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OF THE AstronomicalSociety of the Pacific.

Vol. XXI. San Francisco, California, April 10, 1909. No. 125

ADDRESS OF THE RETIRING PRESIDENT OF THE SOCIETY, IN AWARDING THE TO DR. GEORGE WILLIAM HILL.

By Charles Burckhalter.

The eighthaward of the BruceGold Medal of thisSociety has beenmade to Dr. GeorgeWilliam Hill. To thosehaving understanding of the statutes,and the methodgoverning the bestowal of the medal,it goes without sayingthat it is always worthilybestowed. The statutes requirethat, for each medalawarded, eminent astronomers, - thedirectors of six of thegreatest observatories in theworld, - shallbe askedeach to nominatethree astronomers worthy to receivethe Bruce Medal "fordistinguished services to astron- omy." The six observatorieshave always been Harvard, Lick,and Yerkesin thiscountry, and Berlin,Greenwich and Paris in the Old World. From the nominationsthus made theDirectors of thisSociety, by ballot,choose the Medalist. The listof namesof thosewhom we havethus honored, and in honoringthem we havehonored the Society, is an inspiring one- all householdnames in .The firstaward was to ,a nativeof Nova Scotia; the secondto Arthur Auwers, a German;the thirdto Sir David Gill, an Englishman;the fourthto Giovanni Virginio Schiapa- relli, an Italian; the fifthto Sir William Huggins, an Englishman; the sixthto anotherGerman, Hermann Carl Vogel ; theseventh to Edward C. Pickering,the first native- bornAmerican to receivethe medal; and the eighthto our presentmedalist, also an American. The Bruce Medal, like astronomy,knows nothing of boundaries. 52 Publications of the In a paper in the Popular Science Monthlyfor October, 1908,Professor Pickering calls attention to thefact that there are onlysix Americanswho are membersof three or more of the sevenleading National Scientific Societies (England, France,United States, Russia, Germany, Austria and Italy) and of thesesix, threeare astronomers;and I wish to add, of thethree astronomers our medalistis one. ProfessorPick- ering also pointsout thatevery living Bruce Medalist belongs to threeor moreof thesesocieties. Althoughthis is butthe eighth award of themedal, so great is thecare takenin its bestowal,and withsuch wisdom have theselections been made, that the Bruce Gold Medal is already recognizedas one of thegreatest honors that can be conferred uponan astronomer. The statutesfor the bestowalof the Bruce Medal require thePresident of theSociety at thismeeting to "Announcethe awardand thereasons for making it." Let me heresay that no adequatereview of Dr. Hill's workcan be comprisedin thisaddress ; in his "CollectedMathematical Works" there are eighty-fourmemoirs, every one of whichshows the master mind. And I wishat once,frankly, to confessmy inability to give an opinionof thevalue of Dr. Hill's servicesto astron- omy,and in this,your President feels that he is butone of a large company. Only specialistsin celestialmechanics are competentto appreciateand appraisethe utilityand impor- tanceof his monumentalachievements; the opinionsof those intellectualgiants who are his colaborers,alone can reflectto us a trueestimate of theirvalue. ProfessorNewcomb in his "Reminiscencesof an Astrono- mer"says of Dr. Hill that"he willeasily rank as thegreatest masterof mathematicalastronomy during the last quarterof thenineteenth century." Wedded to his chosenfield of labor, he worked,year afteryear, scarcelyknown to the public, patient,underpaid, his abilityappreciated by only a few of his colleagues,but contentthat he .couldadvance his beloved science. Speakingof thistime, Newcomb says of him: "Here was perhapsthe greatest living master in thehighest and most difficultfield of astronomy,winning world-wide recognition for his countryin thescience and receivingthe salary of a depart- mentclerk." AstronomicalSociety of the Pacific. 53

The list of honors conferredupon our medalist by institu- tions of learning in this and foreigncountries is a long one, among whichI may mentionthe Damoiseau prize,of the Paris academy,the degreeof Doctor of Laws, by CambridgeUniver- sity,England, and the gold medal of the Royal Astronomical Society. Our own recognitionis, of necessity,somewhat tardy, forhe has been nominatedagain and again by eminentastron- omers,for the honorof the Bruce Medal. I have said that only an expert can rightfullyweigh such work as that of our medalist. Such an one is Mr. Henri PoiNCARÉ,who has written,in French,an introductionto the "Collected Mathematical Works of George William Hill." From this, the followingabstract has been prepared by Dr. R. G. Aitken, of the Lick Observatory:- "Dr. Hill," says Poincaré, "is one of the most original figuresin the American scientificworld. Throughout all his works and his calculationshe has remaineda strangerto the feverishlife that has troubledothers; he has conducted his researchesin isolation,formerly in the bureau of the Nautical Almanac,more recentlyat his quiet home in the Hudson Val- ley. This reserve,I was about to say this shyness,has been a happy circumstancefor science,for it has permittedhim to carry his ingeniousand patientresearches to their conclusion withoutsuffering distractions from the constantaccidents of the world outside." Dr. Hill was born in New York, March 3, 1838. His father,an Englishman,came to America in 1820 at the age of eight; his mother,of an old Huguenot family,transmitted to him the traditionsof the earliest colonists of America. He passed his infancyat the farmat West Nyack, about twenty- five miles from New York, which his fatherbought shortly after Hill's birth. "He always loved his home, he returnedto it as often as he could, and when he resigned from the Nautical Almanac Officehe made his permanenthome there,and therehe pursues his studies in tranquility,avoiding, as far as possible, even journeysto New York." At the age of seventeenhe enteredRutger's college, New Brunswick,New Jersey,where his professorin 54 Publications of the was Dr. Strong, a friendof Bowditch, the translatorof "Laplace's MécaniqueCéleste." "Dr. Strong was a manof traditions,one who praisedthe bygonetimes; forhim Euler was the god of mathematics, and afterEuler, decadencebegan ; it is truethat this is a god one may worshipwith profit. With rare exceptionsDr. Strong'slibrary was pitilesslyclosed against all booksof later datethan 1840. Happily,excellent works on celestialmechan- ics werewritten prior to 1840,for example, those of Laplace, Lagrange, Poisson, and Pontécoulant. These were the mastersby whom Hill was introducedto therudiments of the science." Receivinghis degreein 1859 he wentto Harvardto con- tinuehis mathematical studies, but in thespring of 1861joined the NauticalAlmanac staff and spentthe next thirtyyears of his lifein thatservice. Those werethe most fruitful years in pointof scientificproductions. At thistime the Nautical AlmanacOffice was at Cambridge,Mass., under the direction of ProfessorRunkle, the founderof the Mathematical Monthly,in whichprizes were proposedfor the solutionof mathematicalproblems. "One of the firstarticles published," says PoiNCARÉ,"revealed the hand of a masterand easily gainedthe prize. It dealtwith the functionsof Laplace and thefigure of theEarth. The authorwas Mr. Hill, who was just aboutto leavecollege." Hill continuedto make manycontributions on various mathematicalsubjects, generally, however, connected in some way withcelestial mechanics, to the MathematicalMonthly, and to theAnalyst and similarjournals. Whilethe NauticalAlmanac Office remained in Cambridge underRunkle, Hill's workfor it was done at his homein West Nyack. But,in 1877,when New comb assumed charge, "he wishedto undertakea colossaltask, the reconstructionof thetables of all theplanets ; Mr. Hill's partin thistask was themost difficult ; it was thetheory of Jupiterand Saturn, with whichhe beganto busyhimself about 1872." He could not wellcarry on thiswork away from his chief and his colleagues, so was obligedto leave his home,a sacrificereadily made in view of the importanceof the work. But he returnedto his homewhen the bureauwas movedto Washington. AstronomicalSociety of the Pacific. 55

In 1892 he resignedfrom the NauticalAlmanac. For a shorttime thereafter he helda professorshipin ColumbiaUni- versity,but he did notcontinue in thisposition long, and since then"has livedalone with his booksand his souvenirs." Inspectionof the tablesof contentsin the fourvolumes of his"Collected Mathematical Works," published by the Carnegie Institution,of Washington,shows what a widerange his inves- tigationstook, in everydepartment of celestialmechanics. Hill's greatestwork, according to all competentjudges, consistsin his researcheson, and contributionsto, the .This accountof it is basedon Poincaré's introduction. He saysit is thework to whichHill "devotedall theoriginal- ityof his genius." To comprehendHill's work,Poincaré givesa briefaccount of the stateof the theorywhen Hill beganhis work. Two greatworks on thissubject had beencompleted at thistime - Hansen's and Delaunay's- each exhibitingthe resultsof thehighest sagacity and of extremepatience. The methodsadopted by these two investigators are radically different; Hansen, whocompleted his work first, had in mind thepurely utilitarian object of computingaccurate lunar tables, hencehe calculatednumerical co-efficients directly and did not troublehimself to findanalytical expressions. The tablesin actualuse to-dayare basedon Hansen's calculations,and it is probablethat the new theories,more scientific, more satisfac- toryin theirspirit, will not give very different results. Delau- nay, on theother hand, presented his inequalitiesin the form of algebraicformulas. "He givesus, then,not only the theory of the ,but thetheory of any satellitethat revolves, or thatmay revolve about any whatever. From this point of view he leaves Hansen far in the rear. The methodwhich has led himto thisresult constitutes the most important advance that has been made in celestialmechanics since Laplace. To-day,perfected and shortened,it has becomean instrument thatevery one can use, and thathas alreadyrendered good servicein everypart of astronomy." UnfortunatelyDelaunay's seriesconverges only with exces- sive slownessand is thereforeunsuited to numericalcomputa- tions. Hill promptlymastered Delaunay's theoryand made £Ó Publications of the it the subjectof variousmemoirs, but his own methodis very differentand veryoriginal. Delaunay's seriesdepends on fiveconstants, the eccentrici- ties,the inclination, the solar parallax, and a quantitycalled m, whichdepends upon the mean motions. If we supposethe first fourto becomezero, we have a particularsolution of the differentialequations. This solutionwill be verymuch more simplethan the general one sincethe greater part of theine- qualitieswill disappear,the only remaining one beingthe one knownas the"variation." On theother hand this solution does notprecisely represent theMoon's trajectory, but it willserve as a firstapproximation, sincethe neglectedconstants are in effectvery small. The choiceof thisfirst approximation is far more advantageous thanthe Keplerianellipse, since for this ellipse the perigeeis fixed,whereas in thereal orbit it is in motion.The differential equationsare at thesame time simplified, since, the eccentricity and parallaxbeing zero, the is supposedto describea cir- cumferenceof verygreat radius. Mr. Hill again simplified theequations by judiciouslychoosing his variables. He does notexpress them in polarcoordinates, but in the rectangular, and thisis a greatstep in advance. Again his variablesare not referredto fixedaxes, but to axes possessinga uniform rotation,equal to themean angular motion of the Sun. This was a new simplification,since the timeno longerfigures explicitlyin theequations. "But themost important advantage is the following:For an observersituated on thosemoving axes the Moon will appearto describea closedcurve, if the eccentricities,the inclinationand the parallax are zero. As the differentialequations are otherwiserigorous, this is the firstexample of a periodic solution of the problem of three bodies whose existencehas been rigorouslydemonstrated. Recentlyperiodic solutions have assumedcapital importance in celestialmechanics. [Poincaré himselfis one of the fore- mostinvestigators in thisbranch, if indeed he has an equal.] But our medalistwas not contentmerely to demonstrateits existence; he studiedthis orbit (or theseorbits) in detail,and determinedpoint by pointthe closed trajectoriesand calcu- lated the coordinatesof thesepoints to manydecimals." AstronomicalSociety of the Pacific. 57

Not to follow PoiNCARÉ'sdiscussion in too much detail, it may sufficehere to say that in the furthertransformations and solutionsof these equations,Mr. Hill showed as much daring as originality. His solution involved the considerationof an infinitenumber of linear equations. Not only did he not hesi- tate to considerthese, but he also studied determinantsof an infiniteorder, a thing never before attempted1and, happily, his daring was justifiedby his success. If we compare his methods with those of Delaunay, we shall findthat Hill's method by three approximationsgave results for the constant of the motion of the lunar perigee correctcertainly to thirteendecimals, whereas (to quote from Hill's own paper) "Although Delaunay has been at the great pains of computingeight terms of this series,they do not sufficeto give correctlythe firstfour significantfigures [eight decimals] of the quantity sought. ... As well as can be judged frominduction it would be necessaryto prolong the series, in powers of m, as far as m27,in order to obtain an equally preciseresult." This will sufficeto illustratethe great advance made by Hill in the developmentof the lunar theory,for the method thus applied to the motionof the lunar perigee may also be applied to the motionof the node. The main difficultiesare thus con- quered,and the subsequentapproximations are relativelyeasy. Neverthelesspractical difficultiesstill remain,and the fieldis open for new improvementsand new theories. It is unneces- sary in this connectionto do more than merelyto referto the workof anothereminent investigator, Professor E. W. Brown, who has developedmethods of whichmuch is expected. The inequalitiesin the lunar motion so far discussed have been thosedue to the actionof the Sun. They are the ones that would be producedif only the Sun, the Moon, and the existed,and they were reduced to materialpoints. But there are other inequalitiesin the lunar motion,produced in part by the directaction upon the Moon of the otherplanets, and in 1 At least not to Dr. Hill's knowledgeat this time. It appearsthat Adams, of Neptunianfame, had been led by his researchesto a determinantvery nearly identicalwith the one discoveredby Hill. Poincaré also refersto a memoirby M. KoTTERiTzscHin PoggendorfsAnnalen, though he says the methodthere given had "nothingin commonwith that of the AmericanGeometer." 58 Publications of the part,indirectly, by thedisturbing action of theseplanets upon themotion of theEarth about the Sun. Further,the Earthis not spherical,and its equatorialprotuberance exercises an influenceupon the lunar motion. In theplanetary perturbations of theMoon's motion we may furtherdistinguish between the secularvariations and the periodic. Mr. Hill has studiedsuccessively the secular accel- erationof themean motion, that of themotion of theperigee, and theinfluence of thevariations of theecliptic. Nor has he neglectedthe others, and he has also givencareful study to the effectof theoblateness of the Earthand discussedthe results of thependulum determinations of the force of gravityon the Earth'ssurface. But the lunartheory did not absorball of Hill's timeor energy. Time and again he turnedto investigationsdealing withgeneral problems, the generaltheory of planetaryper- turbationsand specialproblems in thisfield. His largestpiece of workof thiskind is the one undertakenfor the Nautical Almanacin connectionwith and as partof Newcomb's"colos- sal task,"referred to above,of thecomplete discussion of the mutualperturbations of Jupiterand Saturn. Laplace had enteredupon this theory which presents very greatdifficulties because of whatis termed"the great inequal- ity,"but his evaluationof theterms of the secondorder was buta roughapproximation. Hansen was morefortunate, and so arrangedhis calculationsthat he was able to estimatethe importanceof termshe neglected,but he carriedout his work completelyonly in the case of Saturn. For Jupiterhe was contentto stop withterms of the firstorder. We may pass over othermemoirs as relativelyunimportant until we come to that of Le Verrier, publishedin 1876. His theoryis indeedcomplete, but his formulaeare entirelyliteral, and so arrangedthat the co-efficients of the inequalities are expressed in termsof correctionsto all of theosculating elliptic elements. In practicethis leads to excessivelylong computationsand thereare apparentlysome outstanding errors in the formulae; besides,the tables are notconvenient to use. But Hill began his investigationsin 1872,before Le Verrier's resultswere publishedand at a timewhen it was uncertainwhen they would AstronomicalSociety of the Pacific. 59

be published.The tablesthen in actualuse were Bouvard's, and thesewere quite inadequate to meetthe needs of astrono- mers. Havinga purelypractical end in view,the construction of accuratetables in the shortestpossible time, Hill did not seek to developa new methodof investigation.He adopted Hansen's, butwith modifications that greatly simplified it. So greatan amountof computingwas involvedin thisresearch thatMr. Hill devotedto it sevenand a halfyears of time, beingrelieved during this timeof all routineduties in the NauticalAlmanac Onice and havingthe services of an assistant to checkby duplicatecomputations all themore important cal- culations.The resultof thislong, and in manyways necessa- rilytedious work is a splendidvolume of theoreticalresearches, and two volumesof accuratetables, one of the motionof Jupiter,the other of themotion of Saturn. The recentprogress of celestialmechanics has receivedand continuesto receiveMr. Hill's constantattention. Hansen's and Delaunay's methods,Gyldén's intermediateorbit, the recentdevelopments in periodicsolutions of the three-body problem[as presentedby Poincaré, Darwin and others],have all beenassimilated, and manyof themanalyzed and discussed in someof his publishedmemoirs. In his concludingsentence Poincaré says: "Thereis there- forenot a singlepart of celestialmechanics to whichhe is a stranger,but his chiefwork, that which will makehis name immortal,is his theoryof theMoon. It is therethat he is not only an able artist,a carefulinvestigator, but an inventor, originaland profound.I do notwish to say thatthe methods whichhe createdare applicableonly to the Moon. I am well persuadedto thecontrary ; I thinkthat those who are engaged in studyof theminor will be astonishedat the relief fromdifficulties they will experienceon theday when,having fathomedtheir spirit, they apply them to thisnew object. But at presentit is forthe Moon thatthey have proved themselves. Whenthey enter upon a widerdomain, we oughtnot to forget thatit is to Mr. Hill thatwe owe so preciousan instrument." We have earnestlyhoped that our medalistwould be with us thisevening, but in a letterto Dr. Hale he said he did not feel strongenough to undertakethe journey; therefore,in 6o Publications of the his absence,I hand to you, Mr. Secretary,for transmission to Dr. Hill, thismedal, the highesthonor this Societycan possiblyconfer upon an astronomer,and withit I ask you to sendthe greetings and bestwishes of theAstronomical Society of thePacific, and thatwe trustthe evening of his lifemay be a longand happyone. March27, 1908.

THE LAWS OF COSMICALEVOLUTION AND THE EXTENSIONOF THE SOLARSYSTEM BEYOND NEPTUNE.

By T. T. T. See. (Abstractof addressdelivered January 30, 1909.)

It has long been consideredsomewhat of a reproachto astronomythat the processes of cosmogonyhave remainedso obscurethat definite laws could not be establishedregarding eventhe mode of formationof thesolar system, while still less was knownabout the laws forthe developmentof othersys- temsin space. In view of thegreat progress of the physical sciencessince the time of Laplace, one is compelledto admit thatthis criticism of theoldest and mostexact of thephysical sciencesis notwholly unjust and withouta certainfoundation. Not onlyhas the failureof researchesin cosmogonyaffected astronomyadversely, but it has also narrowedthe fieldof effortin severalof therelated sciences. This shouldnot, how- ever,occasion surprise among those who studythe history of the physicalsciences. For as cosmogonydepends on the othersciences for its fundamentaldata, any circumstance whichhas affectedthem adversely would also retardthe devel- opmentof cosmogonyitself, and vice versa. In additionto thenatural difficulties inherent in the devel- opmentof a complexand dependentscience like cosmogony, anotherhas arisenfrom the demoralizationof spiritsdue to thefailures of previousinvestigators. Those who have labored foryears without gaining any satisfactory light on thesubject mayeasily convince themselves that there are no definitelaws