiit into t III' lii(\ when -cl at locu-. . ir/i/'i ill/ nil I'liirlnl Inn'i.iiii. ,1 Shi iiiliiril i/iii'il. 'J. ( 'III .til riiiiiii\-< sh 1 1 Ifiinl shiiiiltinl ili(iiii:< ami arrow-. '■',. Till fill i-oil.-i c.i|.|ici| with -liiilitly ciiiM'il metal cml- ;:nil Ill.'Kle III' -e;|su|ieil ileal, >oalNei| ill oil, |iainlei| ail'l \ ai ni-lieij. I . StaiKJai'il cliaiii. .'t. (h-'linary chain- with >|iaie link-. ,1 fmi'ldlili li'iliisil I iisl I'll nil III . :('.» A I'll) Mlllil ( '.-I III 1 1 .1 1,1 i/h ss sili'i r li I'l r iriihl, with |il;ilc l:I:is> IVomI, ,-iti o| iirr -;(mm1 ualelics; ilicse sliolliij lie lulled ii|i separately, eaeli in a Ioo>cly \via|i|'ed parcel of dry clot lies, ill w liicli tliev will never eoiiie to liaiin ; lliey slioiild lie laltelled :iiid rai'elv opened. 'I'lie iniliiediate eii\ elope slioiild lie Ifee i'roni lliilV and dill, ("overs of cliainoi> leailicr slioidd he waslieil liel'ore use. Two spare walcii glasses vhoiild l>e taken lor cacli wateli, and some spare w ateli ke\ s. \\ liet lier I II II I:, t i-hrnin'iui /i rs slioiild lie taken or not depends on 1 lie means a\ a i la Mr lor t ranspoit inn' I lieiii . Tliei'e is ;i irndeiicy to relv on tliein if lliey are taken, and uliere tlie dillimlt of transport has Imiii <_;rcat tliey lia\e 'iro\cd lailiiies. 'I'lr,, ^irisiiKillf I iiiiijKiKto .-< is cases with slinks, i^radiiated li'oiri u" to .ilWl'. 'I'iri) fiiiil.il rii/ii/ii>s.\\\ I', to :.' inches ill dianieler. wllli pleiilN ol' depth, and calihc- lo leliciC llie needle I'roni ils pi\o| u hen not used. 'I'he n lie- shoiihl work sleadil\ and ipiiekl) .'Hid not with Ioiil;' -low o-eillalion. .1 niillliiri/ .u- i>i>iLi I iiiK I'lilils ( .' ', ins. .aerosv ), riirnished with slid iiiLT -eales, and capaMeol' w oik Iiil; w it lioiit I'r.iet iire o\ er t he highest mountain pas> that i-e\pieied. They can lie ohiaiiied jiradnali'd lip to |,'.,un(i feel at nio-i iii--i inmeiii makers. ImiI as siieli lieiij:lit. however, I liei r lecmd- are not lo he depended on. Aneroids are excellent I'ormosl dill'erelit i;il ohserv aliolis. Imt ai'e nmi liulili I'm- iilisiilnli ones ; tlli'V should lieoliseiVed as llllich aspo-silijeill con jiiiiciion with liie lioiliii'j- point thermometers. 'Two :ii |ea>,| ;ire reipiired, liecaii-^e -imnltaneon- ohserv at ions are import an I, and sncli ciltsei', at ion-, taken eviii at distance- ol' -.'oii oi iiiii mile- ap,"il, arc «i|' value, a- tin' area- are ii-iiall\ ver\ lar'_;e i.vcr which the liaro meter ha- neaiK the -Mine heiulit al the -anie ;iiiilnde ;iiid moineiil of I inie. Mercurial Karoinelci - are veiv lialde to vmI Inukeii In the Held. I I'l iliiiin li )■ . A- th: I- apt to 'jel "Ml ol' order, three ;il Ica-t should lie in ll-e il tliev ale elllploved. Tin nil inn h i\< . 'I'll lee -lio it and - Imit lioilin'4 p">iiit I hcrinometcrs, with apparalii- lor lioiliuM them in. 'The reailin^s ol' ;dl these should lie taken al aiiv -talion of oli-er\ al ion . 41) 'I'lirci' iirdiii.'iis I lM'riiit»iiii'l<'i>, wli'n-li ^lioiilil lie i;i;iili,;il('i| iVciin |ii"(ir more licNiw tlic rifc/.iii'j iiinl .iliuvr llif iMiiliiiLr |M(iiit, A st;iiiil;iril llicriiiniiiclcr. '^iiuliiiitril :il llic l\t\\ Oli-c \ Mhnv . .M;i\iiii:i ;iii''y„ t'l ,|- (.|ivri\ .11 ion "if nli jibe's i<\' .1 ii|iit it's ^;ili'Hilrs. ( )||(. Ill' -J ini'li i(l>jc"| o|;i^> (ricir ;i|M'il nil) .iii'l ol In iii;i^nir\ inn |i".u, r, li\ .'I I'i'mIK 'j;iii>i\ iinilvt'i'. :iiilioiiM liclricil for iliis |Hir|io^f lirl'oic liciii^i l;il-ni:ill sc.'llc, s;i\ iVoliI _' lo :M( liiilo lo llic ilji'li ;iliil |o |i|o| |||(> mm Till work iloni' l»\ olliir in-t niiiii'iil- ;!•> il |iroMi'c>.v(.s. >o ili.il il i;ni lie seen .11 ;i '^l.'iiii'i' w lull li;i^ liccn 'lone, ;iii>iii:ill s|MTi.il >«iir\i\^ lor llic inci^-niiiiiciil of ^|o|icv mikI con louring. A i^ooil lilnli I'll is iiiiis/ i lii/ii>rlillil. Il >-llol||c| lie l|sci| willl nil :tinl riiini--licei npon t he uroiiiiil. .\lso ih.il nil llic inlciniil lilliiiLis r.-iii lie rciiio\ei| ami cle;iiiei|, ainl llnit llie\ are sulidK iiiMdc. not iiierelv sohlcreij. Il s||.iiiM lie I'll mislicd willl ;i I'ctlcclor, lo throw ;i cle.ar li'_;hl lorwards and iAy/c//(C, //•,/„• ; a iiio\alilc shade of Tn^hi ^reeii Lihi^'^ i> :i ,i;r<':il ions eiiicncc. as it |ire\eii|s ihe li'_iht lioni da/./linu llic e\es, ;ind eli.'ililcs the olisciver In hike the rc.ldini^ oil :i >c\t:inl wilh '^re;ilcr case. .\ small h.ill of sjiare wick, oil oj the liesi (|iialil\, .'iml w;i\ !;i|iei> shuiild aUo lie taken. .1 I ,-'>'lii< iiiiii l< r I'oc incasiirinu distances sh.nild lie taken when W hi'eled coli\e\alires , i re likcl\ |o lie Used. Ml \s| !;| \ I. will M \ ri'l Ni. I \ - I i;i Ml \ I s. A siiimII cMse of diMwiii'.; iiis| niii.< '. I s. .M:iii|iiuis's scales. I 'fill r.aciors, reclan",iil:ir ami ciri il.ir. .\ '_;r;idii:iteii iiielal nih'r. „' do/en ail isl's |iiiis. Measiiriii!^ I;i|ie. I ■' loot inle. Iieaiii <'oni|iass. 41 'I iVnlii •illl . Si \ I Hi\ i;iiv liciW (■!•_ llMlil.l '(■(• liir uli A «-ki'ii-irniu r;ivc. ..|- liliM'k of |i;i|ifr l«ii' >k(tilniiu I:iiiil>c.'i|M'<, All .•iiii-l'-- loianl. I'li'iity of IS I unliiiMiy \' l^;^ll^|l;^|•(•lll clntli tr;iciiiu |i;i|mt. N,.lr 1mm, ks, inlr.l 7" I I". .M;iim>-rii|it lMi.'k< dl' st r.piio riilcil |i;i|Kr, rip.iUi;i|i Ni/c, ciirli w'uli llti'lCll. :i IcMlIn r .)iiiiliii<4 ; llir jiii^cs ^liuiihl he iiiiliintTci .liHii'iKil liiMik i«r iirdccnliiiLr-^ I'lir llic itiiu'iMr\ , tiiiirs ol' li;ilt<, (•ulii|i;|-.v licMIIII'.;-, ;ilirl'<>iil iilcrl'X ;|1 lulls, Iii'M' ■)'{ s|irinL;s, ilcscripl imi of ciiiinl r\ . uilli liltir illiisi i;ii i\ ,■ ink skctclio, ir |Mi>sililc. ( Hi-cr\ lit ii'li liiHiks, «Miit;iiiiili^ III one cinl llir t lie Mlulitc ;iii'j,'!<', ;i/liiiiit I;, :iiii| iii;i^ic;il rc'^istci's i>\ liiii|ii r;il lire mihI li:ii-ciiiicl ric rfatliiiuts in t-:iiii|i :il I'ci-tain lioiiis. l{iMi'_[li s,iil(|,rii|._r liKiiks, ill w liicli all ul>-i'i\ atioii- can |m' wnrkol •Mil. I'oniis I'lir calciilat \>>w-. I llcill iiiu |'a|ici-. I*('iikiii\t's. Iiiilianililii'i-. I'ciiliuldciN ami iiili-. I"'im' «lra\\iii'4 |irn- an-l li'i|(Iri>^. 11.11.. II., F., ami II. 1*.. iMiniU. Ink |>it\\ ili'i- tliat il" not rri|iiirc viin'iTMi'. I:\ ^aii. I •_' salilc |iainf luii-lii-v Sifin,::, /'<^/'< /■ for lakiii^ ini|>ic>;>ii(ii< i>\ iiist ri|>1 ioii^, i-arviiii^s, I'll'. |;< M (Ks. S|ili(iical ami riaiiical .V^t iun-niiv , li\ ( 'liaii\ cin'l. J \ii|s. Naulif.-il AliiiaiiMf I'lH- turirnt ami InlniT years, ^t lunuly slitclnil ill cloth. ( 'liamlicix" .Mat liciiialica! TaMcs, itrotlnr I'luaritliiii Ituipk. Sliailw cH's ('ai'riiinl;i'. » 42 All Aliiiaiiac. ('('l('>ti;il iii;i|ts, |i;ist('il on ciilicu. Slur Fiiiflt'i-. (Sec ]»;i<,'i' T:!.) Illaiik iiiiips. nil('(l r.tr tlic l.'ititinli' mikI l(»iiLriliiilcs uf tlic pnt- |i(is(m| route. 'I"ln' Ih'sI iii;i|is ol»taiii:il»lc of llir coiiiitiy to Im' visited. Mints to 'riavciifis. Afliiiiralty Manual for tlic use of 'riavi'llers. ( Jallon's ,\rt ol' Ti'avcl. I. Old ami llaiiics' Sliil'ts ami lv\|K'(liciiis of ('aiii|' Life, 'rra\(l, :uiil l<]\i>loratioii. i;.\A.MI NA I loS Ol IN>II!IMI;NI c • Kvcry instniincnt used oiiiilit td Itc (dni|parr(| with a slandanl mihI tcslcil tlioi-oiiif|i!y. iiotli hcrorc coinnu'iiciiiLi' uoi'k ami alter com. pli'tiiii^ tlic journey. Tliis is done at a tiitliiiu' cost at the l\c\v ( Htscrvatory. Instniincnt a! eorrerl i(Mi-> -lioiild lie carcrnlK rccordnl in tlie rei_ristiT. I' \< Ki m;. I It is ditlieiiil |o^i\e 'general nile», Itccaiisc tlic iikmIcs nf Iran s|iort var\ materially in dilVeienl eoiinlries. Im|iii?v slionid lie made liy tlic intending I ra\ellcr at the !{■ lyal < tetiii!a|iliieal Soeiet \ "^ rooms, as to uli;it wonld lie lies! lor liim. Tlic corners ul' all tlic iiislriimcnl eases sjiould lie Inass lioiind ; tlic litliiiLiN >lioMld lu' screwed, and not Liliicd ; and tlu' liovcs slnmld lie lartie emiii'^ii to admit of tile iiisl riimeiils heintr taUen out and re |i laced u it li iicrfccl case. Iiisl riimeiil makers are :i|it to attend overmiicli to conipaet- iiess, making; as much as [lossihh- tio into a Miiall, solid l»o.\, uhicli «':ui easily he |nit on a sludl" ; hut this is not what a t ra\(dlcr wants J>ulk is r.arely a dilliculiy to him, though weiiihl is ; aiidahhoil pieces. The hail's retain the curvature and act ;is .sprinctM. Instruments travel cvccllenliv when packed ii /iitist , liintl>/iil cloths. v.\ III llif |i:ickiii!;; td'slnn's wr r('(|uin' — I. M»'tli:i<'ka^es slioidd evtT exceed T."i His. in uein;lit. I f of a con veiiieiit sli.'ijte. a lioiseor niiile can take a package of this weitj.it on each xide, .and .">'» or tlo lli^. hetweeii tlieiii on the top. IT the tjood> ;ire to lie carried liv porters, no pack.'iire should weiiih more th.in ."iti llis. Heavier pack.'iLCes than the alio\e always rceeixe loiioh treatment ; oliloiej; iioxes travel lle^l. The ai rt i^ht metal, so c.illed '• I ^lirorni '" ca- es, re stro.i^ly recommended, Itiit for :i proloiiLjed iniiinev t hev leipiire i'V woollen cast iirlher -ecnntv ai-ainst to lie protected hv out lireaka"j;e can lie had liy siili ili\ ision. Ai'licles likely to in jury one ;iiiot her should not lie lirouu'hl into close contact. I'"r;iiiile articles should he p.'icked in Miiall separate lio\es or cases, so that should they lie hiokeii, ihev mav not leave a \(iid which will cause .all the contents of their case to juinlile .ilioiii. ('hcinicals and exphisives should he kept separate troiu iithrr things, and het'ore lieini,' packed, impiirv shniild he made as to regulations to which they will have ••; sulunit on ship hoard, ifcc. 'I'o >4uard aij.aiiisl .A////// on ship hoard, in very r.aiiiy countries pas^aue of rivers, tfcc, all pei'ishalile tliintjs should, where piactic- alile, lie enclosed in till and solder<'d, particular care licinn' taken that e\er\tliini; is thorou>_dily dry hel'ore heimr soldered up. The outer Wiiodeli cases should he made oi' iiie hesl dc.ll, closeK lilled, ;;nd varnished or doiihle \arnished, ti-l\ lilted, well made case-., w liich .itVcird ^it at Iroiilile to thieves. (iapim;- p.ack ai^es, with pai'tU exposed conlciits, iii\'|e ridilier\ . Iloxes should he serewed down with /ii-'iss screws, which stand openiiiL; .and re opeiiin'.^ Iieller than I he iron screw >. iron screws should he l,il lowed lielore insertion to ni.ike them unscrew more easily. IT nails are used care shoiihl he taken that tlie\ are iioi driven into the art icies into the lio\ . •">. /:'i "IK 1/111/ "/' >/'(/'•(. -'riiere should he no w.asle space in the packa'.4es. Iity of |.iir|.<.s('s. It ili.i explorer (I. .(■< not liiniM-lf siipcriiiti'iKl tlic packini,' of tlif u^o-mIs. Ik- imist not expect foivsiiilit in tlioe small Inii important parliciil:ii-. •J. 77m i'olnlnip:, n/ fhiljif. As ea<-li packau'e is tinislieij, \\^ eont.'Uts slioiiM lie cMivfnlly catalou'iMMl, an.l the pa.'kaire .listincLl\ numl.ered on several sides, ihc conv«:poii.lini.Miiiinliers l„.in«. cnteivd in the cataloi,rne. I r I, :i- a, I It' tl le LJo'hIs, lie irliciilar-. islicil, itv disliiicLlv iL ciitfri'd