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HOLDEN, M.A., PHOVMBOB IN TBI V. 8, IIAVAIi OBRKBTATOBT. V \eVw ism .nV NEW YORK HENRY HOLT AND COMPANY 1879 ^ ^\ { ' Copyright, 1870, BY HcNRT Holt & Co. PRigg or JoHM A. Grat, Aot., ]8 Jacob Strbbt. KBW YORK. I n PREFACE. The following work is designed principally for the use of those who desire to pursue the study of Astronomy as a branch of liberal education. To facilitate its use by stu- dents of different grades, the subject-matter is divided into two classes, distinguished by the size of the type. The portions in large type form a complete course for the use of those who desire only such a general knowledge of the subject as can be acquired without the application of ad- vanced mathematics. Sometimes, especially in the ear- lier chapters, a knowledge of elementary trigonometry and natural philosophy will be found necessary to the full understanding of this course, but it is believed that it can nearly all be mastered by one having at command only those geometrical ideas which are familiar to most intelli- gent students in our advanced schools. The portions in small type comprise additions for the use of those students who either desire a more detailed and precise knowledge of the subject, or who intend to make astronomy a special study. In this, &3 in the ele- mentary course, the rule has been never to use more ad- vanced mathematical methods than are necessary to the development of the subject, but in some cases a knowl- edge of Analytic Geometry, in others of the Differential Oalculus, and in others of elementary Mechanics, is neces- VI PREFAVB. T Biirily presupposed. The object aimed at has been to lay u broad foundation for furtiier study rather than to at- tempt the detailed presentation of any special branch. As some students, especially in seminaries, may not de- sire so extended a knowledge of the subject as that em- braced in the course in large type, the following hints are added for their benefit : Chapter I., on the relation of the earth to the heavens, Chapter III., on tlie motion of the earth, and the chapter o!i Chronology should, so far as pos- sible, be mastered by all. The remaining parts of the course may be left to the selection of the teacher or student. Most persons will desire to know something of the tele- scope (Chapter II.), of the arrangement of the solar system (Chapter I V. , §§ 1-2, and Part II. , Chapter XL), of eclipses, of the phases of the moon, of the physical constitution of the sun (Part II., Chapter II.), and of the constellations (Part III., Chapter I.). It is to be expected that all will be interested in the subjects of the planets, comets, and meteors, treated in Part II. , the study of which involves no difficulty. An acknowledgment is due to the managers of the Clarendon Press, Oxford, who have allowed the use of a number of electrotypes from Chambers's Descriptive Astronomy. Messrs. Fauth & Co., instrument- makers, of Washington, have also lent electrotypes of instruments, and a few electrotypes have been kindly furnished by the editors of the American Journal of Science and of the Popular Science Monthly. The greater part of the illus- trations have, however, been prepared expressly for the work. u i »Hi.- « »|]i.- ^ > been to lay ' than to at- branch. , may not de- as that em- ing hints are elation of the lotion of the 80 far as pos- } of the conrso r or student. r of the tele- e solar system .), of eclipses, onstitution of constellations d that all will comets, and hich involves lagers of the d the use of a i Descriptive »nt-makerB, of instruments, nished by the n 181 >ON. ng Force — B — Rotation lt» ,he Moon — —Character 168 ravitation — (ravlty with BMion of the 188 iTAlTCT. the Solar 1 818 lOHT. f Light 884 MendarB — 9 Calendar — 10. 845 V0NTENT8. PART II. TIIE SOLAR SYSTEM IN DETAIL I.V aiAPTEU I. PAOB Stbiicturk of the Solak System 267 CHAPTER II. THE HUN. General Sammarf— The PliotoHphere— Sun-Spots and Pacute— The Sun'ii Chromospliere and Corona—Bources of the Sun'a Heat 278 CHAPTER III. THB IKTBRIOn VhKSVn. Motions and Aspects —Aspect and RoUtion of Mercury — The Aspect and supposed Rotation of Venus— Transits of Mercury and Venus — Supposed intramercnrial Planets 810 CHAPTER IV. The Moon 886 CHAPTER V. THE PLANBT MAB8. The Doscriotion of the Planet— Satellites of Mars 884 CHAPTER VI. The Minor Planktb 840 CHAPTER VII. JCFITBR AND HIS BATBI.LITB8. The Planet Jupiter— The Satellites of Jupiter 848 CHAPTER VIII. > BATTRN AND HIB BTBTBIC. Gtoneral Description— The Rings of Saturn— Satellites of Satnm. . 8SS llil 'ife;.. j --: -K-ffm^^p-.sv''.i X VQHTKMti. CHAPTER IX. Tub PI.ANKT Ukaniis— BatuIIitM of Ursniu. UOa CHAPTER X. TiiK Pi.AMvr Nbitonb— Hatellitu of Neptune 800 CHAPTER XI. Tub Phybioal Constitution of tub Plambts 870 CHAPTER XII. MBTEORB. Plionomena and Cauwa of Metoorn — Meteoric Skowera 870 CHAPTER XIII. COMBTB. Aipect of CometB— The Vaporous EnvelopeB— Tlie Physical Con- stitution of Comets — Motion of Comets — Origin of Comets — Remarkable Comets 888 PART III. TIIE UNIVERSE AT LARGE. Introddction *11 CHAPTER I. THB OOnSTBLIiATIOIIB. General Aspect of the Heavens— Magnitude of the BUre— The Constellations and Names of the Stars— Deacriptlon of Con- stellations—Numbering and Cataloguing the Stars 410 CHAPTER n. VARIABT'B AKD TBHPOHABT BTAIW. Stars Regularly Variable— Temporary or New SUrs—Theoiy of Variable Stars **® VONTKNTS. «^ (!HAPTKU III. MIII.TII'I.K MTAIIX. PAOl Character of Doable »nd Multlj.k 8tar»-()rbll« ..f Binary KtarH. . 44M CHAl'TKH IV. MRBUL/K AND CI.IIMTKUC. Dlncovory of Nebulie— (naMlflcatlon of Nebulas and Cluatera— Htar Clu«teni-H|H)ttra of Nebula> and CluHtern-Dlstribuilon of Nebul* and Cluatera on the Surface of tho Celestial _ , „ 457 Sphere CHAl'TEK V. BPKCTRA or FIXKD ilTARB. Charactera of Stellar Spectra— Motion of SUra In the Line of Sight. 4fl8 CHAPTEK VI. MOTIONB AKD DIBTANCKB OF THR BTAR8. Proper Motlona— Proper Motion of the Bun— Dbtances of the Fixed Stan ^'^ CHAPTER VII. CiOIIBTRUCTION 0» THH HBAVRHB *'^ CHAPTER Vlll. COSMOOONT Index «» f "'^jfi^'-Tavi'-'^"^-*'.'''^^'''-^"'^''"'""'^" •' ' '"^~~'-'''''^ I ' ;ieiiiW»;«W«»^«"IMlMratli')M iiiiiiiiliilliiiiJiMii'HliW ASTRONOMY, INTRODUCTION. AsTROKOMT {pKttrfp — a star, and ko'/ios — ^a law) ia the science which has to do with the heavenly bodies, their appearances, their nature, and the laws governing their real and their apparent motions. In approaching the study of th the most ancient of thd sciences depending upon observation, it must be borne in mind that its progress is most intimately connected with that of the race, it having always been the basis of geog- raphy and navigation, and the soul of chronology. Some of the chief advances and discoveries in abstract mathe- matics have been made in its service, and the methods both of observation and analysis once peculiar to its prac- tice now furnish the firm bases upon which rest that great g^up of exact sciences which we call physics. It is more important to the student that he should be- come penetrated with the spirit of the methods of astron- omy than that he should recollect its minntisa, and it is most important that the knowledge which he may gain from this or other books should be referred by him to its true sources. For example, it will often be necepaiy to speak of certain planes or circles, the ecliptic, the equa- tor, the meridian, etc., and of the relation of the appa- rent positions of stars and planets to them ; but his labor will be useless if it has not succeeded in pving him a precise notion of these circles and planes as they exist in ASTRONOMT. the sky, and not merely in the ligures of his text -book. Above all, the study of this science, in which not a single step coald have been taken Avithont careful and painstak- ing observatioB' of the heavens, should lead its student himself to attentively regard the phenomena daily and hourly presented to him by the heavens. Does the sun set daily in the same point of the hori- zon ? Does a change of his own station afFect this and other aspects of the sky 2 At what time does the full moon rise ? Which way are the horns of the young moon pointed ? These and a thousand other questions are already answered by the observant eyes of the an- cients, who discovered not only the existence, but the motions, of the various planets, and gave special names to no lees than fourscore stars. The modem pupil is more richly equipped for observation than the ancient philoso- pher. If one could have put a mere opera-glass in the hands of Hipparohus the world need not have waited two thousand years to know the nature of that early mystery, the Milky Way, nor would it have required a Galilbo to discover the phases of Venus and the spots on the sun. From the earliest times the science has steadily progress- ed by means of faithful observation and soimd reasoning upon the data which observation gives. The advances in our special knowledge of this science have made it con- venient to regard it as divided into certain portions, whioh it is often convenient to consider separately, although the boundaries cannot be precisely fixed. SphArioal and Praotiottl Astronomy. — ^First in logical order we have the instruments and methods by which the positions of the heavenly bodies are determined from obser- vation, and by which geographical positions are also fixed. The branch whidi treats of these is called spherical and practical astronomy. Sf^erical astronomy provides the mathematioal theory, mA practical astronomy (whioh is almost as mudi an art as a soienoe) treats of the applioap tion of this theory. 'J orHm ajt. i' j * ivJ^ t j|B Hj yy ii DIVIBIONS OF TBE SUBJECT. i 8 text book, not a single id painstak- its Btudent a daily and of the hori- fect this and ioes the full >f the young ler questions I of the an- ice, but the icial names to pupil is more sient philoso- i-glass in the ve waited two sarly mystery, a Gauleo to on the sun. idily progress- imd reasoning le advances in made it con- lortions, which , although the irst in logical ) by which the ed from obser- are ako fixed. spherical and r providas the >my (which i» of the appiioar Theorttioal Astronomy deals with the laws of motion of the celestial bodies as determined by repeated observiitiong of their positions, and by the laws according Jo which they ought to move under the influence of their'Inutual gravi- tation. The purely mathematical part of the science, by which the luws of the celestial motions are deduced from the theory of gravitation alone, is also called Celestial Jfechaniet, a term first applied by La Place in the title of his great work JHecanique CeleHe. OownloSil FhysiaB.— A third branch which has received its greatest developments in quite recent times may be called CosmiocH Physics. Physical astronomy might be a better appellation, were it not sometimes applied to celestial mechanics. This brandi treats of the physiqal constitution and aspects of the heavenly bodies as investi- gated with the telescope, the spectroscope, etc. We thus have three great branches which run into each other by insensible gradations, but under which a large part of the astronomical research of the present day may be included. In a work like the present, however, it will not be advisable to follow strictly this order of sub- jects ; wc shall rather strive to present the whole subject in the order in which it can best be undentood. This order will be somewhat like that in which the knowl- edge has been actually acquired by the astroncnners of different ages. Owing to the frequency with which we hAve to use terms expressing angular oieasnra, or ref anring to droles on a sphere, it may be admissible, at the outset, to give an idea of these terms, and to recapitulate some prop* erties of the sphere. Aniwler IbMaiea. — ^The unit of angular measure most used for oonsidaiible ang^ is the degree, 840 of which extend round the eiiele. The reader knows that it is 90" tnuk the horiaon to the aodth, and that two objeele 180? apart are diametrioally opposite. An idea of distanoes of T 4 ASTttOItOMY. a few degrees may l)e obtained by looking at the two Btars which fonn the pointers in the constellation Urm Major (the Dipper), soon to be described. These stars are 5° apart. The angular diameters of the sun and moon are each a little more than half a degree, or 30'. An object subtending an angle of only one minute ap- pears as a point rather than a disk, but is still plainly vis- ible to the ordinary eye. Helmholtz finds that if two minute points are nearer together than about 1' 12', no eye can any longer distinguish them as two. If the ob- jects are not plainly visible— if they are small stars, for instance, they may have to be separated 3', 5', or even 10', to be seen as separate objects. Near the star a Lyra are a pair of stars 3^' apart, which can be separated only by very good eyes. If the object bo nf>t a point, but a long line, it may be seen by a gootl eye when its breadth snbtends an angle of only a fraction of a minute ; the limit probably ranges from 10' to 15'. If the object lie much brighter than the background on which it is seen, there is no limit below which it is neoes- sarily invisible. Its visibility then depends solely on the qnantity of light which it sends to the eye. It is not likely that the brightest stars subtend an an^eofT^^ of a second. So long as the angle subtended by an object is nmU, we may regard it as varying directly as the linear magnitude of the body, and inversely as its distance from the ob- server. A line seen perpendicularly snbtends an wo^ of 1° when it is a little less than 60 times its length dis- tant from the observer (more exactly when it i» 67-8 lengths distant) ; an angle of 1' when it is 8488 lengths distant, and of 1' when it is 206866 lengths distsnt. These numbers are obtained by dividing the number of degrees, minutes, and seoonds, respectively, in the cir- cnmferenoe, by 2 x 814169966, the mtio of the droom^ ference of a circle to tlie radius. »ii*li«i>to- sphere. SYMBOLS AND ABBREVIATIONS. w «IOH8 or THB PLAMRS, STC. I or i The Boa. The Moon. MefCBiy* Veoot. TheEuth. i Man. 21 Jniriter. « Sfttani. S Uruiu. ^ Neptane. The asteraidfl uedltttagnMied by a drele iBdodag n number. whl«h number indicates the older of dlMsovenr, or by thdr BMneB, or br both. fu^iHeeate ' UONB or THB womjko. Spring eigne. (a T Ariee. V Tanme. n Gemini. Sammer { t ® Ctacer. it. Slhw. Virgo. Antamn eigne. Winter rigns. CIO. V3 (13. X ^ Libri 111 Sflorpina t Sagittarina. V3 Caprieorana. Aqoariiuk X Piaeeiw AinDon. 6 Oonjanetioo, or having the ewae loafrltnde or right aaoearion. a Qoadratore, or diflbring fK>° in " - 9 t^poaitioB, or difibring 180* la ■• '• «• inmber,whldi M,or by both, llM. iirt>ina gitUriaa. prieoraua. lOAriiuk AaTBONOMWAL SYMBOLS. Q Ascending node. n Descending node. N. North. 8. South. E. Bust. W. WlBt. " Degree*. ' Minutes of uc ' Seconds of »rc. *> Hours. » Minutes of time. . Seconds of time. L. Mesn longitude of » body g, Mesn nnoinsly. f. True snomnly. R^. or rt. Bight ascension. Dec. or 6. Decllnstion. r True senlth distance, r Apparent «nith distance. ^Dirtance from the earth. J Heliocentric longitude. 6, Heliocentric latitude. X Oeocisffltrlclongltude. a t4eooentric latitude. 2'o,^ Longitude of ascending Undtationol orbit to the eclii.. tie. a Mean anomaly. «>. /[True anomaly. .„ » unit U. An8»»" ^'•»""" *""" ^ i: Mean sidereal motion in * unit «. ^g ^ ^^ ^^^ of time. L Distance from node, or argu r, Radius vector. | ' ^ent for latitude. ^, Angle of e««'»t'*7y„„„ ,,i«, ! «. Altitude. ;:LoSgltude of perihelion C^"" | ^^i^.th. '*^^"^' p. Earth'sBquatorlal radius. familiar with It In reading the pans occur : lAsttars. Rmms. L.tt«n. VuMM. J, ^ Ku A a Alpha Y yC Oamma E ( Bpsllon ZCi »»• H« »»• e « Th«ta li I«» K « K»PP», j^ X Lambda Mm Bfu THE METRIC SYSTEM, Thb metrle .y-tem of weight, and measure, being en>PW«i «« J. volume, the following relations between the unit, of thl. .y.tem mcl ;ied aid th«« of our oidiowy ou. will b. found conrenlent for reference : MRABURRS or LBMOTH. 1 kilometre = 1000 metres ^ 02187 mile. 1 metre = the unit = 89 87 inches 1 millimetre = TiAnr of a metre = 008987 Inch. HKASUREB OF WEIGHT. 1 mlllier or tonneau = 1.000.000 gramme. = 8304-6 pounds. !"•''-••"'-• -ther*""""': i^rg^nr 1 K^Lnme = W»» of a gramme = 001648 grain. The fbllowing rough approximation, may be memoriied : The kUometfe i. a little more than A of a mUe. but leM than | of imile. The mile i. lV\r Ulometies. The kilogramme I. H pound.. The pound i. lew than half a kilogramme. wing employed \n mlU of this Kjtttm and oonTenlent for 3187 mile. 7 incbeii. 8037 incli. 1204-6 pounds. 2 3046 pounds. 15-482 grains. 0- 01548 gntn. aemoriied : He, but Itm tliu | of CHAPTER I. ,„K HK..T.O. -J„B^ -KT„ TO THB $i 1. THB EABTH. U considering the n^ladon of «;« -f^J^.tolTm^^ we iieceBearily l>egin r:f\'^l,t^\ChL:yXhJJ, n'3eS*:f':;St;wn fact, will show th^ this eih'l^u which we live is, at l east approxiuiately, > Klobe whose dimensions are gigantic ^.hM^^^ when compared to onr ordinary aiid daily ide«. of si«». If «^P«^^ Mveral ways known to he nearly | that of a sphere. , I It haa been repeatedly circum- navigated in various directions. II Portions of its swrface, via- ble from elevated positions in the midst of extensive phihis or at sea, Zoeu to be hounded by circles. T^Jl^^^ IWppemnce at all points of the j^TtS^^^:. sorfw* Ta body i. a geometnca' f^,,^^^tSSlXi&^ attribute of a globufar form only. _ . m Fortlier than this we know thrt «*»tui ""»" geodetie surveys have agreed with this general wu g (m aggg B j^y^:F'^''^g^^ ^WW^^' '*^ *'^^ ^N*"* 10 ASTHONOMT. More procisct reasonH will li« apparent later, but those will be Buttioient to base our general considerations npon. Of the aize of the earth we may form a rongh idea by the time re(|uired to travel completely around it, which is now about three months. We find next that this globe Ih completely isolated in space. It neither rests on any thing else, nor is it in contact with any surrounding body. The most obvious proof of this which presents itself is, that mankind have visited nearly every part of its surface without finding any such connection, and that the heavenly bodies seem to perform complete circuits around it and under it with- out meeting with any obstacles. The sun which rose to- day is the same body as the setting sun of yestetrlay, but it has been seen to move (apparently) about the earth from east to west during the day, and it regulariy reap- pears each morning. Moreover, if attentively watched, it will be found to rise and set at different parts of the horizon of any place at different times of the year, which negatives the ancient lielief that its nocturnal joiirney was made through a huge subterranean tunnel. % 1. THS DZUBITAL KOnON AlTD THB CODUnTIAIi PaisiDg now from the earth to the heavens, and vMwing the son by day, or the stars by n^t, the first ]^ienomeiKm whidi fMam our attention is that of the divmal motkm. Wemwt here cantion the reader to carefnllj distin- goiah between apparent and reed motions. For examine, when the phenomena of the dinnuU motion are aet forth as real visible motions, he must be prepared to ^um rab- seqnentiy that this appearaneo^ which is obvioM to all, is yet a oonseqnenoe of a real motion only to be detected by reason. We shall first describe the dinmal motkm as it appewn, and show that all the appearaaoes to » qieotator at any one place may be re proao n lBd by 8a|i|Kiiii% the earth to remain fixed in spaoe, and the wM* otnoave of I 1 1 t 1 r t ii li t si n I si t^ tl HT t( s« it ii. tl tl it t))om] will I npon. Of idoa by the t, which is ely isolftted nor is it in io8t obviooH aikind have ont finding IxMlies seem idor it with- ich rose to- fiterday, but it the earth ulariy reap- )ly watched, )art8 of the year, which joiirney was andTWwing dienomeiM» tlmoikm. folly diatiii- Bxample, re aet forth lewninb- ns toall,i8 d0t6otod by motion as it Aspeotator iipoiog the oeoeaTeof TI/K DIURNAL MOTION. 11 the noavoiiH to turn abont it, and finally it will be shown that we have reason to Iwlieve that tlio solid uarth itself is in constant rotation while the heavens runmin immov- ablo, pruHunting different portions in tnm to the obsorvor. The motion in (piestSon is most obvious in the case of the sun, which appears to make a daily circuit in the heavens, rising in the vast, passing over toward the south, setting in the west, and inovhig around under the earth until it reaches the eastern horizon again. Observations of the stars made through any one evening show that they also appear to perform a similar circuit. Wliatevor stars we see near the eastern horizon will be found constantly rising higher, and moving toward the south, while those in the west will be constantly setting. If we watch a star which is rising at the same point of the horizon where the sun rises, we shall find it to pursue nearly the same ooune in the heavens through the night that the sun follows through the day. Continued obaervations will show, however, that there are some stars which do not let at all — namely, those in the north. Instead of rising and letting, they appear to perform a daily revolution around a point in the heavens which in onr latitudes is neariy half way between the senith and die northern horizon. Thla oen- tral point i» called the pole of the heavens. Near it is situated Polarity or the pole star. It may be recog- nized by the Poinier»t two atars in the oonstelktion Ur»a Mt^cTt famiHarly known aa TKe Dipper. These stars are ahown in Fig. 8. If we wateh any star be- tween the pole and the north horizon, we shall find that instead of moving from east to west, aa the stars generally appear to move, it really appears to move toward tiie east ; but instead of oontinning its motion and setting in the east, we shall find that it gradnally dUres its course upward. If we could follow it for twenty-four hours we should see it move upwards in the north-east, and then pen over toward the west between the zenith ai^ the pole, then sink down in the north-west ; and on the ,TVi'/-*wrpjv..w*'j 3^' II Asrnnmmrr. following night cnrvo itH couno onco nu.^o toward tho east. The arc which it appears to deflcrilH) in a perfect eircle, having tho pole in its centre. The farther ffom the pole we go, the larger the circle which each star aeeina to describe ; and when we get to a distance equal to that between the pole and tho horizon, each star in its rent passage below the pole just grazus 41h) horizon. 8.— rm APPABBMT DiniMAI. MonoM. As a result of this apparent motion, each individual constellation changes its configuration with respeot to the horizon, that part which is highest when the oonstellatitm is above the pole being lowest when below it. This is shown in Figure 4, which represents a supposed omMtel- lation at five different times oi the night. Going farther still from the pole, tiie stars will dip be* 11- THK DIURNAL MOTION. M toward the I a porfeot rther fi-om 1 star seeina [ual to that n its izon. indiTidnal ipeoktothe »tMteIlati(m it. This is led omistel- Hrill dip be> ,„, the .,«ri.on anring a portion o^ ^ " C:^! t »t ,,„„.|,y „crc«>,ng ^''-«Jl,v„«,d on. hJ( Wow r!l""'«» J[t;^S iirwlLn, «.d tUerob^ longer Toni;::.! .Lt. i'* u. *» ^ »« -.'.. -^ - Bets a little to the west of it. % '- rl 1- NORTH Fio. 4. «nm, ♦«','^„'t^^ tSrfLTtoS but they J« p,^t MvotaUon m 1^ "^^ jirtiac from «ch Swerve iiiidi»nged Aeir """"^j, ^„^ «, wm- Uher, tKth the «"»!*■<»> «« *7' '£?S are »WHe »chMg. •ir.'JrJ the Se"'' the ide. thrt thM. UrSttjrSrjTc^-neSl^^"-'^'-- ^ «rW".-flt. 14 ASTBONOMY. apparent explanation, both of this and of the phenomena of the diurnal motion, was offered by the conception of the celestial sphere. The salient phenomena of the heavens, from whatever point of the earth's surface they might be viewed, were represented by supposing that the globe of the earth was situated centrally within an im- mensely larger hollow sphere of the heavens. The vis- ible portion, or upper half of this hollow sphere, as seen from any point, constituted the celestial vault, and the whole sphere, with the stars which studded it, was called the firmament. The stars were set in its interior surface, or the firmament might be supposed to be of a perfectly transparent crystal, and the stars might be situated in any portion of its thickness. About one half of the sphere could be seen from any point of the earth's surface, the view of the other half being necessarily evt off by the earth itself. This sphere was conceived to make a diurnal revolution around an axis, necessarily a purely mathemat- ical line, passing centrally through it and through the earth. The ends of this axis were the poles. The situa- tion of the north end, or north pole, was visible in north- em latitudes, while the south pole was invisible, being below the horizon. A navigator sailing south would so change his horizon, owing to the sphericity of the earth, that the location of the north pole would sink out of sight, while that of the south pole would come into view. It was clearly seen, even by the r' Jents, that the diur- nal motion could be as well represented by supposing the celestial sphere to be at rest, and the earth to ravolve around this axis, as by supposing the sphere to revolve. This doctrine of the earth's rotation was maintained by several of the ancient astronomers, notably by Abistab- oHus and Timoohabis. The opposite view, however, was maintained by Ptolbmt, who could not con<»ive that the earth could be endowed with such a rapid rotation with- out disturbing the motion of bodies at its surface^ We now know that Ptolbict was wrong, and his opponents < THE CELESTIAL SPHERB. 15 phenomena mception of lena of the surface they ing that the thin an ira- B. The vis- lere, as seen alt, and the , was called rior surface, a perfectly lated in any the sphere surface, the off by the ke a diurnal ' mathemat- hrongh the The sitna- le in north- dble, being h would so the earth, tut of sight, idew. It the diur- pposing the to revolve to revolve, intoined by y Abistab* >wev^r, was !ve that the sation with- pface* We opponents < right. Still, so far as the apparent dinmal motion is con- oerncd, it is indifferent whether we conceive the earth or the heavens to be in motion. Sometimes the one concep- tion, and sometimes the other, will make the phenomenA the more clear. As a matter of fact, astronomers speak of the sun rising and setting, just as others do, although it is in reality the earth which turns. This is a form of language which, being designed only to represent the ap- pearances, need not lead us into error. .^ ^ , , The celestial sphere which we have described has long ceased to figure in wrtronomy as a reaUty. We now know that the celestial spaces are pmcticaUy perfectly void ; that some of the heavenly bodies, which appear to l^ on the surface of the oelertial sphere at equM dwtaneesfrom the earth as a centre, are thousands, or even milhons of times farther from the earth than others ; that there is no material oonneetion betwefen them, and that the celestial sphere itself ii» only a result of optical pewotive. But the huiguage and the conception are stiU ret&i»4^ 1»cause they afford the most dear and definite method of repre- sentimr the directioBs of the heavenly bodies fiom the obs«rw, wherever he may be situated. In this respect it sema the same pwpose that the geometnc sphere does in apherical trigono^netry. The stodeiit of this sci- ence knows that there is reaUyno need of supposing a sphere or a spherical trianj^e, because every spherical are is only the representative of an angle between two lines which emanate from the centre, one to each end of the are, whae the angles of the triangle are only those of the philies containing the three lines which are drawn to Lh angle from the centre. Spherical trigonometry m, therefore, in reaMty, only the trigonometry oi s^id angles ; and the purpose of the sphere is only to afford a convenient method of conceiving of such angles. In the same way, althou^ the celestial sphere has no real ex- istence, yet by eonoeiving of it a. a redity, and suppojng eertain Unes of reference drawn upon it, we are enabled to JWga»JM»»'"''^.' fl 16 # A8TR0N0MT. form an idea of tlie relative directions of the heavenly bodies. We may conceive of it in two ways : firetly, as having an infinite radius, in which case the centre of the earth, or any point of its surface, may equally be supposed to be in the centre of the celestial sphere ; or, secondly we may suppose it to be finite, the observer carrying the wn- Fio, 5 — aTARs nam oir thk CBuniAi. vbmhm. tre with him wherever he goes. The iirat assumption wiU probably l)e the one which it is best to adopt. The object attained by each mode of representation is that of having the observer always in the centre of the supposed sphere. J*^. 5 will give the reader an idea of its apphcatjon. He w supposed to be stationed in the centre, 0, ancl to have Mwmd him the bodies py».,<, etc The sphere itself temg supposed at an immense distance, outside of all these bodies, we may suppose lines to be drown fiom each of them directly away from the centre until they waoh the sphere. The points PQBST, etc., in whieh =s& ■ i-.i j * w»^'ifl*"r^^ wh^7h. ''*^''' wLthe oSTrverTon «ie eartVs equator. ^U «ee hi. zenith »«« -•y^^^^^h'^m d-criSi a the earih revolve, on it. «-» .^» TT every point of great circle around tlu. celestial »P^«^ •^"Y^^ „ ^e ^ch wiU be eqniOly dirtant 'TV^'fTm „y ^t of the ewrth's equator ^e'^^Uj;*? ^^ ^ ' them, conceive ihai iU !»««* «f ^ ««* jj S^ irbadf to the called the «*•«« '''^'.^^TZIn^t^iT^ above a to the terrestrial eqnator is t\«* ^J^'T^;"^; "tors lie corresponduM^ point ofihela^The^two^^ ^^ ^^^ f^^'^^^tA^Sandtorr^ ^ belong, to both *~ ?T^^ ^^ from the eqnator Now .appow that the ^^^^T^^Thavinir changed by to 460 of north latitude. ^^^^""^^VX^SSon, 45-, the noiih polej^ now^be 46 *ove ^^ ^^ imMm^L'^-.^W. 80 ASTRONOMY. sphere which Mrill be overywliere 45" distant from the celestial equator. This cirde will thus correspond to the parallel of 45° north upon the earth. If he goes to lati- tude 60° north, he will see the pole at an elevation of 60°, and his zenith will in the same way describe a circle which will be everywhere 60° from the celestial equator, and 80° from the pole. If he passes to the polo, the latter will be directly over his head, and his zenith will not move at FlO. 7.— TBBBnTRUIi AXD (ntUWIAI. all. The celestial pole is simply the point inwiiioh the earth's axis of rotation, if continued out in a straight line of infinite length, would meet the celestial sphere. We thus have a series of circles on the celestial spliere ooire- sponding to the parallels of latitude upon the eartk. Unfortunately the celestial element owresponding to latitude on the earth is not called by that name, but by that of dedmaUon. The d«dinaUon of a tkax is ^ distance north or south from the edestial equator, pre- ■■r »nt from the 'espond to the goes to kti- ivation of 60", a circle which aator, and 80" he latter will 1 not move at in whioh the a straight line sphere. We splbere oorre- n the earth. 9ep but appa. ) pole as the ana oontinaed hut really in of these rae- fphiin the in- fir the Stan, *« appear to ade between between the >n8 what ow. nee between however, in ^ MMfWMn, , n his mind ude on the fnsidored as but a [pliere, tho rational one and the same of the ojtls of the ^ing that one which New York, in tho the Stan in their \dedinatumy KN ; \ the centre of the or the tqmiMalMly andlel of dedination ■phere. nut which appeMr in te earth. Th« itan innud patiha bJMdted the horiion as helaw '^ VIHCim OF TUi 1 1 Kit K. • it ; tho8uter than 90° will bu a Hliurtur tiniu nbuv«' tlit rizon ; those whoso polar-distance* aru lues than i** li longer time. Tho circle iViT drawn aronnd tho pole Pm a centre fo as to graze the horizon is called the circle iif perpetual apparition^ liecauso stars situatKl within it never set. The corresponding circle S U round tho south polo is called tho circle qfperpetvMl disappearance, because stars within it never rise above our horizon. The groat circle passing throu«di the zenith and the pole is the celettial meridian, NPZS. The meridian intersects the Korixon in the meridian line, and the points N and 8 are the north and touthpointg. the prime vertical, £ZW,i» perpendicular to the meri- dian line and to the horizon : its extremities in the hori- zon are the ead and toettpointt. The meridian plane is perpendienlar to the equator and to the horizon, and therefore to their inteiMction. Hence this intersection it the eatt and VMti line, which ia thus determined by the inteneotion of the ]danei of the equator and of the hn-imm. The edUtudt of a htwrenly body ia ita apparent distance above the horison, expreaaed in degreea, minutes, and seconds of aro. hk the cenith the altitude is 90**, which is the greatest poarible attitude. If ^ be any hetTenly body, tho angle ZPA which the oirde P A drawn from the pole to the body makes with the meridian ia ealled the hour angle of the body. The hour angle ia the angle through which the earth has ro- tated on ita axis aince the body was on the meridian. It is ao called becauae it measurea the time which has elapaed linoe the paange of the body over the meri- dian. Thai diameter of the earth which ia coincident with the Qonataat diraotion of the axis of the oekacial aphere is its MM, and interaeots the earth in ita north and aouM poUz, - proach tho north polo, but as the zenith is not a visible point, the motion will be naturally attributed to the pole, which will seem to approach the point overhead. The new apparent position of the pole will change the aspect of the observer's sky, as the higher the pole appears above the horizon the greater the circle of perpetual apparition, and tlterefore the gi-eater the number of stars, wliich never set. If the observer is at the north pole his zenith and the pole itself will coincide : half of the stars only will be vis- ible, and these will never rise or set, but appear to nwve around in circles parallel to the horizon. . The horijcon and equator will coincide. The meridian will be indetw- minate since Z and P coincide ; there will be no eMt and west line, and no direction but south. The sphere in this case is called a paraUd tphere. WMSLMKT UlTI- 1 of an observer V an the observer iuo of the earth, lia zenith '^11 )i|)- is not a visible voted to the pole, overhead. The change the aspect ole appears above )etnal apparition, of stars, wliich is zenith and the rs only will be vis- it appear to move aa. . The horicon an will be indetw- will b« DO eaat and The sphere in this ' .' ' ■ S 8'":V DIUHNAL MOTION IN DIFFKHKNT LATITUDISS. 97 If itiHtuud of tnivt'Uiiig to the nortli the oltnerver shuiild go toward tiie (Hiuatoi*, the nortli pole woiUd seem to ap- proach iiiH horizon. Vt'hon he reached the (Hjuator Itoth poles would be in the horizon, one north and the other Honth. All the Btiirs in buccetwion would then be viHible, and each would bo an equal time above and below the horizon. Fm. 11 The sphere in this case is called a righi (^here, because the diurnal motion is at right angles to the horizon. If now the observer travels southward from the equator, the south pole will become elevated above his horizon, and in the southern hemisphere appearances will be reproduced whidi we have idready described for the northern, except that the direction of the motion will, in one respect, be di£Ferent. The heavenly bodies will still rise in tie east and set in the west, but those near the equator will pass north of the zenith instead of south of it, as in our lati- tudes. The sun, instead of moving from left to right, tliera moves from right to left. The bounding line be- tween the two directions of motion is the equator, where the snn culminates north of the zenith from Haroh till September, and south of it from September till March. If the observer travels west or east of hb first sta- tic, his lenith will still remain at the same angular 28 ASTRONOMY. distance from the north pole as before, and as the phe- nomena caused by the earth's diurnal motion at any place depend only upon the altitude of the elevated pole at that place, these will not be changed except as to the times of their occurrence. A star which appears to pass through the zenith of his first station will also appear to pass through the zenith of the second (since each star re- mains at a constant angular distance from the pole), but later in time, since it has to pass through the zenith of every place between the two stations. The horizons of the two stations will intercept difiEerent portions of the celestial sphere at any one instant, but the earth's rotation will present the same portions successively, and in the same order, at both. § 6. BEI.ATI01T OF TIME TO THB 8FHEBB. As in daily life we measure time aj the revolution of the hands of a clock, so, in astronomy, we measure it by the rotation of the earth, or the apparent revolution of thf celestial sphere. Since the sphere seems to perform one revolution, or 360° in 24 hours, it follows that it moves through 16" in one hour, 1° in 4 minutes, 16' in one minute of time, and 16* in one second of time. The hour angle of a heavenly body counted toward the west (see definition, p. 26) being the angle tlirough which the sphere has revolved since the passage of the body over the meridian, it follows that the time whidi has elapsed eince that passage may be fonnd by dividing the hour angle, expressed in degrees, minutes, and seconds of arc, by 15, when the result will be the required interv^ ex- pressed in hours, minutes, and seconds of timo. If we know the time at which the body passed the meridian, and add this interval to it, we sludl have the time corre- sponding to the hoar angle. If we call it noon when the sun passes the meridian, the hoar angle of the son at any moment, divided by 16, gives the time since noon. Mepear8 to pass ilso appear to each star re- the pole), but the zenith of e horizons of )rtion8 of the arth's rotation y, and in the 8FHEBE. I revolution of measnre it by revolution of ns to perform ollows that it ninutes, 15' in >f time. ted toward the tlirough whidi the body over 'Ja has elapsed ling the hour leconds of arc, id interval ez- timo. If we the meridian, he time cone- it noon when ^le of the sun me since noon, lasnred by the «un, after allowing for certain inequalities hereafter de- "1£re, however, an important remark is to be made^ Really ihe earth does not revolve on its axis m 24 of he ^ZnZ in ordinary life, but in about 4 minutes less than ^hirclre exactly in 23 hours 56 minutes 4.09 seconds ) If wei^te the exact time at which a star crosses the men- i irorri-or setB, ordisappearsbehmd achunney or o^^^^ terr^trial object on one night, we shall find it to do tue rXTnaS minutes 56 seconds earlier on the night follow- thet^^ween two tr«»i.. of the «.n o^ «» »- V. I. * K„ ♦!,«♦ between two transits of tne same siar. rfter d.«ned), mi » .bout 8 'r""f',"XdMded into r^r:-:s*^^-^brcwideai.to 24 tuureat nourvj ««* „.a«tiv like the common JlTrate- that is, it gains about one second m sixminutes, 30 ASTRONOMT. ten seconds in an hour, 3 minutes 56 seconds in a day, two hours in a month, and 24 hours, or one day, in a year. The hours of the sidereal day are counted forward from to 24, instead of being divided into two groups of 12 each, as in our civil reckoning of time. The face of the sidereal clock is divided into 24 hours, and the hour hand makes one revolution in this period instead of two. The minutes and seconds are each counted forward from to 60, as in the common dock. Tho hands are set so as to mark O*" 0" 0» at the moment when the vernal equinox passes the meridian of the observer. Thus, the sidereal time at any moment is simply the interval in hours, min- utes, and seconds which has elapsed since the vernal equi- nox was on the meridian. By multiplying this time by 16, we have the number of degrees, minutes, and seconds through which the earth has turned since the transit of the vernal equinox. The sidereal time of onr common noon is given in the astronomical ephemeris for every day of the year. It can be found within ton or twelve minutes at any time by re- membenng that on March 22d it is sidereal hours about noon, on April 22d it is about 2 honro sidereal time at noon, and so on through the year. Thus, by adding two hours for each month, and 4 minutes for each day after the 22d day last preceding, we have the sidereal time at the noon we require. Adding to it the number of hours since noon, and one minute more for ever fourth of a day on account of the constant gain of the clock, we have the sidereal time at any moment. Eeam/ple. — Find the sidereal time on July 4th, 1881, at 4 o'clock A.1I. We have : i h ■ June 22d, 3 months after March 22d ; tobe X S, 6 July 3d, 12 days after June 22d ; x 4, 48 4 A.M., 16 hours after noon, nearly | of a day, 16 3 This result is within a minute of the truth. 22 51 8IDSBBAL TIME. 81 ids in a day, iay, in a year, orward from pe of 12 each, )f the sidereal e hour hand of two. The ard from to e set so as to emal equinox }, the sidereal n hours, min- e vernal equi- this time by }, and seconds the transit of 1 given in the year. It can y time by re- ) hours about ereai time at J adding two ush day after iereal time at iberof hours nrth of a day we have the 4th, 1881, at h ■ X S, 6 48 r, 16 8 22 61 Th« reader now understands that a sidereal dock is one the Bun, but by ttat of 'f f^. J^",,^ , ki„„ the '""TroX':^'*«WvX ^t'nSL We h.ve poBtiont of the rt«re ^«^ J ;„„ „i ,ho rtars now to .how how he fin^ the ng^ ^^ ^^ ___^_.. S'jr^nl^e.t.^f *i- ^-i.'-- «^-^ for the chapter on »»^'"«™- " . j„^ j, a^ed in an a ™aU ttleaeop; ""^J* «^ ^ !» «xed, the tele- power of the tele«.pe. ^* ".Srir^acay on the »r'rs;te'rr^"^''--a^. Suppose now in ^^ ^^ ^^^^ moment mmBm '''''^X!^tfi^t^r^^^»^^<>- of «.y rtar or again. Then, *p^'***^™ *, y^enit ig about to reach other heavenly ^y»^l^*^f,!i^t instrument at the the meridian ; then directo the *«,""* ^^"^^ time, point where it is about to cross, and notes ^ eMCt^ Shouts, minutes, and •^"^•;;r^J*'^^ti7yi^ "^^ tt te^'haft :4m ^.^n of^'^r^ de- time by 16, he has tne ngni 'T; , ^^ j^ the trouble SS.t^^oH'^n.w^lXtoex^i^. |ij,uj; i i.,i 'I I i i.|ii| i 82 ASTRONOMY. riglit ascensions of tlie heavenly bodies, not in degrees, but in time. The circle is divided into 24 houre, like the day, and these hours are divided into minutes and seconds in the usual way. Then the right ascension of a star is the same as the sidereal time at which it passes tlie meridian. The relation of arc to time, as angular measores, can be readily remembered by noting that a minute or a second of time is fifteen times as great as the corresponding de- nomination in arc, while the hour is 15 times the degree. The minute and second of time are denoted by the initial letter of their names. So we have : 1" =16" 1"=16' 1*=15' 1"'=4» l'=4» 1"=0'.0666. Belation of Time and Longltade.— Considering our civil time as depending on the sun, it will be seen that it is noon at any and every place on the earth when the son crosses the meridian of that place, or, to speak with more precision, when the meridian of the places passeB under the sun. In the lapse of 24 hours, the rotation of the earth on its axis brings all its meridians under the sun in succassion, or, which is the same thing, the sun appears to pass in succession all the meridians of the earth. Henoe, noon continually travels westward at the rate of 15* in an hour, making the circuit of the earth in 24 houw. The difference between the time of day, or local time as it is called, at any two places, will be in proportion to the diflbr- ence of longitude, amounting to one hour for eveiy 16 degrees of longitude, four minutes for every degree, and so on. Vice versa, if at the same real moment of time we can determine the local times at two different places, the difference of these times, multiplied by 15, will give the difference of longitude. in degrees, hours, like minutes and ascension of ich it passes 'nres, can be or a second iponding de- the degree. >y the initial =4"> '=4» ring our civil en that it is hen the sun k with more PMses under ttion of the f the sun in n appears to 'h. Hence, GHANOB OF DA Y. 33 I)fl6« in an lours. The t'm0 as it is » the diflbr. w eveiy 15 Iflgrae, and nt of time «nt phu»s, ', will give Tlie longitudes of places are determined astronomically on this principle. Astronomers are, however, in the habit of expressing the longitude of places on the earth like the right ascensions of the heavenly bodies, not in degrees, but in hours. For instance, instead of saying that Washington is 77" 3' west of Greenwich, we com- monly say that it is 5 hours 8 minutes 12 seconds west, meaning that when it is noon at Washington it is 5 hours 8 minutes 12 seconds after noon at Greenwich. This course is adopted to prevent the trouble and confusion which might arise from constantly having to change hours into degrees, and the reverse. A question frequently asked in this connection is. Where does the day change ? It is, we will suppose, Sun- day noon at Washington. That noon travels all the way round the earth, and when it gets back to Washington again it is Monday. Where or when did it change from Sunday to Monday ? We answer, wherever people choose to make the change, l^avigators make the change occur in longitude 180° from Greenwich. As this meri- dian lies in the Pacific Ocean, and scarcely meets any land through its course, it is very convenient for this purpose. If its use were universal, the day in question would be Sunday to all the inhabitants east of this line, and Mon- day to every one west of it. But in practice there have been some deviations. As a general rule, on those islands of the Pacific which are settled by men travelling east, the day would at first be called Monday, even tiiough they might cross the meridian of 180**. Indeed the Rus- sian settlers carried their count into Alaska, so that when our people took possession of that territory they found that the inhabitants called the day Monday, when they themselves called it Sunday. These deviations have, how- ever, almost entirely disappeared, and with few exceptions the day is changed by common consent in longitude ) ' '° from Greenwich. 84 A8TR0N0MT. g e. DETEBMnrATIOirS of TEBSB8TBIAL LONOI- TUDES. We have remarked that, owing to the rotation of the earth, there is no such fixed correspondence between meridians on the earth and aniong the stars as there is between latitude on the earth and declination in the heavens. The observer can always determine his latitude by finding the declination of his zenith, but he cannot find his longitude from the right ascension of his zenith with the same facility, be- cause that right ascension is constantly changing. To deter- mine the longitude of a place, the element of time as mea- sured by the diurnal motion of the earth necessarily comes in. Let us once more consider the plane of the meridian of a place extended out to the celestial sphere so as to mark out on the latter the celestial meridian of the place. Consider two such places, Washington and San Francisco for example ; then there will be two such celestial meri- dians cutting the celestial sphere so as to make an angle of about forty-five degrees with each other in this case. Let the observer imagine himself at San Francisco. Then he may conceive the meridian of Washington to be visible on the celestial sphere, and to extend from the pole over toward his south-east horizon so as to pass at a distance of about forty-five degrees east of his own meridian. It wonld appear to him to be at rest, although really both his own meridian and that of Washington are moving in consequence of the earth's rotation. Apparently the rtan in their course will first pass the meridian of Washington, and about three hours later will pass his own meridian. Now it is evident that if he can determine the interval which the star requires to pass from the meridiftn of Wash- ington to that of his own place, he will at once have the difference of longitude of the two places by simply turn- ing the interval in time into degrees at the rate of fifteen degrees to each hour. Essentially the same idea may perhaps be more raa^ftiy grasped by considering the star as apparently piassing over LONOITUDE. 85 ion of the earth, in meridians on 'een latitude on The observer the declination |tude from the e facility, be- ing. To deter- >f time as mea- cessarily comes f the meridian >here so as to I of the pkce. San Francisco celestial men- kke an angle of [this case. Let ^. Then he n to be visible I the pole over It a distance of meridian. It arh really both are moving in rentlj the stars f Washington, own meridian. « the interval (Han of Wash- once have the Y simply tum- nte of fifteen more rea(|lly f passmg over gg-feiiiwa^saaajg the snccessive terrestrial meridians on the surface of the earth, the earth being now supposed for a moment to be at rest. If we imagine a straight line drawn from the centre of the earth to a star, this line will in the course of twenty-four sidereal hours apparently make a complete revolution, passing in succession the meridians of all the places |)n the earth at the rate of fifteen degrees in an hour of sidereal time. If, then, Washington and San Francisco are forty-five degrees apart, any one star, no matter what its declination, will require three sidereal hours to pass from the meridian of Washington to that of San Francisco, and the sun will require tluee gdar iiours for the same passage. Whichever idea we adopt, the result will be the same : difference of longitude is measured by the time required for a star to apparently pass from the meridian of one place to that of another. There is yet another way of defining what is in effect the same thing. The sidereal time of any place at any instant being the same with the right ascension of its meridian at that instant, it follows that at any instant the sidereal times of the two places will differ by the amount of the difference of longitude. For instance : suppose that a star in hours right ascension is crossing the meridian of Washington. Then it is hours of local sidereal time at Washington. Three hours later the star will have reached the meridian of San Francisco. Then it will be C hours local sidereal time at San Fran- cisco. Hence the difference of longitude of two places is measured by the difference of their sidereal times at the same ins^ At of absolute time. Instead of sidereal times, we may equally well take mean times as measured by the sun. It being noon when the snn crosses tiie meridian of any place, and the snn requiring three hours to pass from the meridian of Washington to that of San Francisco, it follows that when it is noon at San Francisco it is three o'olodc in the afternoon at Washington.* * The dUtawnoe equivalent Here on the anding offer person who ing the Ion- Phis reward ui who con< tion and a uonometer. tion of the n afforded id methods vention of n its appli. nNwnedu • lie longitude >tais.infM3t. ilagton frmn > longitude is rorddereaL cation, but necessarily limited to places in telegraphic communication with each other. Longitude by Motion of the Moon. — When we de- scribe the motion of the moon, we shall see that it moves eastward among the stars at the rate of a)K)ut thirteen de- grees per day, more or less. In other words, its right as- cension is constantly increasing at the rate of a degree in something less than two hours. If, then, its right ascension can bo predicted in advance for each hour of Greenwich or Washington time, an observer at any point of the earth, by noting the local time at his station, when the moon has any given right ascension, can thence determine the corresponding moment of Greenwich time ; and hence, from the difference of the local times, the longitude of his place. The moon vrill thus serve the purpose of a sort of clock running on Greenwich time, upon the face of which any observer Mrith the proper appliances can read the Greenwich hour. This method of determining longitudes has its difficulties and drawbacks. The motion of the moon is so slow that a very small change in its right ascen- sion will produce a comparatively large one in the Green- wich time deduced from it — about 27 times as great an error in the deduced longitudes as exists in the determi- nation of the moon's right ascension. With such instru- ments as an observer can easily carry from place to place, it is hardly possible to determine the moon's right ascen- sion within five aeoonds of are ; and an error of this amount will produce an error of nine seconds in the Greenwich time, and henoe of two miles or more in his deduced longitude. Besides, the mathematical processes of dedndng from an observed right-ascension of the moon the corresponding Greenwich time are, under ordinary oircumstances, too troublesome and laborious to make this method of value to the navigator. Tmnaportfttioii of Ghxonometers. — ^The transportation of ohronometera affords a simple and convenient method of obtaining the time of the standard meridian at any moment. The observer sets his chronometer as nearly as 38 ASTnONOMT. possible on Greenwich or Washington time, and deter- mines its correction and rate. This he can do at any sta- tion of which the longitude is correctly known, and at which the local time can be determined. Then, wherever he travels, he can read the time of his standard meridian from the face of his chronometer at any moment, and compare it with the local time determined with his transit instrument or sextant. The principal error to which this method is subject arises from the necessary uncertainty in the rate of even the best chronometers. This is the method almost universally used at sea where the object is simply to get an approximate knowledge of the ship's position. The accuracy can, however, be increased by carrying a large number of chronometers, or by repeating the de- termination a number of times, and this method is often employed for fixing the longitudes of seaports, etc. Between the years 1848 and 1855, great numbers of chro- nometers were transported on the Cunard steamers plying between Boston and Liverpool, to determine the difference of longitude between Greenwich and the Cambridge Ob- servatory, Massachusetts. At Liverpool the chronometers were carefnily compared with Greenwich time at a >ocal observatory — ^that is, the astronomer at Liverpool found the error of the chronometer on its arrival in the ship, and then again when the ship was about to sail. When the chronometer reached Boston, in like manner its error on Cambridge time was determined, and the det«inination was repeated when the ship was about to return. Having a number of such determinations made alternately on the two sides of the Atlantic, the rates of the cfaronometers could be determined for each double voyage, and thus the error on Greenwich time could be calculated for the mo- ment of each Cambridge comparison, and the moment of Cambridge time for each Greenwich xiomparison. Longitade by the Bectrio Tdegzmph. — ^As soon as the electric telegraph was introdaced it was seen by American "OMns mmm^fimi^^ l mk^M'^-^^^'' |e, and deter- lo at any sta- liown, and at pen, wherever lard meridian inonient, and |ith Lis transit to which this incertaintj in This is the the object is >f the ship's by carrying a ftting the de- ithod is often seaports, etc. ibers of chro- iamers plying the difference unbridge Ob- chronometers ime at a .'ocal erpool found in the ship, saiL When mer its error letennination rn. Having utely on the shronometers and thus the for the mo- > moment of ion. i soon as the y American LOyOITUDE BT TBLEORAPn. •• astronomers that wo here had a method of determining longitudes wliicli for rapidity and convenience would supersede all others. The first application of this method was mode in 1844 between Washington and Baltimore, under the direction of the late Admiral Charles Wilkes, U. 8. N. During the next two years the method was intro- duced into the Coast Survey, and the difference of longitude between New York, Philadelphia, and Washington was thus determined, and since that time this method has had wide extension not only in the United States, but between America and Europe, in Europe itself, in the East and West Indies, and South America. The principle of the method is extremely simple. Each place, of which the difference of time (or longitude) is to be determined, is furnished with a transit instrument, a clock and a chronograph ; instruments described in the next chapter. Each clock is placed in galvanic communication not only with its own chronograph, but if necessary is so connected with the telegraph wires that it can record its own beat upon a chronograph at the other station. The observer, looking into the telescope and noting the crossing of the stars over the meridian, can, by his signals, record the instant of transit both on his own chronograph and on that of the other station. The plan of making a determination between Philadelphia and Washington, for instance, was essentially this : When some previously selected star reached the meridian at Phil- adelphia, the observer pointed his transit upon it, and as it crossed the wires, recorded the signal of time not only on his own ohron<^praph, but on that at Washington. About eight minutes afterward the star reached the meridian at Washington, and there the observer recorded its transit both on his own chronograph and oa that at Philadelphia. The interval between the transit over the two places, as measured by either sidereal clock, at once gave the difference- of longitude. If the record was in- stantaneous at the two stations, this interval ought to be the same, whether read off the Phihtdelphia or the Wash- 'to ARTHOiTOMY. ington chronogrupli. It was found, however, tliat there wan a difleronce of a Binall fraction of a second, ariHing from the fact tliat electricity re(£uired an interval of time, minute but yet appreciable, to puss between the two cities. The PhiUdelphia record was a little too late in being recorded at Washinj^ton, and the Washington one a little too late in being recorded at Philadelphia. We may illustrate this by an example as follows : Suppose £ to Ih3 a station one degree of longitude eaat of another station, W ; and that at each station there is a clock exactly regulated to the time of its own place, in which case the clock at E will l)e of course four minutes fast of the clock at W ; let us also suppose that a signal takes ft quarter of a second to pass from one station to the other : Then if the obgerver at E sends a nignal to W at exactly noon by his clock 12'' O" COO It will be received at W at * n*" 66"> 0'.25 Showing an apparent difference of time of S" fiiCTS Then if the observer at W sends a signal at noon by his dock la* 0" COO It will be received at E at 12'' 4"" 0".a6 Showing an apparent difference of time of 4" 0*.25 One half the sum of these differences is four minutes» which is exactly the difference of time, or one degree of longitude ; and one half their difference is twenty-live hundredths of a second, the time taken by the electric im- pulse to traverse the wire and telegraph instruments. This is technically called the "wave and armature time." We have seen that if a signal could be made at Wash- ington noon, and observed by an observer anywhere sit- uated who knew the local time of hia station, his longi- tude would thus become known. This principle is often employed in methods of determining longitude other than those named. For example, the instant of the banning that there Olid, ariHiiig •vul of time, ton tlio two too lato in ington ono a slphia. We ngitudo eaat >ii there ig a vn place, in oar minutes lat a signal tttion to tlie la* o-o-.oo .ll*>86">0'.a5 ig S" 59'.76 la'o-'O'.oo la"- 4- c.aa i-O-.M ur minutegf le degree of twenty-liye electric iin- ments. d armature le at Wash- ywhere dt- , his long!- pie is often I other than 9 beginning TUKOttY OF THM bVUKHK. 41 .„. .nOuM, of an «=Up» of .... -J" *^*:^ --"^^J .v.rfm.tlv dotln to p lenoineiion. it this w ooiwrvwu j wo obirve™, and these in.t«iU noted by each in the Wal ttieo? his station, then the difference of thej« W S (subject to small correction, due U, pa«Uax, etc.) will bo the difference of longitude of the two aw ^'^Tho satellites of Jupiter suffer ecUpaes frequently, and the cCuw^rand wihington times of theje phenomena a^ ooZZa and set down in the Nautical Almanac Ob- L'vatSon- of these at any -^*!<>V'L*tf S'rl^^^^^^^^ ence of longitude between tlus rtation and ^^'''^'''V'J wlington^ As, however, they require a larger tele- ZeTnd a higher magnifying power than can Ik, used at t^rtWB meth^ is not a practical one for navigators. 8 7. ItATHBIATIOAL Iggg OP THB 0«L«TIAL In thU •xplanatKm «' »'« » tX%*ir7nt*i^^^^^^^^^^ the heavenly bodies to «*«>«• «°*of the rSlor is necesgarlly pre- Ipherlcal trUnometry on the ~^^ „, f^^on gUpposed. >• 8«i«'!l'5?i*lolJtrrclroWs u follow. : thi sphere i«refen*dto axed pgntoor^^^^^ U Uken as a bwis, A M»"«nt»ie:!r* S^S"u.. bbject from ttie Fm.l4 Pig. 14 be a view of the celestial hemiaphere which is above tiie hraixon, as seen from the eaat Wetiienhave: HER F, the horiaon. P, the pole. Z, the lenith of the observer. ir Jf Z P JJ; the meridian of the observer. P Ji; the latitude of the observer, which call f. ?C'S.1Litf S£»«»*.-' = •«• - ■^'"«»"- Ta,i\M altitude, which call a, za,i\» aenitii diataooa = W" - «• MZS, itoaaimuth, = 180' -anrie 8 Z P. Z P ^ its hour an^e, which call *. The spherical triangle Z P -8, of which the angles are formed by u ASTRONOMY. the xenith, the pole, and the star, is the fundamental triangle of our problem. The latter, as commonly solved, may be put into two forms. I. Givi-n the latitude of the place, the declination or polar dis- tance of the star, and its hour angle, to find its altitude and azimuth. We have, by spherical trigonometry, considering the angles and sides of the triangle Z P 8 : con Z S = coB PZcoB PS + sin P Z sin PS cos P. Bin ZS coa Z = sia PZ eoa PS — coA PZ sin PS cos P. sin ZSain Z = Bin PS sin P. By the above definitions, Z S=90° — a, (a being the altitude of the star). PZ=90° — ^, (^ being the latitude of the place). PS = 90' — d, (6 being the declination of the star, + when north). P = h, the hour angla Z = 180° — t, (2 being the azimuth). Making these substitutions, the equation becomes : sin a = sin f sin 4 + coa f cos 4 cos A. COR a cos • = — COB ik sin ^ + sin f coa ' cos A. cos a sin • = cos J sin A. From these equations sin a and cos a may be obtained separately, and, if the computation is correct, they wul give thi> i.. <3 val'je of a. If the altitude only is wanted, it mayv.1be obiaim 1 f > t*>e first equation alone, which may be transformed in Tarioua . xy^ ained in works on trigonometry. II. Given the latitude of the place, the deelination of a star, and its altitude above the hwison, to find its hour ande and (if its right ascension is known) the sidereal timi when it liaa the given altitude. We find from the first of the above equations. cosA = sin a — 'lin ^ sin dl. or we may use : sin'iA = i cos ^ 008 i COS (f — «t) — sin o cos ^ 000 4 Having thus found A, we have Sidereal time s= A + cr, a being the star's right ascension, and the hour angle A being changed into time by dividing by 16. ni. An interesting form of this last problem arises when we sup- pose a sa 0, which is the same thii^; as supposing the star to be in tal triangle of our tit into two forms. ion or polar dis- ,ude and aumuth. ig the angles and 1 P S cos P. \PScmP. ir, + when north). 6 cos h. tabled separately, It, b. .aval'jeof a. 1 1 r > ♦»•« arst » . < xit^'vined Ion of a star, and e and (if its right the given altitude. le A being ebanged iseswhenwe au ABlRONOMr. 46 the horizon, and therefore Xo be rising or setting ^s • t'blj time between its "«"«. *""':* P^Ji^s interval is caUed the $emir tween this passage and its setting. This mtervai « c~. diurnal are, and by doubling it ^^^— ^ we have the time between the rising and setting of the star or other object Putting a = in the preceding expression for cos h we find for the semi diurnal arc A, _ wn ^ sin j CCS ft — — -^ ^ cos S = — tan ^ tan d, and the arc during which the sUr is above the horiion is 2 *. Prom this formula may be deduced at once many of the ^bbbi^^^^^^^^ results given in the preceding j.^ IB.— cpm um umtM mro- S6Ction8. HAIi ABC& (I). At the poles f = ,^' ^^ r _ »„ftnitv But the cosine of tan * = infinity, and thw^JJw cm A ^^ ^« ^^ ^„^ an angle can never be g'«'*«' .IS" "^^L' e^ . gjr »rthe pole can of A which fulfite the condition. Hence, a siar at w» i~ neither rise nor set . _ ao ♦.n a = whence cos A = 0, . <'>in^* *;«T^ iXihaterer beT *TO. brinj a semicircum. {e^nSrAVhia7en\?'boj£*rh.lf the time above the hori«.n to ^e t^aL-SiSu^tTeJielirve^h^^^^ tude of the observer. Here we except *« I^|«; ^'^^.S *. ^tuok Und tand Ig the star ire aup- to be in «••* = " SSI "■ tan (90° - f) wbMiW,»»?' *"« 46 A8TR0N0MT. negative i, cos h is positive, A < 90% 2 A < 180°. Hence, in north- em latitudes, a northern sUr is more than half of the time above the horizon, and a soi'them star loss. In the southern hemisphere, f and tan f are negative, and the case is reversed. ^ (6). If, in the preceding case, the declination of a body is supposed constant and north, then the greater we make ♦ the greater the nega- tive value of cos h and the greater h itself will be. Considering, m succession, the cases of north and south declination and north and south latitude, we readily see that the farther we go to the north on the earth, the longer bodies of north declination remam above the horiioD, and the more quickly those of south declination set. In the southern hemisphere Uie reverse is true. Thus, in the month of June, when the sun is north of the equator, the dnys are shortest near the aoath pole, and contiDually increase in length as we go north. Examples. (1). On April », 1879, at Washington, the altitude of Rigel above the west hmuon was observed to be 12° 26'. Ite position was : Right ascension = S" 8- 44'-27 = a. Declination = - 8° 20' 86' = «. The latitude of Washington is + 88° 58' 89' = *. What was Uie hour angle of the star, and the sidereal time of ob- servation f lgBina= 9-882478 lg8in#= 9-797879 lgsind= - 9- 161681 — Ig sin ^ sin S = 8-959560 -sin*sin.J= 0-091109 sina= 0-215020 sin a - sin f sin il = 0806129 Igcos^ = Igoos o = IgCOSf 008 d = Ig (ain a — sin ^ sin d, = Igcos A = * -I- 1« = sidereal time = 9-891151 9-995879 9-886580 9-486905 9-599875 , 66° 84' 88' 4^ 26'* 18'.90 6* 8-44'.2T »k85» 2'.47 (2) Had the star been observed at the same altitude in the east, iriut would have been the sidereal timet Ans. a-A = 0k4a-8«*.07. DBTBRMINATION OF LATITUDE. 47 Hence, in north- the time ftbove the hemisphere, f and i body ifl supposed ) greater the nega- I. Cmisidering, in on and north and go to the north on I remain above ths ination set. In the , in the month of dnys are shortest ^h as we go north. ude of Rigel above position was : = a. iidereal time of ob- J8' 18'.90 a'.47 altitude in the east, (8). At what sidereal time does Rigel rise, and at what sidereal time does it set in the latitude of Washington f - tg« - -9-906728 tgd = - 9166801 cos h = A = * -5- 15 = a ^ - 9 078029 88^ 12' 19" 6h 82'» 49*.27 5k 8»'44'.27 rises 23'' 8»" aS'.OO sets 10<' 41"' 88*.fi4 (4). What is the greatest altitude of Rigel above the horicon of Washington, and what is its greatest depression below it r Ans. Altitude=4a' 46' 45" ; depression =89° 26' 67'. (6). What is the greatest altitude of a ater OD the equator in the meridian of Washington f Ans. 51° •' 81". _ (6). The ddcllnatron of the pointer in the Great Bear whioh is nearest the pole is 62' 80' N., at what altitude does it pass abow the pole at Washington, and at what altitude does it pass below it V Ans. 66° 88' 89' above the pole, and 11" 28' »9' when below it. (7). If the declination of a star is 00° N., what length of sidereal time is it above the horiaon of Washington and what length below it during its apparent diurnal drauitf Ans. Above, ai** 68"* ; below. 2'' S". § 8. DETBBMIFATION OF lATTFUDBS ON THE BABTH BY ASTBONOlCKSAIi OBSBBVATIONB. Latitude fivm eireumpolor Uan.— In Pig. 16 let Z represent the zenith of the place of observation, P the pole, and MPZ it the me- ridian, the observer bring at the centre of the sphere. Suppose .Sand iS* to be the two points at which a oircumpolar atar ' crosses the meridian in the d*- scription of its q>pannt diurnal cn-bit Then, since P is midway between 8 and S", ZS + ZB „„ .^ or. Z+Z' = W-f. If, then, we can measure tiie dia- tances Z and Z, we have Z4-Z^ Fie. 16. whidi seeree to determine f. The diataooes ZmA iF can be m«M- , ,. .KS5^- fc " ^SSy 1 48 A8TnoyoMr. l! ! nred by the meridian circle or the sextant— both of which instru- ments are descrilied in the next chapter — and the latitude in then known. Z and Z" must be freed from tlie effects of refraction. In this method no previous knowledge of the star's declination is re- quired, provided it remains constant lietween the upper and lower transit, which is the case for fixed stars. Latitude by Oiroum-ienith Obaerratioiui If two stars 8 and S*, whose declinations 6 and A' are known, cross the meridian, one north and the other south of the xenith, at zenith distances Z 8 and ZS', which call Z and Z', and if wo have measured Z and Z, we can from such measures find the latitude ; for ^ = d + Z and « = <' — Z", whence f = i((d + d') + (z-2r)]. It will be noted that in this meth- od the ktitude depends simply upon the mean of two declinations which ean be determined before- hand, and only requires the diff'er- meg of Moith distances to be ac- curately measured, while the aln solute values of these are unknown. In this oonslsts its capital ad- vantage. This is the method invented by Oapt. Amdrrw Talcott, U.S.A., and now universally adopted in America in Add astronomy, in the practice of the Coast Survey, etc. Latitude liy a Single Altitude of a Star. — In the triangle ZPS(Vig. 14)thesidesareZP=iK)''~f;P5=90'' — a; Z8 = Z = 90" — ri ; ZP8 = A = the hour angle. If we can measure at any known sidereal time the altitude a of the star iS, and if we further know the right ascension, a, and the declination, oth of which instru- «] the latitude in then ict8 of refraction. In tar's declination is re- the upper and lower btions.— If two stars rn, cross the meridian, it zenith distances Z 8 hich call Z and Z', and measured Z and Z, we ich measures find the Dr f = i + Z and « = ence i + d') + (z-2r)]. 9ted that in this meth- tud« depends simply san of two declinations be determined before- nly requires the diff'er- th distances to be ac- wmred, while the ab- oonalsts its capital ad- ipt. Amdrrw Talcovt, icain Add astronomy, Har. — In the triangle ?S=90' — 6i Zti = If we can measure at the star ugh its period served to be at the hmzoniai pahallax. 51 pavaJlm of a hoavo.ilj IkkIj. The parallax first doserilwl "1 the last pairugmpli varies with the jKwition of tlio ob- ■erveron the surface of the earth, and lias its greatest value when the body is seen in the horizon of the ob- server, as may be seen by an inspection of Fig. 19 in which the angle GPS attains its maximum when the Hne 18 IS tangent to the earth's surface, in which case P will appear in the horizon of the observer at 8. IV.— HmunniTAi. pAkaixax The horizontal parallax depends upon the distance of a body m the followmg manner: In the triangle C P 8. nght-angled at S, we have S ^ ^ ^. C8^GPmiCP8. If, then, we put p, the radius of the earth G8\ JS^the distance of the body P from the centre of the »r, the angle 8P G, or the horizontal parallax, we shall have, sin n' P = r sin >r; r i« It wf Tf" *' '''** P^'^^^y «P^«"«J' the quantity p ^^tabsolute y con^nt for aU parts of thi earth, «7ito greatest value w usually taken as tiiat to which 4e hori- r^ nt' "^ ^ ^^«"^- This greatest value fa' « we shall hereafter see, the radius of the equator, a^d h" VS**" 53 ARTRONOMT. corresponding valnc of tho parallax 18 thcroforo called the eqmiton'd /wrhontaf jMUuUfito). When the diatanco /• of the Ixxly i» known, tho wpxa- tonal horizontal parallax can bo found by the firet of the above equationa ; when tho paralUx can be obBerved, the distance r is found from the second equation. IIow this is done will be described in treating the subject of celes- tial measurement. . . , . ^ , ii„. It is easily seen that the equatorial horizontal parallax, or the angle CPS^i'^ the same as the anguUr seim- diameter of the earth seen from the object P. In fact, if we draw the Une PST tangent to «io earth at ^, he angle 5 P 5' will be the apparent angular diameter of tlie earth as seen from i>, and wiU also be donblo the angle CP8 The apparent semi-diameter of a heavenly body is therefore given by the same f ormute as the pindlax its own radius being substituted for that of the earth. If we put, p, the radius of the body in linear measure ; r, the distance of its centre from the observer, expressed in the same measure ; ^w.,„«r • «, its anguhir semi-diameter, as seen by the observer , we shall have, . . P sm « = -• r If we measu^ tbe semi-diameter «, and know the dis- tance, r, the radius of the body will be p = r rin «. Generally tlie angnkr semi-diameters of the heavenly bod^r.^L small that they may be considered the same ^Wr^nl We may theref o.^ say that the apparent Tn^Sr diameter of a heavenly body varies inversely as its distance. oforo called the own, the cqua- the first of the e observed, the ion. IIow thia iibject of celea- izontal parallax, anffular semi- set P. In fact, earth at S', the diameter of the onblo the angle a heavenly body as the parallax, if the earth. If lure ; server, expressed the observer ; kd know the dis- i of the heavenly isidered the same th^the apparent aries inversely as CHAPTER II. ASTRONOMICAL INSTRUMENTa § 1. THE EBFEAOmrO TBLBSOOPB. In explaining the theory and use of the refracting tele- scope, we shall assume that the reader is acquainted with the fundamental principles of the refraction and disper- sion of light, so that the simple enumeration of them will recall them to his mind. These principles, so far as we have occasion to refer to them, are, that when a ray of light passing through a vacuum enters a trans- parent medium, it is refracted or bent from its course in a direction toward a line perpendicular to the sur- face at the point where the ray enters ; that this bend- ing follows a certain law known as the law of sines ; that when a pencil of rays emanating from a luminous point falls nearly perpendicularly upon a convex lens, the rays, after passing through it, all converge toward a point on the other side called a focus : that light is com- pounded of rays of various degrees of refrangibiUty, so that, when thus refracted, the component rays pursue slightly different courses, and in passing through a lens come to slightly dififerent foci ; and finally, that the ap- parent angular ma^itude subtended by an object when viewed from any point is inversely proportional to its distance.* • More exactly. In the cam of a globe, the sine of the angle Is in- venely as the diatanoe of the object, aa shown on the preceding page. t 64 AHTRONOMY. We ■hall tint doscrilM) tho toloncopo in its siniplMt ^H^^B form, showing the principluH upon whidi ^^^^^1 its action depends, leaving out of considora- ^^^^^1 tion tlie defects of aberration which retpiiro ^l^^^l special devices in order to avoid them. In ^^^^^1 the simplest fonn in which we can conceive ^^^^^1 ^ of a telescope, it consists of two lenses of ^^^^H § unequal focal lengths. The puqKMBo of one ^^^^^H ° of these lenses (called the of^ectivc, or object ^^^^M I gkua) is to bring the rays of light from a ^^^^H i distant object at which the telescope is ^^^^H ° pointed, to a focus and there to form an ^^^^H ^ image of the object. The purpose of the ^^^^H M other lens (called the eye-piece) is to view ^^^^H I this object, or, more precisely, to form an- ^^^^^1 other enlarged image of it on the retina of ^^^^^1 ^ tho ^^^^H § The figure gives a representation of the ^^^^H I course of one pencil of the rays which go to ^^^^H S fonn the image ^ 7' of an object / li after ^^^^H & passing through the objective 0'. The " pencil chosen is that composed of all the rays emanating from / which can possibly [ „ fall on tho objective 0'. All these are, ^^^^1 2 by the action of the objective, concentrated ^^^^H 2 at the point T. In the same way each point ^^^^H g of the image out of the optical axis A B ^^^^H % emits an oblique pencil of diverging rays ^^^^H '. which are made to converge to some point ^^^^1^. of the image by the lens. The image of ^^^^H£ the point B of the object is the point A of ^^^^H the image. We must conceive the image of ^^^^H any object in the focus of any lens (or ^^^^H mirror) to be formed by separate bundles ^^^^H of riiya as in the figure. The image thus HHlB formed Inicomcs, in its turn, an object to be viewed by the eye-piece. After the rays meet to form I MAUNIFYIlfU PVWh'Il VF TKI.K8G0PK. 55 in ito ninipleflt »luit upon which mi of conaidera- )n which recjuiro iivoid them. In wo can concoivo of two lunacH of e pur]K)8o of one hjective^ or cijeot of light from a the telescope is lere to form an 3 purpose of the •piece) is to view jely, to form an- on the retina of Mentation of the rays which go to object / B after jtivc 0'. The ipoeed of all the liich can poflBibly '. All these are, ;iye, concentrated lie way each point optical axis A B )f diverging rays irge to some point The image of ig the point A of leive the image of of any lens (or separate bundles The image thus iim, an object to rays meet to form the imago (»f an object, as at /, thoy continue on tlioir course, diverging from /' as if the latter wore a material object reflecting the light. There is, however, this excep- tion : that the rays, insteiwl of diverging in every direction, only fonn a small cone having its vertex at /', and having its angle equal io O F C The reason of this is that only those rays which pass through the objective can form the image, and thoy must continue on their course in straight lines after forming the image. This image can now bo viewed by a lens, or even by the unassisted eye, if the observer places himself behind it in the direction A^ so that the pencil of rays shall enter his eye. For the pres- ent we may consider the eye-piece as a simple lens of short focus nice a common hand-magnifier, a more com- plete description l>« ng given later. Magnifying Fow«r.— To unc^orstand the manner in which the telescope magnifies, we remark that if an eye at the object-glass could view the image, it would appear of the saine size as the actual objt^ct, the iii^ge and the object subtending the same angle, but lyinc^ » opposite direc- tion. This angular magnitude beih^ ihe same, whatever the focal distance at which • > ^ tmage is former', it follows that the size of the inuige vf ties iirectly as thu local length of the object-ghus. But when we view an object with a lens of small focal distance, its apparent magnitude is thr; same as if it were seen at that focal distance. Consequently the apparent angular magnitude will be inversely as the focal distance of the leuF Hence the focal image as seen with the eye-pioce will appear lai^r than it would when viewed from the objective, in the ratio of the focal distance of the objective to that of the eye-piece. But we have said that, seen through the objective, the image and the real object subtend the same angle. Hence the angu- lar magnifyi^T power is equal to the focal distance of the objective, dir h-] by that of the eye-piece. If we simply turn the telescope end for end, the objective becomes the eye-piece and the latter the objective. The ratio is in- 66 ASTRONOMY. verted, and the object is diminished in size in tl:o same ratio that it is increased when viewed in the ordinary way. If we should form a telescope of two lenses of equal focal length, by placing them at double their focal distance, it would not magnify at all. The image formed by a convex lens, being upside down, and appearing in the same position when viewed with the eye-piece, it follows that the telescope, when constructed in the simplest manner, shows all objects in- verted, or upside down, and right side left. This is the case with all refracting telescopes made for astronomical uses. Light-gathering Power.— It is not merely by magnify- ing that the telescope assists the vision, but also by in- creasing the quantity of light which reaches the eye from the object at which we look. Indeed, should we view an object through an instrument which magnified, but did not increase the amount of light received by the eye, it is evident that the brilliancy would be diminished in propor- tion as the surface of the object was enlarged, since a con- stant amount of light would be spread over an increased surface ; and thus, unless the light were faint, the object might become so darkened as to be less plainly seen tlian with the naked eye. How the telescope increases the quantity of light will be seen by considering that when the unaided eye looks at any object, the retina can only re- ceive so many rays as fall upon the pupil of the eye. By the use of the telescope, it is evident that ac many rays can be brought to the retina as fall on the entire object- glass. Tlie pupil of the human eye, in its normal state, has a diameter of about one fifth of an inch ; and by the use of the telescope it is virtually increased in surface in the ratio of the square of the diameter of the objective to the square of one fifth of an inch. Thus, with a two-mch aperture to our telescope, the number of rays collected is one Imndred times as great as the number collected with the naked eye. , 1 ze in tlio Batne the ordinary two lenses of ible their focal Q being upside when viewed elescope, when all objects in- t. This is the or astronomical sly by magnify- but also by in- hs the eye from uld we view an !;niiied, but did by the eye, it is shed in propor- :ed, since a con- er an increased faint, the object lainly seen tiian e increases the g that when the na can only re- »f the eye. By t ac many rays le entire object- B normal state, ch ; and by the d in surface in th& objective to with a two-inch ays collected is r collected with POWER OF TELESCOPE. "With a 5-inch object-glass, the ratio is (( in (< (t (( (( (( n ii^g (( le. This fonc- l for more thaa ^^m^^m half a century after its invention, and after a long and rather acrimonious contest between two schools of astron- omers. Until the middle of the seventeenth century, when an astronomer wished to determine the altitude of a celestial object, or to measure the angular distance be- tween two stars, he was obliged to point his quadrant or other measuring instrument at the object by means of ' * pinnules. ' ' These served the same purpose as the sights on a rifle. In using them, however, a difliculty arose. It was impossible for the observer to have distinct vision both of the object and of the pinnules at the same time, because when the eye was focused on either pinnule, or on the object, it was necessarily out of focus for the others. The only way to diminish this diflSculty was to lengthen the arm on which the pinnules were fastened so that the latter should be as far apart as possible. Thus Tycho Bbahe, before the year 1600, had measuring in- struments very much larger than any in use at the pres- ent time. But this plan only diminished the difficulty and could not entirely 't)bviate it, because to be manageable the instrument must not be very large. About 1670 the English and French astronomers found that by simply inserting fine threads or wires exactly in the focus of the telescope, and then pointing it at the ob- ject, the image of that object formed in the focus could be made to coincide irith the threads, so that the observer could see the two exactly superimposed upon each other. "When thus brought into coincidence, it was known that the point of the object on which the wires were set was in a straight line passing through the wires, and through the centre of the object-glass. So exactly could such a pointr ing be made, that if the telescope did not magnify at all (the eye-piece and object-glass being of equal focal length), a very important advance would still be made in the ac- curacy of astronomical measurements. This line, passing oentrally through the telescope, we call the line of col- Umatim of the telescope, A Bin Fig. 20. If we have sifiEBisiasjc;; IfSJ^WftffS flO A8TB0N0MT. any way of determining it we at once realize the idea ex- pressed in the opening chapter of this book, of a pencil ex- tended in a definite direction from the earth to the heav- ens. If the observer simply sets his telescope in a fixed position, looks through it and notices what stars pass along the threads in the eye-piece, he knows that those stars all lie in the line of collimation of his telescope at that instant. By the diurnal motion, a pencil-mark, as it were, is thus being made in the heavens, the direction of which can be determ'ned with far greater precision than by any meas- urements with the unaided eye. The direction of this line of collimation can be determined by methods which we need not now describe in detail. The Aohromatio Telescope. — The simple form of tele- scope which we have described is rather a geometrical conception than an actual instrument. Only the earli- est instruments of this class were made with so few as two lenses. Galileo's telescope was not made in the form which we have described, for instead of two convex lenses having a common focus, the eye-piece was concave, and was placed at the proper distance inside of the focus of the objective. This form of instrument is still used in opera- glasses, but is objectionable in large instruments, owing to the smallness of the field of view. The use of two con- vex lenses was, we believe, first proposed by Eepleb. Although telescopes of this simple form were wonderful instruments in their day, yet they would not now be re- garded as serving any of the purposes of such an instru- ment, owing to the aberrations with which a single lens is effected. We know that when ordinary light passes through a simple lens it is partially decomposed, the differ- ent rays coming to a focus at different distances. The focus for red rays is most distant from the object-glass, and that for violet rays the nearest to it. Thus arises the ohromatio aberration, of a lens. But this is not all. Even if the light is but of a single degree of refrangi- bility, if the surfaces of our lens are spherical, the rays I? ize the idea ex- , of a pencil ex- ■th to the hear- scope in a fixed stars pass along it those stars all B at that instant, it were, is thus >f which can be in by any meas- tion of this line ithods which we le form of tele- r a geometrical Only the earli- th so few as two ide in the form leo convex lenses as concave, and ' the focus of the 11 used in opera- Lments, owing to use of two con- led by Kepleb. were wonderful not now be re^ such an instru- li a single lens is 17 light passes )0fled, the differ- distances. The the object-glass, it. Thus arises t this is not all. ;ree of refrangi- herical, the rays A CHROMA TIO OBJECT- GLASS. 61 wliich pass near the edge will come to a shorter focus than those which pass near the centre. Thus arises spherical aherratian. This aberration might be avoided if lenses could be ground with a proper gradation of curvature from the centre to the circumference. Prac- tically, however, this is impossible ; the deviation from imiform sphericity, which an optician can produce, is too small to neutralize the defect. Of these two defects, the chromatic aberration is much the more serious ; and no way of avoiding it was known until the latter part of the last century. The fact had, indeed, been recognized by mathematicians and physicists, that if two glasses could bo found having very different ratios of refractive to dispersive powers,* the defect could be cured by combining lenses made of these different kinds of glass. But this idea was not realized until the time of DoLLOND, an English optician who lived during the last century. This artist found that a concave lens of flint glaj98 could be combined with a convex lens of crown of double the curvature in such a manner that the dispersive powers of the two lenses should neutralize each other, being equal and acting in opposite di- rections. But the crown glass having the greater refractive power, owing to its greater cur- vature, the rays would be brought to a focus without dispersion. Such is the construction of the achromatic objective. As now made, the outer or crown glass lens is double convex ; tlie inner or flint one is generally nearly plano-concave. Fig. 31 shows ihe section of such an objective as made by Alvan Glabk & Sons, the inner curves of the crown and flint being nearly equal. * By the r^fraelitie power of a glass is meant its power of bending the rays out of thefar ooane, so as to bring them tn a focus. By its d^pvr- «iw potter is meant its power (rf separating tlie colors so as to form a Vectnun, or to produce chromatic aberration. iiiiiiiriii Fio. 21.— flBonoN or oBntoT- ahim. ms.:-^^ aX-^ mmm 63 ASTRONOMY. ^\ A great advantage of the achromatic objective is that it may be made to correct the spherical as well as the chro- matic aberration. This is effected by giving the proper curvature to the various surfaces, and by making such slight deviations from perfect sphericity that rays passing through all parts of the glass shall come to the same focus. The Secondary Speotrum. — It ia now known that the chromatic aberration of an objective cannot be perfectly corrected with any combination of glasses yet discovered. In the best telescopes the brightest rays of the spectrum, which are the yellow and green ones, are all brought to the same focus, but the red and bine ones reach a focus a little farther from the objective, and the violet ones a focus still farther. Hence, if we look at a bright star through a large telescope, it will be seen surrounded by a blue or violet light. If we push the eye-piece in a little the enlarged image of the star will be yellow in the centre and purple around the border. This separation of colors by a pair of lenses is called a secondary spectrum. Bye-Pleoe.— In the skeleton form of telescope before described the eye-piece as well as the objective was con- sidered as consisting of but a single lens. But with such an eye-piece vision is imperfect, except in the centre of the field, from the fact that the image does not throw rays in every direction, but only in straight lines away from the objective. Hence, the rays from near the edges of the focal image fall on or near the edge of the eye- piece, whence arises distortion of the image formed on the retina, and loss of light. To remedy this difficulty a lens is inserted at or very near the place where the focal image is formed, for the purpose of throwmg the different pencils of rays which emanate from the several parts of the image toward the axis of the telescope, so that they shall all paos nearly through the centre of the eye lens pro- per. These two lenses i>re together called the eye-piece. There are some small differences of detail in the con- struction of eye-pieces, but the general principle is the J TUEOnr OF OBJBXfT-OLASS. ictive is that it 11 as the chro- ng the proper making Bnch it rays passing he same focus. noMOi that the )t be perfectly yet discovered. ' the spectrum, > all brought to s reach a focus le violet ones a t a bright star arrounded by a piece in a little w in the centre ration of colors iotntm. elescope before jective was con- But with such 1 the centre of loes not throw ght lines away L near the edges Jge of the eye- lage formed on this difficulty a gvhere the focal ug the different several parts of 36, so that they the eye lens pro- the eye-piece. »il in the con- principle is die same in all. The two recognized classes are tlio posi- tive and negative, the former being those in which the imago is formed before the light reaches the field lens ; the negative those in wlilch it is fonned between the lenses. The figure shows the positive eye-pieco drawn accurately to scale. / is one of the converging; pencils from the object-glass which forms one point (/) of the focal image / a. This image is viewed by the Jlela lent F of the eye-piece as a real object, and the shaded pencil between F and E shows the course of these rays after de- viation by F. If there were no eye-lmu E an eye properly placed beyond F would see an ima^ at /' a'. The eye-lens E receives the pencil of rays, and deviates it to the observer's eye placed at such a point that the whole incident pencil will pass through the pupil and fall on the retina, and thus be effective. As we saw in the 22.— BBcnoR or a vaarmt BTR-pmnL figure of the refracting telescope, ever; point of the object producet a pencil similar to /, and the whole surfaces of the lensea F and E are covered with rays. All of these pencils paasinK through the pupil ^ to make up the retinal image. This image u refemd by the mind to the distance of distinct vision (about ten inches), and the image A I" represents the dimension of the final image A F' relative to the image a / as fonned by the objective and — y ^ evidently the nujipiif ving power of this particular eye-piece used in combination with this particular objective. More Eicaot Theory of the Ol^jeotive For the benefit of the reader who wishes a more precise knowledge of the optical princi- Sles on which the action of the objective or other system of lenses epends, we present the following geometrical theory of the sub- ject. This theory ft not rigidly exact, but is sufficiently so for all ordinary computations of we focal lengths and sizes of image in the usual combinations of lenses. 1 IV 64 A8TR0N0MY. Oentrea of Oonyenenoe and Divergenoe.—Siinpoge A B, Fig. 28, to be a IcnH or commnation of lonscs on which the light falls from the left hand and passes through to the right. Suppose rays parallel to 7? P to fall on every part of the first surface of tnc glass. After passing through it they are all supposed to converge nearly or ex- actly to the same point If. Among all these rays there is one, and one only, the course of which, after emerging from the glass at Q, will be parallel to its original direction It P. Let li P Qlf be this central ray, which will \hs completely determined by the direction from which it comes. Next, let m take a ray coming from another direction m 8 P, Among all the rays parallel to 8 P, let us take that one which, after emerging from the glass at 7*, moves in a line parallel to its original direction. Continuing the process, let u> suppose isolated rays coming from all parts of a distant object sub- ject to the single condition that the course of each, after passing through the glus or system of glasses, shall be parallel to its original course. These rays we may call cmtml rayt. They have this re- markable property, pointed out by Oauw: that they all converge Fig. 38. toward a single point, i*, in coming to the gloss, and diverge from another point, i*, after passing through the last lens. These points were termed by Gacbs " Hauptpunkte," or principal points. But they will probably be better understood if we call the first one the centre of convergence, and the second the centre of divergence. It must not be understood that the central rays necessarily pass through these centres. If one of them lies outside the first or lost refracting surface, then the central rays must actually pass through it. But if they lie between the surfaces, they will be fixed by the continuation of the straight line in which the rays move, the latter being refracted out of their course by passing through the surface, and thus avoiding the points in question. If the lens or system of lenses be turned around, or if the light passes through them in an opposite direction, the centre of oonveigence in the first case be- comes the centre of divergence in the second, and mee verta. The necessity of this will be clearly seen by reflecting that a return ray of light will always keep on the course of the original ray in the opposite direction. liinpoBe A B, Fig. ic light falls from iposc rays parallel the glass. After rge nearly or ex- tnere is one, and n the glass at Q, R P Q li' he Mh { by the direction ling from another SP, let us take T, moves in a line he process, let us istant object sub- ich, after passing illcl to its original hey have this re- they all converge and diverge from lens. These points cipal points. But 11 the first one the tre of divergence. j» necessarily pass ide the first or last dually pass through rill be fixed by the lys move, the latter irough the surface, le lens or system of hrough them in an n the first case be- tnd mce verta. The ig that a return ray I original ray in the riiKOBY OF oBJBcr-aLAaa. 65 The figure represents a plano-convex lenn with light falling on the convex side. In this case the centre of convergence will be the convex surface, and that of divergence inside the glass on aliout one third or two fifths of the way from the convex to the plane surface, the positions varying with the refractive index of the glass. In a double convex lens, both points will lie inside the glass, while if a glass is concave on one side and convex on the other, < ne of the points will be outside the glass on the concave side. It nust be remembered that the positions of these centres of conver- gance and divergence depend solely on the form and size of the lenses and their refractive indices, and do not refer in any way to the distances of the objects whose images the^ form. Tht principal properties of a lens or objocti^ c, by which the size of imageb «re determined, are as follows : Since the angle 9 P B! made by the u>erging rays is equal U> RP 8, made by the con- verging ones, it fo;Ws, that if a lens form the image of an object, the size of the image will be to that of the object as their respec- tive distances from the cei:t<«« of convergence and divergence. In other words, the object seen from the centre of convergence P will be of the same angular magnitude as the image seen from the centre of divergence P*. By eotyugaU fon of a lens or system of lenses we mean a puIi- of points such that if rays diverge from the one, they will converge to the other. Hence if an object is in one of a pair of such foci, the image will be formed in the ot)i«r. By the rtfraOMt powr of a lens or combination of lenses, we mean its influence in refracting parallel rays to a focus which we may measure by the recipiocai of its focal distaoce or 1 -i-f. Thus, the power of a piece of plain glass is 0, because it cannot bring rays to a focus at alL The power of a convex lens is positive, while that of a concave lens is negative. In the latter case, it will be remembered by the student of optics that the virtual focus is on the same side of the lens from which the rays proceed. It is to be noted that when we speak of the focal distance of a lens, we mean the distance from the centre of diveq^nce to the focus for ])arallel rays. In astronomical language this focus is called the stelhir focus, being that for celestial objects, all of which we may regard as infinitely distant. If, now, we put p, the power of the lens ; /, its stellar focal distance ; fy the distance of an object from the centre of convergence ; . /', the distance of its image from the centre of divergence ; then the equation which determines/ will be 1 1 1 f^f'-f-^' or. f- ffL. . /' = >^. f-f From these equations may be found the focal length, having the distance at which the image of an object is formed, or viee verta. ?*?»*« wfsMTfmi'^m**- , 11 06 ASTRONOMY. 8 9. BSFLEOTDfO TXLBBOOPBS. Ar wo liavo Been, the most enential part of a rafraeting teluHcopo is the objective, which brings all the incident rays from an object tu one focus, forming there an image of tba object. In reflecting telescopes (reflectors) the objective is a m^ror of speculum metal or silvered glass ground tQ the shape of a paraboloid. The figure shoMnt the action of such a mirroi on a bundle of parallel rays, which, after impinging on it, are brought by reflection tu one focus F. The image formed at this focus may be viewed with an eye-piece, as in the case of the refracting telescope. The eye-pieces used with such a mirror are of the kinds already described. In the figure the eye-piece would FlO. 84.— CONCAVS MIRBOR rORMINO AK IMAOC have to be placed to the right of the point F, and the observer's head would thus interfere with the incident light. Various devices have been proposed to remedy this inconvenience, of which we will ifaiiffribe the two most common. Hm Vewtonieii IMeecope. — In this form the rays of light reflected from the mirror are made to fall on a small plane mirror placed diagonally just before they reach the principal focus. The rays are thus reflected out laterally through an opening in the telescope tube, and are there brought to a focus, and the image formed at the point marked by a heavy white line in Fig. 25, instead of at the point inside the telescope marked by a dotted line. )PES. t of a rofracting nil the incident there an image (reflectors) the >r silvered glass 10 figure showt )f parallel rays, by reflection to I focus may be f the refracting are of the kinds )ye-piece would K IMAOB. oint F, and the ith the inddeDt d to remedy this le fhe two most ytm the rays of bo fall on a small ■e they reach the ted out laterally le, and are there led at the point i5, instead of at »y a dotted line. RKFLKCTtNO TSLBBGOPKS. 07 This focal image is then examined by means of an or- dinary eye-piece, the head of the observer being outside of the telescope tube. Tills device is the invention of Sir Isaac Nkwton. HBWTONIAN TBLBSCOPB. FiOw M. CAflERnRAINTAN TKI^BSCOTB. Tlie Oalisegr&iman Teietiooi>e. — In this form a second- ary convex mirror is piaced in the tube of the telescope '-~«.sSB^PIB««WB'"- r.8 AHTHONOMr. |i abont three ■ 'tii« ,jf Uie wiiy from the hirge HptMtuiutn to the fociiH. The riiyH, after l>eing roHeetetl from the largo 8))ecuhiin, fall oa this mirror befoit) reaching the focus, and are reHected back again to the Bpvculuni ; an opening is made in the centre of the latter to lot the ravs imm through. The position and curvature of the secondary mirror are adjusted so that the focus shall be formed just after passing through the opening in the speculum. In this telescope the obsurvet stands behind or under the speculum, and, with the oyo-pieco, looks through the opening in the centre, in the direction of the object. This form of reflector is much more convenient in use than the Newtonian, in using which the observer has to be near the top of the tube. This form was devised by Cabrkorain in 1672. Tho advantages of reflectors are found in their cheap- ness, and in the fact that, supposing the mirrors perfect in tigure, all the rays of the spectrum are brought to one focus. Thus the reflector is suitable for spectroscopic or ]>hotographic researches without any change from its or- dinary fonn. This is not true of the refractor, since the rays by which we now photograph (the blue and violet rays) are, in that instrument, owing to the secondary spectrum, brought to a focus slightly different from that of the yellow and adjacent rays by moans of which we 060* Beflectors have been made as large as six feet in aper- ture, the greatest being that of Lord Robse, but those which have been most successful have hardly ever been larger than two or three feet. The smallest satellite of Satntm {Minuu) was discovered by Sir Wiujam Hersohel with a four-foot speculum, but all the other satellites dis- covered by him were seen with mirrors of about eighteen inches in aperture. With these the vast majority of his faint nebnlsB were also discovered. The satellites of Neptune and TTrantts were discovered by Lassell with a two-foot speculum, and much of the nKFLKcrmn TKijjsropfss. m largo Rpecuium leotud from tlio m reaching the J spvculuin ; an ir to lot the rava of the secondary i Im) formed just tpeculum. behind or nndor x>k8 through the I of tho object, mveniont in uro ) observer has to in 1672. in their chcap- nirrors perfect in brought to one spectroscopic or mge from its or- f ractor, since the ) blue and violet o the secondary ferent from that lans of which we \ six feet in aper- RosBE, but those hardly ever been allest satellite of II4.IAM Hersohel ther satellites dis- at about eighteen t majority of his » were discovered and much of the work of Lord Rohhk has boon doiio with liitt throo-foot mirror, iuHtead of liiw (Hilobmtod nix foot oin\ From tho tinu) of Nkwton till (luito rccontly it wa« usual to make tho largo mirror or objoj-tivo out of Bpcu- Inni motal, a brilliant alloy liublo to tanuHh. Whon tho mirror was onco tiirnishod through cxjKWuro to tho woathor, it could bo ronowod only by a proccBS of jwlish- ing almost equivalent to figuring and polishing tho mirror anew. Consequontly, in such a speculum, after the cor- rect f jrtn and polish wore attained, there was groat diffi- culty in preserving them. In rocont years this difficulty has been largely ovorcomo in two ways : first, by im- provements in the composition of the alloy, by which its liability to tarnish under exposure is greatly diminished, and, secondly, by a plan proposed by Foucault, which C(m8ist8 in making, onco for all, a mirror of ghws which will always retain its good figure, and depositing upon it a thin film of silver which may be removed and restored Mrith little labor as often as it becomes tarnished. In this way, one important defect in the reflector has been avoided. Another great defect has been less success- fully treated. It is not a pntcess of exceeding difficulty to give to the reflecting surface of either metal or glass the correct parabolic shape by which the incident raya are brought accurately to one focus. But to maintain this shape constantly when the mirror is mounted in a tube, and when this tube is directed in succession to various parts of the sky, is a mechanical problem of extreme diffi- culty. However the mirror may be supported, all the unsupported points tend by their weight to sag away from the proper position. "Wben the mirror is pointed near the horizon, this effect of flexure is quite different from what it is when pointed near the zenith. As long as the mirror is small (not greater than eight to twelve inches in diameter), it is foimd easy to support it so that these variations in the strains of flexure have little practical effect. As we increase its diameter up to 48 or mmm 10 AaTRONOMT. 72 inches, the effect of flexure rapidly increases, and special devices have to he used to couuterhalaiice the injury done to the shape of the mirror. § 3. CHBONOMETEBS AND CLOCKS. In Chapter I., § 5, wo described how the right ascen. sions of the heavenly bodies are measured by the times of their transits over the meridian, this quantity increas- ing by a minute of arc in four seconds of time. In order to determine it with all required accuracy, it is necessary that the time-pieces with wliich it is measured shall go with the greatest possible precision. There is no great difficulty in making astronomical measures to a second of arc, and a star, by its diurnal motion, passes over this space in one fifteenth of a second of time. It is there- fore desirable that the astronomical clock shall not vary from a uniform rate more than a few hundredths of a second in the course of a day. It is not, however, necessary that it should be perfectly correct ; it may go too fast or too slow without detracting from its char- acter for accuracy, if the intervals of time which it tellfl off—hours, minutes, or seconds— are always of ex- actly the same length, or, iu other words, if it gains or loses exactly the same amount every hour and every day. The time-piecos used in astronomical observation are the chronometer and the clock. The chronmnMer is merely a very perfect time-piece with a balance-wheel so constructed that changes of tem- perature have the least possible effect upon the time of its oscillation. Such a balance is called a eom^pematum bal- ance. The ordinary house clock goes faster in cold than in warm weather, because the pendulum rod shortens under the influence of cold. This effect is such that the clocl will gain about one second a day for every fall of 3° Cent. {ft" A Fahr.) in the temperature, supposing the pendulum THE ASTRONOMICAL CLOCK. 11 increases, and iterbalanco the [jOCKB. :ho right ascen. d by the times uantity increas- time. In order •, it is necessary lasured shall go ere is no great ■£S to a second passes over this e. It is there- c shall not vary lundredths of a not, however, rect ; it may go from its char- time which it 3 always of ex- 3, if it gains or and every day. observation are rfect time-piece changes of tem- 1 the time of its vr^^ensaMon bal- in cold than in i shortens under li that the clocE rfaUof 3°0ent. g the pendulnm rod to be of iron. Such changes of rate would be entirely inadmissible in a clock used for iistronomical purposes. The astronomical 'Jock is therefore provided with a com- pensation pendulum, by which the disturbing effects of changes of temperature are avoided. There are two forms now in use, the Harrison (grid- iron) and the mercurial. In the gridiron pendulum the rod is composed in part of a number of parallel bars of steel and brass, so connected together that while the expansion of the steel bars produced by an increase of temperature tends to depress the hob of the pendulum, the greater expansion of the brass bars tends to raise it. When the total lengths of the steel and brass bars have been properly Jidjusted a nearly perfect compensation occurs, and the centre of oscillation remains, at a con- stant distance from the point of sus- pension. The rate of the clock, so far as it depends on the length of the pendulum, will therefore be constant. In the mercniial pendulum the weight which f onus the bob is a cylindric glass vessel nearly filled with mercury. With an increase of temperature the steel suspension rod lengthens, thus throwing the centre of osdllation away from the point of suspension ; at the same time the expanding mercury rises in the cylinder, and tends therefore to raise the centre of oscillation. When the lengdi of the rod and the dimensions of the cylinder of mercury are properly proportioned, the centre of osdllation is kept at a constant distance from the point of suspension. Other methods of making tiiis compensa- tion have been used, but these are the two in most gen- eral use for astronomical clot-.ks. Pig. 27.— oRroiRON ■vmmmmmmmmimiitmiiM Hi 78 ABTBONOMT. Ill The Mtreetion of a chronometer (or clock) is the quantity of time (expressed in hours, minutes, seconds, and decimals of a second) which it is necessary to add algebraically to the indication of the hands, in order that the sum may be the correct time. Thus, if at sidereal 0\ May 18, at New York, a sidereal clock or chronometer indicates 23'' 58"' 20* -7, itc correction is + 1»' 89'. 8 ; if af.O'' (siderwl noon), of May 17, its correction was + 1"' 88- -8, its daily rate or the change of its correction in a sidereal day is + 1*0: in other words, this clock is loring 1" daily. For clock Blow the sign of the eorreetion is + ; «' '' fast " " " " " '8 — ; " " gaining " " " " rate loting 18 — 5 is + . A clock or chronometer may be well compensated for temperature, and yet its rate may be gaining or losing on the time it is intended to keep : it is not even necessary that the rate should be small (ex- cept that a small rate is practically convenient), provided only that it IS constant. It is continually necessary to compute the clock cor- rection at a given tims from its known correction at some other time, and its known rate. If for some definite instant we denote the time as shown by the clock (technically "the clock-face") by 2', the true time by T and the clock correction by a T, we have T = T + A r, and i,T = r - T. In alt obserratories and at sea observations are made daily to de- termine A T. At the instant of the observation the time T is noted by the clock; from the data of the observation the time r is com- puted. If these agree, the clock is correct. If they differ, ATia found from the above equations. If by observation we have found A 7» = the clock correction at a clock-time 7», A 7* = the clock correction at a clock-time T, ST =: the clock rate in a unit of time, we have Ar= AT, + d2'(5P-7',) where T — T, must be expressed in days, hours, etc., according as dr is the rate in one day, one hour, etc. :,,.-,. . When, therefore, the clock correction A T. and rate ST have been determined for a certain instont, T., we can deduce the true time from the clock-face 2* at any other Instant by the equation r = T . AT* + dr(7'— !•)• " ^^ dock correction has been deter- mined at two different ttmes, T. and T to be A T. and A T, the rate is inferred from the equation 6T. AT- Ag> the quantity of time nmals of a second) :he indication of the t time. Thus, if at Dck or chronometer '•8; if aiO'' (sidereal , its daily rate or the *-0: in other words, ■ion is + ; is — ; I is — ; is + . ited for temperature, le time it is intended should be small (ex- ), provided only that »mpute the clock cor- n at some other time, it we denote the time ace") by 2\ the true have are made daily to de- >n the time T is noted a ths time T is com- If they differ, LTxs ;k-time T», ck-time 7, me, ITS, etc., according as md rate ^ 7 have been deduce the true time the equation 7* = 7 ction has been deter- , T» and A T, the rate THE ASTRONOMICAL CLOCK, 73 These equations apply only so long as we can regard the rate as comtnnt. As observations can bo made only in clear weather, it is plain that during periods of overcast sky wc must depend on these equations for our knowledge of 7" — i.e., the true time at a clock- time T. The intervals between the determination of the clock correction should be small, since even with the best clocks and chronometers too much dependence must not be placed upon the rate. The follow- ing example from Cbauvemet's Astronomy will illustrate the practi- cal processes : " Example. — At sidereal noon, May 5, the correction of a sidereal clock is— 16"' 47'0; at sidereal noon, May 12, it is — 16'" IS'-SO; what is the sidereal time on May 25, when the clock-face is 11" 13'" 12" -6, supposing the rate to be uniform ? May 5, correction = — IB"" 47'. 30 " 12 , " = -16"' 13' . 50 7 days' rate =r+ 83' "50 dT= + 4'.829. Taking then as our starting-point T^ = May 12, O**, we have for the interval to T= May 25, ll"- 13'« 12'-6, T- To = W^ W 13'" 12"e = 18''-467. Hence we have Ar.,= - 16»l'}«-60 dT(T— To)= + 1" fi'OS AT= - 15" 8'-47 T=n*' 18'»J2;^^60 7»= 10^ SS" 4'. 13 But in this example the rate is obtained for one true sidereal day, while the unit of the interval 18''-467 is a sidereal day as shovn by the clock. The proper interval with which to compute the n\te in this case is W 10^ 68" 4* 18= 18'' -457, with which we find AT»= — Id" IS'. 50 6Ty 18-457= + 1- 4' 98 A 7* = — IS" 8'. 52 T = 11'' 18"' 12' -60 7*- 10'' 68" 4* 08 This repetition wVl 'ot rendered unnece^^sary by always giving the rcte in a vntt of the ek>A. Thus, suppose that on June 8, at 4" 11*" 12'-86 by the clock, we have found the correctiori + 2*" 10* 14; and on June 4, at W ^7*" 49*. 89 we L.* .^ fo>jnd tba correction -i- 2"' 10<-89 ; the rate in cm iuiw of the eloek will be iiT = -^9••7S 84'11'M rr = t- 0'-2868." ■■M .U:, -^ 74 ASTRONOMY. I 4. THE TRANSIT INSTBUMENT. The meridian transit instrument, or briefly the " tran- sit," is used to observe the transits of the heavenly bodieg. Fig. 28.— a tiukbit ihstbiimbnt. and from the times of these transits as read from the clock to determine either the corrections of the clock or the right ascension of the observed body, as explained in Chapter I., §5. [TMENT. briefly the " traii- 3 heavenly bodies. BNT. 18 read from the nB of the clock or y, as explained in THE TnANSIT INaTRUMENT. tS It has two general forms, one (Fig. 28) for use in fixed observatories and one (Fig. 29) for nse in the fiekl It consists essentially of a telescope TT TFiir 28^ mounted on an axis F Fat right angle's to it ^ ^' ^ Pig. 29.-P011TABLE transit mSTRlWKNT. The ends of this axis terrainate in accurately cvlindrio^l Bteel pivots which re«t in metallic bearing FfTI.^ like the letter Y, and hence called the f, ' *^ iWi^aB Bit j w i iff iw Maw '' re AaTltONOMT. These are fastened to two pillars of stone, l)rick, or iron. Two counterpoises W W are connected with the axis as in the plutc, so as to take a largo portion of the weight of the axis and telescope from the Ys, and thus to diniinish the friction npon these and to render the rota- tion about V V more eaay and regular. In the ordinary use of the transit, the line F F is placed accurately level and perpendicular to the meridian, or in the east and west line. To effect this *' adjustment," there are two sets of adjusting screws, by which the ends of F F in the Ys may be moved either up and down or north and south. The plate gives the form of transit used in permanent observa- tories, and shows the observing chair G^ the reversing car- riage R, and the level L. Tl arms of the latter have Y'b, which can be placed over the pivots F F. The line of coUiination of the transit telescope is the line drawn through the centre of the objective perpendic- ular to the rotation axis V V. The reticle is a network of fine spider lines placed in the focus of the objective. In Fig. 30 the circle represents the field of view of a transit as seen through the eye-piece. The seven ver- tical Unes, I, II, III, IV, V, VI, VII, are seven fine spider lines tightly stretched acroes a metal plate or diaphragm, and so adjusted as to be perpendicular to the direction of a star's apparent diurnal motion. This metal plate can be moved right and left by five screws. Tb' hori- zontal wires, guide-wires, a and h, mark the centre of the field. The field iii Illuminated at night by a lamp at the end of the axis which shinep through the hollow interior of the lat- ter, and causes the field to appear bright. The wires are dark against a bright ground. The line of sight is a line joining the centre of the objective and the central one, IV, of the seven vertical wires. &^ TUH TRANSIT INSTUUMKNT. 77 me, brick, or ictcd with tlio )ortiou of the 8, and thus to inder the rota- ;n the ordinary iccurately level e east and west are two sets of In the Ys may id south. The lanent observa- s reversing car- the latter have VV. elescope is the jtive perpendic- lines placed in Id of view of a The seven ver- II, IV, V, yi, no spider lines 08B a metal plate 10 adjusted as to the direction of diurnal motion. 1 be moved right «wfc. Tb' hori- i-vn/reB, a and b, the field. The , the end of the iterior of the lat- . The wires are of »ight is a line B central one, IV, The whole transit is in adjustment when, first, the axis V V is horizontal ; second, when it lies east and west ; and third, when the line of sight and the line of collinia- tion coincide. When these conditions are fulfilled the line of sight intersects the celestial sphere in the meridian of the place, and when T T\9, rotated about V V the line of sight marks out the meridian on the sphere. In practico the three adjustments are not exactly made, since it is impossible to effect them with mathematical precision. The errors of each of them are first made as small as is convenient, and are then determined and allowed for. To find the error of level, we place on the pivots a fine level (shown in position in the figure of the portable transit), and determine how much higher one pivot is than the other in terms of the divisions marked on the level tube. Such a level is shown in Fig. 4 of plate 85, page 86. The value of one of these divisions in seconds of arc can be determined by knowing the length I of the whole level and the number n of divisions through which the bubble will run when one end is raised one hundredth of an inch. If I is the length of the level in inches or the radius of the circle in which either end of the level moves when it is raised, then as the radius of any circle is equal to 57° • 296, 3437' • 75 or 206,264" • 8, we have in thui particular circle one inch = 206, 264" -8 -s- I; 0-01 inch = 2(0^264 -8 -4- 100 Z = a certain arc in seconds, say a". That is, n divisions = a", or one division d = a" -i- n. The error of eoUimation can be found by pointing the telescope at a distant mark whose image is brought to the middle wire. The telescope (with the axis) is then lifted bodily from the Ys and re- placed so that the axis V Fis reversed end for end. The telescope is again pointed to the distant mark. If this is still on the middle thread the line of sight and the line of eoUimation coincide. If not, the reticle must be moved bodily west or east until these conditions are fultiUed after repeated reversals. To find the error of mimuth or the departure of the direction of VV from an east and west line, we must observe the transits of two btars of different declinations d and die thread to be and 0'. If — 6' = « — «', »hea the mid4lc wii» is in the meridian and the azimuth is zero. For if the nziinvitli was not zero, but the west end of the axis w«us tou far south, for example, the line of sight would fall eant <«l the meridian for a south stifkr, and further and further cast tK ftirthcH !«>wth the star was. Hence if the two stars have widel> tliff(ro»t detlinationa 6 and <5', then the star furthest south would lom* ]>ioportion»toly sooner to the middle wire than the otlK''t :««Ki U — 0' wowkl be different from a — u'. The amount of irM» diSereBC« give!> a mm MMMMI 78 A8TR0N0MT. means of deducing tho deviation oi A A from an east and west tine. In a similar way the effect of a given error of level on the time of the transit of a star of declination 6 is found. Methods of Obaerving with the Transit Instrument.— We ]i.)ve »o far asHUiiicd tliat the time of a star's transit over the middle tliread was known, or could be noted. It is neccHsary to speak more in detail of how it is noted. When tho telescope is pointed to any star the earth's diurnal motion will carry the image of the star slowly across the field of view of the telescope (which is kept fixed), as before explained. As it crosses each of the threads, the time at which it is exactly on the thread is noted from the clock, which must be near the transit. The mean of these times gives the time at which this star was on the middle thread, the threads being at equal intervals ; or on the " mean thread," if, as is the case in practice, they are at unequal intervals. if it were possible for an astronomer to note the exact instant of the transit of a star over a thread, it is plain that one thread would be sufficient ; but, as all estima- tions of this time are, from the very natifre of the case, but approximations, several threads are inserted in order that the accidental errors of estimations may be eliminated as far as possible. Five, or at most seven, threads are sufficient for this purpose. In the figure of the reticle of a transit instru- ment the star (the plimet Vemta in this ciise) may enter on the right hand in the figure, and may be supposed to cross each of the wires, the time of its tran- sit over each of them, or over a suffi- cient number, being noted. The method of noting this time may be best understood by referring to the next figure. Suppose that the line in the middle of Fig. 32 is one of the transit- threads, and that the star is passing from the right hand of the figure toward the left ; if it in on this wire at an Pio. 81. THE TRANSIT HfSTRVMENT. 79 %n east and west )r of level on the d. Instrument.— a star's transit >uld be noted, ow it is noted, tar the earth's le star slowly which is kept 38 each of the the thread is the transit. at which this being at equal 18 is the case in note the ^Must ■ead, it is plain r, as all estima- te of the case, iserted in order \y be eliminated en, threads are rpose. In the a transit instra- et Ventw in this ■ight hand in the pposed to cross time of its tran- or over a suffi- noted. The ime may be best . Suppose that of the transit- the right hand this wire at an Fie. 82. exact second by the clock (which is always near the ob- server, beating seconds audibly), this second must be writ- ten down as the time of the transit over this thread. As a rule, however, the transit cannot occur on the exact beat of the clock, but at the seventeenth second (for exam- ple) the star may be on the right of the wire, say at a ; while at the eighteenth second it will have passed this wire and may be at h. If the distance of a from the wire is six tenths of the distance a 5, then the time of transit is to be recorded as — hours — minutes (to be taken from the clock-face), and seven- teen and ^x tenths seconds ; and in this way the transit over each wire is observed. This is the method of " eye- and-ear" observation, the basis of such work as we have described, and it is so called from the part which both the eye and the ear play in the appreciation of intervals of time. The ear catches the beat of the clock, the eye fixes the place of the sti r at me hour, min- the sheet at its the spaces on momy depends ure of an angle, istomary to em- circles.. These ^d as possible, arrow flat band ine radial lines ling engine" at terTftIs may be 5 of the circle le narrow band > said to be di- Fio. 88. vided to 10', r»', y'. The separate diviBJons are numbered consecutively from 0" to 30(>^ or from 0" to 1)0°, etc. The graduated circle has an axiH at itH centre, and to this may be attached the telescope by whicli to view tlie pointti whose angiilar distance is to be dcteriuiued. To this centre is also attached an arm wliicli revolves with it, and by its motion past a certain nuinbur of divi- sions on the circle, determines the angle through which the centre has been rotated. This arm is called the index arm, and it usually carries a vernier on its extremity, by means of which the spaces on the graduated circle are subdivided. The reaijimj of the circle when the index a ' in any position is the number 'agrees, minutes, and seconds w cH correspond to that po- sition ; when the index arm is in an- other position there is a different reading, and the differences of the two readings S' — al ti( rij si: ni te THE EQUATOJtIAL. 87 direction, and observing succossively the stars wliich ^ass through its field of view. It is by this rapid method of observing that the largest catalogues of stars have been formed. § 7. THE EQUATORIAL. To complete the enumeration and description of the principal instruments of jistronomy, we require an account of the eqtiatorial. This terra, properly speaking, refers to a form of mounting, but it is commonly used to in- clude both mounting and telescope. In this class of instruments the object to be attained is in general the easy finding and following of any celestial object whose apparent place in the heavens is known by its right as- cension and declination. The equatorial mounting con- sists essentially of a pair of aaes at right angles to each other. One of these S N (the jpolar oasis) is directed to- ward the elevated pole of the heavens, and it therefore makes an angle with the horizon equal to the latitude of the place (p. 21). This axis can be turned about its own axial line. On one extremity it carries another axis Z D (the declination ha screw. To measure the distance apart, of two objects A and B, wire 1 (the fixed wire) is placed on A and wire 2 (movable by the screw) is placed on B. The number of revolutions and parts of a revolution of the screw is noted, say 10' -267 ; then wires 1 and 2 are placed in coincidence, and this zero-reading noted, say 5' -143. The dis- tance A B is equal to 5'- 124. Placing wires 1 and 2 a known num- ber of revolutions apart, we may observe the transits of a star in the equator over them ; and from the interval of time required for this star to move over say fifty revolutions, the value of one revolution 90 AaTRONOMT. is known, and can alwavs bo used to turn distances measured in revolutions to distances in time or arc. By the filar micrometer we can determine the distance Hput in seconds of arc of any two stars A and B. To completely nx the relative position of A and B, wo require not only this distance, but also the angle which the line A B mulces with some fixed direction in space. We assume as the fixed direction that of the meridian passing through A, Suppose in Fig. 88 A and £ to be two stars visible in the field of the equatorial. The clock-work is detached, and by the diurnal motion of the earth the two stars will cross the field slowly in the direction of the parallel of declination passing through A, or in the direction of the arrow in the figure from E. to W., east to west. The filar micrometer is con- structed so that it can be rotated bodily about the axis of the tele- scope, and a graduated circle measures the amount of this rotation. The micrometer is then rotated until the star A will pass along one of its wires. This wire marks the direction of the parallel. The wire perpendicular to this is then in the meridian of the star. The pontion angle of B with respect to ^1 is theh the angle which A B makes with the meridian A N passing through A toward the north. It is zero when B is north of A, 90* when B is east, IHO when B is south, and 270° when B is west of A. Knowing p, the position angle (NAB in the figure), and i (A B) the distance of B, we can findthe difference of right ascension (A a), and the differ- ence of declination (hi) ot B from A by the formulte, Aa = < sin |>; A6=s$ cotp. Conversely knowing Aa and Ad, we can deduce « and p from these formulae. The angle p is measured while the clock-work keeps the star A in the centre of the field. § 8. THX ZnnTH TBLB800PE. The accompanying figure givn a view of the zenith telescope in the form in wuich it is used by the United States Coast Survey. It consists of a vertical pillar which supports two T$. In these rests the horizontal axis of the instrument which carries the tele- scope at one end, and a counterpoise at the other. The whole in- strument can revolve 180° in azimuth about this pillar. The tele- scope has a micrometer at its eye-end, and it also carries a divided circle provided with a fine level. A second level is provided, whose use is to make the rotation axis horizontal. The peculiar features of the zenith telescope are the divided circle and its at- tached level. The level is, as shown in the cut, in the plane of motion of the telescope (usually the plane of the meridian), and it can be independently rotated on the axis of the divided circle, and set by means of it to any angle with the optical axis of the telescope. The circle is divided from zero (0°) at its lowest point to 90° in each direction, and is firmly attached to the telescope tube, and moves with it. By setting the vernier or index-arm of the circle to any degree and minute as a, and clamping it there (the level moving with it). THE ZKNirU TKLEtiUOPK. n d in rt in the , but ction idian two work two lei of 9W in I con- I tele- ttion. ilong rallel. }tar. which d the , 180 p, the of B, differ- > from L-work Qope in rey. It 1 these le tele- liole in- he tele- divided ovided, peculiar [its at- tlane of I, and it cle, and lescope. ;o 90° in ibe, and y degree with it), -rtiiiijiljli PlO. 89.— THE ZBNITH TELBBCOPE. Of AsTn(tm)MY. niul then rntatiiiK thn tclcscono and tho whnlo NyMtuin nlioiit tliu horizontal axis until tlie bub1>lc of the level ix in tho contro of tlu; lovcl-tubo, tho axiH of the tolcHcopcH will bo directed to the zenith diHtance a. The filar micromotor \* ho adjusted that a motion of itit Hcrow moMurcB differences of zenith distance. Tho uhc of tho ze- nith telescope is for determining tho latitude by Talcott'h method. The theory of this operation has been already given on irngo 48. A description of tho actual process of observation will illustrate the excellences of this method. Two stars, A and B, are selected beforehand (from Star Cata- logues), which culminate, A south of the zenith of tho nluce of ob- servation, B north of it. They are chosen ot nearly eijual zenith dis- tances f* and £*, and so that $* — {* is less than tho breadth of tho field of view. Their right ascensions are also chosen so as to bo alwut the same. The circle is then set to the mean zenith distance of the two stars, and the telescope is pointed so that the bubble is nearly in the middle of the level. Suppose the right ascension of A is the smaller, it will then culminate first. The telescope is then turned to the south. As A passes near the centre of the field its distance from the centre is measured by the micromotor. Tho level and micrometer are read, the whole instrument is revolved 180", and star B is observ«}d in the same way. By these operations wo have determined the difference of tho zenith distances of two stars whoso declinations d* and <)■ uro known. But tp being the latitude, ^ = (J* -f. 4* and ^ = d" — {", whence ^ = !(<)* + ')•) + 1 ({* - «"). The first term of this is known ; tho second is measured ; so that each pair of stars so observed gives a value of tho latitude which depends on the measure of n very small arc with the micrometer, and UN this arc can be measured with great precision, the exactness of the determination of the latitude is equally great. hnruim plane < This the in( which E is a Hv it a silverec plane c g 8. THE SEXTAnr. Tho sextant is a portable instrument by which tho altitude!^ of celestial bodies or tho angular distances between them mny bo measured. It is used chiefly by navigators for determining the latitude and the local time of the position uf tho ship. Knowing the local time, and com]}aring it with a chronometer regulated on Greenwich time, the longitude becomes known and the snip's place is fixed. It consists of the arc of a divided circle urually 00° in extent, whence the name. This arc is in fact divided into 120 equal parts, each marked as a degree, and these are again divided into smaller spaces, so that by means of the vernier at the end of tho index-arm M San arc of 10" (usually) may be read. The index-arm M 8 carries the ind«e-ghu» M, which is a silvered plane mirror set perpendicular to the plane of the divided arc. The (and t\ second to be re telosco{ the sex one dir Tho which thelaat TIIK SRXTANi: W hnrixim-iihiM m is nlt<(» n pliino mirror flxp«l |M!rpcnilttr to tint i»lun<> oif tlio «livi(lf(l <-inU'. TliiH liiHt kIiihh Im 11x1(1 in poHitixn, wliilii (lie llrnt rcvolvcH with the index-unn. Tlu! horizon-gliws in divided into two piirtH, «»f wJiich the lower one is Bilvered, the vippcr Imlf beinjr tran»i»«rent. E iH II tcleBCope of low power i>ointu(l toward the horizon-gluiw. Hy it any object to which it Ih directed can Iw seen through tlic un- Bilvcrcd half of the horizon-glawH. Any other object in the f«anio pUnu can be brought into the same field by rotating the indpx-arm FfO. 40. — THB BKXTAHT. (and the index-glass with it), so that a beam of light from this second object shall strike the index-glass at the proper angle, there to be reflected to the horizon-glass, and again reflected down the telescope E. Thus the images of any two objects in the plane of the sextant may be brought together in. the telescope by viewing one directly, and the other by reflection. The principle upon which the sextant depends is the following, which IS proved in optical works. The artgle between theftnt and the latt directum of a ray which hat suffered two rejUetiont in the tame Vv ASTRONOMY. \ plans M equal to tt^^s the angle whkh tTu two reflecting mrfaeea make with each other. . . , , , a *».;„ ,o„ Sa i.tr In the figure S A is the ray incident upon -4, and this ray is by reflection brought to the direction BE The theorem declares that the angle BE Sis equal to twice D C B, or tvice the angle of theiairrors, since BO mAD Care perpendicular to Band ^. To measure the altitude of a star (or the sun) at sea, the sextant is held in the hand, and the telescope is pointed to the sea-honzon, which appears like a definite line. The index-arm is then moved until the reflected image of the sun or of the star coincidcB with the Fie. 43.— ABTIVTCIAL HOBOOK. imaee of the sea-horizon seen directly. When this occurs the tune isto be D ied from a chronometer. If a star is observed, the reaa- injr of the divided limb gives the altitude directly; if it is the sun or moon which has been observed, the lower limb of these is brought to coincide with the horizon, and the altitude of the centre is found Almanac The an ured by j tant abou vided arc the indes star's imi On shn tho obsei hffrigon, ' liquid, a surface if a A, fror in the di ing E A to an eye With a » angle 8 and if A all celest will equi half the i i mahe y isby eclarea ngle of A. To : is held , which id until nth the THE 8KXTANT. 96 is found by applying the semi-diameter as found Jn the Nautical Almanac to the observed altitude ol the limb. The angular distance apart of a star and the moon can be meas- ured by pointing the telescope at the star, revolving the whole sex- tant about the sight-line of the telescope until the plane of the di- vided arc passes through both star and moon, and then by moving the index-arm until the reflected moon is just in contact with the star's image seen directly. On shore the horizon is broken up by buildings, trees, etc., and tho observer is therefore obliged to have recourse to an artificial harum, which consists usually of the reflecting surface of some liquid, as mercury, contained in a small vessel A, whose upper surface is necessarily parallel to the horizon DAG. A ray of light 8 A, from a star at 8, incident on the mercury at A, will be reflected in the direction A E, making the angle 8AG= A 8' (A 8^ be- ing E A produced), and the reflected image of the star will appear to an eye at £ as far below the horizon as the real star is above it. With a sextant whose index and horizon-masses are at /and H, the angle 8 E 8 may be measured ; but aES = 8AS — A8E, ana it A E'vi exceedingly small as compared with ^ i8, as it is for all celestial bodies, the angle A 8 Emaj be weglected, and 8 B 8' will equal 8 A 8', or double the altitude of the object : hence one half the reading of the instrument will give the apparent altitude. the time the read- it is the these is he centre \ii Hi '■'I §1. CHAPTER III. MOTION OF THE EARTH. ANCIENT IDEAS OF THE PLANETS. It was obBerved by the ancients that while the great mass of the stars maintained their positions relatively to Lh other not only during each diurnal revolution, but ^nth after month and year after year, the«, were vi«. bleto them seven l^eavenly bodi^ which ch^gedth^r positions relatively to the starB and to «««'5^^f «^. J^,^ Siey called planets or wandenng stars. Still calbng the apmi^t crystalline vault in which the sters seem to ^^ the celestial sphere, and imagining it as at rest, ^wt found that the seven planets performed a y^ slow revolution around the sphere from west to e.«t L periods ranging from one month in the case of the mooTto thirtyVars in that of m^n. 1* w- eviden that these bodies could not be «o"«^«'«'l. ^^ ^* ^ not same solid sphere with the stars, because tW could^no then change their positions among the stars. Vanous w^s of acfounting for their motions were therefore pro- xJed One of the earliest conceptions is associated with rnameofPvTHAOOKAS. He is said to have taugM t^t each of the seven planets had its ^^/P^^^^^^^^t^j concentric with that of the fixed stars, and that these len hoUow spheres each performed its own revolution, Se^ndently of theothers. Thisideaof anumber of con- 3c solid^heres was, however, apparently given up without argumci close ex tent wit being » perfect i by the The latl move so it was ( nearer \ were en fixed in use — th( space or These lowed, \ rightly < stars. ] most slo distance case of J We n the eart! scope ha themseh ably grei surface I pared wi stars. 1 tem, it if its sever them thi following to be eij in the o bodies a STB. the great atively to ution, but were visi- iged their r. These ailing the I seem to 18 at rest, id a very t to east, ase of the ras evident set in the could not . Various •efore pro- ciated with taught that iside of and that these revolution, iber of con- f given up THE SOLAR SYSTEM. W without any one having taken tlie trouble to refute it by argument. Although at first sight plausible enough, a close examination would show it to be entirely inconsis- tent with the observed facts. The idea of the fixed stars being set in a solid sphere was, indeed, in seemingly perfect accord with their diurnal revolution as observed by the naked eye. But it was not so with the planets. The latter, after continued observation, were found to move sometimes backward and sometimes forward ; and it was quite evident that at certain periods they were nearer the earth than at other periods. These motions were entirely inconsistent with the theory that they were fixed in solid spheres. Still the old language continued in use — the word sphere meaning, not a soUd body, but the space or region within which the planet moved. These several conceptions, as well as those which fol- lowed, were all steps toward the tnith. The planets were rightly considered as bodies nearer to us than the fixed stars. It was also rightly judged that those which moved most slowly were the most distant, and thus their order of distance from the earth was correctly given, except in the case of Mercury and Venus. We now know that these seven planets, together with the earth, and a number of other bodies which the tele- scope has made known to us, form a family or system by themselves, the dimensions of which, although inconceiv- ably greater than any which we have to deai with at the surface of the earth, are quite insignificant when com- pared with the distance which separates us from the fixed stars. The sun being the great central body of this sys- tem, it is called the Solar System. It is to the motions of its several bodies and the consequences which flow from them that the a oention of the reader is directed in the following chapters. We premise that there are now known to be eight lai^ planets, of which the earth is the third in the order of distance from the sun, and that these bodies all perform a regular revolution around the son. ■ 98 ASTRONOMY. Mercnry, the nearest, performs its revolution in three montlis ; Neptune, the farthest, in 164 yea". First n importance to us, among the heavenly boU es which we see from the earth, stands the sun, the supporter rS and motion upon theearth. At fi«t«ghUUm^ seem curious that the sun and seemmg stars like Ma/rs and^a Cm should have been classified together as plajete bv the ancients, while the fixed stars were considered as forming anoth;r class. That the ancients were acute Z^f to do this tends to impr^ m wHh a favorable sense of the scientific character of their mteUect To any but the most careful theorists and observers, the star-like pknete if we may call them so, would never have seemed rSng in the Ime class with the sun but rather m hat of tie stars ; especially when it ^^^^^^^^ '^' '^^ were never visible at the same time with the sun. iJut Srthe times of which we liave any histenc r^rd there were men who saw that, in a motion from west to rramong the fixed stars, these several ^^^ ^^^^J common character, which was more ««««^*^^.2;^ ^^^^ of the universe than were their immense diforences of aspect and lustre, striking tl^o^g^^.J^'fl^^-^ ,„_ It must, however, be remembered ^^^^J^^^^ consider the sun as a planet. We have /no^^^f *^« "^ dent system by making the sun and the earth /jhaage llrso that the latterl now regarded as one of theei^t wTiknets, while the former has taken the place of the e^Kfientral body of the system In consequence oUhe revolntion of the planets romid Jbe "an «ach of them seems to perform a corresponding circuit m the htvenriund Se celestial sphere, when viewed from any other phmet or from the earth. § 2. AMlfUAL EBVOLTJTIOM OF THE BABTH. To an observer on the earth, the sun seems to pe^o^f Jua^volution among the stars a fact v.hich has b^n Wn from the earUest ages. We now know that this is due sun. tion oi directe it and which In ] of the fixed tent, 1 AB numb 15 dn called exten the p MOTION OF TBE BARTB. 99 is due to the annual revolution of the earth round the Bun. It is to the nature and eflfects of this annual revolu- tion of the earth that the attention of the reader is now directed. Our first lesson is to show the relations between it and the corresponding apparent revolution of the sun, which is its counterpart. In Fig. 43, let S represent the sun, ABC D the orbit of the earth around it, and EFQIl tlie sphere of the Fia. 43.— BRVOLCTioN or thb earth. fixed stars. This sphere, being supposed infinitely dis* tent, must be considered as infinitely larger than the circle A B G D. Suppose now that 1, 2, 3, 4, 5, 6 are a number of consecutive positions of the earth. The line \S drawn from the sun to the earth in the first position is called the radius vector of the earth. Suppose this line extended infinitely so as to meet the celestial sphere in the point V. It is evident that to an observer on the tofO. jQO ASTRONOMY. y ;^Tr«:'aLtI »« .Sana so„„. in other Will ^PP"" , rnvolves around the sun, the latter -''itr=rre^;r.:'r>rt-,. .o.a described. „„„„„i mvolntion of the Let us now study the apparent ^^^'^'^Xe i^ult of «„n produced in the way just mentioned. One result TUE aUN'B APPARENT PATH. 101 >liere 2, it other latter Btars, inrould xactly i from dly in t that ng the irately rse de- ited by iiity in eat cir- pear to ndiffer- iptic is the po- eferred. letry, it a think- ceive of ical line perpen- Rgure is iects the c. This Ets an ex- j, owing hereafter on of the result of this motion is probably familiar to every reader, in the different constellations whicli are seen at different times of the year. Let lis take, for example, the bright star Aide- baran, wliicli, on a winter evening, we may see north- west of Orion. Near the end of February this star crosses the meridian about six o'clock in the evening, and sets about midnight. If we watch it night after night through the months of March and April, we shall find that it is far- ther and farther toward the west on each successive even- ing at the same hour. By the end of April we sliall bare- ly be able to see it about the close of the evening twilight. At the end of May it will be so close to the sun as to be entirely invisible. This showa. that during the months we have been watching it, the sun has been approaching the star from the west. If in July we watch the eastern horizon in the early morning, we shall see this star rising before the sun. The sun lias therefore passed by the star, and is now east of it. At the end of November we will find it rising at sunset and setting at ennrise. The sun is therefore directly, opposite the star. During the winter months it approaches it again from the west, and passes it about the end of May, as before. Any other star south of the zenith shows a similar change, since the relative positions of the stars do not vary. § 3. THE SUV'S AFPASEirF PATH. It is evident that if the apparent path of the sun lay in the equator, it would, during the entire year, rise exactly in the east and set in the west, and would always cross the meridian at the same altitude. The days would always be twelve hours long, for the same reason that a star in the equator is always twelve hours above the hori- zon and twelve hours below it. But we know that this is not the case, the sun being sometimes north of the equator and sometimes south of it, and therefore having a motion in declination. To understand this motion. XOa ASriiONOMY. 8unix«e that on March 19th, 1879, the Bun had been observed with a meridian circle and a Biderca,! clock at the moment of transit over the meridian of Wa«hnigton. Its position would have been found to bo this : Eight Ascension, 23" 55™ 23' ; Declination, 0" 30' south. Had the observation been repeated on the 20th and following days, the results would have been : March 20, R. Ascen. 23" 59™ 2'; Dec. 0° 6' South. 2 J u 0" 2™ 40"; " 0° 17' North. 22' " 0" C™ 19* ; " 0° 41' North. Fio. 44.— THB BOH CROfltniO THB BQUATOB. If we lay these positions down on a chart, we shall find them to be as in Fig. 44, the centre of the sun being south of the equator in the first two positions, and north of it in the last two. Joining the successive positions by a line, we shall have a small portion of the apparent path of the sun on the celestial sphere, or, in other words, a small part of the ecliptic. ^v. * *i. It is clear from the observations and the figure that the sun crossed the equator between six and seven o'clock on the afternoon of March 20th, and therefore that the equa- tor and ecliptic intersect at the point where the sun was at that hour. This point is called the verrud e^mnox, the TUB SUN'S APPAUKNT PATH. IW been it the Itg louth. 1 and ith. rth. •rth. ai find being i north ions by nt path ^ords, a ;hat the lock on le eqna- i was at UKC, the first word indicating the eeason, cxpreflscs the equality of the nights and days which occurs when the sun is on the equator. It will be remembered that this equinox is the point from wliich right ascensions are counted in the heavens in the same way that longitudes on the earth are counted from Greenwich or Washington. The sidereal clock is therefore so set that the hands shall read hours minutes seconds at the moment when the vernal equi- nox crosses the meridian. Continuing our observations of the sun's apparent course for fe c:v m/inflia fiTtm Mftnth 20th ^ while the second six months from March 20th till September 23d, we should find it to be as in llg. 45. It will be seen that Fig. 44 cor- responds to the right-hand end of 45, but is on a much larger scale. The sun, moving along the great circle of the ecliptic, will reach its greatest northern declination about June 2l8t. This point is indicated on the figure as 90° from the vernal equinox, and is called the sum- iner solstice. The sun's right ascension is then six hours, and its declination 23i° north. The course of the sun now inclines toward the south, and it again crosses the equator about September Sad at 104 ASTRONOMY. a point diametrically opposito the vernal equinox. In virtue of the theorem of spherical trigonometry that all great circles intersect each other in two opposite points, the ecliptic and equator intersect at the two opposite equi- noxes. The equinox which the sun crosses on September 22d is called the autumnal equinox. During the six months from Septemher to March the sun's course is a counterpart of that from March to Sep- tember, except that it hes south of the equator. It at- tains its greatest south declination about December 22d, in right ascension 18 hours, and south declination 234°. This point is called the winter soUtice. It then begins to incline its course toward the north, reaching the vernal equinox again on March 20th, 1880. The two equinoxes and the two solstices may be re- garded as the four cardinal points of the sun's apparent annual circuit around the heavens. Its passage through these points is determined by measuring its altitude or declination from day to day with a meridian circle. Since in our latitude greater altitudes correspond to greater declinations, it follows that the summer solstice occurs on the day when the altitude of the sun is greatest, and the winter solstice on that when it is least. The mean of these altitudes is that of the equator, and may therefore be found by subtracting the latitude of the place from 90°. The time when the sun reaches this altitude going north marks the vernal equinox, and that when it reaches it going south marks the autumnal equinox. These passages of the sun through the cardinal points have been the subjects of asti-onomical observation from the earliest ages on account of their relations to the change of the seasons. An ingenious method of finding the time when the sun reached the equinoxes was used by the as- tronomers of Alexandria about the beginning of our era. In the great Alexandrian Museum, a large ring or wheel was set up parallel to the plane of the equator— in other words, it was so fixed that a star at the pole would shine 1 '_^X^ THE ZODIAC. 106 »x. In that all points, te equi- ptumber ircli the to Sep- , It at- )6r 22d, on 234°. )eginB to e vernal y be re- apparent through itude or I. Since I greater xscnrs on and the mean of therefore ace from ide going it reaches al points tion from le change ; the time by the as- f our era. or wheel —in other )uld shine 1 perpendicularly on the wheel. Evidently its plane if extended must have passed through the cast and west points of the horizon, while its inclination to the vertical was equal to the latitude of the place, which was not far from 30°. When the sun reached the equator going north or south, and shone upon this wheel, its lower edge would be exactly covered by the shadow of the upper edge ; whereas in any other position the sun would shine upon the lower inner edge. Thus the time at which the sun reached the equinox could be determined, at least to a fraction of a day. By the more exact methods of modem times, it can be determined within less than a minute. It will bo seen that this method of determining the an- nual apparent course of the sun by its declination or alti- tude is entirely independent of its relation to the fixed stars ; and it could be equally well applied if no stars were ever visible. There are, therefore, two entirely dis- tinct ways of finding when the sun or the earth has com- pleted its apparent circuit around the celestial sphere ; the one by the transit instrument and sidereal clocV, which show when the sun returns to the same position among the stars, the other by the measurement of altitude, which shows when it returns to the same equinox. By the for- mer method, already described, we conclude that it has completed an annual circuit when it returns to the same star ; by the latter when it returns to the same equinox. These two methods will give slightly different results for the length of the year, for a reason to bo hereafter described. • The Zodiac and its Diviaioiia. — The zodiac is a belt in the heavens, commonly considered as extending some 8° on each side of the ecliptic, and therefore about 16° wide. The planets known to the ancients are always seen within this belt. At a very early age the zodiac was mapped out into twelve signs known as the signs of the zodiac^ the names p£ which have been handed down to the present time. Each of these signs was supposed to be the seat of loe AsmoNimr. ■«»»> a conftoUation after whicli it wa« calUui Oommcncmg it the vorruvl ciuinox, tho tt«t thirty dogrc«8 through whchtroHun i,La,orth« region a.no..g the «tar8 m whic it wa8 ou.ul during tlie mH( suit the mui will tim abo sevi the BCV on tw^t 1 son tor isp! is s cell In vei hal isp to soi fai en It lat is, be BU in Tl o\ 1 onuqvirr of riiK Kvuprra. 107 incing rough aro in , waH callod I their i'b eu- •nstella- »nBtella- gns and the Bun I on the m Aries preces- 8 at the equator, li. This kU angle IB called the obliquity of (lie ocliptit , iirid its dotonnination ia very siinpk'. It is onlv necesBary to find by repeated oiwervation tin tiun's greato«t north declination at the eutnnier Bolstice, and its greatest south declination at the winter aolstice. Either of these decliimtions, which must bo equal if the olworvations are accurately made, will give the obliquity of the ecliptic. It has iMjen con- tinually diminishing from the earliest ages at a rate of about half a Becond a year, or, more exactly, about forty- seven seconds in a century. This diminution is due to the gravitating forces of the planets, and will continue for several thousand yearn to come. It will not, however, go on indefinitely, but the obliquity will only oscillate be- tween comparatively narrow limits. The relation of the obliquity of the ecliptic to the Bea- Bons is quite obvious. When the sun is north of the equa- tor, it culminates at a higher altitude in the northern hem- isphere, and more than half of ita apparent diurnal course is above the horizon, as explained in the chapter on the celestial sphere. Hepce we have the heats of summer. In the southern hemisphere, of course, the case is re- veraeu : when the sun is in north declination, less than half of his diurnal course is above the horizon in that hem- isphere. Therefore this situation of the sun corresponds to summer in the northern hemisphere, and winter in the southern one. In exactly the same way, when the sun is far south of the equator, the days are shorter in the north- em hemisphere and longer in the southern hemisphere. It is therefore winter in thft former and summer in the latter. If the equator and the ecliptic coincided— that is, if the sun moved along the equator— there would be no such thing as a difference of seasons, because the sun would always rise exactly in the east and set exactly in the west, and always culminate at the same altitude. The days would always be twelve hours long the world over. This is the case with the planet Jupiter. In the preceding paragraphs, we have explained the 108 ASTRONOMY. apparent annual circuit of the sun relative to the equator, and shown how the seasons depend upon this circuit. In order that the student may clearly grasp the entire subject, it is necessary to show the relation of these apparent move- ments to the actual movement of the earth around the sun. To understand the relation of the equator to the eclip- tic, we must remember that the celestial pole and the celestial equator have really no reference whatever to the heavens, but depend solely on the direction of the earth s axis of rotation. The pole of the heavens is nothing more than that point of the celestial sphere toward which the earth's axis points. If the direction of this axis changes, the position of the celestial pole among the stars will change also ; though to an observer on the earth, unconscious of the change, it would seem a& if the starry sphere moved while the pole remained at rest. Again, the celestial equator being merely the great circle in which the pkne of the earth's equator, extended out to infimty in every direction, cuts the celestial sphere, any change in the direction of the pole of the earth necessarily changes the position of the equator among the stars. Now the positions of the celestial pole and the celestial equator among the stars seem to remain unchanged throughout the year. (There is, indeed, a minute change, but it does not affect our present reasoning.) This shows th•^t, as the earth revolves around the sun, its axis is constantly directed toward nearly the same pohit of the celestial sphere. § 5. THE 8EA80IV8. The conclusions to which we are thus led respecting the real revolution of the earth are shown in Fig. 46. Here S represents the sun, with the orbit of the earth surrounding it, but viewed nearly edgeways so as to be much foreshortened. ABGD are the four cardina positions of the earth which correspond to the cardinal poll In< nor san it i Ag the inc ] sur noi dai son an| wi the thi ilh m( gl« pe tk ator, In ject, love- l the jclip- the the irth'a thing vhich > axis 1 stars sarth, starry a, the ihthe ity in ige in langes w the juator ighout t does •\t, as itantly ilestial tecting ig. 46. i earth I to be ardinal ordinal THE SEASONS. 109 points of the apparent path of the sun ah*eady described. In each figure of the earth J/'S is the axis, iT being its north and S its south pole. Since this axis points in the FlO. 46.— CAV8B8 OF THK 8BA80NB. same direction relative to the stars during an entire' year, it follows that the different lines N S Are all parallel. Again, since the equator does not coincide with the ecliptic, these lines are not perpendicular to the ecliptic, but are inclined from this perpendicular by 23i°. Now, consider the earth as at ^ ; here it is seen that the sun shines more on the southern hemisphere than on the northern ; a region of 23^° around the north pole is in darkness, while in the corresponding region around the south pole the sun shines all day. The five circles at right angles to the earth's axis are the parallels of latitude around wMch each region on the surface of the earth is carried by the diurnal rotation of the latter on its axis. It will be seen that in the northern hemisphere less than half of these are illuminated by the sun, and in the jiouthern hemisphere more than half. This corresponds to our winter solstice. When the earth reaches -ff, its axis JVS is at right an- gles to the line drawn to the sun, so that the latter shines perpendicularly on the equator, the plane of which passes through it. The diurnal circles on the earth are one half no A8TB0N0MT. illuminated and one half in darkness. This position cor- responds to the vemal equinox. ^ a *^ At G the case is exactly the reverse of that at A, the sun shining more on the northern hemisphere than on the southern one. North of the equator more than half the diurnal circles are in the illuminated hemisphere, and south of it less Here then we have winter in the southern and summer in the northern hemisphere. The sun is above a region 23i° north of the equator, so that this position cor- responds to our summer solstice. At D the earth's axis is once more at right angles to a line drawn to the sun. The latter therefore shines upon the equator, and we have the autumnal equinox. In whatever position we suppose the earth, the Une A JV, continued indefinitely, meets the celestiad sphere at its north pole, while the middle or equatorial circle of the earth, continued indefinitely in every direction, marks out the celestial equator in the heavens. At first sight it might seem that, owing to the motion of the earth through so vast a circuit, the positions of the celestial pole ^d equa- tor must change in consequence of this motion. We might say that, in reaUty , the pole of the earth describes a circle in the celestial sphere of the same size as the earth's orbit. But this sphere being infinitely distant, the circle thus de- scribed appears to us as a point, and thus the pole of the heavens seems to preserve its position among the stars through the whole course of the year. Again, we may suppose the equator to have a slight annual motion among the stars from the same cause. But for the same reason this motion is nothing when seen from the earth. On the other hand, the slightest change in the direotim of the axis SIf wUl change, the apparent position of the pole among the stars by an angle equal to that change of direc- tion. We may thus consider the position of the celestial pole as independent of the position of the earth in its orbit, and dependent entirely on the direction in which the axis of the earth points. 1 tic of ex diJ ch is th pi nc th to be la ri| b( cc til di cc es w ni tt si C ai al t< tl b o cor- thc 1 the : the ionth I and jve a 1 cor- to a upon SJV, Eit its f the C8 ont [night igh so equa- tnight vie in orbit, osde- >f the I stars e may unong reason )n the of the o pole direc- elestial in its which CELESTIAL LATITUDE AND LONGITUDE 111 If this axis were perpendicular to the plane of the eclip- tic, it is evident that the sun would always lie in the plane of the equator, and there would be no change of seasons except such slight ones as might result from the small differences in the distance of the earth at different seasons. § e. CELESTIAL LATTTUDB AND LONQITUDB. Besides "the circles of reference described in the first chapter, still another systfem is used in which the ecliptic is taken as the fundamental plane. Since the motion of the earth around the sun takes place, by definition, in the plane of the ecliptic, and the motions of the planets very near that plane, it is frequently more convenient to refer the positions of the planets to the plane of the ecliptic than to that of the equator. The co-ordinates of a heavenly body thus referred are called its celestial Utituds and hmgitude. To show the relation of these co-ordinates to right asocmsion and declination, we give a figiwe showing both co-ordmates at the same time, as marked on the celestial sphere. This figure is supposed to be the celes- tial sphere, having the solar system in its centre. The direction /> ^ is that of the axis of the earth ; IJ\& the ecliptic, or the great circle in which the plane of the earth's orbit intersects the celestial sphere. The point in which these two circles cross is marked 0^, and is the ver- nal equinox from which the right ascension and the longi- tude are both counted. The horizontal and vertical circles show how right ascen- sion and declination are counted in the manner described in Chapter I. As the right ascension is counted all the way around the equator from (^ to 24S so longitude is counted alon ,' the ecliptic from the point 0^, or the vernal equinox, toward J in degrees. The whole circuit measuring 360", this dlhtance will carry us all the way round. Thus if a body ^ in the ecUptic, its longitude is simply the number o^'i^ees from the vernal equinox to its position, meas- lllif !n the direction from / toward J. If it does not lie 1 112 A8TR0N0MY. SipSo ffle»gth of thi. i«rpe«dicukr,me«oredm ^, i» cUedX W«. of .ho IfJy. f'f ™y^ ««^l. nr south whUe the distance of the foot of the per ™^.S^f*m r vomal eqainox i. called '■^^'-f^- botoTof the Botar «T»tem, retatively to the smi, by their ^"X^taat«d«. lU«.intheecUptiewehave . FlO. 47.— CIBCUCB OF THE BPHBBB. plane more nearly fixed than that of the equator On^e Ler hand, it is more convenient totepreeent ^ po«Uon of aU the heavenly bodies ae Been from the ««^^y *^ right ascensions and declinations, because we ««»o* «T; rJhriongitudes and latitudes <^%^;\r^f^ observe right ascension and decimation. If we wisn w dSn/the longitude and l^tjude of a^y as -n from the centre of the earth, we have to fi«*/»f ^^«j^^ ascension and decUnation by observation, and then cbm^ Sr^nMitities to longitude and htitude by tngonometn- oal formnlsB. priB nitii core rate ter, pki the uia' • iut( g iirsi tha the whi mo mo alw cer 1 pec boc sun pla we the iin ^ be per- ude. the heir vea )nthe wition ^ their meas- riOAly rifih to B seen I right shauge unetri- CHAPTER IV. THE PLANETARY MOTIONS. § 1. APPABEnr Ain> beal Monovs of the FLAITETS. DeflnitioiiB. — The solar system, as wo now know it, com- prises so vast a number of bodies of various orders of mag- nitude and distance, and subjected to so many seemingly complex motions, that we must consider its parts sepa- rately. Our attention will therefore, in the present chap- ter, be particularly directed to the motions of the great planets, which we may consider as forming, in some sort, the fundamental bodies of the system. These bodies may, with respect to their apparent motions, be divided into three classes. Speaking, for the present, of the sun as a planet, the first class comprises the »un and moon. We have seen that if, upon a star chart, we mark down the positions of the sun day by day, they will all fall into a regular circle which marks out the ecliptic. The monthly course of the moon is found to be of the same nature, although its motion is by no means uniform in a month, yet it is always toward the east, and always along or very near a certain great circle. The second class comprises Venus and Mereury. The peculiarity exhibited by the apparent motion of these bodies is, that it is an oscillating one on each side «, ? the sun. If we watch for the appearance of one of theae planets after sunset from evening to evening, we shall find i ABTR0N0M7. it to appear above the western horizon. Night after night wiUW arther and farther from the sun untU it attems ar^Sr maximum distance; then it^llappearter^^^^ to the sun again, and for a while to be lost m its rays. A f^w Zs ISer it will reappear to the west of the ^n, fnd ther^^ter be visible in the eastern horizon before Bunrise In the case of Mercury, the time reqmred for oneT»mplete oscillation back and forth is about four Zt?^7and in the case of Venus more than a year and * m third class comprises Jfor«, Jupit^, and Saturn as weU ^a ^at num Jof planets not visible to the na^ed Tye Thfgeneral or average motion of these planets i X'ard the% a complete revolution i- «^« J^^^^ Bphere being performed in times ranging from two years ZZ Z^e^Mars to 164 years in that of Neptnn.. But instead of moving uniformly forward, they^m to have a swinging motion ; first, they move forward or towIrJ Zlt 'through a pretty long arc tb- backw^^ or westward through a short one, then forward through a J^r one, etc. It is only by the excess of the longer aiTS the shorter ones that the circuit of the heavens ^'S'general motion of the sun, moon, and planets among the stars being tcJward the east the motion inth^^ diredlon is called direct; whereas the occasional short Z&Z toward the west are called retro^. During the periods between direct and retrograde motion, the pknete will for a short time appear stationaiy. ^^^ The planets Venm and Mercury are said to be at great- est ^atUm when at their greatest «^"g^;.J«^^^^™ the sun The elongation which occurs with the planet J^tTihe sun, andXrefore visible in the -estei. hon- zon after sunset, is called the eastern elongation, the other ^T^^irslid to be in conjunction with the smi when it is in the same direction, or when, as it seems to pass by the B oppo. tion- apla sun, yond Ai knov and \ cent! plant inFi in tl whi< fari / ight tains itum rays. Bun, afore i for four r and m as laEked lets is lestial years ■kune. em to a-d or kward rough longer eavens planets in this short During in, the igreat- se from planet m hori- le other n when pass by ARRANGEMENT OF THE PLANETS. 116 the sun, it approaches nearest to it. It is said to be in apposition to the sun when exactly in the opposite direc- tion — rising when the snn sets, and vi^ie vecsa. If, when a planet is in conjunction, it is between the earth and the sun, the conjunction is said to be an inferior one ; if be- yond the snn, it is said to be tniperior. Arrangements and Motions of the Planets. — We now know that the sun is the real centre of the solar system, and that the planets proper all revolve around it as the centre of motion. The order of the five innermost large planets, or the relative positions of their orbits, are shown in Fig. 48. These orbits are all nearly, but not exactly, / 48. — ORBIT8 OF THB PLANETS. in the same plane. The planets JUercury and Venits which, as seen from the earth, never appear to recede very far from the sun, are in reality those which revolve inside ,,„ ASTKONOMr. llo , , -♦I, The Dlanets of the third clasB, the orbit of the earth. ?^* Pn^auces from the «un, which perform tl^«y^;j-^„^^^^^^^^^ and ai.;nore are what we now call the J^ "*\ t*. ^f these, the or- aietantfj^the^mH^-^^^^ telescopic planets bits of Mars, Jujnter, ana a ^ ^^^ are shown in t^>« ^f ^^f ^* Jurvisible to'^the naked Samr>., the farthest P^»"«* ^Ll telescopic planets, eye, and ^^ J^^^;^jteX^l ^'^^ ^^^^^ On the scale of l?ig. *» ^"« Wnallv, the moon is a ^ore than two feet m diameter. ^^Z.U. ^^^.^e, and The farther »?!»««* i^^*^^^ '^e go frJm tbe sun, is its orbital motion. TW ^^ f^, ^he double reason the periods of revolution are ^«J««'^;^**J^ribe and moves that the planet has a larger orbit o de^"^^^.^^ ^^ ^^^ xnonj slowly in it« orbit, f^^^^^^^^^trognide motion outerplanetsthattheoccasiomaapp««ntreirog li Jplanets is du. - -y^r<^^W a pU, We first remark that the ^Pl^j^^ , ^^ li^e joining as seen from the «^%» *^^^t Z to be continued the earth and planet. S^I^^lt tbe celestial sphere, onward to infinity, so as to ^^J^^^^J^J^efined by the the apparent motion of t^l^P^''?* ^^temcte the sphere, motion of the point ^^^^^f ^^^^^^^^^^^ dir^t ; if If this motion is toward the east, it wiu oe toward the west, retrograde. g ^ V"/^X*JV ': Cu^clTve "^Itioi^ of the earth poee ^i^^ff^cDEF ^ ^« corresponding posi- in its orbit, wAABtVJ;^ Tt must be remembered that tions of ^-- - fXTi^Tnti^cTnnection, we do irmiraf alir dl^ction in space, but a direction aronr down diroc inovt earth beini evidi great sun I totl ^ the dir the H pel inj eai is ItJ lird cla8B, i the Bun, are jnore se, the or- lic planets Iter comes the naked ic planets. I would be moon is a sentro, and an. inude that outside that I, the Blower 9m the sun, )uble reason } and moves lotion of the grade motion ring Fig- 49- of a planet, i line joining be continued cstial sphere, efined by the its the sphere, be direct ; if pUnet. Sup- riB of the earth spending posi- membered that inection, we do but a direction APPABBNT MOTIONS OF TlIK PLANKTS. 117 around the sphere. In the figure wc are supposed U , when it 1>ecomcs stationary. After- ward it is direct until the two bodies again come into the relative positions I Ji. Let U8 next snpposo that the inner orbit A B CD EF represents that of the earth, and the outer one that of a superior planet, Moth ior instance. We may consider O QPJitohe the celestial sphere, only it should be infi- nitely distant. While the earth is n «yving from ^ to ^ the planet moves from II to 7. This ^ni. tion is direct, the di- rection OQP li being from west to east. While tlie earth is moving from B to D, the planet Is moving from / to Z ; the former motion l)eing the more rapid, the earth now passes by the planet as it were, and the line conjoin- ing tiiem turns in the same direction as the hands of a watch. Therefore, during this time the planet seenu* to describe the arc P Q' in the celestial sphere in the direction opposite to its actuai orbital motion. The lines Z D and MEixe supposed to be parallel. The planet is then really stationary, even though as drawn it would seem to have a forward motion, owing to the distance of these two lines, yet, on the infinite sphere, this distance appears as a point. From the point M the motion is direct until the two bodies once more reach the relative positions B I. When the planet is at JT and the earth at C, the former is in opposition. Hence the retrograde motion of the supe- rior planets always takes place near opposition. Theory of Bpioyand D really have a o lines, rs as a itil the wJ?/ pmer is 3 sope- repre- whioh made planet insteatl of a motion of the earth around the sun, whiclk really causes it. But their theory was, notwithstanding, tlie means of leading Cupeuniuus and others to the percep- tion of the true nature of the motion. We allude to the celebrated theory of epicycles, by which the planetary motions were always represented before the time of Copkk- NiciTB. Complicated though these motions were, it was seen by the ancient astronomers that they could be repre- sented by a combination of two motions. First, a small circle or epicycle was supposed to move around the earth with a regular, though not uniform, forward motion, and then the planet was supposed to move around the oircnm- ference of this circle. The relation of this theory to the true one was this. The regular forward motion of the epicycle represents the real motion of the planet aronnd the sun, while the motion of the planet aronnd the cir- cumference of the epicycle is an apparent one arising from the revolution of the earth around the snn. To ex- plain this we must understand some of the laws of relative motion. It is familiarly known tl)at if an observer in unconscious motion looks upon an object at rest, the object will ap- pear to him to move in a direction opposite that in which he moves. As a result of this law, if the observer is unconsciously describing a circle, an object at rest will appear to him to describe a circle of equaJ size. This is shown by the following figure. Let 8 represent the sun, and A B CDBF the orbit of the earth. T^ us suppose the observer on the earth carried around in this orbit, but imagining himself at rest at 8^ the centre of motion. Suppose he keeps observing the direction and distance of the planet P, which for the present we suppose to be at rest, since it is only the apparent motion that we shall, have to consider. When the observer is at ^ he really sees the planet in a direction and distuioe A P, but imagining himself at 8 he thinks he sees the planet at the point a determined by drawing a line Sa parallel and 120 ASTHONOMV. equal to A P. A* he pam-H from A to B the planet will wjeui to him to move in the opiKwite ilirection fr(»m A to b, the point h Injiiig deter- mined by drawing Sb equal and parallel to B P. As ho reeedes from the planet through the arc BCDy the planet seems to re- cede from him through hcd\ and while he moves from loft to right through DE the planet seeniB to move from right to left through D E. Finally, as he ap- proaches the planet through the arc EFA the planet seems to approach him through EFA, and when he returns to A the pUnet will appear at ^, as in the beginning. Thus the planet, though really at rest, will seem to him to move over the circle ahcdef corresponding to that iu which the obser^'er himself is carried around the sun. Tlie planet being really in motion, it is evident that the combined effect of the real motion of the planet and the apparent motion around the circle a J o <; «/ will bo represented by carrying the centre of this circle P along th.> true orbit of the planet. The motion of the earth being more rapid than that of an outer planet, it follows that the apparent motion of the phmet through a J is more rapid than the real motion of P along the orbit. Hence in this part of the orbit the movement of the planet wUl be retrograde. In every other part it will be direct, because the progressive motion of P will at least overoome, some- times be added to, the apparent motion around the circle. In the ancient astronomy the apparent small circle ahcdef was called the epieyde. In flm ri I lero ifH rei the t forwt tion earth In wo hi by II really to de of iti tho a that of itf incoi all tl was, was 1 mov< bratc the I of m real in 01 part, poset Thej Tl ineqi to b< •I quity Buppc idled UNEQUAL MOTION OF THE PLANETS. 121 I planet >ii fr<»iii ij tlotur- [ual iind recedos the arc 8 to ro- ll hed\ w loft to ) planet t to left u ho ap- )Ug1l tlio iOoinB to EFA, o A the as in the planet, nil seem he circle to that litnBelf is dent that lanot and f will bo ) P along the earth it follows h is more . Hence let will be t, because ne, some- he circle. lall circle t In the ciiHo of \\\v innor planets Mernnnf and Vtrnm \\w rbliition of tlie epiiiyelo to the tr.io orlnt Ih reverHed. Here the epieyelic motion ih tliiit of the plunet annind ifH real orbit— that is, the true orliit of the plunot around tho sun was itself taken for the epicycle, while the forward motion was really duo to tho apparent revolu- tion of tho sun produced by tho aimual motion of tho earth. In tho preceding descriptions of tho planetary motions wo have spoken of them all as eircukr. But it was found by Ilii'J'ARcnus * that none of tho planetary motions were really unifonn. Studying tho motion of the sun in order to determine tho length of tho year, ho observed tho times of its passage through tho equinoxes and solstices with all tho accuracy which his instruments pennitted. He found that it was several days longer in passing through one half of its course than through tho other. This was apparently incompatible with tho favorite thcoiy of tho ancients that all tho celestial motions were circular and uniform. It was, however, accounted for by supposing that the earth was not in the centre 6f tho circle around which tho sun moved, but a little to one side. Thus arose the cele- brated theory of tho eccentric. Careful observations of the planets showed that they also had similar inequalities of motion. The centre of the epicycle around which the real planet was carried was found to move more rapidly in one part of the orbit, and more slowly in the opposite part. Thus the circles in which the planets were sup- posed to move were not truly centred upon the earth. They were therefore called eccentrics. This theory accounted in a rough way for the observed inequalities. It is evident that if the earth was supposed to be displaced toward one side of the orbit of the planet, * HnTAi{CHi7B was one of thie most celebrated astronomers of anti- quity, being frequently spoken of as the father of the science. He is supposed to have made most of his observations at Rhodes, and flour- hdied about one hundred and fifty years before the Christian era. iiu Ti ♦ 522 ASTRONOMY. the latter wonkl seem to move more rapidly when nearest the earth than when farther fron. it. 1^ wa. not untd ^e time of KE..LEK that the eccentric w;ifl shown to be caTable of accounting for the real motion ; and it ,s his discoveries which we are next to descnbe. § 2. KBPLBB'S LAWS OF PLAJTETARY MOTION. The direction of the sun, or its longitude, can be deter- mined from day to day by direct observation If we could also observe its distance on each day, we should, by laying down the distances and directions on a large piece 7paper, through a whole year, be able to trace the curve Sthe earth describes in its annual course, this cour^ C^g, SB already shown, the counterpart of the appa^^t L of the sun. A rough determination of *ae rela- tive distances of the sun at difierent times of the year may be made by measuring the sun's apparent angular diame- ter, becaJe this diameter varies inversely aa Je distan^ of the object observed. Such measur^ would show that the diameier waa at a maximum of 32' 36' on January 1st, Ind a"a minimum of 31' 32" on July 1st of every y«.^ The difference, 64% is, in round numbers, A tbe mean diameter-that is, the earth is nearer the sun onjmu^j 1st than on July Ist by about ^ We may consider ^ as A greater than the mean on the one date, and ^ less TntheSer. This is therefore the actual displacement of the sun from the centre of the earth s orbit. Again, observations of the apparent daaly motion of thT^ among the stars, corresponding to the real dady 'IZoi the'earth round the sun, show t^s motion^o be . least about July Ist, when it amounts to 57 12 _ 34d^ , and greatest about January 1st, when it a^^^^ t° «1 ' ir = 3671'. The difference, 239', is, m round num- bers A the mean motion, so that the range of variation M; proportion to the mean, double what it is in the c^ p L diBtences. If the actual velocity of the earth m its pou] was pose m 01 in lo half long eartl ingi attril the t bit- centi the ( greal A] tion : radi', rouni pose, and day t of it and I geom the a: are ir KEPLER'S Laws. Idd jarest il the ae iu- is his [ON. deter- [f wo Id, by piece curve course parent 3 rela- armay diame- istance »w that ary iBt, y y«ar. Q mean fanuary sider it L^less icement )tion of al daily on to be ^ = 3432', rants to nd num- variation the case rth in its orbit were niiiforin, tlie apparent angular motion round tlie sun would be inversely as its distance from the sun. Actually, however, the angidar motion, as given above, is inversely as the square of the distance from the sun, be- cause (1 + ^V)' = 1 + tV very nearly. The actual ve- locity of the earth is therefore greater the nearer it is to the sun. On the ancient theory of the eccentric circle, as pro- pounded by IIippAKcnus, the actual motion of the earth was supposed to be uniform, and it was necessary to sup- pose the displacement of the sun (or, on the ancient theo- ry, of the earth) from the "ontre to be ^ its mean distance, in order to account for the observed changes in the motion in longitude. We now know that, in round numbers, one half the inequality of the apparent motion of the sun in longitude arises from the variations in the distance of the earth from it, and one half from the earth's actually mov- ing with a greater velocity as it comes nearer the sun. By attributing the whole inequality to a variation of distance, the ancient astronomers made the eccentricity of the or- bit—that is, the distance of the sun from the geometrical centre of the orbit (or, as they supposed, the distance of the earth from the centi-e of the sun's orbit) — twice as great as it really was. An immediate consequence of these facts of observa- tion is Kepleb's second law of planetary motion, that the radii vectored drawn from the sun to a planet revolving round it, sweep over equal areas in equal times. Sup- pose, in Fig. 51, that /.S' represents the position of the sun, and that the earth, or a planet, in a unit of time, say a day or a week, moves from P, to P,. At another part of its orbit it moves from P to P, in the same time, and at a third part from P. to P.. Then the areas SP,P,, SPP„ SP,P, will all be equal. A Kttle geometrical consideration will, in fact, make it clear that the areas of the triangles are equal when the angles at S are inversely as the square of the radii vectores, SP, etc., *^- 1^4 ASTBONOMT. .„ee t„e exprcion ,. the - "V^^™^'" '" *" *"° angle at S w very anaU « J angle * X ^ J O pi * Fig. 51.— law of areas. 1„ the ttoe of K...« *-> ™-™J*.Cr^^ctd! .un'B '»P'l»%*'r''''r J™£'3> S the earth around ing method of deternnmng the l«ttt o ^^ ^^^ thl »m could 7'\''»^:^;tol^edhyTTOHoBKi■>.^ motions of the planet ^ar.,^*^ ^^^ ^ , that Keplee was led to his ceieore j^ ^ motion. He found that "» P^^^J";^^ „p^«>nt the ,r^y elreular orb^ho«-7C*^ ejeu^tions and „h<«rvat.ons. Jf^' ;™f ™ t numher of hypothes^ the triil aud rejection ot a p« ^^^ ^^^^^ he was led to the eondusion .«'»' «« ^^ j^ the analo- toanelU^. having *e™-^^Jtl^; attthe planets, gieaof nature led *»*« ™'°"°r„ „f the same chM, L earth i»«<^V "'°'^„ twX ™ led to enunciate S*C^- "^ -"^" ^"^""^ "'°"°- ,hich were as follow: ,^ ^ „ „ . .^. „.., Ok, ^ 7ve the area mentioned above. 1 KEPLER' 8 LAWS. 195 1 I. Eachplanet moves around the sun in an ellipse, hav- ing the sun in one of its fad. II. The radius vector joining each planet toith the sun, moves over equal areas in equal times. To these be afterward added another showing the rela- tion between the times of revolution of the separate planets. III. The square of the tim^ of revolution of each planet is proportional to the cube of its mean distance from, the sun. These three laws comprise a complete theory of plan- etary motion, so far as the main features of the motion are concerned. There are, indeed, small vai-iations from these laws of Keplkh, but the laws are so nearly correct that they are always cmijloyed by astronomers as the basis of their theorios. Mathematioal Theory of the Elliptio Motion. — The laws of Kkpleb lead to problems of such mathematical elegance that we give a brief synopsis of the most impor- tant elements of the theory. A knowledge of the ele- ments of analytic geoUietry is necessary to understand it. Let us put : a, the semi-major axis of the ellipse in which the pUuut moTflt. In the figure, if (7 is the centre of the el- lipse, and <9 the focus in which the sun is situated, then resented by the angle n 8E, B being the direction of the vernal equinox from which longitudes are counted. n, the mean angular motion of the planet round the sun in a unit of time. The actual motion being variable, the mean motion is found by dividing the As. 01. circumference = 860° by the time of revolution. T, the time of revolution. Tj the distance of the planet from the sun, or its radius vector, a variable quantity. I. The first remark we have to make is that the Mij^ieiUu of the 126 ASTIiONOMY. the ellipse we have : 8B = Bemi-major axis = a, BC= semi minor axis = a V 1 — «', or 5 = a (1 - i «') nearly, when e is very small. very nearly, so that flattening of the orbit is only about ^ or .02 of the major axis. , j^ j ^jjid^ ^ = .093 ; B^t""! -loirs tff'i«tE.^»l"« of tl,, orbit U only a ,cry cioso approxnmtion to the true lorm o.^ I ' ^, It 1. Jnl, leceuar, to »"PP°?° *? ""°jrof tl • Scentricit, loto re?Sto'o?rS.'rS.?er»S?"™„pre«„t«ioo o. the -t' •^,C.X£°o<'t!!iVS:ot .™m «io .- " and the greatest distance is Kefleb : Q vector during such unit ^i^" *^* J^rfg ggcond law. Therefore, jj»"!'^fi5tcto;?r.t:^'ri'?s%hoh, .r» oj^th. dh^ which i. . »• V T^-?.. The time re,m»a to do th«i 1» . 1» ll,i. formula , n,pr»env. Iho r«io of the ciroumfereoee of the circle to its diameter. mg alB( an KEPLER'S LA WS. 1»7 ing called T, the area swept over with the areolar velocity 4CU t\ao\GT. Therefore J C 2' = IT a' ^\ — e' ; 2irfl*Vf — e» = j5 The .luantity 2 t hero represents 860% or the whole circumference, called M. Therefore 2ir and , -. C = a*n Vl - e\ This value of being substituted in the expression for 8, wc have a' rt Vl -"? *^ ^= is IV By Kepleh's third law r is proportioned to a" ; that is, IL is a constant for all the planets. The numerical value of this and a for the earth will both be unity, and the ratio ^ will there- fore be unity for all the planets. Therefore a» = 2" ; o = r*. i^vc^r ATS-trhricrrormined with very great pre- "^V °To find the position of a planet we must kno^^t^e epoch at M^ Wn^irolllfte^^^^^^ Se';s;s^he%a£/Thi^^^^ ^Ssltiorof the planet at this time wc shall have Area of sector PSk _ _r ]\fcaof ^hoie ellipse T (1). 128 ASTRONOMY. The times r and T being both given, the problem is "^uced to t. Jt of c^tUng a given area of the ellipse by a line drawn from the V^„« to some point of its circumference to be found. This is ISn as Slkk'8 problem, and may be solved by analytic geom- TiO. 68. the ratio of i> P to D i*, or of « to b. Hence, Area GPB : area OP'S = b:a. n * «,«« nvn- ftnffle P" B x i a», taking the unit radius as?Se unU of'^fgulaJ^lllsfre!' Hence, putting « for the angle pf G Bvre have Area CPB = - area CP* 5 = J « S « a (2). Again, theareaof the trianglcOPSisequaltoibaseC^f x al- titudePD. AlsoPD = ^-P'AandP'i>= CP' sin « = «sm«. Wherefore, tri) an( It ( or, PD = &sin (8). KSPLBR'8 LAWS. 1S9 By the first principles of conic sections, C 8, the base of the triangle, is equal to a «. Hence Area CP8 = iabeMau, and, from (3) and (8), Area SPB = Jo ft (« — «Bin «). Substituting in equation (1) tliis value of the sector area, and IT a 6 for the area of the ellipse, we have tt — g sin w _ jr 3^;^ ~ 2" or. u — « sin u = 2 T -^. Prom this equation the unknown angle « « ^^*^^:V'^ equation being*a transcendental one, this ««"»"«» ^,^°°«f^"?L but it may be rapidly done by successive approximation, or the value of u may be developed in an infinite series. Next we wi^h to expreiTthepositionof *£« P»"'«*i.^J^^Si?,K by its radius vector -8 P and the angle B 8 ^,^,^|«f *7"^'^''" vMtor makes with the major axis of the orbit. Let us put r, the radius vector SP, /' the angle B 3P, called the true anomaly. Then . „^ r sin/ = P2> = ft sin « (Equation 8), rcos/=8D=CD- 08= P cosu - ae = a(coiu-e), from which r and r can both be determined. By taking the square I^^oTtiS^sums ohhe squares, they give, by suUable reducbon and putting ft' = a' (1 - «'), r = a (1 — « cos u), and, by dividing the first by the second, ft sin « tan/ = a (cos M — «) Vl — g' sin M cos u- e Prtttog, „ before, . for the longtad. of th. peritaUon, th.tr.. '°t'°T;s'.X''^woVS°rpT::iKuti;..,.o th. ..upuc, 180 ASTRONOMY. the inclination of the orbit to the ecliptic has to be taken into ac- counf The orbits of the several large planets do not lie in the Smc plane, but are inclined to each other, and to the ecliptic, by tft^ous imkll anirles. A table giving the values of these angles ;rb"e g™en her&r, from whi?h it^ill be seen that the orbu o Mereurv^haA the greatest inclination, amounting to 7 , and that of f/mSe least, teing only 40'. The reduction of the position of tlHw to «;« ecliptic fs a problem of spherical trigonometry, the solution of which need not be discussed here. 1 fun whi feal rea< trat fiho cov teBl abi of Ion tig iB< foi Bci i'- 1 gr( th( tb ex tis & in nto ac- ) in the jtic, by angles orbit of that of ition of ometry, CHAPTER V. UNIVERSAL GRAVITATION. § 1. NEWTON'S LAWS OP MOTION. The eBtablishment of the theory of universal gravitation furnishes one of the best examples of scientific method which is to be found. We shall describe its leadmg features, less for the purpose of making known to the reader the technical nature of the process than for illus- trating the true theory of scientific investigation, and Bhowing that such investigation has for its object the dis- covery of what we may call generalized facts. The real test of progress is found in our constantly increased abiUty to foresee either the course of nature or the eSects of any accidental or artificial combination of causes. So long as prediction is not possible, the desires of the mves- tiinTtor remain unsatisfied. When certainty of prediction is once attained, and the laws on which the prediction is founded are stated in their simplest form, the work of science is complete. , . The whole process of scientific generalization consists in grouping facts, new and old, under such general laws that they are seen to be the result of those laws, combined with those relations in space and time which we may suppose to exist among the material objects investigated It ib essen- tial to such generalization that a single law shall suffice for grouping and predicting several distinct facts. A law invented simply to account for an isolated fact, however 17 '.'If' II Wi ASTBONOMY. general, cannot be regarded in gcienco as n law of nature. It may, indued, bo true, Imt its truth caniiut lie proved until it is shown that eeverol distinct facts can he accounted for by it better than by any other law. The reader will call to mind the old fable which represented the earth as suppoi'ted on the back of a tortoise, but totally forgot that the support of the tortoise needed to be accounted for as much as that of the earth. To tlie pre-Newtonian astronomers, the phenomena of the geometrical laws of planetary motion, which we have just described, formed a group of facts having no connection with any thing on the earth. Tlie epicycles of Hippakciiits and Ptolkmv were u truly scientilic conception, in that they explained the seemingly erratic motions of the planets by a single simple law. In the heliocentric theory of Coper- MiODS this law was still further simplified by dispensing in great part with the epicycle, and replacing the latter by a motion of tho earth around the sun, of the same nature with the motions of the planets. But Copebnicds had no way of accounting for, or even of describing with rigor- ous accuracy, the small deviations in the motions of the planets around the sun. In this respect he made no real advance upon the ideas of the ancients. Kepleb, in his discoveries, made a great advance in representing the motions of all the planets by a single set of simple and easily understood geometrical laws. Had the planets followed his laws exactly, the theory of planetary motion would have been substiuitially complete. Still, further progress was desired for two reasons. In the first place, the laws of Keplkr did not perfectly represent all the planetary motions. When ob- servations of the greatest accuracy were made, it was found that the planets deviated by small amounts from the ellipse of Kepler. Some small emendations to the motions com- puted on the elliptic theory were therefore necessary. Had this requirement been fulfilled, still another step would have been desirable — namely, that of connecting the 8 t n c 1 tl 1 V Ci 1« f< it f« n tl i t t ii I ^ LAWS OF MOTION. 188 turo. oved intc law of planetary motion to be that around the circum- ference of an ellipse, as asserted in his law, he said all that it seemed possible to learn, supposing the statement per- fectly exact. And it was all that could he learned from the mere study of the planetary motions. In order to connect these motions with those on the earth, the next step wm to study the laws of force and motion here around us. Sm- gukr though it may appear, the ideas of the ancients on this subject were far more erroneous than then- concep- tions of the motions of the planets. We might ahnost say that before the time of Galileo scarcely a single correct idea of the laws of motion was generally entertained by men of learning. There were, indeed, one or two who in this respect were far ahead of their age. Leonardo da Vinci, the celebrated painter, was noted in this respect. But the correct ideas entertained by him did not seem to make any headway in the world until the early part of the seventeenth century. Among those who, before the time of Newton, prepared the way for the theory in question, Galileo, Hutghbns, and Hooke are entitled to especial mention. As, however, we cannot develop the history of this subject, we must pass at once to the gen- eral laws of motion Ldd down by Newton. These were three in number. 184 A8TR0N0M7. Law First : Jl^jery body preserves its stats qf rest or (ff un'tform motion in a right Htm, tnUens it is compelled to change that state by forces impressed thereon. It waft foimorly eiipposcd that a XwAy acted on by no forco tended to come to rest. Here lay one of the great- est difflcultioB which the predecessors of Newton found, in accounting for tlie motion of the planets. The idea that the sun in some way caused these motions was enter- tained from the earliest times. Even I*T0LBMr had a vague idea of a forco which was always directed toward the centre of the earth, or, which was to him the same thing, toward the centre of the universe, and which not .only caused heavy bodies to fall, bat bound the whole nni- versfl together. Kepleb, again, distinctly aifiims the ex- istence of a gravitating force by which the sun acts on the planets ; but he supposed that the sun nmst also exercise an impulsive forward force to keep the planets in motion. The reason of this incorrect idea was, of course, that all bodies in motion on the surface of the earth had practically come to rest. But what was not clearly seen before the time of Kewton, or at least before Gald^eo, was, that this arose from the inevitable resisting forces which act upon all moving bodies around us. Law Second : The aU&raUon of motion is ewr propor- tional to ike mooing force impressed, and is made in the direction qf the right line in which that force acts. The first law might be conddered as a particular case of this second one arising when the force is supposed to van- ish. The accuracy of both laws can be proved only by very carefully conducted experiments. They are now considered as mathematically proved. Law Third : Toevery action there isahoays qfy)08ed an equal reaction / or the mtitual actions of two bodies "wpon each other are always equal, and in opposite directions. That is, if a body A acts in any way upon a body B, B will exert a force exactly equal on ^ in the opposite direction. lat tlu to mo of la/u am OR] seal con obv cull in a peri fort ive circ! law fOH low cen bits sun mol the witl will rest or cf TmpeUed to I on by no ' the j^roat- rTON found, The idea J was ontor- KMY liad a jted toward n the same I which not a whole nni- litns the ox- I acts on the ilso exercise B in motion, irso, that all d practically 1 before the iras, that this ich act upon everjtropor- made in the e acts. icular case of posed to van- oved only by 'hey are now ocame possible to calcu- late the motion of any body or system of bodies when oncu the forces which act on them wore known, and, vice versa, to define what forces were re<^uisite to produce any given motion. The question which presented ifaself to the mind of Newton and his contemporaries was this : Under what lo^ (if force will planets move round the sun in accord- ance with Kepi.kr'b laws t The laws of central forces had been discovered by IIuy- OHENS some time before Newton commenced his re- searches, aad there was one result of them which, taken in connection with Kbpleb'b third law of motion, was so obvious that no mathematician could have had much diffi- culty in perceiving it. Supposing a body to move around in a circle, and putting R the radius of the circle, T the period of revolution, IIuyoiiens showed that the centrifugal force of the body, or, which is the same thing, the attract- ive force toward the centre which would keep it in the circle, was proportional to ^. But by Kepler's third law 7" is proportional to I^. Therefore this centripetal R 1 force is proportional to -^j, that is, to -^. Thus it fol- lowed immediately from Kepler's third law, that the central force which would keep the planets in their or- bits was inversely as the square of the distance from the sun, supposing each orbit to be circular. The first law of motion once completely understood, it was evident that the planet needed no force impelling it forward to keep up its motion, but that, once started, it would keep on forever. The next step was to solve the problem, what law of force will make a planet describe an ellipse around the sun, having the latter in one of its foci ? Or, supposing a planet to move rotmd the sun, the latter attracting it with a force inversely as the square of the distance ; what will be the form of the orbit of the planet if it is not cir- is. ■4 — '- 136 AamONOMT. cnlar ? A solution of cither of these problems was beyond ArpowetoTmathematicians before the time o Newton ; Ind^ttaremained uncertain whether the planets uh>v- wlder the influence of the sun's gravxtation would or wouW not describe elUpses. Unable at first, to reach a raJSLtory solution, Newton attacked the problem m :Sw Section, sUng f-\*^« n-^n^ll^tinl the sun, but of the earth, as explained m the following section. § 2. OBAVTPATION IN THE HEAVENS. The reader is probably familiar with the story of N ew- J^ and the falling apple. Although it has «o authonta- TeToundation, if is strikingly illustrative of the method by wWch New;,k first reached a solution of the problem. fi,e course of reasoning by which he ascended from gra^v- itetion on the earth to the celestial motions was as f^^ . We see that there is a force acting all over the earth by which all bodies are drawn toward its centre This force S f^ar to every one from his infancy, and is property ^ed gravitation. It extends without sensible diminut^n TtheTops not only of the highest braidings, but of the highest mountains. How much higher does it extend? my should it not extend to the moon ? If it does, the moon would tend to drop toward the earth, ]ust as a stone ^Zvm from the hand drops. As the moon moves romid Sr^th in her monthly cou«e, there -ust be some ^rce drawing her toward the earth ; else, by the first law of motionfshe wouldflyentirely away in a straight hue. Why Zuld not the force which makes the apple fall be the ^ioL which keeps her in her orbit ? To answer tlus ^^ion,itwasnotonTynece8sarytocalcuktethemten«ty of the firce which would keep the moon herself in her orbit but to compare it with the intensity of gravity at the S's surface. & long been know, that ^e distanc^^ of the moon was about sixty radu of the earth. If this for© then the I teen were The GliA VITATION OF THE PLANETS. 137 5yond fTON ; mov- ald or aach a >in in aot of owing : New- horita- nethod oblem. a grav- jllows : arthby is force •roperly linution of the extend ? oes, the a stone » round ne force ; law of a. Why I be the iwer this intensity f in her ty at the distance If this force diminished as the inverse square of the distance, then, at the moon, it would be only ^^ as great as at the surface of the earth. On the earth a body falls six- teen feet in a second. If, then, the theory of gravitation were correct, the moon ought to fall toward the earth ^^-^ of this amount, or about ^ of en inch in a second. The moon being in motion, if we imagine it moving ui a straight line at the beginning of any second, it ought to be drawn away from that Une -^ of an inch at the end of the second. When the calculation was made with the correct distance of the moon, it was found to agree ex- actly with this result of theory. Thus it was shown that the force which holds the moon in lier orbit is the same which makes the stone fall, only diminished as the inverse square of the distance from the centre of the earth.* As it appeared that the central forces, both toward the sun and toward the earth, varied inversely as the squares of the distances, Newton proceeded to attack the mathe- matical problems involved in a more systematic way than any of his predecessors had done. Kepler's second law showed that the line drawn from the planet to the sun will describe equal areas in equal times. Newton showed that this could not be true, imless the force which held the planet was directed toward the sun. We have already stated that the third law showed that the force was in- versely as the square of the distance, and thus agreed ex- actly with the theory of gravitation. It only remained to * It is a remarkable fact in the history of science that Newton would have reached this result twenty yec\rs sooner than he did, had he not been misled by adopting an erroneous v alue of the earth's diame- ter. His first attempt to compute the earth's gravitation at the distance of the moon was made in 1665, when he was only twenty-three year« of age. At that time he supposed that a degree on the earth's surface was sixty statute miles, and was in consequence led to erroneous results by supposing the earth to be smaller and the moon nearer than they really were. He therefore did not make public his ideas ; but twenty years later he learned from the measures of Picabd in Prance what the true diameter of the earth was, when he repeated his calculation with entire success. tiC-S'jBfW" paMP ir I: I i!; 138 ASTRONOMY. consider the results of the first law, that of the elliptic motion. After long and laborious efforts, Nkavton was enabled to demonstrate rigorously that this law also re- sulted from the law of the inverse square, and could result from no other. Thus all mystery disappeared from the celestial motions ; and planets were shown to be simply heavy bodies moving according to the same laws tliat were acting here around us, only under very different circum- stances. All three of Kepler's laws were embraced in the single law of gravitation toward the sun. The sun attracts the planets as the earth attracts bodies here around us. Mutual Action of the Flanets. — It remained to extend and prove the theory by considering the attractions of the planets themselves. i3y Newton's third law of motion, each planet must attract the sun with a force equal to that which the sun exerts upon the planet. The moon also must attract the earth as much as the earth attracts the moon. Such being the case, it must be highly probable that the planets attract each other. If so, Kepler's laws can only be an approximation to the truth. The sun, being immensely more massive than any of the planets, overpowers their attraction upon each other, and makes the law of elliptic motion very nearly true. But still the comparatively small attraction of the planets must cause some deviations. Now, deviations from the pure elliptic motion were known to exist in the case of several of the planets, notably in that of the moon, which, if gravitation were universal, must move under the influence of the com- bined atti'action of the earth and of the sun. Newton, therefore, attacked the complicated problem of the deter- mination of the motion of the moon under the combined action of these two forces. He showed in a general way that its deviations would be of the same nature as those shown by observation. But the complete solution of the problem, which required the answer to bo expressed iu numbers, was beyond his power. othJ ticlj sul " ''''^^ tiUHBT'" ATTRACTION OF GRAVITATION. 139 sUiptic )N was Iso re- l result )m the simply at were jircum- aced in 'he sun iS here extend 18 of the motion, 1 to that oon also •acts the probable sr's laws Che sun, planets, id makes still the tist cause re elliptic •al of the ravitation the com- Newton, the deter- combined mend way I as those ion of the pressed in Gravitation Besides in each Particle of Matter. — Still another question arose. Were these mutually attractive forces resident in the centres of the several bodies attracted, or in each particle of the matter composing them ? New- ton showed that the latter must be the case, because the smallest bodies, as well as the largest, tended to fall toward the earth, thus showing an equal gravitation in every separate part. The question then arose : what would be the action of the earth upon a body if the body was attracted— not toward the centre of the earth alone, but toward every particle of matter in the earth 'i It was shown by a quite simple mathematical demonstra- tion that if a planet were on the surface of the earth or outside of it, it would be attracted with the same force^as if the whole mass of the earth were concentrated in ite centre. Putting together the various residts thus arrived at, Newton was able to formulate his great law of uni- versal gravitation in these comprehensive words : *' Every particle of matter m the immeree at^acta every other particle with a f&rce directly as the masses of the two particles, and vrwersely as the square of the distance which separates them.^^ To show the nature of the attractive forces among these various particles, let us represent by m and m' the masses of two attracting bodies. We may conceive the body w to bo composed of m particles, and the other body to be composed of m' particles. Let us conceive that each particle of the one body attracts eadi particle of the other with a force -, . Then every particle of m will be r attracted by each of the m' particles of the other, and therefore the total attractive force on each of these m par- ticles will be 'i Each of the m particles being cquaUy subject to this attraction, the total attractive force between the two bodies will be turn When a given force acts J ASTRONOMY. r „po„ a body. H will pK^oce 1-.™>"„K ^ be ^« ; and couvcrBely the accelerating force acting on the body m will be represented by the fraction -^. § 3. PBOBLEMB OP QBAVITATIOW. The problem solved by I. K^^^^ eBt genemlity, was ^^^^.^^^^^^H^^^^^^ and are given are P^J^.^ ^^ "^'^ ^^^ u^ ^ motion under with certain velocities. W hat wm ^ ^ the influence of t^-r mutual gravi^^aU^j J^^^^^^ tive motiondcH. -^-^^J^jf^^^^^^ of g^avit; will each revolve around tneir commv/ o ^l^hpe, aainthecaseof planetaiT-^^^^ ever, the illative velocity «^«^^^^*^^S,g ^„nd the bodies will separate f^-J^^^^j^f^^^^^ common centre of,g^«;f f /^"^X^ in the case where These curves are found o be ^™^'^ hj^^bolas when the velocity is exac^^ at *^ ^^tr^urvrmay be de- the velocity exceeds it. ^J^LZ^^ the two bodies scribed, the common centre of g^a^ «* ^^ ^^^^^ will be in the focus of the curve ^^^^^^^^^^^.^^ to two bodies, the problem admits of a perfectly ngo mathematical solution. rv^Hem of planetary Having succeeded in solvi^he p^bl^ of p^,^ J motion for the case of *7« ^^'.HfieTa rimilar solu- temporaries very natumlly desired to effee^a «nn mimber of Iwdies , ana nav ug ^^^ two bodies, it was necessary next to try tnai •HI larger the lal to the the body jcts on the lotion, will iiig on the in its great- the masses Bctions, and otion under I their rela- mount, they B of gravity 3. If, how- mit, the two around the ite branches. J case where erbolas when may be de- etwo bodies en restricted jctly rigorous of planetary and his con- i similar solu- em of motion ;ion of a great in the case of that of three. PROBLEMS OF GRAVITATION. HP 141 Thus arose the celebrated problem of three bodies. It is fonnd that no rigorous and general solution of this problem is possible. The curves described by the several bodies would, in general, be so complex as to defy mathematical definition. But in the special case of motions in the solar system, the problem admits of being solved by approxima- tion with any required degree of accuracy. The princi- ples involved in this system of approximation may be com- pared to those involved in extracting the square root of any number which is not an exact square ; 2 for instance. The square root of 2 cannot be exactly expressed either by a decimal or vulgar fraction ; but by incretaing the number of figures it can be expressed to any required limit of approximation. Thus, the vulgar fractions |, |J, fH, etc., are fractions which approach more and more to the required quantity ; and by using larger numbers the errors of such fraction may be made as small as we please. So, in using decimals, we diminish the error by one tenth for eve- ry decimal we add, but never reduce it to zero. A process of the same nature, but immensely more complicated, has to be used in computing the motions of the planets from then- mutual gravitation. The possibility of such an ap- proximation arises from the fact that the planetary orbits are nearly circular, and that their masses are very small compared with that of the sun. The first approximation is that of motion in an ellipse. In this way the motion of a planet through several revolutions can nearly always be predicted within a small fraction of a degree, though it may wander widely in the course of centuries. Then sup- pose each planet to move in a known ellipse ; their mutual attraction at each point of their respective orbits can be expressed by algebraic f ormulie. In constructing these formulsB, the orbits are first supposed to be circular ; and afterward account is taken by several successive steps of the eccentricity. Having thus found approximately their action on each other, the deviations from the pure eUiptic motion produced by this action may be approximately cal- hi • r- 1 1 149 ASTROIfOMT. ciliated. This being done, tlic motionfl will bo more exact- ly duteriiiinod, and the niutnal action can be niui'e exactly calcnlated. Thus, the process can be carried on step by step to any degree of precision ; but an enormous amount of calculation \& necessary to satisfy the requirements of modern times with respect to precision.* As a general rule, every successive step in the approximation is much more laborious than all the preceding ones. To understand the principle of astronomical investiga* tion into the motion of the planets, the distinction be- tween observed and theoretical motions must be borne in mind. When the astronomer with his meridian circle de- termines the position of a planet on the celestial sphere, that position is an obseiTcd one. When ho calculates it, for the same instant, from theory, or from tables founded on tlie theory, the result will be a calculated or theoretical position. The two are to be regarded as separate, no mat- ter if they should be exactly the same in reality, because they have an entii*ely different origin. But it must be re- membered that no position can be calculated from theory alone independent of observation, because all soimd theory requires some data to start with, which observation alone can furnish. In the case of planetary motions, these data are the elements of the planetary orbit already described, or, which amounts to the same tiling, the velocity and di- rection of the motion of the planet as well as its mass at some given time. If these quantities were once given with mathematical precision, it would be possible, from the theory of gravitation alone, without recourse to observa- tion, to predict the motions of the Janets day by day and generation after generation with an^ required degree of precision, always supposing that they are subjected to no influence except their mutual gravitation according to the law of Newton. But it is impossible to determine the elements or the velocities without recourse to observation ; * In the works of the great mathematicians on this subject, algcbruic formolee extending tlmraj^ many pages are sometimns given. and for 1 then mus mat] obse than obsei 80 tr their W mer] he cc aseri futur he de he mi termi oretic will « the d may I throu its pi some comn omer havin struci toler possilj latioi tirelj only vices I way table --"T--t -'—--■-'•■• PROBLEMS OF GRAVITATION. 143 exact- exactly step by amount leuts of general is much ivestiga- ition be- borne in jirclo dc- , sphere, tes it, for anded on leoretical , no mat- , because ast be re- im theory Qd theory ion alone these data iescribed, ty and di- itB mass at nee given I, from the ) observa- ly by day ■ed degree sctedtono ling to the ermine the ^servation ; |ect, algebraic ren. and however correctly they may seeiiiingly be (letcriiiineil for the time being, subHcquent obscrvatiouH alwiiyH bIiow them to have been more or less in error. The reader must understand that no astronomical observation can be mathematically exact. Both the instruments and the observer are subjected to influences which prevent more than an approximation being attained from any one observation. The great art of the astronomer consists in 80 treating and " bining his observations as to eliminate their err. , anu ». • a result as near the > ' at possible. When, by thus bumbining his observati.,-*, the astrono- mer has obtained the elements of the planet's motion which he considers to be near the truth, he calculates from them a series of positions of the planet from day to day in the future, to be compared with subsequent observations. If he desires his work to be more pennanent in its nature, he may construct tables by which the position can be de- termined at any future time. Having thus a series of the- oretical or calculated places of the planet, he, or others, will compare his predictioas with observation, and from the differences deduce corrections to his elements. We may say in a rough way that if a planet has been observed through a certain number of years, it is possible to calculate its place for an equal number of years in advance with some approach to precision. Accurate observations are commonly supposed to conamence with Beadley, Astron- omer Eoyal of England in 1750. A century and a quarter having elapsed since that time, it is now possible to con- struct tables of the planets, which we may expect to be tolerably accurate, until the year 2000. But this is a possibility rather than a reality. The amount of calcu- lation required for such work is so immense as to be en- tirely beyond the power of any one person, and hence it is only when a mathematician is able to command the ser- vices of others, or when several mathematicians in some way combine for an object, that the best astronomical tables can hereafter be constructed. AaTRONOMT. % 4. RESULTS OP GRAVITATION. From what we have said, it wiU Ihj Been that the problem of the motions of the planets under the influence of grav- itation has caUed forth all the skill of the mathematicians who have attacked it. They actually find themselves able to reach a solution, which, so far as the mathematics of the subject are concerned, may be true for many centuries, but not a solution which shall be true for all time Among those who have brought the solution so near to perfec- tion, La Place is entitled to the firstrank, although there are others, especiaUy La Gbangk, who are fully worthy o L named aloVg with him. It will be of interest to state the general results reached by these and other mathema- ^'''mcall to mind that but for the attraction of the planets upon each other, every planet would move around the sun hi an invariable ellipse, according to Kbplebs laws The deviations from this elliptic motion proved bv their mutual attraction are called perturhaiiom. When they were investigated, it was found that they were of two claies, wliich were denominated respectively perwdtc perturbatiom mi seGular variations. The periodic pert^bations consist of oscillations depend- ent upon the mutual positions of the ^ets, and there- fore of comparatively short period. Whenever after a number of revolutions, two planets return to the same nosition in their orbits, the periodic perturbations are of ^e same amount so far as these two planets are concerned. They may therefore be algebraically expressed ««. depend- ent upon the longitude of the two planets, the d«t™;^>ng one and the disturbed one. For instance, the jwrturba- tions of the earth produced by the action of M^cury depend on the longitude of the earth and on that of Jfjr- eZ. Those produced by the attraction of ^^^ /e- pS upon the longitude of the earth and on that of Vervus, and so on. seni the Let ano the one lim mo son RESULTS OF OBAVITATIOir. 145 problem of grav- naticians Ives able cs of the iries, but Among ) perfec- igh there «low that Bi.me of them rciuire tenn of thonsaiu h, and otherH hundreds of thoiiBands of years to perform the revolution. By the combined motion of them all, the centre of the ellipse deBcribcH a somewhat irregular curve. It i8 ov» dent, however, that the distance of the centre froin the sun ian never be greater than the mm of these revolving lines Now this distance shown the eccentricity of the ellipse, which is equal to half the difference between the greatest and least distances of the planet from the sun. The perihelion being in the direction 6'.^, on the opposite Bide of the sun from C, it is evident that the motion of (7 will carry the perihelion with it. It is found m this way that the eccentricity of the earth's orbit has been diminishing for about eighteen thousand years, and will continue to diminish for twenty-five thousand years to come, when it will be more neariy circular than any orbit of our system now is. But before becoming quite circu- lar, the eccentricity will begin te increase again, and so go on oscillating indefinitely. Seoular Aooeleration of the Moon.— Another remark- able result reached by mathematical research is that of the acceleration of the moon's motion. More than a century ago it was found, by comparing the ancient and modern Nervations of the moon, that the ktter moved around the earth at a slightly greater rate than she did m ancient times. The existence of this acceleration was a source of groat perplexity to La Geanob and La Place, because Lv thought that they had demonstrated mathematically that the attraction could not have accelerated or retarded the mean motion of the moon. But on continuing his m- vestigation, La Place found that there was one cause which he omitted to take account of-namely, the secular diminution in the eccentricity of the earth « orbit ^^ which we have just spoken. He found that this change in the eccentricity would slightly alter the action of the ACt'KI.KHATlON Ot TUB MOON. U1 arrics on itB are «<> slow and other» revolution, mtro of the It 18 ovi re from the BO revolving icity of the between the oni the sun. the opposite le motion of )und in this tit has been ars, and will ind years to tan any orbit 5 quite circu- in, and so go ;her remark- is that of the an a century and modem ;d around the id in ancient as a source of .ACE, because lathematically d or retarded inning his in- ras one cause [y, the secular ;h'B orbit, of it this change action of the Bun upon the moon, and that this alteration of action would l>e such that so long as the eccentricity grew smaller, the motion of the moon would continue to be ac- celerated. Computing the moon's acceleration, he found it to be e(iual to ten seconds into the square of the numlxsr of centuries, the law being the same m tliat for the motion of a falling body. That is, while in one century she would 1)6 ten seconds ahead of the place she would have occupied had her mean motion l)een uniform, she would, in two centuries, be forty seconds ahead, in three centuries ninety seconds, and so on ; and during the two thousand years which have elapsed since the observations of Hipi'archus, the acceleration would be mote than a degree. It has re- cently been found that La Place's calculation was not com- plete, and that with the more exact motliods of recent times the real acceleration computed from the theory of gravita- tion is only about six seconds. The observations of ancient eclipses, however, compared with our modem tables, show an acceleration greater than this ; but owing to the rade and doubtful character of nearly all the ancient data, there is some doubt about the exact amount. From the most celebrated total eclipses of the sun, an acceleration of about twelve seconds is deduced, while the observations of Ptolemy and the Arabian astronomers indicate only eight or nine seconds. Tliere is thus an apparent discrepancy between theory and observation, the latter giving a larger value to the acceleration. This diflEerence is now accounted for by supposing that the motion of the earth on its axis is retarded— that is, that the day is gradually growing longer. From the modem theory of friction, it is found that the motion of the ocean under the influence of the moon's attraction which causes the tides, must be accom- panied with some friction, and that this friction must re- tard the earth's rotation. There is, however, no way of determining the amount of this retardation unless we assume that it causes the observed discrepancy between the theoretical and observed accelerations of the moon. ■r-" ! 148 AHTUoNoMir. Tlow tliis uffwt in imnhKHMl will ho won hy ruflvcting that if thu (liiy iHrontinuully growing longiti' without our know- ing it, uiir obflorvutions of tlic nuMin, whicli wu niuy H(ip|M)M! to bo madu at noon, for oxanijtlo, will l)c couHtantly niado a little later, becauHO the interval from one noon to another will be continually growing a little longer. The moon con- tinually moving forward, the ol)6orvation will place her fur- ther and further ahead than she would have been observed had there l)een no retardation of the time of noon. If in the course of ages our noon-dials get to l)e an hour too late, wr nliould find the moon ahead of her calculated place by one hour's motion, or about a degree. The present theory of acceleration is, therefore, that the moon is really accelerated al)out six seconds in a century, and that the motion of the earth on its axis is gradually diminishing at such a rate as to produce an apparent additional ac- celeration which may range from two to six seconds. § 5. REKABKS ON THE THEORY OF OBAVITA- TIOK. The real nature of the great discovery of Newton is so frequently misunderstood that a little attention may be given to its elucidation. Gravitation is frequently spoken of as if it were a theory of Newton's, and very generally received by astronomers, but still linble to be idtimately rejected as a great many other theories have beeu. Not infrequently people of greater or less intelligence are found making great efforts to prove it erroneous. Every prominent scientific institution in the world frequently receives essays having this object in view. Now, the fact is that Newton did not discover any new force, but only showed that the motions of the heavens could be accounted for by a force which we all know to exist. Gravitation (Latin graviteu — weight, heaviness) is, properly speaking, tlio force which makes all bodies here at the surface of the earth tend to fall downward ; and if any one wishes to HU in, th th on to itta of foi tht J gra doi is. exp as I thai line dev tioE on(! it is con| for no unil witi oui HKALITY OF OllAVITATIoy. 14i) cting tliat nir know- tly made a to another moon con- co her f ur- m observed )on. Hin in hour too dated place 'he present on is really id that the dhninishing ditional ac- icouds. OBAVTPA- Jewton is BO tion may be lently spoken ery generally )e ^dtimately B beeu. Not elligence are eouB. Every Id frequently Kow, the fact orce, but only dbe accounted Gravitation )erly speaking, 5 surface of the one wishes to Htibvort the theory of gravitation, he uiust l)Ogin by prov- ing tliftt this force does not exist. This no one would think of doing. What Nkwton did was to show that this force, which, before his time, had been recognized only as acting on the surface of the earth, really extended to the heavens, and that it resided not only in the earth itself, but in the heavenly bodies also, and in each particle of matter, however situated. To put the matter in a terse form, what Nkwton discovered was not (/ra/oitatian, but the nniversality of gravitation. It may bo inquired, is the induction which supposes gravitation universal so complete iis to be entirely beyond doubt ? We reply that within the solar /stem it certainly is. The laws of motion as established by observation and experiment at the surface of the earth nmst be considered as mathematically certain. Now, it is an ooserved fact that tha planets in their motions deviate from ^a-aight lines in a certain way. By the first law of motion, such deviation can be protluced caly by a force ; and the dire, tion and intensity of this force admit of being ilcnlated once that the motion is determined. When thus < siho lated, it is found to be exactly represented by one great force constantly directed toward the sun, and smaller subsidiary forces directed toward the several planets. Therefore, no fact in nature is more firmly estabhshed than is that of universal gravitation, as laid down by Newton, at least within the solar system. We shall find, in describing double stars, that gravita- tion is also found to act between the components of a great number of such stars. It is certain, therefore, that at least some stars gravitate toward each other, as the bodies of the solar system do ; but the distance which separates most of the stars from each othe" rani from our sun is so immense that no evidence of gravitation between them has yet been given by observation. Still, that they do gravitate according to New ■Jj's law can hardly be seri- ously doubted by any one v ho understands the subject. 160 ASTBONOMT. The reader may now be supposed to see the absurdity of supposing that the theory of gravitation can ever be sub- verted. It is not, however, absurd to suppose that it may yet be shown to be tlie result of some more general law. Attempts to do this are made from time to time by iiM:n of a philosophic spirit ; but thus far no theory of the sub- ject having the sUghtest probability in its favor lias been propounded. i • • Perhaps one of the most celebrated of these theories is that of George Lewis Le Sage, a Swiss physicist of the last century. He supposed an infinite number of ultra- mundane corpuscles, of transcendent minuteness and veloc- ity, traversing space in straight lines in all tUrections. A smgle body placed in the midst of such an ocean of mov- ing corpuscles would remain at rest, sino« it would be equal- ly impelled in overy direction. But two bodies would ad- vance toward each other, because each of them would screen the other from these corpuscles moving in the straight line joining their centres, and there would be a slight excess of corpuscles acting on that side of each body which was turned away from the other.* One of the commonest conceptions to account for grav- itation is that of a fluid, or ether, extending through all space, which is supposed to be animated by certain vibra- tions, and forms a vehicle, as it were, for the transmission of gravitation. This and all other theories of the kind are subject to the fatal objection of proposing complicated systems to account for the most simple and elementary facts. If, indeed, such systems were otherwise known to exist, and if it could be shown that they really would produce the effect of gravitation, they would be entitled to recei»tion. But since they have been imagined only to account for gravitation iteolf, and since there is no proof of their existence except that of accounting for it, they * Reference may be made to nn article on the kinetic theories of gravitation by William B. Taylor, in the Smithsonian Report for 1876. i p I fi CAU8B OP GRAVITATION. VSi dity of be sub- it may •al law. I)y \VA:Xi ho sitb- asbeen iories is ; of the f ultra- d veloc- m&. A of mov- e equal - [)uld ad- i would in the aid be a of each 'or grav- ough all in vibra- ismission ;he kind iplicated smentary mown to ly would entitled d only to no proof • it, they theories of Report for are not entitled to any weight whatever. In the present state of science, we are justified in regarding gravitation as an ultimate principle of mattfcv, incapable of alteration by any transformation to which matter can be subjected. The most careful experiments show that no chemical pro- cess to which matter can be subjected either increases or diminishes its gravitating principles in the slightest degree. We cannot therefore see how this principle can ever be referred to any more general cause. CHAPTER VI. THE MOTIONS AND ATTRACTION OF THE MOON. Each of the planets, except Mercury and Vmua, is at- tended by one or more satellites, or moms as they are some- times familiarly called. These objects revolve around their several planets in nearly circular orbits, accompanying them in their revolutions around the sun. Their distances from their planets are very small compared with the distances of the latter from each other and from the sun. Iheir magnitudes also are very small compared with those of the planets around which they revolve. Where there are several satellites revolving around a planet, the whole of thflse bodies forms a small system similar to the solar sys- '^ in arrangement. Considering each system by itself, the satellites revolve around their central planets or " primaries," in nearly circular orbits, much as the planete revolve around the sun. But each system is carried around the sun without any serious derangement of the motion of its several bodies among themselves. Our earth has a single satellite accompanjang it in this way, the familiar moon. It revolves around the earth m a little less than a month. The nature, causes and con- sequences of this motion form the subject of the present chapter. § 1. THE MOOW'B MOTIONS AHD PHASES. That the moon performs a monthly circuit in the heav- ens is a fact with which we are all familiar from child- hood. At certain times we see her newly emerged from MOTION OF THE MOON. 168 OON. is at- 8ome- 1 their r them B from stances Their of the jre are lole of lar sys- r itself, lets <»• planets around motion in this earth in nd con- present lie heav- n child- ed from the snn's rays in the western twilight, and then we call her the new moon. On each succeeding evening, we see her further to the east, so that in two weeks she is oppo- site the sun, rising in the east as he sets in the west. Continuing her course two weeks more, she has approached the sun on the other side, or from the west, and is once more lost in his rays. At the end of twenty-nine or thirty days, we see her again emerging as new moon, and her cir- cuit is complete. It is, however, to be remembered that the sun hsis been apparently moving toward the east among the stars during the whole month, so that during the interval from one new moon to the next the moon has to make a complete circuit relatively to the stars, and move forward some 30° further to overtake the sun. The revolution of the moon among the stars is perfonned in about 27i days,* so that if we observe when the moon is very near some star, we shall find her in the same position relative to the star at the end of this interval. The motion of the moon in this circuit differs from the appareni motions of the planets in being always forward. We have seen that the planets, though, on the whole, mov- ing directly, or toward the east, are affected with an ap- parent retrograde motion at certain intervals, owing to the motion of the earth around the sun. But the earth is the real centre of the moon's motion, and carries the moon along with it in its annual revolution around the styi. To fonn a correct idea of the real motion of these three bodies, we must imagine the earth performing its circuit around the sun in one year, and carrying with it the moon, which makes a revolution around it in 27 days, at a distance only about ^^ that of the sun. In Fig. 55 suppose S to represent the sun, the large circle to represent the orbit of the earth around it, E to bie some position of the earth, and the dotted circle to rep- resent the orbit of the moon around the earth. We must * More exactly. 27* 82166. 154 A8TR0N0MT. imagine the latter to carry this circle with it in its an- nual course around the sun. Suppose that when the earth is at ^ the moon is at M. Then if the earth move to El in 27^ (lays, the moon will have made a complete revolution relative to the stars — that is, it will be at M„ the line E^ J/, being par- allel to EM. But new moon will not have arrived again because the sun is not in the same direction as lie- fore. The moon must move through the additional arc Jf, EM^, and a little more, owing to the continual ad- vance of the earth, before it will again 1)6 new moon. Phasea of the Moon. — The moon being a non-luminous body shines only by reflecting the light falling on her from some other body. The principal source of light is the sun. Since the moon is spherical in shape, the sun can illuminate one half her surface. The appearance of the moon varies according to the amount of her illumi- nated hemisphere which is turned toward the earth, as can bf seen by studying Fig. 56. Here the central globe is the earth ; the circle around it represents the orbit of the moon. TLo rays of the sun fall on both earth and moon from the right, the distance of the sun being, on the scale of the flgure, some 30 feet. Eight positions of the moon are shown around the orbit at A, E, C, etc., and the right-hand hemisphere of the moon is illuminated in each position. Outside these eight positions are eight others showing how the moon looks as seen from the earth in each position. At .4 it is " new moon," the moon being nearly between the earth and the sun. Its dark hemisphere PHASES OF THE MOON. 155 its an- B earth lOve to moon iinplete to the be at ng par- it new arrived n is not n as be- st move inal arc B more, raal ad- jefore it oon. iiminous on her : light is the sun trance of r illumi- earth, as ! central the orbit iarth and g, on the ns of the etc., and inated in are eight the earth ig nearly emisphore is then turned toward the earth, so that it is entirely invisible. At ^'the observer on the earth sees about a fourth of the illuminated hemisphere, which looks like a crescent, as shown in the outside figure. In this position a great deal of light is reflected from the earth to the moon, ren- dering the dark part of the latter visible b} a gray light. Vis. cm. "old moon in This appearance is sometimes called the the new moon's arms.'' At C the moon is said to be in hrr '* first quarter," and one half l»er illmninated hemisphere is visible. At O three fourths of the illuminated hemisphere is visible, and at B the whole of it. The latter position, when the moon is opposite the sun, is called '* full moon." After this, at H, 2>, F^ the same appearances are re- peated in the reversed order, the position D being called the "last quarter." 156 ASTRONOMY. The four principal phases of the moon are, New mo^!" " Fi4 quarter," " Full moon," " Last quarter, which occur in regt.lar and unending succession, at mter- vals of between 7 and 8 days. §2. THE SUN'S DISTURBmO FOBOB. The distances of the sun and planets being so immensely great compared with that of the moon, their attraction STn the JLrth and the moon is at all times very neariy Zal. Now it is an elementary principle of mechan cs th^if two bodies are acted upon by equal and paraM forces no matter how great these forces may be, the bo2 will move relatively to each other as if those orces did not act at all, though of course the absolute moUon of each will be different from what it otherwise would be. If we calculate the absolute attraction of the sun «pon the moon we shall find it to be about twice as great as that of r^rtZ tea-, although it is situated at 400 tim^ the distance, its mass is al^out 330,000 times as great as that of the earth, and if we divide this mass by the square of the distance 400 we have 2 as the quotient. ,.n,„.,^ To those unacquainted with mechanics, the difficulty often suggests itself that the sun ought to draw the moon away f i^m the earth entirely. But we are to remember that thesun attracts the earth in the same way that it at- tracts tSe moon, so that the difference between the sun s attraction on the moon and on the earth is only a smaU fraction of the attraction between the earth and the moon As a consequence of these forces, the moon moves around the earth nearly as if neither of them were attracted by •In this comparison of the attractive forces of the sun "poiLthe moon and upon the earth, the reader will remember that we are 8p«.k- Sr^JSf the a6«««te force, but of what is called the '^'^'^l^"'^' which is properly the ratio of the absolute force to the mass of he SatrST The earth haying 80 times the mass of the moon the s^sltf course attract it with 80 Umes tlfe ateolute force in order to produce the same motion, or the same accelerating force. SUN'H ATTRACTION ON MOON. 1B7 the sun — that is, nearly in an ellipse, having the earth in its focus. But there is always a small difference between the attractive forces of the sun upon the moon and upon the earth, and this difference constitutes a disturbing force which makes the moon deviate from the elliptic orbit which it would otherwise describe, and, in fact, keeps the ellipse which it approxhnately describes in a state of con- stant change. A more precise idea of the manner in which the sun disturbs the motion of the moon around the earth majr be gathered from Fig. 57. Here 8 represents the sun, and the circle F Q ^ JV repre- sents the orbit of the moon. First suppose the moon at N, the posi- tion corresponding to new moon. Then the moon, being nearer to the sun than the earth is, will be attracted more powerfully by it than the earth is. It will therefore be drawn away from the earth, or the action of the sup will tend to separate the two bodies. Pig. 67. Next suppobo the anon at ^the position corresponding to full moon. Here the action of the sun upon the earth will bo more Sowerful than upon the moon, and the earth will in consecjOence be rawn away from the moon. In this position also the effect of the disturbing force is to separate the two bodies. If, on the other hand, the moon is near the first quarter or near Q, the sun will exert a nearly equal attraction on both bodies ; and ince the lines of at- traction E S and Q 8 then convergt' toward 8, it follows that there will be a tendency to bring the two bodies together. The same will evidently be true at the third quarter. Hence the influence of the disturbing force changes back and forth twice in the course of each lunar month. The disturbing force in question may be constructed for any po- sition of the moon in iia orbit in the following way, which is be- lieved to be due to Mr. R. A. Pkoctok : Let 3f be the position of the moon ; let us represent the sun's attraction upon it by the line M 8, and let us investigate what line will represent the sun's attrac- tion upon the earth on the same scale. From Jf drop the perpen- Ui )1 15g A8TR0N0M7. have, Attrmctionon tmrth _ SM Attraction on moon S E ' We have taken the line 8 M it-elf to represent the attraction on the moon, so that we have Attraction on moon = 8M. Multiplying the two equations member by member, we And, Attraction on earth = S Ji x ^-gi- The line S Af is nearly equal to 8 P, so that we may take for an approximation to the required line. sr 8F '8'E = 8P^ SP* {SP+PEf _ =zSP 1 (}^8P) PE the last equation being obtained by the binomial theorm. But the fraction ^ is so small, being less than ^, that lU p«we« above the first will be small enough to be neglected. 8o we shall have for the required hne, ap—^EP. MOON'S N0DK8. 160 , This re shall the bodies together at the quarten. Conaeauentlv, upon the whole, the tendency of the sun's attraction is to diminish the attraction of the earth upon the moon. ction on «1, le for an D' rm. But bs powers } we shall equal to 2 ae scale be ;h we seek »f the sun I. If then le opporite will repre- omposition mple nuin- lie moon is ■bing force the moon. KAnUiBIf ich tends }ay the line ly from the hich draws g 8. MOnOCT OF THS MOOirS NODSI. Among tho changt« which the snn's attraction produces in the moon's orbit, Oiat which interests ns most is the constant variation in the pUne of the orbit. This plane is indicated by tho path which Xu'^ moon seems to describe in its circuit around the celestial sphere. Simple naked eye estimates of the moon's position, continued during a month, would show that her path was always quite near the ecliptic, l)ecause it would be evident to the eye that, like the sun, she was much farther north while passing from the vernal to the autumnal equinox than while de- scribing the other half of her circuit from the autumnal to the vernal equinox. It would be seen that, like the sun, she was farthest north in about six hours of right as- cension, and farthest south when in about eighteen hours of right ascension. To map out the path with greater precision, we have to observe the position of the moon from night to night with a meridian circle. We thus lay down her course among the stars in the same manner that we have formerly shown it possible to lay down the sun's path, or the ecliptic. It is thus found that the path of the moon may be considered as a great circle, making an angle of 5° with the ecliptic, and crossing the ecliptic at this small angle at two oppo- site points of the heavens. These points are called the moon's nodea. The point at which she passes from the south to the north of the ecliptic is called the ascending node; that in which she passes from the north to the south is the descending node. To illustrate the motion of the moon near the node, the dotted line a a may be taken as showing the path of the moon, while the circles show her position at successive intervals of one hour as she is ap- proaching her ascending node. Position number 9 is exactly IfiO ABTnOirOMT. end wo bIio the at the node. H we continue following her course in this way for a week, wo should find that she had moved about 90°, and attained her greatest north lati- tude at 5° from the ecliptic. At the of another week, should find that had returned to ecliptic and crossed it at her descending node. At the end of the third week very nearly, we should find that she had made three fourths the circuit of the heavens, and was now in her greatest south latitude, being 5° south of the ecliptic. At the end of six or seven days more, we should again find her crossing the ecliptic at her ascend- ing node as before. We may thus conceive of four cardinal points of the moon's orbit, 90° apart, marked by the two nodes and the two points of greatest north and south latitude. Motion of the Nodes. —A remarkable prop- r f we g licr ly for dliml iioved tallied h lati- n the end k, wo dX she the SBcd it r node, e third ly, wo iho had ths tho Qavons, in hor ititudo, of the he end m days Id again ing the ascond- )re. We seive of oints of bit, 90° by the the two est north nde. lo Nodes. >le prop- MOONS NO DBS. 161 orty of these points is tliat they are not Hxed, btit are uoiu Btantly moving. The general motion ia a little irregnlar, but, leaving out small irregularities, it is constantly toward the west. Thus returning to our watch of the course of the moon, we should find that, at her next return to the ascending node, she would not describe the lino a a as before, but the line hh nbuut one fourth of a diameter north of it. She would therefore reach the ecliptic more than 1^° west of the preceding point of crossing, and her (tther cardinal points would be found 1^° farther west as she went around. On her noxt return she would dcscribo the lino CO, then tho line dd, etc., indefinitely, each line l)eing farther toward the west. The figure shows the paths in five consecutive returns to tho node. A lapse of nine years will bring the descending node around to the place which was before occupied by the ascending node, and thus wo shall have the moon crossing at a small inclination toward the south, as shown in the figure. A complete revolution of the nodes takes place in 18.6 years. After the lapse of this period, the motion is re- peated in tlie same manner. One consequence of this motion is that the moon, after leaving a node, reaches the saTue node again sooner than she completes her true circuit in the heavens. How much sooner is readily computed from the fact that tho retro- grade motion of the node amounts to 1° 26' 31' daring the period that tho moon is returning to it. It takes the moon about two hours and a half (more exactly O**. 10944) to move through this distance ; consequently, comparing with the sidereal period already given, we find that the return of the moon to her node takes place in 27''. 82166 — O"*. 10944 = 27*. 21222. This time will be important to us in considering the recurrence of eclipses. In Fig. 59 is illustrated the effeot of these changes in the jwBition of the moon's orbit upon lior motion rela- t leu ASTRONOMY. tivo to the equator. E hero ropre«enU the vernal and uve lo mn «H ^ ^j^^ autunnml eqninox, situated 180° apart. In March, 1876, the moon's aucending node cor- responded with the vernal equi- nox, and her descending node with the autumnal one. Conse- quently she was 6° north of the ecliptic when in six hours of right ascension or near the mid- dle of the figure. Since the ecliptic is 23r north of the equator at this point, the moon at- tained a maximum declination of 284°; she therefore passed nearer the zenith when in six hours of right ascension than at any other time during the eighteen years' period. In the language of the almanac, " the moon ran high." Of course when at her greatest distance south of the equator, in the other half of her orbit, she attained a correspond- ing south declination, and cul- minated at a lower altitude than she had for eighteen years. In 1886 the nodes will change places, and the orbit will deviate from the equator less than at any other time during the eighteen years. In 1880 the descending node will be in six hours of right ascension, and the greatest angular distance of the moon from the equator will be nearly equal to that of the sun. *a(K7i=-iw^ PKHiailK OF TIIK MOON. 183 ftl and it dated 1876, lo cor- I eqni- ; node Conso- of the mre of le inid- ice the of the toon at- ition of i nearer t hours at any )ighteen angaage oon ran I at her of the If of her respond- and cul- ude than ears. In ^ places, ate from any other en years, node will ascension, \r distance e equator ^ 4. MOTION OF THB FIBIOBB. If the sun uxurtod no disturbing force on the moon, the latter would move round the earth in an oUipse according to Kki'lek's laws. But the difference of the sun's attrac- tion on the earth and on the moon, though only a small fraction uf the earth's attractive force on the moon, is yet so great as to produce deviations from the elliptic motion very much greater than occur in the motions of the planets. It also produces rapid changes in the elliptic orbit. The most remarkable of these changes are the progressive motion of the nodus just described and a corresponding motion of the pcrigoo. Referring to Fig. 62, which illus- trated the elliptic orbit of a planet, let us suppose it to represent the orbit of the moon. 8 will then represent the earth instead of the sun, and n will be the Xxmax per- igee, or the point of the orbit nearest the earth. But, instead of remaining nearly fixed, as do the orbits of the planets, the lunar orbit itself may be considered as making a revolution round the earth in about nine years, in the same direction as the moon itself. Hence if we note the longitude of the moon's perigee at any time, and again two or three years later, wo shall find the two positions quite different. If we wait four years and a half, we shall find the perigee in directly the opposite point of the heavens. The eccentricity of the moon's orbit is about 0.056, and in consequence the moon is about 6° ahead of its mean place when 90° past the perigee, and about the same dis- tance behind when half way from apogee to perigee. The disturbing action of the sun produces a great num- ber of other inequalities, of which the largest are the eoectian and the variation. Tlie former is more than a degree, and the latter not much lees. The formulee by which they are expressed belong to Celestial Mechanics, and the reader who desires to study them is referred to works on that subject. 1U4 ASTRONOMY. § 5. EOTATION OP THE MOON. The moon rotates on her axis in the same time and in the same direction in which she revolves around tlie earth. In consequence she always presents very nearly the same face to the earth.* There is indeed a small oscillation called the libt-ation of the moon, arising from the fact that her rotation on her axis is uniform, while her revolution around the earth is not uniform. In consequence of this we sometimes see a little of her farther hemisphere first on one side and then on the other, but the greater part of this hemisphere is forever hidden from human The axis of rotation of the moon is inclmed to the ecliptic about 1° 29'. It is remarkable that this axis changes its direction in a way corresponding exactly to the motion of the nodes of the moon's orbit. Let us sup- pose a line passing through the centre of the earth per- pendicular to the plane of the moon's orbit. In conse- quence of the inclination of the orbit to the ecUptic, this line will point 5° from the pole of the ecliptic. Then, suppose another line parallel to the moon's axis of rota- tion. This line will intersect the celestial sphere 1° 29' from the pole of the ecliptic, and on the opposite side from the pole of the moon's orbit, so that it will bo 6i° from the latter. As one pole revolves around the pole of the ecliptic in 18.6 years, the other wiU do the same, always keeping the same position relative to the first. • This conclusion is often a pons aaiwrum, to some who conceive that, if the swne face of the moon ia always presented to the earth, she cannot rotate at all. The difficulty arises from a misunderstaudmg of the difference between a relative and an absolute rotation. It is true that she does not rotate relatively to the line drawn from the earth to hef centre, but she must rotate relative to a fixed line, or a line drawn to a fixed star. line and in i the earth, y the same oscillation le fact that • revolution equence of hemisphere the greater ■om human ned to the it this axis 5 exactly to Let us sup- ) earth per- In conse- jcliptic, this itic. Then, xie of rota- Aero 1° 29' pposite side ; will bo 6i° around the will do the ative to the who conceive a the earth, she iderstauding of tion. It is true }m the earth to w a line drawn THE TIDES. 105 § 6. THE TIDES. The ebb and flow of the tides are produced by the un- equal attraction of the sun and moon on different parts of the earth, arising from the fact that, owing to the magni- tude of the earth, some parts of it are nearer these attracting bodies than others, and are therefore more strongly at- tracted. To understand the nature of the tide-producing force, we must recall the principle of mechanics already cited, that if two neighboring bodies are acted on by equal and- parallel accelerating forces, their motion rel- ative to each other wiil not be altered, because both will move equally under the influence of the forces. When the forces are slightly different, either in magnitude or direction or both, the relative motion of the two bodies will depend on this difference alone. Since the stin and moon attract those parts of the earth which are nearest them more powerfully than those which are remote, there arises an inequality which produces a motion in the waters of the ocean. As the earth revolves on its axis, different parts of it are brought in in succession under the moon. Thus a motion is produced in the ocean which goes through its rise and fall according to the apparent position of the moon. This is called the tidal wme. The tide-producing force of the sun and moon is so nearly like the disturbing force of the sun upon the motion of the moon around the earth that nearly the same explanation will apply to both. iiCt us then refer again to Pig. 57. and suppose i to represent the centre of the earth, the circle FQNxU circumference, M a par- tide of waver on the earth's surface, and 8 either the sun or the "^The entire earth being rigid, each part of it will move under the influence of the moon's attraction as if the whole were concen- trated at its centre. But the attraction of the moon «pon the Darticle M, being different from its mean attraction on the earth, will ffi to m^ke it move differently from the earth. , The *o«e wtadi causes this difference of motion, as already explained, ^llJe'«P«- sented by the line MA. It is true that this same distuibing force is Tcting ujon that portion of the solid earth at if as well, as upon t e water But the elwth cannot yield on account of its ngidity ; the , 166 ASTnONOMT. water therefore tends to flow along the earth's surface from M toward N. There is therefore a residual force tending to make the water higher at N than at M. If we suppose the particle M to be near F, then the point A will be to the left of F. The water will therefore be drawn in an oppo- site direction or toward F. There will therefore also be a force tending to make the water accumulate around F. As the disturb- ing force of the sun tends to cause the earth and moon to separate both at new and full moon, so the tidal force of the sun and moon upon the earth tends to make the waters accumulate both at M and F. More exactly, the force in question tends to draw the earth out into the form of a prolate ellipsoid, having its longest axis in the direction of the attracting boay. As the earth rotates on its axis, each particle of the ocean is, in the course of a day, brought in to the four positions N Q F R, or into some positions corresponding to these. Thus, the tide-producing force changes back and forth twice in the course of a lunar day. (By a lunar day we mean the interval between two successive passages of the moon acrosdthe meridian, which is, on the average, about 24** 48".) If the waters could yield immediately to this force, we should always have high tide at ^and JVand low tides at Q and R. But there are two causes which prevent tliis. 1. Owing to the inertia of the water, the force must act some time before the full amount of motion is produced, and this motion, once attained, will continue after the force has ceased to act. Again, the waters will continue to accumulate as Icng as th^re is any motion in the required direction. The result of this would be high tides at Q and R and low tides at F and N, if the ocean covered the earih and were perfectly free to move. That is, high tides would then be six hours after the moon crossed the meridian. 2. The principal cause, however, which interferes with the regularity of the motion is the obstruction of islands and continents to the free motion of the water. These deflect the tidal wave from its course in so many different ways, that it is hardly possible to trace the relation between the attraction of the moon and the mo- tion of the tide ; the time of high and low tide must therefore be found by observing at each point along the coast. By comparing these times through a series of years, a very accurate idea of the motion of the tidal wave can bo obtained. Such observations have been made over our Atlantic and Pacific coasts by the Coast Survey and over most of the coasts of Europe, by the countries occupying them. Unfortunately the tides cannot be observed away from the land, and heace little is known of the coarse of the tidal wave over the ocean. We have remarked that both the sun and moon exert a tide-producing force. ^^That of the sun is aI>out ^ of that of the moon, ^^tloew and full moon the two forces are united, and 4;he actual force is equal to their sum first thej a hi and new tide the duct moo aftei est 8 new tion, threi uallj T] lems seve: less I plan! wlii( at d tum havt sofi tidei whi( give cons give obse are the cffe< At THK TIDEa. 167 ace from M to make tlie point A will . in an oppo- be a force the disturb- n to separate ;he sun and ilate both at to draw the ; its longest earth rotates rse of a day, ime positions )rce changes Y a lunar day of the moon 48-".) If the . always have there are two ust act some 1 this motion, lased to act. (ig as there is his would be if the ocean That is, high le meridian, res with the ad continents al wave from ly possible to and the mo- t therefore be 3y comparing te idea of the ic and Pacific its of Europe, I tides cannot kaown of the iQon exert a it ^ of that forces are ir sum. At first and last quarter, when the two bodies arc 90° apart, tliey act in opposite directions, tlie sun tending to produce a high tide where the moon tends to produce a low one, and vice versa'. The result of this is that near the time of new and full moon we have what are known as the spring tides, and near the quarters what are called neap tides. If the tides were always proportional to the force which pro- duces them, the spring tides would be highest at full moon, but the tidal wave tends to go on for some time after the force which produces it ceases. Hence the high- est spring tides are not reached until two or three days after new and full moon. Again, owing to the effect of fric- tion, the neap tides continue to be less and less for two or three days after the first and last quarters, when the grad- ually increasing force again has time to make itself felt. The theory of the tides offers very complicated prob- lems, which have taxed the powers of mathematicians for several generations. These problems are in their elements less simple than those presented by the motion? of the planets, owinj* to the number of disturbing circumstances which enter into them. The various depths of the ocean at different points, the friction of the water, its momen- tum when it is once in motion, the effect of the eoast-lines, have all to be taken into account. These quantities are so far from being exactly known that the theory of the tides can be expressed onl^ by some general principles which do not suffice to enable u^ *o prfK?;''t them for any given place. From observation, howevor, it is easy to construct tables showing exactly what tid* c corrsspond 1,o given positions of the sun and moor, at any norl where tlie observations are made. With such tables th j ebb and flew are predicted for the benefit of all who *re interested, but the results may be a little uneert r'n on acccuui < f the effect of the winds upon the motion ov the wat'^r. CHAPTER VII. ECLIPSES OF THE SUN AND MOON Eclipses are a class of phenomena arising from the shadow of one body being cast upon another, and tlius wholly or partially obscuring it. In an eclipse of the sun, the shadow of the moon sweeps over the earth, and the sun is wholly or partially obscured to observers on that part of the earth where the shadow falls. In an eclipse of the moon, the latter enters the shadow of the earth, and is wholly or partially obscured in consequence of being de- prived of some or all its borrowed light. The satellites of other planets are from time to time eclipsed in the same way by entering the shadows of their primaries ; among these the satellites of Jupiter are objects whose eclipses may be observed with great regularity. g 1. THE EABTH'S SHADOW AND PENUHBBA. In Fig. 60 let 8 represent the sun and E the earth. Draw straight lines, DB Fand D' W, each tivngent to the sun and the earth. The two bodies being supposed spherical, these lines will be the intersections of a cone with the plane of the paper, and may be taken to repre- sent that cone. It is evident that the cone B VB' will be the outline of the shadow of the earth, and that within this cone no direct sunlight can penetrate. It is therefore called the earth's shadow cone. Let us also draw the lines D' B P and D B' P' to rep- resent the other cone tangent tc '^e sun and earth. It is thei the So if 1 = the ci r = R = P = 8,t we ha But h Hence The tlic rei byobsi THE EARTH'S SHADOW. 169 )0N I from the r, and tlius of the sun, th, and the ere on that in eclipse of larth, and is •f being de- he satellites psed in the primaries ; jects whose BnTHBBA. i* the earth. !ach timgent ng supposed IS of a cone on to repre- B VB' will 1 that within ; ia therefore 3' P' to rep- earth. It is then evident that within the region V B P and V B' P' the light of the sun will be piirtially but not entirely cut off. Pig. 60.— form op sitadow. DimmmoM of Shadow. —Let us investigate the distance E Ffrom the centre of tlie earth to the vertex of the shadow. Tlie triangles V E B and V 8 D axe similar, having a right angle at B and at D. Hence, VE: En = VS:SD= ES:(81}-EBy. So if we put l—VE, the length of the shadow measured from the centre of the earth. r = ES, the radius vector of the earth, R=8 D, the radius of the sun. p = EB, the radius of the earth, 8, the angular semi-diameter of the sun as seen front the earth, ir, the horizontal parallax of the sun, we have l=z VE=z ES X EB rp 8D - EB~ R^-P But hy the theory of parallaxes (Chapter I., § 7), p = r sin TT £ = r sin 8 Henco, 1 = sin ^' — sm rr The mean value of the sun's angular semi-diameter, from which the real value never differs by more than the sixtieth part, is found by observations to be altout 16' 0' = 960", while the mean value of ir 1 iro ASTRONOMY. is about 8" ■ 8. We find sin 8-An rr = • 00461, and -^^^--^j^- I - 217 Wc tliercforo conclude that tiic mean lengtli of '"• S h™ "(Srfflt on. BXtieth l™ .tan the .new in D».m- earth's centre it ^ill be equal to (l - ?,)p. for this formula gives the radius p when z = 0, and the dian.eter /*ro when ^ = / as it should.* § 2. ECLIPSES OP THE MOOW. The mean distance of the moon from the eavtli is about 60 radii of the latter, while, as we have jnst Been, the length EVoi the earth's ahadow is 217 radu ot the earth. Hete when the moon passes through the shadow she does BO at a point Iobs than three tenths of tl»e way froin E to F. The radius of the shadow here will be HVT of the radius E B oi the earth, a q.antity which we read- ily find to be about 4600 kilometres. The radius of the moon being 1736 kilometres, it will be f tl'^ly.f^.^^'Xi by the shadow when it passes through it withni 28b4 kilometres of the axis i? Fof the shadow. If its least dis- tance from the axis exceed this amount, a portion ot the lunar globe will be outside the limits B F of the shadow cone, and will thoiofonj receive a portion of the direct light of the sun. If ♦ae least distance of the centre of the nfoon .^rom the uxis of the shadow is greater than the sum of the radii of the moon and the shadow-that is, greater than 6336 kilomf.t ea-tho mooa will not enter tlic * It will bo noted that this expression is not. rigorouslv spf^klnp, the greater than K B. " — ~... rtjUMl ^1 :: i M4j:-^?- ' ^-.-.. ' ^ T ECLTPsm or rnK moon. tri *( — sin fl- n Icngtii of ; ill roiiiul nean radius n the figure li from the ,n in Decem- the distance e from the rmula gives 1 2 = / as it •til is about b seen, the [ the eartli. )W she does way from ch we read- idius of the ^ enveloped vithin 2864 its least dis- rtion of the the shadow t the direct jcntro of the er than the ow— that is, lot enter the y spottklng, the from a point on measured in a iieter woiiltl be Duld be a little shadow at all, and there will be no ellipse proper, thongh the brilliancy of the moon must be diminished wherever sho is within the pennmbral region. When an eclipse of the moon occnrs, the phases are laid down in the almanac in the following manner : Supposing the moon to be moving aronnd the earth from below np- ward, its advancing edge first meets the boundary B' P' of the penumbra. The time of this occurrence is given in the almanac as that of " moon entering penumbra." A small portion of the sunlight is then cut off from the ad- vancing edge of the moon, and this amount constantly in- creases until the edge reaches the boundary B' V of the shadow. It is curious, however, that the eye can scarcely detect any diminution in the brilliancy of the moon ifntil she lias almost touched the boundary of the shadow. The observer must not therefore expect to detect the coming eclipse until very nearly the time given in the almanac as that of " moon entering shadow." As this happens, the advancing portion of the lunar disk will be entirely lost to view, as if it were cut off by a rather ill-defined line. It takes the moon about an hour to move over a distance equal to her own diameter, so that if the eclipse is nearly central the whole moon will be immersed in the shadow about an hour after she firt strikes it. This is the time of beginning of total eclipse. So long as only a moderate portion of the moon's disk is in the shadow, that portion will be entirely invisible, but if the eclipse becomes total the whole disk of the moon will nearly always bo plainly visible, shining with a red coppery light. This is owing to the refraction of the sun's rays by the lower strata of the earth's atmosphere. Wo shall see hereafter that if a ray of light D B passes from tlie sun to the earth, so as just to graze the latter, it is bent by refraction more than a de- gree out of its course, so that at the distance of the moon the whole shattow is filled with this refracted liglit. An observer on the mow at all. nacs, be- } scarcely gc of the csent the >ry of the ;th of the the mean tn, and its as for the bhe length retain the stitute the 3 radius of 8. Multi- length of ,r 235,00() distance of iction with the moon ner will be . matter of Btimes pass through the region O VC\ and sometimes on the other side of F. Now, in Fig. ♦•0, still supposing 7? E Ji' to he the moon, let us draw the lines /> />" /" and JJ' li P tan- gent to i)othtlie n»oon and the sun, but crossing each other between these bodies at h. It is evident that outside the space P li B' P' an observer will see the whole sun, no part of the m(x»n being projected ujwn it ; while within this space the sun will be more or less obscured. The whole obscured space may bo divided into three regiotis, in each of which the character of the phenomenon is differ- ent from what it is in the others. Firstly, we have the region B VB' fonning the shadow cone proper. Here the sunlight is entirely cut off by the moon, and darkness is therefore complete, except so far as light may enter by refraction or reflection. To an observer at V the moon would exactly cover the sun, the two bodies being apparently tangent to each other all around. Secondly, we have the conical region to the right of V between the lines B Fand B' V continued. In this region the moon is seen wholly projected upon the sun, the visible portion of the latter presenting the form of a ring of light around the moon. This ring of light will be wider in proportion to the apparent diameter of the sun, the farther out we go, because the moon will appear smaller than the sun, and its angular diameter will dimin- ish in a more rapid ratio than that of the sun. This region is that of annular eclipse, because the sun will pre- sent the appearance of an annulus or ring of light around the moon. Thirdly, we have the region PB VandP'B V, which we notice is connected, extending around the interior cone. An observer hero would see the moon partly projected upon the sun, and therefore a certain part of the sun's light would be cut off. Along the inner boundary B V and B' V the obscuration of the sun will be complete, but the amount of sunlight will gradually increase out to 174 AtiTRONOMY. tliu outer boiimlary B /' Ji' 7", wliorc tlio whole sun is vi8il>lu. This region uf pai-'jiil obseuration is culluil the jtcnumbra. To sliow more clearly t'iic phenomena of solar c('li|iHo, we jircseiit another figure reprcsi-iiting the pentimhra of Fio. fll.— noiTRB or hhadow por MxnvhAit bclifbb. tlie moon tlirown upon the earth.* The outer of the two circles S represents the limb of the sun. The exterior tan- gents which mark the boundary of the shadow cross each other at F before reaching the earth. The earth being a little beyond the vertex of the shadow, there can be no total ccli^)se. In this case an observer in the penumbral region, C or D Oy will see the moon partly projected on the sun, v/hile if ho chance to be sitnated at O he will see an annular eclipse. To show how this is, we draw dotted lines from O tangent to the moon. The angle bolAoen these lines represents the apparent diameter of the moon as seen from the earth. Continuing them to the sun, they show the apparent diameter of the moon as projected upon the sun. It will be seen that in the case supposed, when * Tt will In; noted that nil the HgiircH of eclipses nrc necessarily drawn very much out of proportion. Really the sun is 400 times the distance of the moon, whicli again is 00 times the radius of the earth. But it would lie entirely impossible to draw a figure of this proportion ; wi are therefore obliged to represent the earth as larger than the sun, ani the moon as nearly half way between the earth and sun. th th th sic rei in ec ili .lJ i a *i HiM.j. i | I IA< I ~Z}. O 8UT1 IB tllud the • C('li|)HO. iuil>ra uf E0LIP8B8 OF TUh' HUN. 175 the vortex of the shadow is hotweon the earth and moon, tlie hitter will neccHsarily apjHjar sniallcr tlian the rjui, and the observer will see a portion of the solar disk on all sides of the moon, as shown in Fig. (52. If the moon were a little nearer the eaith than it is rep- resented in the figure, its shadow would reach the earth nn. >f the two ten or tan- cross each firth being can be no pcnumbral ojected on [le will see •aw dotted i bfct*veen the moon 5 sun, they icted upon ised, when iHarily drawn the distance kfth. But it portion ; we the sun, and FlO. 62.— DARK BOOT OF MOON nUMECTBD OH SUN DORINU AN ANNOLAR ECLIP8B. in the neighljorhood of O. We should then liave a total eclipse at each point of the earth on which it fell. It will be seen, however, that a total or annular eclipse of the sun is visible only on a very small portion of the earth's sur- face, because the distance of the moon changes so little that the earth can never be far from the vertex Fof the shadow. As the moon moves around the earth fi-om west to east, its shadow, wliether the eclipse be total or annu- lar, moves in the same direction. The diameter of the shadow at the surface of the earth ranges from zero to 150 miles. It therefore sweeps along a belt of the earth's sur- face of that breadth, in the same direction in which the tal or annular one. The space CD (Kig. «!) over ^vhi(;h the penumbra extends is generally of about one hull the diameter of the earth. Roughly speaking, a partni! eelipso of the su); may sweep over a ]M>rtion of the earth's surface ranging from zero to perhaps one fifth or one sixth of the whole. There are really more eclipses of the eun than «)f tlie moon. A year never passes without at least two of the fonner, and sometimes five or six, while there are rarely mon than two eclipses of the moon, an«l in many years now: ;Af all. But at any one place more eclipses of the moon vill ,c seen than of the sun. The reason of this is that an eclipse of the moon is visible over the entire hemi- sphere of the earth on which the moon is shining, and aa it lasts several hours, observers who are not in this hemi- sphere at the beginning of the eclipse may, by the earth's i-o- tation, be brought into it before it ends. Thus the eclipse will be seen over more than half the earth's surface. But, as we have just seen, each eclipse of the sun can be seen over oidy so small a fraction of the earth's surface as to more than compensate for the greater absolute frequency of solar eclipses. It will be seen that in order to have either a total or ari«^ nular eclipse visible upon the earth, the line joining the centres of the sun and moon, being continued, must strike the earth. To an observer on this line, the centres >il_ to 3400 ler to Hcc I' hcforo- vliich tliu 'illcllliltctl ,!ineri(loH, Htial ino- f^ ■2 w may bo 'h Hurfttco , ISl) over t onu liuit' u parttu) lio eartli'^ une eixth lan of tlic wo of the are rarely lany years : the moon ^hig is that tiro hemi- r, and as it this heini- earth'sro- tho eclipse face. But, ;an be seen irfaco as to ! frequency total or art*-' joining the lued, must the centres i^igjjaftioua i iawm ^em^^m^mymMMuj.. ' -^.-^- f>#''rm ->f .^ .-,'f..^^.,Ai..m L.i'i^.:. . r ,... .,). .■ r ; ?'g^'tP#H!#Piy'^S^' '"' ._J CIHM/ICMH Series. CIHM/ICMH Collection de microfiches. Canadian Instituta for HIatorical Microraproductiona / Inatftut Canadian da microraproductiona hiatoriquaa BEGURRENOK OF EGLlPSBa. 177 of the two bodies will seem to coincide. An eclipse in which this occurs is called a central one, whether it be total or annular. The accompanying figure will perhajis aid in giving a clear idea of the plienoinena of eclipses of both sun and moon. FlO. 63.^COMPARI80N OV SHADOW AND PKNimBRA OF EARTH AKD MOON. A IS THE POSITION OP TUB MOON DUBINO A BOLAK, B DCB- INO A LUNAR ECLIPSE. § 4. THX BXOUBBHNCE OF aOUPSES. If the orbit of the moon around the earth were in or near the same plane with that of the latter around the sun — that is, in or near the plane of the ecliptic — It will be readily seen that there would be an eclipse of the sun at every new moon, and an eclipse of the moon at every full moon. But owing to the inchnation of the moon's orbit, described in the last chapter, the shadow and penumbra of the moon commonly pass above or below the earth at the time of new moon, while the moon, at her full, commonly passes above or below the shadow of the earth. It is only when at the moment of new or full moon the moon is near its node that an eclipse can occur. The question now arises, how near must the moon be to its node in order that an eclipse may occur ? It is found by a trigonometrical computation that if, at the moment of new moon, the moon is more than 18° '6 from its node, no eclipse of the sun is possible, while if it is less &aa 18** • 7 an eclipse is certain. Between these limits an ecUpse auLj occur or fail aocording to the respeotiye dis- taaocB of tiie snn and moon from tibe earth. Half way be- tw«en these limits, or say 16** from the node, it is an even y IW ASTRONOMY. uhance that an eclipBe will occur ; toward the lower limit (13° -7) the chances increase to certainty; toward the upper one (18° • 6) they diminish to zero. The correspond- ing limits for an eclipse of the moon are 9° and 12^° — that is, if at the moment of full moon the distance of the moon from her node is greater than Vi^^ no eclipse can occur, while if the distance is less than 9° an eclipse is cer- tain. Wo may put the mean limit at 11°. Since, in the long run, new and full moon will occur equally at all dis- tances from the node, there will be, on the average, sixteen eclipses of the sun to eleven of the moon, or nearly fifty per cent more. Fra. 64.— mutntias Imiar cdipM at diUMeiit dMuow ftrom ttw nod*. The dark circle* ar« Ike earth's ahadow, tiia ceatre of whieh Is alwajrs in the ecliptie AB. The moon's orbit is represented ^jOD. At (3> the eelipso is central and total, at J'Uls partial, and at K there is baieljr an ecUpse. tut an illustration of these computatioiM, let us investigatQ the lim- its within which a central eclipse of the sun^ total or annular, can occur. To allow of such an eclipse, it is oTident, from an inspec- tion of Fig. 61 or 68 that the actual distance of the moon from the plane of the ecliptic must be less than the earth's racUus, because the line joining the centres of the sun and earth always lies in this plane. This distance must, therefore, be leaa than 6870 kilo- m«trea- The mean distance of the moon being 884,000 kilometres, the sine of the latitude at this limit is jfUHi mi<1 ^^ Utitude itself is 57'. The formula for the latitude u, by sjdwrical trigonometry, rin latitude = sin • sin «, » being the inclination of the moon's orbit (5* 80, ud « the distance of the moon from the node. The value of sin < is ncA far fmn A. while, in a rough calcuhtion, we may suppose the comnaratively small angles « and the latitude to be tlM mow as theb miiQs. We may, therefore, suppose tt=rll latitDdesB t^'. BEVURttENOE OF ECLIPSES. 179 ird the lower limit iinty ; toward thu The correspond- 9" and 12^°— that [6 distance of the 2^** no eclipse can an eclipse is oer- 1°. Since, in the equally at all dis- he average, sixteen I, or nearly fifty per ■ fhmittMiiod*. Tiiedirk jrt In the ecllptie AB. Ttta OMUml andtoUl, at JTUU ua invettigatQ the lim- , total or wmular, can idnit, from an iii8pec> Be of the moon Iram a the earth's radius, I and earth always lies te less than «870]dlo- ig 884,000 kilometns, and the latitude itself iherical trigonometry, h 80, and « the distance a < is not far f mn A, MS the ooapnntiTely e as their uaes. We We therefore conclude that if, at the moment of new moon, the distance of the moon from the node is less than 101° there will be a central eclipsr, of the sun, and if greater than this there will not be such an •«lip«o. The eclipse limit may range half a degree or more on each side of this mean value, owing to the varying distance of the moon from the earth. Inside cf 10 a central eclipse may be re- garded as certain, and outside of 11° as impossible. If the direction of the moon's nodes from the centre of the earth were invariable, eclipses could occur only at the two opposite months of tlie year when the sun had nearly the same longitude as one node. For instance, if the lon- gitudes of the two opposite nodes were respectively 54'^ and 234°, then, since the sun must be within 12° of the node to allow of an eclipse of the moon, its longitude would have to be either between 42° and 66°, or between 222° and 246°. But the sun is within the first of these re- gions only in the month of May, and within the second only during the month of November. Hence lunar eclipses could then occur only during the months of May and No- vember, and the same would hold true of central eclipses of the sun. Small partial eclipses of the latter might be seen occasionally a day or two from the beginnings or ends of the above months, but they would be very small and quite rare. Now, the nodes o " the moon's orbit were act- ually in the above directions in the year 1873. Hence during that year eclipses occurred only in May and No- vember. We may call these months the seasons of eclipses for 1873. But it was explained in the last chapter that there is a refaoigrade motion of the moon's nodes amounting to 19^° in a year. The nodes thus move back to meet the sun in its annual revolution, and th meeting occurs about 20 days eariiw every year than it did the year before. The re- salt is that tibe season of eclipses is constantly shifting, so that each season ranges throngfaout the'whole year in 18*6 yean. Tor instance, the season oorreeponding to that of November, 1873, had moved baok to July and August in 180 ASTRONOMY. 1878, and will wicur in May, 1882, while that of May, 1873, will bo shifting back to November in 1882. It may bo intoreBting to illuBti-ato this by giving the days in which the sun is in conjunction with the nodes of the moon's orbit during several years. AKCuding Node. 1879. January 24. 1880. January 6. 1880. December 18. 1881. November 30. 1882. November 12. 1883. October 25. 1884. Octobers. DeiicondlDg Mode. 1879. July 17. 1880. Jime27. 1881. June 8. 1882. May 20. 1883. May 1. 1884. April 12. 1885. March 25. During these years, eclipses of the moon can occur only within 11 or 12 days of these dates, and eclipses of the sun only within 15 or 16 days. In consequence of the motion of the moon's node, three varying angles come into play in considering the occur- rence of an eclipse, the longitude of the node, that of the sun, and that of the moon. We may, however, simplify the matter by referring the directions of the sun and moon, not to any fixed line, but to the node — t&at is, we may count the longitudes of these bodies from the node instead of from the vernal equinox. We have seen in the last chapter that one revolution of the moon relatively to the node is accomplished, on the average, in 27 • 21222 days. If we calculate the time required for the sun to re- turn to the node, we shall find it to be 346 • 6201 days. Now, let US suppose the sun and moon to start out together from a node. At the end of 346 '6201 days the sun, having apparently performed nearly an entire rev- olution around the celestial sphere, will again be at the same node, which has moved back to meet it. But the moon will not be there. It will, during the interval, have passed the node 12 times, and the 18th paasage will not occur for a week. The same thing will be tme for BECUBRENVB OF ECL/P8E8. 181 ilo that of May, ill 1882. lis by giving tho iritli tlio nodes of mdlng Node. July 17. June 27. June 8. May 20. May 1. April 12. Marcli 25. in can occur oitly 1 eclipses of tho oon's node, three lering the occur- node, that of the lowever, simplify of the sun and lode — that is, we fl from the node have seen in the loon relatively to ige, in 27-21222 for the sun to re- ^•6201 days. lOon to start out 16-6201 days the y an entire rev- i again be at the leet it. But the he interval, have paamge will not lill be true for IS successive returns of the sun to the node ; we shall not lind the moon there at the same time with the sun ; she will always have passed a little sooner or a little later. But at the 10th return of the sun and the 24:2d of the moon, the two bodies will be in conjunction within half a degree of the node. Wo iind from the preceding periods that 242 returns of the moon to the node require 6585 - 357 days. 19 ♦' " sun " " " 6585-780 " The two IxMlies will therefore pass the node within 10 hours of each other. This conjunction of the sun and moon will be the 223d new moon after that from which wo started. Now, one lunation (that is, the interval between two consecutive new moons) is, in the mean, 29-530588 days ; 223 lunations therefore require 6585-32 days. The new moon, therefore, occurs a little before the bodies reach the node, the distance from the latter being that over which the moon moves in 0*-036, or the sun in 0^-i59. We readily find this distance to be 28' of arc, somewhat less than the apparent semidiameter of either body. This would be the smallest distance from either node at which any new moon would occur during the whole period. The next nearest approaches would have occurred at the 35th and 47th lunations respectively. The 36th new moon would have occurred about 6° before the two bodies arrived at the node from which we started, and the 47th about 1^° past the opposite node. No other new moon would occur so near a node before the 223d one, which, as we have just seen, would occur 0" 28' west of the node. This period of 223 new moons, or 18 years 11 days, was called the Saras by the ancient astron- omers. It will be seen that in the preoedinff calcnlalioiu we taftre aMumed the ran aad moon to more uniformly, so that the aucceuive new moon's occurred at equal intervals of 29 -680588 days, and at equal angular distances around the ecliptic. In fact, however, the month- ly uuqnslities in the motioa at the moon cause deviatiomi from lier 182 AaTRONOMY. % mean motion which amount to six doffroei in either direction, while the annual inequality in the motion of the sun in lonsitiide is nearly two degrees. Consequently, our conclusions respecting the point at which new moon occurs may be astray by eight degrees, owing to these inequalities. But there is a remarkable feature connected with the Saros which greatly reduces these inequalities. It is that this period of 6585} days corresponds very nearly to an integral number of revolutions both of the eartli round the sun, and of the lunar perigee around the earth. Hence the inequalities both of the moon and of the sun will be nearly the same at the beginning and the end of a Saros. In fact, ttSSfil days is about 18 years and 11 days, in which time the earth will have made 18 revolutions, and about 11° on the 10th revolution. The longitude of the sun will therefore be about 11° greater than at the Mginning of the period. Again, in the same perioccur 3° 20' west of ' the eclipse depends the sun, the moon, « led may be stated die of any eclipse. whotlior of the buii or of the moon. Tliuii let lis go for- ward 0585 dayu, 7 hours, 42 minuluH, nnd wo shall find unothor cclipsu very Hiniilar to the tirst. Iludnced to years, the interval will be 18 years and 10 or 11 days, according as a 2Utli day of February intervenes four or live times during the interval. This Iwing true of every eclipse, it follows that if we record all the eclipses which occur dur- ing a ])eriod of 18 years, we shall find a new set to begin over again. If the period were an integral number of (lays, each eclipse of the new set would be visible in the same regions of the earth as the old one, but since there is a fraction of nearly 8 hours over the round imniber of days, the earth will be one third of a revolution further advanced before any eclipse of the new set begins. Each eclipse of the new set will therefore occur about one third of the way round the world, or 120° in longitude west of the region in which the old one occurred. The recur- rence will not take place near the same region until the end of three periods, or 54yeanB ; and then, since there is a slight deviation in the series, owing to each new or full moon occurring a little further west from the node, the fourth eclipse, though near the same region, will not necessarily be similar in all its particulars. For example, if it be a total eclipse of the sun, the path of the shadow may be a thousand miles distant from the path of 54 years previously. As a recent example of the Saros, we may cite some total eclipses of the ' '.* well known in recent times ; for instance : 1842, July 8th, l** a.m., total eclipse observed in Europe ; 1860, July 18th, 9^ a.m., total eclipse in, America and Spain ; 1878, Jnly 39th, 4^ p.m., one visible in Texas, Col- orado, and on the coast of Alaska. A yet more reuuurkable series of total edipsee of the 184 AStTttONOMT. mn uro those of thoyeiirB 1850, 1808, 188<), utc, tho dates and regions 1)eing : 1850, August 7tli, 4'' I'.M., in tlie Pacific Ocean ; 18«8, August 17tli, 12'' I'.M., in India; 1880, August 2»th, 8'' a.m., in tho Central Atlantic < )cean and Southern Africa ; iyo4, Septeud)er Uth, noon, in South America. This scries is remarkable for the long duration of total- ity, aiaouuting to some six minutes. Let lis now consider a series of ecliiHies recurring at i-eg- ular intervals of 18 years and 11 days. Since every suc- cessive recurrence of such an eclipse throws the conjunc- tion 28' further toward tho west of the node, the conjunc- tion must, in process of time, take place so far back from tho node that no eclipse will occur, and the series will end. For the same reason there must be a commencement to the series, the first eclipse being east of the node. A new eclijjse thus entering will at first be a very small one, but will be larger at every recurrence in each Saros. If it is an eclipse of the moon, it will be total from its 18th until its 36th recurrence. There will then be about 18 partial eclipses, each of which will bo smaller than the kst, when they will fail entirely, the conjunction taking place so far from the node that the moon does not touch the earth's shadow. The whole interval of time over which a series of lunar eclipBCS thus extend will be about 48 periods, or 865 years. When a series of solar eclipses begins, the penumbra of the finst will just graze the earth not far from one of the poles. There will then be, on the average, 1 1 or 12 partial eclipses of the sun, each lai^r than the preceding one, occurring at regular intervals of one Saros. Then the central line, whether it be that of a total or annular eclipse, will begin to touch the earth, and we shall have a series of 40 or 50 central edipBes. The central line will strike near one pole in the first part of the MRW ; in the equatorial regions about the middle of tlie aoriet, Mid will VtlAUAGTKIlH OF MHJI'SKS. 185 SB, utc, thu datu8 itic Ocuuii ; Ouiitrul Atliiiitic Aiiiurica. durutiun uf tuta]- i rufurriiig at ivg- 8incti every buc- row8 the coiijuuu- tude, the coiiji.iic- ) so far back from he 8erio8 will end. ioinmeucemeiit to the node. A new cry gniall one, but ch Sarofl. If it is ^rom its 13th until 10 about 18 partial ban the last, when Aking place so far touch the earth's iver which a series >out 48 periods, or , the penumbra of ■ from one of the ^, llor 12 partial he preceding one, Sarofl. Then the I total or annular nd we shall have a e centra] line will theMrifls; in the diesMMs, Mid will Icuvu the earth by thu other ]k)Iu ut the end. Tun or twulvu partial u(!li|MuM will follow, and this particular ite- rics will cuHHu. Tliu wbolu iiuinbur in the series will avur- age iHitwuun 00 und 7U, occupying u fuw uenturies over a thoiiHiuid years. $( 6. 0HABA0TBB8 OF B0LIP8B8. Wc have seen that tho |ioMibility of a tutal eclipso of the sun iiriaeH from the occaHional very Hiight excesa of tho apparent anaular diameter of the moon over that of the sun. This excess is so slight that such an eclipse can never last more than a few minutes. It may be of interest to point out the circumstances which favor a long duration of totality. These are : (1) That the moon should be as near as possible to the earth, or, technically speaking, in perigee, because Its angular diameter as Hccn from the earth will then be greatest. (2) That the sun should be near its greatest distance from the earth, or in apogee, because then its angular diameter will be the least. It is now in this position about tne end of June ; hence the most favorable time for a total eclipse of very long duration is in the summer months. Since the moon must bo in perioee and alao between the earth and aun, it follows that the longitude of the perigee must be nearly that of the sun. Hie longitude of the sun at the end of June Iwing 100*, thia is the most favorable longi- tude of the moon's perig«e. (8) The moon must m very near the node in order that the cen- tre of the shadow may fall near the equator. The reason of this con- dition is, that the duration of a total eclipse may be oonaidenibly increased by the rotation of the earth on ita azia. We have seen that the shadow sweeps over the earth from west toward east with a velocity of about 8400 kilometres per hour. Since the earth rotates in the same direction, the velocity relative to the observer on the earth's surface will be diminished by a quantity depending on thia velocity of rotation, and therefore greater, the greater t& velocity. Tb« vebMsity of rotation u greatest at the earth's equator, when it amounts to 1600 kibtowtres per hour, or nearly half the velocity of the moon's shadow. Hence tne duration of a total ecline may, with- in the tropics, be nearly doubled bvthe earth's rot^ion. when all the favorable cireumstances comUne in the way we have just de- scribed, the duration of a total eclipse within the tropica will be about seven minutes and a half. In our latitude thrmudmum du- rati^ thuy luht tliuy nffonl iiiii(jiio opportuiiitiun f«*r iiivuHti^itiii); tho matter wliivh Uch in tlio iiiiiiic<]iuto noighlHirhood uf tho Mini. IJiidor ordiiiury cinniiiiHtniicofl, this inattor Jh rondottid untiroly iiiviHihlo by the ufTiil^'iicu of tliu Holnr ruyA which ilhiiniiiattiouratiiRmpIioro ; hutwlieii a tMHlyeo distant m the moon '\% intorpoMMl Ixjtwoun tlie olMiorvor and tlie Ban, the ray^ of tliu latter are cut off from a region a hundred miles or more in extent. TIiuh an amount of darkness in tho air is secured wliich \» lm{)oHHilile under any other circumstanceH wlien the sun is fur alN>vo tho horizon. Still this durkness is by no iiieaiiH complete, bccausu the sunlight is reflecterho«Hl uf COM, this iiiattor m l^uiicu of tlio »M>lar l)utwhun u iMxlyBo n the olworvor aiul ff from a region a iiiH an amount of impoHHihlu under h iar alM>vo the rt c<»nipleto, becaunu I on wliiuh the sun (larkneiw may )hs i watcli van be road reful to sliado \m few minutes bo- Btars of the first where to look for turps of the land- unt of the investi- ngs to the physical jfore be given in a I. — A phenomenon logons to an eclipse itar by the moon, are nearer than the i from time to time lanots are, however, ry rare occurrence, 1 generally so faint perior light of the rhera are not more istronomy of a well- occnltation of a star igh there are several 1 to pass over a star. (KnmLTArioN oif ntahn. m Rut the moon is so largo and hur angular motion so rapid, that she |>aHHOS over Kome star visible to the naked uye uvery few days. Such phononiona are toniiod oi^eultations of star» hy the nwim. It mast not, however, be supposed that they can l)e observed by the naked eye. In general, tliu moon is so bright that only stars of the first magnitude can Ih) seen in actual contact with her limb, and even then the ("ontact must be with the nnilluminated limb. But with the aid of a telescope, and the pretlictions given in the Ephomcris, two or threu of thefju occultations can be ol)served during nearly every lunation. #' I 1 1^ i,'»' i 'I'. CHAPTER VIII. THE EAKTH. Our object in the preeent chapter is to trace the ^ecte of terr^trial gravitation and to study the changes to v,rn is subject in various places. Since every part any odJ«^ j ^ ^^ ^ow belonging to the « 1- „AS8 Airo MBsrre ot tb» "abth. We begin by ««»» definitioiie »id Bome prmciple. i«- orier to make it ™»'V™* * ,*£fjSoa. A«M II. to trace the effects idy the changes to . Since every part irt as well as every that the earth and Testrial form a sort of which are firmly action. This attrac- mpoBsible to project Tth into the celestial owbelon^ng to the ain upon it forever. p TBS BABTH. 1 some principles re- etc. , ioA aBthe qwmtUy qf this quantity of mat- ht of the body— thiB orce of attraction be- By the inertia of the we muBt apply tott^ aite velocity. Mathe- ^two method* shwdd iment it i» f ««»<* ^»"* MA88 OP THE EARTH. 189 the attraction of all bodies is proportional to their inertia. In other words, all bodies, whatever their chemical consti- tntion, fall exactly the sjune numl»er of feet in one second under tlio influence of gravity, supposing them in a vacu- um and at the same place on tlie earth's surface. Although the mass of a body is most conveniently determined oy its weight, yet mass and weight must not be confounded. The vieight of a body is the apparent force with which it is attracted toward the centre of the earth. As we shall see hereafter, this force is not the same in all parts of the earth, nor at different heights above the earth's sur- face. It is therefore a variable quantity, depending upon the position of the body, while the mass of the body is re- garded as something inherent in it, which remains constant wherever the body may be taken, even if it is carried through the celestial spaces, where its weight wonld be reduced to almost nothing. The unit of mass which we may adopt is arbitrary ; in fact, in different cases different units will be more con- venient. Generally the most convenient unit is the weight of a body at some fixed place on the earth's surface — ^the city of Washington, for example. Suppose we take such a portion of the earth as will weigh one Ulogram in Wash- ington, we may then consider the mass of that particular lot of earth or rock as a kilogram, no matter to what part of the universe we take it. Suppose also that we conld bring all tlie matter composing die earth to the city of 'W^ashington, one kilogram at a time, for the purpose of weighing it, returning each kilogram to its place in the earth immediately after weighing, so that there should he no disturbance of the earth itself. The sum total of the weights thus found would be the mass of the earth, and would be a perfectly definite quantity, admitting of being n kilograms or pounds. We esn readily cal* MM of a vf the earth or any d and determined on that onr nnits of are practically units e should weigh out ishington, and then be less heavy at the iwe take a pound hter at the equator, ih other, and the tea one pound. Since R,y by weights which JO, say Washington, irorld without being hich has any given d in lihis way, have r place, although its ring bdance or any ictusl weights were > weight of the same I part of the earth to t use this sort of an of metal which are >W8 that what we call lat^l bodies around ti« earth. Aoectt^Bg isaey is not simply a tre of the eartii, but lie f oroes Mrinng frrai the attractions of all the separate parts which compose the earth. The question may arise, how do we know that each particle of the earth attracts a stone which falls, and that the whole attraction does not reside in the centre ? The proofs of this are numerous, and consist rather in the exactitude with which the theory represents a great mass of disconnected phenomena than in any one principle ad- mitting of demonstration. Perhaps, however, the most conclusive proof is found in the observed fact that masses of matter at the surface of the earth do really attract each other as required by the law of Newton. It is found, for example, that isolated mountains attract a plumb-line in their neighborhood. The celebrated experiment of Cav- endish was devised for the purpose of measuring the at- traction of globes of lead. The object of measuring this attraction, however, was not to prove that gravitation re- sided in the smallest masses of matter, because there was no doubt of that, but to determine the mean density of the earth, from which its total mass may be derived bj simply multiplying the density by the volume. It is noteworthy that though astronomy affords us the means of determining with great precision the rdaUve masses of the earth, the moon, and all the pknets, it does not enable us to determine the absolute mass of any hea- venly body in units of the weights we use on the earth. We know, for instance, from astronomioal rasearch, that the son has about 828,000 times the mass of the earth, and the moon only ^ of tiiis mass^ bat to know the abso- lute mass of either of them we must know how many kili^rams of matter the eardi contains. To d^ermine this, we mi^ know the mean douity of the earth, and this is something about which direct observation can give us no inf