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' -> 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'