I must own that I am unable rightly to understand the words which you have employed in explaining the "3rd difference." I do not see how, as regards her orbital revolution, all points in the Moon move with increasing or diminishing velocity successively;" for if there were no motion of rotation in our model moon, each of its points would describe a similar orbit in the same time and with the same velocity; but these orbits would not be concentric. Nor do I see that at present "each point moves with the same speed," unless by "speed" you mean 'angular velocity about the earth's centre.' Indeed, your next remark seems to contradict this one, when you say "All those points in the outer hemisphere [move] faster than those in the inner hemisphere, these two hemispheres never changing their relative position of inner and outer." The division is not exactly into two hemispheres, the boundary being not a plane, but a portion of a sphere, of which the centre of the Earth is the centre. But this is of little importance. What is the cause of the difference of speed? Simply the composition of the motion of rotation round the Moon's axis with the motion of translation round the Earth. For the points in the so-called outer hemisphere, the motion of rotation acts to carry every point forward in the same direction as the motion of translation, and therefore to make it move faster than the centre, which is unaffected by rotation. For the points in the so-called inner hemisphere, the motion of rotation acts to carry every point backwards, or in the contrary direction to the motion of translation, and therefore to make it move more slowly than the centre. When the Earth moves round the Sun, two similar terrestrial hemispheres may be determined at any moment, but on account of the Earth's period of rotation being much shorter than her period of revolution, these hemispheres are perpetually composed of different points.

You proceed to say-" 4th, in axial rotation, all points in the circumference of the rotating body must successively intersect the radius vector, whilst in the Moon one only does." I have already shown that this arises from your referring the motion of rotation to a moving instead of a fixed line.

"5th. While axial rotation is independent of the motion of translation, lunar motion depends on it, and ceases with it." The first part of this statement, to which I drew attention at the beginning of my letter, is true; the second part, I have already endeavoured to show you, is incorrect. If you could prove the second part, the controversy would be at an end; but so would the first law of motion likewise. I suspect that your illustration of a system of two balls rigidly connected, here leads you into error; because, in this case, when the motion of translation ceases, the rigid connection also causes the motion of rotation to cease. But there is a motion of rotation in every part of a rigid body which rotates, this part revolving round the axis, and rotating about its own axis, only precisely because this rotation of the smaller part ceases with the rotation of the whole mass, on account of the rigidity of the connection, it is usually disregarded. On a revolving disk mark out two points, then the line joining these points will be directed successively to all points of the compass during a rotation of the disk; and hence, in the only sense which mathematicians can affix to the term, either of these points revolves or rotates (as they are rigidly connected) round the other, with reference to a fixed line, just as much as if the centre of

rotation of the disk were transferred to either of these points, and the other were, therefore, turned about it in the most self-evident manner.

The following contrivance may make these rotations clearer to nonmathematicians :-Cut out two disks of cardboard, and draw one radius in each. On the larger disk make two holes, one in the centre and one near the circumference on this radius, in which two pins can turn easily. Pass a pin tightly through the centre of the other smaller disk, so that when the pin is twisted by the finger and thumb, the disk rotates, and when it is not so twisted, the disk does not rotate, with reference to the fingers with which the motion is naturally compared. Now, first by a crank-like motion of the right arm (the left hand supporting the pin on which the large disk turns) cause the pin of the small disk, which is placed in the second hole of the large disk, to make the large disk rotate without twisting the finger and thumb, and therefore without rotating the small disk. If you start with the two radii in the same straight line, you will find that the angular separation of the two radii passes thorough all angles in one revolution: hence an apparent retrograde rotation of the smaller disk with reference to the larger radius, and a sensible non-rotation with reference to the finger. Next rotate the large disk with the fingers of the left hand, and without touching it with the pin of the small disk, move the small disk round the larger, so as to keep the two radii in a line; you will find it necessary to twist the pin between your finger and thumb to do so, and hence sensibly to rotate the small disk, while the same disk is now at relative rest with regard to the large one. This instrument, which can be immediately constructed, overcomes the fallacy arising from the rigid connection, by showing that when one body is in motion, any other which is relatively at rest to it must have a precisely similar motion.

Hence it appears to me, that instead of allowing persons to refer motion now to one moving line and now to another (as in the two rotations of the Moon and Earth), and now to a fixed line or plane (as all mathematicians do), it would be more advisable to make them understand the relative nature of motion, and show them the importance of referring all motion to one fixed line or plane, in order that their relative motion might be discovered,—as in the case of specific gravities we compare all bodies with water in place of with one another. Motion relative to such fixed lines or planes is often called "absolute motion,” the term "relative" being commonly, but not necessarily, confined to motion relative to some moving point. Hence Herschel (Outlines of Astronomy, p. 55) says, "The relative motion of two bodies is the same as if either of them were at rest, and all its motion were communicated to the other in an opposite direction. Hence, if two bodies move alike, they will, when seen from each other (without reference to other inner bodies, but only to the starry sphere), appear at rest. Hence, also, if the absolute motion of two bodies [i.e. their motions relative to fixed lines or planes] be uniform and rectilinear, their relative motion is so also."

Some of your correspondents object, that if astronomers are correct, then every man in the world and every ship revolves upon its own axis. This is true; and that it is true is shown by the fact that ships sailing round the world from London to Cape Horn lose a day in their reckoning, owing to their having turned on their own axis once in a direction

contrary to that of the Earth; while those ships which sail by the Cape of Good Hope gain a day, from having turned once on their own axis in the direction of the Earth's rotation,-both in addition to their having turned once on their axis for every rotation of the earth. If there were no interposing Earth, the night would be just as much caused by our backing the Sun, as may be readily seen by walking round a table, with your back constantly turned to it, and placing a lamp or other object at a distance to represent the Sun. The case of losing and gaining the day is well exemplified in the difference between the sidereal and the mean solar year, and in the loss of a sidereal day in 9,448,300 solar days, owing to the precession of the equinoxes (Herschel's Outlines of Astronomy, p. 624).

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On p. 262 you say,-"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, and the other axial rotation: that is the difference. It has been argued that one is only a quicker 'rotation' than the other it may as well be said that walking is a quicker pace than standing still." Part of the misconception implied in this quotation arises, I think, from confounding revolution round an axle with rotation round an axis. The axle of the wheel is a separate body, which may or may not rotate; and the wheel which revolves round or in respect to it causes any point in its own inner boundary to describe a line about the outer boundary of the axle. It may happen that both the axle rotates and the wheel revolves, with equal speed, in the same direction, with reference to some external object. In this case there is no relative rotation, or the axle and wheel are at relative rest. This effect is produced by rigidly fastening the wheel to the axle while the waggon is dragged round a circle.

Rotation round an axis is quite different, because the axis is itself an actual or geometrical part of the revolving body, and is, moreover, a geometrical line, having no diameter. There is no slipping or gliding of any part of the rotating body over this axis, as there is of the revolving body over the axle. Rotation consists simply in the fact that the line drawn from any point perpendicular to the axis points successively to all points of the compass, and this is the case with the dragged as well as with the free wheel, and with the axle of both wheels. The dragged wheel and its axle both rotate once in a revolution of the waggon, with respect to a fixed line, and hence the wheel does not revolve on the axle, and would not even if there were no rigid connection, but only some force applied to prevent friction causing the wheel to revolve. The free wheel revolves a certain number of times about its axle, and rotates once more, with respect to a fixed line, in the course of one revolution of the waggon; that once more not appearing as a motion round the axle, because of the rotation of the axle itself.

A mast does not move with respect to a ship in motion, but it does move with respect to the shore; and a man who walks from the stern to the prow moves faster than the mast with respect to the shore. This is precisely one of the cases in which we may apply your words, and say, that "walking is a quicker pace than standing still!" Your words were

incomplete, because you neglected to mention to what the "standing still" was to be referred. A man who "stands still" on the Earth's surface is really moving very fast through space; but if he moves on the Earth's surface, he may move partly in the direction of this motion or partly against it. In this case, again, "walking is a quicker (or slower) pace than standing still." The path described by the top of a mountain round the Earth's surface is longer than that described by its base in the same time, yet we say that both "stand still." In this case, then, "standing still is a quicker (or slower) pace than standing still!" This is a complete reductio ad absurdum of such non-relative phraseology.

My letter has extended to such an unexpected length that I forbear to allude to other matters which I had marked, especially the effects which you seem to attribute to centrifugal force to produce the rotation whence it is derived, or the independent existence which you assign to it when revolution ceases (p. 259). I have also throughout confined myself to the supposition of one body revolving about another, so that the axes of both are parallel, and I have taken the motions of revolution to be circular and uniform. In fact, I have excluded all the particular circumstances of the Moon's motion, which your theory would leave unaccounted for. But 1 cannot conclude without saying, that your machine, consisting of two balls rigidly connected, is very misleading, and does not in any respect represent the real motions of the Earth and Moon. Thus it will not lead any one to the conception of the Earth's rotation being different from that of your central ball, namely, in about the ratio of 271: 1. It will not allow of any librations of the Moon; for which you have, in fact, not accounted at all, either in longitude or latitude. To suppose that there is a special oscillation of the Moon, is to suppose something which the theory of gravitation must be altered to account for; whereas the theory of a uniform rotation of the Moon on its own axis, and its non-uniform revolution round the Earth (another omitted circumstance), together with the inclination of that axis to the Moon's orbit, and its conical revolution in about nineteen years (all with reference to fixed lines and planes), accounts for all of them.

Hence I conclude, that so far from astronomers having introduced confusion by their saying that the Moon rotates on its axis with reference to a fixed plane, it is those who have talked of her having no axial rotation, without mentioning what line they refer the motion to, while they really refer to a revolving line, who have introduced the confusion-a confusion which always arises on all subjects where the absolute is substituted for the relative.

Hoping that you will read this long letter in the spirit of scientific discussion which has actuated the writer, I remain, with due respect, ALEXANDER J. ELLIS.

SYMPATHY WITH CHILDREN." One of the greatest secrets of success in managing the young, is sympathy with them as children. Nothing but this will lead to a proper understanding and appreciation of the motives by which they are governed, or enable us rightly to estimate the efforts they make for improvement.

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Happy is the teacher who can really enter into the feelings and motives of childhood; and fortunate is the teacher who can discriminate between the apparently wrong actions which are caused by such sudden impulses, and those which are the result of deliberate intention to do wrong."-American Journal of Education.

THE MOON CONTROVERSY.

TO THE EDITOR OF THE ENGLISH JOURNAL OF EDUCATION. SIR,-The controversy on the rotation of the Moon about its axis has suggested to me some reflections which I think not unworthy of attention in your Journal. It is a common notion that mathematical studies enable men to become good reasoners. That this common notion is an error, appears from the recent controversy. I have heard that a gentleman of considerable repute among the learned said, that all the answers to Mr. Symons in the Times were also wrong. It is not difficult to see why they were so. In the first place they lost their tempers; they were indignant that anything which they said should be called in question; they wished, in short, to be considered infallible. So their mathematics did not save them from this first essential of good reasoning-a calm, unimpassioned love of truth. Consequently, in the second place, they never got sight of the real question, and so talked to the wind. As both parties agreed in all the facts of the case, it is clear that the question was one of words or definitions, or by what names the facts should be called. What the mathematicians should have done was, to have laid down the definition of the term rotation about an axis, and then shown that their definition described the facts of the case. Their definition would probably have been opposite to a thing called common sense: hereupon they might have called upon their opponents to define what they meant by common sense." This, if honestly followed out, would have raised questions about realities, questions of infinitely greater importance than this mere verbal question, It would have appeared that "common sense" may be of two kinds: that there is a reflecting common sense and an unreflecting common sense, such as that which would have sent Galileo to the scaffold; and that out of their own peculiar department the mathematicians are allied with the mob of unreflecting common sense, and have none of that right to which they have of late shown a disposition to lay claim, for the purpose of putting down some facts and truths which are now bursting on the world. Let, then, youth be taught not to prostrate its understanding before either great names or that unreflecting common sense to which, I am afraid, every one of us is too apt to yield his judgment. I am, Sir, your obedient Servant,

Great Marlow.

66

W. P. GASKELL.

DR. LARDNER ON LUNAR MOTION.

TO THE EDITOR OF THE ENGLISH JOURNAL OF EDUCATION.

PARIS, June 7th, 1856. SIR,-You will much oblige me by rectifying, in your next number, an error into which Mr. Symons has inadvertently fallen, in one of his quotations from the tract on the planets, published by me in the "Museum of Science and Art." Immediately on the publication of the first number of the work, in January 1854, I discovered an oversight which I had committed in hasty writing, and I immediately had the leaf cancelled, and the error corrected, although it was sufficiently apparent by the context that the mistake was little more than a slip of the pen. It appears that Mr. Symons, by a most singular circumstance, after the lapse of more than two years, has come into possession of one of the few copies of the number in question which were put into circulation before the leaf was cancelled and the error corrected; and finding that the erroneous sentence accorded with his curious lunar theory, he has quoted it in italics, mixed with capitals. A reference to the corresponding page in any copy circulated later will show the error into which Mr. Symons has thus unintentionally fallen.

I have written a paper upon the question of the Moon's Rotation, which I trust will convince any person moderately conversant with elementary mechanics, that the Moon has the rotation which Mr. Symons denies to it, and have sent it to London for publication. I hope it will put an end to the Lunar controversy—if controversy that can properly be called in which all those who are considered by the world in general (including Mr. Symons himself) as the highest authorities in science are of one opinion. Your obedient Servant, DION. LARDNER.

[Dr. Lardner (who is living abroad) is quite in error in his last statement in this note. Mr. Symons's position seems to be this :-"The lunar movement is the revolution of a sphere round a distant centre keeping same face towards it; this is not the rotation of a sphere round its own centre, and should not be so called." Very many scientific men are of this opinion; and a very eminent Wrangler is among them.— ED. E. J. E.]