wheels in such a manner, that whilst the frame is turned round, the wheel H turns the same way that the wheel E does, the wheel G turns the contrary way, and the wheel F no way at all.

"Before we explain the principles on which these three different effects depend, it will not be improper to fix some certain criteria for bodies turning or not turning round their own axis or centres, and to make a distinction between absolute and relative motion.

1st, If a body show all its sides progressively round toward a certain fixed point in the heavens, the body turns round its own axis or centre, whether it remains still in the same place, or has a progressive motion in any orbit whatever; for unless it does turn round its own centre, it cannot possibly have one of its sides toward the west at one time, toward the south at another, toward the east at a third time, and toward the north at a fourth. This is the case with the Moon, which always keeps one side toward the Earth, but shows the same side to every fixed point of the starry heaven, in the plane of her orbit, in the time she goes once round her orbit; because in the time that she goes round her orbit, she turns once round her own axis or centre; on the contrary, if a body still keeps one of its sides toward a fixed point of the heaven, the body does not turn round its own axis or centre, whether it keeps in one and the same place, or has a progressive motion in any orbit or direction whatever. This is the case with the card of the compass of a ship, which still keeps one of its points towards the magnetic north, let the ship be at rest or sail round a circle many miles in diameter.

"Both of these cases may be exemplified either by a cube or a globe having a pin fixt into either of its sides to hold it by. We shall suppose a cube, because its sides are flat.—Sit down by a table, and hold the cube by the pin, which may be called its axis, and keep one of its sides toward any side of the room. Whilst you do this, you do not turn the cube round its axis, whether you still keep it in the same place or carry it round any other fixed body on the table; but if you try to keep any side of the cube toward the fixed body whilst you are carrying it round the same, you will find that you cannot do so without turning the pin round (which is fixt into the cube) betwixt the finger and thumb whereby you hold it, unless you rise and walk

round the table, keeping your face always toward the fixed body on the table, and then both yourself and the cube will have turned once round; for the cube will have shown the same side progressively round to all sides of the room, and your face will have been turned toward every side of the room, and every fixed point of the horizon. 162

2d, If a ship turns round, and at the same time, a man stands on the deck without moving his feet, he is turned absolutely round by the motion of the ship, tho' he has no relative motion with respect to the ship; but if, whilst the ship is turning round, he endeavours to turn himself round the contrary way, he thereby only undoes the effect that the turning of the ship would otherwise have had upon himself, and is, in fact, so far from turning absolutely round, that he keeps himself from turning at all, and the ship turns round him as round a fixed axis, although, with respect to the ship, he has a relative motion.

"Fig. 2 is a small plan or flat view of the machine in which the same letters of reference are put to the wheels in it as to those in Fig. 1, for the conveniency of looking at both the

Plan of the Wheels, Frames, &c., of the Mechanical Paradox. figures, in reading the description of them.-WSEN is the immoveable plate; D, the immoveable wheel on the fixt axis in the centre of that plate; E, the thick moveable wheel, whose

162 We give a familiar illustration of the Moon's rotation in note 143, to which the reader is referred.

teeth take into the teeth of the wheel D; and F is one of the thin wheels, over which G and H may be put; and then F, G, and H will make a thickness equal to the thickness of the wheel E, and its teeth will take equally deep into the teeth of them all. The frame that holds these wheels is represented by the parallelogram a b c d; and if it be turned round, it can give no motion to the wheel D, because that wheel is fixt on an axis which is fixt into the immoveable plate.

Take away the thick wheel E, and leave the wheel F where it lies, on the lower plate of the frame. Then turn the frame round the axis of the immoveable plate W SEN (denoted by A in Fig. 1), and it will carry the wheel F round with it. In doing this, F will still keep one and the same side toward the fixt central wheel D, as the Moon still keeps the same side to the Earth; and although F will then have no relative motion with respect to the moving frame, it will be absolutely turned round its own centre g (like the man on the ship whilst he stood without moving his feet on the deck), for the cross mark on its opposite side will be progressively turned toward all the sides of the room.

"But if we would keep the wheel F from turning round its own centre, and so cause the cross mark upon it to keep always toward one side of the room, or like the magnetic needle, to keep the same point still toward one fixed point in the horizon, we must produce an effect upon F resembling what the man on the ship did by endeavouring to turn himself the contrary way to that which the ship turned, so as he might keep from turning at all, and by that means keep his face still toward one and the same point of the horizon. And this is done by making the numbers of teeth equal in the wheels D and F (suppose 20 in each), and putting the thick wheel E between them so as to take into the teeth of them both; for then, as the frame is turned round the axis of the fixed wheel D, by means of the knob n, the wheel E is turned round its axis by the wheel D; and for every space of a tooth that the frame would turn the wheel F, in direction of the motion of the frame, the wheel E will counteract that motion by turning the wheel F just as far backward with respect to the motion of the frame, and so will keep F from turning any way round its own centre, and the cross mark near its edge will be always directed towards one

side of the room.

Whether the wheel E has the same number of teeth as D and F have, or any different number, its effect on F will be still the same.

"If F had one tooth less in number than D has, the effect produced on F, by the turning of the frame, would be as much more than counteracted by the intermediate wheel E, as is equal to the space of one tooth in F; and therefore, while the frame. was turned once round in direction of the letters W SE N on the immovable plate, the wheel F would be turned the contrary way, as much as is equal to the space taken up by one of its teeth; but if F had one tooth more than D has, the effect of the motion of the frame (which is turned F round in the same direction with it), would not be fully counteracted by means of the intermediate wheel E, for as much of that effect would remain as is equal to the space of one tooth in F; and therefore, in the time the frame was turned once round, the wheel F would turn on its own centre, in direction of the motion of the frame, as much as is equal to the space taken up by one of its teeth; and here note, that the wheel E (which turns F) always turns in direction of the motion of the frame.

"And therefore, if an upright pin be fixed into the lower plate of the frame, under the centre of the wheel F, and if the wheel F has the same number of teeth that the fixt wheel D has, the wheel G one tooth less, and the wheel H one tooth more; and if these three wheels are put loosely upon this pin, so as to be at liberty to turn either way, and the thick wheel E takes into the teeth of them all, and also into the teeth of the fixt wheel D, then, whichever way the frame is turned the wheel H will turn the same way, the wheel G the contrary way, and the wheel F no way at all. The less number of teeth G has, with respect to those in D, the faster it will turn backward; and the greater number of teeth H has, with respect to those in D, the faster it will turn forward, reckoning that motion to be backward which is contrary both to the motion of the frame and of the thick wheel E, and that motion to be forward which is in the same direction with the motion of the frame and of the wheel E; so that the turning or not turning of the three wheels, F, G, H, or the direction and velocity of the motions of those that do turn round, depends entirely on the relation between their numbers of teeth, and the number of teeth in the fixt

wheel D, without any regard to the number of teeth in the moveable wheel E." 163 (Vide "The Description and Use of a new Machine called the Mechanical Paradox,” invented by James Ferguson, F.R. S., London, 1764, pp. 4-9; also, Ferguson's "Tables and Tracts relative to several Arts and Sciences," London, 1767, pp. 171, 172; Ferguson's "Select Mechanical Exercises," London, 1773, pp. 46–57.

We now subjoin the long and interesting letter already referred to, written by Ferguson to his friend the Rev. Mr. Cooper, of Glass, regarding the origin of the Mechanical Paradox. 164

FERGUSON'S INTERESTING AUTOGRAPH LETTER on the ORIGIN of the MECHANICAL PARADOX.—

"REVEREND AND DEAR SIR,-I am glad that my last letter came safe to your hands, and do return you my sincere thanks for delivering the one enclosed in it to my sister, begging that you will now repeat the same favour, as it is exactly on a similar occasion, and she may still be in need of a small supply. I thank God that I am now much recovered of my gravel, and in hopes of getting quite well again.

"I herewith send you an account of my Mechanical Paradox, and my three letters to parson Kennedy, who is now very quiet. He has been most sadly trimmed by all the monthly reviewers. My interview with the watchmaker was as follows:One evening I went to a weekly club with a friend, and on our entering the room (or very soon after), the watchmaker began to hold forth violently against a Trinity of persons in the Godhead, wondering at the impudence of the person who broached such an absurd doctrine, and at the weakness and folly of every one who believed it. I happened to sit just opposite to him, with the table between us, and (you may believe) plenty of wine and punch upon it. I gave him a severe frowning look, on which he asked my opinion concerning the Trinity. I told him that all my belief thereof depended upon the opinion I had of the sure knowledge and veracity of the revealer, but that I did

163 We may mention that such results from wheel-work combinations were known long before the year 1750, when Ferguson made the Paradox, as we find similar combinations of wheels used in the orreries and planetary machines of Rowley, Wright, and others.

164 Glass is a parish in the south-western district of Banffshire, the Church and Manse of which are situated about 10 miles to the south of Keith.

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