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nail it to the board, without changing its relative position to the centre, and thus satisfy himself it has no rotation on its own pivot. When he "suddenly" arrests the board, the ball no doubt begins to rotate, and it does so from the centrifugal force acting on the ball, with no centripetal force acting in the plane of its orbit to counteract it. It is obvious that the immense power of this and other forces, exercised by the Earth and Sun respectively on the Moon, is the main cause why she preserves the same face towards us, instead of rotating. It is strange, indeed, that the very persons who object to rigid instruments, which really prove that part of the lunar motion they correctly represent, should be the first to resort to them to illustrate that which it is physically certain they must misrepresent.

At least a score of those who dissent from me, and have done me the favour of communicating their grounds of objection, have striven hard to convert me, by means of a variety of demonstrations and diagrams, which go to prove that if the Moon moved out of her orbit at a tangent ever so short a distance, she must then rotate on her axis to get back to her orbit again. I quite admit it: if her orbit, for example, were a polygon, she would have to rotate at every corner; but her orbit is not a polygon, nor does she ever go out of it, and all such demonstrations are futile and beside the question, and prove only what nobody disputes.

Another gentleman, whose name I am not at liberty to mention, has also put, in a more scientific manner, one of the modes of opposing my view first named by one of the Times' correspondents. He says :—

"I ask you to conceive the Moon's distance from the Earth to be gradually diminished, until the two centres coincide. You might say that the 'impenetrability of matter' renders such a coincidence impossible. You might Bay so, but you are too sound a logician to do so. I ask you, therefore, at the same time that the Moon's angular motion about the Earth continues as at present, to conceive the distance between the two centres to be gradually lessened, until it becomes absolutely nil. Can you fail to recognize, that in this final position the Moon will be rotating about the common axis of itself and the Earth? When did this rotation begin? Had it no existence till the instant when the two centres came together? You cannot suppose such to be the case, if, in your mind's eye, you have distinctly watched the supposed gradual diminution of the lunar orbit. This axial rotation must, therefore, have existed before the centres coincided, which was the primary point to be proved."

The argument here is, that because the transition of the movement on separate axes to the movement on the same axis may be imperceptible and impalpable, there is no difference between them. I cannot admit it. There are an infinite number of physical differences, alike in motion and matter, imperceptible not merely to the naked eye, but with the best achromatic microscope. Decimals, carried to the 20th place, will accurately indicate distinct quantities, but of which even the mind cannot appreciate the difference: does it therefore follow that there is no difference 1 Just as in the case of the polygon, whose sides may be infinitesimally minute, and although altogether undistinguishable from a circle, to the eye, or by measurement, yet utterly distinct from it.

I reserve a definition of the geometrical differences between rotation and lunar revolution till I have cleared the question of unscientific objections. I must also reserve one or two other objections for consideration until I have given Professor Airey's letters, for they touch on some and dispose of others. But I must first endeavour to clear the question of lunar librations, with which it has been embarrassed by Dr. Lardner : a gentleman of whom I desire, nevertheless, to speak with that respect and gratitude which is due to a man who has done so much to simplify science, and facilitate its comprehension by the people.

Dr. Lardner, in his first letter to the Times (dated April 21st), says:—

"That, as my view is in 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 principles, 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 admitted, the reasoning by which it has been established still requires elucidation and development before its conclusiveness can be perceived by ordinary minds."

Quite conscious that my mind is of the ordinary stamp, I read the further and more elaborate effort, in Dr. Lardner's second letter, to enlighten it, with an expectation by no means satisfied by the following series of assertions, which, though pronounced quasi ex cathedrd, have, I believe, failed to convince any "ordinary mind" of my error. I have, for the sake of brevity, interspersed comments in brackets :—

"The same hemisphere of the Moon is not always presented to the Earth's centre. [Mathematically true.] There is libration in longitude, and also in latitude. [There is.] The former is caused by the fact that the velocity of the rotation is not always equal to that of the revolution [this is pure assumption], being sometimes greater and sometimes less. This arises from the circumstance of the velocity of revolution being variable, while that of rotation is rigorously uniform. [Idem.] The libration in latitude arises from the fact that the axis of rotation is not rigorously parallel to that of revolution, but inclined to it at a small angle (1° 30' 10"). [It is alleged by astronomers to be far greater, viz. 6' 37', or thereabouts. The mean inclination of the Moon's equator to the ecliptic is 1° 30' 11"!] The phenomenon of rotation is therefore distinct from and independent of that revolution. The two motions have different velocities, and take place round different axes. They cannot therefore be identical; and this disposes of the question of the rotation, which is demonstratively established. [If mere assertions can do it: certainly not otherwise.] So far, therefore, as the particular case of the Moon is concerned, the argument of Messrs. Hopkins, Symons, and those who agree with them, falls to the ground."

It is in little danger of any such mishap from such very harmless blows. I trust even this short essay will suffice to " demonstrate" that the Moon has no rotation round her own axis; and if so, some other cause must be sought for her librations; but Dr. Lardner speaks as if they could not arise otherwise. It is obvious that they can: meaning, as I presume he does, the complete rotation of the Moon once round her axis. The utmost extent of her longitudinal libration subtends a very acute angle,* and is therefore effected by a rotatory motion of that extent and no more, whereas the angular rotation contended for is one entire revolution of the Moon's body round her centre of 360°. Admitting, therefore, that these librations amount to axial rotations, they are occasional and minute. Is it a fair mode of arguing, therefore, to say that "the Moon's rotation" is thereby demonstrated 1

Professor Hansen thus enlarges on the interesting phenomena which

* May not these librations, however, be caused by the ellipticity and eccentricity of the lunar orbit and her moments of inertia: affected again, according to recent discoveries made by Professor Hansen, by the remarkable fact, that the interior of the Moon is of heterogeneous density, and that her centre of gravity is about eight geographical miles (reckoning fifteen to a degree of the Equator) further from the Earth than the centre of her figure?

result from the position of the Moon's centre of gravity. I quote from his letter to Professor Airy, in the November Paper of the Royal Astronomical Society of 1854 :—

"If, as we here must assume, the opposite hemisphere of the Moon is more dense than the hemisphere turned towards us, it necessarily follows that the mean level of the former will be somewhat depressed, and the mean level of the latter somewhat elevated. If we suppose the moon to be an ellipsoid, the elongation of which lies in the direction of the Earth, then the hemisphere of the Moon which is next the Earth will rise a little more above the mean level, and the opposite hemisphere will sink a little more beneath it. Nay, we may consider it as not impossible that the surface of the opposite hemisphere of the Moon wholly or partially accommodates itself to one and the same level, in a similar way as we find to be the case with the Earth.

"We need not, then, under these circumstances, wonder that the Moon, when viewed from the Earth, appears to be a barren region deprived of an atmosphere, and of all animal and vegetable life; since, if there existed upon the Earth a mountain proportionally high, and, consequently, having an elevation of 216,000 metres, or 29 geographical miles, there would not be recognizable upon its summit the slightest trace of an atmosphere, or of anything depending thereon. We must not, however, conclude that, on the opposite hemisphere of the Moon, the same relations exist; but rather, we should expect, in consequence of the distance of the centre of figure from the centre of gravity, that an atmosphere and animal and vegetable life may there find place. Nearly at the Moon's limbs the mean level must exist; consequently, we might reasonably expect to discover there some trace of an atmosphere."

Now if the Moon rotated on her axis, the centrifugal force would take effect on all portions of her equatorial circumference. It is because she does not rotate, that it takes effect on the side remotest from the Earth, and thus causes the conformation discovered by Herr Hansen—one of those interesting facts which I stated would be invalidated by the rotation of the Moon, and for which I was sneered at by writers evidently ignorant of these discoveries.

Dr. Lardner has been good enough to assist my effort to rescue children from the utter confusion of ideas caused by the astronomical use of the term rotation, applied to the lunar movement, by this happy ad absurdum illustration of it:—

"To take another illustration of this principle: a mountain—the Peak of Teneriffe for example—is moved round the centre of its parallel of latitude, presenting always the same side to that centre. This mountain is not a globe, like the Moon, and has no geometrical line analogous to the Moon's axis; but that does not affect the principle of the question. The same reasoning which proves the Moon to rotate on its axis must establish, with equal conclusiveness, the rotation of the Peak of Teneriffe upon a certain line as an axis of rotation, that line passing through the mass of the mountain in a direction parallel to the terrestrial axis, the time of rotation being 23 hours 56 minutes.

"I can only repeat, that the point requires more clear exposition than it has yet received."

This is doubtless so: and we are all rotating each on his own separate axis, though standing perfectly still side by side; similarly, every atom of the globe, is doing the same ; and we should teach children that the reason why each portion of the tire of a wheel, and each nail in it revolves in exactly the same time as the nave at the centre, is the "extraordinary circumstance" as Vinke calls it in his Elements of Astronomy, that each part and each nail rotates on its own axis in exactly the same period. Yet this precious absurdity is logically incumbent on all who hold the rotation theory as the fit mode of instructing youth in a clear and intelligible manner. Here is Dr. Lardner's affirmation of this very statement in his second letter of elucidation :—

"Your correspondents explain the phenomenon by supposing it [the lunar motion} to be identical with the case of a globe connected with the centre in the same manner in which the parts of the Earth's surface are connected with its axis ; and they consequently identify the motion of revolution with a motion of rotation, of which the axis is a line drawn through the centre of attraction at right angles to the plane of the orbit. [Just so.] Astronomers, on the other hand, would explain the phenomenon by supposing the globe to have a motion of rotation rigorously equal in velocity to that of its revolution, and round an axis [within itself; for this is the established axiom to which Dr. Lardner is pinned] parallel to its axis of revolution. The adequacy of the explanation it to obvioiu, that it may be demonstrated by any clever school-boy who it familiar with the elements of geometry" 111

Its adequacy consists in explaining that it is rigorously equal and parallel to itself: the two motions being alter et idem!

As this is a dry subject, and I am anxious to render it as popular as possible, may I venture, before broaching points unavoidably geometrical, to insert a letter from an ordinary mind off Corfu, the last of the flood with which I have been honoured and deluged from all quarters on this subject, and by no means the least scientific :—

H.M.S. Modeste, Corfu, May 3rd, 1856. Sib,—Having seen several letters in the Timet, which are not remarkable for philosophical politeness or eveu perspicuity, denying your assertion respecting the non-revolution of the Moon about its own axis, I beg to tender for your consideration the following suggestion :—Let the Earth be supposed to have a perfectly smooth and hard surface, on which is a vehicle capable of being moved in any direction with a

fiven velocity ; in the latter, imagine a staff fixed in a nautical position, surmounted y a spherical-shaped body. The Moon being vertical, let the vehicle be put in motion, keeping the former at an altitude of ninety degrees, until both return to the meridian from whence they started. Then the spherical-shaped body must have had a motion, as far as the argument is concerned, identical with that of the Moon. Now, if your uncourteous opponents can prove that the sphere has revolved round its axis (supposing the experiment to have taken place), every mountain, nay, every human being, must revolve round his or her axis every twenty-four hours. If such be the case, there must be some truth after all in the old story, which refers to our friends at the antipodes standing on their heads.

The Moon certainly revolves round its own axis, as far as showing its various parts to a particular fixed star is concerned ; but the same may be said of Mount Etna, or of a balloon which remains suspended over a given point on the Earth's surface for twenty-four hours.

The Moon revolves round an axis, not its own, but that of the Earth, nearly in the same way as an orange, secured to the side of the rim of a moving wheel, revolves round the centre of the latter.

The Moon, in my opinion, no more revolves round its own axis, than it would do if, having approached the Earth, it was comfortably packed in a luggage-train, and sent on a voyage round the world, occupying just one of its own months on the trip.

By way of apology for presuming to offer an opinion on the above subject, I may state that I am the author of Great Circle Tables for the North Atlantic, and other works.—I am, Sir, your obedient Servant, JAMES M. SHAKE, R.N.,

Jelinger Symons, Esq. Master, H.M.S. Modeste.

To close this part of the subject, I have simply to suggest that one of the best practical modes of illustrating the difference between lunar revolution and axial rotation is, I think, to conceive a waggon travelling horizontally round a circular wall, with one wheel dragged and another revolving as usual. The dragged wheel represents lunar motion, the other axial rotation: that is the difference. It has been argued that one has only a quicker " rotation " than the other. It may as well be said that walking is a quicker pace than standing still! Let us now examine the geometrical properties which distinguish these separate motions, as well as those which fail to establish their identity.

PAST II.

Having the honour of acquaintance with the Astronomer Eoyal, I wrote to him in somewhat general terms embodying the views I have here put forth, and requesting the favour of his opinion upon the subject. The following letter is his reply :—

"Eoyal Observatory, Greenwich, April 19<A, 1856.

"Dear Sib,—In reference to the Moon's rotation round her axis: "The consideration of this matter is, I think, very much confused by introducing the notions of balls fixed to revolving boards, and the like, which have no counterpart in the real cosmical system.

"Instead of this, let me ask you to consider it in the manner of a physical astronomer, which is, in fact, the way of referring it exactly to the laws of action, and to the circumstances of nature.

"The Moon is not compelled by any material connection, to accompany the Earth. She is attracted by many bodies, but especially by the Sun and by the Earth—more powerfully by the Sun than by the Earth—and by these two attractions she is (to use a homely expression) 'pulled about,' so as to describe a very complicated orbit, round the Sun (if considered with reference to the moveable body which moves the least), or round the Earth (if considered with reference to a body which moves more) j but she cannot properly be said to revolve round either of these without reference had to the presence of the other.

"Thus you will see that the Moon is truly a body wandering in space, not tied to any orbit, but describing (under the action of the attracting forces) such a course, that her path relative to the earth will not differ enormously from an ellipse, though it does differ from it very considerably.

"All this applies to the motion of the Moon as a mass, or of the Moon's centre, with reference to the Sun and the Earth. Now let us consider the rotation of the Moon round an axis of its own.

"Suppose that a body moving freely in space, whose motion is determined by the attraction of numerous external bodies, and especially by the attraction of two (the Sun and the Earth) rotates. If we wish to determine either the fact or the velocity of its rotation, we must refer the direction of some produced radius of the body to some zero point. We may take the general zero point, or we may try to find a special zero point. Now what shall that special zero point be 1 The principal attractor, i. e. the Sun? Not at all: the Sun's attraction, great though it be, does not affect the Moon's rotation in the least; quick or slow though the first rotation were, equally quick or slow it would remain under any action of the Sun. There is, therefore, no virtue in the Sun which should make it the point of reference for rotation. Ought it to be the Earth 1 A fortiori, no: because the Earth has less power on the Moon than the Sun has. What then is it to be? Clearly the same which is the point of reference for all other rotations; namely, some point (as a star) at a distance so great, that the motion of the Moon in going round the Sun does not sensibly disturb the direction of the line from the Moon to the star.

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