GIFT OF 
 
7L 
 
 RE FOR 
 
 TO THE 
 
 / 
 SMITHSONIAN INSTITUTION, 
 
 ON THE 
 
 HISTORY OF THE DISCOVERY 
 
 OF 
 
 NEPTUNE. 
 
 BY BENJAMIN APTHORP GOULD, JR. 
 
 WASHINGTON CITY: 
 
 PUBLISHED BY THE SMITHSONIAN INSTITUTION. 
 1850. 
 
REPORT 
 
 ON THE 
 
 HISTORY OF THE DISCOVERY 
 
 NEPTUNE. 
 
 BENJAMIN APTHORP GOULD, JR. 
 
 WASHINGTON CITY: 
 
 PUBLISHED BY THE SMITHSONIAN INSTITUTION. 
 1850. 
 
CAMBRIDGE: 
 
 STEREOTYPED AND PRINTED BY 
 
 METCALF AND COMPANY, 
 
 PRINTERS TO THE UNIVERSITY. 
 
REPORT. 
 
 IN a general account of the discovery of the planet Nep- 
 tune, and of the remarkable circumstances which preceded and 
 attended this discovery, it cannot at this late period be expected 
 that any new views should be brought forward, or any material 
 facts cited, which have escaped the notice of the eminent 
 astronomers who have already written upon this subject. The 
 facts are before the world. The numerical data are in all the 
 books. The history is too strange to be forgotten by any one 
 who has once studied it. In the following Report, therefore, it 
 can scarcely be hoped that the facts should be more clearly 
 arranged or more concisely presented than has already been 
 done by Airy, Biot, and Herschel, or that any thing, bearing up- 
 on the history prior to 1847, be narrated, which cannot be found 
 elsewhere. ^ 
 
 The strange series of wonderful occurrences of which I am 
 to speak is utterly unparalleled in the whole history of sci- 
 ence ; the brilliant analysis which was the direct occasion of 
 the search for a trans-Uranian planet, the actual detection of 
 an exterior planet in almost precisely the direction indicated, 
 the immediate and most unexpected claim to an equal share of 
 merit in the investigation, made in behalf of a mathematician 
 till then unknown to the scientific world, and finally the 
 startling discovery, that, in spite of all this, the orbit of the new 
 planet was totally irreconcilable with those computations which 
 had led immediately to its detection, and that, although found 
 in the direction predicted, it was by no means in the predicted 
 
4 REPORT ON THE HISTORY OF 
 
 place, nor yet moving in the predicted orbit. This series of 
 events, together with the since developed theory of Neptune, 
 constitute the subject of my Report. 
 
 The correctness of the prophecy, made l by a British writer 
 within a few weeks of the discovery of the planet, that the 
 future historian of astronomy would find it necessary to change 
 his pen more than once while discussing this subject, will hardly 
 now be called in question by the strongest partisans of any of 
 those illustrious scientists, who have occupied themselves with 
 the theories of Uranus and Neptune; although the most 
 important point at issue is a question very different from the one 
 then anticipated. 
 
 There has never been a more complicated case presented for 
 the sober judgment of the impartial historian ; and the tempta- 
 tions to dwell upon the romance and poetry of the subject are 
 extremely strong. I purpose, however, in the following history 
 of the circumstances which led to, and have been connected 
 with, the discovery of Neptune, to present the subject as simply 
 as possible ; leaving to others the philosophical considerations and 
 poetic fancies which it suggests, and aiming only at clearness 
 and impartiality. I shall arrange the Report as nearly as possi- 
 ble in the chronological order. 
 
 The planet Uranus was discovered by Sir William Herschel 
 on the 13th of March, 1781, and, although at first supposed to be a 
 comet, was before the end of the year recognized 2 as one of the 
 primary planets of our solar system. Circular elements were 
 first computed 3 during the summer of 1781, by Lexell, of St.- 
 Petersburg, at that time in London ; and others were soon after 
 published in Russia, 4 France, 5 and Germany. 6 The computation 
 
 1 London Athenceum, Nov. 21, p. 1191. 
 
 2 Memoires de I' Acad. des Sciences, Paris, 1779, p. 526 ; Berliner Astronomisches 
 Jahrbuch, 1784, p. 215. 
 
 3 Ada Acad. Petrop., 1780, Mem., p. 307. 
 
 4 Nova Acta Acad. Petrop,, I., Hist., pp. 72, 76, 81 ; Acta Acad. Petrop., 1780, 
 Mem., p. 312. 
 
 5 Lalande, Mem. Acad. Sc. Paris, 1779, p. 526. 
 
 6 KlOgel, of Helmstadt, Berl. Ast. Jahrb., 1785, p. 193. Hcnnnert, ibid., p. 205. 
 
THE DISCOVERY OF NEPTUNE. 5 
 
 of a planetary orbit was at that time a most laborious and trou- 
 blesome process, by no means to be compared with the easy 
 methods in use since Gauss gave l to the world,, ^he elegant 
 and simple formulas of the " Theoria Motus"^ No elliptic 
 elements were computed, therefore, until the year 1783, during 
 which year elliptic orbits differing but slightly from each other 
 were published by Mechain, 2 Laplace, 3 Caluso, 4 and Hennert ; 5 
 and in the French and German astronomical Ephemerides for 
 1787 (published in 1784) were tables of the new planet. The 
 name Uranus, originally proposed 6 by Bode, had at that time 
 become almost universal upon the Continent, although in Eng- 
 land the names " Herschel " 7 and " Georgium Sidus " 8 (or 
 simply " The Georgian ") were generally used until within a 
 very few years, the planet being still designated by the latter 
 name in the British Nautical Almanac. The symbol adopted 9 
 with the name Uranus was that of platinum (6), but in 
 England and France the symbol , formed from the discov- 
 erer's initial, is generally used. 
 
 In the mean time Bode, the Astronomer Royal of Prussia, 
 had suggested 10 that Uranus might have been observed by as- 
 tronomers before the discovery of its planetary nature, and 
 consequently that earlier observations might be found by a 
 proper search in the catalogues of fixed stars. This happy idea 
 prompted him to study over the old star-catalogues, and his 
 search was crowned with abundant success. 11 In August, 1781, 
 
 1 March, 1809. 
 
 2 Hist. Acad. Bed., 1782, pp. 41, 49. 
 
 3 Mim. de Bruxelles, T. 5, Hist.,ip. xlix., Mem., p. 43; Journal de Paris, May 
 31, 1783 ; Berl Ast. Jahrb., 1786, p. 247 ; Conn, des Temps, 1786, p. 3. 
 
 4 Ephem. Astron. Medial, 1784, p. 199. 
 
 5 Berl. Ast. Jahrb., 1786, p. 223. 
 
 6 Berl. Gesellschaft Naturforschender Freunde, March 12, 1782, III. p. 350. 
 
 7 Proposed by Lalande, Dec. 22, 1781; v. Mdm. Acad. Paris, 1779, p. 526; 
 Hist. Acad. Berlin, 1782, p. 39. 
 
 8 Proposed by Herschel, Roy. Soc. Philos. Trans., 1783, p. 1. 
 
 9 Proposed by Koehler, of Dresden; Berl. Ast. Jahrb., 1785, p. 191; Nova 
 Acta Petrop., I. 69. 
 
 10 Berl Ast. Jahrb., 1784, p. 218. 
 
 11 See also Bode, Vom neuentdccJcten Planeten, Berl., 1784. 
 
6 REPORT ON THE HISTORY OF 
 
 he discovered 1 that a star 2 (No. 964) in Tobias Mayer's Cata- 
 logue, which had been observed 3 September 25, 1756, was not 
 to be found in the place indicated, and that it had not been 
 mentioned on various occasions when all the other stars of 
 equal magnitude in the same vicinity had been observed. 
 Uranus must have been nearly in that place at the same time, 
 according to the orbits of Laplace and Mechain, 4 and the pre- 
 sumption became thus quite strong, that this supposed fixed star 
 of Mayer was really the planet Uranus. In the same way 
 Bode and Fixlmillner of Kremsmiinster found, 5 in 1784, that 
 a star observed 6 by Flamsteed, December |f, 1690, and 
 called by him 34 Tauri, was in all probability also Uranus. 
 The same discovery seems to have been made and verified 7 
 independently in France by Lemonnier and Montaigne. Ob- 
 servations of the planet were thus obtained, which embraced 
 an interval of more than an entire revolution, and from these 
 two old observations in 1690 and 1756, and the two oppositions 
 of 1781 and 1783, Fixlmillner computed elliptic elements, 8 
 which not only fully satisfied all the four places upon which they 
 were based, but all the observations known. They were as 
 follows : 
 
 Epoch, Jan. 1st, 1784. 
 
 o / // days. 
 
 Mean anomaly, 297 9 25 Tropical period, 30587.37 
 
 Long, perihelion, 167 31 33 Mean distance, 19.18254 
 
 Long. asc. node, 72 50 50 Eccentricity, 0.0461183 
 
 Inclination, 46 20 Mean daily trop. motion, 42".3704 
 
 which, as the event has proved, were very near the truth. 
 
 1 Berl. Ast. Jahrb., I784,p. 219 ; 1785, p. 189 ; Wurm, Geschichte des Uranus, p. 35. 
 
 2 Reduced Conn, des Temps, 1778, p. 195; Fixlmillner, Berl. Acad., 1783, Hist., 
 p. 15 ; Bessel, Fundamenta Astronomic, p. 283. 
 
 3 Mayer, Opera Inedita, ed. Lichtenberg, I. p. 72. 
 
 4 Hist. Acad. Berl, 1782, p. 40. 
 
 5 Hist. Acad. Berl, 1783, p. 15 ; Berl Astr. Jahrb., 1787, pp. 243, 247. 
 
 6 Hist. Ccdestis Britann., II. p. 86, 2d ed. 
 
 7 Mem. Acad. Roy. des Sciences, 1784, p. 353. 
 
 8 Hist. Acad. Berl, 1783, p. 19; Berl. Astr. Jahrb., 1787, p. 249. 
 
THE DISCOVERY OF NEPTUNE. I 
 
 Wurm, of Niirtingen, computed, 1 at nearly the same time, 2 
 elements 3 very similar ; and Fixlmillner, 4 von Zach, 5 and others, 6 
 constructed tables. Those by the former are in the Berlin As- 
 tronomical Almanac for 1789 (published 1786). These tables, 
 as above remarked, satisfied all the observations, and continued 
 to represent the planet's course most satisfactorily until 1788, 
 when a discrepancy between theory and observation became very 
 apparent ; 7 and Fixlmillner was compelled to disregard Flam- 
 steed's observation, and to calculate new elements and tables 8 
 from the oppositions since Herschel's discovery, and the single 
 observation by Mayer. But at a later period, after Gerstner, 9 
 Lalande, 10 Oriani, 11 and Duval 12 had determined the perturba- 
 tions by Saturn and Jupiter, and Delambre had published 13 his 
 tables of Uranus, the discrepancies vanished, and the same ele- 
 ments were made to represent perfectly all the modern observa- 
 tions, and the two former ones of Mayer and Flamsteed. In 1788, 
 Lemonnier discovered 14 that he had also observed 15 Uranus as a 
 fixed star in 1 764 and 1769. These observations of Uranus, made 
 prior to Herschel's discovery of its planetary character, are called, 
 by way of distinction, " ancient observations." Others have 
 since been found, so that we now have twenty in all, viz. : 
 
 1 Geschichte des Planeten Uranus, p. 37 ; Beti. Astr. Jahrb.^1788, p. 193. 
 
 2 Summer of 1784. 
 
 3 See Reggio, Ephem. Astr. Mediol, 1784, p. 199. 
 
 4 Berl Astr. Jahrb., 1789, p. 113. 
 
 5 Ibid., 1788, p. 217. 
 
 6 Caluso, Him. de Turin, 1787, pp. 113, 132, 137; Dom Nouet, Conn, des 
 Temps, 1787, p. 176; Robison, Trans. R. Soc. Edinb., 1788, Vol. I. p. 305. 
 
 7 Wurm, Gesch. des Uranus, p. 38. 
 
 8 Berl Astr. Jahrb., 1792, p. 159. 
 
 9 Ibid., 1792, pp. 214, 219. 
 
 10 M&m. de VAcad. Sc. Paris, 1787, p. 182. 
 
 11 Ephem. Astr. Mediol, 1790, 1791. 
 
 12 Him. Acad. Berl, 1789; Berl. Astr. Jahrb., 1793, 115. 
 
 13 Computed 1789, crowned 1790. Wurm, Geschichte des Uranus, p. 89; La- 
 lande, Astronomic, ed. 3 me , Vol. I. 
 
 14 See Lalande, Acad. Sc. Paris, 1789, p. 204; von Zach, Comm. Soc. Reg. Got- 
 ting., 1789, p. 91. 
 
 15 Conn, des Temps, 1821, p. 339. 
 
REPORT ON THE HISTORY OF 
 
 One 1 in 1690, by Flamsteed, Dec. f . 
 
 One 2 1712, Flamsteed, *g^. 
 
 Four ^ 1715, Flamsteed, j^ti^'"' Apr. ff. 
 
 Two 4 " 1750, " Lemonnier, Oct. 14, Dec. 3. 
 
 One 5 " 1753, " Bradley, Dec. 3. 
 
 One 6 " 1756, " Mayer, Sept. 25. 
 
 One 7 " 1764, " Lemonnier, Jan. 15. 
 
 Two 8 " 1768, " Lemonnier, Dec. 27, 30. 
 
 Six 9 " 1769, Lemonnier, Jan. 15, 16, 20, 21, 22, 23. 
 
 One 10 " 1771, " Lemonnier, Dec. 18. 
 
 Mr. Le Verrier has, however, found reason n to suspect the 
 accuracy of Flamsteed's observation of f^J, 1715, so that 
 there are really but nineteen available ones. 
 
 The best tables of Uranus which existed before the masterly 
 and accurate researches 12 of Le Yerrier, in 1845 and 1846, were 
 those 13 computed by Bouvard in 1821. Bouvard was acquaint- 
 ed with all the ancient observations which we know, excepting 
 three by Flamsteed in 1715. In the introduction to his tables, 
 he announced 14 that he had been utterly unable to find any el- 
 liptic orbit, which, combined with the perturbations by Jupiter 
 and Saturn, would represent both the ancient and the modern 
 observations. The best tables which he could obtain by the 
 
 Hist. Ccel. Brit., II. 86, 2d ed. ; Hist. Acad. Berl., 1783, p. 16. 
 Hist. Ccd. Brit., IE. p. 537. 
 
 Ibid., pp. 549, 551 ; Conn, des Temps, 1820, pp. 409, 410. 
 Conn, des Temps, 1821, p. 339. 
 
 Bradley, Astron. 06s., I. p. 155 ; Bessel, Fund. Astr., p. 283 ; Greenwich Plan. 
 Reduct., I. p. 300. 
 
 6 Hist. Acad. BerL, 1783, p. 8 ; Bessel, Fund. Astron., p. 284; Conn, des Temps, 
 1778, p. 195. 
 
 7 Bouvard, Conn, des Temps, 1821, pp. 341, 342. 
 
 8 Ibid., pp. 341-343. 
 
 9 Rid., pp. 341,343-347. 
 
 10 Ibid, pp. 341, 347. 
 
 11 App. Conn, des Temps, 1849, p. 125. 
 
 12 Comptes Rendus de I 1 Acad. des Sc., XXI. p. 1050; XXII. p. 907. 
 
 13 Tables Astronomiques, Paris, 1821. 
 
THE DISCOVERY OF NEPTUNE. 9 
 
 use of both represented neither of them, in any way at all sat- 
 isfactory. On the other hand, by using modern observations 
 only, he was enabled to find elements which, although they 
 gave errors amounting sometimes to 74" for the ancient obser- 
 vations, still satisfied all the modern ones comparatively well, 
 never differing more than 10" from theory, and generally much 
 less. 
 
 " Such," said he, 1 " is the alternative which the formation of 
 tables of the planet Uranus presents ; if we combine the an- 
 cient observations with the modern ones, the first will be passa- 
 bly represented, while the second will not be represented with 
 the precision which they require ; but if we reject the former, 
 and retain the latter only, the resultant tables will have all de- 
 sirable precision 'for the modern observations, but will be incapa- 
 ble of properly satisfying the ancient ones. We must choose 
 between two alternatives. I have thought it proper to abide by 
 the second, as being that which combines the most probabilities 
 in favor of the truth, and I leave it to the future to make known 
 whether the difficulty of reconciling the two systems result from 
 the inaccuracy of the ancient observations, or whether it depend 
 upon some extraneous and unknown influence, which has acted on 
 the planet." 
 
 He therefore summarily rejected the former observations, and 
 founded his tables upon the latter alone, adducing arguments 
 against the accuracy of the ancient observations, and forgetting 
 how well they harmonized with one another, and had harmo- 
 nized with the elements obtained soon after the discovery of the 
 planet. 
 
 But a very few years after the publication of Bouvard's ta- 
 bles, important differences between theory and observation be- 
 came 2 again manifest, and attracted the attention of astronomers. 
 
 Airy alluded, 3 in 1832, to these discrepancies, in his Report 
 
 2 See Ast. Nachr., IV. p. 56; VI. pp. 195, 209 ; KOnigsberg Observ., 1826 ; Cam- 
 bridge (Eng.) Observ.j 1826, 1830. 
 
 3 Reports of British Association, I. p. 154. 
 
 2 
 
10 REPORT ON THE HISTORY OP 
 
 to the British Association on the Progress of Astronomy, and 
 mentioned that the tables, constructed only eleven years previ- 
 ously, were in error nearly half a minute of arc. 
 
 It is an easy thing to censure Bouvard for the readiness with 
 which he abandoned the ancient observations, now that we 
 know that the discrepancies were caused by the action of an ex- 
 terior planet, and that the maximum of error in the ancient ob- 
 servations amounted 1 only to nine seconds. But the illus- 
 trious Bessel, who, had his priceless life been spared a very little 
 longer, would have seen his suspicions most fully confirmed, 
 spoke 2 as follows, in a public lecture, 3 in the beginning of the 
 year 1840, six years and a half before the discovery of Nep- 
 tune : 
 
 " In my opinion, Bouvard made much too light of the matter ; 
 inasmuch as, after he had found himself unable to reconcile the 
 theory both with the ancient observations and the forty years' 
 series of modern ones, he contented himself with the remark, 
 that the former were not so accurate as the latter. I have my- 
 self subjected them to a more careful investigation, and new cal- 
 culation, and have thereby attained the full conviction, that the 
 existing differences, which, in some cases, exceed a whole min- 
 ute, are by no means to be attributed to the observations." 
 
 Bessel had already 4 pointed out one error in Bouvard' s ta- 
 bles, in the equation depending on the mean longitude of Saturn 
 minus twice that of Uranus. This error was small in compari- 
 son with the discordances to be accounted for, but we thus 
 see how early he was engaged upon the investigation. 
 
 The last sentence of Bouvard's introduction, just quoted, 
 which was, in fact, the first published suggestion that the discord- 
 ance between the theory and observations of Uranus might be 
 due to the influence of an unknown planet, is hardly definite 
 enough to be viewed historically in that light. But the follow- 
 
 1 Proc. Amer. Acad., I. p. 333. 
 
 2 Bessel, Populdre Vorlesungen, p. 448. 
 
 3 In Konigsberg, Feb. 28th, 1840. 
 
 4 Astron. Nachrichten, No. 48, II. p. 441. 
 
THE DISCOVERY OF NEPTUNE. 11 
 
 ing extract from a letter, 1 written November 17, 1834, by 
 the Rev. Dr. T. J. Hussey, of Hayes, to Prof. Airy, now 
 English Astronomer Royal and then Director of the Observa- 
 tory of Cambridge, England, although first published 2 since 
 the discovery of Neptune, is the earliest written allusion to the 
 subject with which I am acquainted. 
 
 " The apparently inexplicable discrepancies between the an- 
 cient and modern observations suggested to me the possibility of 
 some disturbing body beyond Uranus, not taken into account, 
 
 because unknown Subsequently, in conversation with 
 
 Bouvard, I inquired if the above might not be the case : his an- 
 swer was, that, as might have been expected, it had occurred to 
 him, and some correspondence had taken place between Hansen 
 and himself respecting it. Hansen's opinion was, that one dis- 
 turbing body would not satisfy the phenomena ; but that he con- 
 jectured there were two planets beyond Uranus. Upon my 
 speaking of obtaining the places empirically, and then sweeping 
 closely for the bodies, he fully acquiesced in the propriety of it, 
 intimating that the previous calculations would be more labori- 
 ous than difficult ; that, if he had leisure, he would undertake 
 them and transmit the results to me, as the basis of a very close 
 
 and accurate sweep I may be wrong, but I am disposed 
 
 to think, that, such is the perfection of my equatorial^ object- 
 glass, I could distinguish, almost at once, the difference of light 
 
 between a small planet and a star If the whole matter 
 
 do not appear to you a chimera, which, until my conversation 
 with Bouvard, I was afraid it might, I shall be very glad of 
 any sort of hint respecting it." 
 
 As regards the opinion attributed in this letter to Prof. Han- 
 sen, I have the authority * of that eminent astronomer himself 
 
 * Prof. Hansen informs me, in a letter received since this account was written, 
 that Mr. von Lindenau has been engaged upon a history of the discovery of 
 Neptune. To his inquiries upon this subject, Prof. Hansen had replied in the 
 following words, which he has also authorized me to make public: 
 
 " Die mich betreffende Aeusserung in Airy's Aufeatze ( Astr. Nachr., No. 585, 
 
 1 Notices R. Ast. Soc., VII. p. 123. 2 Nov. 13, 1846. 
 
12 REPORT ON THE HISTORY OF 
 
 for stating, that the assertion must have been founded on some 
 misapprehension, as he is confident of never having expressed or 
 entertained that belief. 
 
 In the Notices of the Royal Astronomical Society for Novem- 
 ber 13th, 1846, is a most important publication l by the Astron- 
 omer Royal, entitled, " An Account of some Circumstances his- 
 torically connected with the Discovery of the Planet exterior to 
 Uranus. " In this paper is a series of extracts from letters, be- 
 fore unpublished, which furnish the testimony to a great part of 
 the history anterior to the actual discovery. From the first let- 
 ter 2 in the series, the preceding quotation was made. 
 
 Mr. Eugene Bouvard, nephew of the author of the Tables, 
 wrote as follows 3 on the 6th October, 1837, from Paris, to the 
 Astronomer Royal of England : " My uncle has given me the 
 tables of Uranus to reconstruct. In consulting the comparisons 
 
 s. 136, und R. Ast. /Sbc., VII. p. 123) 1st jedenfalls unrichtig. Ich besitze freilich 
 keine Copien von meinen Briefen an Bouvard, aber ich finde in seinen Briefen 
 an mich, die ich wieder durchlas, als mir Airy's Aufsatz bekannt wurde, keine 
 Spur, dass ich gegen ihn die von Hussey mir in den Mund gelegte Aeusserung 
 gemacht hatte. Auch war dies unmoglich, da ich meine darauf beziiglichen Ar- 
 beiten nicht so weit fortgesetzt habe, urn darUber eine bestimmte Ansicht habcn 
 zu konnen. Ich kann moglicher Weise geschrieben haben, dass vielleicht die bis 
 dahin in der Bewegung des Uranus nicht erklarten Abweichungen von der Theo- 
 rie nicht von einem, sondern von mehreren auf ihn einwirkenden. unbekannten 
 Planeten herrtlhrten ; aber dass Ein storender Korper die Abweichung nicht er- 
 klaren konne, habe ich auf keinen Fall behauptet. Airy, dem ich kurz nach dem 
 Erscheinen seines Aufsatzes darOber schrieb, antwortete, dass er in seinem 
 nachsten Artikel Ober Neptun davon Gebrauch machen wollte. Es ware mir 
 lieb, wenn dieses auch in dem Ihrigen geschahe. Immer ist es meine feste 
 Ansicht gewesen, dass die Anomalien in der Bewegung des Uranus nichts andcrs 
 sind, als die Wirkung eines oder mehrerer oberhalb befindlichen Planeten, und ich 
 befand mich daruber in Opposition mit Bessel und Nicolai, welche beide dieses 
 fur unmoglich hielten. Ersterer ist indess in seinen letzten Lebensjahren von 
 dieser Ansicht abgegangen, und hat sich ernstlich mit der Aufsuchung dieses 
 Planeten durch Rechnung beschaftigt. Ich hatte in den zwanziger Jahren angc- 
 fangen mich mit dieser Aufgabe zu beschaftigen, gab die Arbeit aber wicdcr 
 auf, und habe sie seitdem wegen anderer Untersuchungen ganz aus dem Ge- 
 sicht verloren. Eine Stelle, wo LeVerrier sagt dass die Uranusbcobachtungen der 
 letzten 20 Jahren nothwendig waren, um ein sicheres Eesultat zu erlangen, lasst 
 es mich nicht bereuen, damals meine Arbeit liegen gelasscn zu haben." 
 
 1 Notices R. Astron. Sbc., VII. p. 121. 2 Ibid., p. 123. 3 Ibid., p. 125. 
 
THE DISCOVERY OF NEPTUNE. 13 
 
 which you have made between observations of this planet and 
 the calculations in the tables, it will be seen that the differences 
 in latitude are very large, and are continually becoming larger. 
 Does this indicate an unknown perturbation exercised upon the 
 motions of this star by a body situated beyond ? * I do not 
 know, but this is at least my uncle's idea." 
 
 Prof. Airy remarked in his reply, 1 that the error in latitude 
 was very small ; that it was the errors in the longitude which 
 were increasing with so fearful rapidity. And, a few months 
 after, he showed 2 that the tabular radius-vector of Uranus was 
 much too small. This result of observations at the quadratures 
 was one to which Prof. Airy, both at that time and uniformly 
 since, attached great importance. 
 
 It is from this period, that the definite belief of most astrono- 
 mers in the existence of a trans-Uranian planet appears to date. 
 Numerous mathematicians subsequently conceived the purpose of 
 entering earnestly into laborious and precise calculations, in 
 order to decide whether the assumption of an exterior cause of 
 disturbance were absolutely necessary, and, if so, to determine 
 from the known perturbations their unknown cause. The Astron- 
 omer Royal most justly expresses 3 himself " confident, that it will 
 be found that the discovery is a consequence of what may prop- 
 erly be called a movement of the age ; that it has been urged 
 by the feeling of the scientific world in general, and has been 
 nearly perfected by the collateral, but independent, labors of va- 
 rious persons possessing the talents or powers best suited to the 
 different parts of the researches." 
 
 The problem became, from this time forth, one of the most 
 important questions of Physical Astronomy. Astronomers in 
 various countries busied themselves with it, and spoke of it 
 without reserve. 
 
 =* " Cela tient-il a une perturbation inconnue apportee dans les mouvements de 
 cet astre par un corps situe au-dela ? " 
 
 1 Notices R. Astron. Soc., VII. p. 126. 3 Notices R. Astr. Soc., VII. p. 122. 
 
 2 Astr. Nachr., No. 349, XV. p. 217. 
 
14 REPORT ON THE HISTORY OF 
 
 The first decided opinion, publicly expressed, was, I believe, 
 that of Bessel, in the lecture l delivered at Konigsberg on the 
 28th February, 1840 ; from which I have already quoted. He 
 had, at that time, already repeated the calculations of the elder 
 Bouvard, and convinced himself that the ancient and modern 
 observations could not be reconciled by any modification of the 
 elements ; and that the differences could not be attributed to 
 inaccuracy of instruments, or to methods of observation. 
 
 " From all these investigations," said he, 2 " I have arrived at 
 the full conviction, that we have in Uranus a case to which La- 
 place's assertion * is not applicable ..... We have here to do 
 with discordances, whose explanation can only be found in a new 
 physical discovery ..... Farther attempts to explain them 
 must be based upon the endeavor to discover an orbit and a 
 mass for some unknown planet, of such a nature, that the result- 
 ing perturbations of Uranus may reconcile the present want of 
 harmony in the observations. If the motions of Uranus can 
 actually be explained in this way, the lapse of time will raise 
 this explanation to the rank of evidence, in the same degree in 
 which it will exhibit the influence of this new power. And this, 
 too, must exert influences upon the motion of Saturn, which are 
 indeed smaller, but will, nevertheless, hardly be able to escape a 
 special investigation, and will thus afford an independent confir- 
 mation of the existence of the new planet." 
 
 In continuing his lecture, Bessel dwelt long upon the labor 
 and difficulty of the problem. He spoke of the painful amount 
 of preparatory labor to be performed, alluded 3 to the patience 
 and care with which his young friend and pupil, Flemming, had 
 already reduced all the observations of Uranus, and, with father- 
 ly kindness, he encouraged him in his work. 
 
 The labors of the talented and enthusiastic Flemming were 
 
 * i. e., that the theory of gravitation entirely explained all motions observed in 
 our solar system. 
 
 1 Bessel, Popul. Vorles., p. 408. 3 Ibid., p. 452. 
 
 2 /6eW.,pp. 449, 450. 
 
THE DISCOVERY OF NEPTUNE. 15 
 
 interrupted by his early death. Eessel himself was upon the 
 point of commencing the investigation which he had proposed, 
 when he sank beneath the weight of his labors ; and " the prize 
 toward which he had stretched forth his hand was wrested from 
 him " l by the lingering and painful disease which closed the 
 long series of his brilliant investigations, and removed him from 
 the world. 
 
 Sir John Herschel stated publicly, 2 soon after the discovery of 
 Neptune, that Bessel, while in England, in 1842, conversed with 
 him concerning the probability that the motions of Uranus were 
 disturbed by an exterior planet, and later in the same year, in a 
 letter from Konigsberg, intimated that he was engaged in re- 
 searches relative to these perturbations. Professor Schumacher 3 
 has promised to publish Flemming's reduction of the Uranus-ob- 
 servations in the Astronomische Nachrichten. 
 
 Meantime Mr. Eugene Bouvard, in Paris, had been engaged 
 upon the same labors. In May, 1844, he wrote again 4 to Mr. 
 Airy, that he had not only reduced the observations of Uranus 
 anew, but that he had reconstructed tables, and succeeded in 
 satisfying the modern observations so nearly that the extreme of 
 discordance amounted only to fifteen seconds, while the tables of 
 his uncle gave places nearly two minutes out of the way. 
 
 But it must be observed, on the other hand, that the ancient 
 observations were not represented with any degree of accuracy ; 
 and, as regards the modern ones, fifteen seconds, although but a 
 small angle, a single second of time, is nevertheless five times 
 the amount of error which we are warranted in assuming. 5 
 
 Thus not only the investigations of Bessel previous to 1840, 
 but the entirely independent ones of Bouvard, between 1837 
 and 1844, showed the impossibility of explaining all the observ- 
 ed motions of Uranus by known causes. The Royal Society of 
 
 Schumacher, Introduction to Bessel's Popular Lectures, p. iv. 
 
 Letter to London Atlienwum, Oct. 3, 1846. 
 
 Introduction to Bessel's Lectures, p. iv. 
 
 Airy's Account, letter No. 5, Notices Astr. Soc., VII. p. 127. 
 
 Proceedings Amer. Acad., I. p. 333. 
 
16 REPORT ON THE HISTORY OP 
 
 Sciences of Gottingen had already proposed, as a prize-ques- 
 tion, 1 in 1842, the full discussion of the theory of the motions 
 of Uranus, with special reference to the cause of the large 
 and increasing errors of Bouvard's tables. 
 
 The death of Flemming and the feebleness and illness of 
 Bessel put a stop to their researches in Germany; and from 
 this period we know of but two mathematicians, Messrs. Adams, 
 of Cambridge University, in England, and Le Verrier, in Paris, 
 who busied themselves with the problem. Mr. Adams states 2 
 that his attention was first called to the subject by reading, in 
 the summer of 1841, Mr. Airy's Report on the Progress of As- 
 tronomy. Mr. Le Verrier undertook 3 the investigation at the 
 request of Arago, the Director of the Observatory of Paris. 
 That eminent scientist had doubtless been impressed by the com- 
 putations of Bouvard with the necessity of the research ; and 
 the Astronomer Royal expresses 4 his conviction, that the knowl- 
 edge of Bouvard's labors " tended greatly to impress upon as- 
 tronomers, both French and English, the absolute necessity of 
 seeking some external cause of disturbance." 
 
 Three months before the date of the laskcited letter of Bou- 
 vard, Mr. Adams had informed Mr. Airy, through Prof. Challis, 
 that he was engaged in researches of a precisely similar nature. 
 The letter 5 of Prof. Challis is dated February, 1844, and was 
 written to obtain the reductions of the Uranus-observations 
 which had been made at Greenwich. Airy immediately 6 for- 
 warded the complete series of heliocentric errors of the Uranus- 
 tables, both in longitude and latitude, for all the Green- 
 wich observations from 1754 to 1830. 
 
 During the summer of 1845, Mr. Arago represented, as 
 above stated, to Mr. Le Verrier, then colleague of the venerable 
 
 1 Abhandl d. KOnigl. Gesellschaft, II. p. x. 
 
 2 Adams's "Explanation," p. 3, Naut. Aim., 1851. 
 
 3 Le Verrier's " Recherches," p. 4, Conn, des Temps, 1849. 
 
 4 "Account" Ast. Soc. Notices, VII. p. 127. 
 8 " Account," letter No. 6, p. 128. 
 
 6 Ibid., letter No. 7, p. 128. 
 
THE DISCOVERY OF NEPTUNE. 17 
 
 and illustrious Biot, the transcendent importance of the ques- 
 tion, and urgently pressed him to enter into a full discussion of 
 the subject. Le Verrier did this, and presented his first paper l 
 to the Academy of Sciences on the 10th November, 1845. 
 
 A fortnight previous, Mr. Adams had communicated 2 to Mr. 
 Airy, that, according to his calculations, the observed inequalities 
 of Uranus might be explained by supposing an exterior planet 
 with a mass and orbit as follows : 
 Mean distance (assumed nearly in accord- 
 ance with Bode's law), . . f< ' '.'! Of . . 38.4 
 Mean sidereal motion in 365.25 days, ... 1 30'. 9 
 Mean longitude, October 1st, 1845, . . . 323 34' 
 Longitude of the perihelion, . . . . . 315 55' 
 
 Eccentricity, 0.1610 
 
 Mass, that of the sun being unity, . . . 0.0001656 
 
 With these elements Mr. Adams gave a table of remaining er- 
 rors of longitude for every three years of the modern observa- 
 tions, none of which exceeded two and a quarter seconds. The 
 ancient observations were satisfied within 12", excepting Flam- 
 steed's in 1690, which differed by 44". 4, not having been used 
 in the equations of condition. It was, however, probable, he 
 thought, that a small change in the mean motion of the hypo- 
 thetical planet would reduce this discordance materially. 
 
 Prof. Airy, in replying, desired 3 to know whether this as- 
 sumed perturbation would explain the error of the radius-vector, 
 as he considered that this trial would be an experimentum crucis. 
 For some reason, no answer was received. This is deeply to be 
 regretted, as, had an affirmative answer been given, Airy would 
 undoubtedly 4 have procured the publication of Mr. Adams's re- 
 sults, and the painful discussion concerning priority would have 
 been spared. As the case now stands, the question of priority 
 depends upon another question, whether Adams's communica- 
 tion of his results to the Astronomer Royal can be considered as 
 a publication of them. 
 
 1 Comptes Rendus, XXI. p. 1050. 3 Account," letter No. 12, p. 130. 
 
 2 Airy's Account, letter No. 11, p. 129. 4 Ibid., p. 131. 
 
 3 
 
18 REPORT ON THE HISTORY OF 
 
 The next letter of Mr. Adams, which has been printed, 1 is 
 dated September 2, 1846. Le Yerrier had, in the mean time, 
 not only published 2 the memoir already alluded to, in which the 
 perturbations of Uranus by Jupiter and Saturn are fully devel- 
 oped, calculated, and discussed, but had communicated to the 
 Academy two other most important papers. In one, 3 presented 
 on June 1st, 1846, he proved 4 that the motions of Uranus could 
 not be accounted for, except by introducing the perturbative in- 
 fluence of an unknown planet, for which he assigned an approx- 
 imate place. In the other, 5 he found an orbit, a mass, and a 
 more precise position for the disturbing planet. This was pre- 
 sented on the 31st August. 
 
 Mr. Airy mentions, 6 that on the 29th June, at a meeting of 
 the Board of Visitors of the Greenwich Observatory, at which 
 Sir John Herschel and Prof. Challis were present, he spoke of 
 the extreme probability that another planet would be discovered 
 within a short time ; and assigned, as a reason for this belief, 
 the coincidence between Mr. Le Yerrier' s results and those of 
 Mr. Adams. He had addressed 7 a letter to Mr. Le Yerrier, 
 similar to that sent a year previously to Mr. Adams, to make in- 
 quiries about the radius-vector. Mr. Le Yerrier answered 8 un- 
 der date of June 28, stating that the errors of radius-vector 
 must be accounted for, inasmuch as the equations of condition 
 depended on observations at the quadratures as well as at the 
 oppositions. Concerning the correctness of this inference, 
 however, there appears room for discussion. Le Yerrier called 
 Airy's attention to the fact, that the position in quadrature in 
 1844, deduced by means of his formulas from the two opposi- 
 tions which comprised it, only differed 0".6 from the observed 
 position, which proved, he said, that the error of radius-vector 
 had entirely disappeared. This he considered as one of the 
 strongest arguments in favor of the truth of his results. For, 
 
 1 " Account? letter No. 20, p. 137. 5 Comptes Rendus, XXIII. p. 428. 
 
 2 Comptes Rendus, XXI. p. 1050. 6 "Account," p. 133. 
 
 3 Rid., XXII. p. 907. 7 Letter No. 13, p. 132. 
 
 4 Ibid., p. 911. 8 Letter No 14, p. 133. 
 
THE DISCOVERY OF NEPTUNE. 19 
 
 while in his first researches he only made use of oppositions, 
 the quadratures were represented with all precision. " Le ray- 
 on vecteur," said l he, " s'est trouve rectifie de lui-meme sans 
 que Ton 1'eut pris en consideration d'une maniere directe. Ex- 
 cusez-moi, Monsieur, d'insister sur ce point. C'est une suite du 
 desir que j'ai d'obtenir votre suffrage." 
 
 At Airy's suggestion, 2 Professor Challis had already com- 
 menced 3 a search for the planet on the 29th July, using a modi- 
 fication of a plan which Mr. Airy had drawn up. The date of 
 the letter suggesting this search was July 9 ; that of the 
 general plan was July 13. Le Verrier's memoir, 4 which as- 
 signed 325 as the probable longitude of the planet, was 
 presented to the French Institute, as we have seen, on June 1st. 
 Still, it does not appear that any search whatever had been 
 instituted in the intervening time in any part of Europe or 
 America ; indeed, there is no account of any search having 
 been made excepting by Professor Challis, before the night of 
 September 23. 
 
 It must, indeed, be confessed that astronomers in general did 
 not seem to consider the theoretical results, published by Mr. 
 Le Verrier, as necessarily indicating the physical existence and 
 true position of such an exterior planet. Professor Challis 
 alone the only astronomer who entered into a systematic 
 search for the planet, and the only one excepting "Dr. Galle, the 
 assistant at the Royal Observatory of Berlin, whom we know to 
 have even looked for it has assigned, 5 as a reason which de- 
 terred him from an earlier search, that it was " so novel a thing 
 to undertake observations in reliance upon merely theoretical 
 deductions ; and that, while much labor was certain, success 
 appeared very doubtful." But is there any practical astrono- 
 mer, in a latitude permitting the search, who was not deterred 
 by the same considerations. Even in Paris, that focus of sci- 
 ence, with its many and powerful telescopes, with its numerous 
 
 1 " Account" letter No. 14, p. 134. 4 Comptes Rendus, XXII. p. 917. 
 
 2 Ibid., pp. 135, 136. 6 Ast. Soc. Notices, VII. p. 145. 
 
 3 Ast. Soc. Notices, VII. p. 145. 
 
20 REPORT ON THE HISTORY OF 
 
 eminent astronomers, where Mr. Le Verrier was known and his 
 brilliant genius appreciated, not to allude to the American 
 observatories, furnished with some of the finest and most power- 
 ful telescopes which have left the ateliers of Munich, we have 
 no information that any attempts* were made to test the physical 
 accuracy of Le Verrier's results, or that the planet was even 
 looked for on one single evening. 
 
 A strange contrast to this apathy on the part of other astrono- 
 mers is furnished by the demeanor of Le Verrier himself. Hav- 
 ing fairly arrived at his results, he looked upon them as 
 conclusive. His computations had been an earnest work. He 
 had employed all his analytical powers, and employed them, 
 too, most successfully, to refine the methods which he used, 
 and to narrow the field of his inquiry; all his powers of 
 application and numerical research, to insure precision; and 
 his indomitable perseverance, in carrying out his computations 
 with full vigor, permitted him to omit no possible test of their 
 accuracy. He proved that the observations of Uranus made it 
 necessary to assume the existence of some unknown disturbing 
 body.- For the observations which he adopted as the basis of 
 his calculations, he had assigned, a priori, the limits of error 
 allowable ; and he found that all the observations could be satis- 
 fied within these predetermined limits by the assumption of an 
 exterior planet, moving in a given orbit, and having a given 
 mass. The correctness of his results was dependent upon no 
 empirical assumption. He gave them, therefore, fearlessly to the 
 world, and staked his reputation upon their accuracy. This 
 forms by no means the least part of his claims to the respect and 
 admiration of scientists throughout the world. Had the planet 
 not been found in the predicted place, Le Verrier would alone 
 have borne the mortification. Neptune was discovered in almost 
 precisely the direction assigned, and Le Verrier receives the 
 admiration so justly due him. 
 
 * It is, perhaps, due to the Naval Observatory at Washington, to make some 
 exception in its favor. Astr. Nachr., XXVI. p. 65. 
 
THE DISCOVERY OF NEPTUNE. 21 
 
 The mass and orbit given in the memoir of August 31st are 1 
 as follows : 
 
 Semiaxis major, ...... 36.1539 
 
 Sidereal period, ...... . 217^.387 
 
 Eccentricity, ....... 0.10761 
 
 Equation of the center, ..... 7 44' 44" 
 
 Longitude of perihelion, January 1, 1800, . . 284 5 48 
 
 Mean longitude, ...... 240 17 41 
 
 Precession in 47 years, . . . . ... -; . . 39 20 
 
 Mean sidereal motion in 47 years, . -. f . 77 50 3 
 
 Mean anomaly, 1847, January 1, ,,., : ; . . 34 1 56 
 
 Mean longitude, . . . ^ , . . 318 47 4 
 Mass, ..... .r. . 
 
 The geocentric longitude, resulting from this orbit, for the end 
 of September, 1846, was 325. Le Verrier, in acknowledging 
 the receipt of a memoir, made use of the opportunity thus 
 afforded, 2 to request Dr. Galle to look for the planet. The letter 
 reached Berlin 3 on the 23d September, and Galle, in comply- 
 ing with this request, found, on the same evening, a new planet 
 in longitude 325 53', or within 55' of the geocentric place 4 as- 
 signed by Mr. Le Verrier. 
 
 The remembrance of the enthusiasm excited by this discovery, 
 of the amazement with which the tidings were received, not only 
 by astronomers, but by almost all classes of the community, and 
 of the homage paid to the genius of Le Yerrier, is still fresh in 
 the memory of all. Nations vied with one another in expres- 
 sions of their admiration. 5 Arago, to whom the right of 
 conferring a name upon the new planet was delegated 6 by Le 
 Verrier, gave 7 to the planet the name of that geometer, with 
 the symbol !, deduced from the initials of the same. This 
 
 1 Eecherches, pp. 234-236 ; Comptes Rendus, XXIII. p. 432. 
 
 2 Astr. Nachr., XXV. p. 51. 
 
 3 Berl Vossische Zeitung, Sept. 25th, 1846. 
 
 4 324 58'. 
 
 5 Comptes Eendus, XXIII. pp. 959- 960, etc. 
 
 6 Ibid., p. 662. 
 
 7 Astr. Nachr., XXV. p. 81. 
 
22 REPORT ON THE HISTORY OP 
 
 name and symbol have not, however, generally prevailed,* as they 
 are at variance with the received nomenclature, in accordance 
 with which the names of Roman deities have been uniformly select- 
 ed ; and the name Neptune, which, with its appropriate symbol 
 (tJO, a trident, was originally proposed 1 by the Bureau des Lon- 
 gitudes, and immediately adopted 2 by Gauss, Struve, Encke, 
 and Airy, has become almost universal. 
 
 The first public announcement of Mr. Adams's labors was in a 
 London newspaper, 3 the Athenaeum. In this journal, under date 
 of October 1st, 1846, Sir John Herschel, commenting on the 
 actual detection at Berlin of the long-expected planet, spoke, as 
 before quoted, of Bessel's conversation with him, and subse- 
 quent letter in 1842, and alluded to the fact that Mr. Adams 4 
 had been engaged in an investigation similar to that of Mr. Le 
 Verrier, and with similar results. 
 
 On the 13th November, the Astronomer Royal presented to 
 the Astronomical Society of London the extremely valuable and 
 important Account which has been so often quoted in my Report, 
 and which must ever remain an integral part of this singular 
 history. Professor Challis presented at the same time an ac- 
 
 * It seems quite desirable that astronomers in different countries should be 
 unanimous in the adoption of fixed names and symbols. For each of the two 
 planets of which this Report speaks, two names and two symbols are in use. 
 The great preponderance of authority seems, however, decidedly in favor of the 
 mythological names used in this Report, with their corresponding symbols, (J> and 
 y. Le Verrier, who in 1846 announced 5 his fixed determination to call Ura- 
 nus by the name of its discoverer, according to Lalande's proposition, has now 
 happily abandoned 6 this unpopular idea ; and even the distinguished Arago, who 
 bestowed the name of the French geometer upon the new planet discovered in 
 consequence of his computations, and publicly declared 7 his determination never 
 to call it by any other name than Le Verrier, has yielded to the general usage 
 of astronomers. 
 
 1 Le Verrier, letter to Galle, Oct. 1, 1846, Ast. Nachr., XXV. p. 194. 
 
 2 Gauss, letter to Encke, Oct. 7, 1846, Ibid. Struve, Bulletin Imp.Acad. Peters- 
 burg, Dec. 27, 1846. Challis, Astr. Nachr., XXV. p. 313. 
 
 3 London Athenaeum, Oct. 3, 1846, p. 1019. 
 
 4 See also Challis's letter of Oct. 15, in London Athenceum of Oct. 17, 1846. 
 
 5 Recherches, p. 3, note. 
 
 6 Comptes Rendus, XXVII. pp. 209, 273, et al 7 Ibid., XXIII. p. 662. 
 
THE DISCOVERY OF NEPTUNE. 23 
 
 count l of his search since the end of July ; and Mr. Adams, a 
 brief notice 2 of his computations. These papers are printed 
 together in the Monthly Notices 3 of that Society. 
 
 During the month of November, 1846, Le Verrier published 
 a complete account of all his investigations, giving in detail the 
 processes by which he had arrived at the results previously made 
 known in the Oomptes Rendus. This memoir may justly claim 
 a place among the most remarkable mathematical works of the 
 age. It is entitled RecJiercJies sur les Mbuvements de la Planete 
 Herschel, dite Uranus, and is published as an Addition to the 
 Connaissance des Temps for 1849, occupying 256 pages. Mr. 
 Adams gave in December a similar abstract of his computations, 
 as a supplement of 31 pages to the Nautical Almanac for 1851. 
 
 It is not easy for those who are not versed in the study of 
 Physical Astronomy, to form any adequate idea of the difficulty 
 of the problem which Messrs. Le Verrier and Adams proposed 
 to themselves. The difficulties in the development of the prop- 
 er methods were exceedingly great, as any one might infer from 
 the manner in which even Airy was accustomed 4 to speak of the 
 problem. An investigation must indeed be eminently difficult 
 and complicated, which that distinguished mathematician would 
 regard as unfeasible. Not only the orbit and mass of the 
 suspected planet, but the elements of Uranus also, were to be 
 regarded as unknown quantities. The limits of error of the 
 ancient observations were also undetermined, but must yet exer- 
 cise an important influence on the result. 
 
 Le Vender's memoir consists of three parts. The first of 
 these 5 contains a complete investigation of the theory of Ura- 
 nus, and corresponds to the first paper of the series in the Comp- 
 tes Rendus. It is an investigation of the highest importance, 
 apart from its relation to the problem of a disturbing planet, and 
 is conducted with extraordinary ability. The theory of Uranus 
 may, as Airy has said, 6 be considered as placed now for the first 
 
 1 Ast. Soc. Notices, VII. p. 145. 4 Airy's "Account," passim. 
 
 2 Ibid., p. 149. 5 Recherches, 2, p. 6. 
 
 3 Vol. VII. No. 9. 6 Account, p. 131. 
 
24 REPORT ON THE HISTORY OP 
 
 time upon a satisfactory foundation. The methods used, though 
 essentially those of Laplace, Lagrange, and Poisson, have been 
 modified in many respects, and it may with safety be asserted, 
 that almost all of the modifications are improvements. 
 
 After a brief consideration of the general formulas of pertur- 
 bations, Le Verrier proceeds 1 to what he calls the simultane- 
 ous determination of all the inequalities. The method adopted 
 is such, that the inequalities are all obtained at the same time, 
 and are so mutually dependent, that an error in any one part 
 of the calculation vitiates the entire result. If, then, after the 
 numerical computation is completed, any one part of the result 
 is found to be exact, it may fairly be concluded that the remain- 
 der is exact also. By this method of " simultaneous determina- 
 tion," Le Verrier determined 2 all the perturbations by Saturn, 
 carrying them out to the very smallest sensible terms ; in 
 many cases even to terms whose coefficients amounted only to 
 0".01. The planetary elements 3 on which his researches were 
 based, were taken from Bouvard's tables. 
 
 The whole computation of the perturbations by Saturn having 
 been completed, Le Verrier proceeded to compute 4 a part of the 
 same quantities by the method of Lagrange, in order to test the 
 accuracy of his previous results. The actual steps of this sec- 
 ond and most laborious calculation are not given, but the agree- 
 ment 5 between the amount of the inequalities as obtained by the 
 two different processes is wonderfully close, and reflects no small 
 honor on the author's powers of numerical computation, and on 
 the precision with which all the small terms were taken into 
 account. The agreement of the parts thus doubly computed 
 is a test of the accuracy of the whole work. 
 
 The perturbations by Jupiter, proportional to the first power 
 of the mass, are determined 6 according to the ordinary meth- 
 ods ; and, although not doubly computed, as in the case of 
 Saturn, are carried out with great rigor. 
 
 1 Reckerches, 5, p. 9. 4 Ibid , 23, p. 32. 
 
 2 Rid., 15, p. 16. 5 Ibid., 26-28, pp. 37 -41, etc. 
 
 3 Ibid., pp. 16, 52. 6 Rul, 34, p. 51. 
 
THE DISCOVERY OF NEPTUNE. 25 
 
 In the investigation 1 of the changes proportional to the 
 square of the disturbing force, Le Verrier has neatly availed 
 himself of the method used 2 by Laplace. The parts of these, 
 which are not strictly secular, can without hesitation be rejected, 
 and he has calculated 3 the values of the coefficients for 1800 
 and for 2300, and determined their values for intermediate years 
 by simple interpolation. The discussion is completed by the 
 consideration 4 of the influence of the changes of Saturn's ele- 
 ments, produced by the attraction of Jupiter. This is investi- 
 gated with much care and elegance. The action of Saturn, as 
 reflected through Jupiter, is not of sufficient importance to 
 require computation. 
 
 The theory of Uranus was thus made complete, and needs 
 now only the addition of the perturbations by Neptune, to 
 accommodate it perfectly to the present demands of science. 
 
 In the second part 5 of Mr. Le Terrier's " Researches," the 
 theory of Uranus, thus remodelled, is compared with the whole 
 series of known observations, in order to discover how large the 
 discrepancies between the observed and computed course of the 
 planet actually were. In entering upon this division of his 
 subject, the author shows 6 that the sum of the errors in the 
 perturbations as given by Bouvard considering those pertur- 
 bations only whose value had completely changed during the 
 period through which our observations extend, and making 
 full allowance for the erroneous masses of Jupiter and Saturn, 
 which Bouvard used amounted to twenty-nine sexagesimal 
 seconds. 
 
 Bouvard's tables appear, indeed, throughout this part of the 
 investigation, in a singularly unfavorable light. Le Yerrier has 
 repeated 7 the whole of Bouvard's computations, and subjected 
 his tables to a most searching scrutiny. The result must have 
 surprised astronomers. 
 
 1 Recherches, 48 - 58. 5 Ibid., p. 89. 
 
 2 Mean. C6L, Bowditch's Trans., HI. p. 283. 6 Ibid., 61, p. 90. 
 
 3 Recherches, p. 61 . 7 Ibid , pp. 91 - 99. 
 
 4 Ibid., 50, p. 65. 
 
 4 
 
26 REPORT ON THE HISTORY OF 
 
 The first inaccuracy alluded 1 to by Le Verrier is in the eccen- 
 tricity of the orbit. In order to discover the value of the eccen- 
 tricity which was used in constructing the tables, we can refer 
 directly to Mr. Bouvard's Introduction, 2 or can deduce it from 
 the table 3 of the equation of the center. These, however, do 
 not agree with one another, and in order to decide which of them 
 is erroneous, we must try, if possible, some third method of find- 
 ing the eccentricity assumed. We are enabled to do this by the 
 algebraical expression of the equation of the center, which is also 
 given 4 in the preface. But, strange to say, we obtain in this 
 way still a third value differing from both the others. The ex- 
 ample, which Le Yerrier gives 5 to illustrate this, is the value of 
 the equation of the center for a mean anomaly of 90. 
 
 The first value deduced from the preface is . . 5 93 57.52 
 The second " " " " " .5 92 71.04 
 
 The value given by Table X. is . . . ,-x . 5 93 48.00 
 
 Mr. Le Verrier assures us, farther, that any attempt to decide 
 definitely, by means of the radius-vector, what was the true eccen- 
 tricity on which the tables were based, would lead only to the 
 discovery of new discordances. 
 
 In the secular motion of the mean longitude, Tables I. and II. 
 do not agree. The error in Table II. would, according to 
 Le Verrier, give an error of 21".5 in the computed place for 
 Flamsteed's observation of 1690. In the formation of equations 
 of condition, Bouvard appears 6 to have been equally unfortunate, 
 both as regards the analytical and the numerical parts of the 
 work. Finally, a series of typographical errors is given, 7 sixteen 
 of them being in the single table of the equation of the center, 
 and the majority of them errors of grave importance. 
 
 Le Verrier was thus compelled to repeat the whole work, and 
 the catalogue of errors which he has given is a sufficient indica- 
 
 1 Eecherches, p. 92. 5 Recherckes, p. 93. 
 
 2 Tables, p. ii. 6 Ibid., pp. 93 - 95. 
 
 3 Table X. p. 95. 7 Ibid., pp. 96, 97. 
 
 4 Tables, p. xv. 
 
THE DISCOVERY OF NEPTUNE. 27 
 
 tion of his thoroughness and high standard of precision. And 
 when we remember, that neither Airy * had detected these errors, 
 nor Bessel published any thing concerning them, excepting the 
 notice of the one term already referred to, we cannot but still 
 more admire the searching rigor of Le Verrier's investigations. 
 
 The author has next given an ephemeris of the heliocentric 2 
 and geocentric 3 places of Uranus for several successive days, at 
 the epochs of the ancient observations, and of those modern ones 
 which he had chosen as most appropriate for the comparison. 
 For these latter he did not construct normal places, but selected 
 a series of two hundred and sixty-two of the best observations, 
 some taken in opposition and some in quadrature, and suitably 
 distributed over the interval between 1781 and 1845. These, 
 together with the ancient observations, 4 were then compared 5 
 with the theory. The comparisons were made in right-ascension 
 and declination, and the differences subsequently converted into 
 differences of longitude and latitude. Each of these differences 
 furnishes one equation, but, on account of the smallness of the 
 errors in latitude, 6 the computation was founded upon the longi- 
 tudes alone ; and the mean equations of condition 7 used were 
 those depending upon the longitudes after grouping together all 
 the observations made at the same period. 
 
 By means of these Le Verrier was enabled to solve the im- 
 portant problem, 8 " Is it possible to satisfy the whole of the 
 preceding equations by a proper determination of the values of 
 the unknown quantities which they contain?" He found, that 
 the elements, furnished even by those equations of condition de- 
 rived solely from the modern observations, were entirely incapa- 
 ble 9 of representing the course of the planet since its discovery ; 
 the discordance in the mean positions deduced from ten obser- 
 vations in the years 1781 and 1782 amounting 10 to 20.5 sexagesi- 
 
 1 "Account? p. 126. 6 Eecherches, p. 137. 
 
 2 Recherches, 71, pp. 100 - 110. 1 Ibid., pp. 138 - 141. 
 
 3 Ibid., 73, pp. 112-124. 8 Ibid., p. 142. 
 
 4 Specially reduced, pp. 124-126, 74. 9 Ibid., 79, p. 144. 
 
 5 Recherches, 77, pp. 129 - 136. 10 Rid., p. 143. 
 
28 REPORT ON THE HISTORY OF 
 
 mal seconds. It is impossible to believe that such an error ac- 
 tually exists, and we thus see that the theory is inadequate. 
 
 The same result is obtained by considering the data from two 
 other entirely different points of view. In the first place, the 
 consideration 1 of the equations of condition formed from the 
 observations at eight equidistant epochs, comprehending an inter- 
 val of ninety-eight years, and of the relations between their 
 second differences, shows 2 that the resulting values of the varia- 
 tion of the mean motion are totally incompatible with one another, 
 and that some change in the then existing theory was inevitably 
 necessary. Secondly , from the relation between the correction 
 of the elements, the mean anomaly at the epoch, and the error 
 of the heliocentric longitude, Mr. Le Terrier forms 3 eighteen 
 equations of condition ; and, by an extremely elegant process, 4 
 deduces the amount of discordance between theory and observa- 
 tion, which could, under the worst possible combination of unfa- 
 vorable circumstances, result from the errors to which the obser- 
 vations are subject. He allows 5 to each of the normal places 
 from modern observations a possible error of four seconds, and to 
 the position deduced from Flamsteed's three observations in 1715, 
 a possible error of fifteen seconds. But even by assuming all 
 the errors at their maximum, and all acting to diminish the dis- 
 crepancy, Le Verrier shows that only 92" out of 356" can be ac- 
 counted for, leaving a discordance of 264 sexagesimal seconds 
 still unexplained. 6 
 
 In the third part 7 of the work, which corresponds to the sec- 
 ond memoir 8 in the Oomptes Rendus, the author proceeds to the 
 discussion of the great problem of the cause of the anomalies 
 which he had proved to exist. In introducing this part of the 
 Researches by some remarks on the difficulty of the problem, 
 Le Verrier remarks, 9 that the unforeseen obstacles which he 
 
 1 Recherches, 80. 6 Recherches, p. 150. 
 
 2 Ibid., p. 146. 7 Ibid., p. 151. 
 
 3 Ibid., p. 147, 81. * C. R., XXII. p. 907. 
 
 4 Ibid., 82, 83, pp. 147 - 149. 9 Recherches, p. 151. 
 
 5 Ibid., pp. 149, 150. 
 
THE DISCOVERY OF NEPTUNE. 29 
 
 encountered would more than once have deterred him from the 
 farther prosecution of his labor, had he not been fully impressed 
 with a conviction of its importance. The readers of the work 
 cannot but be struck with this remark, for the impediments en- 
 countered and the apparent contradictions of the results would 
 indeed have dismayed any less gifted mathematician. Entering 
 upon the immediate subject, the author considers l in a cursory 
 manner various hypotheses which have been suggested to ac- 
 count for the apparently anomalous motion of Uranus, and 
 assigns his reasons for rejecting them. Astronomers must als*o 
 be impressed with the argument 2 which his computations had 
 furnished him, against the hypothesis, that a comet might have 
 produced the disturbances in question. It is, that both the 
 series of ancient observations and that of the observations since 
 1820 are alike incapable of according with elements deduced 
 from the motion of the planet for the forty years immediately 
 following its discovery ; in other words, that between 1690 and 
 1845 two perturbations had occurred. As regards the hypothe- 
 sis of an unknown planet, too, he inferred that it could not be 
 within the orbit of Uranus, as its effect on Saturn must, in that 
 case, have been more perceptible. "It is easy to conclude," 
 said 3 he, " that its perturbative action would only be exerted at 
 the particular time when it passed in the neighborhood of Ura- 
 nus, and the small difference which there would be between the 
 periods of revolution of the two bodies would prevent this cir- 
 cumstance from having taken place more than once in the period 
 which the observations of the planet include. This consequence 
 is contradictory to our deduction from the observations. The 
 period comprised between 1781 and 1820 shows no trace what- 
 ever of large perturbations ; and, on the other hand, can neither 
 be connected with the previous nor with the subsequent observa- 
 tions." 
 
 This last was unquestionably written after the researches were 
 completed. The planet of Le Verrier's theory must have acted 
 
 1 Recherches, 86. 2 Ibid., p. 152. 
 
 3 Ibid., p. 153; C. R., XXII. p. 914. 
 
30 REPORT ON THE HISTORY OF 
 
 on Uranus twice during the interval comprised between the 
 observations of 1690 and 1845. But it is now known that 
 only one maximum of perturbation by Neptune occurred within 
 that period, namely, in 1822. With regard to the fact that the 
 observations between 1781 and 1820 were capable of being per- 
 fectly represented by elliptic elements, the question naturally 
 arises, whether osculating elements might not be found, capable 
 of representing the motions of Uranus for any period of forty 
 years, within moderate limits of error. 
 
 In case the perturbing body be exterior to Uranus, Mr. Le Ver- 
 rier showed x that it could not be at any very remote distance, 
 such, for example, as three times the mean distance of Uranus ; 
 for, in that case, the mass to be attributed must be so large as to 
 affect Saturn very much more than the theory of that planet 
 allows us to assume. It is most natural, then, to commence the 
 first rough approximation, by assuming the new distance to be 
 about double that of Uranus, and the more so, as this distance 
 would correspond to the curious empirical formula 2 of Wurm, Ti- 
 tius, and Bode, which represented, though in a very rough way, 
 the distances of the planets from the sun. Le Verrier is entitled 
 to praise for holding himself independent of this " law," which, 
 as Gauss long since showed, 3 is not only totally devoid of that 
 precision which characterizes nature's laws, but fails entirely 
 when legitimately applied to Mercury. As it is, perhaps the in- 
 fluence of this ill-omened formula may have been instrumental in 
 depriving both Le Verrier and Adams of the satisfaction of ar- 
 riving, by theoretical means, at a knowledge of the elements and 
 mass of Neptune, and thus making a physical as well as a math- 
 ematical discovery. 
 
 The problem to be considered was fortunately simplified, in 
 some degree, by the fact, that the perturbations in latitude were 
 very small. Uranus moves in a plane but slightly inclined to 
 the ecliptic, and the same must, therefore, in all probability, be 
 
 1 Recherches, p. 153. 
 
 2 Bode, Bed. Astr. Jahrb., 1791, p. 188. Lalande, Bibliographie, p. 845. 
 
 3 Monatliche Corresponded, VI. p. 504. 
 
THE DISCOVERY OF NEPTUNE. 31 
 
 true of the disturbing body. Le Verrier, then, assuming that 
 the unknown planet moved in the ecliptic, proceeded to investi- 
 gate the following questions 1 : 
 
 " Is it possible that the irregularities of Uranus are due to the 
 action of a disturbing planet , situated in the ecliptic at a mean 
 distance double that of Uranus? And if so, where is this planet 
 situated ? What is its mass f What are the elements of the 
 orbit ivhich it describes?" 
 
 There is, as he says, but one route to follow in the discussion 
 of this question. " It is necessary 2 to form expressions for the 
 perturbations, due to the new body, in functions of its mass and 
 of the unknown elements of the ellipse which it describes ; we 
 must introduce these perturbations into the coordinates of Ura- 
 nus, calculated by means of the unknown elements of the ellipse 
 which this planet describes. Putting the coordinates, thus ob- 
 tained, equal to the coordinates observed, we must take as un- 
 known quantities in the resultant equations of condition, not only 
 the elements of the ellipse described by Uranus, but also the 
 elements of the ellipse described by the disturbing planet whose 
 position we seek." 
 
 Taking, then, not the planet's coordinates, but the elements of 
 its orbit, as the unknown quantities, Le Verrier follows 3 the 
 course thus indicated, using the common formulas, and omitting 
 all consideration of the terms proportional to the time which can 
 be confounded with the mean motion, and of the constants which 
 can be combined with the epoch. The first attempt at a solution 
 was based upon eight equations 4 similar to those 5 to which we 
 have before alluded. These were founded on observations 6 sub- 
 sequent to 1747, all of Flamsteed's observations being neglected. 
 The second differences were combined precisely as before. The 
 six equations 7 thus obtained were reduced, by a dexterous elim- 
 ination 8 of the epoch, eccentricity, and mean motion, to three 
 
 1 C. R., XXII. p. 915 ; Recherches, p. 154. 3 Ibid., 80. 
 
 2 Recherches, p. 154, 88. 6 Ibid., p. 146. 
 
 3 Rid., 90-95. 7 Rid., p. 162. 
 
 4 Rid., p. 162, 96. 8 Rid., p. 164. 
 
32 REPORT ON THE HISTORY OP 
 
 others, 1 whose constant terms were the same as those in the 
 equations above alluded to. From these he obtained 2 ^yVir 
 (m = 2.11) as a first approximation to the mass. But from the 
 numerous obstacles to accuracy which are manifest in the course 
 of this solution, Le Verrier inferred 3 that the interval of ninety- 
 eight years was not sufficient for his purpose, but that it was 
 necessary to extend the interval as far as observations would per- 
 mit, and to form the equations of condition with all possible 
 rigor. 
 
 This is done in the second solution, 4 a solution which is pre- 
 eminently a discussion of limits, and a brilliant combination of 
 ingenuity, of analytical skill, and of laborious calculation, a 
 solution which cannot be adequately described without departing 
 from the popular form prescribed to me in this Report. Account 
 is taken of all the allowable errors in the ancient observations, 
 and " the field of inquiry narrowed with consummate skill." 5 
 The epoch being assumed at the beginning of the present cen- 
 tury, all the values of its mean longitude which, when substi- 
 tuted in the equations of condition, would give a negative mass, 
 as well as those which would give a mass large enough to affect 
 Saturn sensibly, were promptly rejected. 6 The expression ob- 
 tained for the mass is in the form of a fraction ; 7 the numerator 
 and denominator of which were separately examined, the former 
 by an elegant application of Sturm's formula. The result 8 was, 
 that the longitude of the epoch must be included between the 
 limits 97 and 190, or between 263 and 359, in order to 
 render the corresponding mass positive ; and if those masses be 
 rejected which the motion of Saturn forbids us to assume, 
 Le Verrier found 9 that the longitude of the epoch must be com- 
 prised between 108 and 162, or between 297 and 333. 
 But, after laborious calculation, he also found that neither of 
 
 1 Recherches, p. 164. 6 Recherches, p. 169. 
 
 2 Ibid., 100. 7 Ibid-, 104-107. 
 
 3 Ibid.,p. 165. 8 Rid., p. 174. 
 
 4 Rid., 101-124. 9 Ibid., 112, p. 181. 
 
 5 Peirce, Proc. Amer. Acad., I. p. 66. 
 
THE DISCOVERY OF NEPTUNE. 33 
 
 these limitations will allow the places in 1690 and 1747 to be 
 represented with tolerable accuracy, " so that the consequence 
 which would seem to result from the discussion, thus conducted, 
 would be, that it was impossible to represent the course of Ura- 
 nus by means of the perturbative action of the new planet." 
 
 Le Verrier was fortunately undismayed by this result, although 
 many a good mathematician and experienced computer would, 
 for less reason, have abandoned his apparently unprofitable labor 
 in despair. But even Le Verrier, according to Biot, 1 revolved 
 the matter in his mind for three months without advancing a 
 step. He discovered subsequently, as he next states, 2 that, 
 by neglecting two little inequalities of the longitude, in them- 
 selves so small that one would suppose himself warrantable in 
 omitting them with perfect impunity, all the details of the solu- 
 tion became different, and the errors in 1690 were completely 
 changed. He found, 3 still farther, that in spite of the apparent 
 limitations furnished above by the resulting negative value of the 
 mass, the assumption of values outside these limits, for the longi- 
 tude of the epoch, enabled him perfectly to represent the motions 
 of Uranus. He arrived, in short, at the fundamental proposition, 4 
 " that there was in the ecliptic but one single region in which 
 the perturbing planet could be placed, so as to account for the 
 motions of Uranus ; that the mean longitude of this planet on the 
 1st of January, 1800, must have been betzveen 243 and 252." 
 Continuing his numerical computations, and calculating, 5 for each 
 one of the eighteen normal places, the values of the perturba- 
 tions corresponding to different hypotheses as to the longitude of 
 the epoch, Le Verrier found 6 " that all the observations of Ura- 
 nus could be represented by means of the perturbative action of 
 a planet whose mean longitude was 252 on the 1st of January, 
 1800, and whose eccentricity and perihelion longitude are deter- 
 mined by the formulas" already given. The corresponding 
 longitude in 1847.0 would be 325, and, as above stated, Le Ver- 
 
 1 Journal des Savans, Jan., 1847. 4 Ibid., 114, p. 185. 
 
 2 Recherckes, p. 181. 5 Ibid., 116, pp. 187, 188. 
 
 3 Ibid., p. 182. 6 Ibid., p. 193. 
 
34 EEPORT ON THE HISTORY OF 
 
 rier closed his memoir of June, 1846, by expressing his hope 
 that astronomers might detect the planet. 
 
 The fourth book 1 contains a more precise determination of the 
 elements of the orbit, and, though the processes are given more 
 in detail, is essentially the same as the memoir of August 31, in 
 the Comptes JRendus. 2 In this book the limits of the real and 
 the apparent place of the disturbing body are computed with 
 still greater precision. The hypotheses as to the mean dis- 
 tance vary 3 from 36.2 to 39.2; those as to the longitude of 
 the epoch, from 234 to 270. For these different hypoth- 
 eses, six in number, the coefficients of the equations of condi- 
 tion are minutely computed. 4 Thirty-three equations of condi- 
 tion are formed 5 by comparison with observed geocentric places, 
 From these equations six unknown quantities, and among them 
 the elements of the orbit of Uranus, are eliminated, and 
 their places supplied by the unknown quantities representing 
 the corrections of the mean distance, and of the longitude 
 of the epoch. The resulting 6 thirty-three are solved 7 by the 
 method of least squares, and give, 8 as the final solution, the 
 orbit on page 21 of this Report. 
 
 Le Verrier, in thus assigning definitely the planet's position in 
 the heavens, expressed 9 the belief, that its disc would be large 
 enough to indicate its planetary character to the attentive ob- 
 server. Uranus, at the distance 19, has an apparent diameter 
 of 4". Assuming a position and mass of the new planet con- 
 formable to his computations, and its density the same as that 
 of Uranus, Le Verrier inferred, that its apparent diameter in op- 
 position must be about 3". 3, and its specific brilliancy about one 
 third that of Uranus. 
 
 The orbit and mass thus obtained represent all the ancient 
 observations within eight seconds, except Flamsteed's in 1690, to 
 
 1 Recherches, p. 196. 6 Ibid., 135, p. 231. 
 
 2 C. R., XXIII. p. 428. 7 Ibid., 136, p. 233. 
 
 3 Recherches, p. 198. 8 Ibid., 137-141, pp. 234-236. 
 
 4 Ibid., 130, pp. 203-220. 9 Ibid., p. 237; C. R., XXIII, p. 430. 
 
 5 Ibid., 131, p. 222. 
 
THE DISCOVERY OF NEPTUNE. 35 
 
 which Le Verrier allows l but little weight, being content with 
 representing it within 20".* 
 
 The author next determines 2 the extreme limits within which 
 the disturbing planet is necessarily situated, allowing 3 a large 
 error to all the observations, five seconds, for instance, to the 
 modern ones, and twenty-five to Flamsteed's in 1690. The re- 
 sult is, 4 that the mean distance must be between the limits 35.04 
 and 37.9. Assuming the corresponding times of revolution as 
 the extreme limits of the period, he finds, 5 by an ingenious geo- 
 metrical process, the limits of the planet's place, which they give 
 for 184T. 
 
 The fifth and last part 6 of the Researches corresponds to the 
 memoir presented to the Academy of Sciences on the 5th of Oc- 
 tober, after the discovery of the planet, but was evidently written 
 before the welcome news had reached Mr. Le Verrier. In this 
 he endeavors to deduce from the perturbations of the latitude 
 the position of the place in which the new planet must move. He 
 infers 7 that the observations in latitude concur with those in 
 longitude in indicating the existence of a disturbing planet, that 8 
 the plane of this planet's orbit must be inclined about 4 38' to 
 that of the orbit of Uranus, and that 9 a single observation of 
 the orbit of this new body would suffice to make known approxi- 
 mately the plane in which it moves. These deductions as to the 
 plane of the orbit Le Yerrier submitted with much diffidence, 
 on account of the smallness of the perturbations in latitude from 
 which they must be made. 
 
 Mr. Adams's investigation is of a totally different nature, 
 though characterized by remarkable ability and mathematical 
 
 * In order to do justice to this observation, it must be stated that Neptune, 
 
 with the mass 19540, used by Prof. Peirce, satisfies 10 this observation within a 
 single second. 
 
 1 Recherches, p. 238. 6 Ibid., p. 250. 
 
 2 Ibid., 143-148. 7 Rid., p 251. 
 
 3 Ibid., p 240, 143. 8 Ibid., p. 251. 
 
 4 Ibid., p. 240, 144. 9 Ibid., p. 252. 
 
 5 Ibid., pp. 242, 247. 10 Proc. Amer. Acad., I. 333. 
 
36 KEPORT ON THE HISTORY OF 
 
 power. In it the question of limits is neither discussed, nor in 
 the least degree involved, but the problem is directly proposed, 
 "To find the most probable orbit and mass of the disturbing 
 body which has acted on Uranus." The question, whether it be 
 necessary to assume the existence of such a body, is not dis- 
 cussed. The labors of Bouvard were supposed to have set this 
 question at rest. Mr. Adams states, 1 that his first solution was 
 attempted in 1843, assuming the orbit to be a circle, the dis- 
 tance, nearly* in conformity with " Bode's law," 2 twice that of 
 Uranus, and taking solely the modern observations into account. 
 The errors of the tables were taken 3 from Bouvard's equations 
 of condition as far as the year 1821, and, for later dates, from 
 Schumacher's Astronomische Nachrichten and the Reductions 
 of the Greenwich Planetary Observations. 
 
 Mr. Adams inferred 4 from his results that a good general 
 agreement between theory and observation might be obtained, 
 and therefore commenced 5 a more accurate investigation, the 
 results of which were communicated to the Astronomer Royal in 
 the letter 6 of October, 1845, above 7 referred to, and to Prof. 
 Challis, some time 8 in the month of September. 
 
 Flamsteed's observation of 1690 was entirely rejected. 9 The 
 chief inequalities given by Bouvard were recomputed, 10 without 
 the detection of any error, excepting the one pointed out by 
 Bessel. Airy's mass of Jupiter u was introduced in the place of 
 the one used by Bouvard. Those inequalities depending on the 
 square of the disturbing force, which had been pointed out by 
 Hansen, were also recomputed. 12 The differences between the 
 calculated and observed heliocentric longitudes were converted 13 
 
 1 "Explanation? 3, p. 4. 8 " Account? p. 128, letter No. 6. 
 
 2 Bed. Astr. Jahrb., 1791, p. 189. 9 Rid., p. 130. "Explanation? 25. 
 
 3 Expl, 4. 10 " Explanation? p. 5, 7. 
 
 4 Ibid., 3. " Mem. R. Astr. Soc., X. p. 47. 
 
 5 Ibid., 10 et seq. 12 Expl, p. 6. 
 
 6 "Account? p. 129. 13 Ibid., p. 7, 9. 
 
 7 Expl, p. 15. 
 
THE DISCOVERY OF NEPTUNE. 3T 
 
 into differences of mean longitude. Observations made near op- 
 position were selected, 1 when possible, and the series of mod- 
 ern observations divided into groups, from which were deduced 2 
 normal places at intervals of three years. The equations of con- 
 dition formed from these served as the basis of the entire com- 
 putation. 
 
 Assuming, then, as a first hypothesis, 3 the mean distance of 
 the new body to be twice that of Uranus, Adams computes 4 the 
 value of the fundamental perturbations. The values obtained 5 
 are almost identical with those 6 of Le Verrier. But with regard 
 to the influence of the third and fourth terms, Adams does not ap- 
 pear to have experienced the inconveniences by which Le Verrier 
 states that he was so much impeded. 7 Both mathematicians 
 agree, 8 however, in the rejection of the small perturbations of the 
 second order, dependent on the square of the eccentricity of the 
 disturbing planet. Taking the mean opposition in 1810, as the 
 epoch, Mr. Adams has elegantly arranged 9 his equations of 
 condition in such a way, that they separate themselves into two 
 groups, each having 10 but five unknown quantities. The coeffi- 
 cients are then readily computed by summation. By eliminat- 
 ing n the unknown quantities in each group, it will, therefore, 
 be comparatively easy, for any assumed value of the mass and 
 longitude at the epoch, to obtain a correction by the method of 
 least squares. But as the unknown quantities are the same in 
 all the equations of each group, these equations may be added, 12 
 and the substitution of the elements of Uranus, already approxi- 
 mately known, gives values for the mass and epoch. The 
 equations of condition drawn from the ancient observations, 
 omitting that in 1690, are the snued 3 for the determination of 
 
 Explanation, p. 5, 6. 9 Expl, 13. 
 
 Ibid., pp. 6, 7. 10 Ibid., $ 14. 
 
 Ibid., p. 8, 12. " Ibid., 15-20. 
 
 Ibid., 12. '2 2bid., 21, p. 13. 
 
 Ibid., p. 9. 13 Rid., 23. 
 Recherches, p. 161, 95. 
 Ibid., p. 181. 
 8 Adams, Expl 11, 12; Le Verrier, RechercJies, pp. 158,201. 
 
30 REPORT ON THE HISTORY OF 
 
 the eccentricity and perihelion longitude. This determination is 
 very skilfully made, and the elements l furnished by the whole 
 analysis are those communicated 2 to the Astronomer Royal in 
 October, 1845. 
 
 Mr. Adams next proceeded 3 to repeat the whole investigation, 
 assuming the mean distance to be 37.25, the first hypothesis 
 having been 38.36. For the time when the computations on 
 this hypothesis were made, no date is assigned. They were 
 probably made during the summer of 1846, as the resulting ele- 
 ments were communicated 4 to Mr. Airy on the 2d of September 
 of that year. They are as follows 5 : 
 
 Epoch, . ..^ 1810.328 
 
 Mean longitude, 264 50' 
 
 Longitude of perihelion, .... 298 41' 
 
 Eccentricity, , '..... . . . 0.120615 
 
 Mean longitude, Oct. 6, 1846, . . . 323 2' 
 Mass, . . ^ 
 
 The theoretical corrections of the mean longitude on each 
 hypothesis are next 6 given, the parts due to the changes in 
 the elements of Uranus and to the action of the hypothetical 
 planet being written separately, and these corrections are 
 compared with observation. 
 
 The modern observations of Uranus are thus admirably rep- 
 resented 7 down to the year 1840. The ancient ones were 
 represented within tolerable limits, excepting the observation in 
 1771, where the discrepancy amounts to 11". 8 on the first, 
 and 12". 8 on the second hypothesis. Le Verrier has not given 
 any comparison of this observation with his final orbit. 8 
 
 In the observations since 1840, however, Mr. Adams did not 
 find so satisfactory an agreement. The differences, as deduced 
 from the three oppositions immediately preceding the investiga- 
 tion, were 9 by the two hypotheses, 
 
 1 Expl, 31. 6 Rid. t 49-51. 
 
 2 12. Ast. Soc. Notices, pp. 129, 151. 7 Ibid., 52. 
 
 3 Expl, 32, et seq. s Recherches, 142, p. 238. 
 
 4 12. Ast. Soc. Not., VII. pp. 138, 151. 9 Expl, 53, p. 28. 
 
 5 Expl, 47, 48, p. 25. 
 
THE DISCOVERY OF NEPTUNE. 39 
 
 I. II. 
 
 1843, 7!ll 5.77 
 
 1844, 8.79 7.05 
 
 1845, 12.40 10.18 
 
 The errors for these three years are, thus, about one fifth less 
 in the second hypothesis. The first had assumed that the ratio 
 of the mean distances of the two planets was equal to sin. 30, 
 the second that it was sin. 31. Mr. Adams roughly inferred 1 
 from this consideration, that the true ratio would be about equal 
 to sin. 35, which gives a mean distance of 33.42. 
 
 The application of the Rule of Three to a problem so compli- 
 cated as that which the orbit of the disturbing planet presented 
 was, we are bound to believe, intended merely as a rude means 
 of conjecture. It appears, therefore, surprising, that this infer- 
 ence should be dwelt upon as one of the merits of Mr. Adams's 
 investigation, and as tending to show that the solution at which 
 he arrived corresponded to the orbit and place of Neptune. 
 Even had Mr. Adams intended to apply the rule of simple pro- 
 portion, it is impossible that he should have founded it upon the 
 comparison of three oppositions alone. Le Verrier has shown 2 
 that the assumption of even 35 as the mean distance would lead 
 to intolerable discordances. Peirce has further proved 3 that an 
 important change in the character of the perturbations takes 
 place near the distance 35.3. It is, therefore, evident that no 
 claims can be based upon the rough inference alluded to. And 
 it is but just to assume that Mr. Adams would disclaim any in- 
 tention to dwell upon this point, although stress has been laid 
 upon it 4 by one of the most eminent of his countrymen. 
 
 After giving the formulas for the corrections of the tabular 
 radius-vector of Uranus upon each of his hypotheses, Mr. Adams 
 closes by stating, 5 that, on account of the shortness of the pe- 
 
 1 Expl, p. 29 ; " Account? p. 139. 
 
 2 Recherches, p. 240. 
 
 3 Proc. Amer. Acad., I. 66 . 
 
 4 Herschel, Outlines ofAstr.,~p. 517, note. 
 
 5 Expl, 60. 
 
40 REPORT ON THE HISTORY OF 
 
 riod (98 years, the observation in 1690 having been rejected), 
 and the smallness of the perturbations in latitude, all his at- 
 tempts to fix the plane of the orbit had been unsatisfactory. 
 
 I have thus endeavored to give an account of the origin and 
 progress of the theory of Uranus up to the discovery of Nep- 
 tune, and the publication of those computations which had led 
 astronomers to suspect its existence and direction. Of the discus- 
 sion concerning priority which unfortunately arose, it is not neces- 
 sary to speak. There cannot be the slightest doubt of the fact, 
 that the several investigators were entirely independent of one 
 another ; but as many persons, especially those not professedly 
 devoted to the pursuit of science, attach importance to the ques- 
 tion of priority, I have endeavored to state the facts as im- 
 partially as possible, and thus to give the data by which any 
 one may be enabled to judge for himself. The discussion which 
 subsequently arose was of such a nature as to throw the con- 
 troversy between the partisans of the French and of the English 
 geometer entirely into the background. 
 
 Before passing from the discovery of Neptune to the subse- 
 quent development of the theory, it is proper to allude to the 
 charts published by the Berlin Academy at the instance 1 of 
 Bessel ; inasmuch as these furnished the means by which Nep- 
 tune was discovered at Berlin upon the first search. They have 
 also directly contributed, within the last four years, to the detec- 
 tion of several members of our solar system. The chart 2 em- 
 bracing the region in which Neptune was found had been drawn 
 up with great accuracy by Dr. Bremiker, the eminent coadjutor 
 of Prof. Encke, in the Berlin Astronomical Almanac. Although 
 sometime printed, it had been published but a short time. Had 
 Prof. Challis been in possession of this map, he would probably 3 
 have discovered Neptune on the 4th of August, as he observed 
 it as a fixed star on that day, 4 while sweeping for the planet. 
 
 1 Abhandl d. Kdnigl Akad. Berl., 1824, p. iii.; Astr. Nachr., IV. pp. 297, 437. 
 
 2 Hour XXI. 
 
 3 Astr. Nachr., XXV. p. 102 ; Ast. Soc. Not., VII. p. 146. 
 
 4 Ast. Soc. Not., VII. p. 148. 
 
THE DISCOVERY OF NEPTUNE. 41 
 
 He observed it again, August 12. He also noted it September 
 29, as seeming to have a disc ; but the news of the discovery 
 at Berlin on the 23d arrived before the next fair evening. 1 
 
 The slowness of the planet's motion of course rendered it im- 
 possible to find the true orbit, till after the lapse of considerable 
 time, but elements upon the hypothesis of a circular orbit were 
 computed within the first month by Adams, 2 Galle, 3 and Binet. 4 
 These agreed very nearly with one another, and coincided espe- 
 cially in showing the distance from the sun to be about 30. Mr. 
 Adams afterwards computed 5 a second circular orbit, which gave 
 the same result. 
 
 Mr. Valz, of the Marseilles Observatory, endeavored 6 early in 
 the year 1847 to deduce the form of the orbit from the small arc 
 described by the planet since its discovery. He found himself, 
 however, unable to obtain a reliable value for the eccentricity, 
 but assigned & as the result of his computations. At the same 
 time he requested attention to the fact, that, in a letter to Arago 
 concerning the perturbations of Halley's comet, published 7 Sept. 
 12, 1835, he had expressed his belief in the existence of a 
 planet exterior to Uranus. 
 
 The means of obtaining elliptic elements was afforded by the 
 fortunate discovery of an ancient observation of Neptune. This 
 discovery was made independently by two astronomers, Mr. 
 Walker in "Washington, and Dr. Petersen in Altona, each of 
 whom computed an approximate ephemeris for Neptune, and 
 came to the conclusion that no observer except Lalande had 
 catalogued in the vicinity of the planet. 
 
 The modes of research employed by Dr. Petersen and Mr. 
 
 1 Ast. Soc. Notices, VII. p. 147. 
 
 2 Astr. Nachr., XXV. p. 106. 
 
 3 Ibid., XXV. pp. 192, 311 ; Bericht d. KOnigl. Preuss. ATcad., 1846, p. 280; Ast. 
 Soc. Not., VII. p. 148 ; Astr. Nachr., XXV., p. 311. 
 
 4 Comptes Rendus, XXIII. p. 798. 
 
 5 Ast. Soc. Not., VII. p. 148. 
 
 6 Comptes Rendus, XXIV. p. 638. 
 
 7 Ibid., I. p. 130. 
 
 6 
 
42 REPORT ON THE HISTORY OF 
 
 Walker were essentially different. Dr. Petersen 1 compared a 
 chart of the stars of the Histoire Celeste, in a zone extending 
 from 2 south to 4 north latitude, and from 14 h - to 17 h< right- 
 ascension, directly with the heavens, and found that three stars 
 observed by Lalande were no longer visible. On a second com- 
 parison, he saw that the recorded places of two of these were 
 probably vitiated by a typographical error of one minute of 
 time. The third star had been observed May 10th, 1795, and 
 was entered 2 as follows in the Histoire Celeste : 
 
 Mag. Middle Thread. Zen. Dist. 
 
 7.8 14 h - ll m - 23 s -.5 60 7 19" 
 
 Dr. Petersen then calculated from Galle's circular elements the 
 position of Neptune for the time of this observation, and found 
 an agreement sufficiently close to convince him that the missing 
 star was Neptune. 
 
 Walker had, in the United States, arrived at the same result 
 by a totally different investigation. 3 He made no use of the 
 telescope. He had at hand a large collection of observations, 
 from which he had already computed elements of sufficient accu- 
 racy to show that Lalande had only swept in the neighborhood 
 of Neptune upon the 8th and 10th of May, 1795. For these 
 nights he computed a locus of Neptune, by assuming different 
 eccentricities, upon the two hypotheses of the present true anom- 
 aly being greater or less than 180. The catalogue of La- 
 lande's stars within this locus was then subjected to a rigid scru- 
 tiny. The stars which have been since observed, those more 
 than 15' north or south of the planet's computed path, and 
 those below the ninth magnitude, being rejected, there remained 
 but one star in the list, and that was less than a minute north of 
 the computed declination of Neptune for that right-ascension. 
 This discovery was made on the 2d of February, and on the 
 next day Mr. Walker communicated to the astronomers of the 
 
 1 Astr. Nachr., XXV. pp. 291, 303. 
 
 2 Histoire Ctteste, p. 158, obs. 8. 
 
 3 Washington Union, Feb. 9, 1846; Proc. Am. Phil. Soc., IV. p. 318; Proc. 
 Amer. Academy, I. p. 57 ; Astr. Nachr., XXV. p. 383. 
 
THE DISCOVERY OF NEPTUNE. 43 
 
 Washington Observatory his conviction that the missing star was 
 Neptune. Upon the 4th, the weather first became clear, and 
 Prof. Hubbard of the Observatory had the gratification of find- 
 ing l that the star was no longer in the place where Lalande had 
 seen it. 
 
 It was thus rendered a moral certainty, by the independent 
 labors of the American and the German astronomer, that Nep- 
 tune had been observed by Lalande, and that the eccentricity of 
 the orbit was very small. This enabled Mr. Walker to compute 
 elliptic elements 2 during the month of February, 1847, which 
 represented the motion of the planet with great accuracy ; and 
 Mr. Adams also communicated 3 elliptic elements to Prof. Schu- 
 macher, under date of May 19th. By means of the epheni- 
 erides of Neptune published 4 by these two astronomers, the 
 planet was followed without difficulty through the opposition of 
 184T. 
 
 Mr. Walker and Dr. Petersen had both immediately written 
 to Le Yerrier to inform him of their researches, and Le Verrier 
 communicated to the French Academy of Sciences, at the same 
 meeting, 5 the results of the investigations at Washington and at 
 Altona. 
 
 But in speaking of Mr. Walker's elements, he remarked 6 that 
 " the small eccentricity which appeared to result from Mr. 
 Walker's computations would be incompatible with the nature of 
 the perturbations of the planet Herschel." 
 
 The hypotheses of Walker and Petersen received a striking 
 confirmation from the examination of Lalande's MSS. by Mr. 
 Mauvais, of the Royal Observatory, Paris. Mauvais discovered 7 
 that Lalande had observed the star on the 8th, as well as on 
 
 Proc. Amer.Acad., I. p. 64 ; Proc. Am. Phil Soc., IV. p. 318. 
 Proc. Amer. Acad, I. p. 67 ; Proc. Am. Phil. Soc., IV. p. 319 ; Astr. Nachr., 
 XXV. p. 383. 
 
 Astr. Nachr., XXV. p. 399. See also R. Ast. Soc. Not., June 11, 1847. 
 Astr. Nachr., XXV. pp. 51, 241. 
 March 29, 1847. 
 Comptes Rendus, XXIV. p. 531. 
 7 Comptes Rendus, XXIV. p. 641 ; Astr. Nachr., XXVI. p. 97. 
 
44 REPORT ON THE HISTORY OF 
 
 the 10th of May. But in consequence of the non-accordance of 
 the two observations, he had only inserted the latter in the His- 
 toire Celeste, and annexed to it the (:) which he used to denote 
 a doubtful observation. Both Mauvais and Peiree, after a care- 
 ful reduction of both observations, found l that the discordance 
 corresponded precisely to the motion of Neptune in two days, 
 so that the question of identity is now beyond dispute. 
 
 Meantime, 2 Prof. Peirce had taken a remarkable step. From 
 the distance, 30, and consequent angular motion, without any 
 hypothesis as to the character of the orbit, he arrived at the 
 startling conclusion, 3 " that the planet Neptune is not the planet 
 to which geometrical analysis had directed the telescope ; that 
 its orbit is not contained within the limits of space which have 
 been explored by geometers searching for the source of the dis- 
 turbances of Uranus ; and that its discovery by Galle must be 
 regarded as a happy accident." 
 
 This conclusion, paradoxical as it might at first have appeared 
 to many, was announced with a candor and moral courage only 
 equalled by that of Le Verrier in his original prediction of the 
 planet's place. The reasoning 4 by which Peirce defended his 
 position deserves, even at this late day, the most careful consider- 
 ation. It is so clear and convincing, that it would seem unneces- 
 sary to dwell upon it, were it not that even now astronomers 5 of 
 high eminence do not hesitate to dispute the ground there taken ; 
 although the arguments adduced against it are rather of a popu- 
 lar than of a mathematical character. 
 
 Peirce first alluded 6 to the two fundamental propositions of Le 
 Verrier, viz. : 
 
 1st. That the planet's mean distance must be between 35 and 
 37.9; 
 
 1 Comptes Rendus, XXIV. p. 666 ; Proc. Amer. Acad., I. p. 149. 
 
 2 March 16, 1847. 
 
 3 Proc. Amer. Acad., I. p. 65. 
 
 4 Ibid., pp. 65-68. 
 
 5 Herschel, Outlines of Astronomy, 776, p. 516. 
 
 6 Proc. Amer. Acad., I. p. 66. 
 
THE DISCOVERY OF NEPTUNE. 45 
 
 2d. That the mean longitude for January 1st, 1800, must 
 have been within the limits 243 and 252 ; 
 and announced that he had convinced himself that, although 
 neither of these was inconsistent with the observations made upon 
 Neptune since its discovery, yet that the two combined were ut- 
 terly irreconcilable with observation ; that if the first proposi- 
 tion were assumed as true, the mean longitude in 1800 must 
 have been at least forty degrees from the limits of the second ; 
 and that, if we adopted the second proposition, the time of 
 revolution must be less by forty years than the inferior limit giv- 
 en by the first. 
 
 " It is not, however," continued Peirce, " a necessary conclu- 
 sion that Neptune will not account for the perturbations of Ura- 
 nus ; for its probable mean distance of about 30 is so much less 
 than the limits of the previous researches, that no inference from 
 them can be extended to it. An important change, indeed, in the 
 character of the perturbations, takes place near the distance 
 35.3 ; so that the continuous law by which such inferences are 
 justified is abruptly broken at this point, and it was hence an 
 oversight in Mr. Le Verrier to extend his inner limit to the dis- 
 tance 35. A planet at the distance 35.3 would revolve about the 
 sun in 210 years, which is exactly two and a half times the period 
 of revolution of Uranus. Now if the times of revolution of two 
 planets were exactly as 2 to 5, the effects of their mutual influ- 
 ence would be peculiar and complicated, and even a near approach 
 to this ratio gives rise to those remarkable irregularities of motion 
 which are exhibited in Jupiter and Saturn, and which greatly 
 perplexed geometers until they were traced to their origin by 
 Laplace. This distance of 35.3, then, is a complete barrier to 
 any logical deduction, and the investigations with regard to the 
 outer space cannot be extended to the interior. 
 
 " The observed distance, 30, which is probably not very far 
 from the mean distance, belongs to a region which is even more 
 interesting in reference to Uranus than that of 35.3. The time 
 of revolution which corresponds to the mean distance 30.4 is not 
 168 years, being exactly double the year of Uranus, and the 
 
46 REPORT ON THE HISTORY OF 
 
 influence of a mass revolving in this time would give rise to very 
 singular and marked irregularities in the motions of this planet. 
 The effect of a near approach to this ratio in the mean motion is 
 partially developed l by Laplace in his theory of the motions 
 of the three inner satellites of Jupiter. The whole perturbation 
 arising from this source may be divided into two portions or ine- 
 qualities, one of which, having the same period with the time of 
 revolution of the inner planet, is masked to a great extent behind 
 the ordinary elliptic motions, while the other has a very long pe- 
 riod, and is exhibited for a great length of time under the form of 
 a uniform increase or diminution of the mean motion of the dis- 
 turbed planet." 
 
 Peirce closed this most important paper by showing, 2 that, if 
 the period of Neptune were more than 166|, and less than 169| 
 years, the conclusion was inevitable, that its period was precisely 
 twice as long as that of Uranus. 
 
 Walker has since found 3 the period of Neptune to be 164.6 
 years. The two periods are therefore not precisely commensu- 
 rable, but they approach commensurability so nearly that some of 
 the terms 4 of the perturbations of Uranus require careful investi- 
 gation, which, according to the theories of Le Verrier and Ad- 
 ams, would be merged in the longitude of the epoch and other 
 elements of the elliptic motion. 
 
 In reference to the apparent inconsistency of the assertion, 
 that the discovery was " a happy accident," Peirce still farther 
 showed 5 that the problem was susceptible of several solutions, de- 
 cidedly different from one another, and from those of Le Verrier 
 and Adams, and equally complete with theirs. " The present 
 place of the theoretical planet," to quote from his communica- 
 tion 6 of May 4th, 1847, to the American Academy, " which 
 
 1 Mtcanique Ctleste, Bowditch's Transl., I. p. 656 ; IV. p. 126. 
 
 2 Proc. Amer. Acad., I. p. 68. 
 
 3 Smithsonian Contributions, II. App. I. p. 6. 
 
 4 Proc. Amer. Acad., I. p. 334. 
 
 5 Boston Courier, April 30, 1847. 
 
 6 Proc. Amer. Acad., I. p. 144. 
 
THE DISCOVERY OF NEPTUNE. 47 
 
 might have caused the observed irregularities in the motions of 
 Uranus, would, in two of them, be about one hundred and twen- 
 ty degrees from that of Neptune, the one being behind, and the 
 other before, this planet. If Le Verrier or Adams had fallen 
 upon either of the above solutions instead of that which was ob- 
 tained, Neptune would not have been discovered in consequence 
 of geometrical prediction. The following are the approximate 
 elements for the three solutions, at the epoch of January 1st, 
 1847 : 
 
 I. II. III. 
 
 Mean longitude, . . fc. 319 79 199 
 
 Longitude of perihelion, . . 148 219 188 
 
 Eccentricity, . . . 0.12 0.07 0.16 
 
 In each of these the mass is 0.0001187, and the period of side- 
 real revolution double that of Uranus." 
 
 Continuing the investigation, Prof. Peirce arrived at another 
 singular result. He found himself unable to reconcile the ob- 
 served motions of Neptune with the assumption, that it was the 
 chief source of the unexplained irregularities in the motion of 
 Uranus. This result was in all candor acknowledged to the 
 Academy, but with the remark 1 that he considered " it would 
 be presumptuous in him to claim for his investigations a free- 
 dom from error which the greatest geometers have not escaped, 
 especially in the face of the vastly improbable conclusion to 
 which his analysis tended." He subsequently 2 succeeded, as is 
 well known, in entirely explaining the motions of Uranus by the 
 action of Neptune, using the mass ^fa. The failure of the first 
 attempt was in consequence of his not having carried the devel- 
 opment of the disturbing function to a sufficient number of 
 terms, but having contented himself in the first approximation 
 with the number of terms used 3 by Le Verrier in his formulas. 
 The numerical formulas of Le Verrier's Researches can only 
 
 1 Proc. Amer. Acad., I. p. 145. 
 
 2 Ibid , I. p. 332. 
 
 3 Recherches, 94. 
 
48 REPORT ON THE HISTORY OF 
 
 apply to mean distances within the limits assigned by that geom- 
 eter. 
 
 During the summer of 1847, Peirce determined the approxi- 
 mate perturbations of Neptune by the other planets, and com- 
 municated * them to Walker in November. Walker, using these 
 values, deduced a pure elliptic orbit, from the discussion of 689 
 observations, including those of Lalande. This orbit, 2 together 
 with the normal places on which it was founded, and with the 
 expressions for the heliocentric coordinates, was presented to the 
 American Academy, December 7th. At the same meeting, 
 Peirce communicated 3 the corresponding values of the entire 
 perturbations of Neptune up to the terms depending on the 
 cubes of the eccentricities. He used the masses of the planets 
 assumed by Le Verrier in his theories of Mercury and Uranus, 
 with the exception of the mass of Uranus, for which he took La- 
 ment's determination. 4 The results were given both in the 
 usual form and in that which Le Verrier had adopted in his 
 theory of Mercury. Peirce also gave 5 the particular values of 
 the perturbations of the true anomaly and radius-vector, for 
 the epoch of Lalande's observations, and for every three months 
 from October, 1846, to January, 1851. Through the labors of 
 Peirce and Walker, the elliptic orbit of Neptune, with the numer- 
 ical values of its perturbations by all the other planets, was thus 
 known at the beginning of December, 1847, with such accuracy, 
 that an ephemeris 6 founded on these data satisfied the observa- 
 tions of September, 1848, within one and a tenth seconds of 
 time in righteiscension, and six and a half seconds in declina- 
 tion. 7 
 
 Applying anew Peirce's second values for the perturbations, 
 
 1 Proc. Amer. Acad., I. p. 285. 
 
 2 Ibid. ; Proc. Amer. Phil Soc., IV. 378. 
 
 3 Proc. Amer. Acad., I. pp. 287-295. 
 
 4 Mem. R. Ast. Soc., XL p. 59. 
 
 5 Proc. Amer. Acad., I. pp. 294, 295. 
 e Astr. Nachr., XXVII. p. 347. 
 
 7 Ibid., XXIX. p. 191. 
 
THE DISCOVERY OP NEPTUNE. 49 
 
 and using still later observations, Walker presented, 1 March 6, 
 1848, a second pure elliptic orbit, which has represented 2 the 
 course of Neptune so well up to the present time, as to render a 
 nearer approximation unnecessary, if indeed it were possible. 
 
 The perturbations of Uranus by Neptune were also communi- 
 cated 3 by Prof. Peirce on March 6th. This geometer had 
 found 4 Neptune capable of entirely accounting for all the 
 motions of Uranus, provided a mass of about goooo ^ e adopted. 
 
 The determination of the mass has been a problem of some 
 difficulty. Soon after the discovery of the planet, Mr. Lassell, 
 of Liverpool, discovered 5 a satellite. Observations of this 
 satellite, made at Pulkowa, at Liverpool, and at Cambridge, 
 have given masses * for Neptune differing very considerably 
 from one another. The question must be expected to remain 
 unsettled for some time. Peirce seems to have provisionally 
 adopted the mass (19540) deduced by him from Cambridge ob- 
 servations alone. 
 
 The following table 6 of differences between the observed and 
 calculated longitudes of Uranus is taken from Peirce's commu- 
 nication, and shows how well the motions of this planet now ac- 
 cord with theory. The first column contains the discrepancies 
 which exist when the action of Neptune is not taken into consid- 
 eration. The second and third give those which would remain, 
 did the theoretical planets of Le Verrier and Adams actually 
 exist, and act upon Uranus. The third column contains the dis- 
 crepancies, after allowing for the influence of Neptune, suppos- 
 ing the mass to be that deduced by Peirce from the Cambridge 
 observations alone. 
 
 * Struve found from Pulkowa observations 14596- Peirce, from English and 
 American observations, igygo 5 from American observations alone, isko- 
 
 1 Proc. Amer. Acad., I. p. 331. 
 
 2 Smithsonian Contrib. to Knowl., II., Appendix I. p. 5. 
 
 3 Proc. Amer. Acad., I. p. 334. 
 
 4 Ibid., p. 332. 
 
 5 Astr. Nachr., XXVI. p. 165. 
 
 6 Proc. Amer. Acad., I. p. 333. 
 
 7 
 
)0 REPOKT ON THE HISTORY OF 
 
 RESIDUAL DIFFERENCES BETWEEN THE THEORETICAL AND OB- 
 SERVED LONGITUDES OF URANUS. 
 
 
 Date. 
 
 Without any ex- 
 terior Planet. 
 
 By Le Vender's 
 Theory, with 
 
 By Adams's 
 Theory, with 
 
 Introducing Nep- 
 tune with Peirce's 
 
 
 
 
 Mass = 5335- 
 
 Mass = ^. 
 
 Mass = jgig. 
 
 
 1845, 
 
 + 6.5 
 
 // 
 
 + 10.2 
 
 // 
 
 Q 
 
 
 1840, 
 
 + 0.7 
 
 + 2.2 
 
 + 1.3 
 
 1.1 
 
 
 1835, 
 
 4.5 
 
 0.8 
 
 1.2 
 
 + 2.0 
 
 
 1829, 
 
 7.8 
 
 2.2 
 
 + 2.0 
 
 + 0.8 
 
 
 1824, 
 
 7.6 
 
 5.4 
 
 + 1.7 
 
 2.0 
 
 
 1819, 
 
 + 3.8 
 
 + 0.4 
 
 2.2 
 
 + 1.0 
 
 
 1813, 
 
 + 4.5 
 
 0.9 
 
 1.0 
 
 0.3 
 
 
 1808, 
 
 + 3.8 
 
 + 0.8 
 
 0.0 
 
 0.4 
 
 
 1803, 
 
 3.4 
 
 + 0.8 
 
 + 1.6 
 
 + 0.8 
 
 
 1797, 
 
 6.7 
 
 1.0 
 
 0.5 
 
 + 0.3 
 
 
 1792, 
 
 7.8 
 
 + 0.3 
 
 1.1 
 
 + 0.3 
 
 
 1787, 
 
 + 2.0 
 
 1.2 
 
 0.2 
 
 0.5 
 
 
 1782, 
 
 + 20.5 
 
 + 2.3 
 
 0.0 
 
 3.0 
 
 1 
 
 [1769, 
 
 + 123.3 
 
 + 3.7 
 
 + 1.8 
 
 6.0 
 
 I 
 
 1756, 
 
 + 230.9 
 
 4.0 
 
 4.0 
 
 + 4.0 
 
 1 
 
 1715, 
 
 + 279.6 
 
 + 5.5 
 
 6.6 
 
 + 8.7 
 
 1 
 
 1690, 
 
 + 289.0 
 
 19.9 
 
 + 50.0 
 
 + 0.8 
 
 Beside that solution of the " inverse problem of the perturba- 
 tions of Uranus," which Le Verrier and Adams obtained, we 
 have seen that, using their data, there are a number of other 
 solutions, one of which corresponds to the orbit and mass of Nep- 
 tune. Had Le Verrier been in possession of observations inter- 
 mediate to those which he used, he would not have arrived at so 
 harmonious results. The fact, however, that Neptune does not 
 correspond to his solution, cannot in the least detract from the 
 merit or intrinsic value of his investigations. These may be 
 looked upon as a complete discussion of two distinct problems. 
 
 In the first place, Le Verrier demonstrated the existence of a 
 disturbing planet. He solved this problem completely by prov- 
 ing, not only that it was impossible to represent the motions of 
 Uranus without the assumption of some unknown disturbing 
 body, but that the perturbations were of that analytical form 
 
THE DISCOVERY OF NEPTUNE. 51 
 
 which belongs to the influence of an exterior planet. In so far 
 as this goes, Le Verrier may be considered the discoverer of 
 Neptune. 
 
 In his solution of the second problem, that of the orbit and 
 mass, he was not so successful. But, inasmuch as, by using all 
 the observations within his reach, he found an orbit and mass 
 capable of accounting for the observed motions of Uranus, he 
 must be, in the opinion of mathematicians, legitimately entitled 
 to all the admiration which he would have received had such a 
 planet actually moved in that orbit. He omitted, it is true, the 
 consideration of the terms depending on a near approach to com- 
 mensurability ; but this, although certainly a defect, cannot be 
 considered as an error in the theory, for, within the limits where 
 he had reason to suppose that the orbit was situated, these terms 
 are almost uniformly negligible. His laborious and elegant re- 
 searches have been crowned with brilliant success, and Mr. Le 
 Verrier himself rewarded by the consciousness of having been 
 the immediate occasion of the discovery of Neptune. And al- 
 though the agreement of Neptune's direction at the time of 
 discovery with the direction of the theoretical planet was but 
 accidental, it almost seems as though the heavens strove to 
 show themselves propitious, so happy was the accident, so won- 
 derful the coincidence. 
 
 In order to show as clearly as possible the relative positions of 
 Neptune, and of the theoretical planets of Le Verrier and 
 Adams, I have computed the following table, which gives the 
 true longitude and the radius-vector of each of the three, for 
 every tenth year of the two centuries immediately succeeding 
 Flamsteed's first observation of Uranus. It will be observed 
 from this table, that the longitude of the planet of Le Vender's 
 theory coincided with that of Neptune in 1840 ; and that Nep- 
 tune would be in conjunction with Adams's planet about the year 
 1856. The closest agreement of the radius-vectors was not far 
 from the year 1830, and the greatest discordance in the years 
 from 1710 to 1720, at which time the distance of Neptune was 
 
52 
 
 REPORT .ON THE HISTORY OF 
 
 about midway between that of Uranus and that of the two hy- 
 pothetical planets. 
 
 TRUE LONGITUDE. 
 
 RADIUS- VECTOR. 
 
 Date. 
 
 Neptune. 
 
 Le Verrier. 
 
 Adams. 
 
 Neptune. 
 
 Le Verrier. 
 
 Adams. 
 
 1690, 
 
 34l!l 
 
 
 
 65.1 
 
 81?3 
 
 29.92 
 
 39.09 
 
 40.69 
 
 1700, 
 
 4.4 
 
 79.2 
 
 94.5 
 
 29.84 
 
 39.63 
 
 41.30 
 
 1710, 
 
 26.7 
 
 93.0 
 
 107.3 
 
 29.79 
 
 40.89 
 
 41.66 
 
 1720, 
 
 49.1 
 
 106.5 
 
 119.8 
 
 29.77 
 
 40.04 
 
 41.74 
 
 1730, 
 
 70.8 
 
 120.2 
 
 132.0 
 
 29.80 
 
 39.85 
 
 41.48 
 
 1740, 
 
 93.8 
 
 134.0 
 
 145.5 
 
 29.86 
 
 39.40 
 
 41.09 
 
 1750, 
 
 116.0 
 
 148.2 
 
 158.9 
 
 29.95 
 
 38.71 
 
 40.39 
 
 1760, 
 
 138.0 
 
 163.0 
 
 172.6 
 
 30.04 
 
 37.84 
 
 39.46 
 
 1770, 
 
 160.1 
 
 178.9 
 
 187.3 
 
 30.10 
 
 36.93 
 
 38.37 
 
 1780, 
 
 181.8 
 
 195.1 
 
 202.7 
 
 30.22 
 
 35.68 
 
 37.15 
 
 1790, 
 
 203.5 
 
 212.7 
 
 219.2 
 
 30.28 
 
 34.58 
 
 35.91 
 
 1800, 
 
 225.9 
 
 231.4 
 
 236.8 
 
 30.30 
 
 33.57 
 
 34.74 
 
 1810, 
 
 246.8 
 
 251.2 
 
 255.5 
 
 30.28 
 
 32.78 
 
 33.75 
 
 1820, 
 
 268.5 
 
 270.9 
 
 275.2 
 
 30.23 
 
 32.64 
 
 33.06 
 
 1830, 
 
 290.2 
 
 291.4 
 
 295.5 
 
 30.15 
 
 32.29 
 
 32.77 
 
 1840, 
 
 312.0 
 
 312.0 
 
 315.9 
 
 30.02 
 
 32.63 
 
 32.91 
 
 1850, 
 
 334.2 
 
 332.0 
 
 335.9 
 
 29.96 
 
 3332 
 
 33.47 
 
 1860, 
 
 356.4 
 
 351.0 
 
 355.0 
 
 29.87 
 
 34.26 
 
 34.37 
 
 1870, 
 
 18.7 
 
 8.9 
 
 14.1 
 
 29.81 
 
 35.43 
 
 35.49 
 
 1880, 
 
 41.1 
 
 25.8 
 
 29.9 
 
 29.77 
 
 36.48 
 
 36.73 
 
 1890, 
 
 63.4 
 
 39.4 
 
 45.7 
 
 29.78 
 
 37.99 
 
 37.97 
 
 Mr. Adams has taken no personal part in the controversies 
 which have arisen since the discovery of Neptune, but has con- 
 tinued to devote himself to the pursuit of science. Mr. Le Ver- 
 rier has published 1 several articles in the Oomptes Rendus, in 
 order to defend his claim to be considered the actual discoverer 
 of Neptune, by showing that this planet might have been brought 
 within the limits of his theory. In England, Sir John Herschel 
 has taken 2 similar ground in favor of Mr. Adams. 
 
 1 Comptes Rendus, 1848, Sept. llth, Oct. 2d, etc. 
 
 2 Outlines of Astronomy, pp. 309, 509-512, 516 -518. 
 
THE DISCOVERY OF NJEPTUNE. 53 
 
 The argument of Le Verrier is, 1 that, since he was obliged to 
 found his computations upon irregularities, which, on account of 
 the probable inaccuracy of the observations, were uncertain by a 
 tenth part of their whole value, it would be very natural that 
 this want of precision should affect the positions thence deduced 
 for the disturbing planet, and that these positions should them- 
 selves be wrong by their tenth part. 
 
 I should not have alluded to this reasoning had not Mr. Le 
 Verrier published it, and it will perhaps even now be considered 
 unadvisable to endeavor to refute it. According to this argu- 
 ment, an error of 3.0 would be allowable in the mean distance 
 30, of 4.0 if the mean distance were 40, etc., errors which 
 would make the attraction of the planet to be exerted in a direc- 
 tion totally different from the true one. But even this allow- 
 ance would not correct the error of the radius-vector in 1710 
 and 1890. 
 
 As a rejoinder to the last argument, Mr. Le Verrier says, 2 that, 
 " when there are perturbations, he can tell where Neptune is," 
 but to demand that he " should do it long after the perturbative 
 action has disappeared, is simply to exact of him what is impossi- 
 ble, a species of miracle." Yet at the time of the early ob- 
 servations, when the radius-vector of Neptune differed from that 
 of the theoretical planet by ten times the radius of the earth's orbit, 
 Uranus was, according to Mr. Le Vender's theory, undergoing 
 a perturbation by Neptune. 3 It is unnecessary to allude in this 
 connection to Le Verrier's deduction 4 from his computations, that 
 the small eccentricity of Neptune's orbit would be incompatible 
 with the nature of the perturbations of Uranus. 
 
 With regard to the mass, Le Verrier uses a somewhat differ- 
 ent argument. Assuming the mass to be, as Peirce has found, 
 T 5 ^ of what had been predicted, he shows 5 that this corresponds 
 to an error of but a fifth in the diameter of Neptune. This is 
 very true, no schoolboy will deny it, but it was the mass, 
 
 1 Comptes Rendus, XXVII. p. 273. * Comptes Rendus, XXIV. p. 531. 
 
 2 Ibid., p. 275. 5 jbid., XXVII. p. 277. 
 
 3 SeeHerschel's Outlines of Astronomy, p. 517, 776. 
 
54 REPORT ON THE HISTORY OF 
 
 not the diameter, which he sought. He has farther alluded ] to 
 the fact, that, even now, astronomers are in doubt concerning 
 the exact mass of Uranus, inasmuch as the masses deduced from 
 its action on Saturn and from the periods of the satellites do not 
 harmonize with each other ; and has thence inferred that the 
 same discordance should be expected between the masses of Nep- 
 tune as deduced from the perturbations of Uranus and from the 
 satellite observations. It is an interesting question, and one 
 which still remains open, whether the discrepancies between the 
 two computations of the mass of Uranus might not be reconciled 
 by a proper investigation of the influence of Neptune upon Sat- 
 urn. This is one of the most important questions, connected 
 with Neptune, which remain undecided, and it is earnestly to be 
 hoped that some one of the three illustrious geometers who 
 have labored so faithfully upon the perturbations of Uranus by 
 Neptune may now investigate the Saturn-perturbations produced 
 by the new planet. 
 
 The only other point of Mr. Le Verrier's argument to which I 
 will allude is that in which he says, 2 " The orbit calculated by 
 Mr. Walker, from a position in 1795, and the small arc observed 
 since the discovery, can very well be erroneous by many de- 
 grees, either in 1887 or in 1757, and if I have admitted the 
 positions which it has given for these epochs, it is solely by cour- 
 tesy, and because it presents for me no inconvenience." Of this 
 I may be permitted to say, that Mr. Walker's laborious and ac- 
 curate investigations have given us the orbit of Neptune to a 
 very high degree of precision, and deserve the gratitude and ad- 
 miration of astronomers, not such an imputation as this. It 
 would be contrary to all probability should the place given by 
 Mr. Walker's orbit for those years be false by two minutes.* 
 
 * Walker's orbit represents the course of Neptune as well as can be possibly 
 desired from 1795 to 1848, an interval of 53 years. The error in 1887 or 1757 
 would, according to the doctrine of chances, be to that in either of the years above 
 named in the ratio 65 2 : 27* = 5".8 : 1".0, and the resulting error, therefore, less 
 than six seconds. 
 
 1 Comptes Rendus, XXVII. p. 278. z Ibid., p. 327. 
 
THE DISCOVERY OF NEPTUNE. 55 
 
 We have seen that it represented the observations so perfectly 
 during the opposition of 1848, as to need no correction whatso- 
 ever. 
 
 The argument 1 of Sir John Herschel is mostly based upon 
 the circumstance, that the directions of the real and of the hy- 
 pothetical planet were nearly identical at the time of Neptune's 
 discovery, and upon the fact that the radius-vectors were not 
 very different at that time. But surely it cannot be considered 
 as an analogy between the two orbits, that the perihelion of one 
 was so near the aphelion of the other. 
 
 Reasoning like this seems, however, utterly inapplicable to 
 researches of such nicety and analytical refinement as character- 
 ize those upon the perturbations of Uranus. It would allow to 
 these investigations no other merit than the success with which 
 Neptune's apparent place was approximately predicted. It is 
 an effort to show that the uncertainty of the calculations was so 
 great, that Neptune's perturbative influence may be included 
 within their limits. 
 
 The debt of gratitude which astronomy owes to Le Verrier 
 and to Adams may not be thus diminished. The arguments, 
 which tend to prove that Neptune is the planet of their theory, 
 can only be based upon the supposition of error in that theory, a 
 supposition which I am unwilling to admit. Investigations con- 
 ducted with the care and precision which characterized these 
 must not be so lightly dealt with. The combined labors of Le 
 Yerrier and Peirce have incontrovertibly proved, that, by re- 
 ducing the limits of error assumed for the modern observations 
 to 3", there can be but two possible solutions of the problem 
 There are two different mean distances of least possible error, 
 one of which is 36, and the other 30. The one is included with- 
 in the theory and limits of Le Verrier, and corresponds with 
 Adams's solution; the other is the orbit of Neptune. 
 
 This simple view of the case a view which it seems to me 
 impossible for those not interested in the matter to avoid taking 
 
 1 Outlines of Astronomy, pp. 511, 516. 
 
56 HISTORY OF THE DISCOVERY OF NEPTUNE. 
 
 reconciles all the computations and observations, as well as the 
 discords and contentions. It does not detract in the slightest 
 degree from the well-earned fame of the illustrious geome- 
 ters, who had arrived at a solution of the problem, and I am 
 not aware that it has ever been opposed by mathematical rea- 
 soning. 
 
 CAMBRIDGE, July, 1849. 
 
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