A POPULAR INQUIRY 
 
 THE MOON'S ROTATION 
 
 ON HEE AXIS. 
 
 BY 
 
 JOHANNES VON GUMPACH. 
 it 
 
 WITH NUMEROUS ILLUSTRATIVE DIAGRAMS. 
 
 LONDON: 
 
 BOSWORTH AND HARRISON, 215, REGENT STREET. 
 MDCCCLVI. 
 
G. J. PALMER, SAVOY STREET, STRAND. 
 
PREFACE. 
 
 IT is with much diffidence that the Author submits the 
 present Essay to the Public. To solve a scientific problem 
 in a popular manner ; to render an intricate astronomical 
 question misunderstood by the majority of Astronomers 
 themselves generally intelligible; to oppose the universal 
 current of scientific opinion ; and to prove the most eminent 
 Philosophers and the greatest Mathematicians of this and the 
 past century to have been in error respecting a subject, to 
 which their attention and the exercise of their highest powers 
 have so long been directed, is an attempt not lightly to be 
 undertaken, and suggestive of serious reflections. These 
 reflections presented themselves to the Author with the greater 
 force, as his proof of the Moon's wow-rotation, and his expla- 
 nation of the phenomena on which the prevailing doctrine of 
 the rotation of our satellite is based, involve ulterior conse- 
 quences amounting to little short of a revolution in modern 
 
 974165 
 
IV PREFACE. 
 
 Astronomy. But, however presumptuous the bare thought may 
 be considered, the Author has ri6 cause to shrink from its re- 
 sponsibility ; as his results are chiefly derived from a law of 
 motion, founded on and deduced from established laws of 
 motion, and of which every-day life offers a thousand illustra- 
 tions, though it is for the first time by him applied to Astro- 
 nomy. Of so unobtrusive a character as not likely to attract 
 particular attention, it affects, nevertheless, and in no slight 
 a degree, the theory of gravitation as at present understood, 
 and exposes a fundamental error of modern Astronomy, in 
 consequence of which Astronomers ascribe the corresponding 
 effects of their Theory, as physical irregularities, to the 
 courses of the heavenly bodies; whereas these computed irre- 
 gularities, so far as they are connected with that error, attach 
 to the Theory. Their existence ceases with its source. Hence 
 the recognition of the error will tend materially to simplify the 
 theory of the solar system, and more especially the lunar theory. 
 It is from the very nature of a Science, in which one 
 error not only engenders a hundred others, but moreover 
 stands in the way of a thousand truths, that the law alluded 
 to derives whatever importance may attach to it. Its 
 results and consequences, except so far as they are directly 
 connected with the problem of the Moon's rotation, will not 
 be further traced in the present Essay, the aim of which is 
 but a twofold one : to enlighten the Public on the real merits 
 of a question, which is exciting so general an interest and 
 is understood by so very few ; and to attempt its solution in 
 a popular manner, with the view of submitting the latter to 
 
PREFACE. V 
 
 the scientific judgment of Astronomers. How far the Author 
 may have failed or succeeded in his object, it will be for 
 Astronomers and the Public to decide. 
 
 One feature he has to submit to the more especial consi- 
 deration of Mathematicians. They have, by analysis, proved 
 the modern theory of the Moon's rotation, in its minutest 
 details, even to the participation of the velocity of rotation 
 in the secular inequalities of the lunar orbit, from the laws 
 of gravitation. The Author proves that theory to be un- 
 tenable in every point of view, showing that it does not account 
 for the phenomena of libration, to explain which is its object ; 
 that it is based on an error of more than two days, as regards 
 the assumed period of rotation ; nay, that, in accordance with 
 the phenomena, which the Moon actually presents, her rota- 
 tion is a bare physical impossibility. It will, therefore, be 
 for Mathematicians to take this result into consideration. 
 Resting as it does on empirical facts, it is of vital importance 
 to Analysis as well as to the Theory of Gravitation. 
 
 The documentary evidence relating to the discussion is 
 contained in the Notes, taken, in preference, from popular or 
 elementary astronomical works of a recent date, and composed 
 by the most distinguished Astronomers of various countries. If 
 the reader, acquainted with foreign astronomical literature, 
 should notice the omission of two of the best works extant on 
 Popular Astronomy, namely, ARAGO'S "Astronomic Popu- 
 laire" and KEYSER'S " De Sterrenhemel : " the reason is, that 
 
 * 
 
 the third volume of the former work, which will contain the 
 matter relative to the subject of this Essay, has as yet not 
 
VI PREFACE. 
 
 been published ; and that the Author has been unable to 
 procure the Treatise of the Astronomer of Leyden, in time to 
 avail himself of it on the present occasion. In none of the 
 numerous astronomical works, which have come under his 
 notice, has the subject of the Moon's rotation been treated in 
 anything like a logical and consistent form, or been fully en- 
 tered into as to its real merits and bearings. Hence, he was 
 obliged to have recourse to so many different sources, in order 
 to collect the evidence, which he was anxious to place before 
 the reader upon each single point of his inquiry. 
 
A LIST OF ASTRONOMICAL WORKS QUOTED IN THE 
 PRESENT ESSAY. 
 
 AIRY, G. B., Astronomer Royal, Six Lectures on Astronomy. 2nd Edition. London, 
 1851, 8vo. 
 
 BARLOW, P., of the Royal Military Academy, Woolwich, A new Mathematical and 
 Philosophical Dictionary. London, 1814. 8vo. 
 
 BEER, W. und J. H. MJEPLER, Der Mond nach seinen kosmischen und individuellen 
 Verh'dltnissen, oder Allgemeine vergleichende Selenographie. Berlin^ 1837, 
 4to. 
 
 BIOT, J. B., Membre d 1' Institut de France, Professeur d' Astronomic au College Im- 
 perial de France, &c., Traite elementaire d 1 Astronomie physique. 2de 
 Edition. Paris, 1811, 3 vols. 8vo. 
 
 COMTE, AuausTE, Traitt philosophique d' Astronomie populaire. Paris, 1845, 8vo. 
 
 DELAMBRE, Trsorier de 1' Universite de France, Professeur d' Astronomie au College 
 Royal de France, &c., Astronomie Theorique et Pratique. Paris, 1814, 3 vols. 
 4to. Histoire de V Astronomie au dix-huitieme Siecle, Paris, 1827, 4vo. 
 
 DELAUNAY, CH., Membre de 1' Institut, Professeur a 1' Ecole Poly technique, &c. 
 Cours elementaire d' Astronomie. 2nd Edition. Paris, 1855, 8vo. 
 
 FERGUSON'S Astronomy explained upon Sir Isaac Newton's principles. Edinburgh, 
 1811, 2 vols. 8vo. 
 
 FRANOOEUR, L. B., Professeur de la Faculte" des Sciences de Paris, &c. Uranographie, 
 4e. Edition. Paris, 1828, 8vo. 
 
 GALILEO GALILEI, nobile Fiorentino (died 1642), Opere. Milano, 13 vol. 8vo. 
 
 GRANT, ROB., F.R.A.S., History of Physical Astronomy. London, 1852, 8vo. 
 
 GREGORY, OL., Teacher of Mathematics, Cambridge, A Treatise on Astronomy. Lon- 
 don, 1802, 8vo. 
 
 GUMMERE, JOHN, A.M., Member of the American Philosophical Society, &c., An Ele- 
 mentary Treatise on Astronomy. 4th edition, Philadelphia, 1851, 8vo. 
 
 HERSCHEL, SIR JOHN, A Treatise on Astronomy. London, 1851, 12mo. Outlines of 
 Astronomy. London, 1849, 8vo. 
 
 HIND, J. R., F.R.A.S., The Illustrated London Astronomy, London, 1853, 8vo. An 
 Astronomical Vocabulary, London, 1852, 12mo. 
 
 HUMBOLDT, A. VON, Cosmos. London, 1849 52, 4 vols. 8vo. 
 
 JOHNSTON, A. KEITH, F.R.S.E., Atlas of Astronomy, edited by J. R. Hind. Edin- 
 burgh, 1855, 4to. 
 
Vlll 
 
 KEPLER, J. (died 1630), Epitome A sir onomiffi Coper nicanw. Francofurti,-1635, 8vo. 
 LALANBE, JER., de 1' Academic des Sciences de Paris, Directeur d P Observatoire de 
 
 1' Ecole royale militaire;, &c., Astronomie, 3e edition. Paris, 1792, 3 vols. 4to. 
 LAPLACE, le Marquis de, Pair de France, &c.y Exposition du Systeme du Monde. 6e. 
 
 edition. Paris, 1836, 2 vols. 8vo. 
 LARDNER, DION., D.C.L., formerly Professor of Natural Philosophy and Astronomy in 
 
 University College, London, Handbook of Astronomy, London, 1853, 8vo. 
 
 Handbook of Natural Philosophy, Mechanics, London, 1855, 8vo. Popular 
 
 Astronomy, London, 1855, 8vo. 
 LITTROW, J. J., Director der Sternwarte in Wien, Die Wunder des Himmels, Stuttgardt, 
 
 1836, 3 Bde, 8vo. 
 MIDLER, J. H., Professor der Astronomic und Director der Sternwarte zu Dorpat, &c. 
 
 Popul'dre Astronomic, 4te Auflage, Berlin, 1849, 8vo. 
 NARRIEN, JOHN, F.R.S., &c., Practical Astronomy and Geodesy, London, 1845, 8vo. 
 
 An Historical Account of ilie Origin and Progress of Astronomy. London, 
 
 1832. 
 NEWTON, SIR ISAAC (died 1678). Philosophic Naturalis Principia Mathematica. 
 
 Glasguae, 1833, 2 vols. 8vo. The Mathematical Principles of Natural 
 
 Philosophy, to which is added his System of the World. New York, 8vo. 
 NICHOL, J. P., LL.D., Professor of Astronomy in the University of Glasgow, The 
 
 Planetary System : its Order and Physical Structure. London, 1851, 8vo. 
 NICOLLET, J. N., Memoire sur la Libration de la Lune, lu a V Academic des Sciences 
 
 le 7 Decenibre, 1818, in the " Connaissance des Temps," Annee, 1822, p. 228 
 
 283. Paris, 1819, 8vo. 
 
 QUETELET, A., Directeur de 1' Observatoire Royal de Bruxelles, &c., Elements d' Astro- 
 nomie, 3e edition. Paris, 1847, 8vo. 
 SANTINI, GIOVANNI, Professore di Astronomia nell' J. R. universita di Padova, &c., 
 
 Elementi di Astronomia, edizione seconda. Padova, 1830, 2 vols. 4to. 
 SCHUBERT, FR. TH., Mitglied der Kays, und Konigl. Akademie der Wissenschaften zu 
 
 St Petersburg, &c. Populare Astronomic, St. Petersburg, 1810, 3 Bde 8vo. 
 SMYTH, CAPT. (now Admiral) W. H., R.N., K.S.F., D.C.L., &c., A Cycle of Celestial 
 
 Objects. London, 1848. 2 vols. 8vo. 
 SOMERVILLE, MARY, On the Connexion of the Physical Sciences, 8th edition. London, 
 
 1849, 8vo. 
 VINCE, REV. S., Plumian Professor of Astronomy in the University of Cambridge, &c., 
 
 A Complete System of Astronomy, 2nd, edition. London, 1823, 3 vols. 4to. 
 WOODHOUSE, ROBT., Lucasian Professor of Mathematics in the University of Cambridge, 
 
 A Treatise on Astronomy Theoretical and Practical. Cambridge, 1821, 2 
 
 vols. 8vo. 
 
ON THE MOOFS ROTATION. 
 
 1. SCIENCE has its misconceptions ,'atod delusions, fw well 
 as Popular Opinion, and the learned 'are ' no " less' proile 'tD 
 deceive themselves than are the generality of men. Imperfec- 
 tion, being the stamp of humanity, attaches equally to 
 Genius of the highest order, and to Common Sense of the 
 lowest degree. If the latter be unable to raise itself to the 
 lofty views of the former, the former sometimes finds it difficult 
 to descend to the plain grounds of the latter. Both have their 
 defects and their advantages, both their claims and their 
 rights. Whenever, therefore, Science teaches a doctrine, or 
 maintains a theory of a popular character, which falls within 
 the sphere of ordinary comprehension, and yet proves incom- 
 prehensible to the ordinary understanding, such a theory 
 respecting which the ideas of Science may fairly be supposed 
 not to possess a clearness, of which its language in convey- 
 ing them is deficient becomes the legitimate object of a 
 popular inquiry. 
 
 2. Seldom, probably, on the wide field of astronomical 
 speculation, has a question been raised which, at first sight, 
 appears of so simple a nature and so easy of solution as the 
 question: Does the Moon, or does she not, rotate on her axis1 
 From our very childhood we have beheld the faithful satellite 
 
 B 
 
2 ON THE MOON'S ROTATION. 
 
 of the Earth with the same familiar smile playing about the 
 same familiar face, unchanging in its expression, unaltered 
 in its features ; and whatever vicissitudes of fortune we may 
 have experienced, she, at least, has never turned her back upon 
 us. That modern Astronomy, whilst measuring the orbits of 
 stars at distances from us too vast to conceive, and thousands 
 of millions of times exceeding the distance of the Moon, should 
 not have been able to place beyond doubt the rotation of a 
 heavenly body, which the telescope brings within reach almost 
 of our outstretched hands, sounds like a paradox. Yet such 
 is the case ; and the problem, so far from presenting no diffi- 
 culties, involves considerations which have taxed the greatest 
 practical .sl^lVMd.t'he highest mathematical powers of more 
 than oi\e 'Astronomer '6f this and the past century. " To de- 
 termine,'' the immortal author of the Principia remarks, "the 
 exact position of the Moon's axis in regard to the fixed stars, 
 and the variation of this position," two of the principal ele- 
 ments on which the solution of our question depends " is a 
 problem worthy of an Astronomer" l It need not, therefore, 
 
 1 " In like manner is the moon re- moon's orbit, or to the difference be- 
 
 volved about its axis by a motion twixt its mean and true motions, and 
 
 most equable in respect of the fixed this is the moon's libration in longi- 
 
 stars, viz, in 27 d. 7 h. 43', that is, in tude ; but it is likewise affected with 
 
 the space of a sidereal month ; so a libration in latitude, arising from 
 
 that this diurnal motion is equal to the inclination of the moon's axis to 
 
 the mean motion of the moon in its the plane of the orbit, in which the 
 
 orbit; upon which account the same moon is revolved about the earth, for 
 
 face of the moon always respects that axis retains the same position 
 
 the centre about which this mean to the fixed stars nearly, and hence 
 
 motion is performed, that is, the ex- the poles present themselves to our 
 
 terior focus of the moon's orbit view by turns, as we may understand 
 
 nearly; and hence arises a deflection from the example of the motion of 
 
 of the moon's face from the earth, the earth, whose poles, by reason of 
 
 sometimes towards the east, and the inclination of its axis to the 
 
 other times towards the west, accord- plane of the ecliptic, are by turns 
 
 ing to the position of the focus illuminated by the sun. To deter- 
 
 which it respects; and this deflec- mine exactly the position of the 
 
 tion is equal to the equation of the moon's axis to the fixed stars, and 
 
ON THE MOON'S ROTATION. 3 
 
 surprise that with respect to the real bearings and the true 
 merits of a subject, comprehending problems of this kind, 
 much ignorance should prevail even among men of scientific 
 acquirements and extensive general information ; but that the 
 majority of Astronomers themselves should participate in that 
 ignorance, is a somewhat startling circumstance, abundant proof 
 of which will come to light in the progress of our inquiry. 
 
 3. The opinion that the Moon, whilst revolving round the 
 Earth at the same time, and in precisely the same period rotates 
 on her axis, is of comparatively recent origin, and opposed to 
 the views of the Ancients. True, the striking fact that she 
 always presents the same identical hemisphere towards us, was 
 remarked from the earliest times; but, though the genius of 
 the Greek and Arab Astronomers may have been fully capable 
 of realizing the modern idea, it is by no means singular, as 
 Mr. Grant thinks, that it should not have suggested itself to 
 them. 2 For, independently of other considerations, the Ptole- 
 mean system accounted in the most satisfactory manner for all 
 the phenomena then known to Astronomy ; and there conse- 
 quently existed no necessity for a different explanation. 3 The 
 
 the variation of this position, is a astronomers until the revival of 
 problem worthy of an astronomer." science in modern times." GRANT, 
 NEWTON, The System of the World. History of Physical Astronomy, ip.72. 
 American Ed. of the Principia, p. 3 " Dans les idees des anciens as- 
 535. tronomes sur le mouvement, il n'y 
 2 " Another important subject aurait pas eu la (que la lune tourne 
 which soon afterwards engaged the touj ours vers la terre la meme portion 
 attention of geometers was the libra- de sa surface), une preuve qu 'elle 
 tion of the moon. It had been re- tourne sur elle-meme ; tout au con- 
 marked, from the most ancient times, traire, on en aurait deduit 1'absence 
 that the moon turns the same side de toute rotation de 1'astre autour de 
 
 towards the earth throughout the son centre Si 1' on fait mou- 
 
 entire course of every revolution. A voir la lune autour de la terre, 
 
 curious consequence of this fact is, comme si elle etait attachee a une 
 
 that the motion of that body round barre rigide dirigee vers le centre 
 
 the earth is equal to her motion de ce dernier corps et mobile au- 
 
 round her axis. It is singular, how- tour de ce centre, la lune tournera 
 
 ever, that this conclusion does not necessairement toujours la meme 
 
 appear to have suggested itself to face vers la terre; et Ton n'a pas 
 
 B 2 
 
ON THE MOON'S ROTATION. 
 
 modern theory could not well originate till after the intro- 
 duction of the Copernican refprm, and then only in con- 
 nexion, either with mistaken views or with the more perfect 
 results of telescopic observation. Hence, we find two of 
 the greatest Astronomers of their age, Galileo 4 and Kep- 
 ler, 5 opposed to that theory. It has been asserted that they 
 were so, " merely because they hesitated to call the certainly 
 somewhat peculiar kind of rotation of the lunar globe by this 
 name." 6 Their own words prove the contrary. Kepler assigns 
 even a reason, though only one of speculative philosophy, for 
 
 besoin d' imaginer qu' elle tourne 
 sur elle-meme, pour rendre conipte 
 des apparences." DELAUNAY, Cours 
 Elem. d'Astronomie, p. 392. 
 
 4 In his letter to Sig. Alf. A. 
 di Udine, which treats of the Moon's 
 librations, Galileo, after alluding to 
 certain conjectures he had formed 
 with regard to our satellite, continues 
 thus: "Da queste congetture sveg- 
 liato mi venne, e gia molto tempo, 
 pensiero di por mente se da qualche 
 piu sensata, e certa osservazione io 
 potessi venire in notizia, se per 
 av ventura il gloho Lunare senza mu- 
 tazione alcuna riguardasse sempre il 
 globo Terrestre, in maniera, che pro- 
 dotta una linea retta dal centre della 
 Luna al centro della Terra, questa 
 passasse perpetuamente per lo mede- 
 simo punto della superficie della 
 Luna; il che sarebbe sicuro argomento, 
 che la Luna non avesse in se stessa4n- 
 clinazione, o titubazione alcuna, 
 ma sempre riguardasse la Terra coll' 
 istessa parte della sua faccia." He 
 now describes the optical pheno- 
 mena which, under the above suppo- 
 sition, he imagined to himself the 
 Moon ought to present: "Queste 
 sono le mutazioni, che io per con- 
 
 gettura m* immaginava doversi 
 scorgere. Da questo pensiero 
 spinto ;" he then goes on to say, " in- 
 cominciai ad osservare minutamente 
 se vestigio alcuno di apparente 
 mutazione si potesse da qualche 
 macchia raccorre ; ed in questo mi fu 
 favoravole la Natura." Nothing, 
 therefore, can be more positive than 
 is his opinion against the Moon's 
 rotation on her axis. GALILEO, 
 Opere, Vol. V. p.p. 26, 29. 
 
 5 " Luna vicissim non gyratur cir- 
 ca sui corporis axem, maculis id 
 arguentibus. Cur autem hoc? nisi 
 quia circa lunam nullius amplius 
 planeta circumire cernitur; nullum 
 igitur habet luna planetam, cui 
 motum inferat, gyratione sui corpo- 
 ris : gyratio igitur in luna, ut super 
 vacua, fuit omissa." IOA. KEPLEBI 
 Epitome Astron. Copern. p. 555, 
 556. 
 
 6 " Wenn Kepler und Galilao dem 
 Monde eine Kotation absprachen, so 
 kommt dies nur daher well sie An- 
 stand nahmen, die allerdings eigen- 
 thiimliche Umdrehung der Monds- 
 kugel Rotation zu nennen." BEER 
 und M^DLEE, Selenographie, S. 12. 
 
ON THE MOON'S ROTATION. 5 
 
 his opinion. The Moon, he argues, being attended by no 
 satellite to which to impart motion by its own rotation, has 
 consequently not been endowed with rotation. We ourselves 
 attach no weight to this argument, but is there a known law 
 of nature, to which modern Astronomers had ever pointed in 
 order to explain the mechanical necessity of the Moon's rota- 
 tion? As to their proofs of "the fact" we shall examine them 
 hereafter. The now prevailing opinion was first established 
 by the authority of Newton, one of those men of transcendent 
 genius and vast powers of mind and intellect, who appear but 
 once in a long course of centuries. A mere assumption on his 
 part, barely countenanced by a doubtful analogy 7 the view, 
 expressed by the author of the Principia, gradually gained 
 ground, and ere long was almost generally adopted. Yet it 
 never ceased to be questioned by individual Astronomers, and 
 about the middle of the last century gave rise to a very warm 
 controversy. 8 So great, however, proved the preponderance of 
 Newton's authority, and so little capacity to grapple with the 
 question was evinced by the opposition, that the dispute ended 
 in the complete discomfiture of the latter. From that time the 
 Newtonian theory, having been more firmly established than 
 ever by the labours of the great French Mathematicians of the 
 last and the present century, we meet but in rare instances 
 with a doubt in its correctness, timidly expressed by some Astro- 
 nomer of more independent judgment than real ability and 
 moral courage, 9 until, what in its origin was a simple assertion, 
 
 7 " L' analogie devait naturelle- the former name, that Littrow alludes 
 ment porter les astronomes a re- in Note 21 (comp. 25), and Schu- 
 chercher si la lune n' a point un bert in Note 43. For the reasons 
 mouvement de rotation sur son axe stated in the text, it led to no re- 
 eomrne la terre, independamment suits, and is devoid of all further in- 
 d'un mouvement de translation dans terest to us. 
 
 1'espace." QDETELET, Elements d 'As- 9 "From the most careful obser- 
 
 tronomie, p. 112. vations upon the moon, it appears 
 
 8 It is to this controversy or ra- that she has always very nearly the 
 ther dispute, for it hardly deserves same side turned towards the earth ; 
 
ON THE MOON'S ROTATION. 
 
 is, at the present day, universally received as an elementary 
 fact in modern Astronomy. 10 
 
 4. Nor is this all. Tt was ascertained by Sir William Her- 
 schel, the elder of this illustrious name, that also the satellites 
 of Jupiter continually present the same hemisphere to their pri- 
 mary. From this he concluded that they turn on their axes like 
 our Moon, and consequently in periods equal to their periods of 
 revolution round Jupiter; and Sir John Herschel thinks that 
 there is apparent reason for such an inference. 11 But a simi- 
 lar reserve does not distinguish the opinions of other astrono- 
 mical writers. First, the plausibility of Sir William Herschel's 
 conclusion is raised into certainty. 12 Next, he is represented 
 as having proved the conclusion, instead of the fact from 
 which it was derived. 13 Then the conclusion is laid down as 
 
 hence, if she turn upon her axis, the 
 time of rotation must be equal to 
 that of the moon's synodic revolu- 
 tion," 0. GREGOBY, A Treatise on 
 Astronomy, p. 299. 
 
 10 "It is now received as an ele- 
 mentary fact in astronomy, that our 
 moon turns on its axis in the precise 
 time in which she revolves around 
 the earth. It is by reason of this 
 arrangement that she always presents 
 the same side or face to the earth, 
 for of her other face we know as little 
 as a telescope in Britain can reveal 
 concerning our antipodes." NTCHOL, 
 The Planetary System, p. 52. Com- 
 pare also the various notes, expres- 
 sing the same opinion. 
 
 11 "By assiduous observation it 
 has been ascertained that they (the 
 satellites of Jupiter) are subject to 
 marked fluctuations in respect of 
 brightness, and that these fluctua- 
 tions happen periodically, according 
 to their position with respect to the 
 sun. From this it has been con- 
 
 cluded, apparently with reason, that 
 they turn on their axes like our moon, 
 in periods equal to their respec- 
 tive sidereal revolutions about their 
 primary." SIB JOHN HEBSCHEL, 
 A Treatise on Astronomy, p. 298. 
 
 12 "From the periodical variations 
 in the light of the satellites (of Jupi- 
 ter), Sir William Herschel ascer- 
 tained that each of them presents 
 always the same face to the planet 
 about which it revolves, and hence 
 it follows that, like the moon, each 
 of them performs a revolution on its 
 axis in the time of a revolution in 
 its orbit." NARBIEN, Practical Astro- 
 nomy, p. 210. 
 
 13 "Sir William Herschel has 
 shewn, from his observations, that 
 the periods of rotation and revolu- 
 tion of the (fourth) satellite (of 
 
 Jupiter) are equal The other 
 
 satellites have been found to exhibit 
 appearances similar to those of the 
 fourth, and in the case of each of 
 them also, the periods of rotation 
 
ON THE MOON'S ROTATION. 
 
 a universal rule, 14 and lastly, it is placed before us in the 
 light of a general law of nature, the source of further deductions 
 and consequences. 15 Nay, the philosophical necessity of that 
 law is demonstrated, and its nature shown to be such that, as a 
 convincing proof of its existence, we might indulge the hope of 
 seeing, in due time, a baby-Moon make its appearance in the 
 heavens, were it not that in the life of the Sun's progeniture, 
 as in that of man's, everything is subject to accident, and 
 more or less involved in mystery} 5 
 
 and revolution are equal This 
 
 curious fact had been already esta- 
 blished in the case of the moon and 
 the fifth satellite of Saturn, and 
 there is good reason to suppose that 
 it forms an essential characteristic 
 of the movements of all the second- 
 ary bodies of the planetary system." 
 GRANT, History of Physical Astro- 
 nomy, p. 249. 
 
 " The distance and minuteness of 
 Jupiter's satellites render it extremely 
 difficult to ascertain their rotation. 
 It was, however, accomplished by 
 Sir William Herschel from their 
 relative brightness. He observed 
 that they alternately exceed each 
 other in brilliancy, and, by compar- 
 ing the maxima and minima of their 
 illumination with their positions 
 relative to the sun and to their pri- 
 mary, he found that, like the moon, 
 the time of their rotation is equal to 
 the period of their revolution about 
 Jupiter. Miraldi was led to the same 
 conclusion with regard to the fourth 
 satellite, from the motion of a spot 
 on its surface." MRS. SOMEBVILLE, 
 On the Connexion of the Physical 
 Sciences, p. 79. 
 
 14 " The periods of rotation of the 
 moon and the other satellites are 
 
 equal to the times of their revolu- 
 tions, consequently these bodies 
 always turn the same face to their 
 primaries." MRS. SOMEBVILLE, On 
 the Connexion of the Physical 
 Sciences, p. 77. 
 
 15 " But this (our moon turning 
 on its axis in the precise time in 
 which she revolves around the 
 earth) appears the law of all the 
 satellites , it holds, at least, with all 
 whose rotation we have yet disco- 
 vered, in every instance the period 
 of rotation corresponding with the 
 period of revolution. There cannot 
 be a doubt that an arrangement 
 so singular rests on some great, 
 though yet unknown ordinance of 
 Nature, but on the mere surface of 
 the case how utterly does such a fact 
 break down the idea, that the time 
 of the rotation of an orb is in any re- 
 spect related to its distance from the 
 Sun." NICHOL, The Planetary Sys- 
 tem, p. 52. 
 
 16 " It is abundantly plain, I think, 
 that what has already been demon- 
 strated contains the solution of the 
 problem of the satellites. Thrown 
 off by the primary planets, as these 
 had been thrown off from the Sun, 
 their revolutions must correspond in 
 
ON THE MOON'S ROTATION. 
 
 5. If we inquire : Of what importance to astronomical Science 
 is the solution of our problem ? the indirect answers, returned 
 by different Astronomers, differ 'materially and show anything 
 but lucidity to prevail in their views upon the subject. Whilst 
 Delambre gives it as his opinion, concerning one of the prin- 
 cipal elements of our question, that Astronomy might well have 
 rested satisfied with the assertions of Kepler in regard to it, 1T 
 Madler attaches the utmost weight to the determination of a 
 similar element, considering it of the highest consequence with 
 respect to a future history of the origin of our solar system. 18 
 
 direction with the rotation of their 
 central globes; they must lie near 
 the plane of the respective equators 
 of the planets,* and they must ro- 
 tate on their axes as they revolve 
 in their orbits." 
 
 " I have not chosen to refer further 
 to the exceptional case of the satellites of 
 Uranus. Philosophically speaking, this 
 exception must be regarded as an acci- 
 dent, which means a disturbance impressed 
 on one part of a grand scheme by some 
 operation or interference, with the nature 
 of which we are yet wholly unacquainted. 
 The satellites present further a curious 
 anomaly, or rather peculiarity. In so far 
 as we know, they all rotate on their axes, 
 like our moon, in the exact period of a 
 revolution in their orbits. This mode of 
 rotation is evidently that of the original 
 ring ; but why the satellites have pre- 
 served that period is a mystery" 
 NIOHOL, The Planetary System, p 
 241. 
 
 17 Mayer s' arrete a 129' pour 1' in- 
 clinaison (de 1'equateur lunaire sur 
 1'ecliptique). Lalande trouve 143'. 
 Cassini avait donne 2 . . . Kepler 
 avait dit seulement que cetteinclinai- 
 son etait tres petite et la difference 
 absolument nulle entre le lieu du 
 
 rioeud de 1' equateur et celui de 1' 
 orbite. Le probleme paraissait aussi 
 completemeut resolu qu' il le faut 
 pour 1' astronomic, qui peut s' en 
 tenir aux deux assertions de Kepler 
 Le reste n' interesse guere que les 
 geometres. DELAMBRE, Histoire de 
 I' Astron. au 18 siecle. p. 432. 
 
 18 Man sieht dass diese Schwan- 
 kungen nicht eigentlich der Mond- 
 kugel selhst zuzuschreiben sind, und 
 dass sie sogar fur unsern Anhlick 
 wegfallen wiirden, wenn die Un- 
 gleichheiten des Umlaufs zugleich 
 der UmdreJiung angehorten. Man 
 kann indess die Frage aufwerfen, 
 oh in der Umdrehung selbst nicht 
 kleine, hisher nur noch nicht be 
 merkte, Ungleichheiten stattfinden, 
 oh es also nicht ausser der oben 
 betrachteten und als optisch zu be- 
 zeichnenden Libration, noch eine 
 eigenthiimliche physische gebe ? La- 
 place, und noch erschopfender Pois- 
 son, haben diesen Gegenstand theo- 
 retisch, Nicollet, Bouvard und Arago 
 praktisch auf dem Wege der Beo- 
 bachtung untersucht; doch 1st er 
 noch nicht zur letzten Entscheidung 
 gefiihrt: er verdient es im vollen 
 Maasse, dass man das Aesserste 
 
ON THE MOON'S ROTATION. 
 
 Others again, like Littrow, state the entire question of the 
 Moon's rotation to rest on a mere contention about words, 19 so 
 as to offer no interest whatever in a scientific point of view. 
 The latter opinion is as erroneous, as the former is visionary. 
 To doubt the importance of our problem, would be to doubt 
 
 damn setze ihn zu erforschen, denn 
 so wenig dies auf dene rsten An- 
 blick scheinen mochte er 1st von 
 der hochsten Wichtigkeit fur eiue 
 klinftige Gescliichte der Entstehung 
 des Sonnensy stems. MIDLER, Popul. 
 Astronomie S. 163. 
 
 19 " Die andere Seite des Mondes 
 1st von der Erde abgewendet und 
 wird ewig von ihr abgewendet blei- 
 ben, gleichsam als wenn die feste 
 Masse des Mondes durcb eine Stange, 
 die durch den Mittelpunkt des Erde 
 und des Mondes gent, mit ihr im- 
 veranderlich verbunden ware, und 
 an dieser Stange in jedem Monat um 
 uns herum gefuhrt wiirde. 
 
 " Diese Sonderbarkeit in der Be- 
 wegung des Mondes, die wir auch 
 bey den Monden aller iibrigen Plane- 
 ten wieder finden werden, hat zu 
 einem langen und heftigen Streite 
 unter den Gelehrten Anlass gegeben. 
 Newton war es, der jene Erschei- 
 nung zuerst durch folgende Worte 
 ausdrtickte: 'Der Mond zeigt uns 
 immer dieselbe Seite, also drebt er 
 sich um seine Axe.' Andere Astro- 
 nomen wollten daraus gerade das 
 Gegentheil schliessen, dass er sich 
 namlich nicht uui seine Axe drehe, 
 eben weil er uns immer dieselbe Seite 
 zuwendet. 
 
 " Ohne uns hier bei der Geschichte 
 jener Discussionen langer aufzuhal- 
 ten, wollen wir nur bemerken, dass 
 die Sache in letzter Instanz sich 
 mit eiiiem Wortstreite endet, urid 
 dass man, je nachdem man das Wort 
 
 DreJien in dem einen oder dem andern 
 Sinne nimmt, eben so gut sagen kann, 
 der Mond drehe sich, als er drehe 
 sich nicht. Etwa so, wie man von 
 dem Tische, auf dem man schreibt, 
 ganz mit demselben Kechte sagen 
 kann, er babe eine, und er habe 
 keine Bewegung. Er hat keine, 
 weil er seinen Ortgegen die nachsten 
 Umgebungen, gegen die Wande des 
 Zimmers nicht andert; und er hat 
 eine, weil er auf der Erde 1st, die, 
 wie leder weiss, sich um die Sonne 
 und um sich selbst bewegt, und mit 
 der sich daher auch der Tisch be- 
 wegen muss. 
 
 " Das Missverstandniss bei dem 
 Monde kam daher, weil wir densel- 
 ben aus der Erde, aus dem eigentli- 
 chen Mittelpunkt seiner Bewegung, 
 betrachten. Denken wir uns aber ein 
 Auge ausser der Mondsbahn, z. B. 
 in der Sonne, so wird dasselbe zur 
 Zeit unsres Neumondes die uns im- 
 mer verborgene Seite, zur Zeit des 
 Vollmonds aber, zugleich mit uns, 
 die uns immer sichtbare Seite des 
 Mondes sehen, und fur einen solchen 
 Beobachter in der Sonne wiirde es 
 daher keinem weitern Zweifel un- 
 terliegen, dass der Mond sich in der 
 That, und zwar in derselben Zeit um 
 seine Axe dreht, in welcher er um die 
 Erde geht, oder dass die Kotation 
 desselben seiner Revolution, sein 
 Jahr seinem Tage gleich ist." LIT- 
 TROW, Die Wunder des Himmels Bd. 
 I. S. 326, 327. 
 
10 ON THE MOON'S ROTATION. 
 
 the importance of the mechanical economy of the solar system ; 
 for without a definite solution of the one, we can never hope to 
 arrive at an ultimate knowledge of the other. 
 
 6. The erroneous view of Littrow, to which we have just 
 alluded, he supports by arguing that, according to the meaning 
 we attach to the term rotation, we may with equal propriety 
 maintain that the Moon does, and does not, rotate on her axis. 
 A laxity of expression, such as is here implied to be in use 
 among Astronomers, must, however well adapted it might be 
 for the demonstration of Jesuitical dogmas, prove fatal in the 
 pursuit of any scientific inquiry. We have, therefore, to deter- 
 mine the precise sense, which the reader is to attach to our 
 question, before we proceed. Littrow compares the Moon's 
 rotatory motion with that of "the table on which we write," and 
 of which it may with equal truth be said, that it is in motion 
 and that it is at rest. " It is at rest," he goes on to remark, 
 " inasmuch as its position does not change with regard to the 
 immediately surrounding objects ; it is in motion, since it ne- 
 cessarily participates in the motion of the Earth, which, as 
 everybody knows, revolves round the Sun, and at the same 
 time tunis about her own axis." But precisely because 
 everybody knows this, and the motion of the Earth is univer- 
 sally admitted, the illustration on its very surface betrays itself 
 to be a fallacy. And the fallacy consists in this, that it sup- 
 poses rotatory motion to be inseparable from progressive cir- 
 cular motion an opinion very generally entertained, and which 
 has even led to the assertion, that both Astronomers and Mathe- 
 maticians never have used, and never do use, the term rotation 
 in any other sense, 20 so that, virtually, our question would in- 
 
 20 " To the Editor of the Times. about her own axis ; but it has just 
 
 " Sir, No idea was more remote been pointed out to me that Mr. 
 
 from my mind a few minutes ago Symons, in the postscript to his let- 
 
 than that of requesting your inser- ter of yesterday, has referred to a 
 
 tion in The Times of any remarks of Mr. Hopkins, but without using the 
 
 mine on the discussion which has distinctive initials which might have 
 
 recently appeared in your columns clearly shown what person of that 
 
 respecting the moon's rotary motion name he intended to designate. The 
 
ON THE MOON S ROTATION. 
 
 11 
 
 volve an inaccurate or ambiguous mode of expression, rather 
 than a positive fact. Such, however, is by no means the case ; 
 and Madler very justly observes, that the Moon's rotation is a 
 real motion on her own axis, altogether distinct from that of her 
 
 gentleman for whom this honour 
 was intended is, I presume, Mr. 
 W. Evan Hopkins, the author of 
 a work On Geology and Magnetism, 
 by which, as well as by certain other 
 writings, his name has become asso- 
 ciated with geology. As the author 
 of various geological memoirs, and 
 as having recently presided over the 
 Geological Society of London, my 
 own namT has likewise become as- 
 sociated with the same science ; and 
 thus it has not unfrequently hap- 
 pened that some of Mr. Evan Hop- 
 kins's speculations have been most 
 unduly attributed to myself; and I 
 believe that I even enjoyed, for 
 some time, though most unwittingly, 
 the honour, with some persons, of 
 having written the work above re- 
 ferred to, although the scientific 
 views there advocated are diame- 
 trically opposed to those which I 
 entertain. Under these circum- 
 stances I think it right to state that 
 I am not the person alluded to by 
 Mr Symons. I do not choose to 
 incur the risk of being saddled with 
 opinions which I regard with a sort 
 of scientific horror, from whatever 
 quarter they may emanate ; nor do 
 I assuredly wish to undertake the 
 chivalrous task of supporting Mr. 
 Symons against Newton, Laplace, 
 and their whole host of followers. 
 That gentleman is certainly fighting 
 against fearful odds odds which 
 might, one would have supposed, 
 have suggested to him the thought 
 that a little more exalted apprecia- 
 
 tion of the powers of the most dis- 
 tinguished philosophers, and a little 
 more diffident appreciation of his 
 own, together with a little elemen- 
 tary knowledge of dynamics, might 
 have saved him from this dire col- 
 lision with such antagonists. Mr. 
 Symons has yet to learn the mean- 
 ing of the term ' rotary motion ' in 
 the sense which every astronomer 
 and mathematician has ever used it. 
 The * fact that she (the moon) re- 
 volves round the earth, turning the 
 same hemisphere towards it,' does 
 constitute her rotary motion about 
 her own axis in the only sense in 
 which the term has any dynamical 
 significance, and the only sense in 
 which astronomers have ever used 
 it, notwithstanding the astounding 
 boldness with which Mr. Symons 
 asserts the contrary. 
 
 "It is not surprising that philo- 
 sophical errors should be as various 
 as they are, but it is somewhat curi- 
 ous that the same error should some- 
 times appear with almost as regular 
 a periodicity as that of the moon 
 herself. This view of Mr. Symons 
 does not now appear for the first 
 time, and, I doubt not, will appear 
 again ; but it would seem reasonable 
 to suppose that each successive re- 
 appearance of such an error would 
 occur in regions more and more re- 
 mote from educated minds. How is 
 it, then, that on this occasion this 
 periodical error assumes so bold a 
 front in one who, by profession, is a 
 teacher even of teachers? I leave 
 
12 
 
 ON THE MOON S ROTATION. 
 
 revolution round the Earth. 21 With still greater precision Santini 
 states, that, "independently of the Moon's progressive motion, in 
 virtue of which she performs her Evolution round the Earth from 
 west to east, she has a motion of rotation about an axis which 
 
 this fact, however, for the specula- 
 tion of others. Your obedient ser- 
 vant, 
 
 " W. HOPKINS. 
 
 " Cambridge, April 15." 
 " The Times" April 19, 1856. 
 
 Mr. Hopkins, certainly, does not 
 express himself very clearly, it being 
 impossible to clearly express con- 
 fused ideas ; but Mr. Symons, as 
 appears from his reply in The Times 
 of the 21st of April, has misunder- 
 stood him in supposing that he means 
 to identify the Moon's rotation on 
 her axis with her revolution round 
 the centre of her orbit. He only 
 holds the former motion to be the 
 necessary consequence of the latter, 
 and. therefore, inseparable from it. 
 True, in reality, this is identifying 
 the Moon's rotatory and progressive 
 motion; but not so in Mr. Hopkins' 
 opinion. 
 
 The remark of Mr. Hopkins would 
 seem to derive some strength from a 
 letter addressed by Mr. Airy to Mr. 
 Symons, on the 19th April, 1856 
 (the English Journal of Education, 
 pp. 263, 264), and in which the 
 Astronomer Boy al defines the Moon's 
 rotation in a manner, implying its 
 identity with that of her progressive 
 motion. But in his subsequent let- 
 ter of the 5th May (ib. pp. 267, 268), 
 though expressing himself somewhat 
 vaguely, he states that, " every point 
 in the moon's body does describe a 
 ring round the earth, that earth 
 being a travelling centre. Every point 
 in the moons body does also describe 
 
 a ring round that axis of the moon 
 which is perpendicular to the moon's 
 orbit, that axis being a travelling 
 axis" And he further states that 
 "a body consisting of firmly con- 
 nected pa'rts, if it rotate at all round 
 any centre whatever, must (by geo- 
 metrical necessity) so rotate that all 
 points move in concentric rings rela- 
 tive to that centre." Consequently 
 every point in the lunar globe, if it 
 rotate at all, must describe concen- 
 tric lines about its axis of rotation, 
 situated within that globe, and to be 
 considered as at rest, with regard to 
 its rotating parts. And admitting 
 this, Mr. Airy admits the error of 
 his own definition of the Moon's 
 rotation, as given in his first letter, 
 (compare 9 and 30.) 
 
 21 " Der Erde wendet er (der Mond) 
 stets nahe dieselbe Seite zu ; er 
 wiirde, wenn sein Umlauf ein con- 
 centrischer Kreis ware und weder 
 seine Bahn noch sein Aequator eine 
 Neigung hatten, er auch keine Sto- 
 rungen von der Sonne erlitte, dies in 
 aller Strenge thun, wie ein Mensch 
 der um einen Pfeiler so herum geht, 
 dass er diesem stets das Gesicht 
 zuwendet. 
 
 " Indess ist die Bewegung des 
 Mondes nicht vollig so beschaffen, 
 und eben dies macht, dass man den 
 Streit iiber Eotiren oder Nichtrotiren 
 der Mondkugel nicht als einen blossen 
 Wortstreit zur Seite liegen lassen 
 kann, sondern ganz bestimmt und 
 nothweudig eine wirkliche Rotation 
 des Mondes um seine Axe annehmen 
 
ON THE MOON'S ROTATION. 
 
 13 
 
 passes through her centre . . . The Moon then, has a rotatory 
 motion about an axis, passing through her centre, and the angular 
 velocity of which is equal to the mean velocity of her revolution 
 round the Earth." 22 And these opinions express the actual doc- 
 trine of Astronomy, as will appear from the sequel. What we, 
 therefore, shall have to ascertain is, whether, leaving the Moon's 
 revolution in her orbit out of consideration, and assuming her 
 to be stationary in the heavens with regard to the Earth, she 
 will yet, in virtue of a distinct motion of her own, rotate on 
 her axis ? Thus, once more to revert to Littrow's illus- 
 tration, the question is not, whether the table on which we 
 
 muss. Denn es giebt keinesweges, 
 wie Einige wohl geglanbt haben 
 einen Punkt ausserhalb der Erde 
 (etwa den zweiten Brennpunkt der 
 Bahn) urn den der Mond in gleichen 
 Zeiten gleiche Winkel bescbriebe, 
 und dem er stets genau dieselbe 
 Seite zuwendete, weder was die 
 Meridiane noch was die Parallelen 
 betrifft. Die oben erwahnten Storun- 
 gen bewirken, dass auch um den 
 zweiten Brennpunkt die Winkelbe- 
 wegung ungleich 1st, wenn gleich in 
 geringerem Maasse als um die Erde. 
 Die Winkelbewegung des Mondes um 
 seine Axe ist aber gleichfb'rmig, also 
 mit keiner Ansicht des Mondlaufes, 
 man wahle den Standpunkt wo man 
 wolle, identiscb, und mithin ein 
 selbststandiges Phanomen." BEER 
 und MIDLER, Selenographie S. 12. 
 
 22 " Abbiamo gia esposto che la 
 Luna osservata con forti cannocclriali 
 presentava delle grandi irregolarita 
 alia sua superficie, le quali annunzi- 
 avano 1 'esistenza in essa di alte 
 montagne e crateri. Se ora diligen- 
 temerite si prendano a seguire le 
 posizioni di queste irregolarita, non 
 tarderemo ad accorgerci che la Luna 
 rivolge sempre verso la terra la 
 
 medesima faccia, giacche i suoi 
 monti e crateri conservano costante- 
 mente quasi sempre la stessa posi- 
 zione rapporto al suo centro appa- 
 rente. Kisulta da cio che essa oltra 
 il suo moto di traslazione, in virtu di 
 cui da occidente in oriente compie la 
 sua rivoluzione intorno alia terra, e 
 animata eziandio da un moto di rota- 
 zione intorno ad un asse che passa 
 per il suo centro, la quale si compie 
 in un tempo uguale a quello della 
 sua rivoluzione periodica. In fatti 
 se dal centro della terra al centro 
 della Luna fingiamo continuamente 
 coudotta una linea retta, qualora non 
 fosse la Luna animata da alcun moto 
 di rotazione, dovrebbe questa linea 
 incontrare la sua superficie in punti 
 sempre differenti, dimodoche in una 
 rivoluzione sinodica essa 1' avrebbe 
 tagliata luogo una circonferenza. Se 
 adunque questa linea incontra la 
 Luna sempre in un medesimo punto, 
 e forza che di quanto questa si avanza 
 nella sua orbita intorno alia terra 
 verso oriente, di altrettauto quello 
 proceda pure verso oriente intorno al 
 centro della Luna, seguendo cosi il 
 moto della linea nello spazio. IS 
 dunque animata la Luna da un 
 
14 ON THE MOON'S ROTATION. 
 
 write, have a general movement proper to the Earth ; as the 
 question is not, whether the Moon have a progressive movement 
 round the Earth : but the question is, whether the table have 
 a particular movement proper to itself; as the question is, 
 whether the Moon, independently of her motion round the Earth, 
 have a particular motion, proper to the several parts of her 
 globe with regard to an axis within herself? This is the true 
 point at issue ; and if the illustration of the Vienna Astronomer 
 did apply to the Moon which it does not it would be proof, 
 the most positive and undeniable, against her rotation. 
 
 7. The extreme confusion of views and ideas which we find 
 prevailing in regard to our subject, may in most instances 
 be traced to a similar confusion of ideas respecting rotatory 
 and progressive motion, as their common source. For this 
 reason it will be necessary for the reader to bring clear and well 
 defined notions concerning the distinctive nature of both kinds 
 of motion, to the further prosecution of our inquiry. An " Axis 
 of Rotation" Barlow states, is " the line about which a body 
 really revolves when it is put into motion ;" and " Rotation" 
 according to him, is " the motion of the different parts of a 
 solid body about an axis, called the axis of rotation, being thus 
 distinguished from the progressive motion of a body about 
 some distant point or centre : thus the diurnal motion of 
 the Earth is one of a motion of rotation, but its annual motion 
 one of revolution" 23 To quote another authority ; "rotation," 
 
 movimento di rotazione intorno ad tion. In the first place, an axis is 
 
 un asse condotto per il suo centre, not of necessity an axis of rotation, 
 
 la cui velocita angolare sara uguale as here implied ; but any line passing 
 
 alia sua media celerita intorno alia through the centre of a circle at 
 
 terra." SANTINI, Elementi di Astro- right angles to its plane, or any dia- 
 
 nomia, Vol. I. pp. 223, 224. meter of a globe is, in Astronomy, 
 
 23 See Barlows "New Mathera. called an axis. Secondly, the term is 
 
 and Phil. Diet. ;" articles " Axis " and not applicable to planets only, but 
 
 " Kotation." .Mr.fmd,inhis'Astron. to the heavenly bodies generally. 
 
 Vocabulary,' defines an ' axis ' to be And thirdly, even an axis of rotation 
 
 the imaginary line upon which a has not necessarily its north and 
 
 planet revolves. It is the line join- south poles. The terms " rotation," 
 
 ing the north and south poles.' A and " motion, progressive," as dis- 
 
 very loose and unsatisfactory defini- tinguished from rotatory motion are 
 
ON THE MOON'S ROTATION. 
 
 15 
 
 an eminent Astronomer informs us " is the relative motion of 
 the several parts of a body about another part, which is sup- 
 posed to be at rest, and is called its axis .... The straight 
 line, which either really is at rest, or else, apart from its pro- 
 gressive motion, is supposed to be so, is termed the axis of 
 rotation." 24 . . . . "Rotatory and progressive motion are en- 
 tirely distinct, and a heavenly body may have the one without 
 the other, or combine them both in numberless ways." 25 . . . . 
 
 not included in the "Vocabulary." 
 Sir John Herschel very properly de- 
 fines the axis of the Earth as " that 
 diameter about which it revolves." 
 (" A Treat, on Astron." p. 54.) 
 
 24 "DieRotationkeinn stattfinden, 
 wenn gleich der ganze Kb'rper, als 
 Punkt betrachtet, oder wenn sein 
 Schwerpunkt in Ruhe ist : sie ist die 
 relative Bewegung der verschiedenen 
 Theile des Korpers gegen einen Theil 
 desselben, der in Huhe angenommen 
 wird, die Axe . . . Das Kennzeichen 
 der blossen Rotation ist also das, dass 
 irgend ein Theil des Korpers in vb'l- 
 liger Ruhe ist, und sie besteht in der 
 relativen Bewegung der iibrigen 
 Theile um diesen ruhenden Theil . . . 
 Die grade Linie, die sich wirklich in 
 Euhe befindet, oder die durch Ver- 
 nichtung der progressiven Bewegung 
 in Ruhe versetzt wird, heisst die 
 Axe der Umdrehung. Wenn also, 
 wie es gewohulich der Fall ist, die 
 Bewegung des Korpers aus fort- 
 schreitender und umdrehenderzwsara- 
 mengesetzt ist, d. h. wenn kein Theil 
 des Korpers in Ruhe ist, aber auch 
 nicht die Bewegungen aller Theile 
 gleich und parallel sind, so muss 
 man, um die progressive Bewegung 
 kennen zu lernen, von der Rotation 
 abstrahiren, und umgekehrt." SCHU- 
 BEET, Popul. Astronomie. Bd. Ill, s. 
 319, 320. 
 
 25 Drehung um eine Axe und/ortf- 
 schreitende Bewegung im Raum sind 
 ganz von einander unabhangig : ein 
 Weltkorper kann diese oder jene 
 allein haben, oder beyde auf un- 
 zahlige Arten mit einander kombi- 
 nirt. Im letztern Falle sehn wir 
 eine aus beyden zusammengesetzte 
 Bewegung, von der wir die progres- 
 sive trennen miissen, um in dem was 
 iiber bleibt, die Rotation kennen zu 
 lernen. Diese Trennung geschieht 
 am leichtsten, wenn wir uns in die 
 Sonne oder einen andern so entfern- 
 ten Punkt versetzen, dass die pro- 
 gressive Bewegung von selbst ver- 
 schwindet, oder die ganze Bahn des 
 Monds um die Erde zum unmerk- 
 lichen Punkt wird. Die Bewegung 
 der Flecken, die sich alsdenn auf der 
 Oberflache des Monds zeigt, wird 
 seine Rotation bestimmen. Nun ist 
 offenbar, dass in diesem Falle der 
 Mond dem Auge in der Sonne eine 
 Seite nach der andern in einem 
 Monate zeigt : in seiner obern 
 Konjunktion (dem Vollmonde) sieht 
 man in der Sonne dieselbe Seite, die 
 der Erde bestandig zugekehrt ist, 
 die entgegengesetzte aber, wenn er 
 zwischen der Erde und der Sonne 
 steht (im Neumonde). Er hat also 
 wirklich eine Rotation, deren Periode 
 mit dem Monate einerlei ist." SCHU- 
 . III.s.81. 
 
16 
 
 The distinctive mark of rotation is, that some part or other of 
 a body be at perfect rest, whilst ; its other parts are in relative 
 motion about that part." 26 . . * The term progressive motion, 
 on the contrary, is applicable only to a body, viewed as one 
 indivisible whole, not to its single parts .... When all parts of 
 a body have the same motion, being carried through space with 
 equal velocities and in parallel directions, it is simply progres- 
 sive." 26 . . . . " When both kinds of motion, as is usually the 
 case, are combined, we must, in order to arrive at the progres- 
 sive motion, entirely disregard the rotatory motion, and vice 
 versa" All this is so plain and intelligible, as to require 
 no further explanation on our part. The distinction, then, 
 between rotatory and progressive motion, and in perfect har- 
 mony with the definitions of both, as given by Mr. Evan 
 Hopkins ( 68 ), consists mainly in this, that progressive motion 
 is the revolution of all the parts of a body, viewed as one indi- 
 visible whole, about a centre situated without that body ; whilst 
 rotatory motion is the rotation of the several parts of a body, 
 not viewed as one indivisible whole, on an axis, forming part 
 of, and situated within, that body itself. 
 
 8. There are, however, two other marks of distinction be- 
 tween rotatory and progressive motion, deserving of special 
 notice. The former would seem to have a strictly uniform 
 velocity, 26 whilst that of the latter is apparently not uniform ; 
 
 26 "B eider progress! ven Be wegung Bewegung, bei der alle Theile des 
 1st bloss von dem ganzen Korper, bewegten Kb'rpers dieselbe Bewe- 
 nicht von seinen einzelnen Theilen gung haben, also mit gleicher Ge- 
 die Rede; man kann daher, wie wir schwindigkeit nach parallelen Rich- 
 es bei der elliptischen Bewegung der tungen fort gehen, ist lloss progres- 
 Planeten gethan haben, den Korper siv : denn alle grade Linien die man 
 selbst wie einenPunkt ansehen, oder durch jede zwei Punkte des Korpers 
 die Bewegung seines Schwerpunkts ziehn kann, riicken zwar mit dem 
 statt der des ganzen Korpers betrach- Korper fort, aber ohne ihre Richtung 
 ten : und dies setzt voraus, dass alle zu avdern, ohne sich zu drehn, es 
 Theile sich genan auf gleiche Art findet also gar keine Rotation statt." 
 bewegen, so dass man die Bewegung SCHUBERT, Popul. Astronomic. Bd. 
 des ganzen Korpers kennt, wenn die III. S. 318, 319. 
 Bewegung eines Punkts z. B. des * " If a body be started in motion, 
 Schwerpunkts bekannt ist. ... Eine and if no force act upon it, that 
 
ON THE MOON'S ROTATION. 
 
 17 
 
 and whilst progressive motion is indifferent as to change of 
 direction, and even when that direction has assumed a definite 
 character, is more yielding to external influences, rotatory 
 motion has a remarkable tendency to maintain, with regard to 
 its axis, an invariable direction in space under any circum- 
 stances. 27 Thus, the daily progressive motion of the Earth 
 
 body will continue in motion in the 
 same direction, and with the same 
 
 velocity Contrivances have 
 
 been made for the purpose of spin- 
 ning tops and wheels in the ex- 
 hausted receivers of air pumps. 
 They go on spinning and spinning, 
 and the motion seems as if it would 
 
 never come to an end If it be 
 
 true that each part of a body would, 
 if unconstrained, move steadily in a 
 straight line, and if (by the connex- 
 ion of the parts) each part is con- 
 strained to move in a circle, then it 
 appears by mathematical investiga- 
 tion, that the body will revolve 
 with a uniform velocity; but if it 
 were not true, then the body 
 would not move in a circle with a 
 uniform velocity If a pendu- 
 lum be properly constructed, mounted 
 with a steel edge (like that of the 
 best balances), moving on a flat plate 
 of hard agate, and if it be set in 
 the exhausted receiver of an air- 
 pump, it will go on for twenty-four 
 or thirty hours, without the action 
 of anything to keep up its mo- 
 tion." AIEY, Six Lectures on Astro- 
 nomy, p. 193 194. 
 
 27 " It is remarkable, as a mecha- 
 nical fact, that nothing is so perma- 
 nent in Nature as the axis of rota- 
 tion of anything which is rapidly 
 whirled. We have examples of this 
 in everyday practice .... In all 
 
 these circumstances there is this 
 wonderful tendency in rotation to 
 preserve the axis of rotation un- 
 altered. 
 
 "Now, from all these circum- 
 stances, we see sufficient reason to 
 explain how, whilst the earth is 
 going round the sun, its axis of ro- 
 tation should remain parallel to it- 
 self without being disturbed, that is 
 to sa*y, that the position of the axis 
 has no respect whatever to the sun. 
 Whatever the position of the axis of 
 rotation be, the earth will travel 
 through space, keeping that axis of 
 rotation in the same position with 
 regard to the distant stars." AIRY, 
 Six Lectures on Astronomy, p. 67, 
 68. 
 
 "In my former lecture I called 
 your attention to the motion of a 
 quoit and a top, in order to show 
 you the strong tendency which rota- 
 tory motion has to maintain the 
 position of its axis unaltered. It is 
 the quoit whose motion has the 
 most striking analogy to the motion 
 of the earth. The quoit is not im- 
 peded by contact with the floor as 
 the top is. This has been made the 
 subject of mathematical investiga- 
 tion, as well as of experiment, and 
 the result of both is, that the earth, 
 if revolving round the sun, would 
 carry its axis of rotation always 
 parallel to one line, as we see it." - 
 C 
 
18 ON THE MOON'S ROTATION. 
 
 in her orbit varies, as to angular velocity, from 57' 12" to 1 
 r 10", but in her rotatory motion Astronomers have not been 
 able to discover, from observation, the very slightest variation, 
 not so much as the fraction of a second of time. 28 And again, 
 whilst the progressive motion of the Earth, whether we suppose 
 its primitive direction to have been rectilinear or circular, has 
 assumed the elliptic form, and still, in this definite form, 
 continues subject to perturbations ; her axis of rotation in- 
 variably maintains the same direction in space, liable 
 only to almost imperceptible changes. A striking illustration 
 of this characteristic of rotation is offered by a common 
 spinning top, made to rotate on a table. In whatever manner 
 you lift the table up, at whatever inclination you hold it (so as 
 not to disturb the balance of the top), in whatever direction 
 you carry it about, the direction of the axis of the spinning 
 top will be affected by no movement of the tabled 
 
 9. A few familiar illustrations will serve still further to eluci- 
 date the nature and distinction of rotatory and progressive mo- 
 tion. If we suppose a carriage to be so supported as to leave 
 one of its wheels just free of the ground, the wheel will rotate 
 on its axis, so soon as motion shall have been imparted to 
 it. The axis being at rest, each point in the wheel, in virtue 
 of its rotatory motion, will describe about the former a circle, 
 the diameter of which is equal to double the distance of such a 
 
 AIRY, Six Lectures on Astronomy, p. mesure au temps, et aux revolutions 
 
 79. des corps celestes." LAPLACE, Ex- 
 
 as "Enfin, le jour lui meme, est pos. du Syst. du Monde, tome ii. p. 
 
 assujetti par le emplacement de 1' 243. 
 
 ecliptique, combine avec 1'action du " " Man lasse einen Kreisel auf 
 
 soleil et de la lune, a de tres petites einem Tische sich drehen, wiihrend 
 
 variations indiquees par la theorie, man den Tisch aufhebt, forttragt, 
 
 mais qui seront toujours insensibles oder was immer fur eine Bewegung 
 
 aux observateurs. Suivant cette mit demselben macht, der Kreisel 
 
 theorie, la rotation de la terre est wird wofern er nicht etwa herab- 
 
 unifonne, et la duree moyenne du fallt fortfahren sich um seine Spitze 
 
 jour peut etre supposee constants, in derselben Bichtung zu drehen." 
 
 resultat tres important pour 1'astro- MIDLER, Populaire Astronomie, 
 
 nomie, puisque cette duree sert de s. 50. 
 
ON THE MOON'S ROTATION. 19 
 
 point from its centre. All circles, thus described by all the 
 points in the wheel, will be concentric to, or coincide with 
 each other. This is, setting aside the motion of the Earth, 
 simple rotation, unaccompanied by any progressive motion; 
 and its characteristic is, that every point in the wheel continu- 
 ally changes its direction with regard to the axis, being at rest. 
 But if we suppose the carriage, to which the wheel belongs and 
 with a shoe or brake attached to the latter, put into motion, the 
 axis of the wheel and the wheel itself will move in the direc- 
 tion in which the carriage is being drawn, and each point in 
 the wheel, whilst maintaining the same direction relative to the 
 axis, will describe straight lines coinciding with, or parallel to, 
 each other and to the plane on which the wheel is dragged 
 along. This is simple progressive motion, unaccompanied by 
 any rotation ; and its characteristic is, that the axis continually 
 changes its direction with regard to the surrounding objects, 
 whilst every point in the wheel maintains the same direction 
 with regard to the axis. 
 
 Let w, in Fig. 1, represent the wheel rotating on its fixed 
 axis A ; s s' a portion 
 of space in which the 
 points ABC shall indi- 
 cate the direction of the 
 corresponding points a 
 b c in the axle being 
 the axis of rotation as 
 well as the direction of 
 the points a b c in the 
 circumference of the 
 wheel, before it is put 
 into motion. Then, 
 being put into motion, m ' " 
 
 the points a b c in the axle will maintain the same direc- 
 tion, and continue to regard the corresponding points ABC 
 in space ; but the points a b c in the circumference of (and 
 
 c 2 
 
20 ON THE MOON'S 'ROTATION. 
 
 any other points in) the wheel will change their direction, both 
 with regard to space and to the. axis of rotation, and the point 
 a, at first corresponding to '. the points A and , will in 
 a' and a" successively correspond to the points B and Z>, c 
 and c. Thus with every other point. And the circles des- 
 cribed by the points a and a 7 , m and m', n and n' 9 o and 0', and 
 any other points equi-distant from the centre of the axle A, will 
 coincide, the one with the other, as m m', n ri\ whereas the 
 circles described by the points a, m, n, o, and any other point 
 not equi-distant from the centre of the axle, will be concentric 
 to each other as m m' and n ri. It is, therefore, a change of 
 direction, not only with regard to space, but both with regard 
 to space and to the axis of rotation, which determines the ro- 
 tatory motion of the wheel. 
 
 In Fig. 2, let the same wheel be represented with its rotatory 
 ^motion arrested by the shoe *, and drawn along the plane and 
 in the direction of p q. The points a b c in the circumference 
 of the wheel will undergo no change of direction with regard 
 to the corresponding points a b c in the axle ; but the whole of 
 these points will, 
 each in a similar 
 manner, change 
 their direction with 
 regard to space or 
 to the surrounding 
 objects, represent- 
 ed by the points 
 ABC, so that the 
 points c and c, at 
 first corresponding 
 to c, will in suc- 
 cession correspond 
 to B and A. Thus FlG ' 2 ' 
 
 with every other point. And the points m n o in the wheel will 
 no longer describe circles, but straight lines, m m", n n", o o", 
 
ON THE MOON'S ROTATION. 21 
 
 those described by all points equi-distant from the plane p q 
 coinciding with each other, as m m' m", n n n" 9 and those, not 
 equi-distant from the plane p q, being parallel to each other, as> 
 m m" and n n". 
 
 Remove now the shoe from the wheel, and, the carriage con- 
 tinuing in motion, we shall have an illustration of progressive 
 and rotatory motion combined, the axis participating exclusively 
 in the former, but every point in the wheel partaking of both. 
 Hence, whilst the axle continues to describe a straight line in 
 the direction of the progressive motion, all the various points 
 in the wheel will describe lines parallel to each other no longer, 
 but a sort of combination of the straight line and the circle 
 (called a cycloid), of symmetrical form, and in the combined 
 proportion of the circumference of the wheel to its diameter, 
 and the distance of each point from the centre of the axle, the 
 lines approaching to straight lines as the distance decreases. 
 In Fig. 3, the wheel w, whilst moving along the plane and in 
 the direction q p, shall in w' have performed one complete re- 
 volution about its axle A. When the latter has described the 
 straight line A A', having arrived in A', the wheel will have per- 
 formed a quarter of a revolution about its axis, and the point m 
 will be in m, n in n, o in o, each point having described the 
 respective curve m m', n n, o d. When the axle has arrived 
 
 FIG. 3. 
 
 in A", the wheel will have completed half a revolution about its 
 axis, and the point m will be in m", n in n", o in o", each point 
 
22 
 
 having respectively described the curves m m' m", n ri n" 9 o o' o". 
 When the axle has arrived in A'", the wheel will have accom- 
 plished three quarters of a resolution about its axis, and the 
 point m will be in m" 9 n in n"', o in d", each point having re- 
 spectively described the curves m m' m" m"', n ri ri' ri", o o d' o'". 
 When the axle has arrived in A'"', at a distance from A exactly 
 equal to the circumference of the wheel, the latter will have 
 performed a complete revolution about its axis, each point 
 having returned to its first position with regard to the same, 
 and the points m, n, o, having respectively described the 
 cycloids m m' m" m"' m"", n n' n" ri" ri'", o o o" o 1 " o"". In the 
 course of that revolution, moreover, the point m and every 
 other point in the wheel has changed direction, not only in a 
 relative sense, both with regard to space and to the axle, as 
 described in the explanation of Fig. 1, but also in an absolute 
 sense, with regard to space, as described in the explanation of 
 Fig. 2. In other words, combined rotatory and progressive 
 motion presents all the phenomena of rotatory motion, indepen- 
 dent of progressive motion ; as well as of progressive motion 
 independent of rotatory motion. 
 
 10. If, instead of considering the motion of a wheel, we had 
 chosen for illustration that of a solid spherical body, f. i. a bil- 
 liard-ball, and assuming that a uniform rolling motion had been 
 imparted to it, we should have arrived at precisely similar re- 
 sults ; only that the axis, which in the wheel appears in the 
 tangible shape of the axle, in the billiard ball assumes the form 
 of an imperceptible line, equal in length to the diameter of the 
 ball, and passing through its centre parallel to the plane of the 
 billiard-table, and at right angles to the path traced by the 
 point of contact between the table and the ball. Indeed, we 
 may view the wheel in the light of a diametrical section of a 
 solid ball, and every other corresponding circular section of the 
 ball in the light of a wheel of decreasing diameter, and rolling 
 along a plane of increasing elevation. 
 
 11. The preceding illustrations offer two peculiar features: 
 
ON THE MOON'S ROTATION. 
 
 23 
 
 rotatory motion is there the result of progressive motion com- 
 bined with friction, consequently of a dependent character, and 
 subject to the laws of mechanical contact ; and progressive 
 motion has been referred to a plane. Continuing to view ro- 
 tatory motion in the same character, we will first change our 
 supposition with regard to the latter point, and assume the 
 carriage- wheel to pass over very uneven ground. We shall, 
 therefore, have to substitute for the rectilinear surface, a curvi- 
 linear one, as p q in Fig. 4. Let s s' represent a portion of 
 
 FIG. 4. 
 
 space. The wheel w shall in each of its positions w', w", w'" 
 have performed a complete revolution about its axis. The 
 straight line A a passes through the centre of the wheel and its 
 point of contact with the plane p q. The lines B a', c a", D a"\ 
 pass through the centre of the wheel, and are parallel to A a. It 
 is evident, the line a e being a diameter of the wheel, that, when 
 the point of contact a, after a complete revolution about the axis 
 of rotation, shall in a', a", and a"' have again become the point of 
 contact, its diametrically opposite point e must in e, e", and e" 
 
24 
 
 have performed corresponding revolutions, being still the point 
 diametrically opposite to a in a, a", and a". It is further 
 evident that, if we suppose the' rotatory motion of the wheel 
 to have been arrested, and the motion of the wheel from w to 
 w'" to have been of a simply progressive nature, so that the 
 point a would have remained the point of contact throughout, 
 its diametrically opposite point e must in e' f e", and e" still be 
 diametrically opposite to it. But this identical point e, which 
 at first regarded the point A in space, afterwards, and in conse- 
 quence of an apparent forward movement at a great rate of 
 velocity, regarded in succession the points B, c, D, until the 
 wheel arrived in w', when it looked to a point m in advance 
 of D. Then, by an apparent retrograde movement at a similar 
 rate of velocity, it looked in succession again to D, c, B, A, 
 until the wheel came to w", when it regarded a point n behind 
 A ; and once more by a forward movement, it regarded in suc- 
 cession the points A, B, c, D, until the wheel arrived in w"'. 
 On the contrary, the centre of the wheel has steadily progressed 
 with regard to space, corresponding to A in w, to B in w 7 , to 
 c in w", and to D in w"'. Consequently in measuring pro- 
 gressive motion, we have only to regard the centre of a body, 
 and to lose sight of the rotatory motion, if combined with it, 
 altogether ( 7, note 28) ; for, in progressive motion, when the 
 body moves upon a curve, no point in the surface of the body 
 will at any given moment point in a direction which shall be 
 parallel to its direction at the preceding or the following mo- 
 ment. But if a point e in the surface of the body, produced 
 through its centre, be at any time perpendicular to the tangent 
 of the curve, upon which the body moves, or, which is the 
 same thing, cuts the curve in the point of contact a : it will do 
 so always. 
 
 Thus, if a spherical body A, Fig. 5, be made to move, with- 
 out rotation, upon the surface of a much larger spherical body 
 B, and in the direction a a: a straight line, drawn from 
 the centre of B through the centre of A and their mutual 
 
ON THE MOON'S ROTATION. 25 
 
 point of contact, will meet the 
 
 surface of A in the point a, 
 
 and, when produced, a corre- 
 sponding point in space s. 
 
 Then, supposing A to have 
 
 successively arrived in A' and 
 
 A": the point a will not be 
 
 in b and &', pointing to s' 
 
 and s" in a direction, with re- 
 gard to space, parallel to a s ; 
 
 but it will be in a and a", still, 
 
 as in , in a straight line, drawn 
 
 from the centre of B through 
 
 that of A and their mutual 
 
 point of contact. Because, if a 
 
 were no longer in that straight 
 
 line, the point of contact could 
 
 be no longer the same ; and 
 
 the point of contact having changed, there must have been 
 rotation. Indeed, if we suppose the point #, when A is at 
 A' and A'', to be in b and b', in the lines s" b' and s b parallel 
 to s a : it must, in virtue of a rotatory movement, have de- 
 scribed the arcs a b and a' b', which is contrary to our propo- 
 sition ; and it must have described them, moreover, in a direc- 
 tion opposed to that of the progressive motion of A, which is 
 contrary to the laws of mechanics, and therefore impossible. 
 
 12. As has been remarked before, we have thus far illustrated 
 rotatory motion only, as being the result of progressive motion, 
 combined with friction. Under this supposition, the lines de- 
 scribed by any point in a wheel, in consequence of its com- 
 bined rotatory and progressive motion, never vary, whatever be 
 the velocity of the progressive motion. But many are the 
 circumstances of common life in which rotatory motion is a 
 free one, i. e. independent of progressive motion, though both 
 
26 ON THE MOON'S ROTATION. 
 
 take place simultaneously. In such cases, the lines described 
 by any point in the rotating body, will be modified by the rate 
 of its progressive velocity, and' thence assume corresponding 
 and varying forms. Let us, for instance, suppose ourselves to 
 be on board a steam- vessel, gliding along on the smooth sur- 
 face of a river at a uniform speed. There shall be a large coil 
 of rope on deck, with a person walking round it, likewise at a 
 uniform pace. He will appear to perform a circular movement ; 
 and relative to the immediately surrounding objects, he does 
 so. But if we imagine his feet, at every step he takes, to 
 touch the water's edge, and to leave a track behind him ; that 
 track will form a curve, more or less differing from a circle, ac- 
 cording to the respective speed, at every moment, of the steam- 
 vessel and the man, walking round the coil of rope. If the 
 latter greatly exceed the former, so that, whilst the man accom- 
 plishes a whole circuit round the coil of rope, the steam-vessel 
 advances less than one-third of that linear distance (the circum- 
 ference of the coil of rope) : the track, which we imagine 
 the man to leave on the sur- 
 face of the water, will be a 
 cycloid approaching the form 
 represented in Fig. 6. On the 
 contrary, if the speed of the 
 steam-vessel greatly exceed 
 
 that of the man walking round the coil of rope; the path 
 of the latter, relative to the surface of the water, will de- 
 scribe a more or less undulating curve, such as represented 
 in Fig. 7, and in 
 which the ascend- 
 ing portion of the 
 cycloid will be 
 somewhat shorter _, ,_ 
 
 JflQ. 7. 
 
 and more steep 
 
 than its descending line, according to the varying proportion of 
 
ON THE MOON'S ROTATION. 
 
 27 
 
 the respective rates of velocity. The form of the cycloid, repre- 
 sented in 9, Fig. 3, is the form of transition from that re- 
 presented in Fig. 6 to that represented in Fig. 7. 
 
 13. With regard to the motion of the heavenly bodies, when 
 combining rotatory with progressive motion, it partakes of the 
 character illustrated in the preceding paragraph. The two 
 kinds of motion, as we have seen in 6, 7, 8, are entirely 
 distinct; but whether rotatory motion is so absolutely inde- 
 pendent of progressive motion as Astronomers usually suppose, 30 
 is a question which we need not enter into ; it being not our 
 object to inquire into the mechanical economy of the solar 
 system, but simply to ascertain a fact relative to the motion of 
 the Moon. What, however, can admit of no doubt is, that the 
 heavenly bodies are possessed of the capability of perfect 
 freedom of motion, and that, unless restrained by their own 
 inertia* 1 or determined by external influences, there is no 
 diameter in such a body upon which it might not rotate, and 
 
 30 "La rotation des planetes est 
 ahsolument independante de leur 
 revolution : une planete peut suivre 
 son orbite par un mouvement de 
 translation d' Occident en orient, 
 sans tourner sur son axe; et elle 
 peut tourner sur un axe quelconque 
 en sens contraire et avec une vitesse 
 quelconque." LALANDE, Astronomie 
 torn, iii., p. 277. 
 
 31 a " Materia vis insita est potentia 
 resistendi, qua corpus unumquodque, 
 quantum in se est, perseverat in statu 
 suo vel quiescendi vel movendi unifor- 
 miter in directum. 
 
 " Haec semper proportionalis est 
 suo corpori, neque differt quicquam 
 ab inertia massae, nisi in modo 
 concipiendi. Per inertiam materiae 
 fit, ut corpus omne de statu suo vel 
 quiescendi vel movendi difficulter 
 deturbetur. Unde etiam vis insita 
 nomine significantissimo vis inertias 
 
 dici possit. Exercet vero corpus 
 hanc vim solummodo in mutatione 
 status sui per vim aliam in se 
 impressam facta; estque exercitum 
 illud sub diverso respectu et Ke 
 sistentia et Impetus : resistentia, 
 quatenus corpus ad conservandum 
 statum suum reluctatur vi impresses ; 
 impetus, quatenus corpus idem, vi 
 resistentis obstaculi difficulter ce- 
 dendo, conatur statum obstaculi 
 illius mutare. Vulgus resistentiam 
 quiescentibus et impetum moventi- 
 bus tribuit : sed motus et quies, uti 
 vulgo concipiuntur, respectu solo 
 distinguuntur ab invicem; neque 
 semper vere quiescunt quae vulgo 
 tanquam quiescentia spectantur." 
 NEWTONI, Philos. Nat. Principia 
 Math. ; Definitio III., vol. I., pp. 3, 4. 
 31 b l ner tia or inactivity is the 
 total absence of all power in the 
 body to change its state. If the 
 
28 ON THE MOON'S ROTATION. 
 
 110 direction in space in which it might not progress. In a 
 general sense, we may view the revolution of a heavenly body 
 about the centre of its primary, for about their mutual point of 
 gravity falling within the body of that primary ( 18, 19), in 
 the light of one spherical body moving upon the surface of 
 another spherical body of much larger dimensions (11, Fig. 5), 
 and whose circumference shall be represented by the orbit of 
 the satellite. Only, we must remember that there is here no 
 mechanical contact at least not in the common acceptation of 
 the term between the two bodies; consequently no mecha- 
 nical friction ; and, therefore, if the satellite rotate on its axis, 
 no constant proportion between the velocity of rotation and 
 progressive motion ; the cycloid, described by any point in the 
 rotating body with regard to space (comp. 21), varying in 
 form according to the varying rate of the velocity of its rotatory 
 and progressive motion. Thus, in reverting to Fig. 5, 11, we 
 may regard the circumference a a' a of the larger body B as 
 the orbit of a satellite A, revolving, without rotation, about its 
 primary, occupying the centre of B. The results, under this 
 supposition, will be precisely the same as though B were a 
 solid body. For the satellite A having once assumed a position 
 of its several parts, as in A, not with regard to space, bearing, 
 as it does, in no way whatever upon the same, but with regard 
 to its own orbit, which determines that position in space, as we 
 have seen in 11 : it will, in the course of its revolution, and 
 in virtue of the law of inertia, maintain the same position 
 of its several parts, not with regard to space, but with 
 regard to the orbit in which it revolves about its primary. 
 Hence it follows that the same diameter of a heavenly body, 
 revolving, without rotation, about a distant centre, will 
 always be perpendicular to a tangent to its orbit, at the 
 point of contact. 
 
 body be at rest, it cannot put itself change must be produced from some 
 
 in motion ; if the body be in motion external cause independent of the 
 
 it can neither change that motion body." LARDNER, Handbook of Nat. 
 
 nor reduce itself to rest. Any such Philosophy, p. 26. 
 
ON THE MOON'S ROTATION. 29 
 
 14. It is possible to conceive that a satellite which revolves 
 about its primary, and at the same time rotates on its own axis, 
 may combine rotatory and progressive motion in such a manner 
 as to render it difficult for a spectator in the primary to dis- 
 tinguish the former from the latter, and to decide whether the 
 satellite really do rotate or not. For, though rotatory motion, 
 as we have seen ( 9, Fig. 1), is attended by a change of 
 direction, not only with regard to space, but also with regard 
 to the axis of rotation ; yet if the periods of rotation and of 
 revolution should happen to be exactly equal, we can imagine 
 circumstances under which the satellite, as seen from the 
 primary, would never undergo any very material changes of 
 aspect. How, then, in such a case, is rotatory motion to be 
 arrived at ? It is evident that the difficulty would be overcome 
 if the spectator in the primary could translate himself to a 
 point in space directly facing one of the poles of the axis of 
 the assumed rotation of the satellite ; when he would see every 
 distinctive spot on its surface describe concentric lines about 
 the axis of rotation ( 9, Fig. 1). This he cannot do. But it 
 is equally evident that to an observer, on the surface of the 
 satellite, at some distance from one of the poles of the axis of 
 rotation, if rotation there be, the pole, in the course of a revo- 
 lution, would appear to describe a kind of ellipse, whilst in 
 reality he himself would describe such an ellipse about the 
 pole ; and that if the spectator in the primary could, in some 
 manner or other, avail 
 himself of the assistance 
 of the observer in the 
 satellite, he might gain 
 the desired information. 
 And this he can do. Let 
 
 us imagine a straight line 
 
 . . . FIG. 
 
 joining the centres of the 
 
 primary p, Fig. 8, and the satellite s, pass through the station 
 a of the observer in the latter. Then, whether one of the poles 
 
30 
 
 of the axis of rotation of s be in r r r", or any other point in 
 the surface of s, the point a will, under the assumed circum- 
 stances, appear to a spectator in- the centre of p to describe, in 
 the course of a revolution of s, a kind of ellipse similar to the 
 ellipse which the pole appears to the observer in a to describe. 
 And hence it follows that, if the satellite have rotation, the angle 
 formed by its axis of rotation and a straight line joining the 
 centres of the satellite and the primary, must, in the course of 
 a revolution of the former, continually change ; and, on the 
 contrary, if that angle be constant, that there cannot possibly 
 be rotation without a continual change in the aspect of the 
 hemisphere, presented by the satellite to its primary. 
 
 15. Goethe remarked of his own countrymen, that they 
 certainly understood the art how to render themselves unin- 
 telligible. We fear that remark applies not to us Germans 
 alone. Confused ideas are ever expressed in confused lan- 
 guage ; and a confused mode of expression is always the sign 
 of confused ideas. If we cannot make ourselves intelligible to 
 others, let us rest assured, that we are neither intelligible to 
 ourselves nor master of our subject ; and that the obscurity in 
 which we imagine that subject to be involved, is but the re- 
 flexion of our own want of understanding. What can be more 
 simple and easy of comprehension than rotatory motion, pro- 
 perly defined ? Yet " rotatory motion," we are told by a writer 
 in the Athenceum (1856, p. 461), a journal which deservedly 
 holds a high rank in the scientific world, u is a curious thing. 
 " When a wheel revolves, we all know that the whole revolves 
 " about the centre, but how many of our readers know that every 
 " point in the wheel revolves about every other point ? We do 
 " not mean that there is a separate agent of revolution in every 
 " point of the wheel. We mean that the line which joins any 
 u two points of the wheel changes direction as the wheel revolves 
 " and change of direction in any line is revolution of each end 
 " about the other, and of every point in the line about all the 
 " rest. So that every motion of revolution about a fixed axis is 
 
ON THE MOON'S ROTATION. 
 
 31 
 
 " an infinite number of motions of revolution about axes which 
 " themselves revolve." In case the reader should not fully com- 
 prehend the writer's theory respecting rotatory motion, one or 
 two familiar illustrations of Dr. Lardner's may assist him. We 
 allude to those of " Wyld's Great Globe in Leicester Square," 
 and " the Peak of Teneriffe," contained in a letter lately pub- 
 lished in The Times, and which we transcribe in the appended 
 note. 32 To render the latter illustration perfectly plain, let E, 
 
 32 " To the Editor of The Times. 
 
 " SIR, Considering that the pro- 
 position advanced by your corre- 
 spondent, the 'Inspector of Schools,' 
 is in direct contradiction to the 
 conclusions of all the more eminent 
 astronomers of the present and the 
 last age, and that it relates not to 
 a point of abstruse mathematical 
 physics but to one depending on the 
 most elementary mechanical prin- 
 ciples, it would be wonderful indeed 
 if it were not completely erroneous. 
 Now, although it certainly is so, it 
 is very evident from the matter of 
 Mr. Symons's letters, as well as from 
 the various answers which they have 
 elicited, that however universally 
 the moon's rotation has been ad- 
 mitted, the reasoning by which it 
 has been established still requires 
 elucidation and development before 
 its conclusiveness can be perceived 
 by ordinary minds, and I think it 
 would be more rational to supply 
 such elucidation than to attempt to 
 pooh-pooh the question. I am the 
 more free to say this, as I have 
 attempted myself to elucidate the 
 point in more than one of my ele- 
 mentary works and public lectures, 
 and I confess that something more 
 
 seems to be required to satisfy com- 
 mon minds. 
 
 " Although a daily paper is not 
 the place for such an exposition 
 and I shall not, therefore, attempt it 
 I will suggest a point of view in 
 which the question may be regarded, 
 and in which its elucidation would 
 be useful. 
 
 "If a terrestrial globe take for 
 example Wyld's Great Globe in 
 Leicester Square be placed with its 
 axis parallel to the axis of the earth, 
 it will be carried round the centre of 
 the parallel in which it is placed in 
 twenty-three hours fifty- six minutes, 
 presenting, like the moon, always 
 the same hemisphere to the centre 
 of the parallel. Now, the same 
 reasoning which proves the moon's 
 rotation must equally demonstrate 
 that Wyld's globe rotates on that 
 diameter which is parallel to the 
 terrestrial axis once in twenty three 
 hours fifty-six minutes, Can it be 
 expected that common understand- 
 ings will readily admit this upon 
 the force of the received demon- 
 strations ? 
 
 "To take another illustration of 
 this principle : a mountain the 
 Peak of Teneriffe for example is 
 
32 ON THE MOON'S ROTATION. 
 
 Fig. 9, represent the Earth ; 
 e e the equator; p j)' the 
 axis of rotation, with its 
 poles p and p' ; T the Peak 
 of Teneriffe (its propor- 
 tionate size being greatly 
 exaggerated) ; i i f its axis 
 of rotation, parallel to the 
 terrestrial axis p p'; / /' its 
 parallel of latitude. From 
 Dr. Lardner's proposition 
 it would follow that, sup- 
 posing we are stationed at a safe distance from the Peak at the 
 moment its summit has reached its greatest elevation, we should 
 see it, independently of the rotation of the Earth, move slowly 
 round ; after six hours strike its head through the plain ; after 
 six more hours reach what is commonly believed to be the 
 usual position of its base, now exposed to the view of the 
 heavens ; after six further hours reappear above ground ; and 
 after a total lapse of twenty-three hours fifty-six minutes, re- 
 sume its first position in order to instantly leave it again on a 
 similar errand. But it is not the Peak of Teneriffe alone that 
 performs these rotatory pranks ; Wyld's Great Globe, in Lei- 
 cester Square, nay, you yourself, good reader, perform them 
 
 moved round the centre of its pa- line as an axis of rotation, that line 
 rallel of latitude, presenting always passing through the mass of the 
 the same side to the centre. This mountain in a direction parallel to 
 mountain is not a globe like the the terrestrial axis, the time of ro- 
 moon, and has on geometrical line tation being twenty-three hours fifty- 
 analogous to the moon's axis ; but six minutes. 
 
 that does not affect the principle of " I can only repeat that the point 
 
 the question. The same reasoning requires more clear exposition than 
 
 which proves the moon to rotate on it has yet received, 
 its axis must establish with equal " Your obedient servant, 
 
 conclusiveness the rotation of the "DiON. LARDNER." 
 
 Peak of Teneriffe upon a certain " April 21." 
 
ON THE MOON'S ROTATION. 
 
 33 
 
 once in every twenty-three hours fifty-six minutes ! 33 Can it 
 be expected that common understandings will readily admit 
 this upon the force of the received demonstrations ? Nor will 
 Reason admit such theories upon any demonstrations. They 
 are false teachings, proclaimed only in the name of Science. 
 Science disowns them. We will not do Dr. Lardner the in- 
 justice to think that he has, for one moment, contemplated the 
 palpable conclusions, to which his own propositions lead. The 
 writer in The Atlienceum states expressly "he does not mean 
 that there is a separate agent of revolution in every point of 
 the wheel." But it is difficult to find out what they do mean. 
 They deny and affirm the same thing at the same time. " The 
 Peak of Teneriffe and the reader," is their mode of arguing, 
 
 33 The Peak of Teneriffe would 
 seem to have a decidedly " astrono- 
 mical" turn, since it, apparently, 
 regulates its movements according 
 to sidereal time. This, however, is 
 a mere presumption on our part, 
 and as Dr. Lardner by no expression 
 of his justifies our inference, we 
 would not hold him responsible for it. 
 But we must hold him responsible 
 for what lie actually does state, 
 namely, that the Peak of Teneriffe 
 rotates on an axis of its own in 
 twenty-three hours, fifty-six minutes 
 (of course with respect to the centre 
 of the Earth) ; and as this precise 
 time is repeated with so much solemn 
 emphasis, whilst everybody knows 
 that the Earth herself takes an entire 
 day to rotate, it is but fair to con- 
 clude that the author meant to dis- 
 tinguish the respective periods of 
 rotation, as he distinguishes the re- 
 spective motions. However this 
 may be, there are his plain words, 
 stating that the Peak of Teneriffe 
 rotates on an axis of its own in 
 
 twenty-three hours, fifty-six minutes, 
 whilst we know the Earth to rotate 
 on an axis of her own in twenty- 
 four hours. Hence, as every indivi- 
 dual point on the Earth's surface is 
 said to do as the Peak of Teneriffe 
 does, it follows that the Earth rotates 
 at one and the same time in two 
 different periods, and that in half-a- 
 year from any given moment of mid- 
 day, and in any geographical posi- 
 tion whatever, we shall have noon 
 and midnight at the same physical 
 instant; in other words, we shall 
 have the Sun simultaneously right 
 in our face before us, and diametri- 
 cally opposite right at our back be- 
 hind us; so that there will be the 
 most dazzling glare of a summer's 
 day and the blackest darkness of a 
 winter's night in the same place at 
 the same time ! The reader will par- 
 don the tone into which the gambols 
 of the Peak of Teneriffe have be- 
 guiled us. Difficile est non satyram 
 scribere. 
 
34 ON THE MOON'S ROTATION. 
 
 " do not actually rotate on their own independent axes once in 
 twenty-three hours fifty-six minutes, and yet they do so ; only 
 that none but we men of Science* are capable of comprehending 
 this, and we have not, as yet, succeeded in giving to our reason- 
 ing sufficient elucidation and development to render its conclu- 
 siveness perceptible by ordinary minds! " (See note 32.) There 
 lies the proof of their own incapacity ; but there also lies the 
 mischief. They delude others into a belief of their superior 
 knowledge, and make them feel ashamed of what they imagine 
 to be a want, on their own part, of a sufficient power of com- 
 prehension. Hence, deceived by some vague impression that 
 they really have comprehended, or else from mere vanity and 
 against their own better judgment, thousands of persons are 
 induced to confess that they do understand what they do not 
 understand ; and thus many absurdities maintain their scientific 
 credit, though Science blushes and Common Sense revolts at 
 them. 
 
 16. To revert to the remarks of The Athenceum. The theory 
 adopted by their author and Dr. Lardner is easily traced to its 
 source a confused idea regarding motion. The line, we are 
 told, which joins any two points of the wheel, changes direc- 
 tion as the wheel revolves. Undoubtedly it does. Let in Fig. 
 1 0. w represent the wheel ; s 
 its axle, and at the same time 
 the Sun ; a b any line in the 
 wheel ; e and m any two 
 points in that line, and at the 
 same time e the Earth ; m the 
 Moon. When the wheel is 
 made to revolve about its axis, 
 the line a b, and consequently 
 also the line e m will change 
 direction in space ; so will 
 
 the Earth and the Moon in 
 
 Fia. 10. 
 
ON THE MOON'S ROTATION. 35 
 
 revolving together round the Sun. But we are further told 
 that change of direction in any line is revolution of each 
 end about the other, and of every point in the line about 
 all the rest. When the line a b shall have moved to a' b', 
 we find that the two ends of that line a! and b' still occupy ex- 
 actly the same position, both with regard to each other and 
 with regard to the axis or any point whatever in the whole 
 wheel, they occupied in a and b ; and that the same is the case 
 with the two points e and m. Consequently, they have simply 
 changed direction in space and with regard to their axis, in 
 virtue of the rotatory movement of the wheel, of which they 
 form part ; but they have not turned round themselves, much 
 less about every other line or point in the wheel. But if we 
 take e to represent the Earth and m the Moon, we shall then 
 find the Moon to be, say in m" 9 and to have changed her posi- 
 tion both with regard to the Earth and the Sun (or any other 
 point in the system represented by the wheel), in consequence 
 of her having revolved round the Earth by a motion altogether 
 distinct from that of the Earth's motion round the Sun. And 
 this shows clearly that to use the words of the writer in The 
 AthencBum in his sense for the ends of the line a b, or the 
 points e and m to revolve about each other a separate agent of 
 revolution is necessary. Tt is, however, simply impossible for 
 the two ends of a line, joining any two points in a wheel, to 
 revolve one about the other, or for one point to revolve about 
 all the rest ; considering that every point in a wheel occupies 
 & fixed position with regard to every other point (excepting the 
 axle), whereas the movement in question is dependant on a 
 change of the position of any given point, not with regard to 
 space, but with regard to all the other points in the wheel. To 
 render it conclusive to any sensible person, that the point b 
 which in virtue of the wheel's rotation on its axis describes the 
 circle b b f b, at the same time, whilst tracing this path, turns 
 round the point 0, describing the circle a a a, concentric with 
 
 D 2 
 
36 ON THE MOON'S ROTATION. 
 
 the former, will require a development of reasoning (see Note 
 32) of which we do not hold Reason capable. If the motion 
 of a wheel about its axis is to te made the subject of investi- 
 gation, we must look upon the wheel in the light of one whole 
 indivisible body, the motion of which will be one of rotation. But 
 if any particular point in the wheel is to be considered, we must 
 regard that point as altogether distinct from the wheel, and its 
 motion as one about a distant centre, consequently as one of 
 revolution. 
 
 17. For any two bodies, whether heavenly bodies or not, and 
 whether connected with each other by some rigid link or alto- 
 gether free, to simultaneously revolve about each other, is an 
 impossibility in itself. Either A revolves about B, at rest with 
 regard to A ; or B resolves about A, at rest with regard to B. 
 Both cannot at the same time, A about B and B about A, revolve 
 about each other. Mere change of direction in space does not 
 constitute this movement. Let us imagine a kind of turn-stile, 
 both ends of the turning-pole forming circular frames, large 
 enough to. freely admit the body of a boy ; suppose two boys 
 to get into these frames, and to run in the same direction, with 
 the view of overtaking, or which is the same thing in this case, 
 the motion being a circular one, with the view of turning round 
 each other. They continually will change direction in space ; 
 yet they will never accomplish their object, because they do 
 not change direction relative to each other and to the centre 
 round which they move. The supporting-pole being shifted 
 from the middle towards one of the ends of the turning-pole, 
 will in no manner alter the case. Not till it is taken away 
 altogether, and one of the boys has assumed a stationary posi- 
 tion, will the other boy succeed in his attempt. 
 
 18. Take another instance. Fig. 11 shall represent two 
 solid balls A and B, connected by a metal rod c ; and the latter 
 be provided with a number of small holes, to admit the pointed 
 top of a stand, D. Now, in this connecting rod there is a 
 
ON THE MOON'S ROTATION, 37 
 
 diagonal line (with a hole supposed to correspond to it), 
 which, whether the balls be of equal weight or not, will ex- 
 
 Fm. 11. 
 
 actly balance them when horizontally placed on the stand. 
 The centre of this diagonal line is termed the centre of 
 gravity of the two balls. When turned round, they will both 
 of them revolve about this centre; but though they con- 
 tinually change direction in space, they will not turn about 
 each other. If the balls be of equal weight, their centre of 
 gravity will lie at equal distances from both i. e. in the middle 
 of the connecting-rod. If A be heavier than B it will lie 
 nearer to A, and vice versa ; and if the weight of A with regard 
 to B exceed a certain proportion, their mutual centre of gravity 
 will fall within the heavier body A itself. But then its very 
 weight also will arrest its progressive motion. Assuming now 
 the motion of the other ball B to continue ; we may view it 
 according to one or two suppositions, under corresponding 
 aspects. If we suppose the force by which B is propelled, and the 
 strength of its connection with A to be sufficient to turn A about 
 itself, we must regard the whole system ABC (which may be 
 compared to a revolving wheel dashed to pieces with the ex- 
 ception of one perfect spoke left standing), as one body ro- 
 
BS ON THE MOON'S ROTATION. 
 
 tating upon its own axis. If, on the contrary, we suppose the 
 socket of the connecting-rod to; be free of A, and to slide 
 around it, in virtue of the leaning tendency of B towards A, 
 combined with the movement of the latter body ; then B will 
 turn round A, and its motion be a progressive one. 
 
 19. Setting aside the connecting-rod, precisely the same 
 relation as that just described exists between any two heavenly 
 bodies. Take a double star. It is just as impossible for the 
 two stars to turn round each other, as though they were con- 
 nected by an iron rod. In whatever point of the intervening 
 space their mutual centre of gravity falls, they both revolve 
 about that centre, and the motion of each is a progressive one. 
 Whenever their mutual centre of gravity falls within one of the 
 two bodies themselves, this body will be stationary with regard 
 to the other; and the latter alone will, in virtue of its progres- 
 sive movement, revolve round the former. Thus, the common 
 centre of gravity of the Moon and the Earth falls within the 
 body of the latter, that of the Earth and the Sun within the 
 body of the Sun. Nobody maintains that whilst the Moon is 
 revolving round the Earth, the Earth also revolves round the 
 Moon ; and whilst the Earth is revolving round the Sun, that 
 the Sun at the same time revolves both round the Moon and 
 the Earth. Yet, upon the principles of motion advocated by 
 Dr. Lardner and others, such ought to be the case. 
 
 20. Upon the subject of real and apparent motion we need 
 touch but slightly. In our days we presume no reasonable 
 person doubts the fact that the Earth, in the space of a year, re- 
 volves round the Sun, which is the real motion, whilst, in the 
 same period, the Sun and the stars seem to revolve round the 
 Earth, which is the apparent motion ; and that on the one hand 
 the Earth, in the course of a day, rotates on her own axis ; on 
 the other, that the Moon, in the space of a month, revolves 
 round the Earth, which are the real motions, whilst both the Sun 
 and the Moon in a period of twenty-four hours, seem to re- 
 volve round the Earth, which is the apparent motion. Daily 
 
ON THE MOON'S ROTATION. 39 
 
 experience, whether gained on board a steam-vessel or in a 
 railway-carriage, has rendered every one familiar with the phe- 
 nomena of apparent, as distinguished from real, motion ; and 
 no one is likely to question the delusion to which our senses, 
 in this respect, are subject. 
 
 21. To divest motion of its relative character, and to regard 
 it in an absolute point of view, in other words, to trace the path 
 of moving objects, more especially of the heavenly bodies, in 
 space, without any fixed comparison, presents far greater diffi- 
 culties. Indeed, soaring high as modern Astronomy does, this 
 is a problem she is still unable to solve. Let us, for one 
 moment imagine, if we can, the whole luminous world of stars, 
 whose splendour bursts upon our view in a cloudless night, 
 to be swept away from the heavens, as far as the utmost 
 boundaries of the Milky Way. Let us further imagine our- 
 selves to remain stationed in the centre of the immeasurable 
 vacuity thus created by fancy the most perfect darkness would 
 then close in upon us, and it is only when we further imagine 
 our sight to become endowed with the power of our largest 
 telescopes, that the darkness would again seem to expand, and 
 that, floating in space, small specks of the most varied forms, 
 and at the most varying distances from each other, would ap- 
 pear; the nearest as conglomerations of numberless points 
 of light each point a Sun diffusing a soft brilliancy around ; 
 the most distant as dim spots of vapour, fainting away from 
 view. Each one of these specks is, what is usually but im- 
 properly called a galaxy, a system of myriads of worlds, ap- 
 parently crowded together, but, in reality, separated by dis- 
 tances as vast as those which separate the individual stars of 
 our own, a similar system, comprising all the heavens visible 
 to the naked eye in either hemisphere, and, as viewed from one 
 of the neighbouring systems, appearing in space like a speck^ 
 such as we have described. Now, we have every reason to as- 
 sume, that these floating isles of countless Suns have a motion, 
 peculiar to them as a whole, and that, consequently, each indi- 
 
40 ON THE MOON'S ROTATION. 
 
 vidual Sun, independently of its own proper motion, participates 
 in that of the entire system; as the Earth, independently of her 
 motion, participates in that of thfe Sun, and the Moon again, in- 
 dependently of her motion, does in that of the Earth. In order, 
 therefore, to trace the Moon's absolute or real path in space, it 
 would be necessary to have a knowledge of that vast stellar 
 system to which our own solar system, a mere point in the 
 former, belongs, as well as of the motion of the latter. But 
 such a knowledge we do not possess. Whether it will ever 
 be acquired by man with regard to our stellar system, is highly 
 improbable. As yet Astronomy has not succeeded in determin- 
 ing, with any degree of precision, the motion of our Sun within 
 the immediate cluster of stars among which he performs his 
 vast course, though the genius of Sir William Herschel has 
 laid the first foundation for a future solution of this important 
 problem, the conclusions he arrived at having been confirmed 
 by the subsequent labours of Argelander of Bonn, and Otto 
 Struve of Pulkowa. Hence all motion, subject to our measure- 
 ment, is, in the present state of astronomical science, of a rela- 
 tive kind. Disregarding, however, the motions of our solar and 
 stellar systems, unknown to us, motion relative to the Sun, is 
 frequently described as, and not seldom mistaken for, absolute 
 motion or real motion with regard to space. These terras, 
 therefore, whenever they are employed, must be accepted in a 
 qualified sense. 
 
 22. Among the larger heavenly bodies, composing the solar 
 system, there is none whose orbit is subject to so many ap- 
 parent perturbations, and consequently presents so many ap- 
 parent irregularities, as does our satellite. Into all of these 
 irregularities, however, we have no occasion to enter. To trace 
 the general path of the Moon, relative to the Sun and the Earth, 
 and to point out certain phenomena she offers in reference to 
 the latter, will satisfy our present purpose. Simple as the 
 former problem is, there appears, even among Astronomers, 
 some confusion of ideas in regard to it. Thus Madler states 
 
ON THE MOON'S ROTATION. 41 
 
 that, the velocity of the Earth 
 in her orbit thirty times ex- 
 ceeding that of the Moon, 
 she describes, within the solar 
 system, a cycloid represented 
 in Fig. 12, a, a, a", marking 
 the points of her inferior, 
 b, b', b", those of her supe- 
 rior conjunction. 84 But this 
 is an error; and Sir John 
 Herschel truly denotes the 
 Moon's real path to be an 
 undulated curve, only that 
 we have to take the word 
 " real " in the qualified sense FlG - 12 - 
 
 of "relative to the Sun." Fig. 13A will give the reader 
 a tolerably correct idea of this curve, except that the 
 undulations are exaggerated. E E' represents a quadrant of 
 the Earth's orbit ; a a, b b', c c, the Moon's path, in which 
 the rising portion of each wave, owing to the Moon's 
 retrograde motion relative to the Earth's direct movement, is 
 somewhat shorter, and consequently somewhat more steep 
 in its ascent, than the descending portion of the same 
 wave. The former corresponds to the Moon's motion, Fig. 13s, 
 
 34 "Ware die (linear gemessene) cloide des Erdmondes erbalt also 
 
 Fortriickung des Mondes grosser die in der Figur dargestellte Form ; 
 
 als die der Erde in Beziehung auf a, a', a' sind die Punkte der untern 
 
 die Sonne, so wiirden wir eine Conjunktion b, b', b" die der obern. 
 
 Cycloide mit Durchschliugungen Dabei bezeichnen die kleinen wech- 
 
 erbalten. Allein die Erde riickt selsweise leeren und vollen Kreise 
 
 durcbschnittlich in jeder Minute 240 den Ort der Erde in den verschie- 
 
 Meilen, der Mond nur etwa 8 denen Punkten ibrer Babn, die 
 
 Meilen fort. Es folgt hieraus, dass den Hauptpunkten der Mondbahn 
 
 er in Beziehung auf die Sonne nie entsprecben." MIDLER, Popul. 
 
 retrograd werdeu kann. Die Cy- Astronomic, S. 152. 
 
42 
 
 ON THE MOON'S ROTATION. 
 
 FIG. 13. 
 
 from b to d 
 through c ; the 
 latter to her mo- 
 tion from d to b 
 through a. 
 
 23. Both the 
 orbits of the 
 Earth and the 
 Moon have just 
 been assumed to 
 be perfect cir- 
 cles. In reality, 
 however, they 
 are of an oval or 
 elliptical form, 
 to represent 
 which in a man- 
 ner intelligible to the eye, we have again considerably to ex- 
 aggerate the eccentricity of both orbits, i e. their deviation 
 from a perfect cir- 
 cle. The ellipse, 
 Fig. 14, is such an 
 exaggerated repre- 
 sentation, c is its 
 centre. The line A 
 D, the longest 
 straight line that 
 can be drawn be- 
 tween any two 
 points in the cir- 
 cumference of the ellipse, and therefore passing through its 
 centre, is termed its greater or major axis ; the line B E, the 
 shortest that can be drawn through the centre between any two 
 points in the circumference, and consequently forming right 
 angles with the major axis, is called the lesser or minor axis. 
 
 FIG. 14. 
 
ON THE MOON'S ROTATION. 43 
 
 The two points F and F' in the line of the major axis, at equal 
 distances from the centre c, those distances increasing or decreas- 
 ing with the increasing or decreasing eccentricity of the ellipse, 
 are termed the foci of the ellipse. In one of these foci the Sun 
 and the Earth are placed. The eccentricity of the lunar orbit is 
 proportionally more than four times greater than that of the 
 terrestrial (commonly but very erroneously called the solar) 
 orbit. In A the Earth is nearest to the Sun or in her perihelion, 
 the Moon nearest to the Earth or in her perigee ; in D the Earth 
 is at her greatest distance from the Sun or in her aphelion, the 
 Moon at her greatest distance from the Earth, or in her apo- 
 gee. The direction of the major axis of the Moon's orbit, 
 however, or which is the same thing, the Moon's perigee and 
 apogee, do not always coincide with the corresponding points 
 of the Earth's orbit; but are subject to a change of position 
 with regard to the latter. Supposing they coincided at any 
 given time ; then, rather more than two years later, the Moon's 
 perigee will have advanced from A to B, its apogee from D to 
 E, as regards direction ; thus performing a complete revolution 
 about the Sun in 8 years 310 days and nearly 14 hours. 
 
 Nor does the plane of the Moon's path round the Earth co- 
 incide with the plane of the Earth's path round the Sun or the 
 ecliptic ; but is inclined to it at a mean angle of 5 8' 48", this 
 angle varying from 5 to 5 18'. That point of the ecliptic in 
 which our satellite descends from the northern into the southern 
 portion of her orbit, or from above to below the ecliptic, is 
 called her descending node ; and that point in which she again 
 ascends from below to above the ecliptic, or from the southern 
 into the northern portion of her course, her ascending node. 
 These nodes also are subject to a change of position, in conse- 
 quence of which they retrograde, with regard to the direction 
 of the Moon's motion in her orbit nearly 20 in every year; 
 thus completing a whole circuit in 18 years 218 days and 
 nearly 22 hours. 
 
 24. Such are, apart from numerous minor points, the chief 
 
44 
 
 ON THE MOON'S ROTATION. 
 
 features in the Moon's motion round the Earth; leaving us still to 
 describe certain phenomena, which she presents to the terrestrial 
 observer in the course of each of her revolutions. It has already 
 been stated that our satellite continually keeps the same hemi- 
 sphere turned towards us. Strictly speaking, however, such is not 
 the case ; for at times we perceive portions of her surface near 
 her borders, which are hidden from our sight at other times. The 
 knowledge of this circumstance, for which we are indebted to Ga- 
 lileo, was one of the first-fruits of the discovery of the telescope. 35 
 Let Fig. 15A represent the Moon's surface as usually seen, or 
 
 FIG. 15. 
 
 more correctly speaking as seen when in a mean state of 
 libration ; and Fig 15s the opposite hemisphere, turned away 
 from the earth. Then a crescent of the latter becomes some- 
 times visible on the western borders of the Moon, a cor- 
 responding crescent of the former disappearing on her eastern 
 border, as shown in Fig. 15c; and vice versa, as shown 
 in Fig. 15D. At other times a crescent of the opposite 
 
 35 Galileo was by blindness, which 
 darkened the last years of his old 
 age, prevented from following up his 
 investigations regarding the Moon's 
 libration ; and it is touching to read 
 the simple and resigned language 
 in which he speaks of that fearful 
 calamity in his letter to Di Udine. 
 " lo voleva," are his words, " con 
 
 piu accurate osservazioni andar ri- 
 trovando altre particolarita, non solo 
 nelle macchie reali, antiche, ed am- 
 plissime vedute coll'occhio libero, 
 ma nelle piccole adombrazioni de- 
 pendenti dalle eminenze, e cavita, 
 delle quali ne e numero grandissimo 
 nella faccia Lunare, e che col solo 
 Telescopic sono visibile, ed osserva- 
 
ON THE MOON'S ROTATION. 
 
 45 
 
 hemisphere presents itself to our view on the Moon's 
 northern border, a corresponding crescent of her hemi- 
 sphere, turned towards us, receding from it on her south- 
 ern border, as shown in Fig. 15E ; and vice versa, as 
 shown in Fig. 15F. These apparent oscillations of the lunar 
 globe have been termed the Moon's librations ; that on her 
 eastern and western borders, the libration in longitude; and 
 that 011 her northern and southern borders the libration in 
 latitude? 6 Their period is a month. But the Moon's libration 
 has also been observed, in very diminished proportions, to take 
 place daily. This daily variation is called the diurnal or pa- 
 rallactic libration. Of a fourth, the physical libration, as dis- 
 tinguished from the three other kinds of libration, to which a 
 mere optical character is ascribed, we shall speak hereafter 
 
 bill, per le quali, seiidone sparse 
 per tutto, si scopriranno altre mu- 
 tazioni in confermazione di questa, 
 che possiamo quasi chiamare Titu- 
 bazione della Luna verso di noi: 
 ma dalla fortnna mi e stato tolto il 
 poter cio eseguire, essendomi da 
 circa sei mesi in qua caduta una 
 flussione negli occhi, che mi toglieva 
 1'uso del Telescopic, la qual flussione, 
 sono adesso piu di due mesi, che 
 ando a terminare in una total cecita, 
 avendomi coperte le luci con den- 
 sissime cateratte." This letter was 
 written on the 20th February, 1637, 
 consequently nearly five years be- 
 fore his death. GALILEO, Opere, vol. 
 V., pp. 32, 33. 
 
 36 "The moon's rotation on her 
 axis is uniform, but since her mo- 
 tion in her orbit (like that of the 
 sun) is not so, we are enabled to 
 look a few degrees round the equa- 
 torial parts of her visible borders, 
 
 on the eastern or western side, ac- 
 cording to circumstances; or, in 
 other words, the line joining the 
 centres of the earth and moon fluc- 
 tuates a little in its position from its 
 mean or average intersection with 
 her surface to the east or westward. 
 And, moreover, since the axis about 
 which she revolves is neither exactly 
 perpendicular to her orbit nor holds 
 an invariable direction in space, her 
 poles come alternately into view for 
 a small space at the edges of her 
 disk. These phenomena are known 
 by the name of librations. In con- 
 sequence of these two distinct kinds 
 of libration, the same identical point 
 of the moon's surface is not always 
 the centre of her disk, and we there- 
 fore get sight of a zone of a few 
 degrees in breadth on all sides of 
 her border beyond an exact hemi- 
 sphere." SIR JOHN HERSCUEL, Out- 
 lines of Astronomy, p, 262. 
 
46 ON THE MOON'S ROTATION. 
 
 when the entire subject of the Moon's librations will have to 
 engage our special attention. 
 
 25. Having thus laid before the reader all the preliminary 
 information requisite for a proper understanding of our ques- 
 tion, we may now proceed to consider that question itself. In 
 doing so, however, our only object, in the first place, will be to 
 disencumber it of those general arguments and illustrations, by 
 which Astronomers have so greatly contributed towards ren- 
 dering it obscure to themselves and unintelligible to others. 
 Indeed, strange and almost incredible as it may appear, it is 
 nevertheless the simple truth, that the real merits of the 
 problem of the Moon's rotation have been as little compre- 
 hended, generally, by the Learned as by the Unlearned ; and 
 that Sir Isaac Newton's proposition relative to it has been 
 entirely misunderstood by the far greater majority of his own 
 followers in Science. The author of the Principia, finding 
 that, under the assumption of the Moon's uniform rotation on 
 her axis in the precise time of her periodic revolution, and of 
 an inclination of her axis of rotation to the plane of the ecliptic, 
 her librations might be accounted for in a more perfect manner 
 than had then been attempted, he, pointing to the analogy of 
 the rotation of the Sun and the planets, asserted the fact, and 
 then stated its necessary consequence to be, that the Moon must 
 always turn nearly the same face towards the Earth ( 26). 
 But this proposition has since been exactly reversed. The 
 Moon always presenting the same face towards the Earth: 
 thence, we now hear it generally argued, it follows of necessity 
 that she must rotate on her axis ; just as though one person 
 had advanced the proposition that man, being a living thing, 
 had consequently motion ; and another person were to argue 
 from it that a living thing, having motion, must consequently 
 be a man. And the worst is, that in its corrupt form, the pro- 
 position is usually ascribed to Newton himself; and has thus 
 usurped all the weight attaching to his immortal name. In so 
 
ON THE MOON'S ROTATION. 47 
 
 characteristic a manner is this sometimes done, f. i. by the late 
 Astronomer Imperial of Vienna, that we cannot refrain from 
 quoting his own words. "It was Newton," he writes, "who 
 first expressed that phenomenon (the Moon's constantly pre- 
 senting the same face towards the Earth) in the following 
 terms : ' The Moon always turns the same face towards us : 
 consequently she rotates on her axis.' Other Astronomers have 
 drawn the very opposite conclusion from that fact ; namely, 
 that the Moon does not rotate on her axis, precisely because 
 she always keeps the same face turned towards us " (see Note 
 18). The inverted commas would convey the impression that 
 the words thus marked by Littrow were the very words of 
 Newton. And that impression, as to their sense at least, is a 
 very general one. Hence, the Moon's rotation is in almost all 
 the later astronomical works whether of a popular or a scien- 
 tific character with which we are acquainted, represented as 
 virtually, if not solely, resting on that one circumstance : the 
 constant direction of the same face of our satellite towards the 
 Earth ; whilst the determining elements of the problem are lost 
 sight of, and the Moon's libration is treated as a feature merely 
 incidental to the assumed/hctf of her rotation. 
 
 26. The seventeenth proposition and fifteenth theorem of 
 the third book of Newton's " Mathematical Principles of Natu- 
 ral Philosophy," reads thus : " That the diurnal motions of 
 " the planets are uniform, and that the libration of the Moon 
 "arises from the diurnal motion. The proposition is proved 
 "from the first law of motion, and Cor. 22, Prop. Ixvi., Book 1. 
 "Jupiter, with respect to the fixed stars, rotates in 9h. 56m.; 
 "Mars in 24h. 39m. ; Venus in about 23h. ; the Earth in 23h. 
 "56m.; the Sun in 25|d. ; and the Moon in 27d. 7h. 43m. 
 " These things appear by the phenomena. The spots in the 
 " Sun's body return to the same position upon the Sun's disk, 
 "with regard to the Earth, in about 27|d. ; and therefore with 
 "regard to the fixed stars the Sun rotates in about 26<|d. But 
 "because the day of the Moon, uniformly revolving about her 
 
48 
 
 ON THE MOON'S ROTATION. 
 
 "axis, is menstrual, she will always turn the same face nearly 
 "towards the upper focus of her orbit ; and, on account of the 
 "position of that focus, will tnence a little deviate from the 
 "Earth. This is the Moon's libration in longitude. For the 
 "libration in latitude arises from the Moon's latitude and the in- 
 " clination of her axis to the plane of the ecliptic. This theory 
 "of the Moon's libration Mr. N. Mercator, in his Astronomy, 
 "published at the beginning of the year 1676, has more fully 
 "explained from my letters." 37 A description of a practical 
 mode of illustrating the theory, from which it derives anything 
 but further support, is virtually all respecting it that we learn 
 from Mercator's work. 38 It rests solely on the grounds, if 
 grounds they be, contained in the preceding proposition. 
 
 s? PRAEPOSITIO xvii. THEOREMA 
 xv. Planetarum motus diurnos uni- 
 formes esse, et librationem lunae ex 
 ipsius motu diurno oriri. 
 
 "Patet per motus legem 1, et 
 Corol. 22, Prop. Ixvi. Lib. 1. Jupiter 
 utique respectu fixarum revolvitur 
 9h. 56m.; Mars 24h. 39m.; Venus 
 23h. circiter; Terra 23h. 56m. ; Sol 
 25d. ; et Luna 27d. 7h. 43m. Haec 
 ita se habere ex phaenomenis mani- 
 festum est. Maculae in corpore 
 Soils ad eundem situm in disco Soils 
 redeunt 27d. circiter, respectu 
 terrae ; idioque respectu fixarum Sol 
 revolvitur 25d. circiter. Quoniam 
 vero Lunae circa axem suum uni- 
 formiter revolventis dies menstruus 
 est, hujus fades eadem ulteriorem 
 umbilicum orbis ejus semper respi- 
 ciet quamproxime, et propterea pro 
 situ umbilici illius deviabit hinc inde 
 a Terra. Haec est libratio lunae in 
 longitudinem ; nam libratio in lati- 
 tudinem orta est ex latitudine lunae 
 et inclinatione axis ejus ad planum 
 eclipticae. Hanc librationis lunaris 
 
 theoriam D. N. Mercator in Astro- 
 nomia sua, initio anni 1676, edita, 
 ex literis meis plenius exposuit." 
 NEWTONI Phil. Nat. Principia Math. 
 Vol. ii. p. 51 54. Compare also 
 note 1, p. 3, and the following note 
 p. 38. 
 
 38 The passage from Mercator's 
 Instit. Astron. p. 286, will also be 
 found in Le Seur and Jacquier's 
 edition of the Principia, and an 
 abstract of it in Vince's Astronomy, 
 which we will here transcribe. " Sir 
 I. Newton," we read there, " proposes 
 the following method of representing 
 the libration of the moon in latitude 
 and longitude. Take a common 
 globe, and elevate the pole to the 
 zenith, so that the equator may co- 
 incide with the horizon, and let the 
 ecliptic represent the moon's orbit. 
 Conceive the centre of this globe to 
 represent the plane of the earth, and 
 the surface of the globe the sphere 
 in which the moon revolves. Take 
 two small spheres, having each a 
 meridian, and suspend each by a 
 
ON THE MOON S ROTATION. 
 
 49 
 
 In this proposition our warmest admiration of Newton's 
 genius fails to discover a trace of its presence. The theory of 
 the Moon's rotation was not his own. Far too vaguely ex- 
 pressed, it betrays its own weakness. Its argument is faulty, 
 and its logic imperfect. Does it follow that what is true of the 
 planets should be true also of their satellites ? Is it a neces- 
 sary consequence that, because rotation is uniform in bodies 
 whose rotatory velocity exceeds their progressive velocity liun- 
 
 string from one of its poles. Let 
 one of these represent a fictitious 
 moon carried uniformly round the 
 earth, having its equator coinciding 
 with the horizon of the globe, and 
 revolving uniformly about its axis in 
 the same time in which it revolves 
 about the earth; then the same 
 meridian of the moon will always 
 pass through the earth, and the 
 moon would not be subject to any 
 libration. Let the other sphere, re- 
 presenting the true moon, be carried 
 in the ecliptic with its proper angu- 
 lar motion about the earth, having 
 its axis and meridian parallel to 
 those of the other moon. Then, as 
 the true moon moves from the pe- 
 rigee to its apogee, preceding the 
 fictitious moon, the meridian will 
 appear towards the left of its disk, 
 and the spots will appear to move 
 towards the east by as many degrees 
 as there are between the longitudes 
 of the true and fictitious moons, or 
 by the equation of the orbit ; when 
 the true moon moves from apogee 
 to perigee, the meridian of the true 
 moon will appear towards the right 
 of the disk, and the spots will ap- 
 pear to move towards the west ; 
 thus representing the libration in 
 
 longitude. When the true moon 
 moves from its ascending node to 
 its greatest north latitude, the north 
 pole of the moon will disappear, and 
 the south pole, with the spots about 
 it, will come into view ; and as the 
 moon leaves this northern limit they 
 will begin to disappear, and when 
 the moon has reached its greatest 
 southern latitude, the northern pole, 
 with the spots about it, will be 
 brought into view, and appear fur 
 thest upon its disc; thus represent- 
 ing the libration in latitude." 
 VINCE'S Complete System of Astro- 
 nomy, vol. I., pp. 206, 207. Comp. 
 the preceding note 37, and note 1, 
 page 3. 
 
 This illustration, though it gives 
 a good practical explanation of the 
 Moon's libration, does injustice to 
 Newton's own view regarding rota- 
 tion; in as much as it would lead 
 us to infer that, like Mr. W. Hopkins 
 (see Note 19), he had regarded the 
 rotatory motion of the Moon as in- 
 separable from, and therefore in 
 reality identical with, her progres- 
 sive motion. But the nature of his 
 proposition and the very object of his 
 illustration alone suffice to disprove 
 such an inference. 
 
50 ON THE MOON'S ROTATION. 
 
 dreds or thousands of times, it should be uniform also in bodies 
 whose rotatory velocity is equal to their progressive velocity ? 
 Are we to take it for a proof, because the rotation of the planets 
 is a uniform one, that therefore the satellites rotate on their 
 axis ? As we have already had occasion to remark, the Moon's 
 rotation, as proposed by Newton, rests solely on the very 
 doubtful analogy of the rotation of the Sun and the planets 
 an analogy so weak in itself that the philosophical argu- 
 ment of Kepler ( 3, and Note 5,) almost suffices to upset it 
 and is in reality little more than a naked assertion. Nor is 
 this all. The proposition embodies a threefold fallacy. Sup- 
 posing the lunar axis to be so inclined to the plane of the 
 ecliptic as nearly to coincide with it, and at right angles to a 
 line joining the centres of the Earth and the Moon: then, 
 although the periods of the Moon's rotation and revolution be 
 perfectly equal, she would yet, in the course of each revolution 
 in her orbit, present her entire surface to our view, and we 
 should in succession see the hemisphere now constantly hidden 
 from us, as well as the hemisphere now constantly turned to- 
 wards the Earth. On the contrary, if we suppose the lunar 
 axis to lie in the direction of the line joining the centres of the 
 Earth and the Moon, and again nearly to coincide with the 
 plane of the ecliptic : then, whether the Moon rotate on her 
 axis in a hundred hours or in a hundred years, or whether she 
 rotate at all, she would yet invariably keep nearly the same 
 face turned towards the Earth. True, in the case of rotation, 
 the spots on her face, the face itself remaining the same, would 
 in a corresponding and visible manner change their position 
 with regard to the axis ; and, no doubt, the proposition of 
 Newton was framed with a view to the fact that the spots 
 change their position, apparently, but little : since, however, it 
 is silent upon this point, we have to consider it in the form in 
 which it is submitted to us. Again, supposing the lunar axis 
 of rotation, at whatever angle it be inclined to the ecliptic, to 
 maintain an invariable position with regard to the Earth ; it 
 
ON THE MOON'S ROTATION. 51 
 
 would, as we have shown in 14, be impossible for the Moon 
 to rotate and at the same time keep the same face turned to- 
 wards the Earth. So far, then, as the Newtonian proposition is 
 based on its own demonstration of the Moon's rotation on her 
 axis, it is utterly untenable. But, though that demonstration 
 be at fault, it does not follow that the proposition should be so ; 
 that the Moon should not rotate ; and that the phenomena of 
 her libration should not arise from her rotation. What we 
 had to show in this place was, that, in its origin the theory 
 of the Moon's rotation rests on a mere assertion and an 
 untenable argument. The consistency it may have gained by 
 subsequent researches, will be a subject for special considera- 
 tion hereafter. 
 
 27. We need hardly observe that the list of those few 
 Astronomers who, though adopting Newton's proposition, 
 have not misapprehended its bearings, includes the name 
 of Sir John Herschel. 3 9 Others, like Admiral Smyth (9- 3 ), Mr. 
 Hind, 40 and Professor Nichol,( 9 ) without inverting it, at the 
 
 39 " The lunar summer and winter precisely the same interval, whence 
 arise from the rotation of the moon it occurs that only one-half of the 
 on its own axis, the period of which moon can ever be seen from the 
 rotation is exactly equal to its si- Earth." -HiND, in Johnston's Atlas 
 dereal revolution about the earth, of Astronomy, p. 5. 
 and is performed in a plane 1 30' 40b "The moon rotates upon her 
 II" inclined to the ecliptic, and axis in exactly the same interval 
 therefore nearly coincident with her that she requires to perform a revo- 
 own orbit. This is the cause why lution round the earth, i. e. in one 
 we always see the same face of the siderial period of 27d. 7h. 43m. In 
 moon, and have no knowledge of consequence of this she always pre- 
 the other side. This remarkable sents the same side towards us ; for 
 coincidence of two periods, which at the number of degrees passed over 
 first sight would seem perfectly dis- by a lunar meridian, in virtue of 
 tinct, is said to be a consequence of the rotation, in a certain time, say 
 the general laws to be explained twenty-four hours, is exactly equal 
 hereafter." SIR JOHN HERSCHEL to the average number traversed by 
 A Treatise on Astronomy, pp. 230 the moon in one day, as seen from 
 231. the earth, and hence, the same me- 
 
 4oa The moon revolves round our ridian is always directed towards 
 globe in a period of 27d. 7h. us." HIND, Illustr. London As- 
 43m., and rotates upon her axis in tronomy, p. 54. 
 
 E 2 
 
52 ON THE MOON'S ROTATION. 
 
 same dine omit every allusion to the position of the lunar 
 axis; and, consequently, our preceding remarks in Refutation 
 of the proposition in question, apply to the mode in which 
 they state it with still greater force. 400 
 
 28. Setting this proposition now aside, and taking a general 
 view of the general arguments and illustrations, which have 
 been urged both for and against the Moon's rotation, we are 
 struck by the startling fact, that two diametrically opposite 
 conclusions have been drawn by Astronomers from the same 
 identical astronomical phenomenon. That the Moon always 
 presents the same face towards the Earth whilst revolving round 
 her, is an indubitable proof, the majority argue, that she rotates 
 on her axis. That the Moon does not rotate on her axis is in- 
 dubitably proved, argue the minority, by her always, whilst 
 revolving round the Earth, presenting the same face towards 
 her. How is this logical paradox to be explained? In the 
 simplest manner possible. Astronomers generally, with very 
 few exceptions, are in ignorance of the real merits of the 
 problem ; or if they are not, we can but say they have perfectly 
 succeeded in hiding their knowledge from themselves and the 
 world. Unconsciously they set out by begging the question 
 either way. Those who assert the Moon's rotation, unknowingly 
 assume the fact in tacitly and, no doubt, in most cases uncon- 
 sciously assuming that there is a diameter in the lunar globe 
 inclined to the ecliptic, which maintains the same direction in 
 space ; and then proceed to argue on the ground of that silent 
 assumption, which of itself implies rotation ( 51). And those 
 who deny the Moon's rotation, pursue precisely the same course 
 in the opposite direction, by tacitly and unknowingly to them- 
 selves assuming that there is in the lunar globe no such 
 
 c Von Humboldt also considers consequently, they always present 
 
 it highly probable that the period of the same face to the latter." A. v. 
 
 rotation of all the secondary planets HUMBOLDT, Cosmos, vol i. p. 83 ; iii. 
 
 is equal to that of their revolution p. 477 
 round their primary, adding that, 
 
ON THE MOON'S ROTATION. 53 
 
 diameter as we have just described. Thus, no one compre- 
 hending himself, no one comprehends the other ; and the 
 inevitable consequence is that confusion of ideas in regard to 
 our subject, which we shall find reflected in the arguments on 
 both sides. The source of all this confusion, undoubtedly, 
 must be sought in the faulty demonstration and vague ex- 
 pression of the Newtonian proposition, which, on the one hand, 
 has misled his followers to identify, as it were, the rotatory and 
 progressive motion of the Moon ; and, on the other hand, has 
 betrayed their antagonists into a belief that the theory of the 
 Moon's rotation rests on no other ground, save the constant 
 direction of the same hemisphere of our satellite towards the 
 Earth. 
 
 29. We have before had occasion to remark that the majority 
 of modern Astronomers have inverted the proposition of Newton 
 regarding this subject. Newton asserted a thing to explain a 
 fact ; they seize the fact to prove the assertion. For what they 
 maintain is, that from the circumstance of the Moon always 
 presenting the same face towards us, it follows, both that she 
 rotates on her axis, and that she does so in precisely the same 
 time which she takes to perform a revolution in her orbit. But 
 even if we supply a most essential element, omitted in their 
 enunciation of this proposition, namely, that whilst the Moon 
 keeps the same face constantly turned towards the Earth, every 
 feature in that face also maintains nearly an invariable posi- 
 tion with regard to the axis of assumed rotation, or to any 
 plane passing through the Moon's centre ; the proposition is, 
 nevertheless, wholly and altogether untenable, as has been 
 sufficiently shown in 26, but will be more fully demonstrated 
 in 43 48. It could only be true if we were to identify 
 the Moon's rotatory and progressive motion, in assuming that 
 the very fact of her revolving round the Earth with the same 
 face always turned towards her, constituted her rotation (see 
 Note 19). But if such were the case, the velocities of both 
 motions must, of necessity, be as inseparably connected with 
 
54 
 
 each other i. e. as identical as the motions themselves ; and 
 the very assumption, consequently, instead of upholding, de- 
 stroys the proposition, which is based on a difference in the 
 respective velocities of both motions (see Notes 38, 20, 36). 
 Moreover, we have already in 6 shown the assumption 
 to rest on an erroneous impression, and not to be the tenet 
 of Astronomy, though it may be the opinion of some few indi- 
 vidual Astronomers. Consistently, therefore, with the empirical 
 fact, that the spots on the Moon's surface, turned towards 
 us, maintain nearly their invariable positions, as seen from the 
 Earth, and whatever be the direction of the Moon's axis of 
 assumed rotation ; so long as that axis does not vary relatively 
 to the centre of the Earth, it is impossible for our satellite 
 always to present the same face towards us and to rotate at the 
 same time (see 14). Yet the fact alluded to has been simply 
 advanced as a proof of the Moon's rotation by Laplace ( 59 ), 
 Nicollet, 41 Comte, 42 Santini ( 2l ), Schubert, 43 Woodhouse, 44 Lard- 
 
 41 " La surface de la Lome offre un surtout noter, a ce sujet, la parfaite 
 grand nombre de taches permanentes identite qui existe ainsi entre la 
 que Ton a observees et decrites avec duree de la rotation de la lune et 
 soin. Ces taches nous montrent que celle de sa revolution siderale autour 
 la Lune dirige toujours vers nous, a de la terre. Sous ce rapport, la 
 peu pres, le meme hemisphere ; elle seconde periode aurait pu des long- 
 tourne done sur elle-meme, dans un temps faire prevoir la premiere, 
 temps gal a celui de sa revolution parceque 'un phenomene continu, 
 autour de la Terre." NICOLLET, souvent sign ale depuis les Grecs, 
 Mem. sur la libr. de la lune, p. 228. annon9aitessentiellementleuregalite 
 
 42 " Ces determinations (de la duree effective : il consiste en ce que la lune 
 desrotations du soleil et des planetes) tourne toujours vers la terre le meme 
 ne sauraient etre accomplies envers hemisphere, comme 1'indique 1' inva- 
 les satellites, vu 1' extreme petitesse riable retour des memes configura- 
 de leurs diametres apparents, excepte tions interieures du disque avec les 
 pour le seul cas qui nous offre un memes phases mensuelles." COMTE, 
 veritable interet, celui de la lune, Traite philos. d' Astron. populaire. 
 dont la rotation est aussi pleinement p. 286. 
 
 appreciable que celle du soleil ; tt " Die Bestimmung der Rotation 
 
 elle s'accomplit en 27 jours, , des Monds^ ist eine sehr einfache 
 
 autour d'un axe presque perpendi- Sache. Da er der Erde, dem Mit- 
 
 culaire a 1'orbite lunaire. II faut lelpunkt seiner Bahn,immer dieselbe 
 
ON THE MOONS ROTATION. 
 
 55 
 
 ner, 45 and numerous other Astronomers and astronomical 
 writers. Schubert, indeed, holds the deduction to be so evi- 
 dent, that he finds it difficult to conceive how any Astronomer 
 could ever have been blind to its truth. 
 
 30. To demonstrate this truth, the fact usually adduced is, 
 that the Moon, in the course of each of her revolutions round 
 the Earth, presents in succession every point of her surface to 
 all the points of the compass. Thus Biot (s 6 ), Francoeur ( 62 ), 
 Delambre, 46a Delaunay, 465 Santini( 21 ), Schubert, 4 ? Gummere, 48 
 
 Seite zukehrt, so dreht er sich genan 
 in derselben Zeit eines siderischen 
 Monats, da er einen Umlauf um die 
 p]rde macht, zugleich einmal um 
 eine Axe, die auf seiner Bahn sek- 
 recht steht. Diese Folge ist so 
 evident, dass es unbegreiflich sein 
 wiirde, wie demohnerachtet Astrono- 
 men daraus geschlossen haben, dass 
 der Mond sich gar nicht um eine 
 Axe drehe, wenn nicht die ganze 
 Sache am Ende auf einem Wortstreit 
 beruhete." SCHUBERT, Populaire 
 Astronomic, Band. III. S. 81. 
 
 44 " One map of the same hemi- 
 sphere has always served to repre- 
 sent the moon's face ; in other words, 
 the same face of the moon is always 
 turned towards us. This is a curi- 
 ous circumstance, and the immediate 
 inference from it is that the moon 
 must revolve round its axis, with an 
 angular velocity equal to that with 
 which it revolves round the earth." 
 WOODHOUSE, Treatise on Astro- 
 nomy, Vol. I. p 634. 
 
 45 " While the moon moves round 
 the earth thus in its monthly course, 
 we find, hy observation of its appear- 
 ance, made even without the aid of 
 telescopes, that the same hemisphere 
 is always turned towards us. We 
 recognize this fact by observing that 
 
 the same marks are always seen in 
 the same positions upon it. Now, 
 in order that a globe which revolves 
 in a circle around a centre should 
 turn continually the same hemi- 
 sphere toward that centre, it is ne- 
 cessary that it should make one re- 
 volution upon its axis in the time it 
 takes so to revolve. For let us sup- 
 pose that the globe, in any one posi- 
 tion, has the centre round which it 
 revolves north of it, the hemisphere 
 turned toward the centre is turned 
 toward the north. After it makes a 
 quarter of a revolution, the centre is 
 to the east of it, and the hemisphere 
 which was previously turned to the 
 north must now be turned to the 
 east. After it has made another 
 quarter of a revolution the centre 
 will be south of it, and it must be 
 now turned to the north. In the 
 same manner, after another quarter 
 of a revolution, it must be turned to 
 the west. As the same hemisphere 
 is successively turned to all the points 
 of the compass in one revolution, it 
 is evident that the globe itself must 
 make a single revolution on its axis 
 in that time." LARDNER'S Handbook 
 of Astronomy, p. 188, 189. (Com- 
 pare the same author's Popular 
 Astronomy t p. 35, 36.) 
 
ON THE MOON'S ROTATION. 
 
 46 a "Si la lime lie townait pas sur 
 elle-meme, le point a regarderait 
 constarnment le merne point du ciel, 
 la ligne A a deviendrait B a' parallele 
 
 a la ligne A a T. Mais ce point a un 
 mouvemeut de rotation yi=a' B a ; 
 il paraitra done en a ; car le mouve- 
 nient de rotation de la lune, comme 
 celui du soleil, se fait d' Occident en 
 orient et suivant 1' ordre des signes." 
 DELAMBHE, Astronomic Thcor. et 
 Prat. torn. III. p. 64. 
 
 4Gb Puisque la rune tourne touj ours 
 la meme face vers la terre, le rayon 
 du globe lunaire qui, a un instant 
 quelconque, est dirige vers le centre 
 de la terre, se deplace en restant 
 constamment dirige vers ce meme 
 point: done ce rayon ne reste pas 
 parallele a lui-meme, ce qui veut 
 dire que la lune tourne autour de son 
 centre, en meme temps qu'elle se 
 incut autour de la terre. Si la lune 
 se transportait de L en L', sans tour- 
 ner sur elle-meme, son rayon L a 
 viondrait prendre la position paral- 
 
 lele L' b. et le point de sa surface que 
 Ton voyait d'abord en a au centre de 
 aon disque se trouverait ensuite en 
 b', ou on le verrait pres d'un des 
 bovds de ce disque. L' observation 
 indiquant que 1' on a vu un instant 
 quelconque au centre du disque de 
 la lune parait toujours dans la meme 
 position centrale, il faut que la lune, 
 en meme temps qu 'elle va de L en 
 L.', tourne sur elle-meme de maniere 
 a donner au rayon L a la direction 
 L' a' : cela ne peut se faire evidem- 
 ment qu 'autant que la lune tourne 
 autour d'un axe perpendiculaire au 
 plan de son orbite, et que Tangle b 
 L' a, dont elle tourne au tours de cet 
 axe, est egal a Tangle L T i/ qu'elle 
 decrit en meme temps autour de 
 la terre." DELAUNAY, Cours. eUm. 
 d 'Astronomic, p. 393. 
 47 " Im Friihlinge, wenn die Sonne im 
 Widder erscheint, ist die hintre Seite 
 des Vollmonds vom Mittelpunkt aus 
 nach der Seite der Waage gerichtet, 
 die der Erde zugekehrte Seite nach 
 dem Widder; beimNeumonde haben 
 beide Seiten die entgegengesetzte 
 Stellung gegen den Mittelpunkt des 
 Monds: sie haben also einen Um- 
 lauf urn diesen Mittelpunkt gemacht, 
 oder der Mond hat sich umgedreht. 
 Der Mond dreht sich namlich in 
 einem siderischen Monat um eine 
 Axe, die der Axe seiner Bahn paral- 
 lel ist, zugleich aber beschreibt jene 
 Axe selbst in eben der Zeit eine 
 Ellipse un die Axe der Bahn." 
 SCHUBERT, Popul. Astronomic, Bd. 
 III. s. 83. 
 
 48 " The moon revolves with a 
 uniform motion, from west to east, 
 about an axis nearly perpendicular 
 to the plane of the ecliptic, in the 
 same time that she makes a revolu- 
 tion in her orbit. 
 
ON THE MOON'S ROTATION. 57 
 
 Lardner ( 45 ), Vince, 49 Airy ( J 9), and many others. 50 In order to 
 Let E be the centre of the earth, comes to w, if it did not revolve 
 
 a a' a part of the moon's orbit, a and 
 a 1 two successive positions of the 
 moon's centre, and a' D a line paral- 
 lel to a E. Then, since nearly the 
 same surface of the moon is always 
 turned towards the earth, that point 
 in the surface, which is not at e when 
 the moon's centre is at a, will be at 
 e\ or nearly so, when the centre is at 
 a'. Assuming the point to be exactly 
 at ef, it must during the interval, 
 have moved about an axis perpendi- 
 cular to the plane of the orbit, 
 through the angle E a' D, which is 
 equal to a E a' the angular motion 
 in the orbit. Hence the angular 
 motions about the axis and in the 
 orbit being equal, the moon must re- 
 volve on her axis in the same time 
 that she makes a revolution in her 
 orbit." GUMMERE, Elem. Treatise 
 on Astronomy,}). 109, 110. 
 
 4a " If the angular velocity of the 
 moon on its axis were about equal to 
 its angular motion about the earth, 
 the libration in longitude would not 
 take place. For if E be the earth, a b 
 c d the moon at v and w, and a v c be 
 perpendicular to E b v d ; then a b c 
 is that hemisphere of the moon at v 
 next to the earth. When the moon 
 
 about its axis, b w d would be paral- 
 lel to b v d, and the same face would 
 not be towards the earth. But if 
 the moon, by. revolving about its 
 axis in the direction abed, had 
 brought b into the line E w, the same 
 face would have been towards the 
 earth, and the moon would have 
 revolved about its axis through the 
 angle w E c, the angle which the 
 moon has described about the earth. 
 When the moon returns to the same 
 point of its orbit, the same face is 
 observed to be towards the earth, 
 and therefore the time of the revo- 
 lution in its orbit is equal to the 
 time about its axis." VINCE, Com- 
 plete System of Astronomy, Vol. I. 
 p. 204, 5. 
 
 50 " To the Editor of the Times. 
 Sir, A Mr. Jelinger Symons in- 
 quires in a letter printed to-day in 
 The Times, ' the grounds upon which 
 almost all school astronomy books 
 assert that the moon rotates on her 
 axis.' It is a question which any 
 intelligent mathematical tutor can 
 explain to his pupil ; and were it not 
 that the gentleman who shows his 
 
58 ON THE MOON'S ROTATION. 
 
 render the demonstration as striking as possible, let in Fig. 16, 
 
 FIG. 
 
 want of comprehension of the sub- 
 ject signs himself ' Her Majesty's 
 Inspector of Schools,' and that, 
 therefore, his understanding subjects 
 taught in school books is a matter of 
 some importance, it would be im- 
 proper to discuss the subject in The 
 Times. 
 
 " A body is said to have no rotary 
 motion when any line drawn in it 
 continually points in the same direc- 
 tion in space. If the moon had no 
 rotation, a line drawn from her 
 centre to any point on her surface 
 would continually point towards the 
 same place in the heavens, i. e. to- 
 wards the same fixed star. A body 
 is said to have a rotary motion about 
 an axis, when any line drawn through 
 that axis and at right angles to it 
 gradually turns round, so as to point 
 
 successively to all points of the 
 heavens lying in a great circle. This 
 is the case with the moon. A line 
 may be conceived to be drawn from 
 her centre to any point in her equa- 
 tor, and to be directed towards some 
 fixed star. In a short time the same 
 line will no longer point to the same 
 fixed star, and in about 13 days 16 
 hours will point in exactly the oppo- 
 site direction. This shows that the 
 line has revolved through half a cir- 
 cumference. In 27 days, 7 hours, 
 and 43 minutes it will have made a 
 complete revolution, and will point 
 to the same fixed star as at first. To 
 a person in the part of the moon 
 from which the earth is visible the 
 earth will always appear in the same 
 direction, but the fixed stars will 
 appear to revolve round the moon 
 
ON THE MOON'S ROTATION. 59 
 
 E represent the Earth ; M, M', M", M///, the Moon in different posi- 
 tions in her orbit ; a b c her hemisphere turned away from the 
 Earth; the stars*,/,/, with regard to which, on account of their 
 great distance, the diameter of the Moon's orbit may be consider- 
 ed as a point some fixed spot in the heavens. Then, it is argued, 
 if the Moon did not rotate on her axis, the point b in her sur- 
 face would constantly be directed towards the same point of the 
 heavens ; 51 and, in the course of each periodic revolution, she 
 would present in succession every portion of her entire surface 
 to the Earth. But, it is further argued, we find the contrary to 
 be the case ; for when the Moon has proceeded from M to M', 
 the point b no longer looks towards the same star, having 
 changed its direction by 90, whilst the Moon, which should now 
 present the hemisphere bed towards the Earth, still keeps the 
 same hemisphere a b c turned away from her ; consequently, 
 the point , having changed its direction from a to a', the Moon 
 must have turned round herself from the position a b c, into 
 
 the position ^ Arrived in M", she must in a similar manner 
 
 >s 
 have turned round herself from her first position a b c, into 
 
 once in twenty-seven days, 7 hours, the case of the non-rotation of the 
 43 minutes, as they do to us here in latter, constantly intersect different 
 a little less than twenty-four hours, points of her surface (see Note 21), 
 The sun will appear to a person in which is, however, hut another form 
 the moon to revolve in ahout two of the argument under consideration, 
 days longer i.e., in about 29 days, and identical with it, as to their corn- 
 just as the sun appears to us to re- mon bearing. 
 
 volve in about four minutes longer The same remark applies to Que- 
 thari the time apparently occupied by telet, who, after stating that the 
 the stars in a complete revolution. Moon really does rotate on her axis^ 
 " I am, your obedient servant, adds, ' Ainsi, considere par rapport a 
 " A CAMBRIDGE WRANGLER." un corps fixe dans 1'espace, et non 
 " April 8. par rapport a la terre, le globe lunaire 
 " The Times" April 9, 1856. fait une revolution sur son axe dans 
 51 " Santini's argument is, that a le meme temps qu'il emploie a par- 
 straight line, joining the centres of courir son orbite." QUETELET, EU- 
 the Earth and the Moon, would, in ments d' Astronomic, p. 112. 
 
60 ON THE MOON'S ROTATION. 
 
 that of ,/> ,,q v ; arrived in M'", into that of ^ and returned to 
 
 M'", she must, in revolving round the Earth, have at the same 
 time performed a complete revolution about herself. Now, it is 
 not to be denied that to the impression of the eye, because 
 that impression has been misguided by the manner of repre- 
 senting the Moon's change from one position into the other, as 
 though it were a sudden one, and performed on the same spot 
 (comp. 37 and Note 52), the demonstration appears con- 
 clusive enough ; but to the understanding its utter fallacy is 
 still more apparent. We have only to bear in mind the first 
 law of motion, the law of inertia (see 13, Note 31), which 
 makes it the characteristic of progressive circular motion, and 
 consequently of the motion of the heavenly bodies in general, 
 that, in the course of each revolution, and in the absence of all 
 rotation, they should present every point in their surface to all 
 the points of the heavens. As we have remarked before, Astro- 
 nomers have, unknowingly to themselves, begged the question, 
 in assuming the invariable direction in space of some point in 
 the lunar globe an assumption which is tantamount to as- 
 suming the Moorfs rotation and then have proceeded to argue 
 upon the basis of the assumed fact. Hence their demonstration 
 amounts to nothing, and proves only what nobody denies, 
 namely, that the Moon revolves round the Earth. To prove that 
 she rotates on her axis as well as revolves in her orbit, it would 
 have been indispensable to show, not only that the Moon as 
 one body, or any point in its surface, performs complete 
 circuits with regard to space ; but, moreover, that every point 
 in the lunar globe performs such circuits both with regard to 
 space and to the axis of rotation (see the preceding 7 
 and 9). 
 
 Surely the difference between rotatory and progressive 
 motion is not an unintelligible one. Or were it, perchance, 
 
ON THE MOON'S ROTATION. 
 
 61 
 
 the same thing for a man to walk round the Isle of Wight, 
 and to turn round upon his heels ? No more than the former 
 would prove the latter performance, two acts altogether distinct 
 the one from the other, no more does the Moon's revolution 
 round the Earth prove her rotation on her axis, two motions 
 equally distinct the one from the other (comp. 6 8). 52 Yet 
 
 52 Let us imagine a railroad to go 
 round the Earth, so that the plane 
 of its " orbit " or course shall pass 
 through the centre of our planet. 
 Each separate mile of such a rail- 
 way we should look upon as forming 
 a perfect plane. It does not. It 
 forms a curve, the portion of a circle, 
 whose diameter is equal to the di- 
 ameter of the Earth; because every 
 point in the railway rests perpen- 
 dicularly on the Earth's radius. But 
 so large is this circle, that any two 
 contiguous points in its circumfer- 
 ence may, without a sensible error, 
 be regarded as a perfect plane ; and 
 even in an arc of the extent of a 
 mile, the human eye would fail to 
 discover the slightest deviation from 
 a perfect plane. Now, the circum- 
 ference of the Moon's orbit is still 
 sixty times greater. We may, there- 
 fore, look upon every mile in it as a 
 perfect plane; and thus following, 
 in our imagination, the Moon's 
 course mile by mile, we shall per- 
 form a complete revolution round 
 the Earth, under the impression of 
 having continually travelled on a 
 perfect plane. The same impres- 
 sion would a railway journey round 
 the Earth leave on our mind. And 
 this impression, not misguided by 
 imperfect illustrations (comp. the 
 text p. 59), but produced by reality, 
 is, as to the character of the motion, 
 
 the true one : showing it to be 
 simply and exclusively progressive. 
 And would any sensible man assert 
 that a railway-train, in performing 
 such a circuit round the Earth, and 
 because neither engine, tender or 
 carriage, fascinated by the staring 
 look of some fixed star,* started off 
 the rail in the direction of that star, 
 but soberly and steadily kept "the 
 same face " turned towards the 
 Earth's centre, had therefore and 
 
 * " To the Editor of ike Times. SIR, 
 American naturalists tell of a certain 
 owl who had so obstinate a habit of staring, 
 that the sportsman need only engage his 
 attention for a few minutes and then walk 
 steadily round him, and the deluded victim 
 will quickly wring his own neck and fall 
 a prey to perseverance in his own view of 
 the subject. < 
 
 " Without drawing a comparison be- 
 tween that sapient bird and Her Majesty's 
 Inspector of Schools, let the fate of the 
 former serve as a warning to the latter, 
 for if, in support of his favorite theory, he 
 attempt to imitate the Moon's motion by 
 carrying between his finger and thumb one 
 ball round another so as to present always 
 the same face to it, the second revolution 
 will infallibly sprain his wrist, and doom 
 him to dictate to an amanuensis his next 
 answer to the ' science ' and ' sarcasms ' of 
 the public. Yours obediently, 
 
 " MOONCALF." 
 
 "Lincoln's Inn, April 10." 
 
 The Times, April 11, 1856. 
 
62 ON THE MOON'S ROTATION. 
 
 it is not always unconsciously that they are confounded, in 
 defiance of common sense, fact, and the first principles of 
 science : these very principles are sometimes misunderstood, 
 and therefore misapplied, not, it is true, for the purpose, but 
 nevertheles to the effect of, identifying rotatory and progressive 
 motion. Thus Delaunay, 53 in all seriousness, lays down the 
 rule that, in order to ascertain whether the Moon turns round 
 herself or not, we are to take any straight line within the lunar 
 globe, and ascertain whether that line successively changes its 
 direction in space. The rule is most vaguely and imperfectly 
 expressed ; but it is the same as that, upon which the demon- 
 stration under discussion is based. Though we have already 
 shown it to rest on an error, we may still be permitted to point 
 out some of the consequences to which it leads, and some of 
 the suppositions it embodies. In the first place, it supposes 
 the impossibility of progressive circular motion without a 
 simultaneous and corresponding amount of rotatory motion ; 
 as though it were possible for a heavenly body or a person to 
 perform two distinct and differing acts'(comp. 29) by one 
 and the same identical act ; as though it were possible for a 
 
 simultaneously turned about itself toutes les autres lignes droites que 
 
 on an axis of its own, parallel to the Ton pourrait considerer a 1'interieur 
 
 terrestrial axis (comp. note 32)? de la lune, on pourra dire que cet 
 
 Certainly, as we value our life, we astre n'est anime d'aucun mouve- 
 
 would never think of travelling by a ment de rotation autour de son 
 
 railway-train conducted or conduct- centre. Si, au contraire, on recon- 
 
 ing itself upon Dr. Lardner's prin- nait que certaines lignes tracees a 
 
 ciples of motion. 1'interieur de la lune prennent suc- 
 
 53 " Pour voir si la lune tourne sur cessivement differentes directions 
 
 elle-meme, il faut prendre une ligne dans 1'espace, on devra en conclure 
 
 droite quelconque, a son interieur, que la lune tourne sur elle meme; 
 
 et voir si cette ligne change de et il ue sera pas difficile de voir 
 
 direction avec le temps. Si cette autour de quel diametre s'effectue 
 
 droite ne change pas de direction ; cette rotation. Or, c'est precisement 
 
 si elle reste toujours parallele a elle- ce dernier cas qui se presente." 
 
 meme, malgre le mouvement de DELAUNAY, Cours elem. d'Astronomie 
 
 transport de la lune autour de la pp. 392, 393. 
 terre ; et si Ten est de meme de 
 
ON THE MOON'S ROTATION. 63 
 
 person or a heavenly body to be in two different places or 
 positions (comp. again 29) at one and the same physical 
 moment. In the second place, the rule laid down by M. 
 Delaunay supposes that there is no free motion in space ; as 
 though all the empirical facts on which Astronomy is founded 
 were mere fancies of the imagination. And to what conse- 
 quences do these suppositions lead ? The rule in question is 
 of a general character; and if it be true with regard to the 
 Moon, it must be equally true with regard to the Earth. 
 Let us then assume her to be, at the moment of noon, at 
 any given point of her orbit, at any given date. Half a year 
 subsequently she will have completed half a revolution round 
 the Sun, and according to Delaunay's own demonstration 
 (Note 46), still present towards him the same point of her 
 surface. Consequently, at that point it will be noon. But at 
 the same physical instant the Earth shall actually have per- 
 formed, from the time of the given epoch, 182^ revolutions 
 about herself. Consequently at the point in question it will be 
 midnight. Now, as it can at no given place be noon and mid- 
 night at the same time, it follows that a rule, which involves a 
 contradiction of this kind, must rest on a fallacy. And the 
 fallacy consists in this, that, in the third place, it supposes the 
 centre of the Moon's orbit to be her axis of rotation ; as though 
 our tiny satellite were almost as huge a body as the Sun, with 
 the Earth embedded at her centre, and with the diameter of her 
 orbit for a diameter of her solid sphere. 54 Under whatever 
 aspect, therefore, we view the whole argument here discussed 
 and so commonly advanced in support of the Moon's rotation, it 
 proves utterly untenable. 
 
 31. An equally common argument is, that if we wish to con- 
 vince ourselves in a striking manner of the movement in ques- 
 tion, we have but to place ourselves in imagination somewhere 
 beyond the Moon's orbit, f. i. in the centre of the Sun. We shall 
 
 54 The body of the Sun is of so be a hollow sphere, the Earth placed 
 vast a size, that, if we suppose it to in its centre, and the distance of the 
 
64 
 
 ON THE MOON'S ROTATION. 
 
 then see the Moon, in revolving round the Earth, present in suc- 
 cession every portion of her surface to the Sun, and we cannot 
 possibly any longer doubt her rotating on her axis. Thus Lalande, 55 
 Biot, 56 Francoeur ( 62 ), Littrow ( 18 ), Schubert ( 24 ), Ferguson, 5 ? 
 
 Moon from the Earth to be nearly 
 twice as great as it is: the Moon 
 would yet have sufficient space with- 
 in the circumference of the Sun's body 
 to revolve round the Earth. The 
 mean linear distance of the Moon 
 from us is about 237,600 miles, the 
 mean diameter of her orbit conse- 
 quently about 475,200 miles; the 
 diameter of the sun's body about 
 882,000 miles. 
 
 55 " La Lune presente toujours a 
 la Terre a-peu-pres la meme face; 
 mais nous sommes au-dedans de son 
 orbite: si nous etions places a une 
 tres grande distance au-dela de 
 1'orbite lunaire, nous verrions suc- 
 cessivement tous les points de sa 
 circonference ; done la Lune tourne 
 sur son axe, c'est-a-dire qu'elle a un 
 mouvement de rotation egal a son 
 mouvement de revolution". LA- 
 LANDE, Astronomie, torn. III., p. 308. 
 
 36 " L'observation suivie des taches 
 de la lune prouve que cette astre 
 nous presente toujours a-peu-pres la 
 meme face. Ainsi pendant qu'il fait 
 une revolution autour de la terre, 
 son hemisphere oppose rpond suc- 
 cessivement a tous les points du ciel 
 que se trouvent dans le plan de son 
 orbite. II a done reellement un 
 mouvement de rotation sur lui-meme. 
 
 "On se convaincra de 1'existence 
 de ce mouvement, d'une maniere 
 encore plus frappante, en rapportant 
 les positions successives de la lune a 
 un point tres eloigne comme le soleil ; 
 
 car la face qu'elle nous presente 
 est tantot tourne'e vers cet astre, et 
 tantot lui est opposee. Un obser- 
 vateur place dans le soleil, verrait 
 done la lune tourner sur elle-meme 
 dans le cours d'une revolution syiio- 
 dique. Cette rotation vue des etoiles, 
 serait de meme duree que la revo- 
 lution siderale de la lime : par 
 rapport aux points equinoxiaux, ce 
 serait la revolution tropique". BIOT, 
 Traite elem. d' Astron. physique, 
 tome II., p. 404. 
 
 57 " That the Moon turns round 
 her axis in the time that she goes 
 round her orbit is quite demon- 
 strable; for a spectator at rest, with- 
 out the periphery of the moon's 
 orbit, would see all her sides turned 
 regularly towards him in that time. 
 She turns round her axis from any 
 star to the same star again in 27d' 
 8h.; from the sun to the sun again 
 in 29 days : the former is the length 
 of her sydereal day, and the latter 
 the length of her solar day. A body 
 moving round the sun would have a 
 solar day in every revolution, with- 
 out turning on its axis, the same as 
 if it had kept all the while at rest, 
 and the sun moved round it; but 
 without turning round its axis it 
 could never have one sydereal day, 
 because it would always keep the 
 same side towards any given star." 
 FERGUSSON, Astronomy, vol. I., pp. 
 228, 229. 
 
ON THE MOON'S ROTATION. 
 
 65 
 
 Goodeve, 58 and others. It is instructive to see how fertile 
 in resources and inexhaustible in expedients to deceive 
 itself the human mind is, when once a preconceived notion 
 has taken root in it. Our refutation of the former argu- 
 ment ( 30) applies with equal force to the present one, 
 
 58 " To the Editor of the Times. 
 SIR, It is a novel thing to find 
 an inspector of schools arguing in 
 favour of an exploded fallacy, and 
 allowing the public, in their turn, to 
 inspect the results of his own imper- 
 fect education. 
 
 "There can he no question hut 
 that Mr. Hind and the astronomers 
 are in the right. 
 
 "If Mr. Symons will regard his 
 ingenious apparatus from a little dis- 
 tance, he will observe every portion 
 of the surface of the ball during each 
 revolution of the bar, and will see 
 that the ball by virtue of its con- 
 nexion with the revolving bar, does 
 really and truly perform a rotation 
 upon its own axis. 
 
 " Upon this Mr. Symons will re- 
 tire into a distant part of the country 
 and betake himself vigorously to the 
 study of applied mechanics. 
 
 " I am, Sir, your obedient servant, 
 
 "T. M. GOODEVE, M.A. 
 " Professor of Natural Philosophy in 
 " King's College, London." 
 
 " King's College, April 8. 
 The Times, April 8. 
 
 Professor Goodeve, who, as a man 
 of consistency, would, no doubt, on 
 the occasion presenting itself, act up 
 to the advice which he so conside- 
 rately tenders to others, may per- 
 haps be glad to have his attention 
 directed to the following passage in 
 Madler's " Popular Astronomy :" 
 " The nights of the Moon," we read, 
 
 page 168, "are of a twofold kind. 
 Those of the opposite hemisphere 
 are perfectly dark: none of the 
 larger heavenly bodies appear above 
 the horizon : the stars and the 
 planets (excepting the Earth) shine 
 forth in undiminished brilliancy. 
 Nowhere within the whole planetary 
 system is there a place to be found 
 better fitted for making the nicest 
 astronomical observations and for 
 solving the most difficult problems 
 concerning the constitution of the 
 universe, than the opposite hemisphere 
 of the Moon" We confess we do not 
 ourselves see the connection between 
 the darkness of lunar nights and a 
 higher development of the mental 
 faculties; nor would we altogether 
 vouch for the strict accuracy of the 
 information conveyed in the quota- 
 tion; but if anybody may be sup- 
 posed to know anything about what 
 nobody has ever seen, the Moon's 
 opposite hemisphere, it is the 
 Author of the " Selenography," or a 
 Special Description of the Moon's 
 Surface ; from whom Professor Good- 
 eve may even learn what sort of 
 people one is likely to meet with 
 there, beings with "more than 
 eagle-eyes," and " of uncommon elas- 
 ticity of motion;" what is the ar- 
 rangement of their calendar; of 
 what construction are their astro- 
 nomical clocks; and as to their 
 Observatory the most perfect in 
 the solar system Madler feels pretty 
 
 F 
 
ON THE MOON S ROTATION. 
 
 for the simple reason that a fact remains the same, whether 
 viewed from a greater or a lesser distance. But let in Fig. 
 17 s represent the Sun, E the Earth, M the Moon; a E c 9 
 
 Fio. 17. 
 
 a E' c, &c., the same hemisphere of the Earth ; a M c, 
 a M' <?, &c., the same hemisphere of the Moon. Our eye 
 occupies the centre of the Sun. In viewing the phenomena as 
 they present themselves thus to our notice, what is the 
 conclusion a reasoning person would arrive at? If he were 
 to see the Moon with regard to the Earthy in their common 
 revolution round the Sun, remain stationary stationary relative 
 to the Earth, and not to some gratuitously imagined wholly 
 
 confident that it is not likely to attached to it (Popul. Astronomic, 
 have the worst set of Astronomers S. 202). 
 
ON THE MOON'S ROTATION. 67 
 
 indifferent spot in the wide heavens so as always to be in the 
 position M with regard to her primary, and yet present to the 
 Sun in succession every portion of her surface : in that case 
 he would conclude that she rotated on her axis, and did not 
 revolve round the Earth. But actually, seeing that, in the 
 course of their common revolution round the Sun, the Moon 
 did revolve round the Earth, and but once in the exact period 
 of their revolution presented every part of her surface to the 
 Sun, whilst keeping invariably the same hemisphere towards 
 the Earth, he would as certainly conclude that she simply re- 
 volved round her primary, and did not rotate on her axis. Or, 
 if he were to draw a different deduction, he could then, in 
 viewing from a point z beyond the orbit of the Earth, on seeing 
 our planet offer the contrary phenomena to those presented by 
 the Moon, have also, by parity of reason, from the former to 
 draw the contrary inferences. Moreover, an observer in the 
 Sun would see the Moon, according to Biot's own words ( 56 ), 
 rotate on her axis in the course of a synodical revolution, equal 
 to 29 days 12h. 44m.; which is in contradiction to the as- 
 tronomical doctrine, stating her period of rotation to be 27 days 
 7h. 43m. (comp. 59). 
 
 As regards the principal argument, a familiar illustration 
 may serve still further to show its futile nature. Suppose we 
 are stationed on some eminence on the banks of a river, as 
 the steam-vessel mentioned in 12 is passing by : who, in 
 seeing the man walk round the coil of rope, would assert 
 that he, in so doing, were turning round himself? Or, in 
 seeing the railway-train, alluded to in note 52, fly past, who 
 would maintain that it turned about itself in thus hurrying 
 along the iron plane f Rotatory motion, indeed, must be " a 
 curious thing" ( 18), since it has put such curious ideas into 
 men's heads. And does not Schubert ( 24 ) actually wish us to 
 view the Moon from such a distance that her very orbit shall 
 appear reduced to an imperceptible point ? Nevertheless, by 
 observing from that distance the motion of the spots on the 
 
 F 2 
 
68 ON THE MOON'S ROTATION. 
 
 Moon's disk the disk itself being altogether invisible we are 
 to convince ourselves of the Moon's rotation on her axis. 
 
 32. Most infelicitous also in his illustration is the immortal 
 Author of the Mecanique celeste. 5 He places us in the centre 
 of the lunar globe, which we are to imagine transparent. 
 Then, he states, we shall see the Earth and the visual ray turn 
 round us ; and as that ray intersects the lunar surface nearly 
 in the same point, it is evident, he concludes, that this point 
 must turn round us in the same time and in the same direction 
 as the Earth does. Laplace has, at the moment, simply lost 
 sight of the circumstance that the Earth never does rise and set 
 for the lunar spectator in general ; in other words, that, inde- 
 pendently of small oscillations, she remains always stationary 
 relative to the Moon. Whilst the Sun and the stars rise and 
 set once every month, our planet presents no other material 
 changes to the view of the lunar observer, save those of the 
 spots on her surface, indicative of the Earth's rotation on her 
 axis, and those corresponding to the Moon's libration. 60 The 
 argument adduced by Laplace is the strongest he could well 
 have urged against the Moon's rotation. 
 
 59 "Le disque lunaire presente un 1'observateur." LAPLACE, Exposition 
 
 grand nombre de taches invariables du Systeme du Monde, tome I., p. 54. 
 
 que Ton a observees et decrites avec 60 " Die t Erde hat fiir jede gegebene 
 
 soin. Elles nous montrent que cet Mondgegend einen bestimmten mitt- 
 
 astre dirige toujours vers nous, a peu lern Ort am Himmel, und bewegt 
 
 pres le meme hemisphere ; il tourne sich nur innerhalb eines beschrank- 
 
 donc sur lui-meme, dans un temps ten Baumes, in dessen Mitte dieser 
 
 egal a celui de sa revolution autour Punkt liegt. Dieser Raum ist ein 
 
 de la terre ; car si Ton imagine un spharisches Eechteck von 15 46' 
 
 observateur place au centre de la Lange und 13 14' Breite, welche 
 
 lune suppose'e transparente, il verra ausserte Grenzen sie indess selten 
 
 la terre et son rayon visuel se erreicht. Unter-und aufgehen kann 
 
 mouvoir autour de lui, et comme ce sie nur fiir diejenigen|Mondgegenden, 
 
 rayon traverse toujours au meme deren Horizont dies Rechteck durch- 
 
 point a peu pres la surface lunaire, schneidet, was beilaufig fiir ^ der 
 
 il est evident que ce point doit Mondoberflache der Fall ist ; ' sehen 
 
 tourner en meme temps et dans le sie bestandig, ' n i e ." MIDLER, 
 
 meme sens que la terre, autour de Popul. Astronomic, S. 168, 169. 
 
ON THE MOON'S ROTATION. 
 
 69 
 
 33. Two more remarkable illustrations are offered by 
 Schubert, remarkable because that eminent Astronomer had 
 so clear a perception of the distinct nature of rotatory and 
 progressive motion, and yet, upon the very grounds of this 
 distinction, urges the most apparent fallacies in support of the 
 Moon's rotation. In the first place, namely, he makes use of 
 the illustration of a billiard-ball, rolling from the right cushion 
 of the table towards the opposite cushion, as combining the two 
 kinds of motion ; and then adds that " the Moon presents the 
 very same phenomena." 61 To show how erroneous this remark 
 is, we need only refer back to the explanation given in 11 
 13. If Schubert's assertion were correct, and the Moon's orbit 
 being regarded in the light of a billiard-table, she ought, as 
 her diameter is about 2,155 miles and her progressive motion 
 from 35 to 36 miles in a minute, 62 to rotate on her axis once in 
 about every eight hours. 
 
 61 "DieBewegung eines Korpers 
 1st bloss fortschreitend,wenn alle seine 
 Theile in paralleler Richtung mit 
 gleicher Geschwindigkeit fortgehn, 
 so dass sie gegeneinander immer 
 dieselbe Stellung behalten ; so bald 
 aber die relative Lage der einzelnen 
 Theile geandert wird, so bald einige 
 Theile sich langsamer oder schneller 
 als andere und nach verschiedenen 
 Riehtungen bewegen, so findet eine 
 Rotation statt: wenn z. B. eine 
 Billards-Kugel vom westlichen nach 
 dem ostlichen Rande der Tafel ge- 
 stossen wird, so wiirdeihre Bewegung 
 bloss progressiv sein, wenn alle 
 Punkte der Kugel immer auf gleiche 
 Art nach Osten liefen, so dass der 
 vorderste Punkt immer vorn, der 
 oberste immer oben bliebe, &c. Da 
 aber der obere Punkt sich offenbar 
 schneller, und der untere langsamer 
 als der Mittelpunkt bewegt, da jener 
 
 nach Osten riickt, dieser einen Au- 
 genblick still steht, da jeder Punkt 
 bald b'stlish bald westlich, bald 
 oberhalb bald unterhalb vom Mittel- 
 punkt lieget ; so findet eine Rotation 
 statt, die Rugel rollt. Der Mond 
 zeigt uns dieselben Phanomene." 
 SCHUBERT, Popul. Astronomie, Bd. 
 III., S. 82. 
 
 62 Dr. Lardner states the Moon's 
 progressive motion in her orbit to 
 be only 3| miles, in consequence of 
 a striking arithmetical miscalcula- 
 tion, rather unaccountable in a 
 professional Astronomer, who may 
 be fairly supposed to have some 
 general notion of the linear value of 
 the Moon's diameter and her hourly 
 progressive motion, independently of 
 the rule of three. His own words 
 are ; " Since the time taken by the 
 moon to make a complete revolution, 
 or 360 of the heavens, is 27d. 7h. 
 
70 ON THE MOON'S ROTATION. 
 
 34. In the second instance Schubert advances the fallacious 
 proposition that the Mpon's revolution round the Earth is at- 
 tended by the same phenomena, as though the Earth revolved 
 round the Moon in the same direction. Under the latter sup- 
 position, he states, we should evidently, in the course of each 
 month, obtain a view of the Moon's entire surface, provided 
 she did not rotate on her axis ; but as she continually presents 
 to us the same face, it follows that whilst revolving round the 
 Earth, she must at the same time and in the same period, turn 
 on her own axis. Like a man, he then proceeds to illustrate 
 his proposition, who carries a bundle on his shoulders which 
 he wishes to conceal from us, as the Moon would seem to do 
 her opposite hemisphere ; and who, if we walk round him in 
 a circle, must, in order to accomplish his end, necessarily turn 
 round himself in the same time. Those, he adds, who have 
 failed to convince themselves and others of this, cannot have 
 sufficiently considered wherein the distinction between rotatory 
 and progressive motion really consists. 63 Most evidently, our 
 
 44m. or 655h. 73, it follows that mitted by Dr. Lardner is, that the 
 
 her mean apparent motion per day Moon's apparent motion per hour, 
 
 is 13 10' 35", and per hour is 32' at the rate of 13 10' 35" per day, is 
 
 42", which is a little more than her equal to 32' 56" instead of 32' 42". 
 
 mean apparent diameter. The rate Secondly, if the linear value of 1" is 
 
 of the moon's apparent motion on I 5 3 mile, that of 1' will he 69 miles 
 
 the firmament may therefore be re- instead of 6-g, miles; consequently the 
 
 membered by the fact, that she Moon's real motion in her orbit per 
 
 moves over the length of her own hour, and the approximate length of 
 
 apparent diameter in an hour. Since her diameter 69x32'9=2270 miles, 
 
 the linear value of 1" at the moon's instead of 227 miles ; and, therefore, 
 
 distance is ri6 mile, the linear her orbitual motion per minute 37 
 
 value of 1' is 6*9 miles, and conse- miles, instead of 3 miles. More 
 
 quently the real motion of the moon correctly, it is between 35 and 36 
 
 per hour in her orbit is miles per minute. The real diameter 
 
 6-9x32-9=227 miles. of the Moon, Dr. Lardner himself, 
 
 Her orbital motion is therefore at and on the very same page, states to 
 
 the rate of 3'8 miles per minute." be 2,153 miles. 
 Handbook of Astronomy, pp. 186, M " Wir wissen, dass der Mond in 
 
 187. 27 Tagen um die Erde lauft, und 
 
 The first arithmetical error com- dass dies dieselben Erscheinungen 
 
ON THE MOON S ROTATION. 
 
 71 
 
 Author, in advancing this argument, cannot have sufficiently 
 considered wherein consists the real difference between our 
 walking round another person, stationary with regard to us, and 
 that person walking round us, stationary with regard to him ; 
 between the Ptolemean and the Copernican system of the 
 world. If the Sun revolves round the Earth, his argument is 
 evidently correct; if the Earth revolves round the Sun, it is 
 evidently fallacious. 
 
 35. Another favourite subject of illustration, adduced by 
 Francoeur 64 , Madler ( 20 ), and others 65 , is a man walking round 
 
 geben muss, als wenn die Erde nach 
 eben der Eichtung um den ruhenden 
 Mond liefe. In diesem letztern Fall 
 aber wiirden wir doch offenbar in 
 einem Monat den Mond von alien 
 Seiten selin, wenn er sich nicht 
 drehte. Da wir nun bestandig dieselbe 
 Seite sehn, so folgt dass der Mond 
 sich in eben der Zeit, da die Erde 
 um ihn, oder er um die Erde lauft, 
 um eine Axe dreht, so wie jemand 
 der mir etwas auf seinem Biicken 
 verbergen will (wie der Mond seine 
 hintere Seite), wenn ich im Kreise 
 um ihn herum gebe, sich nothwen. 
 dig in eben der Zeit umdrehen muss. 
 Diejenigen die sich oder andere 
 hievon nicht iiberzeugen konnten, 
 hatten nicht hinlanglich iiberlegt, 
 worin eigentlich progressive und 
 rotatorische Bewegung sich unter- 
 scheiden." SCHUBERT, Popul. Astro- 
 nomic, Bd. III., S. 81, 82. 
 
 64 "En observant les taches tres 
 remarquables du disque de la lune, 
 on recommit que I'hemisphere qui 
 nous regarde est toujours le meme. 
 Si Ton se transporte par la pensee 
 dans le soleil, on verra la terre, 
 autour de laquelle tourne la lune: 
 lorsque celle-ci est au dela de notre 
 
 globe, le soleil a 1'aspect de la meme 
 face que nous voyons ; et quand elle 
 est en de9a, entre le soleil et nous, il 
 voit 1'autre hemisphere, puisque la 
 face vue de la terre est demeuree la 
 meme. Ainsi, du soleil, on voit tour 
 a tour les divers points de la surface 
 lunaire. En accomplissant la revo- 
 lution entiere de son orbite, la lune 
 a en meme temps execute precise- 
 ment un tour entier sur son axe, 
 puisqu'elle a presente au soleil tous 
 les points de sa sin-face. C'est ainsi 
 qu'un hornme qui, le visage con- 
 stamment dirige vers un arbre, en 
 ferait le tour entier, aurait aussi fait 
 un tour sur lui meme, puis qu'il 
 aurait vu toute la campagne qui 
 1'environne." FRANCOS UR, Urano- 
 graphie, p. 91. 
 
 65 To the Editor of the Times. 
 SIR, Mr. J. Symons labours under 
 a difficulty arising from utterly con- 
 fused notions concerning rotary 
 motion. I should be wasting your 
 valuable space were I thoroughly to 
 investigate a complete remedy for 
 an infirmity so ordinary and unin- 
 teresting. I will, therefore, simply 
 propose the following practical re- 
 medy : Let your correspondent 
 
72 ON THE MOON'S ROTATION. 
 
 a tree, a pillar, or a circular table, keeping his face constantly 
 turned towards the centre. Why that good man should be 
 made to walk, in a mcst unnatural manner, sideways, it is 
 difficult to conjecture, unless* it is done under an unconscious 
 impression of early childhood connected with the full face of 
 " the man in the Moon," and in order to render the illustration 
 all the more applicable. Psychological phenomena of this 
 kind, though generally suffered to pass unobserved, are as 
 frequent as they are interesting ; and those who can realize 
 them, will be able the more readily to account for the many and 
 strange delusions, to which the Moon's presumed rotation on 
 her axis has given rise. One not the least striking of these 
 delusions is involved in the argument in question ; for he who 
 uses it must have been able to persuade himself that, in walking 
 round the table, he was turning upon his own heels : two acts 
 as distinct as any two acts well can be, and impossible to be 
 performed in accordance with the terms of the proposition. 
 The fallacy, which marks this illustration, proceeds again from 
 confused ideas regarding rotatory and progressive motion, so 
 generally entertained. 66 To prove the fallacy, we had but to 
 
 walk round a circular table, with his Cumberland Street." The Times, 
 
 face always turned towards its centre, April 9, 1856. 
 
 and by observing that tbe objects M Another illustration of this con- 
 
 which originally appeared on his fusion of ideas regarding rotatory 
 
 right will appear on the completion and progressive motion we have in 
 
 of one-half of his perambulation to the following letter, which appeared 
 
 be on his left, he will probably be in The Times of the 9th April, and 
 
 able to convince himself that he has reads thus : 
 
 been turning round a vertical axis. " To tlie Editor of the Times. 
 " Before dismissing your patient, SIR, A gentleman, whose services 
 I beg to advise him not to form any as inspector of schools Her Majesty 
 ' ingenious theories ' about the form has had the good fortune to secure, 
 of the Moon before going through a having experienced, no doubt, con- 
 medicinal course of elementary ge- siderable difficulty in conveying an 
 ometry, otherwise those theories accurate conception of the moon's 
 might astonish the world as much rotary motion to unlearned audi- 
 as the letter to which I am replying ences, has had recourse to the in- 
 astonished A WRANGLER. 5, Great genious expedient of publishing in 
 
ON THE MOON'S ROTATION. 
 
 73 
 
 show the distinctive nature of both motions [ 6 9) ; nay, it 
 would have sufficed to point out the fact that the man of the 
 illustration is neither the tree, nor the pillar, nor the table. 
 
 your columns an unqualified denial 
 of that scientific fact that he may be 
 thereby enabled to give the proofs 
 upon which it is based a more irre- 
 sistible aspect. Great merit is due 
 to the originality of this method of 
 instruction, but it is just possible 
 that your more superficial readers 
 have not fathomed the deep pur- 
 poses of his letter, and that thus a 
 result the exact reverse of the writer's 
 intention may have been produced 
 in their minds. You will therefore, 
 perhaps, kindly grant a little addi- 
 tional space to the complete enun- 
 ciation of the conclusions to be 
 drawn from the statements of Mr. 
 Symons. He says that the Moon 
 ' performs precisely the same motion 
 in relation to the Earth that a point 
 on the tire of a wheel does to the 
 box or axle, or that . the round end 
 of the minute hand of a watch does 
 to the pivot in the centre.' And 
 further, that ' it is easy to construct 
 a small instrument similar to this, 
 by fixing a ball on one end of a 
 strip of wood to represent the earth, 
 and fastened by a pivot serving as 
 its axis, and on the other end a 
 smaller ball, also fastened by a 
 pivot. If the strip of wood be 
 turned round on its pivot at the end 
 representing the Earth, the small 
 ball will exactly represent the Moon, 
 and will present the same face 
 (through the whole of its revolution) 
 to the large ball. 
 
 "Now, this is all perfectly true, 
 
 and proves that the Moon's motion is 
 just the same as if she were rigidly 
 connected with the Earth, and the 
 latter were supposed to revolve on 
 her axis once in a month ; whence, 
 of course, it follows that if the 
 earth revolved on her axis once in 
 a month, the moon must do so like* 
 wise; in reference, that is, to fixed 
 space, and not to the supposed rod 
 joining the Earth and Moon. 
 
 " Precisely the same relative phe- 
 nomena would result, if (with the 
 rigid connexion before supposed) 
 the moon's axis were fixed in space, 
 and the moon were to revolve round 
 that axis once in a month, carrying 
 the earth with her. The object of 
 the concluding paragraph of the 
 letter, in which we are informed 
 that * the error is not only prevalent 
 and unaccountable, but it interferes 
 with some very interesting and 
 ingenious theories, based on the 
 oblate surface of the Moon on this 
 side, and her convexity on the other, 
 which are occasioned by the fact 
 that she is centrifugal without rota- 
 tion,' is not sufficiently obvious to 
 justify me in trespassing upon your 
 valuable space with any comments 
 thereupon. I am, Sir, your obedient 
 servant, CAM. Oxford and Cam- 
 bridge University Club, Pall Mall, 
 April 8." 
 
 Now what is the process of reason- 
 ing here pursued ? It is this : " the 
 Moon would rotate on her axis, if 
 she formed an integral part of the 
 
74 
 
 ON THE MOON'S ROTATION. 
 
 36. Of a similar nature is an illustration, which makes a 
 man, sideways again " as the (man in the) Moon does," walk 
 round a chalk-spot, marked on the floor, and gradually get closer 
 
 Earth, and if the Earth turned on 
 her axis once in a month instead of 
 once in twenty-four hours ; therefore 
 of course, she does rotate on her 
 axis; that is to say: she revolves 
 round the Earth; that is to say: 
 this is just the same thing as though 
 she turned about herself, because it 
 is just the same thing, whether the 
 Earth revolves round the Moon, or 
 whether the Moon revolves round 
 the Earth" (compare 34). 
 
 A similar logic distinguishes the 
 following letter, which appeared in 
 The Times of the same day. 
 
 "To the Editor of the Times. Sir, 
 I have just read, in your impression 
 of this morning, a letter headed 
 'The Moon has no rotary motion,' 
 which, suffered to pass without a 
 reply, may have some tendency to 
 shake public confidence in the teach- 
 ings of science, or at all events in the 
 accuracy of modern writers. It is 
 for the sake of showing that this 
 accuracy has been called in question 
 through a misconception on the part 
 of your correspondent that I trouble 
 you with these remarks. To proceed 
 at once to the subject of his letter, it 
 may be well to state distinctly that 
 the works to which he refers are 
 quite correct in stating that the moon 
 rotates about its own axis in the 
 same tune as it revolves about the 
 earth. At the same time your cor- 
 respondent's illustration of the mo- 
 tion of the Moon is quite correct; its 
 motion with reference to the Earth 
 
 being very nearly the same as that of 
 a ball fixed on the tire of a wheel re- 
 volving about its axis with reference 
 to another ball placed at the centre 
 of the wheel; his mistake lies in 
 supposing that this is not a motion 
 of rotation. To make this clear, 
 suppose that a mariner's compass is 
 fixed on the edge of a wheel which 
 is placed in a horizontal position, 
 and made to revolve about its axis. 
 In this case the needle of the com- 
 pass will always point in the same 
 direction, viz. towards the north 
 and the index card that is fixed to it 
 will be carried round by the motion 
 of the wheel, without any rotation 
 about its own axis. But this is a 
 very different motion from that of 
 the Moon ; and, in fact, if the Moon 
 moved round the earth in a manner 
 similar to that just described, all 
 the parts of its surface would be in 
 succession visible from the latter. 
 
 " In conclusion, I may be allowed 
 to remark that your correspondent 
 would have done well to reflect that 
 it requires some considerable degree 
 of mathematical knowledge to see 
 through and decide upon questions 
 such as the one he has raised; that 
 it was more likely he should himself 
 be in error than that all the Astro- 
 nomical writers with whom he was 
 acquainted should agree in making 
 a false statement of so serious a 
 nature as he suggests; and that he 
 might with greater propriety have 
 submitted his doubts to some quali- 
 
ON THE MOON S ROTATION. 
 
 75 
 
 to it, until he finds himself moving upon the spot itself. Assum- 
 ing that he will be admitted, in the last instance, to turn round 
 himself: the author of this illustration thereupon imagines to 
 have settled the question, in consequence of his having demon- 
 strated to evidence the perfect identity of rotatory and pro- 
 gressive motion ; or, he triumphantly asks, should anyone per- 
 chance be able to explain, at what particular distance from the 
 
 fied person privately, before casting 
 in so public a manner an imputation 
 on the accuracy of writers of acknow- 
 ledged ability. 
 
 "Hoping you will consider the 
 importance of the subject an apology 
 for intruding so far upon your space, 
 " I am, Sir, your obedient servant, 
 " April 8. " S." 
 
 Here we are given to understand 
 that the Moon does rotate, and that 
 she does not rotate on her axis ; in- 
 asmuch as it is only through a mis- 
 conception that some people will not 
 allow non-rotation to be rotation ; to 
 render which perfectly clear, it is but 
 necessary to say, that the Moon's mo- 
 tion, certainly, is very different from 
 that of a mariner's compass, for if it 
 were not, her entire surface must be 
 seen in succession from the Earth. 
 
 But nothing daunted by the latter 
 verdict, " CANTABRIGIENSIS," after 
 having brooded a fortnight or longer 
 over the idea of the mariner's com- 
 pass, writes to The Times (25th April, 
 1856), to inform Mr Symons that the 
 compass need only to be placed on 
 the extremity of one of the arms of 
 a turn-stile, instead of on the edge of 
 a wheel, in order to prove on the 
 25th of the month what it had dis- 
 proved on the 8th. Here is the letter. 
 " To the Editor of the Times. Sir, 
 
 Mr. Jelinger Symons will perhaps be 
 able to understand the nature of the 
 motion which puzzles him by the 
 following illustration : 
 
 Let him take a common compass 
 and place it at the extremity of one 
 of the arms of a turnstile. When 
 the turnstile has gone half round let 
 him look at the compass, and he will 
 find that the northern end of the 
 needle points to the south of the 
 card. By the time the turnstile has 
 got all the way round, the needle will 
 again (as at first) point to the north 
 of the card. Now, here it is very 
 plain that either the needle has 
 moved on its axis round the card or 
 the card has performed a revolution 
 on that which is the common axis of 
 itself and the needle. Mr. Symons's 
 eye will inform him that the former 
 is not the case, and consequently, it 
 is to be hoped that his reason will 
 no longer question the latter. 
 
 " The fact seems to be that Her 
 Majesty's Inspector of Schools be- 
 lieves rotary motion can only be per- 
 formed about a fixed axis, or, at least, 
 that the phrase should be confined 
 to that particular case. 
 
 " I am, Sir, your obedient servant, 
 
 " CANTABRIGIENSIS." 
 We will leave the two writers to 
 settle the disputed point between 
 
76 
 
 ON THE MOON S ROTATION. 
 
 spot, rotatory motion can be said to commence ? 67 The expla- 
 
 themselves, referring the reader re- 
 garding the illustration of the ma- 
 riner's compass to 40. 
 
 67 " To the Editor of the Times. 
 Sir, When I was at Cambridge we 
 used to say of a man who seemed 
 incapable of learning mathematics 
 enough for a Junior Opt., ' Oh, he is 
 a kind of a man who can't under- 
 stand that the moon revolves on her 
 axis.' We certainly never expected 
 that the time would come when this 
 view of the lunar theory was to be 
 taught by authority, and school- 
 masters reported against to the Privy 
 Council if they refused to teach it, 
 as we may infer they will be, within 
 Mr. Jelinger Symons's dominion at 
 any rate, from his letter in The Times 
 to-day. 
 
 "I don't know what Lord Gran- 
 ville's views about the moon may be ; 
 but, for the protection of Mr. Sy- 
 mons's inspectees, allow me to offer 
 two very simple illustrations out of 
 two hundred which might be easily 
 offered for the purpose of making 
 this matter clear, perhaps even to 
 the inspector himself, although " some 
 very interesting and ingenious theo- 
 ries," which he wishes to impart to 
 the pupils, will be thereby knocked 
 on the head. 
 
 " In the next school he goes to let 
 him make a chalk spot on the floor 
 and walk round it at any distance 
 he likes, keeping his face steadily to 
 it, as the moon does to the earth, 
 and as he welks let him go closer 
 and closer to the spot, saying all the 
 time, ' Now, boys, you see I don't 
 turn on my axis,' till at last he finds 
 himself moving upon the spot itself; 
 and let him then ask them whether 
 
 e is turning on his axis or not. I 
 believe it will be clear to the stu- 
 pidest boy in the school. Assuming 
 it to be so, he must next proceed to 
 explain to them and himself at what 
 particular distance from the spot ro- 
 tation is to be said to begin. Per- 
 haps he has a supplementary theory 
 to settle this point. 
 
 " By way of another illustration 
 let him take a ball with one side 
 white and the other black, and put it 
 on the right hand of the inkstand, 
 with the white side facing it. Then 
 let him lift it up, and, taking care 
 not to turn it in his hand, put it 
 down on the left hand of the ink- 
 stand. I rather think he will find 
 the dark side now facing the ink- 
 stand ; and if he wants to make the 
 light side face it he must turn the 
 ball half round on its axis. The 
 ball is the moon, and the inkstand 
 is the earth. And, having done 
 that, perhaps it will occur to Mr. 
 Symons that the moon could not 
 present the same face to the Earth 
 if it did not turn half round on its 
 axis in exactly the same time that 
 it performs half its revolution or 
 orbit round the earth, or once round 
 in one revolution in its orbit, and he 
 may perhaps cease to wonder that 
 this revolutionary theory should have 
 received the sanction of such high 
 authority as he mentions; and I 
 suppose he might have added that of 
 every astronomer in the world who 
 ever thought it worth while to ex- 
 press an opinion on such an ex- 
 tremely simple matter. 
 
 " Yours obediently, 
 
 "Aprils. "E. B. D." 
 
 The Times, April 9, 1856. 
 
ON THE MOON'S ROTATION. 77 
 
 nation demanded by him is easily given. Rotation begins at 
 no distance whatever from " the spot," i. e. from its axis the 
 author having, by the truly Jesuitical gradatim-process, con- 
 verted the centre of revolution into an axis of rotation ; but only 
 can take place upon the spot itself, i.e. upon the axis of rota- 
 tion. So long as the man continued walking round the chalk- 
 spot, he was walking round the spot,- the moment he occupied its 
 very place, he could though, if he choose, he might, upon the 
 spot and by an independent act, turn about himself- no longer 
 turn round the spot, as the Moon could no longer turn round 
 the Earth from the moment she had trodden her underfoot. A 
 gradual transition from progressive into rotatory motion, under 
 the circumstances of the case, is simply an impossibility, with- 
 out the man being transformed into the chalk-spot ; or, speaking 
 of the Moon and the Earth, without the former being metamor- 
 phosed into the latter. For, even supposing that the Moon had 
 "got so close" to the Earth, as to revolve round her face to 
 face: she would still continue to revolve round her, and, in the 
 course of each one of her revolutions round the Earth at the 
 same time, and, of necessity, have to turn about four times 
 round herself, in virtue of the mechanical contact of both sur- 
 faces, and according to the proportionate dimensions of their 
 respective circumferences ; nothing than which can more clearly 
 show the distinct nature of rotatory and progressive motion, 
 and the futile character of the argument in question. To ren- 
 der the illustration applicable, its author would have to reduce 
 the diameter of the Moon's orbit to " a chalk-spot" in the centre 
 of the Earth, to convert the elliptical motion of our satellite 
 into a spiral one, thus gradually to bring the Moon and the 
 Earth face to face, and lastly to transform the Earth into the 
 Moon, or to unite their two bodies into one. 
 
 37. In his second illustration,( 6r ) the author substitutes the 
 instanter for the gradatim principle of demonstration. We 
 are to take up the Moon, in the shape of a ball, at any point of 
 her orbit ; carry her, as, according to the apocryphal tradition, 
 
78 
 
 the angel carried Daniel from Judaea into Babylonia, right across 
 the Earth, represented by an inkstand, to the opposite point of 
 her orbit ; and then put her down in a wrong position, in order 
 that, by turning her round into the right one, we may convince 
 ourselves and others, that she has turned round herself. The 
 author is infelicitous in the choice of his illustrations. They 
 do not apply. It is not precisely in the manner he adopts 
 with regard to the ball and inkstand, that the Moon is carried 
 round the Earth. Let him place the inkstand close before him ; 
 let him, with an outstretched arm and hand, take up the ball as 
 by him described ; let him thus, in imitation of the Moon's 
 actual progressive motion in her orbit, carry it half round the 
 inkstand and put it down on the opposite side : he will then 
 find that the white side of the ball, which faced the inkstand 
 when he took it up, still faces it when he puts it down again ; 
 and that, consequently, as the ball cannot well be supposed, 
 unobserved by him and o/* itself, to have turned half round in 
 his hand, whilst being carried half round the inkstand, the 
 experiment tends to prove the very contrary opinion, regarding 
 the Moon's rotation, to that entertained by himself. 
 
 38. This differing opinion, which denies the Moon's rotation, 
 has, as we have already stated, in modern times become almost 
 extinct among Astronomers completely so ( 4) and as far 
 as our knowledge goes, Mr. Evan Hopkins, ( 68 ) and Mr. Jelin- 
 
 68 "To the Editor of the Times. have had the courage, for many years, 
 Sir, As the author of the work al- to question the correctness of the 
 luded to in support of Mr. Symons' statement, that 'the moon has an 
 argument on the moon's motion, I axis of rotation.' ' Rotation, accord- 
 beg leave to solicit your kind permis- ing to engineers, means the act of 
 sion to insert this letter. turning on an axis or centre, situated 
 
 "I am one of the many who have within the moving body, and ' revo- 
 
 studied the laws of, and the terms lution' means the act of turning 
 
 employed in, dynamics who have not round a centre situated beyond the 
 
 been able to understand how the moving body. Hence, according to 
 
 term 'rotation' can be applied to the the above definitions, the governor 
 
 moon's motion. I am, also, one of balls of a steamengine, crank pins, 
 
 the few who have presumed, and teeth of the cogwheels, and all bodies 
 
ON THE MOON'S ROTATION. 79 
 
 ger Symons, (^ are the two solitary public representatives of 
 
 presenting the same faces towards 
 the centres of revolution, cannot, 
 under such conditions, turn round 
 on axes situate within themselves. 
 
 " In elucidating the cause of the 
 apparent misapplication of the term 
 ' rotation/ I respectfully request Dr. 
 Lardner to make a distinction be- 
 tween changes produced by revolu- 
 tion round a distant centre, such as 
 *the Peak of Teneriffe' round the 
 earth's centre, and the movement of 
 a body on a centre situated within 
 itself, as the argument rests on the 
 distinction of the terms and their 
 respective movements. I have the 
 honour to be, Sir, Yours respectfully, 
 
 " April 22. EVAN HOPKINS." 
 
 " The Times" April 23rd, 1856. 
 
 All this is perfectly correct, except 
 that the question of the Moon's rota- 
 tion is not, as here implied, in any 
 way dependent on the truth or error 
 of the general arguments which have 
 been urged either for or against her 
 rotation ; but, as will be seen here- 
 after, rests on special merits, upon 
 which those arguments do not even 
 touch. 
 
 69 " THE MOON HAS NO ROTARY 
 MOTION." 
 
 " To the Editor of the Times. 
 Sir, May I request the favour of a 
 small space in your columns to in- 
 quire the grounds upon which almost 
 all school astronomy books assert 
 that the moon rotates on her axis ? 
 This theory is positively stated in 
 Schcedler's Book of Nature, a work 
 of authority in Germany, and it has 
 just received the high sanction of 
 Mr. J. R. Hind, in his edition of 
 
 Keith Johnston's New Atlas of Astro- 
 nomy. The Moon is there said to 
 1 revolve round our globe in a period 
 of 27 days 7 hours and 43 minutes, 
 and to rotate upon her axis in pre- 
 cisely the same interval, whence it 
 occurs that only one-half of the moon 
 can ever be seen from the earth.' 
 
 "On the contrary, if the moon 
 turned at all on her axis, a little 
 consideration will show that all her 
 surface would be successively shown 
 to the earth, and that it is because 
 she has no rotary motion at all that 
 one side only is seen by us. She 
 performs precisely the same motion 
 in relation to the earth that a point 
 on the tire of a wheel does to the box 
 or axle, or that the round end of the 
 minute hand of a watch does to the 
 pivot in the centre. It is easy to 
 construct a small instrument similar 
 to this, by fixing a ball on one end 
 of a strip of wood, to represent the 
 Earth, and fastened by a pivot serv- 
 ing as its axis, and on the end a 
 smaller ball, also fastened by a pivot. 
 If the strip of wood be turned round 
 on its pivot at the end representing 
 the Earth the small ball will exactly 
 represent the Moon, and will present 
 the same face (through the whole of 
 its revolution) to the large ball ; but 
 if the small ball be made to rotate 
 on its axis ever so little it will im- 
 mediately present a change of face 
 to the larger ball, and so would the 
 Moon to the Earth. 
 
 " If the Earth moved round the 
 Moon, then the rotation imputed to 
 her in the same period as the Earth's 
 revolution round her would be neces- 
 
80 
 
 ON THE MOON S ROTATION. 
 
 it at the present moment. It remains, therefore, to be seen, 
 whether the arguments advanced 'by them in support of their 
 view, are of a more conclusive character than those we have 
 
 sary to her presenting the same face 
 to us. The error is not only preva- 
 lent and unaccountable, but it inter- 
 feres with some very interesting and 
 ingenious theories based on the oblate 
 surface of the Moon on this side and 
 her convexity, on the other, which 
 are occasioned by the fact that she 
 is centrifugal without rotation. I am, 
 Sir, your obedient servant, 
 
 " JELTNGEB SYMONS, 
 " Her Majesty's Inspector of Schools." 
 ' The Vineyard, near Hereford, 
 
 April 5." 
 
 " To the Editor of the Times. 
 Sir, I ventured to ask for proofs 
 that the moon rotates on her own 
 axis, and I am favoured to-day with 
 seven letters, referring me to facts 
 each of which results either from the 
 Moon's revolution round the Earth, 
 or her cycloidal revolution round the 
 Sun. These are no proofs that she 
 rotates on her axis, which I appre- 
 hend is in her centre, but which 
 ' Cam' confuses with the centre of 
 her orbit. Her rotation round her 
 own axis would be a distinct and an 
 additional motion to that which the 
 Moon has ; and a reference to an 
 orrery, or to the simpler instrument 
 I mentioned, will afford demonstra- 
 tion that this is the fact ; nor will it, 
 
 I think, be refuted by Science or 
 affected by sarcasms. I am, Sir, 
 your obedient servant, 
 
 " JELINGER SYMONS, 
 
 II Her Majesty's Inspector of Schools." 
 
 " April 9." 
 
 "To the Editor of the Times. 
 Sir, This subject excites so much 
 interest, and I have received so many 
 communications on both sides of the 
 question, that I very reluctantly tres- 
 pass once more on your courtesy, 
 praying to be allowed to support my 
 statement more fully that the Moon 
 has no rotation on her axis. My 
 opponents produce elaborate proofs 
 of the palpable fact that she revolves 
 round the Earth, turning the same 
 hemisphere towards it. This is not 
 axial rotation, such as the astrono- 
 mical books attribute to the Moon. 
 Nor is the difference one in terms 
 only, but in distinct kinds of motion, 
 governed by different laws, and pro- 
 ducing widely different results and 
 phenomena. It is one with which 
 every mechanic is familiar. 
 
 " The rotation of a body round a 
 centre distant from it, like the gover- 
 nor-balls of a steam-engine, cannot 
 be performed without progressive 
 motion round its own orbit; but ro- 
 tation round its own axis can. Ar- 
 rest the Earth in its revolution round 
 the Sun, and it will still rotate on its 
 axis, for it has axial rotation. Arrest 
 the Moon in her revolution round 
 the Earth and she will be motionless, 
 because she has no axial rotation. 
 To put this question to a practical 
 test, let an orrery be constructed so 
 that the Moon shall rotate on her 
 axis, as we are told she does, in the 
 same period in which she revolves 
 round the Earth. Let this period 
 be ten minutes. I affirm that at the 
 
ON THE MOON'S ROTATION. 
 
 81 
 
 had occasion to comment upon in the preceding paragraphs. 
 But, as has been remarked before ( 25), the two opposite 
 opinions are based upon the same identical fact, namely, the 
 
 expiration of five minutes after the 
 orrery is put in motion the Moon 
 will present to the Earth exactly the 
 opposite hemisphere of her surface 
 to that which she presented to it at 
 starting. If I am right in this the 
 question is settled, and the Moon 
 has no axial rotation ; for at what- 
 ever rate she rotates a similar result 
 will ensue. 
 
 " The practical effects of the Moon 
 having no axial rotation seem to he 
 chiefly these and they are not un- 
 important : The projectile force of 
 the centrifugal motion of the moon 
 will affect that side which is the 
 most remote from us, causing her 
 convexity on that side, together with 
 the ascertained fact that her centre 
 of gravity is on the other side of the 
 centre of her circumference, and the 
 probability that there are humidity, 
 atmosphere, vegetation, and, possi- 
 bly, inhabitants there. If the Moon 
 had axial rotation the projectile force 
 would take equal effect on the whole 
 of her surface ; she would be a sphe- 
 roid oblate at the poles, and these 
 phenomena, discovered by the Ger- 
 man astronomers, would be deprived 
 of the support they derive from her 
 axial immobility. 
 
 " If I am in error in my humble 
 effort to correct what appears to me 
 to be a fallacy well worth removing, 
 I shall be grateful to anyone who 
 sets me right by relevant reasoning ; 
 but, at present, I confess that I am 
 conscious only of the mistake of sup- 
 posing that a question of science 
 
 might now, even in England, be 
 treated with candour and debated 
 with temper. I am, Sir, your obe- 
 dient servant, 
 
 " April 12. JELINGEE SYMONS. 
 
 "P.S. Mr. Hopkins apprises me 
 that he has already supported the 
 same view as myself in his well- 
 known work on Geology." 
 
 " To the Editor of the Times. 
 Sir, May I venture to trespass 
 again on your indulgence for the 
 privilege of replying to the letter of 
 so eminent a mathematician as Mr. 
 W.Hopkins? He was little likely 
 to be mistaken for his equally dis- 
 tinguished namesake, whom I had 
 designated as a writer on Geology ; 
 though I believe him to be skilled 
 in dynamics, and a great authority 
 thereon. 
 
 " Your correspondent is ' astound- 
 ed at my boldness' in questioning 
 an astronomical dictum, sanctioned 
 by great philosophers. Be it so ; the 
 time has gone by for a poped om in 
 Science. Plain people are concerned 
 in the truth of her dogmas. Nor 
 will even the ' scientific horror' of 
 Mr. W. Hopkins deter the heretical 
 tendencies of investigation, now that 
 minds are at work and thumbscrews 
 banished. 
 
 " Inspectors of schools, who are often 
 censured for failing to effect plain 
 instruction in common things, are 
 especially interested in the distinct- 
 ness and correctness of the informa- 
 tion on such subjects given in books, 
 G 
 
ON THE MOON S ROTATION. 
 
 constant direction of the same face of the Moon towards the 
 Earth : and it follows by no means that a fact which does not 
 prove one thing, should prove the contrary. " If the Moon 
 
 which philosophers indite for schools. 
 Their accuracy, therefore, is a ques- 
 tion of more importance than my 
 audacity in endeavouring to clear up 
 a practical point in elementary in- 
 struction, which Mr. Hopkins admits 
 is in periodical doubt. 
 
 "Mr. Hopkins maintains that if 
 the Moon in her orbit turns the same 
 hemisphere constantly to the Earth, 
 that is rotary motion round her own 
 axis. Will he permit me to suggest 
 to him the difficulties which occur to 
 the prof anum vulgus in thus defining 
 the Moon's motion? In the first 
 place, how is the Earth's motion to 
 be distinguished from the Moon's 
 motion ? They are clearly different ; 
 for the latter always presents the 
 same hemisphere to its centre (the 
 Earth), while the Earth presents dif- 
 ferent hemispheres to its centre (the 
 Sun). That kind of rotation which 
 takes place on an axis within the 
 rotating body the Moon clearly has 
 not, or she must present all her sides 
 successively to the Earth, or to any 
 point within her orbit. I do not un- 
 derstand Mr. Hopkins to dispute this. 
 If he does, I must refer him to 
 'E. B. D.'s" tin moon, and the ex- 
 periment whereby that urbane person 
 has so ingeniously confuted himself. 
 The motion which Mr. Hopkins terms 
 the rotation of the Moon on her own 
 axis is precisely that which Mr. 
 Hopkins, when he stands still, is 
 performing round the axis of the 
 Earth. Like the Moon, that scienti- 
 
 fic gentleman's body presents the 
 same part of it always to the Earth's 
 centre, and all sides of himself in 
 succession to the Sun ; and we are 
 all of us, according to his definition, 
 rotating each round his own axis. 
 Now, if Mr. Hopkins is not too much 
 horrified at the idea, will he be so 
 good as to spin himself round on his 
 heel? However shockingly unscien- 
 tific, it is still physically possible to 
 do so. Will he be kind enough to 
 tell me what that additional motion 
 is to be called ? ' Axial rotation ' is 
 already bespoken and appropriated 
 to his former and lunar motion. 
 There must be another name for 
 another movement, so utterly and 
 entirely distinct from the former one 
 that either may take place without 
 the other. If any of your unscien- 
 tific readers doubt this let them hold 
 a ball firmly between finger and 
 thumb. By turning the body round, 
 with the ball at arm's length, it will 
 perform what I iised to consider, in 
 my Cambridge days, revolution round 
 a distant centre, and it will present 
 the same side, just as the Moon does, 
 towards that centre. This is one 
 motion. Now, turn the ball round 
 with the finger and thumb, and how- 
 ever slowly this is done it will present 
 a change of face to the holder. Nor 
 will this latter movement be anywise 
 affected by stopping the revolution 
 of the arm. But' if this movement 
 of the ball be as I humbly submit 
 ' axial rotation' its former revolu- 
 
ON THE MOON'S ROTATION. 
 
 83 
 
 turned at all on her axis," Mr. Symons argues, " a little consi- 
 " deration will show that all her surface would be successively 
 " shown to the Earth, and that it is because she has no rotatory 
 
 tion cannot be; neither can it be 
 properly so called, for distinct mo- 
 tions must have distinct names. 
 
 " If the astronomers are right in 
 saying that the Moon presents the 
 same face to us because she rotates 
 on her own axis once in the same 
 time that she revolves round the 
 Earth, let us also teach that the rea- 
 son why each part of the felloe of a 
 wheel presents its face to the box at 
 the centre is because such parts rotate 
 on their axes in the same time that 
 they revolve round the box; and the 
 same reason will apply likewise to 
 every nail in the tire, just as much 
 as to the Moon. 
 
 " If, as Mr. Hopkins implies, the 
 difference between us is one of terms, 
 it is high time that astronomers 
 should adapt and apply them with 
 that precision and clearness which 
 Science demands. If this contro- 
 versy has no other result, I shall 
 heartily rejoice at the humble part I 
 have taken in it. It has aroused a 
 spirit of instructive and earnest in- 
 quiry such as no trivial or settled 
 point could possibly have sustained; 
 and I am in doubt whether I owe 
 more, in the maintenance of my 
 ground, to the aid of my adherents 
 or the abuse of my opponents. With 
 renewed thanks for your courtesy to 
 myself, I am, Sir, your very obedient 
 servant, 
 
 "April 19. JELINGER SYMONS " 
 " The Times" April 9th, llth, 13th, 
 2 1st, 1856. 
 
 In reference to Mr. Symons's first 
 letter, we need hardly state to those 
 who are at all acquainted with Ger- 
 man Literature in general, and with 
 " Schoedler's Book of Nature" in par- 
 ticular, that, although a very excel- 
 lent and deservedly popular school- 
 book, it has no better claim to " au- 
 thority" in Germany than, f. i., those 
 equally excellent little volumes in 
 Weale's " Rudimentary Series," which 
 treat on the same subjects, have in 
 England. 
 
 Since the above was written we 
 have seen Mr. Symons's paper on 
 " Lunar Motion," in " The English 
 Journal of Education" for the 1st of 
 June, 1856, pp. 253275; but as it 
 leaves the question of the Moon's 
 rotation in precisely the same posi- 
 tion, we have no occasion further to 
 revert to it. Mr. Airy's two letters, 
 embodied in this paper, and in con- 
 tradiction with each other, would 
 seem to have impressed Mr. Symons 
 with the erroneous idea, very gene- 
 rally entertained, that the question 
 really rests on a mere inaccuracy of 
 expression on the part of Astrono- 
 mers. (Compare 6 and Note 19.) 
 
 A paper, also, on " the Moon Con- 
 troversy," printed for private circula- 
 tion, has been communicated to us, 
 from which we perceive that its au- 
 thor, Mr. H. Perigal, jun., shares in 
 the opinion of Mr. Evan Hopkins 
 and Mr. Symons ; and from the state- 
 ment of the latter (p. 275) it would 
 appear that they have many scientific 
 
 G 2 
 
84 ON THE MOON'S ROTATION. 
 
 " motion at all that one side only is seen by us." This proposi- 
 tion, however, is an erroneous qne. To convince ourselves of 
 it, we have but to imagine the Moon's axis of rotation to lie in 
 a straight line, produced to the centre of the Earth ; in which 
 case, whether she rotate or not, nearly the same identical hemi- 
 sphere would always be turned towards the Earth, as is the case 
 now. The fact itself is no more a proof against the Moon's 
 rotation, than the contrary fact nearly her entire surface being 
 successively seen from the Earth would be a proof for her 
 rotation. It would only amount to a strong presumption in the 
 latter case, as it does in the former ; for, though it is highly 
 improbable that any diameter of the Moon but an axis of rota- 
 tion should be constantly directed towards some fixed spot in the 
 heavens, still, we can as readily conceive this to be possible, as 
 it is possible that she should rotate and yet always present 
 nearly the same face towards us, without any material change 
 in the relative position of the various features of that face, 
 with regard to the axis of rotation, being observable from the 
 Earth. 
 
 39. To prove what we have just advanced, namely, that the 
 Moon, in the course of each revolution round the Earth, might 
 in succession present every portion of her entire surface to our 
 view, without any rotatory movement on her own axis: let, in Fig. 
 18, E represent the Earth, M the Moon, a b c the opposite hemi- 
 lere, s some fixed star. A plane is to be supposed to pass 
 
 supporters, who have lent to him Athenceum of March 31, we have not 
 
 their " invaluable aid in this contro- seen. The " Censor" concludes his 
 
 versy." We can but regret, then, remarks upon it with the words, 
 
 that Mr. Symons should so little have "We abide by our opinion that a 
 
 availed himself, of that aid as to nose cannot first point north and 
 
 show hardly a trace of it in his paper. then south without a right-about 
 
 The letter on the Doctrine of the face," and we must confess the su- 
 
 Moon's rotation, addressed by Mr. perficiality betrayed by these few 
 
 Mac Elshender to the Astronomical words, with regard to our problem, 
 
 Censor of The Athenaum, quoted by reflects little credit on the scientific 
 
 Mr. Symons, and noticed in The character of The Athenaum. 
 
ON THE MOON'S ROTATION. 85 
 
 through the centres of the Earth and of the Moon in her different 
 positions. It is evident that, if the Moon rotated, every spot 
 on her surface must 
 constantly change its 
 position with regard 
 to that plane ; and 
 that, to an observer 
 in s, each spot would 
 appear to describe 
 concentric circles, and 
 to an observer in E 
 perpendicular lines, 
 on the lunar disk. 
 But each spot is to 
 remain perfectly sta- 
 tionary with regard to 
 the plane described ; 
 in other words, the 
 Moon is to be sup- 
 posed not to ro- 
 tate on her axis. Yet, 
 
 in revolving round the 
 
 FIG. 18. 
 Earth, and keeping 
 
 the point b in her surface constantly directed towards the 
 same point of the heavens, she will in succession present every 
 portion of her entire surface towards the Earth ; first the he- 
 misphere a d c; then bed; thereupon a b c ; afterwards 
 dab; and once more a d c. According to Mr. Symons' argu- 
 ment, this would be a proof of the Moon's rotation ; accord- 
 ing to Astronomers, it is a proof of her non-rotation. 
 
 Let us then see wherein, according to both parties, rotation 
 really consists. The Moon's motion, as just described, would 
 resemble that of a man walking round a table, with his face 
 constantly turned towards some remote object in the room, we 
 will say a looking-glass, and presenting every part of his body 
 
86 ON THE MOON'S ROTATION. 
 
 in succession towards the centre of the table. Astronomers, on 
 the contrary, make him keep his face constantly turned towards 
 the centre of the table, and present every part of his body in 
 succession to the looking-glass. In the former case the mau 
 has to walk forwards, in performing half the circuit of the 
 table, and backwards in accomplishing the other half. In the 
 latter case he has to walk sideways the entire distance round. 
 Here, then, we have two practical definitions of "rotation." 
 According to Mr. Symons, walking sideways round a table is 
 non-rotation : but walking half forwards, half backwards 
 round it, is rotation. According to Astronomers, on the con- 
 trary, walking half backwards, half forwards round a table, is 
 non-rotation, but walking sideways round it is rotation. In 
 the former case they contend that, in walking round the table, 
 a person does not turn round himself; in the latter case they 
 maintain that he does. Presenting, in the course of circular 
 progressive motion, one's face continually towards the centre 
 of the table, and one's face and back alternately to the looking- 
 glass, involves, or even " constitutes" rotation on one's axis; 
 but presenting one's back and face alternately towards the cen- 
 tre of the table, and one's face continually towards the looking- 
 glass, does not. What could place the fallacy of every one of 
 the general arguments, w r hich have been urged both for and 
 against the Moon's rotation, in a more striking light, than the 
 contradictory consequences to which they lead ? Illustrations, 
 under all circumstances, unless perfectly applicable, are apt to 
 mislead our judgment, by exercising an undue influence upon 
 our senses instead of assisting our under standing . 
 
 40. We may be permitted to exemplify the truth of 
 this remark. In Fig. 19, let w represent a wheel, re- 
 volving horizontally about its axis A ; w N E s a mariner's 
 compass, placed on the edge of the wheel ; p the polar star. 
 The axle A may at the same time represent the Earth, the 
 compass-box the Moon. When the wheel has revolved from 
 
ON THE MOON'S ROTATION. 87 
 
 w to w", the point N in the compass-box will still occupy 
 
 FIG. 19. 
 
 its first position with regard to the wheel ; but no longer so 
 with regard to the axle and the magnetic needle, the latter 
 having remained stationary with regard to space, whilst the 
 compass-box, stationary with regard to the wheel, made half a 
 circuit with regard to space. In other words : whilst the 
 Moon was performing half a revolution round the Earth, the 
 straight line s A joining both centres has invariably passed 
 through the same identical point s in the Moon's surface, which 
 proves that no rotation has taken place. But let us now, 
 whilst the wheel is making another revolution, and leaving the 
 compass resting on its edge, impart to the latter (in a contrary 
 direction to the wheel's motion) such a motion that the point N 
 
88 ON THE MOON'S ROTATION. 
 
 in the compass-box shall continue to correspond to the direc- 
 tion of the magnetic needle, and consequently continue to 
 regard the same point p in space. In virtue of this motion, 
 the compass-box turning round on the edge of the wheel will 
 perform a complete circuit with regard to the wheel in precisely 
 the same time which the wheel takes to revolve about its axis, 
 and present to that axis successively every point in its circum- 
 ference. Apparently the analogy between this combined 
 motion and the combined motion of the Moon's rotatory and 
 progressive motion, is a perfect one ; and therefore, it has been 
 made use of to demonstrate the Moon's rotation, our own eyes 
 being called upon to witness the compass-box actually turning 
 round on its own axis (see Note 66). But not only does the 
 assumed analogy not exist ; our senses also deceive us, our own 
 eye, producing an erroneous impression on our mind, misleads 
 our judgment. To prove the former point, it suffices to direct 
 attention to the circumstance, that the Moon is asserted to 
 rotate on her axis in the same direction in which she revolves 
 round the Earth ; and that the compass-box must of necessity 
 be turned round in a contrary direction to that of the wheel's 
 motion, in order to produce the described result. As to the 
 second point, it is as easily proved. A moment's clear reflec- 
 tion will show us, that in turning the compass-box round on 
 the edge of the revolving wheel, we merely paralyze the effect 
 of the wheel's motion upon the compass-box with regard to 
 space ; but do not impart to the latter any motion with regard 
 to its own axis, i. e. do not impart to it an independent rotatory 
 motion. To convince ourselves of this, we have but to lift the 
 compass just off the wheel's edge, and, keeping the point N 
 steadily directed towards the same point in space, hold it over 
 the place it previously occupied, the wheel revolving all the 
 time. Astronomers themselves allow that this kind of motion 
 involves no rotation on the part of the compass-box or of the 
 Moon, which it represents ; for it is identical with the motion 
 described in the- preceding paragraph, and there proved to be 
 
ON THE MOON'S ROTATION. 89 
 
 simply progressive. And this is further proved by the fact that, 
 in turning round the compass on the edge of the wheel, the 
 point N, though changing direction with regard to every point 
 in the wheel as well as with regard to the axis A of the wheel, 
 yet does not change direction either with regard to space or 
 with regard to its own axis : whereas, according to 9, rotatory 
 motion is determined by a change of direction, both with re- 
 gard to space and the own axis of rotation. In imparting to 
 the compass-box, placed on the edge of a revolving wheel, such 
 a motion as to keep a certain line N s, passing through its 
 centre, constantly directed towards the same point in space, we 
 offer but a perfect illustration of the proposition, demonstrated 
 in 39, namely, that a satellite without rotating on its axis, 
 may yet, in the course of each revolution round its primary, 
 present every portion of its surface to that primary. What ren- 
 ders the illustration apparently so complicated is the introduc- 
 tion of a new element, the magnetic needle, representing a 
 certain diameter of the satellite's body, constantly directed to- 
 wards the same point in space. 
 
 If we would really impart to the compass- box a rotatory mo- 
 tion on its axis, we must either turn it round after having lifted 
 it off the edge of the wheel, and independently of the movement 
 corresponding to that of the latter, which way we like ; or leav- 
 ing it resting on the wheel's edge, we must turn it round in the 
 same direction in which the wheel revolves. In this case, 
 when the wheel has described the arc w w', and the compass- 
 box has arrived in w', the point s will no longer correspond to 
 s', but to some point a, and the line s' A, joining the centres of 
 the wheel and the compass-box, will in succession have passed 
 through every point in the arc a s', showing that, independently 
 of its movement as connected with the wheel, the compass-box 
 has been turned about itself, i. e., about its own axis, the point 
 s having described the independent arc s' a. In the opposite 
 case, we are at liberty to turn the compass-box either way ; and 
 it is evident that in turning it in the direction of the wheel's 
 
90 ON THE MOON'S ROTATION. 
 
 movement, we may impart to it such a motion that, in following 
 that movement, the point s shall continually be directed to- 
 wards the centre A. And here again, the advocates of the 
 Moon's rotation will tell us, with an air of triumph, that we are 
 but proving the very thing they are contending for ; and here 
 again their judgment is misled. Here again, they confound 
 rotatory and progressive motion from their similar effects with 
 regard to space. It is certain, that in the same plane the same 
 motion cannot produce two different changes of direction with 
 regard to the centre of motion ; and that the same change of 
 direction with regard to the centre of motion cannot indicate 
 two different motions. Whether the compass-box in the same 
 period turn round itself on the spot w, or whether it turn round 
 the axis A ; the point N will have performed the entire circuit 
 of the heavens, and return to the same fixed star p at the same 
 moment. But, after performing a quarter of a revolution, the 
 point N will, in the former case, be in w, its direction with re- 
 gard to the centre A differing by an angle of 90 from its first 
 direction, whilst in the latter case, its direction with regard to 
 the centre A will have undergone no change whatever. After 
 half a revolution, the difference will be 180; in other words, 
 the point N, in the former case, will now directly face the centre, 
 whilst in the latter case, the diametrically opposite point s will 
 still do so. At the completion of each revolution only, the co- 
 incidence of direction confined to that physical moment, will 
 occur again. One of those changes of direction is characteristic 
 of rotatory, the other of progressive motion : when both are 
 combined, both changes of direction also must necessarily take 
 place ( 9) ; which in our illustration is not the case 7 . In 
 
 70 Moreover, the Moon is not, over the compass-box as to allow the 
 
 speaking in the sense of analogy, a latter &free horizontal motion with 
 
 compass-box, for there is this es- regard to it ; whereas the diameter 
 
 sential difference between the real of the Moon, corresponding to the 
 
 object and the object representing it magnetic needle, is & fixed line with- 
 
 for the purpose of illustration, that in the lunar globe, a motion round 
 
 the magnetic needle is so suspended which of the several parts of that 
 
ON THE MOON'S ROTATION. 91 
 
 imparting to the compass-box the motion here in question, we 
 therefore simply carried it " in following the movement of the 
 wheel," round the wheeTs centre; and, consequently, imparted 
 to it only progressive, but no rotatory motion. If any rotatory 
 motion had accompanied the progressive motion, the point N of 
 the compass-box when arrived in w', according to the direction 
 in which we turned it, would either have been in some point b 
 in arrear of s', or else in some point a, in advance of s' ; for, 
 in order to arrive at the amount of rotatory motion, when com- 
 bined with progressive motion, we must disregard the latter 
 altogether, and with reference to our illustration, consider the 
 compass-box as having remained stationary in w' ( 7 and 
 note 23). 
 
 41. To return to the argument of Mr. Symons, who has been 
 misled by Astronomers themselves to look upon the general 
 fact of the Moon's always turning the same face towards the 
 Earth, as the sole ground on which the theory of her rotation 
 rests : whereas that fact is only one incidental to the theory. If 
 the spots which we observe on the, to us, visible hemisphere of 
 our satellite were subject to no changes of position whatever, 
 except those arising from a change of her respective position in 
 her orbit with regard to the Earth, or from the lunar parallax, 
 such a fact would undoubtedly and conclusively disprove the 
 Moon's rotation on her axis ; and no Astronomer, deserving of 
 the name, would, as Dr. Lardner asserts 71 , or ought to, think of 
 
 globe, in the plane of the diameter else as is compatible with clearness, 
 
 is an impossibility, and can only "The same hemisphere of the 
 
 take place in a direction at right Moon is not always presented to the 
 
 angles to the latter. Earth's centre. There is libration in 
 
 71 " To the Editor of the Times. longitude, and also in latitude. The 
 
 SIR, As the dispute about the former is caused by the fact that the 
 
 Moon's rotation has excited such ge- velocity of the rotation is not always 
 
 neral interest, and the popular faith equal to that of the revolution, being 
 
 in astronomical authorities seems sometimes greater and sometimes 
 
 to waver, perhaps you will consider less. This arises from the circurn- 
 
 the following worth the space it will stance of the velocity of revolution 
 
 occupy. I have rendered it as con- being variable, while that of rotation 
 
92 
 
 ON THE MOON S ROTATION. 
 
 explaining the phenomenon, by supposing the lunar globe to 
 have a motion of rotation rigorously equal in velocity to that of 
 its revolution, and round an axis parallel to its axis of revolu- 
 
 is rigorously uniform. The libration 
 in latitude arises from the fact that 
 the axis of rotation is not rigorously 
 parallel to that of revolution, but in- 
 clined to it at a small angle (1 30' 
 10"8.) The phenomenon of rotation 
 is therefore distinct from and inde- 
 pendent of that of revolution. The 
 two motions have different velocities, 
 and take place round different axes. 
 They cannot therefore be identical, 
 and this disposes of the question of 
 the rotation which is demonstratively 
 established. 
 
 " So far, therefore, as the particu- 
 lar case of the Moon is concerned, 
 the argument of Messrs. Hopkins, 
 Symons, and those who agree with 
 them, falls to the ground; but the 
 general question raised by them re- 
 mains untouched. That question is 
 this : Suppose a globe to revolve 
 in a circle round a centre of attrac- 
 tion so as to present the same hemi- 
 sphere always to the centre, will 
 such a globe have a motion of rota- 
 tion or not? Your correspondents 
 say No. Astronomers universally 
 say Yes. It is no answer to the 
 question to say that no such case 
 exists, or ever could exist in the 
 physical universe, unless the re- 
 volving body were fixed by a ma- 
 terial connexion to the centre of 
 revolution, as bodies on the surface 
 of the Earth are to the terrestrial 
 axis. This only goes to show that 
 the question is merely one of curious 
 speculation and of no practical im- 
 portance whatever. It is well under- 
 
 stood that, under the physical con- 
 ditions in which the bodies of the 
 solar system are placed, uniform 
 circular motion round a centre of 
 attraction is impossible. The orbit 
 must in such cases be an ellipse, 
 and the motion of revolution must, 
 therefore, be variable, increasing in 
 velocity as the distance decreases, 
 and vice versa. The motion of axial 
 rotation must, on the contrary, be 
 rigorously uniform, and consequently 
 the libration in longitude, observed 
 in the case of the Moon, is a phy- 
 sical necessity, and becomes an 
 incontrovertible proof of a rotation 
 distinct from and independent of 
 revolution. 
 
 " But let us accept, for argument's 
 sake, the imaginary, though physi- 
 cally impossible, supposition of a 
 globe revolving round a centre of 
 attraction so as to present the same 
 hemisphere rigorously to that centre. 
 Your correspondents explain this 
 phenomenon by supposing it to be 
 identical with the case of a globe 
 connected with the centre in the 
 same manner in which the parts of 
 the Earth's surface are connected 
 with its axis ; and they consequently 
 identify the motion of revolution 
 with a motion of rotations of which 
 the axis is a line drawn through the 
 centre of attraction at right angles 
 to the plane of the orbit. Astrono- 
 mers, on the other hand, would ex- 
 plain the phenomenon by supposing 
 the globe to have a motion of rota- 
 tion rigorously equal in velocity to 
 
ON THE MOON'S ROTATION. 93 
 
 tion ; since such an explanation would involve the absurdity of 
 elucidating that which is impossible. Nor is, as Dr. Lardner 
 further contends, the question, viewed under the above aspect, 
 merely one of curious speculation, and of no practical im- 
 portance whatever, as will appear from the sequel. Like the 
 majority of Astronomers and astronomical writers, he would 
 seem to have had no clear perception of the real merits and 
 those elements of our problem, which constitute its leading 
 features and on which its solution altogether depends. We 
 will, therefore, now proceed to explain these features to the 
 reader, as clearly as we may be able and the nature of the 
 subject admits of. True, from the Chair ( 58 ), and out of the 
 Chair ( 71 ), the main opinion held by Mr. Evan Hopkins and Mr. 
 Symons respecting the Moon's rotation, and held before them 
 by Kepler and Galileo ( 3), has authoritatively been de- 
 nounced as an exploded fallacy, and summarily disposed of as 
 a settled question, demonstratively established in favour of the 
 prevailing view; 72 but, as Mr. Symons justly remarks (fy), the 
 
 that of its revolution and round an question in popular terms, and illus- 
 
 axis parallel to its axis of revolution. trated it by popular arguments. It 
 
 The adequacy of this explanation is would have been much easier and 
 
 so obvious, that it may be demon- more conclusive to have applied to 
 
 strated by any clever school-boy who it the machinery of mathematical 
 
 is familiar with the elements of physics, but that would not so well 
 
 geometry. suit the half-million readers and 
 
 "In fine, the question is purely hearers of The Times. I remain, 
 
 speculative and of no practical in- Sir, your obedient servant, DION. 
 
 terest whatever. The supposition of LARDNER. Paris, April 25." The 
 
 Messrs. Hopkins and Symons would Times, April 27, 1856. 
 
 explain the supposed but physically 72 Dr. Lardner, in his first letter 
 
 impossible phenomenon, but it would to The Times (see note 32) insists 
 
 require the admission, of a connexion himself on the necessity of a clearer 
 
 between the revolving body and the exposition of the theory of the 
 
 centre of revolution even more in- Moon's rotation than has been 
 
 conceivable than the supposition hitherto given by Astronomers, in 
 
 itself. order to render it intelligible to " or- 
 
 " All Astronomers will perceive that dinary minds," "judging it to be," as 
 
 I have here purposely discussed the he does, " more rational to supply 
 
94 ON THE MOON'S ROTATION. 
 
 time for a Popedom in Science has gone by ; and we must take 
 those apodictical verdicts just for what they are worth. And 
 worth they are nothing. 
 
 In casting a retrospective glance at the general arguments 
 by which the theory of the Moon's rotation is usually sup- 
 ported on the part of Astronomers, the reader will readily 
 admit the justice of our introductory remark, that, with regard 
 to an intelligible subject upon which Science had failed to 
 render herself intelligible, it was but fair to assume that her 
 ideas did not possess a clearness, of which her language, in 
 conveying them, proved deficient. 
 
 42. The real problem of the Moon's rotation on her axis, 
 which we shall now have to consider in its details, may be 
 divided into three distinct questions ; namely, the preliminary 
 question, whether, in accordance with the phenomena which 
 the Moon actually presents, it is at all possible for her to 
 rotate ; and, if so, under what general suppositions ? the main 
 question, whether, under the suppositions resulting from the 
 preceding inquiry, there is a diameter in the lunar globe in- 
 variably directed towards the same point in space ? and the 
 concluding question, if such be not the case, which is the true 
 explanation of the phenomena of the Moon's libration ? 
 
 such elucidation than to attempt to statements as this, and the " 23 
 
 pooh-pooh the question ;" and in his hours 56 minutes," relating to the 
 
 very next letter (see note 71), by period of rotation of the Peak of 
 
 way of such a rational supply of Teneriffe (compare note 33), would 
 
 desirable information, virtually does convey the impression that he may 
 
 " pooh-pooh the question," in " dis- be relied on for correctness to the 
 
 posing M of it as " demonstratively very fraction of a second, is liable 
 
 established " by his mere dictum, in- to commit such arithmetical errors, 
 
 eluding an essential error. A late crowded together in one single page, 
 
 Professor of Astronomy, who can as we have had occasion to point 
 
 ascribe the Moon's libration in lati- out in note 62, and whose principles 
 
 tude solely to the inclination of her of motion lead to the most extrava- 
 
 axis of rotation to the ecliptic at an gant consequences, is little justified 
 
 angle of 1 30' 10" 8 ; who, though to speak in so authoritative a tone, 
 he evidently, by such numerical 
 
ON THE MOON'S ROTATION. 95 
 
 43. There are three principal axes, about which the Moon 
 (or any heavenly body) may be supposed to rotate. A common 
 double-cross, Fig. 20, 
 will represent them, 
 as to their respective 
 position relative to 
 each other. They are 
 those of a system of 
 three lines having 
 one common point of 
 intersection, and at 
 right angles to each 
 other. The first axis 
 
 a a' shall be that diameter of the lunar globe M, which co- 
 incides with a straight line, produced, joining the centres of 
 the Moon and the Earth E. The second axis b b' will then be 
 the Moon's diameter, at right angles to the former, but situ- 
 ated in the same plane. The third axis c c is the lunar 
 diameter, at right angles to both the two other diameters, and, 
 consequently, perpendicular to their common plane, as the 
 axis b b' is perpendicular to the common plane of the axes 
 a a and c c, and the axis a a perpendicular to the common 
 plane of the axes b b' and c c. Whichever diameter we may 
 suppose to be the axis of rotation, rotation can only take place 
 in a direction either towards the right or towards the left in 
 facing the axis of rotation at right angles to it. Thus, about 
 the axis a a', rotation must take place (assuming the Moon to 
 rotate) in the direction either of c b' or c b ; about the axis 
 b b'y in the direction either of c a' or c a ; about the axis c c in 
 the direction either of b' a b or b' a b. 
 
 We have before remarked ( 23), that the plane of the 
 Moon's path round the Earth, in which lie the axes a a' and 
 b b', and perpendicular to which is the axis c c', does not co- 
 incide with the plane of the ecliptic, but is inclined to the 
 latter at a mean angle of 5 8' 48". The consequence is that a 
 
96 ON THE MOON'S ROTATION. 
 
 straight line joining the centres of the Earth and the Moon 
 coincides with the plane of the ecliptic only at the moments 
 when the Moon is in one of her nodes ; its deviation from it 
 increasing or decreasing as the Moon approaches or recedes 
 from her greatest northern or southern latitude; the Moon's 
 " latitude" in fact, being identical with this deviation, and the 
 Earth always revolving in the plane of the ecliptic her own 
 path this somewhat complicates the phenomena of the Moon's 
 assumed rotation, as seen from the Earth, and which we purpose 
 to describe. But they are further complicated by the direction 
 of the axis of rotation, which may be constant with regard to 
 the Earth, or with regard to some fixed point in space, or with 
 regard to neither ; that is to say, it may have a slow motion of 
 its own, as the terrestrial axis of rotation has. Now, our de- 
 scription cannot follow the Moon's course step by step ; nor 
 have regard to every intermediate station of the possible di- 
 rection of the Moon's axis of rotation. We will, therefore, 
 consider the phenomena which the Moon would present, as 
 referred to her three principal axes of assumed rotation in the 
 course of one revolution round the Earth, and AS SEEN FROM 
 THE EARTH'S CENTRE, under the following suppositions : A the 
 Moon's non-rotation ; B the Moon's rotation : a the axis of 
 assumed rotation being invariable, as to direction, with regard 
 to the Earth's centre; b, its being so, with regard to some 
 fixed point in space : 1. the Moon being in one of her nodes ; 
 2. the Moon being at her greatest northern, and 3. at her 
 greatest southern latitude. 
 
 44. Let us take the first axis a a'. We will assume its di- 
 rection to be constant with regard to the Earth ; and the Moon 
 to be in one of her nodes. Whether she rotate or not, our 
 axis will mainly coincide with the straight line, produced, 
 joining the centres of the Earth and the Moon ; but not exactly. 
 It will, with regard to the latter, be subject to certain apparent 
 oscillations, proportionate to, and arising from the eccentricity 
 of the lunar orbit. Let, in Fig. 21, the ellipse represent the 
 
ON THE MOON'S ROTATION. 97 
 
 curve upon which the Moon performs her revolution round the 
 
 FIG. 21. 
 
 Earth; c its centre; d" d its major axis; d' eT'its minor 
 axis ; E one of its foci, in which the Earth is placed ; M 
 the Moon; a b c her opposite hemisphere; d b that lunar 
 diameter which in M shall be perpendicular to a tangent to 
 its orbit at the point of contact, and therefore will be 
 always so ( 13), as its position is not affected by the 
 Moon's rotation upon the axis a, this axis being identical with 
 the very diameter in question. Then, when the Moon is in M 
 and M" or in her apogee and perigee, the diameter d b=d" b will 
 perfectly coincide with the line, produced, joining the centres of 
 the Earth and the Moon, and coinciding on its part with the 
 major axis of the lunar orbit. Consequently, there will be no 
 apparent oscillation, i. e. we shall, in those two points of her 
 orbit, see the same identical hemisphere of our satellite directed 
 towards us. But when she is in the intermediate stations M' 
 and M"', or when the lunar diameter d b coincides with the 
 minor axis of the Moon's orbit, produced, it will be different; 
 
 H 
 
98 ON THE MOON'S ROTATION. 
 
 for the line joining the centres of the Moon and the Earth de- 
 viates from the latter by a certain angle, determined by the 
 semi-axis itself and the lunar eccentricity. Thus, when the 
 Moon has arrived in M', this angle E c subtends the arc 
 e d' of the Moon's surface ; in other words, our line of vision, 
 according to our assumption ( 43), coinciding with the line 
 joining the centres of the Moon and the Earth : we shall now 
 see a crescent of the eastern border of the opposite lunar hemi- 
 sphere, corresponding to the arc e d', which we did not see 
 when the Moon was in M ; and a corresponding crescent of her 
 western border, we then saw, will now have receded from our 
 view. This change of surface, presented to us by the Moon, 
 having taken place gradually from M to M', until in M' it reached 
 its maximum ; it will, during the Moon's revolution from M' to 
 M", as gradually take place in the opposite sense, until in M" it 
 reaches its minimum, when we shall see again exactly the same 
 hemisphere we saw when the Moon was in M. But during her 
 progress from M'' to M'", it will once more gradually increase 
 until it reaches its maximum in M"', when a crescent of the 
 western border of the opposite lunar hemisphere, corresponding 
 to the arc e d'", which was previously invisible to us, will have 
 appeared into view, whilst a corresponding crescent of the 
 Moon's eastern border, previously visible, has receded from our 
 sight. In the course of her return from M'" to M, the maximum 
 of the change of surface, presented to us by the Moon, will again 
 gradually decrease until in M it is restored to its minimum ; 
 and thus the Moon, during each of her revolutions round the 
 Earth, will appear to oscillate about her centre, from her apogee 
 to the point of intersection with the minor axis of her orbit 
 towards the west, from thence towards her perigee towards the 
 east, in the same direction from her perigee to the other point 
 of intersection with the minor axis of her orbit, and from thence 
 to her apogee again towards the west. 
 
 45. We have here supposed the Moon to revolve in one of 
 her nodes, or in the plane of the ecliptic. We will now sup- 
 
ON THE MOON'S ROTATION. 99 
 
 pose her to be at 
 her greatest north- 
 ern or southern la- 
 titude. The phe- 
 nomena just de- 
 scribed will be 
 affected only in 
 a trifling degree, 
 consequent on this 
 assumption ; but 
 other and similar 
 phenomena will 
 present them- 
 selves. The Moon, 
 with regard to the 
 line joining her 
 centre and that of 
 the Earth, will be 
 subject to certain 
 apparent oscilla- 
 tions, proportion- 
 ate to, and arising 
 from, her latitude, 
 and the direction 
 of which will be at 
 right angles to the 
 direction of her 
 
 apparent oscilla- 
 
 FIG. 22. 
 tions, arising trom 
 
 the eccentricity of her orbit. In Fig. 22 let E represent the 
 Earth, M and M" the Moon in her ascending and descending- 
 nodes, M' the Moon at her greatest northern, and M"' at her 
 greatest southern latitude ; a c her axis of assumed rotation 
 (a a 43) ; b c d the Moon's opposite hemisphere ; e e' her 
 orbit ; a a the ecliptic. Then a c, according to the law of 
 
 H 2 
 
100 ON THE MOON'S ROTATION. 
 
 inertia ( 11), will always maintain the same position with 
 regard to itself. Hence, if . any diameter in the lunar 
 globe be inclined to the ecliptic at a certain angle, that 
 angle will not vary, and, consequently, a' c' and a" c" will 
 be parallel to a c and a c". When the Moon is in M, we 
 shall see the hemisphere bad, and our line of vision, directed 
 towards the Moon's centre, will pass through the point a of 
 the lunar surface. But when the Moon has reached her 
 greatest northern latitude in M' this will no longer be the case. 
 Our visual ray, directed towards her centre, will pass, not 
 through the point a'=a, but through a point e below the latter, 
 and we shall see the hemisphere g e f. In other words, we 
 shall now see a crescent of the southern border of the opposite 
 hemisphere, corresponding to the arc d f, which we did not see 
 when .the Moon was in M; and a similar crescent of her 
 northern border, corresponding to the arc g b, which we then 
 saw, will have receded from our view. A similar change of 
 surface, presented to us by the Moon, will take place, but in 
 the opposite sense, during her progress from M' to M" ; when 
 we shall see again exactly the same hemisphere b" a' d"= bad, 
 we saw in M ; our line of vision, directed towards the Moon's 
 centre, passing through the same point a of her surface. When 
 the Moon has reached her greatest southern latitude in M'", this 
 will once more be different. The line in question will now in- 
 tersect a point & over a, and we shall see the hemisphere 
 fe'g r . In other words, we shall now see a crescent of the 
 northern border of the opposite hemisphere, corresponding to 
 the arc b'" f, which we did not previously see, whilst a similar 
 crescent of her southern border, corresponding to the arc d'" g f , 
 previously visible to us, has receded from our view. A similar 
 change of surface presented to us by the Moon will, in the 
 opposite sense, take place as she completes her revolution in 
 returning to M ; when, once again, we shall see exactly the 
 same hemisphere bad. And thus, as all these changes take 
 place gradually, the Moon will, in the course of each revolution 
 round the Earth, appear to oscillate about her centre from her 
 
ON THE MOON'S ROTATION. 101 
 
 ascending node, when the amount of this apparent oscillation 
 is at its minimum or=0, to her greatest northern 'lilftud'e, wheu 
 it reaches its maximum, towards the south; from her ^gre^te^t. , , , 
 northern latitude to her descending node, when ii; is agaiii r&>-> 
 duced to its minimum, towards the north; from her descending 
 node to her greatest southern latitude, when the apparent os- 
 cillation a second time reaches its maximum, towards the north; 
 and from her greatest southern latitude to her ascending node, 
 when it is restored to its minimum, again towards the south. 
 
 46. As the line of the Moon's apsides, or the line joining her 
 apogee and her perigee, has a direct movement, performing a 
 complete circuit in 8f years, and the line of the nodes has a 
 retrograde movement, performing an entire revolution in 18f 
 years ( 23), the two kinds of apparent oscillation to which the 
 Moon is subject, and described in the two preceding para- 
 graphs, will perform complete revolutions with regard to each 
 other, their respective maxima and minima coinciding at the 
 
 end of periods of =2 ||| or very nearly every 
 
 three years. In the case of the Moon's non-rotation, the cor- 
 responding changes of surface presented towards the Earth 
 will be readily perceptible through the telescope, without any 
 measurement ; but in case of her rotation it would be difficult 
 to discern them, except by means of angular measurement. 
 
 47. Independently of these apparent oscillations, the Moon 
 under the suppositions stated at the beginning of 44, and 
 whether she rotate or not, will always present the same 
 hemisphere towards the Earth, and we shall never see the 
 opposite hemisphere. If she do not rotate, every feature of 
 her visible surface will maintain the same relative position, and 
 appear stationary to us ; but if she rotate, every spot on that sur- 
 face will, in the course of each period of rotation, whatever tha^ 
 period be, seem to describe a circle round the pole of the axis 
 of rotation, which is directed towards the Earth and is situated 
 in the centre of the Moon's visible disk; so that all the 
 
102 
 
 ON THE MOONS ROTATION. 
 
 circles thus described will be concentric, as shown in Fig. 23. 
 : t ' 48, If we suppose the axis a a' 
 
 of the Moon, instead of being con- 
 stant as to direction with regard to 
 the Earth, to be so with regard to 
 some fixed point in space, the ap- 
 parent oscillation, described in 
 44 and 45 will take place all the 
 same ; but if the Moon do not ro- 
 tate, the former alone, and if she 
 do rotate, the latter as well, will 
 be difficult of perception save by 
 
 horary measurement.? 3 For in the case of rotation, not only 
 
 FIG. 
 
 73 The measure of motion is TIME. 
 In employing it in the instance re- 
 ferred to in the text, we shall find 
 that any point of the lunar surface, 
 supposing the Moon's rotatory move- 
 ment to be directed towards the 
 centre of the Earth, and to be of uni- 
 form velocity, will appear later into 
 view when the Moon is at her 
 greatest northern latitude, and earlier 
 when she is at her greatest southern 
 latitude, than it did when she was 
 in her ascending and descending 
 nodes. The reason is, that our 
 visual ray strikes the moon's surface 
 in different points, according to her 
 different positions. Let us revert to 
 Fig. 22, and erect the line p e per- 
 pendicular to the ecliptic and falling 
 into the centre of the Earth. When 
 the Moon is in M and M", the line 
 of vision, passing from the Earth's 
 centre to the upper surface of the 
 Moon, will touch it at the points b 
 and b" respectively, forming with 
 the perpendicular p an angle of 
 90 less the Moon's mean apparent 
 semi-diameter = 15' 32", or an angle 
 
 of about 89 44' 28". But when the 
 Moon is in M', her orbit being in- 
 clined to the ecliptic at a mean 
 angle of 5 8' 48," the visual ray will 
 with the perpendicular p form an 
 angle of only 89 44' 28"-5 8' 48"= 
 84 35' 40", and touch the Moon's 
 upper surface at a point g, nearer to 
 the Earth's centre than b'=b, the 
 line b' d' b d being parallel to p e ; 
 so that, in virtue of the Moon's ro- 
 tation, the point b' will have to de- 
 scribe the arc b' g before it becomes 
 visible to an eye at the centre of the 
 Earth ; and, consequently, will be 
 seen, now that the Moon is in M', so 
 much later than it was when the 
 Moon was in M. On the contrary, 
 when the Moon is in M'", the visual 
 ray will, with the perpendicular 
 p, form an angle of 89 44' 28"+ 
 58'48"=94 53' 16", and touch the 
 Moon's upper surface in the point 
 f, more distant from the Earth's 
 centre than b'"=b; so that, in con- 
 sequence of the Moon's rotatory 
 motion, the point b'" will, by a time 
 corresponding to the arc f" b", ap- 
 
ON THE MOON'S ROTATION. 103 
 
 will all the spots on the Moon's surface be in constant 
 motion, but as that entire surface will in succession be turned 
 towards the Earth, the concentric lines (Fig. 23) will gradually 
 change into perpendicular ones, 
 as shown in Fig. 24 ; as gradu- 
 ally change again into concen- 
 tric lines ; once more into per- 
 pendicular ones ; and once 
 again, on the Moon's com- 
 pleting a revolution, into con- 
 centric ones. In case of non- 
 rotation, on the contrary, the 
 
 spots on the Moon's surface will 
 
 i -i i FIG 24 
 
 constantly preserve their rela- 
 tive positions and appear stationary on the lunar disk ; the 
 Moon herself, however, presenting in succession every portion 
 of her surface towards the Earth (see 39 and Fig. 18). 
 
 Under any circumstances, therefore, the Moon's rotation on 
 the axis a a is utterly irreconcileable with the phenomena 
 which her visible disk actually presents, and, consequently, im- 
 possible. 
 
 49. In substituting the second axis b b' for the axis a a, 
 the resulting phenomena will, in almost every particular, cor- 
 respond to those described in 4448. The only difference 
 will be, that if we suppose the Moon to rotate, and the axis of 
 rotation to be constant, as to direction, with regard to the 
 Earth, the spots on the Moon's surface will appear to describe 
 perpendicular, instead of concentric, lines across her visible 
 disk ; and, if we suppose the Moon to rotate, and the axis of 
 rotation to be constantly directed towards the same point in 
 
 pear sooner into view ; and, conse- is at her greatest latitude, gradually 
 quently, now that the Moon is in decreasing as she progresses towards 
 M'", will he seen so much earlier one of her nodes, in order to increase 
 than it was when the Moon was in again in the same gradual manner 
 M". The difference in time will as she approaches towards the great- 
 reach its maximum when the Moon est northern or southern latitude. 
 
104 ON THE MOON'S ROTATION. 
 
 space, that the change of the lines in question will take place 
 in a different order, returning after each revolution, to the per- 
 pendicular, instead of the concentric form. And the same will 
 in every material sense, be the case with regard to any axis 
 situated in the same plane as the axes a a' and b b', and what- 
 ever be its intermediate angle, relative to the line joining the 
 centres of the Earth and the Moon. Consequently, the Moon's 
 rotation on the axis b b', as well as on that of a a or any 
 intermediate axis in the same plane, is utterly irreconcileable 
 with the phenomena, which her visible disk actually presents, 
 and, therefore, impossible. 
 
 50. The same we find, at first sight, to be the case, if we 
 suppose the third axis c c to be the axis of the Moon's rota- 
 tion ; for, the angle formed by that axis and the line, joining the 
 centres of the Moon and the Earth, being,- apart from slight 
 periodic oscillations ( 45, 45), a constant one, rotation can- 
 not take place, without the Moon's presenting to us in succes- 
 sion every portion of her surface ( 14). A second and cor- 
 responding reason is, that under the supposition in question 
 the poles of the axis must, in virtue of it, perform a complete 
 circuit with regard to space, in the course of each revolution of 
 the Moon round the Earth. That circuit, it is true, would not 
 be a large but a very small circle of the heavens about the 
 poles of the ecliptic, corresponding to the lunar orbit ; and for 
 this reason it might possibly be argued, that the entire circle 
 being but a vanishing point in space, with regard to the distance 
 of the fixed stars, it ought to be considered only as such, and 
 that consequently, the direction of the axis c c' must be, of ne- 
 cessity, constant with regard to space. But such an argument 
 would be a very erroneous one, inasmuch as we are to consider 
 assumed realities actually visible from the centre of the Earth, 
 not imagined appearances, altogether invisible from even the 
 nearest of the fixed stars (compare 31) ; and, at the best, its 
 effect would be to convert the centre of the Earth into the 
 Moon's axis of rotation, and her orbit into the circumference of 
 her solid globe (compare 36). The axis c c, therefore, 
 
ON THE MOON'S ROTATION. 105 
 
 looked upon as the Moon's axis of rotation, cannot be supposed 
 to maintain an invariable direction with regard to space. In 
 every other sense the consequences resulting from that sup- 
 position, will be essentially identical with those previously 
 described, only that, if we assume the Moon to rotate, the 
 spots on her surface will appear to move across her disk in a 
 horizontal, instead of in a perpendicular or in a concentric 
 direction. 
 
 51. Only one more supposition, if we are to admit the mere 
 possibility of the Moon's rotation on her axis in accordance 
 with the phenomena actually observed, is thus left open to us ; 
 namely, that this axis, neither coinciding with, nor perpendicular 
 to, the ecliptic, be inclined to it at a certain angle ; moreover, 
 as rotation in our case depends altogether on that angle being 
 variable with regard to the line joining the centres of the Earth 
 and the Moon ( 14), that the axis of assumed rotation be con- 
 stantly directed towards the same point in space ; and lastly, 
 that the two periods of the Moon's rotation on her axis and of 
 her revolution round the Earth, be of exactly equal duration. 
 And under this threefold supposition, it is evident that rotation 
 must take place if the Moon, in a general sense, is always to 
 present the same face, without any material periodic alteration 
 of its features, towards the Earth ; and, this periodic alteration 
 of her features being actually confined within very narrow limits, 
 that the angle, at which we have to suppose the Moon's axis of 
 assumed rotation to be inclined to the ecliptic, cannot but be a 
 very small one. 
 
 The reader may convince himself of the truth of our pre- 
 ceding remarks by a very homely illustration. Let him take up 
 his stick or his umbrella, and, in his natural way, walk round 
 an oval table ; he will then, naturally, keep his side-face con- 
 stantly turned nearly towards the centre of the table, as the 
 Moon, in revolving round the Earth, naturally always presents 
 her side-face nearly towards the centre of her orbit. If the 
 reader hold his stick, in any way he please, steady with regard 
 to himself, so that both its ends perform a circuit corresponding 
 
106 ON THE MOON'S ROTATION. 
 
 in direction to his own, f. i. if he hold it perpendicular to the 
 plane of the table, so that one end will describe an ellipse with 
 regard to the floor, the other with' regard to the ceiling ; he will 
 be able to continue his natural mode of walking, at the same 
 time that he continues to present his side face invariably to- 
 wards nearly the centre of the table. But if he point the stick 
 f. i. towards one of the upper corners of the room, and, in con- 
 tinuing to walk, keep it pointed in the same direction ; he will 
 then find that he will either have to walk in an unnatural man- 
 ner round the table, half backwards, half forwards, and so as 
 to present every part of his face and head towards the centre of 
 the table, or else, in order to keep the same side-face turned 
 towards it, that he will be obliged to turn his head round the 
 stick. The reader's stick represents the Moon's axis, directed 
 towards some fixed spot in the heavens; and the experiment 
 illustrates the necessity of the Moon's rotation on her axis, 
 under the conditions, stated at the commencement of this para- 
 graph. 
 
 52. Having thus answered our preliminary inquiry, as to the 
 general suppositions, under which, in accordance with the 
 phenomena the Moon actually presents, it is possible for her 
 to rotate ; we will proceed to consider the main question, in- 
 volving the solution of our problem, namely : whether there is 
 in the lunar globe a diameter, which being inclined at a small 
 angle to the ecliptic, is invariably directed towards the same 
 point in space. To propose this question is to assume that, 
 since the times of Newton, his theory regarding the Moon's 
 rotation has gained consistency by subsequent researches 
 ( 26) ; and already in an early part of this inquiry ( 3), we 
 have intimated that such is the case. 
 
 The first development to that theory was given by one of 
 Newton's own contemporaries, and one of the greatest Astro- 
 nomers of his time, Dominic Cassini. He discovered that the 
 inclination of the lunar equator to the ecliptic, which he fixed 
 at 2 40', is nearly constant ; that the nodes of the equator 
 always coincide with the opposite nodes of the lunar orbit ; 
 
ON THE MOON'S ROTATION. 107 
 
 and that the Moon's axis of rotation is, consequently, not 
 always directed towards the same point in space, but has a 
 comparatively slow motion, equal to that of her nodes, i. e. 
 performing a complete circuit in about 18| years ( 46), which 
 is also the period of the variation of the lunar equator with 
 regard to the ecliptic. The period of rotation he found to be 
 exactly equal to the Moon's siderial revolution. 74 
 
 More than half a century elapsed before the researches of 
 Cassini were further pursued. At that period the subject of 
 the Moon's libration began to engage the attention of several 
 eminent Astronomers and Mathematicians. Tobias Mayer, of 
 Gottingen, undertook a series of careful observations of the 
 position of the lunar mountains " Manilius," " Dionysius," and 
 "Censorinus," and from these observations, by a new and more 
 exact method of calculation, first employed by Euler, found the 
 inclination of the lunar axis of rotation to the ecliptic to be 
 only 1 29', but the nodes of the lunar equator and orbit, in 
 accordance with Cassini's theory, nearly to coincide. Soon 
 afterwards Lalande arrived at results confirming the latter point. 
 The inclination of the lunar axis of rotation to the ecliptic, 
 however, he found to be 1 43'. 75 . 
 
 74 Gleich alien uns naher bekann- tions, pensa quo pour avoir un juste 
 
 ten Welt-Korpern bewegt sich der milieu, il fallal t supposer 1'obliquite 
 
 Mond um eine durch seinen Mittel- =129', quoique Cassini 1' eutdonnee 
 
 punkt gehende Axe. Theorie und de 240' . . . Enfin je dois ajouter 
 
 Erfahrung bestatigen das nachfol- qu' ayant observe les latitudes de 
 
 gende, zuerst von Cassini aufgestellte Manilius, avec un excellent micro- 
 
 Gesetz : Die Neigung des Monda- metre, au mois d' Octobre 1763, ou 
 
 quators gegen die Ebene der Erd- la Lune se trouvait a lafois dans ses 
 
 bahn ist constant ; seine Knoten noeuds, dans ses apsides et dans ses 
 
 fallen mit den entgegengesetzten syzygies, j'ai trouve 1' inclinaison de 
 
 Knoten der Mondsbahn stets in 1' equateur lunaire de 1 43' par des 
 
 dieselben Punkte, und die Mond- observations qui me paraissent en- 
 
 kugel dreht sich wahrend eines side- core plus sures que celles de Mayer." 
 
 rischen Umlaufs vollig gleichformig LALANDE, Astronomic, torn. III. 
 
 einmal um ihre Axe." BEER und p. 326. 
 
 MIDLER, Selenograpliie, S. 1.1. Mayer's excellent paper on the 
 
 74 " Mayer, en pesant le degre de subject of his observations is con- 
 precision de ses differentes observa- tained in the " Kosmographische 
 
108 ON THE MOON'S ROTATION. 
 
 At this time d'Alembert's investigation of the Moon's physical 
 libration, undertaken spme time previously, having proved un- 
 satisfactory, the French Academy of Sciences offered a prize 
 for a complete theory of the Moon's libration. The prize 
 (1764) was awarded to the Memoir of Lagrange. He supposed, 
 with Newton, the Moon to have been originally in a fluid state ; 
 he confirmed the theory of the latter, regarding the elongation 
 of the Moon's axis directed towards the Earth, as the physical 
 cause of our continually seeing the same face of the Moon ; he 
 thus, under the supposition that the periods of the Moon's 
 rotation and revolution nearly coincided at the beginning, 
 arrived at the existence of a slight physical libration of the 
 lunar globe in longitude, and a corresponding slight periodic in- 
 equality in the velocity of the Moon's rotation. The actually 
 observed phenomena of the coincidence of the nodes of the 
 lunar orbit and equator he failed to deduce from analysis ; but in 
 a subsequent memoir he showed that, on the stated coincidence 
 being assumed to have taken place at the beginning, it would, 
 in virtue of the law of gravitation, continue to the end. Thence it 
 would follow, and his analysis further showed, that the Moon's 
 rotation, though independent of the periodical, participates in 
 the secular, inequalities of her revolution; in consequence of 
 which the opposite hemisphere of our satellite will ever be 
 hidden from our view, as it is now and always has been. 76 
 
 Nachrichten und Sammlungen auf found this inclination to be t45' 
 
 das Jahr 1748." Niirnberg 1750, and that Lalande obtained 29' for 
 
 4. S. 52 183. From twenty-seven it (History of Physical Astronomy, 
 
 observations of " Manilius," he ob- p. 73), is altogether very inaccurate 
 
 tained 130' for the inclination of the in recording the history of the theory 
 
 lunar axis of rotation ; from nine ob- of libration. 
 
 servations of " Dionysius," nearly the TO " En employant toutes les atten- 
 
 same ; but from twelve observations tions que vous venons de recom- 
 
 of " Censorinus," a somewhat differ- mander, et les appliquant a un grand 
 
 ing result, which induced him to fix nombre d' observations fort exactes, 
 
 the inclination at 129' (compare on confirme avec la plus grande cer- 
 
 loc. cit. p. 317 325). Mr. Grant, titude la Constance de 1'inclinaison 
 
 who erroneously states that Mayer de 1* equateur lunaire sur 1' eeliptique 
 
ON THE MOON'S ROTATION. 
 
 109 
 
 These results of the analytical researches of Lagrange were 
 fully confirmed by those of Laplace and Poissoii. The former 
 found that the planes passing through the Moon's orbit and 
 equator, and a third plane passing through her centre parallel 
 to the ecliptic, having nearly always the same lines of inter- 
 section, the secular variations of the ecliptic will neither alter 
 the coincidence of the nodes of those planes, nor their mean 
 inclination, which the attraction of the Earth will maintain 
 
 et le parallelisme de ses noeuds avec 
 les noeuds moyens de 1'orbite. Ces 
 resultats peuvent etre represented par 
 la construction suivante: par le 
 centre delalune concevez trois plans, 
 1'un representant 1' equateur de la 
 lune, le second le plan moyen de son 
 orbite, le troisieme parallele a 1' 
 ecliptique. Ce dernier qui sera 
 toujours compris entre les deux 
 autres, passera par leur commune 
 section. II fera, avec le premier, un 
 angle de 1 67 ; c'est 1' inclinaison 
 de 1' equateur lunaire sur 1' ecliptique; 
 avecle troisieme, un angle de 5, 7222 ; 
 c'est 1'inclinaison moyenne de 1'orbite 
 sur le meme plan. Ces resultats, 
 qui forment une des plus belles 
 decouvertes de 1' astronomic moderne, 
 ont ete trouves pour la premiere fois 
 par Dominique Cassini; et Ton doit 
 a M. Lagrange d' avoir montre qu'ils 
 sont une consequence de 1' attraction 
 que la terre exerce sur le spheroide 
 lunaire." BTOT, Traite elem. d' 
 Astron. physique, tome II. p. 411,412. 
 (Compare LAPLACE, Exposition du 
 Systeme du monde, tome 1. p. 58. 
 
 " Nous avons vu qu'il existe dans 
 le moyen mouvement de la lune, des 
 variations seculaires qui doivent y 
 produire a la longue des change- 
 ments tres considerables. Si le 
 
 mouvement de rotation de la Lune 
 restait toujours le meme, et ne parti- 
 cipait point a ces inegalites, il ces- 
 serait bientot de contre-balancer 
 exactement le mouvement de revo- 
 lution, et la lime, se detournant peu- 
 a-peu pour nous, on devrait, par le 
 suite des siecles decouvrir succes- 
 sivement tous les points de sa surface ; 
 mais la theorie a prevu d'avance que 
 cela ne doit point arriver. Le mouve- 
 ment de rotation de la lune, inde- 
 pendant des inegalites periodiques 
 qui affectent le mouvement de revo- 
 lution, est assujetti aux memes ill- 
 egal ites seculaires. Ces mouvements 
 s'altereront done peu-a-peu par les 
 memes periodes en se contre-balan- 
 9ant toujours, et la face de la lune 
 opposee a celle qu'elle nous presente* 
 nous sera eternellement cachee. Par 
 uiie suite des memes lois, la con- 
 stante d'inclinaison de 1'equateur 
 lunaire sur 1' ecliptique et le paralle- 
 lisme de ses noeuds avec les noeuds 
 moyens de 1'orbite, subsisterons tou- 
 jours malgre le developpement des 
 inegalites seculaires. Ces beaux 
 theoremes resultent encore de 1'ana- 
 lyse de M. Lagrange." BIOT, Traite 
 elem. d' Astron. physique, Tome. II. 
 p. 412. Compare GRANT'S History 
 of Physical Astronomy, p. 73 75. 
 
110 ON THE MOON'S ROTATION. 
 
 constant. But, varied in form, independent in character, and 
 valuable in an astronomical point of view, as all these results 
 appear to be, they are not so. They are interesting and beautiful 
 illustrations of the same process of analytical reasoning ; no 
 more. What truths they contain, may be traced to the corre- 
 sponding empirical elements thrown into that process; but its 
 foundation being an erroneous hypothesis, the process itself 
 cannot possibly lead to establish, or serve to confirm, empirical 
 facts. Hence the illustrious author of the Mecanique celeste 
 is forced himself to acknowledge, that the results obtained by 
 him in harmony with actual observation are irreconcileable 
 with the hypothesis upon which his labours are based, and to 
 ascribe the phenomena in contradiction with his theory to "the 
 " high mountains which are observed on the Moon's surface and, 
 " which, doubtlessly, have exercised a considerable influence in 
 " their regard ; the more so as the Moon is but slightly flattened 
 " at her poles, and her mass is a very small one." 7 ? 
 
 ! " 7a " Le phenomene singulier de (Compare BIOT, Traite elem. d'Astr. 
 la coincidence des noeuds le 1'equa- Phys., tome II., p. 412.) 
 teur de la lune avec ceux de son 77 b " Observons ici que les phe- 
 orbite, est encore une suite de 1'at- nomenes precedens ne peuvent pas 
 traction terreste. C'est ce que La- subsister avec 1'hypothese dans la- 
 grange a fait voir le premier, par une quelle la lune primitivement fluids 
 tres belle analyse qui 1'a conduit a et formee de couches de densites 
 1'explication complete de tous les quelconques, aurait pris la figure 
 mouvemens observes dans le sphe- qui convient a leur equilibre : ils in- 
 ro'ide lunaire. Les plans de 1'equa- diquent entre les axes du spheroide 
 teur et de 1'orbite de la lune, et le lunaire, de plus grandes differences 
 plan mene* par son centre parallele- que celles qui ont lieu dans cette 
 ment a 1'ecliptique, ont toujours a hypothese. Les hautes montagnes 
 fort peu pres la meme intersection: que Ton observe a la surface de la 
 j'ai reconnu que les mouvemens lune, ont sans doute, sur ces phe. 
 seculaires de 1'ecliptique n'alterent nomenes, une influence tres sensible 
 ni la coincidence des noeuds de ces et d'autant plus grande, que son 
 trois plans, ni leur inclinaison mo- aplatissement est fort petit, et sa 
 yenne que 1'attraction de la terre masse peu considerable." LAPLACE, 
 maintient constamment la meme." Exposition du Systbne du Monde, 
 LAPLACE, Exposition du Systems du tome II., p. 250. (Compare NICOLLET, 
 Monde, tome II., pp. 249, 260. Memoire, p. 279.) 
 
ON THE MOON'S ROTATION. Ill 
 
 Of a different nature are the latest researches of Bouvard 
 and Nicollet, in continuation of those of Cassini, Mayer, and 
 Lalande, being founded, not upon speculative theories, but 
 upon a long series of the most accurate and careful observa- 
 tions, undertaken at the request of Laplace. ( 7 7) They fully 
 corroborate the result obtained by the latter, namely, that the 
 theory of the primitively fluid state of the Moon is irrecon- 
 cileable with the deductions, drawn by analysis from those 
 actual observations ; and they further confirm the accuracy of 
 Mayer's calculations relative to the inclination of the lunar 
 axis of rotation to the ecliptic, for which Nicollet found the 
 angle to be about 1 30'.? 8 These calculations, the French 
 Mathematician remarks, prove to evidence the Moon's rotation ; 
 and the ground he assigns is, that the selenocentric longitudes 
 and latitudes of all the various spots on the Moon's surface, as 
 appears from repeated observations, differ between themselves, 
 
 ? 8 The following are the exact and the remaining 50 by himself, 
 results of Nicollet, compared with The first column (I.) supposes the 
 those of Mayer. They are founded Moon's semi-diameter to be that of 
 on 174 observations of " Manilius," Burckhardt's Tables ; the second 
 124 of which were made by Bouvard, column (II.) supposes it to be 2" less. 
 
 Mayer. Nicollet 
 
 Distance of the Moon's node I II 
 
 from that of her equator . 3 45' 1<> 17' 55" 9' 19" 
 Inclination of the equator to 
 
 the ecliptic 1 29' 1 30' 6" 1 28' 42" 
 
 Selenographic Latitude of 
 
 "Manilius" 14 33' 14 23' 2" 14 26' 48" 
 
 Selenographic Longitude of 
 
 the same 9 2' 8 46' 47" 8 48' 54" 
 
 Inequality of the libration in 
 
 longitude 4' 45", 65 
 
 Correction of the Moon's semi- 
 diameter 0' 0", 66360 
 
 According to this result, the dence in his calculations to war- 
 Moon's semi-diameter, as adopted rant such an alteration. NICOLLET, 
 by Burckhardt, should be dimi- Memoire sur la libr. de la lune, pp. 
 nished by about 1"; but Nicollet 259,271272. 
 himself places no sufficient confi- 
 
112 
 
 and vary according to time. Hence, he concludes, the Moon 
 must rotate on an axis, inclined ; to the plane of the ecliptic; 
 and since the latitudes, compared with each other, present but 
 slight changes, the axis of rotation can be inclined to the 
 ecliptic only at a small angle, which so far agrees with the 
 result of our preliminary inquiry ( 51). 
 
 53. Thus, modern Astronomy would seem to have established 
 " the fact " of the Moon's rotation beyond a reasonable doubt ; 
 unless Astronomers, indeed, should have greatly miscalculated 
 or rested their computation upon some fundamental error un- 
 known to them. Under ordinary circumstances, a supposition 
 of this kind might have been unhesitatingly dismissed ; but in 
 reference to the particular subject of our inquiry, we have found 
 the general arguments and illustrations advanced by Astrono- 
 mers in support of their theory to be so utterly untenable, that we 
 may, without presumption, proceed to submit also their special 
 proofs to a closer scrutiny. And this the more so, as there are 
 many apparent contradictions and errors embodied in that 
 theory, whilst some of the very elements, composing it, would 
 seem to be destructive of its truth. Such being the case, the 
 reader will probably first inquire what amount of reliance, then, 
 may be placed on the exactitude of the observations and the 
 analytical calculations to which we have alluded ? 
 
 We need hardly state that the poles of the Moon's axis of 
 assumed rotation are no tangible points. Moreover, from their 
 position, they are so extremely difficult of angular measurement 
 
 T 9 " Nous sommes en etat de mettre de 1'ecliptique. Les latitudes, com- 
 
 en evidence la rotation de la Lune : parees entre elles, ne presentent que 
 
 en calculant tin grand nombre de de petits changements ; 1'axe de ro- 
 
 longitudes et de latitudes seleno- tation differe done peu de celui de 
 
 centriques, d'apres les observations 1'ecliptique, ou, ce qui est la meme 
 
 d'une tache faites a diverses epoques, chose, 1'equateur lunaire est peu in- 
 
 on trouve que ces longitudes et la- cline sur le plan de 1'ecliptique." 
 
 titudes different entre elles et varient NICOLLET, Mem. sur la librat. de la 
 
 avec le temps; la Lune tourne done lune, p. 253. 
 autour d'un axe incline au plan 
 
ON THE MOON'S ROTATION. 
 
 113 
 
 that this measurement is effected in an indirect manner by re- 
 peatedly measuring some well-defined spot on the lunar disk, 
 situated near its centre. Nor is even this unattended with con- 
 siderable difficulties. The apparent diameter of the Moon's 
 disk subtends but the very small mean angle of 31' 3"; and the 
 combined motion of the Earth in her orbit and about her own 
 axis, together with the Moon's motion in her orbit and her 
 varying parallax, render a complicated correction of the actual 
 observations necessary at the very outset. 80 The observation 
 itself refers to the difference of right-ascension and declination 
 of the chosen spot and one of the borders of the lunar disk, a 
 measurement liable to an error of seldom less than 1", and often 
 amounting to 6" or 1" of an arc with regard to each single ob- 
 servation, chiefly owing to the more or less rugged appear- 
 ance of the lunar border, the size and varying apparent form 
 of the spot, and other circumstances. 81 And such an error, 
 
 80 " On sait combien de difficult^ 
 sont propres a ce genre d'observa- 
 tion; 1'arc de 90 du globe lunaire 
 n'est vu de la Terre que sous un 
 angle de 15 a 17 minutes, la moindre 
 erreur dans 1'observation geocen- 
 trique, a une influence considerable 
 sur les determinations rapportees 
 au centre de la Lune; d'une autre 
 part, la tache n'est pas un point 
 mathematique, sa forme est peu 
 reguliere, les moindres nuages ou le 
 trop grand eclat de la Lune aug- 
 mentent les dimcultes de la bien 
 apercevoir ; on la compare aux bords 
 de la Lune, qui sou vent sont ondu- 
 lans, mal termines, e'cbancres meme ; 
 on concoit done que toutes ces cir- 
 con stances sont autant de causes 
 defavorables qui, se portant dans les 
 donnees et se multipliant par les 
 calculs, alterent les resultats que Ton 
 en deduit ; on ne peut done esperer 
 
 d'attenuer et de vaincre tous ces 
 effets, qu'en fondant nos recherches 
 sur un grand nombre d'observations." 
 NICOLLET, Mem. sur lei librat. de 
 la Lune, p. 235. 
 
 81 .... Ces considerations m'ont 
 d'abord inspire quelques defiances 
 sur 1'emploi des deux bords dans les 
 calculs, et je me suis convaincu 
 qu'elles devaient etre prises en con- 
 sideration ; en effet, les distances de 
 la tacbe aux deux bords en ascension 
 droite, ou aux deux bords en de- 
 clinaison, etant mesurees et corrigees 
 des diverses causes qui les alterent, 
 si Ton en fait la somme, elle doit etre 
 egale au diametre des tables calcule 
 pour 1'instant de 1'observation. Or, 
 j'ai fait un grand nombre de ces 
 comparaisons, et j'ai trouve que la 
 difference n'etait pas moindre d'une 
 seconde, et qu'elle s'elevait a 6 ou 7". 
 Si dans ces differences, il faut faire 
 
114 
 
 ON THE MOON'S ROTATION. 
 
 trifling as the reader may think it, assumes a very considerable 
 proportion in the further process of calculation, when the cor- 
 rected observations, in their geocentric form, are converted into 
 selenocentric places or positions of the observed spot on the 
 lunar surface, divided into selenographic parallels (corre- 
 sponding to our geographical parallels) of longitude and lati- 
 tude, as seen from the centre of the Moon ; the mere formulae 
 for the various computations, necessary to determine the Moon's 
 rotation, occupying in Biot's Astronomy nearly six closely 
 printed pages (ii. 423 428). For when we are told f. i. that 
 the libration in longitude can reach a maximum of 7 55' to- 
 wards either side, we are to understand these to be seleno- 
 graphical degrees. A great number of the most careful 
 observations therefore, and their combination by a method of 
 calculation, called the method of equations of condition, can 
 alone insure the least possible amount of error. 82 But when- 
 
 la part de la petite correction que 
 peut encore exiger le diametre de la 
 Lune, on ne peut disconvenir que le 
 reste ne provienne de la difficulte de 
 faire les observations. Pour mettre 
 plus d'uniformite dans les dpnnees, 
 et surtout pour rendre les resultats 
 independans d'une nouvelle cause 
 d'erreur, j'ai done rejete un des 
 bords de la Lune, toutes les fois que 
 j'ai trouve une observation double." 
 NICOLLET, Mem sur la librat. de 
 la Lune, p. 236. 
 
 82 "Pour determiner la position 
 de 1'axe et la duree de la rotation 
 de la lune .... on observe a la 
 machine parallactique, les differences 
 d' ascension droite et de declinaison 
 entre un des bords de la lune et une 
 tache determinee. Comme on con- 
 nait aussi par observation ou par le 
 calcul, le demi-diametre apparent du 
 disk, on deMuit de ces donnees la 
 
 declinaison et 1'ascension droite 
 entre la tache et le centre, et par 
 suite, la difference de longitude et de 
 latitude. Mais avant d'affectuer ces 
 calculs, il faut faire subir aux 
 observations une correction par- 
 ticuliere. D'apres ce que nous 
 avons dit sur 1'usage de la machine 
 parallacttque, on concoit que pour 
 observer la difference des passages 
 ou des declinaisons entre la tache et 
 les bords du disque, on tourne le 
 micrometre de maniere que la tache, 
 dans son mouvement en ascension 
 droite, suive le fil parallele a 1'equa- 
 teur, et alors les autres fils perpen- 
 diculaires a celui-la represented des 
 cercles horaires; mais ces suppo- 
 sitions ne sont pas rigoureusement 
 vraies ; car la lune ayant un mouve- 
 ment propre qui la porte succes- 
 sivement sur differens paralleles, ce 
 mouvement se compose avec celui 
 
ON THE MOON'S ROTATION. 
 
 115 
 
 ever such a calculation is founded on a sufficient number of 
 such observations, as is actually the case with reference to the 
 Moon's presumed rotation, the possible amount of error is 
 reduced to very small dimensions, and the general result may 
 be accepted with the most perfect reliance in its correctness. 
 If, therefore, the result obtained by Mayer, Lalande, and 
 Nicollet be notwithstanding an erroneous one, the error, in all 
 human probability, cannot attach either to the observations or 
 to the analysis, but is rather indicative of there being something 
 wrong in the Theory of Astronomy. 
 
 54. For this reason, it may be desirable before entering into 
 the real merits of the question, to notice a theory which offers 
 a second explanation of the constant direction of the same face 
 of the Moon towards the Earth. It was first given by Newton 
 
 de la rotation diurne du ciel, et, par 
 consequent, sa route apparente est 
 oblique a 1'equateur et aux paralleles, 
 excepte dans les instans ou elle 
 atteint ses plus grandes declinaisons. 
 Le parallel apparent decrit par la 
 tache et sur lequel on regie le mi- 
 crometre, differe done du parallel 
 vrai, et a cause de la rapidite du 
 mouvement propre de la lune, cette 
 difference peut devenir fort sensible. 
 Un autre effet analogue resulte du 
 changement qu' eprouve la parallaxe 
 de hauteur a mesure que la lune 
 s'eleve sur I'horison ; car dans le cas 
 meme ou la lune n'aurait aucun 
 mouvement en declinaison, le seul 
 changement de la parallaxe rendrait 
 sa marche apparente inclinee a 
 1'equateur; il faut done corriger, 
 dans les observations, les effets de 
 ces deux causes, et c'est une chose 
 tres-facile. 
 
 " Les positions geocentriques de 
 la tache etant ainsi connues, on 
 
 les convertit en positions seleno- 
 centriques 
 
 " Pour ne rien omettre de ce qui 
 peut contribuer a 1'exactitude de ces 
 calculs, je ferai remarquer qu' il faut 
 employer une parallaxe differente 
 pour le centre du spheroide lunaire 
 et pour les taches, qui sont a sa 
 surface ; car a cause de la grandeur 
 de la parallaxe de la lune, cette 
 difference est sensible 
 
 "Enfin, afin d'eviter 1'effet des 
 petites erreurs que les observations 
 coinportent toujours, il faudra en 
 calculer un tres-grand nombre, et les 
 combiner par la methode des equa- 
 tions de condition. Ce concours est 
 tres-necessaire, car les erreurs des 
 positions geocentriques s'agrandis- 
 sent dans une enorme proportion, 
 quand on transforme celles-ci en se- 
 lenocentriques," BIOT, Traite elem. 
 cFAstron. physique, tome IL, p. 409 
 41L. 
 
 i 2 
 
116 
 
 ON THE MOON'S ROTATION. 
 
 and adopted by Laplace 83 , Biot 8 *, Madler 85 , Narrien 8fi and 
 the majority of Astronomers. 
 
 83 "II nous reste a expliquer la 
 cause de la libration de la lune, et du 
 mouvement des noeuds de son equa- 
 teur. La lune, en vertu de son 
 mouvement de rotation, est un peu 
 aplatie a ses poles; mais 1'attraction 
 de la terre a du allonger son axe 
 dirige vers cette planete. Si la lune 
 etait homogene, et fluide, elle pren- 
 drait pour elre en equilibre, la forme 
 d'un ellipsoide dont le plus petit axe 
 passerait par les poles de rotation : 
 le plus grand axe serait dirige vers 
 la terre, et dans le plan de 1'equateur 
 lunaire; et 1'axe moyen situe dans 
 le meme plan, serait perpendiculaire 
 aux deux autres - 
 
 " On congoit aisement que si le 
 grand axe de la lune s'ecarte un peu 
 de la direction du rayon vecteur qui 
 joint son centre a celui de la terre, 
 1'attraction terrestre tend a le rame- 
 ner sur ce rayon ; de meme que la 
 pesenteur ramene un pendule, vers 
 la verticale. Si le mouvement de 
 rotation de ce satellite cut etc primi- 
 tivement assez rapide pour vaincre 
 cette tendance ; la duree de sa rota- 
 tion n'aurait pas ete parfaitement 
 egale a la duree de sa revolution, et 
 leur difference nous eut decouvert 
 success! vement tons les points de sa 
 surface. Mais dans 1'origine, les 
 mouvements angulaires de rotation, 
 et de revolution de la, lune ay ant ete 
 peu differens ; la force avec laquelle 
 le grand axe de la lune s'eloignait de 
 son rayon vecteur, n'a pas sum pour 
 surmonter la tendance du meme axe 
 vers ce rayon, due a la pesenteur ter- 
 restre qui, de cette maniere, a rendu 
 
 ces mouvements rigoureusement 
 egaux; et de meTne qu'un pendule 
 ecarte par une tres petite force, de la 
 verticule, y revient sans cesse en 
 faisant de chaque c6te, de petites 
 oscillations; ainsi, le grand axe du 
 speroide lunaire doit osciller de 
 chaque cote du rayon vecteur moyen 
 de son orbite. De la resulte un 
 mouvement de libration dont 1'eten- 
 due depend de la difference primitive 
 des deux mouvements angulaires de 
 rotation et de revolution de la lune. 
 Cette libration est tres petite, puisque 
 les observations ne 1'ont point fait 
 connaitre." LAPLACE, Exposition 
 du Sy steme du Monde, torn. II. p. 247 
 248. 
 
 84 " Les pbenomenes que presente 
 la rotation du spheroide lunaire prou- 
 vent que la lune n'est pas sphe'rique : 
 elle doit avoir la forme d'un el- 
 lipsoide, ayant son plus grand axe 
 constamment tourne vers la terre, 
 et dirige dans le plan de 1'equateur 
 lunaire, son plus petit axe dirige 
 suivant les p61es de rotation ; enfin 
 le troisieme axe perpendiculaire aux 
 deux autres et intermediaire entre 
 eux pour la longueur. Ces beaux re- 
 sultats sont encore une decouverte de 
 M. Lagrange." BIOT, Traite elem. 
 d'Astron. Physique, tome. II., p. 413. 
 
 85 " Die absolut genaue Ueberein- 
 stimmung der mittleren Rotations 
 uud mittleren Umlaufszeit des 
 Mondes die sogar, wie Laplace 
 gezeigt hat, fur alle Zeiten statt- 
 findet, so d ass das Menschengeshlecht 
 nie die andre Seite des Mondes 
 gesehen bat, noch sehen wird ist 
 
ON THE MOON'S ROTATION. 
 
 117 
 
 The thirty-eighth proposition and nineteenth problem of the 
 Third Book of Newton's Principia reads thus : " To find the 
 " figure of the Moon's body. If the Moon's body were fluid like 
 " our sea, the force of the Earth to raise that fluid in the nearest 
 
 nach aller Wahrscheinlichkeit nicTit 
 zufallig. Als Erde und Mond sich 
 ans dem formlosen Chaos zu bilden 
 anfingen, fugten sich die einzelnen 
 Weltatome nicht vollig concentrisch 
 um den anfanglichen Mittelpunkt 
 des Mondes, sondern nach der Seite 
 der Erde zu in etwas starkerem Maas- 
 se als nach der entgegengesetzten. 
 Diese Seite ward mithin die scJiwe- 
 rere, und wie gering auch das 
 Ubergewicht immerhin sein mochte, 
 es veranlasste eine Neigung dieser 
 Seite, sich bestandig der Erde zuzu- 
 wenden. Es fragt sich nun, ob die 
 Rotation des Mondes gleich an fangs 
 so beschaifen gewesen, wie wir sie 
 jetzt finden, oder ob ihre Periode 
 eine von der Umlaufsperiode ver- 
 schiedene nur nicht zu stark ver- 
 schiedene war, die aber nach und 
 nach durch jene iiberwiegende Nei- 
 gung der schwereren Mondhalfte zur 
 Erde auf die jetzige zuriickgebracht 
 worden? In letzterm Falle hatten 
 anfangs grosse Schwankungen statt- 
 gefunden, die aber allmahlig ahn- 
 lich wie die eines in starke Bewe- 
 gung gesetzten Pendels sich bis zum 
 Unmerklichen verminderten. Da 
 sie indess nicht ganz vernichtet 
 werden kb'nnen, vielmehr ihre gegen- 
 wartige Grosse, wenn sie anders ex- 
 istiren, vender aufanglichen abhangt 
 und auf sie zuriickschliessen lasst, so 
 ist es nunmehr Sache der beobach- 
 
 tenden Praxis, diese physische Libra- 
 tion zu entdecken." M^EDLEB Popul. 
 Astronomie, S. 164. 
 
 86 " The attraction exercised by the 
 Earth on the Moon, combined with 
 the rotation of the latter on her axis, 
 is shown to have caused a small 
 elongation of the diameter which, 
 when produced, passes through the 
 Earth with a corresponding contrac- 
 tion of that diameter of the Moon's 
 equator which is at right angles to 
 the former; and to this is ascribed 
 the constant presentation of nearly 
 the same face of the Moon to the 
 Earth; for, though foreign pertur- 
 bations may cause the former of 
 these diameters to deviate from the 
 line joining the centres of the Moon 
 and Earth, the attraction of the latter 
 will presently bring it back; thus, it 
 will be made to oscillate about its 
 mean position within certain limits 
 of very small extent : and this ten- 
 dency of the same face of the Moon 
 towards the Earth causes her, with 
 respect to any fixed point in space, 
 to make one revolution on her axis 
 during the period of one revolution 
 in her orbit, about the Earth : and 
 subsequently to Newton's time, it 
 has been proved that, from the same 
 cause, the nodes of her equator and 
 those of her orbit retrograde with 
 equal velocities." NABRIEN, An His- 
 torical Account, p. 449450. 
 
118 ON THE MOON'S ROTATION. 
 
 " and remotest parts would be to the force of the Moon, by which 
 " our sea is raised in the places under and opposite to the Moon, 
 " as the accelerative gravity of tHe Moon towards the Earth to 
 " the accelerative gravity of the Earth towards the Moon, and the 
 " diameter of the Moon to the diameter of the Earth conjunctly ; 
 " that is, as 39,788 to 1, and 100 to 365 conjunctly, or as 1081 to 
 " 100. Wherefore, since our sea, by the force of the Moon, is 
 " raised to 8-f feet, the lunar fluid would be raised by the force of 
 " the Earth to 93 feet ; and upon this account the figure of the 
 " Moon would be a spheroid, whose greatest diameter produced 
 " would pass through the centre of the Earth, and exceed the 
 " diameters perpendicular thereto by 186 feet Such a figure, 
 " therefore, the Moon affects, and must put on from the begin- 
 " ning. Q. E. J. 
 
 " COR. Hence it is that the same face of the Moon always re- 
 " spects the Earth; nor can the body of the Moon possibly rest 
 " in any other position, but would return always by a libratory 
 " motion to this situation ; but those librations, however, must be 
 a exceedingly slow, because of the weakness of the forces which 
 " excite them ; so that the face of the Moon which should be 
 " always obverted to the Earth, may, for the reason assigned in 
 " proposition seventeen, be turned towards the other focus of the 
 " Moon's orbit, without being immediately drawn back, and con- 
 " verted again towards the Earth." 
 
 Here, then, we have a physical cause of attraction assigned 
 for the constant direction of the same face of the Moon towards 
 the Earth ; and as this physical necessity, which at first sight 
 we might take to be the very cause of the Moon's rotation (^j, 
 will, upon closer inspection, be found to be irreconcileable 
 with it : it is in reality irreconcileable also with the whole 
 theory of that rotation itself. Newton does not express himself 
 very clearly. The next lemma relates to his explanation of 
 the precession of the equinoxes. Into this subject we need not 
 enter. We only wish to direct attention to the circumstance 
 
ON THE MOON'S ROTATION. 119 
 
 that, whilst the Earth, according to 
 the "fluid" theory, and owing to her 
 velocity of rotation, would form a re- 
 gular spheroid, such as Fig. 25, taken 
 from the Lemma of Newton, which it 
 serves to illustrate : the Moon, on ac- 
 count of the coincidence of her periods 
 of revolution and assumed rotation, 
 and because she always presents the FIG. 25. 
 
 same face towards the Earth, would form an ellipsoid, such 
 as represented in Fig. 26, 
 and in which the axis of 
 assumed rotation would 
 not pass through the 
 centre c, but through the 
 focus F, farthest from 
 the Earth. If this, how- 
 ever, were the case; if, 
 
 as Madler ( 85 ) states, the "~ FIG. 26. 
 
 side a c a were heavier than the opposite side ; or if, accord- 
 ing to Laplace ( 80 ), the elongated axis of the lunar globe, 
 however small the elongation be 87 , were a kind of pendulum, 
 which, in consequence of the attraction of the Earth, constantly 
 returns to the vertical of the line, joining the centres of the 
 Earth and the Moon : the consequences, certainly, would have 
 been what Newton and Laplace state, namely, physical libra- 
 tions, the existence of which practical Astronomy has not been 
 able to discover (comp. 55) ; but these, too, would be the 
 only consequences. In other words, the Moon's rotation, and 
 the inclination of her axis to the ecliptic, would, compatibly with 
 
 87 " Les observations de Mayer sur cette oscillation (physique), la diffe- 
 
 la libration de la lime, et celles que rence dont elle depend, doit etre tres 
 
 MM. Bouvard et Nicollet viennent petite." LAPLACE, Exposition du 
 
 de faire sur le meme objet, a ma Systeme du Monde, torn. II. p. 564 ; 
 
 priere, n'ayant point fait reconnaitre compare also note 86. 
 
120 ON THE MOON'S ROTATION. 
 
 those physical librations, reduced to an imperceptible degree, 
 be an impossibility. For the tendency of the supposed attrac- 
 tion of the Earth upon the lunar ellipsoid would have been a 
 double one, trifling in a longitudinal, more considerable in a la- 
 titudinal sense ; but both uniting to produce the same effect. In 
 the former sense, the terrestrial attraction, whether we suppose, 
 with Laplace, the period of the Moon's rotation and revolution to 
 have originally differed but little, or not, that of rotation being 
 necessarily the shorter period, would gradually have diminished 
 the rotatory velocity, until the two periods should have become 
 absolutely equal, as Laplace very truly concludes. He over- 
 looks, however, the important fact, that the attraction would 
 have continued, and, consequently, that its effects would not 
 have been arrested, but would next have converted the rotatory 
 motion into a libratory one, the vibration^ of the elongated 
 part of the ellipsoid gradually diminishing as previously the 
 velocity of rotation had diminished until they reached that 
 minimum of which modern Astronomy is in search, the last 
 perceptible trace of rotation, and, therefore, utterly incompati- 
 ble with the co-existence of rotation itself. In the latitudinal 
 sense, and simultaneously with the effect produced in longitude, 
 the supposed terrestrial attraction would have gradually reduced 
 the inclination of the axis of assumed rotation, until it should 
 have become perpendicular to the lunar orbit; thus, again, 
 rendering the Moon's rotation, in accordance with the pheno- 
 mena she actually presents, an impossibility ( 14). The ne- 
 cessary consequences of the Newtonian theory, therefore, are 
 at variance with empirical facts; nor have Astronomers suc- 
 ceeded in discovering, from observation, the oblate form of the 
 lunar globe, which that theory demandsj any more than its 
 physical libration ; 88 and, lastly, the conclusions, analysis has 
 
 88 Der Mond 1st eine Kugel, und BEER und MIDLER, SelenograpUe, 
 
 die genauesten Messungen haben S. 10. 
 
 uns noch keine Abplattung derselben Der Mond hat keine irgend wahr- 
 
 im Sinne der Erdabplattting gezeigt. nehmbare Abplattung, dagegen aber 
 
ON THE MOON'S ROTATION. 121 
 
 drawn from a series of the most careful observations, are equally 
 opposed to it ( 52) : reasons sufficient, we think, to have war- 
 ranted us in ascribing to the hypothesis of the primitively fluid 
 state of the Moon an erroneous character ; and now to justify 
 us, to revert to the position in which we left our problem at the 
 conclusion of the preceding paragraph, without entering further 
 into the more speculative researches of Lagrange, Laplace, 
 Poisson and others, based as those researches are upon an un- 
 tenable theory. 
 
 55. Nearly every Astronomer, in stating the perfect coinci- 
 dence of the two epochs of the Moon's rotation and revolution, 
 at the same time characterises it as a " singular," a " strange," 
 a "curious, a "remarkable" fact. The greatest probability is 
 against the supposition that this coincidence should have taken 
 place from the beginning, ( 92 ) the chances being infinite to 
 one, 89 and no philosopher would willingly have recourse to it. 90 
 Mr. Grant's opinion, considering it as the interesting result of 
 Supreme Intelligence, 9 . 1 is not even calculated to do credit to 
 
 eine wiewohl ausserst geringe und dert wird, um seine Rotation der pro- 
 
 nur durch die Theorie gefundene gressiveu Bewegung genau gleich zu 
 
 Verlangerung gegen den Erdkorper machen, ist so unwahrscheinlich, 
 
 hin, die kaum 1,000 Fuss betragt. dass kein Philosoph diesen einzelnen 
 
 Wir konnen also den Mond, wenn Fall, gegen den es unzahlige andre 
 
 wir von semen physischen Ungleich- gleich mogliche giebt, annebmen 
 
 beiten absehen, vollig als Kugel be- wird, so lange noch dio geringste 
 
 trachten. MIDLER, Popul. Astro- Mbglicbkeit ist, die Sache auf eine 
 
 nomie, S. 161. andere Art zu erklaren. SCHUBERT, 
 
 89 Un des phenomenes des plus Popul. Astronomie, Bd. iii. S. 344, 
 
 singuliers du systeme solaire, est 345. 
 
 1'egalite rigoureuse que Ton observe 91 Laplace is surprised tbat New. 
 
 entre les mouvements angulaires de ton should have failed to notice that 
 
 rotation et de revolution de chaque in order to assure the constant 
 
 satellite. II y a l'infini contre un a equality of the motions of rotation 
 
 parier qu'il n'est point 1'efFet du and revolution, it was not absolutely 
 
 hazard. LAPLACE, Exposition du necessary that at the origin they 
 
 Systeme du Monde, torn. ii. p. 564. should have been exactly equal. . . . 
 
 )0 Dass der Mond wirklich grade It is natural enough, indeed, to sup- 
 den Stoss erhalten habe, der erfor- pose that the illustrious author of 
 
122 
 
 ON THE MOON S ROTATION. 
 
 the Pulpit, since it does not show upon what principle the coin- 
 cidence in one doubtful case should be a manifestation of 
 Supreme Intelligence rather tHan the wow-coincidence in at 
 least six certain cases. Thus, religion explains the assumed 
 coincidence as little as the theory of attraction does ( 54) ; 98 
 and the circumstance remains, indeed, an unaccountable one. 
 But there are other circumstances in the lunar system not less 
 singular, as is very justly remarked by Admiral Smyth, 93 whom 
 this striking feature, in its general bearing, has not escaped. 
 The coincidence f. i. of the nodes of the lunar equator with the 
 opposite nodes of the orbit, 94 is as strict as that of the periods of 
 
 the Principia did not feel any anxiety 
 to repudiate the original equality of 
 the motions of rotation and revolu- 
 tion a relation which, although 
 perhaps difficult to explain by the 
 doctrine of chances, becomes very 
 interesting and suggestive when it 
 is considered as the result of Supreme 
 Intelligence. GRANT, History of 
 Physical Astronomy, p. 75. 
 
 92 On voit done que la theorie de 
 la pesanteur explique d'une maniere 
 satisfaisante, 1'egalite rigoureuse des 
 deux moyens mouvemens angulaires 
 de rotation et de revolution de la 
 lune. II serait centre toute vraisem- 
 blance, de supposer qu' a 1'origine, 
 ces deux mouvemens ont ete par- 
 faitement egaux ; mais pour 1'expli- 
 cation de ce phenomene, il suffit que 
 leur difference primitive ait ete tres 
 petite ; et alors 1'attraction de la terre 
 a etabli la parfaite egalite que Ton 
 observe. LAPLACE, Exposition du 
 Systeme du Monde, tome ii. p. 248. 
 
 93 The Moon turns upon her axis 
 precisely in the same time as she 
 takes to revolve round the Earth; 
 and this is the reason why she always 
 
 presents to us very nearly the same 
 face. If the rotatory motion were 
 absolutely uniform, while the motion 
 of translation is subject to secular 
 variations, she would, in the course 
 of ages, successively present to the 
 Earth all the points of her surface. 
 But it is proved by theory that the 
 terrestrial attraction on the lunar 
 spheroid communicates to the Moon's 
 axical rotation the secular inequali- 
 ties of her orbital motion, so that 
 one of the lunar hemispheres must 
 be for ever concealed from the inha- 
 bitants of the Earth. The question 
 is intricate, since the lunar system 
 displays a very singular coincidence 
 of effects, totally dependent on each 
 other. The nodes of the Moon's 
 equator tally with those of her orbit: 
 the one is in motion from the action 
 of the Earth upon the Moon, and 
 the other arises from the action of 
 the Sun upon the Moon; and the 
 former is assisted by the constitution 
 of the Moon's body. SMYTH, A Cycle 
 of Celestial Objects, vol. I. p. 120 
 121. 
 94 Ainsi, les intersections de I'equa- 
 
ON THE MOON S ROTATION. 
 
 123 
 
 rotation and revolution, which Laplace has even proposed as 
 the basis of a universal meridian; 95 and here, again, theory 
 shows that oscillations must exist, which observation fails to 
 discover. 96 
 
 teur lunaire avec 1'ecliptique, ou ses 
 noeuds, coincident toujours avec les 
 noeuds moyens de 1'orbe lunaire, et 
 comme eux, ils ont un mouvement 
 retrograde dont la periode est de 
 6730,J39.108. Dans cet intervalle, 
 les deux poles de 1'equateur, et de 
 1'orbe lunaire, decrivent de petits 
 cercles paralleles a 1'ecliptique, en 
 comprenant son pole entre eux, de 
 maniere que ces trois poles soient 
 constamment sur un grand cercle de 
 la sphere celeste. LAPLACE, Exposi- 
 tion du Systeme du Monde, torn. i. 
 p. 58, 59. 
 
 95 L'egalite des mouvemens de ro- 
 tation et de revolution de la lune, 
 fournit a 1'astronome qui veut en 
 decrire la surface, un meridien uni- 
 versel donne par la nature, et facile 
 a retrouver dans tous les temps; 
 avantage que n' a point la geogra- 
 phie dans la description de la terre. 
 Ce meridien est celui qui passe par 
 les poles de la lune, et par 1'extremite 
 de son grand axe toujours a fort peu 
 pres dirige vers nous. Quoique cette 
 extremite ne soit distingue par au- 
 cune tache, cependant on peut en 
 fixer la position a chaque instant en 
 considerant qu' elle coincide avec la 
 ligne des noeuds moyens de 1'orbite 
 lunaire, quand cette ligne coincide 
 elle-meme avec le lieu moyen de la 
 lune. La situation des principales 
 taches de sa surface, a ainsi et de- 
 termiuee aussi exactement que celle 
 de beaucoup de lieux remarquables 
 de la terre. LAPLACE, Exposition du 
 
 me du Monde, torn. ii. p. 252, 
 
 253. 
 
 96 Quand la nature assujettit les 
 moyens mouvemens celestes, a des 
 conditions determinees; ils sont tou- 
 jours accompagnesd'oscillations dont 
 1'etendue est arbitraire; ainsi, 1'ega- 
 lite des moyens mouvemens de rota- 
 tion et de revolution de la lune, est 
 accompagnee d'une libration reelle 
 de ce satellite. Pareillement, la 
 coincidence des noeuds moyens de 
 1'equateur et de 1'orbite lunaire, est 
 accompagnee d'une libration des 
 noeuds de cet equateur, autour de 
 ceux de 1'orbite ; libration tres petite, 
 puis qu'elle a echappe jusqu' id aux 
 observations. On a vu que la libra- 
 tion reelle du grand axe de la lune 
 est insensible, et nous avons observe 
 dans le chapitre vi., que la libration 
 des trois premiers satellites de Jupi- 
 ter, est pareillement insensible. II 
 est tres remarquable que ces libra- 
 tions dont 1'etendue est arbitraire et 
 pourrait etre considerable, soient ce- 
 pendant fort petites; ce que Ton 
 peut attribuer aux memes causes qui, 
 dans 1'origine, ont etabli les condi- 
 tions dont elles dependent. Mais 
 relativement aux arbitrages qui tien- 
 nent au mouvement initial de rota 
 tion des corps celestes, il est naturel 
 de penser que sans les attractions 
 etrangeres, toutes leurs parties en 
 vertu des frottemens et des resist- 
 ances qu' elles opposent a leurs 
 mouvemens reciproques, auraient 
 pris a la longue, un etat constant 
 
124 ON THE MOON'S ROTATION. 
 
 The slowness of the assumed rotatory motion of the Moon 
 on her axis, is a no legs remarkable circumstance. In Mercury 
 its velocity exceeds that of the -progressive motion not far from 
 100, in Venus more than 200, in the Earth almost 400, in Mars 
 more than 600, in Jupiter nearly 10,000, and in Saturn more 
 than 20,000 times. True, this rate of velocity whatever Pro- 
 fessor Nichol may think to the contrary (Note 14), would seem 
 to decrease with the decreasing distance of the planets from 
 the Sun ; and the very proximity of the Moon to her primary, 
 therefore, might be urged in explanation of the slowness of her 
 rotation. But the subject admits of a different and more philo- 
 sophical view. If we regard rotation as the effect of a cause, 
 and which we are rather necessitated to do than merely war- 
 ranted in doing, then, of whatever vital nature and mechanical 
 force that cause may be, its power, and, consequently, the 
 capability of the heavenly bodies to rotate, would seem to 
 decrease with the decreasing ratio of the rotatory in excess of 
 the progressive velocity, and to reach its minimum, when both 
 compensate each other, i. e., rotation, in virtue of the laws of 
 motion, would seem to become impossible, so soon as the rota- 
 tory force of any heavenly body fails to be in excess of its 
 progressive force, thus merging, as it were, into the latter. In 
 fact, if we decompose the two forces, where combined, and 
 consider the rotatory force by itself, its very existence ceases, 
 and, consequently, also its effect, i. e., rotation must cease, from 
 the moment that the periods of rotation and revolution fall 
 into coincidence. The Moon's rotation, thus viewed, would of 
 itself be impossible ; but even setting aside this argument there 
 remains, as deduced from general considerations, so little pro- 
 
 d'equilibre, qui ne peut exister qu' la terre, comme on s'en est assure 
 
 avec un mouvemeut de rotation uni- paries observations les plus precises : 
 
 forme, autour d' un axe invariable ; le meme resultat s'etend a la lune, 
 
 en sorte que les observations ne et probablement a tous les corps 
 
 cloivent plus offrir dans ce motive- celestes. LAPLACE, Exposition du 
 
 ment, que les inegalites dues a ces Systtme du Monde, tome ii. p. 251, 
 
 attractions. (Test ce qui a lieu pour 252. 
 
ON THE MOON'S ROTATION. 
 
 125 
 
 lability in favour of the prevailing opinion, as to amount almost 
 to impossibility. 
 
 56. We have now, however, to descend from generalities to 
 particulars ; and, with the view of answering the question, 
 whether the Moon does or does not rotate ? to compare the 
 astronomical theory and its logical consequences, upon every 
 special point, with the results of actual observation, and the 
 real phenomena which the Moon presents. 
 
 57. According to the almost unanimous opinion of Astrono- 
 mers, the three kinds of libration to which our satellite is 
 subject the diurnal libration, the libration in longitude, and 
 the libration in latitude are all of a purely optical nature, in 
 no way affecting the Moons real motion of rotation. gi Indeed, 
 
 9781 Toutes ces causes ne produisent 
 qu' une libration apparente, dans le 
 globe lunaire; elles sont purement 
 optiques, et n' affectent point son 
 mouvement reel de rotation. Ce 
 mouvement peut cependant etre as- 
 sujetti a de petites inegalites; mais 
 elles sont trop peu sensibles pour 
 avoir ete observees. LAPLACE, Ex- 
 position du Systeme du Monde, tome 
 i. p. 57. 
 
 975 En adoptant cette expression 
 (de libration), qui peint bien les ap- 
 parences observees, il ne faut pas lui 
 donner un sens positif, car le pheno- 
 mene lui-meme n'a rien de reel; 
 ce n'est qu'un resultat compose de 
 plusieurs illusions optiques. 
 
 Pour con^evoir et demeler ces illu- 
 sions, ramenons-nous a des termes 
 precis. Imaginons un rayon visuel 
 mene du centre de la terre au centre 
 de la lune. Le plan par ce centre 
 perpendiculairement a ce rayon, 
 coupe le globe lunaire suivant une 
 circonference de cercle qui est pour 
 nous le disque apparent. Si la lune 
 
 n'avait point de mouvement de rota- 
 tion reel, c'est-a-dire si chaque point 
 de sa surface restait invariablement 
 dirige vers le ineme point de ciel, son 
 seul mouvement de revolution, autour 
 de la terre, nous decouvrirait succes- 
 sivement tous les points de cette sur- 
 face : le rayon visuel la rencontrerait 
 successivement en differens points, 
 que nous verrions passer les uns 
 apres les autres au centre apparent 
 du disque lunaire. Le mouvement 
 reel de rotation contrarie les effets 
 de cette rotation apparente, etramene 
 constamment vers nous la meme face 
 du globe lunaire: voila pourquoi 
 nous ne decouvrons jamais la face 
 opposee BIOT, Traite elem. d J As- 
 tron. physique, tome ii. p. 406. 
 
 97 c Les causes des trois librations 
 partielles que nous venons de faire 
 connaitre, ne produisent qu' une li- 
 bration totale et apparente dans le 
 globe lunaire ; ces causes sont pure- 
 ment optiques ; elles n' affectent point 
 le mouvement reel de rotation de la 
 lune, et disparrissent entierement 
 
126 
 
 ON THE MOON S ROTATION. 
 
 they would altogether disappear, Madler states, ( 1? ) if the in- 
 equalities of the Moon's revolution in her orbit were partici- 
 pated in also by her rotatory niotion. But already upon this 
 first point we find Astronomers in contradiction with themselves. 
 For they admit that, in consequence of the inequalities alluded 
 to, the Moon, assuming her rotation to be uniform, would in 
 the course of centuries, notwithstanding the theory of Newton, 
 present in succession every portion of her surface to the Earth. 98 
 
 quand on rapporte ce mouvement au 
 centre de cette planete. NICOLLET, 
 Mem. sur la librat. de la Lune, p. 
 229. 
 
 98a Le moyen mouvement de la 
 lune est assujetti a de grandes in- 
 egalites seculaires qui s'elevent a 
 
 plusieurs circonferences 
 
 L'attraction terrestre, en ramenant 
 sans cessevers nous, le grand axe de 
 la lune, fait participer son mouve- 
 ment de rotation aux inegalites secu- 
 laires de son mouvement de revolu- 
 tion, et dirige constamment le meme 
 hemisphere vers la terre. La meme 
 theorie doit etre etendue a tous les 
 satellites dans lesquels on a observe 
 1'egalite des mouvemens de rotation 
 et de revolution autour de leur pla- 
 nete. LAPLACE, Exposition du Sys- 
 tbne du Monde, tome ii. p. 248, 249. 
 
 98 b Das Schwanken des Mondes 
 ist durchaus nothwendig, wenn er 
 uns bestandig dieselbeSeite zukehren 
 soil. Es scheint zwar beim ersten 
 Anblick, dass eben das erreicht 
 wiirde, wenn die Kotation des Monds 
 mit seiner mittlern Bewegung um 
 die Erde genau ubereinkame, da 
 wir dann von der hintern Seite des 
 Monds von Zeit zu Zeit nur so viel 
 crblicken wiirden, als die periodischen 
 Gleichungen des Monds von wenigen 
 
 Graden betragen, so dass am Ende 
 des Monats oder der Periode dieser 
 Gleichungen, der Erde immer die- 
 selbe Seite wieder zugekehrt ware. 
 Allein bei naherer Untersuchung 
 findet sich hierin ein Widerspruch : 
 denn es entsteht natiirlich die Frage, 
 mit welcher mittleren Bewegung die 
 Kotation iibereinkommen sollte. 
 Wir wissen, dass die mittlere Bewe- 
 gung des Monds seit 10,000 Jahren 
 immer schneller geworden ist, und 
 noch iiber 20,000 Jahre zunehmen 
 wird. Da diese Aenderungen nicht 
 periodisch sind, oder vielmehr Pe- 
 rioden von ungeheurer Lange haben 
 so ist ofFenbar, dass, wenn die Kota- 
 tion einer bestimmten mittlern Be- 
 wegung, die der Mond zu einer ge- 
 wissen Zeit hatte, gleich war, sie da- 
 von immer niehr abweichen musste, 
 so dass endlich ein betrachtlicher 
 Theil der hintern Mondscheibe zum 
 Vorschein kommen wiirde, welches 
 aber nicht der Fall ist. Soil also der 
 Mond der Erde bestandig dieselbe 
 Seite zukehren, so muss seine Rota- 
 tion an der Ungleichheit der mittlern 
 Bewegung Theil nehmen ; oder da 
 jede freiere Rotation ihrer Natur nach 
 gleich f o rmig ist, so muss eine be- 
 sondre Kraft dieselbe Seite des Monds 
 zur Erde immer zuriick fiihren. 
 
ON THE MOON'S ROTATION. 
 
 127 
 
 This, however, being contrary to the empirical facts of the case, 
 they explain the latter by stating, that the excess of the velocity 
 of rotation is counteracted by the physical attraction of the 
 Earth upon the elongated part of the lunar spheroid ( 54) > 
 which thus causes the Moon's rotatory motion to participate in 
 the secular inequalities of her revolution ( 98 ), by actually dis- 
 turbing its uniform velocity, and retarding it." 
 
 58. One contradiction will generally lead to another. So 
 here. At the same time, that Astronomers, driven to extremities 
 in order to keep their theory in accordance with empirical 
 facts, subject the Moon's rotatory velocity to secular inequali- 
 ties, they assume its strictest uniformity upon the ground both 
 of analogy and of the laws of mechanics 100 : a uniformity, 
 which they contend, is affected neither by the elliptical ine- 
 qualities of the Moon's revolution, nor by those, resulting from 
 the perturbations of her orbit l01 , or the periodical inequalities 
 of her revolution. Surely, a more inconsistent means to up- 
 
 Diese Kraft 1st die Attraktion der 
 Erde auf den verlangerten Durch- 
 messer des Mond-Aequators, und die 
 Sch wankungen desselben bringen das 
 merkwiirdige Phanomen hervor, das 
 sich den Erdbewohnern seit Jahrtau- 
 senden zeigt. SCHUBERT, Popul. 
 Astronomic, Bd. iii. S. 347, 348. 
 
 90 Da der Mond die Figur einer 
 Ellipsoide hat, so geschieht seine 
 Eotation wirklich um eine freie Axe. 
 Diese Rotation wlrd aber durch die 
 Attraktion der Erde auf den erha- 
 bener Mond-Aequator gestort; und 
 hiedurcb entsteht eben das Schwan- 
 ken des Monds. SCHUBERT, Popul. 
 Astronomic, Bd. iii. S. 348. 
 
 100 " Quant an mouvement de ro- 
 tation de la lune sur elle-meme, il 
 est naturel d'admettre qu'il est uni- 
 forme, comme la rotation de la terre; 
 d'alleurs les lois de la mecanique in- 
 diquent qu'il doit en etre ainsi." 
 
 DELAUNY, Cours 6lem. tf Astronomie, 
 p. 395. 
 
 101 j) er M on( j ro ti r t um seine Axe 
 vollhommen genau in derselben Zeit, 
 in welcber er um die Erde lauft, 
 jedocbist diese Rotation gleichformig 
 und nimmt weder an den elliptischen, 
 noch an den durch die Storungen 
 er zeugten Ungleichbeitendes Mond- 
 umlaufes Theil. Die Axe der Mond- 
 kugel macht mit der Ekliptik einen 
 unveranderlichen Winkel von 88 
 31' ] 5", und der Aequator neigt sich 
 also gegen diese um 1 28' 45". 
 Dagegen ist die Neigung des Mond- 
 aquators gegen seine eigene Bahn 
 veranderlich, wie diese selbst, und 
 kann von 6 29' bis 6 47' gehen. 
 Der niedersteigende Knoten des 
 Mondaquators in der Ekliptik fallt 
 stets mit dem aufsteigenden der 
 Mondbahn in der Ekliptik zusam- 
 men ; diese drei Ebenen bilden also 
 
128 
 
 ON THE MOON'S ROTATION. 
 
 hold a failing theory, than that alluded to, could not well have 
 been devised. For, a.uniform velocity being one of the cha- 
 racteristics of rotation (26) : it would have been hazardous to 
 deprive it of this characteristic under any circumstances ; but 
 to represent it as unaffected by the powerful influences of the 
 Moon's periodical perturbations, which cause her apsides to 
 perform an entire circuit of the heavens in less than nine years 
 ( 46), and yet to be sensitive of a secular perturbation, almost 
 imperceptible, and to which no greater force is ascribed, than 
 that of drawing the Moon nearer towards the Earth by about 
 nine feet in a century 102 , is only the more difficult to reconcile 
 
 nur zwei gemeinschaftliche Durch- 
 schnittspunkte und haben eine 
 Knotenlinie mit einander gemein. 
 Cassini hat diese Bestimmungen 
 durcli Beobachtung ermittelt; die 
 spatern franzosischen Astronomen, 
 namentlich Laplace, haben sie 
 durch die Theorie erwiesen." 
 M^EDLER, Popul. Astronomic, S. 
 161, 162. 
 
 102 Was endlich die erwahnte 
 Secular-Ungleichheit betrifft, so ist 
 diese so klein, dass sie auch dem 
 aufmerksamsten Beobachter entge- 
 hen wiirde, wenn er sie allein aus 
 den in Laufe seines Lebens 
 gemachten Mondsbeobachtungen 
 schliessen sollte. Denn die ganze 
 Naherung des Mondes zur Erde, 
 welche hierdurch bewirkt wird, ist 
 9 Fuss in einera Jahrhundert ! Dies 
 direkt wahrzunehmen ist absolut 
 unmoglich." BEER und MIDLER. 
 Selenographie, S. 7. 
 
 103 < sj 1'orbite decrite par la tache 
 autour du centre de la lune est plane, 
 trois observations doivent suffire 
 pour determiner son plan, et par 
 suite la position de 1'equateur lunaire 
 qui lui est parallele. Mais ici st 
 presente une nouvelle particularite. 
 
 " Si Ton emploie trois observa- 
 tions faites dans un intervalle de 
 terns peu considerable, par exemple, 
 dans 1'espace de quinze jours, on 
 trouve que la longitude du noeud 
 ascendant de 1'equateur lunaire est 
 egale a la longitude moyenne du 
 noeud ascendant de 1'orbite ; et 
 comme en repetant ce calcul, a des 
 epoques quelconques, mais toujours 
 sur des observations voisines, on 
 trouve toujours cette egalite, il en 
 faut conclure que la trace de 1'equa- 
 teur lunaire sur 1'ecliptique est con- 
 stamment parallele a la ligne des 
 noeuds moyens de 1'orbite ; par con- 
 sequent, cette trace a un mouvement 
 retrograde egal a celui des nosuds ; 
 et aiiisi 1'axe de rotation du spero'ide 
 lunaire decrit une surface conique 
 autour de 1'axe mene par son centre 
 perpendiculairement a 1'ecliptique. 
 Quant a 1'inclinaison de 1'equateur 
 lunaire sur 1'ecliptique, il parait 
 qu'elle est constante ; car en la cal- 
 culant ainsi par des observations 
 voisines, on lui trouve dans tous les 
 terns, la meme valeur, qui est 1, 67. 
 BIOT, Traite Elem. d Astronomic 
 Physique, tome ii. pp. 409, 410. 
 
ON THE MOON'S ROTATION. 129 
 
 with reason, as Mathematicians know how to reconcile it with 
 the laws of gravitation (? 6 ). Analysis is a most invaluable 
 instrument, when resting upon facts to explain a theory ; but 
 a very dangerous one, when made to rest upon theory to ex- 
 plain facts. 
 
 59. The contradictions, we have noticed, augur not well for 
 the correctness of the astronomical views regarding the Moon's 
 rotation. But this view embodies another contradiction, in- 
 volving consequences of a more positive nature. Astronomers 
 are unanimous in stating, in accordance with the theory of 
 Newton, that the period of the Moon's rotation strictly coin- 
 cides with that of her revolution : which ? her sidereal, her 
 tropical, or her synodical revolution ? The common answer 
 is : the sidereal revolution ; though many Astronomers say 
 the synodic revolution, (9) or pass over this very essential 
 point in complete silence, whilst others omit to distinguish 
 between the tropical and sidereal revolution the difference 
 between which is only a few seconds by omitting to add 
 the seconds to the time of rotation. Biot, however, informs us, 
 ( 56 ), that, as seen from the Sun, this time is equal to a synodi- 
 cal revolution, or 29 days 12h. 44' 2", 87; as seen from the 
 stars, equal to a sidereal revolution, or 27 days 7h. 43' 11", 5; 
 and as seen from the point of the vernal equinox, equal to a 
 tropical revolution, or 27 days 7h. 43' 4", 7. Here, then, we 
 are in presence of a threefold contradiction. For the Moon's 
 rotation is not a relative, but an absolute and a uniform motion, 
 which, whether seen from the stars, the Sun, the Earth, or any 
 other point in space, can have only one definite period, and 
 possibly cannot have three differing periods. This is so self- 
 evident, as to require no explanation. Yet, we would wish to 
 make the subject perfectly clear to the reader. 
 
 For this purpose, we will first consider the Moon's rota- 
 tion apart from her revolution. Let, in Fig. 27, M represent 
 the Moon, s the Sun, E the Earth, v the point of the Vernal 
 Equinox, all supposed to be stationary ; A the star Aldebaran, 
 
 K 
 
130 ON THE MOON'S ROTATION. 
 
 z any other star or fixed point 
 in space. The lines ss, EC, &c., 
 produced, shall join the centre of 
 the Moon and that of the Sun, 
 the Earth, &c. Then, if at any 
 given moment the point s in the 
 lunar surface, in virtue of the 
 Moon's rotation, pass the line of 
 intersection ss, at the same phy- 
 sical instant will the other points 
 e, V) a, and z pass their respective 
 lines of intersection Ee, vv, A, 
 27. and zz\ because the Moon is 
 one solid spherical body, in which the least movement of ro- 
 tation (or any general movement whatever) is common to all 
 the particles composing it. No one point in the Moon's sur- 
 face, therefore, could make a rotatory movement without every 
 other point participating in it, any more than we could turn 
 one point in the circumference of a wheel round its axle, with- 
 out turning round all the other points. And in the same man- 
 ner and for the same reason, when the point s in the lunar 
 surface, after some definite period subsequent to passing the 
 line of intersection ss we will say, after 2,500,001 seconds of 
 time shall have completed a whole circuit, and again pass the 
 line s.9: then, at the same physical instant also, will each of 
 the other points e, v, a, 2, have completed an entire circuit, and 
 again pass their respective lines of intersection Ee, vv, Aa, and 
 zz ; so that, whether seen from the Earth, the Sun, the point of 
 the Vernal Equinox, the star Aldebaran, or any other star or 
 point in space whatever, the Moon will have once rotated on 
 her axis in the same definite period of 2,500,001 seconds of 
 time, and that without the difference of so much as the smallest 
 fraction of a second relative to any two or more points in space. 
 Hence, the Moon's period of rotation being a definite one, it 
 
131 
 
 cannot possibly be subject to any difference of duration, whether 
 we refer it to some star or any other fixed point in space ; or 
 whether we refer it to some point moving itself through space. 
 For the Moon's own progressive motion, or that of the Earth or 
 Sun, can no more alter the definite period of her assumed ro- 
 tation, than the time of the revolution of a wheel is altered, 
 whether we regard it from a lesser or greater distance, or 
 whether we look at it in walking about or standing still. 
 
 60. Having determined this point, we have to ask: which, 
 then, is the real period, Astronomy assigns to the Moon's ro- 
 tation ? The authority of Newton and the most eminent of his 
 followers, leaves no doubt upon the subject : it is the sidereal 
 month, or the period which elapses between two successive re- 
 turns of the Moon to the same star=27 days 7h. 43' 11", 5; 
 and which Astronomers further state to be the true period of 
 her revolution round the Earth. 103 This being a point of im- 
 portance, more especially with regard to our problem, inas- 
 much as the two periods of the Moon's rotation and revolution 
 are said to coincide, and as there would be no such coincidence, 
 if an erroneous duration should have been ascribed by Astro- 
 nomers to the Moon's real period of revolution round the 
 Earth, we shall have to inquire into the correctness of their 
 statement. Let, in Fig. 28, E represent the Earth, M the Moon, 
 s and t two stars. The straight lines s a E, s a" E' and t a' E' 
 shall pass through the centres of the Moon and the Earth ; the 
 former line shall be parallel to the latter; and E a" parallel to 
 E, a" s. Then if we suppose the Earth to be stationary in E, 
 and the Moon to revolve round the Earth in the indicated di- 
 
 103 ) er W ahre Umlauf des Mondes leicht die tagliche, stundliche u. s. w. 
 
 um die Erde wahrt 27t. 7h. 43' 11", mittlere Bewegung des Mondes ab- 
 
 5. Nach Verlauf dieser Zeit steht leiten, so wie die Zeit, welche er 
 
 der Mond, in Bezietmng auf Lange, gebraucht, um cine gegebene Anzahl 
 
 wieder bei demselben festen Punkte von Graden zu durchlaufen. Diese 
 
 des Himmels (demselben Fixsterne), wdhre Umlaufszeit fiihrt auch den 
 
 und er hat also 360 seines Laufes Namen der siderischen" MJEDLEB, 
 
 zuriickgelegt. Hieraus 1'asst sich Popul. Astronomie, S. 152. 
 
 K 2 
 
132 ON THE MOON S ROTATION. 
 
 FIG .28. 
 
 rection, she will have performed one complete revolution 
 round the Earth, when the diameter a b, after coinciding with, 
 and passing the line s E, shall again coincide with it. The 
 interval of time, elapsing between the moments of two succes- 
 sive coincidences, would be equal to the period of a mean lunar 
 month, or a sy nodical revolution of the Moon=29 days 12h. 
 44' 2", 87. But the Earth, instead of being stationary, revolv- 
 ing round the Sun, as the Moon does round the Earth, she 
 will, whilst the Moon performed a complete revolution round 
 her, have performed about a thirteenth part of such a revolution 
 round the Sun, and at the moment, when the Moon completed 
 her circuit, no longer be in E, but have advanced to E ; the 
 lines joining the centres of the Earth and the Moon, point- 
 ing neither any longer to the star s, but to a different star t. 
 Consequently the line joining the star s and the Earth's centre 
 will also no longer coincide with the line s E, but form a con- 
 siderable angle E s E' with it ; and the Moon's diameter a" b"= 
 a b, if we suppose the Earth stationary in E', will pass that line 
 much earlier than it would have done had the Earth remained 
 stationary in E. Relative, therefore, to the line joining the 
 
ON THE MOON'S ROTATION. 133 
 
 fixed star s, and the travelling centre of the Earth, the Moon 
 performs one complete apparent revolution in a shorter time 
 than she does a real revolution with regard to the centre of her 
 orbit. 
 
 This shorter period is by Astronomers termed a sidereal 
 month, and the period of 27 days 7h. 43' 11", 5, which it com- 
 prises, the true period of the Moon's revolution round the 
 Earth ; though the corresponding revolution is evidently not a 
 complete one with regard to the Earth. For, when the Moon 
 has arrived in M", she has still to revolve through an arc, equal 
 about to the arc b" b ', being a portion of her path round the 
 Earth, which it takes her 2 days 5h. 10' 51", 3 to perform, be- 
 fore she returns to the same point of her orbit, i. e. before she 
 actually completes a revolution. And the period of an incom- 
 plete revolution cannot possibly be the true period of a revolu- 
 tion. Or, if we suppose the Earth to have remained stationary 
 in E, would any observer in .9, having at any given moment 
 seen the Moon in a straight line with the Earth, maintain that 
 she had performed a revolution on her having arrived only in 
 M'" (E II" a" being parallel to E' b" a") ? would he assert the 
 time which the Moon took to perform that portion of the cir- 
 cuit, corresponding to the arc b c b"', to be the true period of 
 her revolution ? To use a familiar illustration. Suppose a 
 clock placed near the right edge of a mantle-piece, with the 
 minute-hand at the moment of noon, pointing to some particular 
 spot on the ceiling. Let it be very slowly moved towards the 
 left edge of the mantle-piece, so that, at five minutes to one 
 o'clock, the minute-handle shall point to the same spot on the 
 ceiling to which it pointed at noon, when the clock was stand- 
 ing near the right edge of the mantle-piece : would any one, 
 upon this account, assert that the minute-hand had made a com- 
 plete circuit of the dial ; and that 55 minutes were the true 
 duration of an hour ? Yet, this is what Astronomers virtually 
 do assert, r in stating the space of 27 days 7h. 43' 11", 5 to be the 
 true period of the Moon's revolution round the Earth. 
 
134 
 
 ON THE MOON S ROTATION. 
 
 Their error, however, being connected with the true principle 
 that, from the great distance of the stars, the entire diameter 
 of the Earth's orbit subtends an angle, the two sides of which 
 may be considered parallel to each other, we will trace it to 
 its source, in order to leave no doubt regarding it. For the 
 sake of greater simplicity, we will, in Fig. 29, suppose the 
 Moon M to revolve exactly twelve times round the Earth E, 
 whilst the latter revolves round the Sun s, i. e. in the course of 
 
ON THE MOON'S ROTATION. 135 
 
 a solar year. The stars a, b, c, d, e,f, g, h, i, k, I, m, shall in- 
 dicate the direction of the same diameter of the lunar globe 
 coinciding with a line joining the centres of the Earth and the 
 Sun ; and shall, consequently, mark also the exact moment of 
 the completion of each revolution of the Moon round the Earth. 
 A single glance at the figure will show us that the astronomical 
 view must rest on an error ; in as much as the visual ray, pass- 
 ing from the star a to the centre of the Earth (accompanied 
 by the Moon), in the monthly positions of her orbit, diverges, 
 if we take the perpendicular a a g for the normal line, only to 
 a certain degree from it; then converges till it coincides with it; 
 diverges again; converges once more, and finally coincides again 
 with the normal line, the divergence towards either side being 
 of equal extent. This clearly indicates that the apparent du- 
 ration of a lunar revolution, in the course of a year or the 
 period of a return of the Earth, accompanied by the Moon, to 
 the same star, is subject, as measured from that star, to fluc- 
 tuations between certain limits, and compensating each other ; 
 so that the mean or the true sidereal period of the Moon's re- 
 volution round the Earth, will be exactly equal to her synodical 
 period, as, indeed, it could not possibly be otherwise (compare 
 59). 
 
 And this conclusion is as fully borne out if we regard the 
 visual rays proceeding from the star a' as parallel to the nor- 
 mal line a a and to each other, as a" a", a"' a'", &c. First, 
 they depart from, then return to it; again depart from, and 
 again return to it. The Earth be in E a. The Moon M a will 
 at the point n enter, with regard to the star a', the opposite half 
 of her orbit, leaving it again at the point n'. In this part of the 
 orbit, the Earth will interpose between the Moon and the star 
 a' ; in the other portion the Moon between the star and the 
 Earth. When the latter has arrived in E b, having revolved 
 through an arc of 30, the Moon, at her full in M a, passes in 
 M' b' the Star a by 30 of her orbit, or about 2j days before 
 she is again at her full in M b. When the Earth has arrived in 
 
136 ON THE MOON'S ROTATION. 
 
 E c, having revolved through an arc of 60, the Moon passes in 
 M' c 1 the star a!" by 60 .or about 4^ days before she is again at 
 her full in M c. When the Earth has arrived in E d, the Moon 
 passes in M' d' the star a'" by 90 or about 6| days before she 
 is again at her full in M d. But the very moment the observer 
 in a! has gained this maximum of time he loses it again. The 
 Earth is now passing into the opposite half of her orbit ; the 
 visual ray of the observer in a' returns towards its normal line ; 
 and when the Moon arrives in M" d" ; she is seen from the 
 star a as much later with regard to the full Moon M d, as she 
 was in M' d' earlier, namely, about 6f days. When the Earth 
 has arrived in E e, that difference has decreased to about 4^ 
 days ; when in E y, to about 2J days ; and in E g it is fully 
 and exactly compensated. The return of the Earth and the 
 Moon from E g and M g to E a and M a offers but a repetition 
 of the same process. 
 
 The view of Astronomers, on the other hand, implies the 
 assumption that the visual ray d a E a continues to diverge or 
 to progress in the same direction, in consequence of which, 
 since Sun, Earth, and Moon do not revolve round the star ', 
 they have to make the star, in theory, revolve round the Sun, 
 but so as to keep always one sidereal month in arrear of the 
 Earth in her orbit. In other words, they suppose an observer 
 in a, who is seeing the Moon pass the Earth in M a, to wait in 
 a till he sees her pass the Earth again in M" ", about 2 days, 
 5h. 11' earlier than he would have done, had he at once started 
 with the Earth for the star b. But he only starts now, after 
 having waited in a about 27 days 7h. 43', for the star 5; where 
 he arrives to see the Moon pass the Earth again about 2 days, 
 5h. 1 1' earlier than he would have done, were he not a sidereal 
 month in arrear of the Earth. And thus from star to star. 
 The result is that, when Earth and Moon return to their original 
 positions in E a and M a, after having performed a complete 
 revolution round the Sun, our observer is still at some star z, 
 whence it will take him yet more than 27 days to reach the 
 
ON THE MOON'S ROTATION. 137 
 
 star a again, i. e. the time which he lost at the beginning by 
 waiting in a to see the Moon pass the Earth in M" b", before he 
 started on his journey. The space of twelve lunar months, in 
 which we have here assumed the Earth to perform a complete 
 revolution round the Sun, and the Moon to return with the 
 Earth to the same star, is actually about 354 days ; that of 
 twelve sidereal months about 327 days ; and the difference of 
 about 27 days is exactly the time, which we have found the 
 sidereal observer still to require, in order to accomplish his 
 return to the same star. It is evident, therefore, that the Moon, 
 actually taking as she does 354 days to return to the same 
 point in space, and to perform a complete revolution round a 
 fixed centre (considering the Sun to be immoveable), with re- 
 gard to a fixed star, she can, in 327 days, only have performed 
 a portion, and that 327 days cannot possibly be the true period, 
 of that revolution. But, if the entire period of 327 days 20h. 
 38' 18", comprising twelve sidereal months, is not the true 
 period of twelve revolutions of the Moon round the Earth, 
 neither can the period of one sidereal month, or 27 days, 7h. 
 43' 11", be the true period of one revolution. Yet Astronomers 
 contend that the observer of their theory, when, after starting 
 from the star a, in the direction b, c, &c., has arrived in z only, 
 he has performed a complete circuit, i. e. has returned to , and 
 that the time of 327 days, it has taken him to reach z, is the 
 true time of 354 days, which he will require to reach a again. 
 Take two clocks, both indicating the true time of twelve o'clock. 
 The one shall continue to go, the other shall stop for an hour, 
 and then continue to go. At the expiration of twelve hours, the 
 former clock will again indicate twelve o'clock, and its handles 
 have performed, the minute-hand twelve complete circuits, the 
 hour-hand one complete circuit, of the dial. The other clock 
 will indicate eleven o'clock, and its handles have performed, the 
 minute-hand, eleven complete circuits, the hour-hand eleven- 
 twelfths of a complete circuit, of the dial. Who would main- 
 tain that eleven hours were the true time of twelve hours ? and 
 
138 ON THE MOON'S ROTATION. 
 
 that the eleven-twelfth part of a circumference were the true 
 measure of a circumference ? 
 
 In order to complete our demonstration of the error of the astro- 
 nomical view, let us not refer the period of the Moon's revolu- 
 tion round the Earth to the star , which the Earth has passed, 
 but to the star c, as much in advance of the Earth in E b as the 
 star a is in arrear of her. In other words, let us suppose our 
 observer in a, after seeing the Moon pass the Earth in M , 
 instead of waiting in a to see her pass the Earth a second time, 
 hurry on to the star c, so as to perform the two months' journey 
 in one month; but subsequently travelling at the same rate 
 with the Earth. He will then see the Moon pass the Earth 
 about 2 days later than he would have done from the star b, 
 an interval of 31 days 17h. 54' 54" elapsing between the two 
 events. In d again he will see the Moon pass the Earth about 
 2 days later than he would have done in b, and so on. Con- 
 sequently, upon precisely similar principles, the duration of the 
 sidereal month is found to be 31 days 17h. 54' 54" and 27 days 
 7h. 43' 11". If the latter, therefore, be the true period of the 
 Moon's revolution round the Earth ; the former, for the same 
 reasons, must equally be that true period an impossibility. 
 The true period is the mean of these two relative periods, 
 namely, 29 days I2h. 44' 2", 87 ; and thus the synodical revo- 
 lution of the Moon round the Earth is, once more, shown to be 
 of exactly equal duration with the true, i. e. the mean, sidereal 
 month, and at the same time to be the true period of the Moon's 
 revolution round the Earth ; the same as the true number of 
 entire days, contained in the solar year, is 365, though when 
 travelling in an easterly direction round the Earth, we make 
 the number to be 366, and 364 when travelling round the 
 Earth in a westerly direction. 104 
 
 104 Mr. H. Perigal, Junr., in his 69), very justly observes, that the 
 paper on the Moon Controversy, to additional sidereal day, which the 
 which we have already alluded (Note year contains more than the number 
 
ON THE MOON S ROTATION. 
 
 139 
 
 61. Astronomers have, therefore, assigned an erroneous period 
 to the Moon's revolution round the Earth, namely, 27 days 7h. 
 43' 11", 5, instead of 29 days 12h. 44' 2", 87. Consequently, 
 they must also have assigned an erroneous period to her as- 
 
 of solar days, is due to the Earth's 
 revolution ; and that, if the Earth 
 rotated in a direction, contrary to 
 her movement in her orbit, the year 
 would contain one sidereal day short 
 of the number of solar days With 
 equal justice he applies the same 
 remark to the Moon. But it is only 
 an assertion on his part, which he 
 does not support by any proof or 
 argument, when he adds, that the 
 Moon always presents the same 
 hemisphere towards the Earth, be- 
 cause she does not rotate; and that 
 the astronomical explanation of the 
 Moon's librations is a fallacy no 
 longer available, the real cause of 
 these librations having yet to be 
 discovered. 
 
 Some of Mr. Perigal's other re- 
 marks are so true and so much to 
 the point, that we may be permitted 
 to transcribe them here, as his paper 
 reached us too late to insert them in 
 their proper place. "The contro- 
 versy regarding the Moon's rota- 
 tion," he states " is occasioned by 
 the misuse of technical terms, and of 
 equivocal definitions based upon the 
 erroneous idea that, if a body turns 
 round, it can only do so by rotating 
 about its own axis ; whereas, in fact, 
 it may turn round by revolving 
 about a centre or axis more or less 
 distant from itself. Thus there are 
 two ways of turning round ; rotation 
 round an interior axis or centre, 
 like the diurnal rotation of the 
 
 Earth ; and revolution round an ex- 
 terior centre, like the Earth in its 
 orbit. Thus, a grindstone rotates on 
 its axle, as a whole; while each of 
 its particles revolve round the same 
 axle, all describing concentric circles 
 larger or smaller, according to the 
 relative distance of each particle 
 from the centre common to all of 
 them as well as to the axle itself. 
 The preposterous notion that each 
 particle also moves round every 
 other particle is nonsensical so- 
 phistry. It has been asked, What 
 are the grounds for asserting that 
 the Moon rotates on her axis ? The 
 only grounds assigned are arbitrary 
 definitions of rotation and revolu- 
 tion, which we say do not properly 
 and strictly define either of them, 
 and, moreover, are not definitions, 
 but fallacious and untenable propo- 
 sitions. The Moon question is con- 
 venient for discussing these defini- 
 tions ; and their amendment is the 
 real point at issue. 
 
 " It is not denied that the assumed 
 rotation of the Moon is logically in- 
 ferred from the dogmatic definitions 
 of rotation and revolution, referred 
 to by a " Cambridge Wrangler." 
 Good Mathematicians seldom err in 
 their deductions from correct data: 
 but, not unfrequently, they base their 
 arguments and calculations on falla- 
 cious assumptions ; as in this ques- 
 tion of the Moon's imaginary rota- 
 tion." 
 
140 ON THE MOON'S ROTATION. 
 
 sumed rotation, since both periods are said exactly to coincide. 
 Indeed, even if the sidereal period of 27 days 7h. 43' ll", 5 
 had been the "true" period of the Moon's revolution, it could 
 yet possibly not have been the period of her rotation ; for it is 
 not to the stars that the Moon always presents the same face, 
 but, in the course of her revolution round the Earth, to the 
 Earth. In M a (Fig. 29) the Moon is at her full. When the 
 Earth has arrived in E g, the Moon, according to the astro- 
 nomical theory ought to be in M' </', between the star a and the 
 Earth^ presenting the same hemisphere o'o towards the star ; 
 but she is actually in M g as full Moon, with the Earth inter- 
 posing between the star and her, and consequently presenting 
 towards the Earth the opposite hemisphere to what she should 
 do in accordance with the astronomical theory. To render this 
 
 more clear,let, in Fig. 
 30, E represent the 
 Earth, M the Moon, 
 o a" her orbit ; a, b, c, 
 d as many points in 
 her surface, each two 
 separated by a dis- 
 tance corresponding 
 to the time of 2 days 
 5h. 11' rotation. Then, 
 if at any given mo- 
 Fl - 30 * ment the Moon be at 
 
 the point a" of her orbit in a straight line with its centre and 
 the centre of the Earth, and the point a in the lunar globe be 
 at the same moment facing the centre of the Earth, this point 
 will face it again in about 27^ days, and when the Moon, in 
 294 days, returns to the point a" of her orbit, the point b in her 
 surface will be opposed to the Earth's centre ; and instead of 
 the hemisphere d a </, which we saw before, we shall now see 
 the hemisphere e b e. After another revolution, the point c will 
 face the Earth, and the hemisphere / c f be turned towards 
 
ON THE MOON'S ROTATION. 141 
 
 her ; so that in the course of one single year the Moon, if she 
 rotated on her axis in 27 days 7h. 43' 1 1", 5, must of necessity 
 present in succession every portion of her entire surface to- 
 wards the Earth. Astronomers have disregarded this prominent 
 fact, a consequence of their erroneous view regarding the true 
 duration of the Moon's revolution round the Earth ; but that 
 Mathematicians, under the circumstances of the case, should 
 have been able to prove the astronomical theory from the laws 
 of gravitation* only serves to illustrate the truth of our previous 
 remark regarding the dangers of Analysis, when made to rest 
 upon Theory to explain facts. 
 
 62. To return to the period of the Moon's rotation. If a 
 second time we suppose the duration assigned to it by Astro- 
 nomers to be correct, a second time it proves to be irrecon- 
 cileable with the constant direction of the same face of the 
 Moon towards the Earth, on account of the Precession of the 
 Equinoxes, which modern Astronomy ascribes to a slow re- 
 trograde motion of the node of the Earth's equator on the 
 ecliptic. It affects the course of the Earth with regard to the 
 stars, performing a complete circuit of the heavens in about 
 25,600 years, whilst the Moon's period of assumed rotation is 
 fixed by Astronomers with reference to the stars. Hence the 
 difference of only a few seconds of time between the two 
 periods of the Moon's sidereal and tropical revolution the 
 duration of 27 days 7h. 43' 4", 7, assigned to the latter ( 59 
 and Note 56) being likely erroneous ; and hence also the ne- 
 cessity of the Moon turning in succession, and in the course of 
 about 25,600 years, every portion of her surface towards the 
 Earth, if the time of her rotation had regard to the stars. We 
 see no occasion, however, to demonstrate this more fully, as it 
 only shows what our preceding demonstration has already done, 
 namely, that the period assigned by Astronomers to the Moon's 
 assumed rotation is, and must of necessity be, an erroneous 
 one. 
 
 63. Thus far the results of our inquiry into the main question 
 
142 ON THE MOON'S ROTATION. 
 
 we have proposed to ourselves have been almost exclusively of 
 a negative character.. Still they are not altogether without 
 their value, nor without their bearing upon the final solution 
 of the problem. We will now pass on to the consideration of 
 the three different kinds of libration, commencing with diurnal 
 or parallactic libration. 105 
 
 If we place a very small globe of two or three inches in 
 diameter before us, and, turning our head to the right side, look 
 at it and mark some particular spot in its centre, and, in a 
 straight line with it, a similar spot on its right and left border ; 
 if we then turn our head to the left side, we shall find that the 
 spot on the right border, we saw before, disappears behind the 
 visible disk of the globe ; that the spot in its centre moves to 
 the right ; that another spot on the left moves into the centre ; 
 that the spot we saw before on the left border of the globe 
 moves upon the disk, and that another spot, previously invisible, 
 appears from behind the opposite hemisphere upon the left 
 
 IDS Une troisieme illusion pro- n'appercevrait pas du centre de la 
 
 vient de ce que 1'observateur est terre. A mesure que la lune s'ele~ve 
 
 place a la surface de la terre, et non sur 1'horison, ces points s'approchent 
 
 a son centre. C'est vers ce centre que du bord supe'rieur du disque, et enfin 
 
 la lune tourne toujours la meme disparaissent, tandisque d'autres se 
 
 face, et le rayon visuel, mene de la decouvrent vers son bord inferieur ; 
 
 lune, rencontrerait toujours sa sur- le meme effet se continue pendant 
 
 face au meme point, abstraction faite tout le terns que la lune est visible, 
 
 des inegalites precedentes. II n'en et comme la partie de son disque qui 
 
 est pas de meme du rayon visuel parait en haut a son lever, se trouve 
 
 mene' de la surface de la terre ; ce en bas a son coucher ; c'est a ces 
 
 rayon fait un angle sensible avec le deux instans que la difference est 
 
 precedent, a cause de la proximite sur-tout sensible. Ainsi dans son 
 
 de la lune ; cet angle, a 1'horison, rnouvement diurne, le globe lunaire 
 
 egale la parallaxe horisontale; par parait osciller autour du rayon 
 
 suite de cette difference, le contour vecteur, mene de son centre au 
 
 apparent du spheroid e lunaire n'est centre de la terre. On a design e ce 
 
 pas le meme pour le centre de la phenomene par le nom de libration 
 
 terre et pour 1'observateur place' a diurne" BioT,Traitel&n.d?Astron. 
 
 sa superficie. Celui-ci, lorsque la physique, tomelL, p. 407. (Compare 
 
 luiie se leve, decouvre vers son bord LAPLACE, Expos, du syst. du monde, 
 
 superieure, quelques points que Ton tome I., p. 57. 
 
ON THE MOON'S ROTATION. 143 
 
 border. In a precisely similar manner it is that, in conse- 
 quence of the Earth's rotation on her axis, we look every day 
 at the Moon, and precisely similar are the phenomena we 
 witness. 
 
 The reason of these phenomena is neither, as Astronomers 
 assert, that the Moon always turns the same face towards the 
 centre of the Earth ( 105 ), which is not strictly the case, nor 
 alone her proximity to us ( l05 ), for if the Earth were as small 
 a body as the Moon, or smaller, we should not perceive the 
 phenomena in question ; but the greater size of the Earth, ro- 
 tating on her axis, in connection with the Moon's proximity. 
 In consequence of this, whilst the Moon, as seen from the 
 Earth, subtends a mean angle of 3l' 7", the Earth, as seen from 
 the Moon, subtends a mean angle of 1 54' ; so that, if we take 
 two diameters at right angles to a line joining the centres, we 
 are enabled from the extremities of the Earth's diameter to look 
 round the corresponding extremities of the Moon's diameter ; 
 thus from both positions obtaining a view of somewhat more 
 than a perfect hemisphere of our satellite in the course of each 
 daily revolution about the terrestrial axis of rotation. Let in 
 Fig. 31 E represent the Earth, M the Moon. When any point 
 on the Earth's surface, we will here suppose any point of the 
 equator, and that point to be our own position, arrives in e' 
 we shall see the Moon 
 rise. Our visual ray 
 will meet the lunar sur- 
 face in b and b', and 
 the point d' occupy the 
 centre of the visible 
 disk, comprising the 
 hemisphere b d b'. 
 
 When we arrive in e, 
 
 f ^ FIG. 31. 
 
 m consequence of the 
 
 Earth's rotation, the Moon will set. Our visual ray will meet 
 the lunar surface in c and c ; the points b and a have gradually 
 receded from our view ; the point d' has apparently turned 
 
144 
 
 ON THE MOON'S ROTATION. 
 
 round to d" ; the point d has taken its place in the centre of the 
 visible disk, which now comprises the hemisphere c d c ; the 
 point b, previously on the left; border of the lunar globe, has 
 advanced upon it to near d', and the point c, first invisible, 
 has advanced into the place of the latter. Thus, we shall have 
 obtained a view of two small crescents of the Moon's surface, 
 corresponding to the arcs a b' and a c, beyond a perfect hemi- 
 sphere a d a ; and whilst we ourselves were actually turning 
 round from e' to e, the Moon appeared to us to turn round from 
 c to b' and from c to b. lo6 This apparent movement, conse- 
 quently, takes place both in a longitudinal, and, on account of 
 
 106 d g'ji n ' ex i s tait aucune des deux 
 librations dont nous venous de 
 parler, il y aurait un des rayons de 
 la lune qui resterait constamment 
 dirige vers le centre de la terre, un 
 observateur place en ce point verrait 
 toujours 1'extremite de ce rayon oc- 
 cuper exactement le centre du disqne 
 de la lune. Mais un observateur 
 place a la surface de la terre ne se 
 trouve pas dans les memes condi- 
 
 tions. Admettons, pour simplifier, 
 qu'en vertu du mouvement diurne, 
 la lune passe au zenith meme du point 
 A, d'ou on 1'observe. Aux diverses 
 heures de la journee, le rayon L a de 
 le lune, que nous supposons toujours 
 dirige vers le centre T de la terre, 
 doit paraitre prendre successivement 
 des positions diffe'rentes, telles que 
 L a, i! d, L" a". Lorsque la lune est 
 en L, peu de temps apres son lever, 
 le point a parait un peu a 1'orient du 
 centre c du disque lunaire; lorsque 
 la lune est au zenith, en L, ce point 
 parait en a! au centre du disque ; et 
 lorsque la lune est en L", peu de 
 temps avant de se coucher, le meme 
 point parait en ", un peu a 1'occi- 
 dent du centre c" du disque : le point 
 a doit done sembler osciller chaque 
 jour de part et d'autre de sa position 
 moyenne. Les taches de la lune 
 eprouveront evidemment un mouve- 
 ment d'oscillation analogue, quelle 
 que soit la position de 1'observateur 
 sur la terre : c'est ce mouvement que 
 Ton noinme la libration diurne" 
 DELAUNAY, Cours Mem. d'Astronomie, 
 pp. 397, 398. 
 
ON THE MOON'S ROTATION. 145 
 
 the inclination of the terrestrial axis of rotation, also in a lati- 
 tudinal sense differing in different latitudes and in the same 
 direction ; for which reason Astronomers improperly ascribe to 
 it an oscillatory character ( 105 , l06 ) . It is of a purely optical 
 nature, as is proved by the fact that, notwithstanding the 
 Moon thus seemingly rotating on her axis through a certain 
 arc every day, this arc never multiplies ; the same face of the 
 Moon continuing to regard the Earth. 
 
 The maximum of this diurnal libration of the Moon can 
 amount to 32'' ( 113 ), or to 2 3' ( 114 ), selenographically speaking. 
 
 64. The libration in longitude has first originated the theory 
 of the Moon's rotation ( 25 26). The general phenomena 
 which constitute it have been described by Newton (Note 1), and 
 consist in the Moon presenting to our view, periodically, small 
 crescents of her opposite surface, alternately upon her eastern 
 and western border, which periodically recede again from our 
 sight, as though she were oscillating to and fro about her centre, 
 with regard to a line joining that centre and the centre of the 
 Earth. Hevelius, the father of Selenography (died 1687), ex- 
 plained these phenomena by supposing that the Moon always 
 kept the same face turned, not towards the centre of the Earth, 
 but towards the centre of her orbit ; 107 Madler, however, justly 
 remarks that there is no point in her whole orbit except, 
 he erroneously adds, the centre of the Earth to which the 
 
 107 " Hevelius expliqua la libration velius, 1'axe de rotation toujours 
 
 en longitude, en supposant que la perpendiculaire a 1'ecliptique." 
 
 Lune presente toujours la meme face NICOLLET, M&n. sur la librat. de la 
 
 au centre de 1'orbe lunaire, dont la Lune, pp. 229, 230. 
 
 Terre occupe un des foyers. Newton, It is true that the practical illus- 
 
 dans une lettre addressee a Mercator, tration of the Moon's rotation which 
 
 en 1675, perfectionna 1'explication Newton suggests in his letters to 
 
 d' Hevelius, en la ramenant a 1'idee Mercator, implies the supposition 
 
 simple d'une rotation uniforme de la mentioned by Nicollet ; but he could 
 
 Lune sur elle-meme, pendant qu'elle not have read the Principia in attri- 
 
 se meut inegalement autour de la buting to Newton that supposition 
 
 Terre; mais il supposait avec He- as his real view of the subject. 
 
 L 
 
146 ON THE MOON'S ROTATION. 
 
 Moon always does present the same face. Consequently, the 
 supposition of HeveV was an erroneous one. Otherwise the 
 explanation offered by him, would have been the most simple 
 that could possibly have been given ; and it is difficult to un- 
 derstand the logic of Nicollet ( lo7 ), in stating that Newton 
 reduced that explanation to a still more simple idea by the 
 introduction of a new element into the lunar theory, that of 
 the Moon's rotation. The question, however, is not one of 
 simplicity, but of truth. 
 
 If we assume the Moon to rotate on her axis in precisely 
 the same time that she revolves in her orbit ; if we further 
 assume that orbit to be a perfect circle, and consequently the 
 velocity of both motions to be strictly uniform : the phenomena 
 of the libration in longitude could not possibly take place. But 
 the Moon's orbit being an ellipse, and her angular velocity in 
 her orbit, therefore, alternately exceeding and falling short of 
 her mean velocity, whilst the velocity of rotation, unaffected by 
 the form of the orbit, continues the same, it is evident that 
 the Moon, under this supposition, would present phenomena 
 similar to those which are actually observed. 108a Hence has 
 
 iosa "Supposons que la rotation de ment la meme, et ces deux effets se 
 
 la lime soit sensiblement uniforme ; surpassent tour-a-tour. Les points du 
 
 c'estadire, qu'elle ne participe pas globe lunaire doivent done paraitre 
 
 aux inegalites periodiques : cette tourner tantot dans un sens, tantot 
 
 supposition est au moms la plus dans un autre, autour de son centre, 
 
 naturelle que Ton puisse faire, et et 1'apparence qui en resulte est la 
 
 la theorie a prouve qu'elle est ri- meme que si la lune avait un petit 
 
 goureuse. Alors une des causes qui balancement de part et d' autre du 
 
 produisent la libration devient evi- rayon vecteur mene de son centre a 
 
 dente ; car le mouvement de revolu- la terre. C'est ce que Ton nomme 
 
 tion participant aux inegalites pe- la libration en longitude" BIOT, 
 
 riodiques, est tantot plus lent, tantot Traiteeltm. 'd'Astron. physique, tome 
 
 plus rapide. La rotation apparente II., p. 406. (Compare LAPLACE, 
 
 qu'il occasionne ne peut done pas Expos, du syst. du monde, tome I., 
 
 toujours contrebalancer exactement pp. 54 56.) 
 la rotation reelle, qui reste constam- 108 b " Nothing can place the doc- 
 
ON THE MOON'S ROTATION. 147 
 
 sprung the theory of Newton, 108b adopted by all modern As- 
 tronomers, though by none of them ever considered as to its 
 real bearings. Let us do so. 
 
 That, in accordance with the empirical fact of the Moon 
 keeping always nearly the same face turned towards the Earth, 
 the periods of her revolution round the latter, and of her rota- 
 tion must be assumed to strictly coincide, Newton himself has 
 stated (*), and is generally recognized as a fact; and that 
 an essentially erroneous duration has, by the author of the 
 Principia and his followers, been ascribed to both these 
 periods, we have shown to evidence in a preceding paragraph. 
 The error, however, proves only against Astronomers, not 
 against their theory of the Moon's rotation. But a first ques- 
 tion, vitally affecting that theory, and which yet would seem 
 never to have presented itself to them, arises ; namely, whether 
 the period of the actual phenomena of libration in longitude 
 coincide with the periods of the Moon's revolution and assumed 
 rotation? For it is evident that if those phenomena be the 
 effect and the result of rotation, they cannot possibly have any 
 other period but that of rotation, in this instance the common 
 period of rotation and revolution ; and that, if they differ, the 
 phenomena of libration in longitude cannot possibly, in ac- 
 cordance with the main features of the case, be the result of 
 rotation. And the two periods do differ. Observation has 
 established beyond doubt the fact, that the period of the libra- 
 tion in longitude is that of the lunar apsides. It reaches its 
 maxima at the two extreme points B and E of the minor axis 
 
 trine of Astronomy regarding the others (Note 20) would render the 
 
 distinctive nature of the Moon's ro- theory impossible db initw. Those 
 
 tatory motion, as compared with her who hold it, can hardly ever have 
 
 progressive motion, in a clearer light read the proposition of Newton 
 
 than the very theory of Newton. It (Note 37). Littrow evidently had 
 
 is based upon that distinction. The not done so ( 25) 
 opinion of Mr. W. Hopkins and 
 
 L 2 
 
148 ON THE MOON'S ROTATION. 
 
 of the Moon's orbit, Fig. 32, gradually decreasing from B to D, 
 
 the Moon's apogee ; 
 as gradually increas- 
 ing again from D to 
 E ; decreasing from E 
 to A, the Moon's pe- 
 rigee; and increasing 
 again from A to B ; so 
 that at the Moon's 
 apogee and perigee 
 the librationin longi- 
 FlG ' 32 ' tude is = 0,109 i. e. 
 
 reaches its minima. If, therefore, the libration in longitude be 
 the effect of the Moon's rotation on her axis, we may look upon 
 one of those two points when it becomes=o, as we would upon 
 a spot on the lunar surface, indicating by its periodical return 
 to the same plane, passing through the centres of the Earth and 
 the Moon, perpendicular to her orbit, the period of the Moon's 
 rotation ; in precisely the same manner as we measure the period 
 of rotation of a uniformly revolving wheel, by the periodical re- 
 turn of some point in its circumference to a normal or a fixed 
 straight line, passing through the centre of its axle. But we 
 have already seen that the lunar apsides have a motion of their 
 own, performing, as compared with the Moon's return to the 
 same point of her orbit, a complete revolution in 8 years, 310 
 days, 13h. 48', 58", in the same direction in which the Moon 
 
 109 j)ie hauptsachlichste Un- zusammen (was beilaufig alle 3 
 
 gleichheit in der Langc des Mondes Jahre geschieht) so steht der Mond 
 
 1st = Null im Perigao und Apogao, in mittlerer Libration ; abgesehen 
 
 und folglich ist auch die Libration von den kleinen noch iibrig blei- 
 
 in Lange desto kleiner, je naher er benden Ungleichheiten des Laufs, 
 
 diesen Punkten steht. Befindet sich und der parallactischen Libration." 
 
 ferner der Mond im Knoten seiner BEER und MIDLER, Selenograplde, 
 
 Bahn, so fallt die Libration der S. 14, vgl. S. 86. 
 Breite hinweg; und trifft beides 
 
ON THE MOON'S ROTATION. 149 
 
 revolves round the Earth. 110 The consequence is that they 
 gain upon the Moon in the course of each revolution about 3^, 
 and that it will take the Moon, after completing her circuit, 
 about six hours more to reach her apogee or her perigee again. 
 Astronomers have termed this longer period an anomalistic 
 month, or the period of the Moon's anomalistic revolution, and 
 they erroneously assign to it a duration of 27 days I3h. 18' 
 37", 4 instead of about 29 days 18h. This error, however, does 
 not affect our argument, for, whether we compare the true 
 periods of the Moon's revolution in her orbit and of the revo- 
 lution of her apsides, or those assigned to both by Astrono- 
 mers, there will, in either case, appear the difference of about 
 six hours between them, and, consequently, the libration in 
 longitude cannot possibly be the result, and as such indicative, 
 of the Moon's rotation. Or, if it were, she must, of necessity, 
 in the course of each period of about 8 years 310 days, present 
 in succession every portion of her entire surface towards the 
 Earth, which she does not do : for, if we refer to Fig. 30, p. 140, 
 and let a be the point in the lunar surface, corresponding to 
 one of the minimum-points of the Moon's libration in longitude, 
 as seen from the centre of the Earth when the Moon is at the 
 point " of her orbit ; then, on her next return to this point, 
 the point a on her surface will already have passed it six hours 
 earlier, or by an arc of about 3j, so that we shall see a cres- 
 
 110 "DieobenbezeichnetenPerioden er schon in 8 Jahren, 310t. 13h. 68', 
 
 beziehen sich sammtlich nur auf den 53", um den ganzen Himmel herum- 
 
 mittleren Zustand, wie er fur die lauft. Die Bewegung ist eine 
 
 gegenw'drtige Zeit stattfindet, da eine direkte. also im Sinne der Mond- 
 
 Menge von Ursachen dazu beitragen bahn selbst, deshalb braucht der 
 
 sie sammtlich veranderlich und den Mond langere Zeit, sein Perigaum 
 
 Mondlauf zu einem aussert verwi- wiederzuerreichen,alsergebrauchte, 
 
 ckelten zu machen. Die erste dieser um seinen siderischen Umlauf zu 
 
 Ursachen ist die elliptische Form vollenden. Man nennt diese Zeit 
 
 der Mondbahn, verbunden mit dem den anomalistischen Umlauf und sie 
 
 Umstande, dass der Ort der Erd- betragt 27t 13h. 18', 37", 4. 
 
 nahe (Perigaum) selbst, und zwar MIDLER, Popul'dre Astronomie, s. 
 
 sehr rasch veranderlich ist, so dass 154. 
 
150 
 
 ON THE MOON'S ROTATION. 
 
 cent of Sy of her opposite hemisphere upon her eastern border 
 which we did not see before. And thus, after each further re- 
 volution of the Moon in her orbit, a further crescent of 3^, in- 
 visible on the completion of the preceding revolution, will 
 appear into view, every point of her surface passing in succes- 
 sion the line joining the centre of the Earth and the Moon, in 
 the course of about 110 revolutions. 
 
 65. A second question, of equal importance regarding the 
 Newtonian theory, is, Whether the orbital points of the maxima 
 and minima of the lunar libration in longitude, as deduced 
 from that theory, correspond with those established by observa- 
 tion ? Astronomers have generally assumed with Newton that 
 such is the case, and have demonstrated it. We will literally 
 quote the words of Madler, 111 merely changing the conventional 
 cyphers employed by him, to adapt them to Fig. 33. " Let E 
 " represent the Earth in one of the foci of the elliptical orbit of 
 
 111 " MIDLERS Popul. Astrono- 
 mie, s. 162. Delambre, in perfect 
 accordance with him, but by a dif- 
 ferent process of reasoning, says, 
 "Voyons maintenant ce qui doit 
 r^sulter du mouvement inegal de la 
 lime dans son orbite. Le move- 
 ment de rotation parait uniforme, il 
 doit etre egal au mouvement moyen 
 de la lune. Soit M ce mouvement 
 pour un terns quelconque ; le mouve- 
 ment vrai du centre de la lune sera 
 
 M 2 e sin M, en ne considerant 
 que le terme principal de 1'equation 
 
 du centre. Que la lune se soit 
 avancee de A a B, Tangle A T B sera 
 M 2 e sin u; menons B a' paral- 
 lele a T A, nous aurons a' B T=M 2 
 e sin u; prenons T B a=2 e sin u t 
 nous aurons a' B a=M 2 e sin u -f 
 2 e sin W=M, 1'arc a' a sera le mouve- 
 ment de rotation de la lune. Le 
 point a ne nous paraitra plus re- 
 pondre au centre de la lune, ce sera 
 le point b; le point a sera done 
 moins avance en longitude, d'un 
 arc b a =2 e sin u, ou plus exacte- 
 ment, d'un arc b =somme de toutes 
 les inegalites de mouvement en lon- 
 gitude. Nous supposons les inega- 
 lites soustractives ; si elles etaient 
 additives, le point a paraitra plus 
 avance; si elles etaient nulles, le 
 point a se retrouverait au centre." 
 DELAMBRE, Astronomic, tome iii. p. 
 63, 64. 
 
ON THE MOON'S ROTATION. 
 
 151 
 
 " the Moon. The Moon be in M", and d" be a point on her 
 " surface directly facing the Earth, and as seen from her occu- 
 " pying the centre of the lunar disk. Having performed one 
 " quarter of her revolution and rotation, the Moon will be in 
 " M'", and the point d" has turned round to the extent of J of 
 
 FIG. 33. 
 
 the circumference of the lunar globe. But the angle d" E e y 
 exceeding a right angle by the full sum of the equation of the 
 centre : the central line, joining both bodies, does not meet 
 the point cT, but a point e, to the west of d" ; and e d" is 
 now the (maximum of) libration. When the Moon has ad- 
 vanced to M, the point d" will have completed simultaneously 
 half a revolution, and half a rotation ; it, therefore, directly 
 faces the Earth again, and there is now no libration. In M' 
 the point d" has made f of a rotation, whilst the Moon herself 
 has not as yet performed f of her orbit (Madler represents 
 the Moon's angular velocity as her real velocity), in conse- 
 quence of which a point e, to the east of d', now occupies the 
 centre of the lunar disk, and e d' is the Moon's libration on 
 
152 ON THE MOON'S ROTATION. 
 
 " the side opposite to the former." All this most perfectly 
 agrees with the phenomena actually observed, but does it agree 
 with Newton's theory ? We will quote Madler again, who, in 
 the following few words to the point, sums up his preceding 
 explanation of the theory of elliptical motion : " Elliptical mo- 
 " tion," he states in his " Popular Astronomy," p. 89, " conse- 
 " quently reaches its greatest rate of velocity in the Earth's pe- 
 " rihelion (or the Moon's perigee), which rate is slowest in her 
 "aphelion (or in the Moon's apogee). Between these two 
 " points it constantly increases ; constantly decreasing in the 
 " opposite half of the orbit. The mean velocity takes place 
 "about halfway between the points named, i. e. at the two 
 " extremities of the minor axis." According to the Newtonian 
 theory, therefore, the mean libration of the Moon in longitude 
 takes place at those points of her orbit, at which her velocity of 
 revolution is greatest and least, namely, in the Moon's perigee 
 and apogee. But, according to the same theory, the periods of 
 the Moon's rotation and revolution coincide as well as the uni- 
 form velocity of the latter, and the mean velocity of the former. 
 They can, therefore, coincide only in those two points of the 
 lunar orbit at which the velocity of the Moon's revolution is at 
 its mean rate. This may at first sight be considered para- 
 doxical, as many other truisms, f. i. the well known problem of 
 Aristotle's wheel have been; but the explanation is simple: the 
 two motions being united in one body, and the mean coinci- 
 dence of their respective velocities being the necessary conse- 
 quence of the coincidence of their periods, the two motions can 
 have but one common starting point at one common rate of 
 velocity. That starting point is of necessity one of the points 
 of contact between the two motions, or between the ellipse, re- 
 presenting the Moon's motion of revolution, and the inscribed 
 circle representing her motion of rotation,^, e. one of the extre- 
 mities of the minor axis of her orbit, and where observation 
 shows that the velocity of both motions actually is the same. 
 Astronomers, on the contrary, gratuitously assume the Moon's 
 
ON THE MOON'S ROTATION. 153 
 
 perigee to be the starting point, 112 a point where no contact, but 
 the maximum of difference between the two velocities takes 
 place. In other words, they suppose the Moon to have pro- 
 gressed in her orbit from M' to M" before she began to rotate, so 
 that her motion of revolution is constantly 90 in advance of 
 her motion of rotation, implying the revolving Moon to be in 
 M", whilst the rotating Moon is still in M'. Hence, as the 
 Moon's mean libration in longitude necessarily corresponds to 
 her mean velocity of revolution, Astronomers, by making the 
 former coincide with the maxima and minima of her velocity 
 of revolution, make the latter, in their turn, coincide with her 
 mean velocity of revolution a positive contradiction in terms. 
 But without incurring this contradiction, which expresses an 
 impossibility, the Newtonian theory is not in accordance with 
 the actual phenomena, and, consequently, does not explain 
 them ; for, if we take one of the real points of contact between 
 the two motions of the Moon's revolution and rotation for a 
 starting point ; the maxima of the lunar libration in longitude 
 would take place in her apogee and perigee, where actually 
 the minima occur, and the minima would occur at the two 
 
 112 " Man lasse neben dem wirk- tleren geworden 1st. Von hier ab 
 
 lichen in der elliptischen Bahn urn- wird der Vorsprung des wahren 
 
 kreisenden Korper noch in Gedanken Korpers fortwahrend geringer, bis er 
 
 einen zweiten im Kreise umlaufen, endlich im Aphelio (oder Apogao) 
 
 gebe beiden die gleiche Umlausfzeit zu Null wird und er denselben Ort 
 
 und lasse sie gleichzeitig im Perihel einnimmt, den der fingirte durch 
 
 (oder Perigao) stehen. Von diesem seine gleichformige Bewegung 
 
 ausgehend, wird der fingirte Korper. erreicht hat. Von hier ab ist der 
 
 aufangs hinter dem wahren zuriick- wahre Korper im Nachzuge, und 
 
 bleiben, da jener sich mit der mittle- der Unterschied wird immer grosser 
 
 ren, dieser mit der grossten Ge- bis die Geschwindigkeit in der wah- 
 
 sehwindigkeit bewegt. Dieses Voran- ren Bahn abermals die mittlere 
 
 eilen des wahren Korpers wird geworden ist, und nun vermoge des 
 
 zunehmen, so lange er einen Ueber- immer schnelleren Laufes der wahre 
 
 schuss der Geschwindigkeit zeigt, Planet dem fingirten wieder naher 
 
 und es wird sein Maximum da kommt, und im Perihel (oder Peri- 
 
 erreichen, wo die Geschwindigkeit gao) ihn wieder einholt." 
 
 in der wahren Bahn gleich der mit- LEE, Po^ul. Astronomic, s. 99. 
 
154 ON THE MOON'S ROTATION. 
 
 extremities of the minor axis, where actually the maxima take 
 place. 
 
 The latter may amount to 4? 20", 113 or, selenographically 
 speaking, to about 15 50'. 114 Astronomers, though they as- 
 cribe the Moon's libration in longitude to a physical difference 
 of her real motions of rotation and revolution, yet at the same 
 time insist on its purely optical nature ; a contradiction, which 
 does not betray very clear ideas on the subject. 
 
 66. The phenomena of the Moon's libration in latitude are 
 in every respect similar to those in longitude, only that the 
 former libration is perpendicular to the latter, the corresponding- 
 crescents of the opposite hemisphere of the Moon alternately 
 appearing and disappearing upon her northern and southern 
 border. There would seem to exist some confusion of ideas 
 regarding this libration among astronomical writers, arising 
 probably from the manner in which it is viewed or represented 
 by some Astronomers. Thus Biot Il5a evidently takes it in the 
 
 113 " L' amplitude totale de la li- " Die grosste Neigung des Mond- 
 bration en longitude, pourune tache aquators gegen seine Bahn kann 6 
 situee au centre du disque de la 47' betragen, was das Maximum der 
 lune, est d'environ 4' 20", ce qui Libration in Breite ergieht. 
 
 fait a peu pres 7 du diametre appa- " Die grosste parallactische Libra- 
 rent de ce disque. Pour la meme tion endlich ist der Winkel, unter 
 tache, la libration en latitude a une welchem der Erdhalbmesser vom 
 amplitude totale d'environ 3' 35", et Mondcentro aus gesehen warden 
 la libration diurne, seulement de kann, und dieses iibersteigt nie 
 32". Ces trois librations, existant 1 1, 5." BEER und MIDLER, Se- 
 simultanement, il en resulte, pour lenographie, s. 13. 
 chaque tache de la lune, un mouve- . U5 " Les taches de la lune ne 
 ment d'oscillation complexe, qui est conservent pas toujours la meme 
 celui que Ton observe en realite." elevation au dessus du plan de son 
 DELAUNAY, Cours Eletn. d'Astrono- orbite ; quelques-unes meme, par 
 mie, p. 398. 1'effet de la rotation, passent d'un 
 
 114 " Die Summe sammtlicher Un- cote de ce plan au cote oppose. Ces 
 gleichheiten des Mondlaufes in circonstances indiquent un axe de 
 Lange kann auf 7 55' gehen, und rotation qui n'est pas exactement 
 so gross ist also auch das Maximum perpendiculaire au plan de 1'orbe 
 der Libration in Lange, sowohl der lunaire ; or, selon que cet axe nous 
 ostlichcn als westlichen. presente sa plus grande on sa plus 
 
ON THE MOON'S ROTATION. 
 
 155 
 
 restricted sense of the effects of the Moon's assumed rotation, 
 as distinct from the similar effects of her change of latitude ; 
 whilst both effects combined are usually understood by the 
 term in question. Newton himself regarded it so (37), though 
 in another place (*) his mode of expression would imply the 
 combined effect to be due to the Moon's rotation alone. And 
 thus it has been erroneously understood by Schubert, 115b De- 
 launay, 116 Lardner, 11 * and others. Laplace 118 considers each 
 
 petite obliquite, il doit nous decou- 
 vrir successivement les deux poles 
 de rotation du spheroide lunaire ; de 
 meme que 1'axe de la terre presente 
 successivement ses deux poles au 
 soleil dans les deux solstices. Nous 
 devons done appercevoir dans cer- 
 tains terns quelques-uns des points 
 situes vers ces poles, et les perdre de 
 vue ensuite, lorsqu'ils se rapprochent 
 du bord apparent ; c'est ce que Ton 
 nomme la libration en latitude. 
 Elle est peu considerable, ce qui 
 indique que 1'equateur de la lime 
 difieretres peu du plan de son orbite." 
 BIOT, Traite EUm. dAstron. 
 physique, tome ii. p. 406-7. 
 
 ash j) as Schwanken nach der 
 Breite konnte nicht statt linden, 
 wenn die Axe des Monds genau auf 
 seiner Bahn senkrecht ware; dies 
 ist aber nicht der Fall : der Aequator 
 des Monds fallt nicht mit seiner 
 Bahn zusammen, sondern erhebt 
 sich nordlich und siidlich iiber sie, 
 und der ganze Begen betragt 10 
 Grad. Es tritt also bald der Nord, 
 bald der Siid-pol des Monds, nebst 
 den hinter ihm liegenden Flecken } 
 in die uns sichtbare Scheibe, und 
 die Flecken riicken iiberhaupt um 
 10i von Norden nach Siiden hin 
 und her." SCHUBERT, Popul. Astro- 
 nomic, bd. iii. s. 84. 
 
 116 " L'axe de rotation de la lune, 
 au lieu d'etre exactement perpen- 
 diculaire au plan de son orbite, est 
 un peu incline sur ce plan : il se 
 transporte parallelement a lui-meme 
 en faisant avec la perpendiculaire 
 au plan de 1'orbite un angle d'environ 
 6 37'. II est facile de voir que cette 
 seule circonstance suffit pour occa- 
 sionner une libration des taches. 
 Pour nous en rendre compte, pre- 
 nons la lune dans deux positions 
 diametralement opposees sur son 
 orbite, en L et en i/. 
 On voit tout de suite 
 que, lorsque la lune est 
 en L, on ne peut pas 
 apercevoir son pole p, 
 et Ton aper9oit sans 
 peine le pole oppose q 
 tandisque, lorsque la 
 lune est venue en if, 
 le pole p est devenu 
 visible, et le pole q est 
 devenu invisible a son 
 tour. Un point a de 
 1'equateur lunaire pa- 
 rais sait, dans le pre- 
 mir cas, audessus du 
 centre du disque ; lors- 
 que la lune a fait un 
 demi-tour autour de 
 la terre, pour venir en L', et que, par 
 consequent, elle a fait a peu pres un 
 demi-tour sur elle-meme, autour de 
 
156 
 
 effect se 
 
 ON THE MOON'S ROTATION. 
 
 sparately ; Santini, 11 ^ Lalande, and the majority of 
 
 son axe p q, le point a de 1'equateur 
 lunaire est venu se placer en a, e'est- 
 a-dire au-dessousdu centre du disque. 
 Ainsi, par suite de 1'obliquite de 1' 
 axe de rotation de la lime par rap- 
 port au plan de son orbite, les taches 
 de sa surface doivent eprouver un 
 mouvement d'oscillation, dirige per- 
 pendiculairement au plan de 1'orbite, 
 c'est-a-dire a peu pres perpendicu- 
 lairement au plan de 1'ecliptique." 
 DELAUNAY, Cours Elem. d'Astrono- 
 mie, p. 396. 
 
 a? When the north pole leans 
 towards the earth we see a little 
 more of that region, and a little less 
 when it leans the contrary way. 
 This variation in the northern and 
 southern regions of the moon visible 
 to us is called the libration in lati- 
 tude." LARDNER, Handbook of As- 
 tronomy, p. 190. Compare Note 71. 
 
 118 " Le globe lunaire a une autre 
 libration en latitude, perpendiculaire 
 a celle en longitude, et par laquelle 
 les regions situees vers les poles de 
 rotation de ce globe, disparaissent et 
 reparaissent alternativement. Pour 
 concevoir ce phenomene, supposons 
 1'axe de rotation, perpendiculaire a 
 1'ecliptique. Lorsque la lime sera 
 dans son noeud ascendant, ses deux 
 poles seront au bords austral et 
 boreal de 1'hemisphere visible. A 
 mesure qu'elle s'elevera sur 1'eclip- 
 tique, le pole boreal et les regions 
 voisines disparaitront, tandis que les 
 regions voisines du pole austral se 
 decouvriront de plus en plus, jusqu' 
 au moment ou 1'astre parvenu a sa 
 plus grande hauteur boreale, com- 
 mencera a revenir vers 1'ecliptique. 
 Les phenomenos precedens se repro- 
 
 duiront alors dans un ordre in vers ; 
 et lorsque la lune par venue a son 
 noeud descendant, s'abaissera sous 
 1'ecliptique, le pole boreal presentera 
 les pheuomenes que le pole austral 
 avait offerta. 
 
 " L'axe de rotation de la lune n'est 
 pas exactement perpendiculaire a 
 1'ecliptique; et son inclinaison pro- 
 duit des apparences que Ton peut 
 concevoir en supposant la lune mue 
 sur le plan meme de 1'ecliptique, de 
 maniere que son axe de rotation 
 reste toujours parallele a lui meme. 
 II est clair qu' alors chaque pole sera 
 visible pendant une moitie de la re- 
 volution de la lune autour de la 
 terre, et invisible pendant 1'autre 
 moitie, en sorte que les regions qui 
 su sont tres voisines seront alterna- 
 tivement decouvertes et cachees." 
 LAPLACE, Exposition du systems du 
 monde, tome i. p. 56, 57. 
 
 119 < p er conce pi re come segna la 
 librazione in latitudine, rappresenti 
 T il centre della terra, T E il piano 
 dell' ecclittica, T L 1'orbita lunare in- 
 clinata oil' ecclittica circa 5, T c il 
 piano dell' equatore lunare inclinato 
 
ON THE MOON'S ROTATION. 157 
 
 Astronomers consider both effects conjointly. United, their 
 maxima amount to 3' 35" ( 113 ), corresponding to about 13 34' 
 (, 114 ) of which 10 36' or the united maxima of the Moon's 
 greatest latitude of 5 ] 8' are due to her latitude, and the re- 
 maining 2 58' to the assumed inclination of her axis of rota- 
 tion to the ecliptic at an angle of about 1 29'. Astronomers 
 fall here into the same contradiction, pointed out at the conclu- 
 sion of the preceding paragraph ; for, whilst ascribing the 
 Moon's libration in latitude partly to the physical cause of her 
 assumed rotation, they at the same time vindicate its purely 
 optical character. 
 
 Let, in Fig 34, E represent the Earth, M the moon ; e e 
 her orbit ; a a the ecliptic ; b d be perpendicular to the eclip- 
 tic. Then r r will be the Moon's axis of supposed rotation, 
 for the present assumed to be constant as to direction in 
 space, and inclined to the ecliptic at a constant angle of 
 about 1 29' = the arc b r ; 1 1 the lunar equator, likewise 
 inclined to the ecliptic at a constant angle of about 1 29'= 
 the arc a /; and whether the Moon be in M, M,' M", or 
 M'", r' r will at all times be parallel to itself, as will also be /' /. 
 
 solo 2. Sia la Luna (mentre passa sottende 7 nel centre della Lima 
 
 al meridiano) in L avendo una lati- verso il sud. Questo punto, e quindi 
 
 tudine boreale di 5, e ravvolgasi in- tutti gli altri hanno avuto un moto, 
 
 torno all' asse L K perpendicolare in virtu di cui sonosi transportati 
 
 all' equatore lunare. Coudotta A G L dal nord al sud di un arco uguale a 
 
 paralella a T c, rappresentera questa 10 del disco lunare, vale a dire le 
 
 la traccia dell' equatore lunare, ed macchie sembrano essersi avanzate 
 
 un suo punto G apparira distante verso il sud di una nona parte del 
 
 dal centre apparente di una quan- semidiametro apparente. In seguito 
 
 tita G D che sottende verso il nord la latitudine della Luna ritornando 
 
 un angolo G L D=3. Dopo 14 15 indietro per gli stessi gradi, le 
 
 giorni la Luna ritornando al meridi- macchie tornano pure al nord per gli 
 
 ano in M con una latitudine australe stessi gradi mostrandosi quel moto 
 
 di 5, la traccia dell' equatore e nella apparente oscillatorio dal nord al sud 
 
 linea Q M H parallela a T c, il punto che abbiamo definite per librazione 
 
 G trovasi in Q distante dal centro in latitudine" SANTINI, Elementi di 
 
 apparente F di un arcs=Q F che Astronomia, Vol. i. p. 225. 
 
158 
 
 If we now wish to 
 consider the libra- 
 tion due to the 
 Moon's assumed 
 rotation, distinct 
 from that due to 
 her latitude ; we 
 may,with Laplace, 
 regard the Moon 
 as moving in the 
 plane of the eclip- 
 tic. The position 
 M shall correspond 
 to the Moon's 
 greatest northern 
 latitude, and her 
 north pole r be 
 averted from the 
 Earth : we then 
 see her south pole 
 advanced upon 
 her disk, and a 
 crescent of her op- 
 posite hemisphere 
 beyond the south 
 pole, answering 
 to the arc r f d' = 
 
 1 29', which we 
 FIG. 34. , , I 
 
 should not have 
 
 seen but for the Moon's supposed rotation, since otherwise 
 the point / would have been in d. In the Moon's position 
 E, answering to her descending node, we see the same he- 
 misphere, we should have seen, did she not rotate ; with this 
 difference, however, that the point r, in the latter case, would 
 
ON THK MOON'S ROTATION. 159 
 
 have been in b, and the point r' in a. In the position M", 
 answering to the Moon's greatest southern latitude, her south 
 pole r is averted from us; we see the north pole advanced 
 upon her disk, and a crescent of her opposite hemisphere 
 beyond the north pole, corresponding to the arc r' d"= l' 29" 
 which we should not have seen but for the Moon's assumed 
 rotation, since otherwise the point r' would have been in d". 
 In the Moon's position E, now corresponding to her ascending 
 node, we see again the same hemisphere, we should have seen, 
 did she not rotate ; with this difference, that, in the latter case, 
 the point r" would have been in b, and the point r in d ' . 
 Thus, the Moon's libration in latitude, due to her assumed 
 rotation, reaches its maxima at her greatest northern and 
 southern latitude ; being = 0, or at its mean, in her nodes. The 
 same is the case w r ith her libration due to her latitude, answer- 
 ing to the arcs a' e and e' a". The maxima and minima, there- 
 fore, fall together ; the libration gradually increasing from to 
 6 47', as the Moon progresses from one of her nodes to her great- 
 est northern and southern latitude; and as gradually decreasing 
 again, as she progresses from the points of her greatest latitude 
 towards one of her nodes. If Astronomers were consistent in their 
 mode of explaining the Moon's libration, they ought again to have 
 supposed the revolving Moon to be 90 in advance of the rotating 
 Moon ; but here they rationally take the point of contact the 
 mean latitude and the mean libration for the starting-point. 
 
 67. The general phenomena of libration in latitude, as de- 
 duced from the theory of Newton, perfectly correspond with 
 those of actual observation ; but they will not bear a closer in- 
 vestigation, whilst the theory itself as utterly fails, as we have 
 seen that of the libration in longitude to do. In the first place, 
 the same vital question presents itself, does the period of the 
 libration in latitude coincide with the periods of the Moon's 
 revolution and assumed rotation ? For we have already shown 
 the impossibility, in the contrary case, of libration being the 
 
160 ON THE MOON'S ROTATION. 
 
 result of rotation. And not being the result of rotation, libra- 
 tion cannot possibly prove or be reconcileable with it. 
 
 Observation shows that the period of the Moon's libration in 
 latitude is that of her nodes ( 10 9, 118 , 11 9), which we have found 
 to perform a complete retrograde circuit of the heavens, with 
 regard to the period of the Moon's revolution in her orbit, in 
 the course of 18 years 218 days 21h. 22, 46". The consequence 
 is, that the nodes, in each revolution, go to meet the Moon in 
 her orbit about H ; and that the Moon thus reaches again 
 the same node about 2f hours before she returns to the same 
 point of her orbit. Astronomers term this shorter period a 
 draconic revolution of the Moon, and erroneously assign to it a 
 period of 27 days 5h. 5' 36", 12 instead of about 29 days lOh. 
 But, whether we take the true periods of the Moon's libration 
 in latitude and of her revolution, or as determined by Astro- 
 nomers, their difference will in either case amount to about 2f 
 hours; and consequently, the libration in latitude cannot pos- 
 sibly, any more than the libration in longitude, be the effect, 
 and as such indicative, of the Moon's rotation. For, if it were 
 otherwise, she must of necessity, and in the course of every 
 period of about 18 years 219 days, or of about 230 revolutions, 
 present in succession her entire surface to our view ; which she 
 does not. 
 
 68. If we were to consider the apparent motion of each point 
 of the lunar disk visible to us, under the supposition of the 
 
 120 Die Neigung der Monds- hung auf die Mond-und Erdbahn, 
 
 bahn gegen die Ekliptik betragt um den Himmel herum. Daraus 
 
 5 8' 89'' und ist gleichfalls ver'an- folgt, dass der Mond nicht voile 
 
 derlich. Sie kann bis zu 5 0' sinken 360 zu durchlanfen hat, um wieder 
 
 und auf 5 18' steigen. Der Durch- zuni aufsteigenden Knoten zu ge- 
 
 schnitts punkt dieser Bahn mit der langen, dass also der draconitische 
 
 Ekliptik ist einer raschen Ver'an- Umlauf, wie man diese Periode 
 
 derung unterworfen ; er geht inner- genannt hat, nur 27 1 5h. 5' 36", 
 
 halb 18 Jahren 218t. 21h. 22' 46", w'ahrt." MIDLER, Popul. Astro- 
 
 und zwar gegen die Ordnung der nomie, S. 154. 
 Zeichen, also retrograd in Bezie- 
 
ON THE MOON'S ROTATION. 161 
 
 Moon's rotation about an axis, inclined to the ecliptic and fixed 
 with regard to space, combined with her motion in latitude and 
 longitude, and with the effects of the Earth's rotation and revo- 
 lution, and of the inclination of her axis, we should find that 
 motion to be for each point an exceedingly varied and compli- 
 cated one. There is, however, no necessity for us to enter 
 into these complications. We will trace only the general path 
 of the two principal points, the assumed poles, having at the 
 same time the by far simplest motion. For it is evident that, 
 if we suppose the north pole r to appear into view in r" (E) on 
 the Moon's eastern border ; to advance, in about a week, upon 
 the disk to o ; and, in another week, to disappear again on the 
 western border in r ; it must, during that fortnight, have traced 
 the path r" o r\ and that, in the course of the next fortnight, 
 the south pole must trace the similar path r o r". But, in 
 reality, the points of the assumed poles, during each revolution 
 of the Moon round the Earth, trace nearly the perpendicular ( 118 ) 
 paths b o, o by and a o, o' a respectively m ; which indicates a 
 phenomenon of that purely optical nature, Astronomers them- 
 selves claim for the Moon's libration in latitude, and which is 
 as irreeoncileable with her rotation as is the difference between 
 the two periods of rotation and libration. 
 
 121 "Findeteine Libration in Lange Meridiane bleiben zwar ebenfalls 
 
 statt, so werden die Paralleleii gar mehr oder minder offne Ellipsen, 
 
 nicht, und die Meridiane auch nur aber ihr sichtbarer Theil ist von den 
 
 in so fern geandert, dass Rand und unsichtbaren nicht mehr durch den 
 
 Mitte von andern Meridianen einge- ersten Meridian getrennt. 
 nommen werden als vorhin ; die " Sind die Punkte nach Norden 
 
 optische Ansicht des ganzen seleno. verschoben, so erblioken wir den 
 
 graphischen Gradnetzes bleibt die- Siidpol, und noch etwas u'ber diesen 
 
 selbe. hinaus, wahrend der Nordpol und 
 
 " Findet hingegen eine Libration seine Umgegend uns verdeckt sind 
 
 in der Breite statt, so erscheinen die und so umgekehrt bei einer nach 
 
 Parallelen und der Aequator als Siiden gerichteten Verschiebtmg." 
 
 Ellipsen, einer der Pole riickt vom BEER und MIDLER, Selenographie, 
 
 Rande auf die Scheibe, der andere S. 12, 14. Compare also Notes 115, 
 
 hingegen auf die hintere Seite. Die 116, 118, 119. 
 
 M 
 
162 
 
 ON THE MOON'S ROTATION. 
 
 69. In an indirect manner, our main question, Whether there 
 is a diameter in the lunar globe constantly directed towards the 
 same point in space ; in other words, whether the Moon rotates 
 or not ? has thus virtually been solved. But Astronomy returns 
 us also a direct answer to that question. The lunar axis of ro- 
 tation does not maintain the same direction in space, but per- 
 forms a complete revolution in about 18 years 219 days, as a 
 necessary consequence of the coincidence of the nodes of the 
 Moon's assumed equator and of her orbit. 122 It has been no- 
 ticed by Delaunay, 123 that otherwise the lunar axis must change 
 its inclination to the plane of the Moon's orbit; but such a 
 change would be in the most perfect accordance with our 
 empirical knowledge of rotatory motion, whilst the rapid 
 change of the axis with regard to space is contrary to that 
 knowledge, " there being nothing so permanent in nature as 
 
 122 " Le resultat des observations de 
 Mayer, et des miennes, d'accord avec 
 celui de Cassini, est que le noeud de 
 1'equateur lunaire est toujours aux 
 memes points du ciel que le lieu 
 moyen du noeud de la Lune; on 
 peut conclure meme des observations 
 d'Hevelius, faites il y a plus de cent 
 ans, qu'il coincidait alors aussi bien 
 qu'aujourd'hui, du moins sensible- 
 ment, avec le noeud de 1'orbite 
 lunaire ; en sorte qu'il faut lui sup- 
 poser un mouvement de revolution 
 en 18 ans, centre 1'ordre des signes, 
 egal a celui des noeuds de la Lune. 
 Si leur mouvement moyen etait dif- 
 ferent, on ne verrait pas la meme 
 coincidence par des observations 
 faites il y a plus de 100 ans, par 
 celles de Cassini, par celles de 
 Mayer, et par les miennes, qui m'ont 
 donne le meme resultat." LALANDE, 
 Astronomic, tome iii., p. 326. (Com- 
 pare also Notes). 
 
 123 NOUS venons de dire que 1'axe 
 
 de rotation de la Luiie se transporte 
 parallelement a lui-meme. Si ce 
 parallelisme se conservait constam- 
 ment, et sans aucune alteration, 
 il en resulterait necessairement 
 un changement d'obliquite de cet 
 axe par rapport au plan de 1'orbite 
 lunaire, puisque le plan de 1'orbite 
 change peu a peu de direction dans 
 1'espace. Mais 1'observation a fait 
 voir que la direction de 1'axe de ro- 
 tation de la lune change en meme 
 temps que celle de ce plan, de telle 
 maniere que Tangle forme par 1'axe 
 et le plan reste toujours le meme et 
 qu'en consequence la libration en 
 latitude conserve toujours la meme 
 amplitude. 
 
 " Voici en quoi consiste le change- 
 ment progressif de direction de 1'axe 
 de rotation de la lune, phenomene 
 dont la decouverte est due a Do- 
 minique Cassini. Si, par le point h, 
 ou se trouve le centre de la lune a 
 un instant quelconque, on mSne une 
 
ON THE MOON'S ROTATION. 
 
 163 
 
 an axis of rotation." Why the Moon's axis of assumed ro- 
 tation should follow the retrograde motion of her nodes, rather 
 than the direct motion of her apsides or her own motion in her 
 orbit, Astronomers do not attempt to explain, though Mathe- 
 maticians acccord the "fact'' with the laws of gravitation. 
 Both, however, overlook its evident consequence, which alone 
 would be fatal to the theory of the Moon's rotation. For, as we 
 have already demonstrated, the latter, in accordance with the 
 actual phenomena the Moon presents, is only possible under 
 the supposition of her axis of assumed rotation maintaining the 
 same direction in space (apart from the motion of the solar 
 system) ; or, in other words, maintaining its parallelism in 
 space with regard to the Sun. From what we have said before, 
 it must be plain to every one, that the very slightest continuous 
 change of this parallelism would produce a corresponding 
 change of the surface, presented by the Moon towards the 
 Earth ; and that, consequently, if the Moon's axis of rotation 
 
 ligne L u perpendiculaire au plan de 
 1'e'cliptique A B c D, puis une ligne 
 
 L v perpendiculaire au plan de 
 1'orbite lunaire N L N' L', 1'axe de 
 rotation L x de la lime se trouve 
 
 toujours dans le plan des deux 
 lignes L u, L v, et il est place, par 
 rapport a ces deux lignes, comme la 
 figure 1'indique. L' angle u L x est 
 egal a 1 28' 45", Tangle v L u est 
 d'ailleurs egal en moyenne a 5 8' 48" : 
 en sorte que Tangle v L x est de 
 6 37' 33". On sait que, abstraction 
 faite de la nutation de Torbite lunaire, 
 la ligne L v tourne autour de L u un 
 mouvement retrograde, en ddcrivant 
 un c6ne de revolution qu'elle parcourt 
 en 18 ans f ; il resulte de ce qui 
 vient d'etre dit que Taxe de rotation 
 L x tourne en meme temps autour 
 de L u, en decrivant e*galement un 
 cone de revolution, dans le meme 
 sens et avec la meme vitesse." 
 DE LA UN AY, Cours tltin. d' Astronomic, 
 pp. 396, 397. 
 
 M 2 
 
164 
 
 ON THE MOON S ROTATION. 
 
 performs a complete revolution in the course of about 18| years, 
 the Moon herself must of .necessity present in succession every 
 portion of her surface towards the Earth, in the course of the 
 same period. 
 
 Let Fig. 35 represent two circular time-pieces A and B. We 
 
 Fio. 35. 
 
 may imagine them to be two watches of different sizes, lying 
 on the table, and the one placed upon the other ; or we may 
 consider them to be two clocks, so fixed against a wall that the 
 smaller one shall admit of being turned on their common centre, 
 in a direction contrary to the movement of their handles. In 
 either case, their motion shall be perfectly uniform ; their 
 centres shall coincide; the division of the dials shall exactly 
 correspond ; and, at any given moment, the minute-hands of 
 both shall point to = 12. It is evident, under these suppo- 
 sitions, that, so long as the position of neither clock is changed, 
 the direction of both rninute-hands will continue to coincide ; 
 that they will precisely indicate the same moment of time ; that 
 
ON THE MOON'S ROTATION. 165 
 
 the point a of the hand of B will always face the point a' of the 
 hand of A ; and that if we imagine an eye to be placed in the 
 former, it would never see, of the circular part a b c d of the 
 latter, the semi- circumference b c d turned always from it. But 
 let there now a slow motion be imparted to the axis of A (or, 
 which is the same thing, to A itself, as joined to that axis), 
 amounting every hour to a space, answering to nearly one 
 quarter of the distance which marks the epochs of two suc- 
 cessive minutes, being the interval of one minute ; and thus 
 completing the entire circuit of the dial of B in about 230 hours 
 or complete revolutions of the minute-hands. Again, it is evi- 
 dent that, notwithstanding this distinct motion of A, which in 
 the strictest sense corresponds to the similar distinct motion of 
 the Moon's axis of rotation in space, or of the lunar body itself, 
 as joined to that axis, and notwithstanding that the direction 
 of^the minute hands of both clocks no longer coincides : they will 
 still continue to indicate precisely the same time, both pointing, 
 at the same physical instant, to the same division of each dial; 
 which proves that when the minute-hand of B completed a 
 perfect revolution at the point = 12 of the dial B, the minute- 
 hand of A had completed a perfect revolution at the point 
 0= 12 of the dial of A, although the latter point in succession 
 corresponded to the points 11, 10, 9, 8, &c., of the fixed dial B, 
 and, consequently, on completing a true revolution about its 
 axis, the minute-hand of A had not completed a revolution re- 
 lative to some fixed spot in space, because the axis of A itself 
 had changed its position with regard to space. And in virtue 
 of this change of position of the axis of A with regard to B, 
 which answers to the change of position of the Moon's axis of 
 assumed rotation with regard to the Earth, the points a and a 
 of the minute-hands will cease to face each other, a apparently 
 gaining more and more upon a ; and thus an eye, supposed to 
 be placed in a, will gradually, as the point 12 of A retrogrades, 
 with regard to space, to 12' A = 11 B, 12" A = 10 B, 12'" A= 9 B, 
 &c., obtain a view of the entire circumference of the circular 
 
166 ON THE MOON'S ROTATION. 
 
 part a b c d of the minute-hand of A. And in the like manner, 
 as we have already stated, the JMoon, if she rotated on her axis 
 in precisely the same period in which she revolves round the 
 Earth, must of necessity and in succession present to our view 
 every portion of her surface, in consequence of the motion of 
 her axis of assumed rotation, and in the course of the exact 
 period of this motion. But it is an empirical fact, that the 
 Moon does not do so. Astronomers themselves, therefore, 
 teaching us, on the one hand, that the periods of libration 
 differ from the common period of the Moon's revolution and 
 assumed rotation, and, on the other hand, that her axis of as- 
 sumed rotation has a motion of its own, instead of maintaining 
 its parallelism in space with regard to the Sun : they furnish us, 
 in the shape of these facts, with the most decisive proofs against 
 their own theory. 
 
 70. In order, however, to leave no doubt on the subject, we 
 may still be permitted to point out that the Moon's rotation, 
 in accordance with the phenomena she actually presents, is 
 a physical impossibility. For it will readily be admitted to be 
 impossible for any heavenly body, to turn about itself at the 
 same time in five different periods and in two different directions ; 
 to be at the same moment in numerous different places ; and to 
 rotate without an axis of rotation to rotate upon. Yet, the 
 astronomical theory supposes nothing less. In looking upon 
 the lunar librations as the effects of rotation, we should, ac- 
 cording to that theory, require a first Moon, rotating in the 
 same period in which she actually revolves round the Earth, 
 i. e. in 29 days 12h. 44' 2" 9 ; a second Moon, rotating in the 
 "true" period of her revolution round the Earth, i.e. in 27 
 days 7h. 43' 11", 5; a third Moon, rotating in the period of 
 her tropical revolution, i. e. in 27 days 7h. 43' 4", 7 ; a fourth 
 Moon, rotating in the period of her anomalistic revolution, i. e. 
 in 27 days 13h. 18' 37", 4 ; a fifth, rotating in the period of her 
 draconic revolution, i.e. in 27 days 5h. 5' 36 ; Jive more Moons 
 preceding the former by 90 ( 65). These ten Moons, of 
 
ON THE MOON'S ROTATION. 167 
 
 which five rotate and five do not, must be supposed to revolve 
 in their orbits from west to east, in order to explain the phe- 
 nomena of libration in longitude. We now require an eleventh, 
 and preceding her by 90, a twelfth Moon to revolve in their 
 orbits, with the nodes, in a contrary direction, i. e. from east to 
 west ; the former rotating from west to east, and carrying the 
 axis of rotation with her ( 69). Four additional Moons are 
 thus needed to represent the four rotating Moons, left without 
 an axis of rotation to rotate upon ; and we lastly require Jive 
 more Moons, rotating in as many different periods in a direc- 
 tion perpendicular to the first ( 66), in order to explain the 
 phenomena of libration in latitude. The impossibility, there- 
 fore, of the Moon's rotation, in accordance with the phenomena 
 she actually presents, is at least a twentifold one. We cannot 
 deny it to be a singular circumstance, that Astronomers should 
 have overlooked the striking contradictions involved in their 
 theory. But it is more singular still that Mathematicians, by 
 an analytical process, should have proved the laws of gravita- 
 tion to lead of necessity to bare physical impossibilities ; and 
 when, in the face of these impossibilities, they arrived at the 
 delicate result that the Moon's velocity of rotation partici- 
 pated in the secular inequalities of her orbit; it may well 
 be said of them, "They strained at gnats and swallowed 
 camels." There must be something fundamentally wrong, 
 either in their process of calculation or in the theory of gra- 
 vitation, or in both. 
 
 71. Having thus determined the principal point of our pro- 
 blem, there remains to be answered only the question What, 
 then, is the true explanation of the phenomena of the Moon's 
 libration ? We have, virtually, answered it already. Libration 
 in longitude is the natural and necessary consequence, consi- 
 dering the eccentricity of the lunar orbit, of the law of motion, 
 established in 13 ; the former offering a striking proof of the 
 latter, the latter offering the most perfect explanation of the 
 former. Let, in Fig. 36, E represent the Earth, M the Moon. 
 
168 
 
 ON THE MOON'S ROTATION. 
 
 FIG. 36. 
 
 According to the theory of attraction, the Moon, in the course 
 of her revolution round the Earth, always presents towards the 
 centre of the latter the same diameter. Consequently the same 
 point d of the lunar globe should, as in M and M", face the 
 Earth likewise in M' and M'". But actual observation shows 
 that, in the two latter positions, the points e and e respectively 
 regard the terrestrial centre. Hence the arcs e d and e' d" 9 
 corresponding to the maxima of libration in longitude, had to 
 be accounted for. Newton did so by assuming the Moon to 
 rotate on her axis, at a uniform rate of velocity, and in the 
 same period, in which she revolves in her orbit. We have seen 
 that this hypothesis does not account for the phenomena in 
 question, and is, moreover, utterly untenable. If, on the con- 
 trary, we view those phenomena in connexion with our law of 
 motion, they no longer require any further explanation, or any 
 supposition to explain them whatever. They explain them- 
 selves. The point d of the lunar surface is in d", d'" and every 
 other point of the lunar orbit, in reality found to be where, ac- 
 
ON THE MOON'S ROTATION. 169 
 
 cording to the law of motion, it should be found ; and the 
 libration in longitude, the minima and maxima of which occur 
 in M and M", and M' and M'" respectively, in the most perfect 
 harmony between Theory and Observation as fully described 
 in 44, comp. 24 is simply a necessary consequence of 
 the eccentricity of the lunar orbit. Its maxima, therefore, are 
 mainly equal to double the equation of the Moon's centre or 
 12 84' 25", 4, and the effects of the diurnal libration in longi- 
 tude = 2 3'. It is of a purely optical nature, and its period 
 that of the lunar apsides. 
 
 72. The Moon's libration in latitude we have found to be 
 somewhat more complicated than that in longitude ; and so is 
 its explanation. This libration is composed of two elements, 
 the one due to the Moon's latitude, the other ascribed by 
 Astronomers to her rotation. We will first consider the for- 
 mer. According to the theory of attraction, the point e=e of 
 the lunar surface M, Fig. 37, should always regard the centre 
 of the Earth E, as in M' and M'". Consequently, when the 
 Moon is in one of her nodes as in M and M", that point ought 
 to occupy the place of the point a = a \ and if Astronomers 
 were consistent, they would, from the theory of attraction ex- 
 pecting the libration, due to the Moon's latitude, to be double 
 what it really is, have had to explain the striking difference 
 between Observation and Theory. But, finding the assumed 
 equator of the Moon actually to maintain its inclination to the 
 ecliptic at an invariable angle, they content themselves to state 
 the fact, without attempting either to account for it, or to ex- 
 amine into its bearings. Like the Moon's real motion in 
 longitude, a strong proof against the theory in question, 
 it is again in the most perfect accordance with the law regu- 
 lating that motion. The parallelism of the lunar diameter a c 
 in M, M', and any other latitude of the Moon, flows from it 
 as a necessary consequence. So far, then, as the Moon's 
 libration in latitude is due to her latitude, as fully described 
 in 45, the astronomical explanation of it, though in contra- 
 
170 
 
 ON THE MOON'S ROTATION. 
 
 diction with the theory of attraction, is perfectly correct. If 
 we regard it apart from diurnal libration, its maxima amount 
 to double the Moon's greatest latitude, or 10 36'. It is of 
 a purely optical nature, and its period is that of the lunar 
 nodes. 
 
 73. We have already stated, that the Moon's diurnal libra- 
 
ON THE MOON'S ROTATION. 171 
 
 tion is attended, not only by longitudinal, but also by lati- 
 tudinal effects, differing in different degrees of geographical 
 latitude. If the terrestrial axis were perpendicular to the 
 ecliptic, such would not be the case ; but being inclined to the 
 latter plane, at the considerable angle of 23 27' 35", 8, it is 
 evident that, in consequence of the Earth's rotation, the terres- 
 trial observer, with regard to a straight line joining the north 
 and south poles of the Earth and the centre of the Moon, is in 
 the course of every revolution of our planet about her axis, at 
 one time as much above that line, as twelve hours later he will 
 be below it, or vice versa, and that the linear distance between 
 those two points will increase in the same proportion as the 
 southern or northern geographical latitude of his position de- 
 creases, being greatest at the equator, where it will be about 
 5,600 miles. It is also evident that a diurnal libration in 
 latitude must thus result, similar to the diurnal libration in 
 longitude, resulting from similar causes; and, further, that 
 this libration will be least when the Moon is in her nodes, in- 
 creasing as she advances in her orbit towards her greatest 
 northern or southern latitude, at which points it will reach its 
 maxima, thus coinciding, in its effects, with the Moon's libra- 
 tion due to her latitude alone. Again, it is evident that, in 
 consequence of the terrestrial axis being constantly directed 
 towards the same points in space, every spot on the lunar 
 globe will appear to describe a different path upon her visible 
 disk every day in the year; and that, consequently, the real 
 period of this libration, though of a diurnal character, will be 
 that of the return of the lunar nodes to the same point of the 
 ecliptic, i. e. about 18 years 219 days. For this reason it is 
 evident, moreover, that in the course of each single month the 
 combined libration in latitude will appear perpendicular to the 
 libration in longitude, and the point b (E) of the lunar globe 
 seem to describe the path b o, o b, whilst, considering its par- 
 ticular libration, it will apparently trace upon the lunar disk a 
 very small ellipse, or part of one ; and that, in the course of 
 
172 ON THE MOON'S ROTATION. 
 
 each period of about 18S years, inconsequence of the combined 
 libration in latitude, it will, the same as the point a, seem to 
 describe, in the shape of small cycloids, a larger ellipse in such 
 a manner that, when the point c, after tracing over the Moon's 
 visible disk the apparent path r o r" in the course of half the 
 period, disappears in r^, the point a appears in /', in order to 
 trace, in the course of the remaining half the period, the path 
 r o r". Lastly, it is evident that, if we were to disregard the 
 real causes of these apparent phenomena, as resulting from the 
 motion of our own point of observation on the surface of the 
 Earth, we should be led to attribute them in a physical sense to 
 the motion of the Moon, just as, on looking at a globe in the 
 manner described in 63, we should be led to imagine the 
 observed spots to move upon its surface, if we forgot that we 
 were moving our own head, and made those spots only appear 
 to move. And this is the case of Astronomy, with regard to the 
 Moorfs libration in latitude, so far as it is the effect of the 
 Earth's rotation on an axis inclined to her orbit and fixed in 
 reference to space, combined with the Moon's apparent motion 
 in latitude. For Astronomers ascribe that effect, which in every 
 detail answers to the latitudinal effect of the Moon's assumed 
 rotation on an axis, inclined to the ecliptic at an angle of 
 about \y, to the lunar globe itself; overlooking that it arises 
 from the motion of their own terrestrial station. True, they at 
 the same time represent it as being of a purely optical cha- 
 racter ; but, as we have already remarked, they do so in most 
 palpable contradiction with their own theory of the Moon's 
 physical rotation. 
 
 74. We need hardly state, that analytical calculations based 
 upon actual observations of the Moon, in which the elements 
 in question have been disregarded, must give for a result these 
 very elements. If, therefore, the results obtained by Mayer and 
 Nicollet were perfectly correct, double the inclination of the 
 lunar axis of assumed rotation to the ecliptic, as found by them, 
 or about 3, would also be the correct amount of the maxima of 
 
173 
 
 the Moon's libration in latitude, dependent on the Earth's 
 diurnal motion. But, though this amount does not materially 
 differ from the truth, it must not be considered as exact. On 
 comparing the results in question (Note 78), we find, between 
 them, the whole lunar globe wavering, as it were, in space to 
 the extent of ^ to J. For the selenographic longitude and 
 latitude of " Manilius " are, the former 9 2' according to Mayer, 
 8 47' according to Nicollet; the latter 14 23' according to 
 Nicollet, 14 33' according to Mayer. Cassini computed the 
 inclination of the lunar axis of assumed rotation to the ecliptic 
 at 2 40'; Lalande at 1 43'; Mayer at 1 29'; Nicollet at 
 1 30' 6". Moreover, from observations of different spots, 
 Mayer found it to be different: 11 to 12' less from " Censori- 
 nus," 1 to 2' more from " Dionysius " two other lunar moun- 
 tains, as compared with " Manilius." Mayer obtained 3 45' 
 for the distance of the node of the Moon's equator from that of 
 her orbit (which should have been found = 0), as deduced 
 from his observations of " Manilius," and -f- 17 J as deduced from 
 those of " Censorinus ;" whilst Nicollet obtained 1 1 7' 55" for it. 
 
 All these calculations, which, owing to the limited elements 
 on which they are based, could not but lead to unsatisfactory 
 and more or less contradictory results, will now have to be made 
 again under an entirely different point of view, and from ob- 
 servations extending over a period of nineteen years. 
 
 Our explanation resolves the " singular " coincidence of the 
 nodes of the lunar equator and orbit, and the apparent retrograde 
 revolution of the Moon's axis of assumed rotation in the period 
 of the revolution of her nodes, into a necessary consequence of 
 the causes, by which the Moorus apparent libration in latitude 
 is produced ; it removes all the obscurity in which the whole 
 subject of libration has been thus far enveloped, all the theo- 
 retical contradictions and fallacies to which it has given rise, 
 all the practical difficulties by which it has been surrounded 
 and shows it to be, indeed, of that purely optical nature, which 
 
174 ON THE MOON'S ROTATION. 
 
 the lunar phenomena so clearly indicated, and Astronomers, 
 themselves, in opposition to their own theory, were forced to 
 claim for it. 
 
 A full discussion of the law of motion, to which the solution 
 of our problem is chiefly due, we must reserve for a more fitting 
 place. Its importance, though little likely to strike the general 
 reader, we cannot well conceal from ourselves, involving, as it 
 does, in its direct and indirect consequences, nothing less than 
 a revolution in modern Astronomy. 
 
 75. In conclusion we may be permitted a few remarks in 
 reference to an apprehension expressed (Notes 66, 71), lest the 
 Public should lose their confidence in the teaching of Astrono- 
 mical Science, if that Science could be supposed to have been 
 in error upon such a point, as the Moon's rotation on her axis. 
 This is placing little confidence in the judgment and the good 
 sense of the English Public. We entertain a far higher 
 opinion of both. Infallibility alone could guard Astronomers 
 from falling into error; and no reasonable person will attribute 
 infallibility to them, nor expect it to attach to their doctrines. 
 Where the liability to err exists no longer, there exists no 
 longer the possibility of progress. Astronomy, undoubtedly, 
 has accomplished much in modern times, much, if we look 
 back upon the past, yet little if we look forward to the future. 
 In reference to the vast course open before her, she has but 
 just entered upon the very threshold of Truth and Knowledge. 
 We do not speak of a knowledge of the Universe, but simply of 
 the limited knowledge of our own Solar system a mere vanish- 
 ing point in space. Supposing Astronomy to have been born 
 four or five thousand years ago, what is she in time of the 
 Solar system ? A babe, scarcely two months old, the first sol- 
 anniversary of whose birthday future generations may expect 
 to celebrate some two hundred centuries hence. However far, 
 therefore, she may be in advance of her age as far as Genius 
 is in advance of Common Sense, and Science is in advance of 
 
ON THE MOON'S ROTATION, ] 75 
 
 General Information still, a wide space of time has to elapse 
 before she can hope to arrive at a period of maturity. In the 
 mean while the Astronomer must err, if Astronomy is to pro- 
 gress. And they will do so, as long as there shall remain a 
 truth for the one to conquer, and an aim for the other to 
 pursue 
 
 " Es irrt der Mensch, so lang' er strebt." 
 
 So far, then, from lessening, the result of our inquiry should 
 tend to strengthen the confidence of the Public in the teaching 
 of Astronomy ; because a source of error has been removed, 
 and Astronomical Knowledge has been extended. 
 
 Nor can the discovery of a new truth in any way affect the 
 merits of those, who have held the old error. The name of 
 Ptolemy will live for ever, though his system has almost been 
 forgotten ; and the name of Sir Isaac Newton will retain its 
 immortality, though some of his theories are sure to perish. 
 Truth is frequently to be arrived at only through error ; and no 
 Science has ever been brought to perfection by one man, or 
 in one age. The coming generations, too, will have their 
 triumphs ; and future centuries are destined to produce their 
 great men, as well as the past ones have done. Their Aristarch, 
 Archimedes, Eratosthenes, Hipparch, Ptolemy ; their Albateg- 
 nius and Ulugh Beigh; their Copernicus, Tycho, Galileo, 
 Kepler, Sir Isaac Newton ; their Sir William Herschel, La- 
 place, Gauss, Bessel, Arago, among many others, will appear in 
 due time and place. But whatever be the future progress of 
 Astronomy, the memory of those illustrious persons, and of 
 their no less illustrious successors of the present day men, 
 the ornament of Science and the pride of their country, 
 whose merits require not our praise will live in the history 
 of Astronomy, so long as there shall be an astronomical Lite- 
 rature. With truth it can be said of them: "Their merits 
 are their own ; their errors those of the Science of their 
 
176 
 
 Astronomy is usually regarded by the Public as a pure 
 Science, i.e. as a Science, whicli;confers no material benefits on 
 a nation. This is a great mistake. We will not ask the reader 
 what he would do without his Almanac ? It has become almost 
 necessary to his existence, and certainly does constitute a great 
 convenience to him. The Nautical Almanac, from which most 
 others are compiled, does more. It saves millions worth of British 
 manufacture and colonial produce, and hundreds of human lives 
 every year. Astronomy is the guiding star of Navigation; 
 Navigation the soul of Commerce ; and Commerce the chief 
 source of British wealth and power. A large share of this 
 wealth and power, therefore, is indirectly due to Astronomy. 
 But the most important influence Astronomy is destined to 
 exercise upon Civilisation ; and it is earnestly to be hoped that 
 the time may not be far distant, when astronomical and reli- 
 gious instruction will be considered inseparable. Next to Reli- 
 gion, no science is so fit as is Astronomy, to raise the loftiest 
 and purest thoughts in the human heart ; to elevate the mind, 
 and to enlarge the intellect ; to remove national and individual 
 prejudices ; to secure peace on Earth ; and to promote the 
 love of God and men. We may learn the Power, and the 
 Wisdom, and the Goodness of our Maker from the sacred pages 
 of His written Word ; but if we would fully comprehend them, 
 we must peruse and study His wondrous works. " The heavens 
 declare the glory of God; and the firmament showeth His 
 handiwork." 
 
 To revert once more to the preceding Essay. It may serve 
 as a warning both to Science and to Common Sense. True, 
 the view taken by the latter of the Moon's presumed rotation 
 on her axis, has proved to be the correct one ; but does the 
 result in any way constitute a triumph of Common Sense over 
 Science ? Far from it. Common Sense had not so much as com- 
 prehended the problem, upon which she pronounced her judg- 
 ment ; she had no more idea of its real merits than had " the 
 
ON THE MOON'S ROTATION. 177 
 
 Man in the Moon" himself. By one of those rare chances, 
 which will perhaps never again occur in the whole course of 
 astronomical history, she happened to be in the right. Let 
 her, therefore, not exalt herself at a success, to which she has 
 no claim. Where Genius has lost her way, mere Common 
 Sense will never discover the right path. An error, in which 
 a Newton and Laplace have shared, is no empty imagining, 
 which Common Sense has merely to look at, in order to expose 
 its fallacy ; it can but be the misconception of a truth that will 
 be conquered. And the more simple it proves when once dis- 
 covered, the more difficult it not unfrequently is of discovery. 
 Let Common Sense, therefore, endeavour to master an astro- 
 nomical problem before she attempts to solve it ; and she will 
 then arrive at a more just appreciation of her own powers, 
 and of those of Science. 
 
 On the other hand, let Science never turn a deaf ear to the 
 voice of Common Sense. She has been entrusted by Provi- 
 dence with a high and responsible mission ; and as it is her 
 noblest privilege to instruct mankind, so it is her first duty to 
 examine herself. When she has failed to render her doctrines 
 intelligible to men of talent and undoubted attainments, let her 
 be assured that such doctrines stand in immediate need of 
 a closer investigation. Astronomy, as a practical and mathe- 
 matical Science, has made vast strides since Tycho and Kep- 
 ler ; but as a philosophical Science, she has made comparatively 
 little progress since the days of Newton. Instead of pursuing 
 the path traced out for her by the author of the Principia, she 
 has followed one requiring less genius than talent ; and since the 
 brilliant analytical researches of Laplace, the Science of Celes- 
 tial Mechanics, to which he reduced Astronomy, has assumed 
 somewhat of the character of a mechanical Science. Not that 
 we would have Astronomy calculate less, but think more. 
 Analysis, like the telescope, should be an instrument in her 
 hands, not the guide of her intellect. No living truths are 
 
 N 
 
178 ON THE MOON'S ROTATION. 
 
 deducible from dead cyphers. The Principia gave birth to the 
 Mecanique celeste; the 4 MecaniqjUe celeste could never have 
 called into existence the Principia. Let Astronomers follow, 
 as Sir William Herschel has done, the path traced out by Sir 
 Isaac Newton. It is a path leading to immortality. 
 
 THE END. 
 
\0 
 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 MAR 15 194E 
 
 REC'D LD 
 
 MAR 2Q1958 
 
 LD 21-100m-9,'47(A5702sl6)476 
 
974165 
 
 Grr 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY 
 
E^EH^^B