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"There is another reason why such a point of reference should be taken. Rotation produces certain calculable dynamical effects ; as oblateness. But this oblateness will be calculated correctly only by referring the rotation to a point as distant as a star.

"Having thus come to the point that rotation is to be referred to something very distant, as a star, and not to anything near, as the Sun or Earth, there is little difficulty in ascertaining from observable appearances whether the Moon does or does not rotate. Does she always turn the same part of her face to the same star 1 No. Then she does rotate. If her face has been once turned to a star, how long time elapses before it is turned to the same star again 1 27^ days. Then 271 days is the time of her rotation.

"I would only again beg you to lay aside the supposed illustrations by pieces of wood and circular boards, and to consider the thing as nature presents it, varying my expressions as you may find best for your understanding.—I am, dear Sir, faithfully yours,

"(Signed) G. B. AIRY. . "Jelinger Symons, Esq.

"P.S. If you will do me the favour to call any morning at the Observatory, I will show you a revolving or conical pendulum in which the revolution is really not accompanied by rotation ; and you will remark the difference of its appearance from that of the Moon."

Nothing can be more lucid than this explanation of the mode in which astronomers use the term "rotation." I have already stated the grounds on which I respectfully deprecate Professor Airy's objection to the use of rigid instruments as illustrations of the point at issue, and which the above-named eccentricities of lunar motion do not affect. The objection, however, would perhaps have less weight in the Professor's mind if the illustration were taken from a ship sailing round the world from north to south. As the Earth is an oblate spheroid, the ship's course would, like the Moon, describe an ellipse, and she would pursue her elliptical pathway round the Earth's centre, with nearly as much freedom as the Moon pursues hers; nor would the ship's deflections from her course, or the heeling over and oscillations of her hull, caused by tides, winds, and waves, very materially differ from the librations of the Moon "pulled about" by Sun and Earth. Her masts would point successively to different stars. Thus, if the Moon rotates on her axis, so does the ship on hers: though such a mode of expression would certainly be esteemed a great mistake in any physical science; and I apprehend, that if I wished a ball or wheel to revolve precisely as the Moon does, in any piece of mechanism, the most certain way of having the whole of it spoiled would be to tell the machine-maker to make it "rotate on its own axis." Nevertheless, we have the high authority of the Astronomer Royal that in astronomical science it is otherwise; and as my sole object is to adapt the explanation of elementary facts in astronomical books for popular instruction to the comprehension of ordinary minds, and certainly not to alter the phraseology of professors, I willingly bow to the fiat of the Astronomer Royal, as far as it affects the arcana of science and the language of savans. For the mathematical calculations of astronomers, dealing with infinite distances,* I can readily conceive why there may be no necessity that their definitions of circular motions should distinguish between those of lunar revolution and axial rotation, and I can also understand why it may be inconvenient to do so; but I do not the less on that account venture to submit that such distinction does exist, both in lunar and sublunary motions, and that for the purposes of popular education it is essential to distinguish them. It is the fact, moreover, and I think it proves the utter unfitness of scientific phraseology in this matter for such educational purposes, that the definitions of axial rotation assume as axioms the question at issue; and fail also to disprove the difference between the two motions.t For example, it is very true that it is a necessary condition of rotation that a radius produced from the rotating body shall point successively to all quarters of the horizon, or, as Professor Airy suggests, to different fixed stars at immeasurable distances. But stated in these terms, such generic property of the rotation of a body round its own axis, is also the generic property of the revolution of a body, which, like the Moon, revolves with the same face turned towards its distant centre of revolution. Figure C in the diagram illustrates this: and also the angular difference and velocities between the radii so produced in the two cases. The radiusya, in the case of lunar revolution, will, while the Moon's centre has passed along the arc ff, have passed from star y to star 2; in the case of axial rotation the same radius f a' will have passed still further round the horizon, and having at the Moon's circumference described the distinct and additional arc a' a", point beyond S, and have reached 6. It is therefore obvious that rotation and lunar revolution have the property in common (differing only in angular velocity) which Professor Airy attributes to the former. I humbly submit that this affords but slight presumption that the two motions are therefore identical; scarcely more so than it would to assert that two triangles were equal, because their sides were equal.Earth, tlmt Earth being a travelling centre. Every point in the Moon's 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.

It has been remarked that the question of rotation is one which must be determined exclusively with relation to space. I venture to submit that it has no necessary relation to space whatever. If the Moon were alone in chaos, the question of her rotation or non-rotation would be nowise altered. The rotation of a body is determinable by the relative movements of different parts of its own body, quite independently of reference to space or to any external point. So is revolution round a

* For example, viewed with relation to euch distances that the diameter of the Moon would subtend no perceptible angle, there would be no assignable difference between the aspects presented by the two movements, and no necessity for denning it.

T "If a sphere, whose centre moves freely in a plane, has no rotatory motion, its radii always remain parallel to themselves."—" A Wrangler!"

Lemma.—" If a circular dish be laid upon the surface of this paper, and any one of the points of its circumference be brought in succession opposite to the four sides of the paper, the dish itself must have been constrained to revolve about its centre, whether that centre be fixed or otherwise."

"If a rigid body is moved from one position to another, by whatever means and path, preserving any numbers of lines joining fixed points within it parallel to their first positions, it does not rotate. But if any one or more such lines are inclined to their first positions, the body has rotated round some axis," &c.

[It would indeed be an easy matter to prove anything by similar postulates, if they were but conceded!]

distant centre, except in as far as that centre itself may be called external to the revolving body; being of course the centre of its orbit. References to the relation which rotating bodies bear to space complicate the question at issue, and are a fertile source of misconceptions. Mr. Perigal has so clearly defined this branch of the question, in bis paper " On the Misuse of Technical Terms," that I beg to quote it here. He says :—

"The velocity and other trigonometric functions of revolution and of rotation may be very properly expressed in terms of the angles which the revolving or rotating radius makes with a fixed lime, given in position, whenever the revolution or rotation is likewise about a fixed centre. Due consideration will show equally clearly, that when the centre is not fixed, but itself describes an orbit, the orbital revolution, or axial rotation, ought to be treated as relative to ike revolving radius, or deferent, in which that centre is located, and with which it traverses space, and not to a fixed line, as in the former case. For instance, supposing the Sun were stationary in space, without any other motion than a circular movement of rotation about its own centre, such axial rotation might be very properly referred to a line fixed in space. If the Earth revolves round the Sun, then such orbital revolution may be referred to the same fixed line. If the Earth, while so revolving, has a second movement of rotation about its own centre of gravity, then such axial rotation should be referred to the line joining the centres of Earth and Sun ; not to the first-mentioned fixed line. Again, assuming that the Earth revolves about the Sun, if the Moon revolves about the Earth, and simultaneously were to rotate about her own centre of gravity, her orbital revolution about the Earth ought to be referred to the line joining the centres of Earth and Sun, while her axial rotation should be referred to the line joining her own centre to that of the Earth. And so on for as many such epicyclic movements as may be in combination; the rotations or revolutions in such cases being about moving centres are relative to the various lines of centres, and must not be confounded with the absolute motion ultimately resulting from the operation of all the different movements acting together.

"Although, in the present discussion, the argument is confined (for the sake of avoiding complexity) to the effects of motion in circular orbits, yet it would be equally applicable to motions in ellipses or other curves, as may be thus shown. Let the Moon's centre describe any orbit whose polar equation is r=f(9); r being=a when 9 = 0. In a, and its production, let there be taken a'=a— e and a" =a + e for the same values of 9; then, for any other values of 9, the three points in the same straight line which joins the centres of the Earth and Moon will describe congenital curves, which may be termed elliptic, or other conchoids, according as the curvilinear directrix (either of the three curves) is an ellipse, or other curve, having for their equations—

rW-e=/(0)-e, r=/(0), and »•''=)• +e=/(6) + e; consequently, no titration will ensue under such circumstances; because there is no rotation in the case, the motion being orbital only."

"* Before proceeding to show the really essential geometrical differences in kind, as well as degree, between the two motions, I may be permitted to remark that the fact that the Moon thus turns the same radius all round the horizon, and different faces to any fixed object outside it, has been the chief stumbling-block of those who believe that she must rotate on her own axis to produce this result. It is caused by simple revolution.* So that the hemisphere of the Moon, which at one part of her orbit faces the north, must at 180°, or when at the opposite side of it, face the south; but in point of fact this is precisely the same kind of motion, or turning round, which would be imparted to a ball, a (See Figure C again), through which the arm of a lever (y, a, F) were passed, having its fulcrum at F. Thus, the outside of the ball a would, as the lever moved it round the circle

* The ordinary delusion is that some additional turning is requisite; the fact being, that the Moon simply keeps one side towards the centre of her orbit.

g, face all points in the compass : but would any one dream of describing the motion of such a ball as rotating round an axis within it, or, what is the same thing, round its own axis ?*

The following letters will appropriately introduce the consideration of the actual differences and distinct properties which, in my humble opinion, clearly distinguish the two movements.

"May 3rd, 1856.

"Dkab Sir,—I think of publishing the best letters which have been written, some of which, like your own, have not yet been published. I should feel greatly gratified by your kind permission to add yours to the number.

"It would materially tend usefully to elucidate the subject, if you were kind enough also to tell me your opinion on these points :—

"1. The Moon's motion being that which you clearly describe in your letter, keeping the same hemisphere always towards the centre of her orbit, must not every point in her body describe concentric rings (allowing for the ellipticity of her orbit) round the Earth, no point in the Moon describing concentric rings round her own centre 1

"2. Must not the point most distant from us in the Moon move, during her revolution, with more velocity than the point nearest to us in the same plane, though at equal distances from her centre; the former point having a larger arc to traverse than the latter in the same time 1

"3. Is it not a necessary condition of the rotation of a body round its own axis, that all points in the body should move in concentric rings round its own centre of rotation? Is it not also a similar condition of such axial rotation, that all points in the rotating body, having the same radius from the centre and in the same plane, should move at equal speed, and that their velocity of rotation is in proportion to their distance from the centre 1

"4. If so, is it not a misuse of terms to say that the Moon rotates on or round her own axis?

"Though it is true that to a fixed star, or any point outside the Moon's orbit, she presents similar faces to those which her axial rotation would present, these, though common to each motion, do not, I venture to submit, prove their identity.—I am, dear Sir, with great respect, yours very truly, "Jelinger Symons.

"professor Airy."

"Royal Observatory, Greenwich, "May 5th, 1856. "Dear Sir—In reply to your questions of May 3— "1. Every point in the Moon's body does describe a ring round the

* I have been favoured on this point with manifold diagrams, several experiments, and sundry suggestions ;—ex. gr., " May I request you to put your finger into a hole in a felloe of a wheel, and see if you can (standing of course outside its circumference) turn it round without allowing the said hole to rotate round the said finger." For the reason above given, certainly not. If I place my elbow at the nave, or centre of the wheel, I obviously can; which brings us to the real difference between the two motions. Again, I have been frequently asked, triumphantly, whether, if a string were tied to the Moon, and held at the other end at the Sun, it would not be twisted round the former. I have no doubt it would ; but this proves no more than that the Moon moves round her orbit: it is the same property as above admitted—common to her revolution and to axial rotation; either would coil the string.

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"2. This question cannot be answered very simply. All motion (except purely angular rotation) is Relative, and the point of reference must be stated; thus—

"With reference to the Earth, the invisible half of the Moon moves with the greatest velocity;

"With reference to the Sun, the half of the Moon invisible to us moves sometimes quickest and sometimes slowest;

"With reference to the Moon's centre, the two halves of the Moon move with equal speed.

"3. A body consisting of firmly-connected parts, if it rotate at all round any centre whatever, must (by geometrical necessity) so rotate that all points move in concentric rings relative to that centre, and the speed of each particle relative to that centre will be proportional to the distance from that centre. But this does not at all determine where that centre is.

"4. Nobody scruples to say that the Earth revolves round its axis, because, if we lay down in a plan the motion of any point of the Earth, conceiving the Earth's axis to be fixed, it shows distinct circular daily rotation with reference to that axis. But if we lay down the motion of the same point in a plan, conceiving the Sun to be fixed, it exhibits no sign of daily rotation. In like manner, we say that the Moon revolves round its axis, because, if we lay down in a plan the motion of any point of the Moon's body, conceiving the Moon's centre to be fixed, it shows distinct circular monthly rotation with reference to that centre. But if we lay down the motion of the same point in a plan, conceiving the Sun to be fixed, it exhibits no sign of monthly rotation. It is true that if the motion of any point of the Moon be laid down, conceiving the Earth to be fixed, it shows distinct circular monthly rotation round the Earth; but this does not contradict what has been said in reference to the apparent motion of a point, when the Moon's centre is conceived to be fixed, and does not invalidate the inference that the Moon rotates round her own axis.

"I may add generally—

5. "Suppose it to be known that a point or body does rotate, toith reference to any specified axis, then the point or body may be said, with equal propriety, to rotate round any other axis parallel to the former, provided that this statement be accompanied with a statement of the motion of translation of the new axis relative to the former.

"You are at liberty to publish my letters, if you think it desirable: but I doubt the advantage of doing so, for I have said nothing which is not extensively known.—I am, dear Sir, faithfully yours,

"Jelinger Symons, Esq." "G. B. AIRY.

I beg now again to refer the reader to the three diagrams in Fig. A, and by aid of the admissions in the above letter, and the application of a few dynamical laws, which I will divest as much as possible of technical phrases, I hope that I shall be able to prove that, by the motion represented in Fig. 1, the Moon, whose motion is indicated by the point of the arrow, would rotate on her own axis backwards, that is, from east to

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