A Mechanical Paradox

"Make four wheels, one wheel as thick as the other three. Cut teeth in all the wheels. Put the thin wheels in one axis. Set the thick wheel to them so that its teeth may take into those of the three thin ones. Turn the thick wheel. One of the thin wheels turns one way, one the other and the third no way at all."

How can this be possible?

This "paradox" was invented by James Ferguson, and apparently used in a disagreement with an atheist clockmaker to prove the existence of God!

If he (Ferguson) could make a device to meet the criteria, then the clockmaker would believe that God existed. At least that's the story. I came across it for the first time in while browsing in the BHI library, and made a working copy of the machine when I returned home.

 

 
 

 
 

 

 

 

 

 

 



Ferguson's Paradox ~ The Solution
Click here for the original text of the chapter from Ferguson's "Select Mechanical Exercises"
The following is an extract from "Wheelwright of the Heavens - the Life and Work of James Ferguson, FRS" by John R. Millburn

Before leaving this subject, it is appropriate that one of Ferguson's orrery-type models should be mentioned, namely, the Mechanical Paradox. Although this was a straightforward piece of mechanical hardware, it owed its origin to a theological discussion. Ferguson related the story in a very long letter to a friend, the Rev Mr Cooper of Glass, Banffshire, written many years later. The original letter does not appear to have survived, but Henderson printed it in full from a copy said to be in Ferguson's own hand; it had previously been printed in the Horological Journal for 1858. The essence of the story was as follows. One evening Ferguson went to a weekly gathering (probably a dining or drinking club), where one of the other people present, a watchmaker, 'began to hold forth against a Trinity of persons in the God-head, wondering at the impudence of the person who broached such an absurd doctrine'. Ferguson, who was sitting just opposite to him, 'gave him a severe frowning look', whereupon the watchmaker asked his opinion concerning the Trinity. Ferguson suggested that they should talk about the watchmaker's business instead, and asked him whether he understood how one gear wheel turned another. 'I hope I do, said he'.

Then, said I, suppose you make one wheel as thick as other three, and cut teeth in them all, and then put the three thin wheels all loose on one axis, and set the thick wheel to them, so that its teeth may take into those of the three thin ones; now turn the thick wheel round: how must it turn the others? Says he, your question is almost an affront to common sense; for everyone who knows anything of the matter must know that, turn the thick wheel which way you will, all the other three must be turned the contrary way by it. Sir, said I, I believe you think so. Think! says he, it is beyond a thought - it is a demonstration that they must. Sir, said I, I would not have you be too sure, lest you possibly be mistaken; and now what would you say if I should say that, turn the thick wheel whichever way you will, it shall turn one of the thin wheels the same way, the other the contrary way, and the third no way at all. Says he, I would say there was never anything proposed that could be more absurd, as being not only above reason, but contrary thereto. Very well, says I. Now, Sir, is there anything in your ideas more absurd about the received doctrine of the Trinity than in this proposition of mine? There is not, said he; and if I could believe the one, I should believe the other too.

Ferguson then said that he could make such a machine, and would bring it along to show to the assembled company the following week. He did so, and asked the watchmaker to explain it. The watchmaker turned it to and fro, took it to pieces and put it back together again, and confessed that he was thoroughly perplexed. 'The thing is not only above all reason, but it is even contrary to all mechanical principles'.

For shame, Sir, said I, ask me not how it is, for it is a simpler machine than any clock or watch that you ever made or mended; and if you may be so easily non-plused by so simple a thing in your own way of business, no wonder you should be so about the Trinity; but learn from this not for the future to reckon every thing absurd and impossible that you cannot comprehend.

The wheelwork of Ferguson's 'Mechanical Paradox', as it came to be called, was a simplified version of the arrangement commonly employed in orreries to produce parallel motion by three equal gears, plus a slow advance or regression. Some years later, he converted his basic model into an orrery, in which the three motions represented the parallelism of the Earth's axis, the advance of the Moon's apogee, and the regression of its nodes. In 1764 he published a tract describing it, illustrated by an engraving. This version of the model was shown to the Royal Society on 21 November 1765. The original model presumably consisted of the two parallel plates and wheelwork, without the Sun ball, Earth ball, and the orbit rings seen at the right-hand side of the plate; it must, however, have had some way of showing the relative movements of the three thin gears, so the latter were probably mounted on coaxial tubes carrying individual pointers above the upper plate. The date of its conception is uncertain, but was probably around 1750.....

The 'mechanical paradox' orrery, from Plate V in Ferguson's Select Mechanical Exercises, 1773.

Anyone who has followed the arguments in Chapter 4 on whether or not the Moon rotates, will realize that the Mechanical Paradox is concerned with matters of definition: It is, of course, quite true, as the watchmaker claimed, that two spur gears meshing with each other must turn 'in opposite directions'. In Ferguson's device, the thick wheel necessarily turned all the three thin wheels in the opposite direction, relative to the plates between which the wheels were mounted; but the arrangement of the wheelwork as a whole was such that it could only be put into motion by turning the plates themselves round a central fixed gear. One of the thin gears, having the same number of teeth as the thick one (and the central fixed gear below the Sun), remained parallel with itself, i.e. it turned 'no way at all'; the other two, having slightly more and slightly fewer teeth, exhibited a slow advance and regression respectively in comparison with the 'parallel' gear.
My Model

The above extract explains the operation of the "paradox". I was sufficiently interested when I first saw it to construct a model for myself.

The major difficulty in construction is that the 3 thin wheels are all the same diameter, but have different tooth counts, so it is not possible to mesh them all perfectly with the thick wheel. In Ferguson's drawing of the later orrery, he made the thick wheel of variable diameter, like three wheels fixed together.

In the earlier model it was a single wheel, but the wheels were made of soft wood to take up any uneven meshing.

I made the model of brass, and found that it works well with slight adjustment of the tooth shapes. The circular base is perspex.

I hope this is all clear. It might be thought to be a complete waste of time, but it was a good exercise in wheel cutting when I needed the practice, and it gave an insight into James Ferguson, a fascinating character who was influential in the 18th century scientific world. If you have bothered to read this far, it has obviously done something for you too!

Historical Background of the Orrery

The orrery (a planetarium) was invented by George Graham about 1710; the first example was constructed by the London instrument maker John Rowley. It was a device of arms and balls and gears, run by clockwork, that showed how the planets and their satellites moved around the sun as time passed; the Earth typically took about ten minutes to go round once. It should be called a 'Graham', after its inventor, but John Rowley made a copy for Charles Boyle, the fourth Earl of Orrery, and ingratiatingly named it in his honour. (Boyle was described later that century as “one of the literary ornaments of the reign of Queen Anne”; he was a relative of the more famous Robert Boyle, he of Boyle’s law.) The orrery became a popular amusement and teaching device; no progressive educational establishment was without one.

The wheelwork of Ferguson's 'Mechanical Paradox', as it came to be called, was a simplified version of the arrangement commonly employed in orreries to produce parallel motion by three equal gears, plus a slow advance or regression.

James Ferguson was born in Keith, Scotland in 1710, he was a self-taught scientist and astronomer. During 1743 in London he established himself as a successful lecturer and author; publishing books and pamphlets on scientific subjects. He designed several astronomical clocks and orreries, mainly for use in his lectures. He was elected a Fellow of the Royal Society in 1763. He became a friend of Dr. Benjamin Franklin, who was in Britain at that time to lobby Parliament about tax grievances. Ferguson designed a variant of Dr. Franklin’s famous three-wheelcd clock for which he received acclaim. The brass plaques, on the large orrery, with engraved zodiac signs are copies of the original drawings by James Ferguson.

The portrait of James Ferguson F.R.S. is from the frontpiece of the posthumous second edition (1778) of his 'Select Mechanical Exercises', incorporating a portrait based on the Townsend print published in 1776.

The principle of operation (from the above site), is Ferguson's own description. It is represented to view by Fig, 1. of plate v. in which, A is called the immoveable plate, because it lies still on a table whilst the machine is at work. B C is a moveable frame, to be turned round an upright axis a (fixed into the center of the immoveable plate) by taking hold of the knob n, which is fixed into the index h. On the said axis is fixed the immoveable wheel D, whose teeth take into the teeth of the thick moveable wheel E, and turn it round its own axis, as the frame is turned round the fixed axis of the immoveable wheel D; and in the Same direction that the frame is moved. The teeth of the thick wheel E take equally deep into the teeth of the three wheels F, G, and H; but operate on these 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 turns 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 centers; and to make a distinction between absolute and relative motion. 1. If a body shews all its sides progressively round towards a certain fixed point in the heavens, the body turns round its own axis or center, 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 center, 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 shews 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 Center. 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 center, whether it keeps in one and the same place, or has a progressive motion in any orbit or direction what-ever. This is the case with the card of the compass in a ship, which still keeps one of its points toward the magnetic north, let the ship be at rest, or sail round a circle of many miles diameter. Both these cases may be exemplified either by a cube or a globe, having a pin fixed into either of its sides to hold it by: we shall suppose a cube, because its sides are flat. Sit down at 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 fixed 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 shewn 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.2. 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. though 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 him-self; 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. 10 for the conveniency of looking at both the figures, in reading the description of them. W S E N is the round immoveable plate: D the immoveable wheel on the fixed axis in the center of that plate: E the thick moveable wheel whose 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 abcd; and if it be turned round it can give no motion to the wheel D, because that wheel is fixed on an axis which is fixed into the great 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 S E N (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 fixed central wheel D, as the Moon still keeps the same side toward 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 center g (like the man on the ship whilst he stood without moving his feet on the deck) for the cross mark on its side next S will be progressively turned toward all the sides of the room.But, if we would keep the wheel F from turning round its own center, 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 round 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 center, 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, whilst the frame was turned once round, suppose in direction of the letters WSEN on the immoveable 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 in number than D has, the effect of the motion of the frame (which is to turn 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 center, 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 center of the wheel F, and if the wheel F has the same number of teeth that the fixed 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 fixed 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 of 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 fixed wheel D, without any regard to the number of teeth in the moveable wheel E.Having solved the paradox, and described the cause of the different effects which are produced upon the three wheels F, G, and H, we shall now proceed to shew some uses that may be made of the machine. This machine is so much of an ORRERY, as is sufficient to shew the different lengths of days and nights, the vicissitudes of the seasons, the retrograde motion of the nodes of the Moon's orbit, the direct motion of the apogeal point of her orbit, and the months in which the Sun and Moon must be eclipsed. On the great immoveable plate A (see fig. 1) are the months and days of the year, and the signs and degrees of the zodiac so placed that when the annual index b is brought to any given day of the year, it will point to the degree of the sign in which the sun is on that day. This index is fixed to the moveable frame B C, and is carried round the immoveable plate with it, by means of the knob n. The carrying this frame and index round the immoveable plate, answers to the Earth's annual motion round the Sun, and to the Sun's apparent motion round the ecliptic in a year. The central wheel D (being fixed on the axis a, which is fixed in the center of the immoveable plate) turns the thick wheel F round its own axis by the motion of the frame; and the teeth of the wheel E take into the teeth of the three wheels F,G,H, whose axes turn within one another, like the axes of the hour, minute, and second hands of a clock or watch, where the Seconds are shewn from the center of the dial-plate. On the upper ends of these axes are the round plates J, K, L; the plate J being on the axis of the wheel F, K on the axis of G, and L on the axis of H. So that, whichever way these wheels are affected, their respective plates, and what they support, must be affected in the same manner; each wheel and plate being independent of the others. The two upright wires M and N are fixed in-to the plate I; and they support the small ecliptic O P, on which, in the machine, the signs and degrees of the ecliptic are marked. This plate also supports the small terrestrial globe e on its inclining axis f; which is fixed into the plate near the foot of the wire N. This axis inclines 23½ degrees from a right line, supposed to be perpendicular to the surface of the plate J, and 66 ½ to the plane of the small ecliptic O P which is parallel to that plate. On the Earth e, is the crescent g, which goes more than half way round the Earth, and stands perpendicular to the plane of the small ecliptic O P, directly facing the Sun Z : its use is to divide the enlightened half of the Earth next the Sun from the other half which is then in the dark; so that it represents the boundary of light and darkness, and therefore, ought to go quite round the Earth; but cannot, in a machine, because in some positions;. the Earth's axis, would fall upon it. The Earth may be freely turned round on its axis by hand. within the crescent which supported by the crooked wire w, fixed to it, and into the upper plate of the moveable frame BC. In the plate K are fixed the two upright wires Q and R; they support the Moon's inclined orbit ST in its nodes, which are the two opposite points of the Moon's orbit where it intersects the ecliptic OP. The ascending node is marked W . to which the descending node is opposite, below c, but hid from view by the globe c . The half, W Tc of this orbit is on the north side of the ecliptic OP, and the other half ,cSW is on the South side of the ecliptic. The Moon is not in this machine: but when she is in either of the nodes of her orbit in the heavens, she is then in the plane of the ecliptic: when she is at T in her orbit, she is in her greatest north latitude; and when she is at S, she is in her greatest south latitude. In the plate L is fixed the crooked wire UU, which points downward to the Small ecliptic OP, and shews the motion of the Moon's apogee therein, and its place at any given time. The ball Z represents the Sun, which is supported by the crooked wire XY, fixed into the upper plate of the frame at X. A straight wire W proceeds from the Sun Z, and points always toward the center of the Earth e; but toward different points on its surface at different times of the year, on account of the obliquity of its axis, which keeps its parallelism during the Earth's annual course round the Sun Z; and therefore must incline sometimes toward the Sun, at other time from him, and twice in the year neither toward nor from the sun, but sidewise to him. The wire W is called the solar ray. As the annual index b shews the Sun's place in the ecliptic for every day of the year, by turning the frame round the axis of the immoveable plate A according to the order of the months and signs, the solar ray does the same in the small ecliptic OP for, as this ecliptic has no motion on its axis, its signs and degrees still keep parallel to those on the immoveable plate. At the same time, the nodes of the Moon's orbit ST (or points where it intersects the ecliptic O P) are moved backward, or contrary to the order of signs, at the rate of 19 ½ degrees every Julian year; and the Moon's apogeal wire UU is moved forward, or according to the signs of the ecliptic, nearly at the rate of 41 degrees every Julian year; the year being denoted by a revolution of the Earth e round the Sun Z; in which time the annual index b goes round the circle of months and signs on the immoveable plate A. Take hold of the knob n, and turn the frame round thereby; and in doing this, you will perceive that the north pole of the Earth e is constantly before the crescent g, in the enlightened part of the Earth toward the Sun, from the 20th of March to the 23rd of September; and the south pole all that time behind the crescent in the dark; and, from the 23rd. of September to the 20th of March, the north pole is constantly in the dark, behind the crescent, and the south pole in the light before it which shews that there is but one day and one night at each pole, in the whole year; and that, when it is day at either pole, it is night at the other. From 20th of March to the 23rd of September, the days are longer than the nights, in those places of the northern hemisphere of the Earth which revolve through the light and dark, and shorter in those of the southern hemisphere. From the 23rd of September to the 20th of March, the reverse.There are 24 meridian semicircles drawn on the globe, all meeting in its poles; and as one rotation or turn of the Earth on its axis, is performed in 24 hour, each of these meridians is an hour distant from the other, in every parallel of latitude. Therefore, if you bring the annual index b to any given day of the year, on the immoveable plate, you may see how long the day then is at any place of the Earth, by counting how many of these meridians are in the light, or before the crescent, in the parallel of latitude of that place; and this number being subtracted from 24 hours, will leave remaining, the length of the night And if you turn the Earth round its axis, all those places will pass directly under the point of the solar ray, which the Sun passes vertically over on that day, because they are just as many degrees north or south of the equator, as the Sun's declination is from the equinoctial.At the two equinoxes, viz. on the 20th of March and 23rd of September, the Sun is in the equinoctial, and consequently has no declination. On these days, the solar ray points directly toward the equator, the Earth's poles lie under the inner edge of the crescent, or boundary of light and darkness; and, in every parallel of latitude, there are twelve of the meridians, or hour-circles, before the crescent, and twelve behind it; which shews that the days and nights then are each twelve hours long at all places of the Earth. And, if the Earth be turned round its axis, you will see that all places on it go equally through the light and the dark hemispheres. On the 21st of June, the whole space within the north polar circle is enlightened, which is 23½ degrees from the pole, all around; because the Earth's axis then inclines 23½ degrees toward the Sun; but the whole space within the south polar circle is in the dark; and the solar ray points toward the tropic of Cancer on the Earth, which is 23½ degrees north from the equator. On the 20th of December the reverse happens, and the solar ray points toward the tropic of Capricorn, which is 23½ degrees South from the equator.If you bring the annual index b to the beginning of January and turn the Moon's orbit ST by its supporting wires Q and R till the ascending node (marked W ) comes to its place in the ecliptic OP, as found by an Ephemeris or by Astronomical Tables, for the beginning of any given year; and then move the annual index by means of the knob n, till the index comes to any given day of the year afterward, the nodes will stand against their places in the ecliptic on that day. And if you move the index onward, till either of the nodes comes directly against the point of the Solar ray, the index will then be at the day of the year on which the Sun is in conjunction with that node. At the times of those new Moons which happen within seventeen days of the conjunction of the Sun with either of the nodes, the Sun will be eclipsed: and at the times of those full Moons, which happen within twelve days of either of these conjunctions, the Moon will be eclipsed. Without these limits there can be no eclipse either of the Sun or Moon; because in nature, the Moon's latitude, or declination from the ecliptic, is too great for the Moon's shadow to fall on any part of the Earth, or for the Earth's shadow to touch the Moon. Bring the annual index to the beginning of January, and set the Moon's apogeal wire UU to its place in the ecliptic for that time, as found by Astronomical Tables; then move the index forward to any given day of the year, and the wire will point on the small ecliptic to the place of the Moon's apogee for that time. The Earth's axis f inclines always toward the beginning of the sign Cancer on the small ecliptic OP. And, if you set either of the Moon's nodes, and her apogeal wire, to the beginning of that sign, and turn the plate A about, until the Earth's axis inclines toward any side of the room (suppose the north side) and then move the annual index round and round the immoveable plate A, according to the order of the months and signs upon it, you will see that the Earth's axis and beginning of Cancer will still keep toward the same side of the room, without the least deviation from it ; but the nodes of the Moon's orbit ST will turn progressively towards all the sides of the room, contrary to the order of signs in the Small ecliptic OP, or from, east, by south, to west, and so on: and the apogeal wire UU will move the contrary way to the motion of the nodes, or according to the order of the signs in the small ecliptic, from west, by south, to east, and so on quite round. A clear proof that the wheel F, which governs the Earth's axis and the small ecliptic, does not turn any way round its own center; that the wheel G, which governs the Moon's orbit O P, turns round its own center backward, or contrary both to the motion of the frame B C and thick wheel E; and that the wheel H, which governs the Moon's apogeal wire UU, turns years and four-fifths of a year, and the nodes in eighteen years and a half. Notwithstanding the difference of the numbers of teeth in the wheels F, G, and H, and their being all of equal diameters. they take tolerably well into the teeth of the thick wheel E, because they are made of soft wood. But, if they were made of metal, the wheel E in Fig.1. ought to be made of the shape of E (seen edgewise) in Fig. 3. with very deep teeth, and the wheels F, G, and H, in Fig I. of diameters proportioned to their respective numbers of teeth, as F, G, and H, in Fig 3. And then the teeth of these three wheels would be of equal size, with those of the wheel E wherein they work; and the motions would be free and easy, without any pinching or shake in the teeth. Return to previous page

Extract from James Ferguson's
"Select Mechanical Exercises" 1773

The Description and Use of a New Machine

called the MECHANICAL PARADOX:

The vulgar and illiterate take almost every thing, even the most important, upon the authority of others, without ever examining it themselves. Although this implicit confidence is seldom attended with any bad consequences in the common affairs of life, it has nevertheless, in other things, been much abused; and in political and religious matters, has produced fatal effects. On the other hand, knowing and learned men, to avoid this weakness, have fallen into the contrary extreme: some of them believe everything to be unreasonable, or impossible, that appears so to their first apprehension; not adverting to the narrow limits of the human understanding, and the infinite variety of objects, with their mutual operations, combinations, and affections, that may be presented to it.

It must be owned, that credulity has done much more mischief in the world than incredulity has done, or ever will do; because the influences of the latter extend only to such as have some share of education, or affect the reputation thereof. And since the human mind is not necessarily impelled, without evidence, either to belief or unbelief; but may suspend its assent to, or dissent from, any proposition, till after a thorough examination ; it is to be wished, that men of literature, especially philosophers, would not hastily, and by first appearances, determine themselves with respect to the truth or falsehood, possibility or impossibility of things.A person who has made but little progress in the mathematics, though in other respects learned and judicious, would be apt to pronounce it impossible that two lines, which were no where two inches asunder, may continually approach toward one another, and yet never meet, although continued to infinity: and yet the truth of this proposition may be easily demonstrated. And many, who are good mechanics, would be as apt to pronounce the same, if they were told, that although the teeth of one wheel should take equally deep into the teeth of three others, it should affect them in such a manner. That in turning it any way around its axis, it should then turn one of them the same way, another the contrary way, and the third no way at all.

On a very particular occasion, about eighteen years ago, I contrived a small machine of this sort, which has been shewn and explained to many; and which I shall here describe, and explain some of the uses it has been applied to. It is represented to view by Fig, 1. of plate v. in which, A is called the immoveable plate, because it lies still on a table whilst the machine is at work. B C is a moveable frame, to be turned round an upright axis a (fixed into the center of the immoveable plate) by taking hold of the knob n, which is fixed into the index h.

On the said axis is fixed the immoveable wheel D, whose teeth take into the teeth of the thick moveable wheel E, and turn it round its own axis, as the frame is turned round the fixed axis of the immoveable wheel D; and in the Same direction that the frame is moved.

The teeth of the thick wheel E take equally deep into the teeth of the three wheels F, G, and H; but operate on these 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 turns 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 centers; and to make a distinction between absolute and relative motion.

1. If a body shews all its sides progressively round towards a certain fixed point in the heavens, the body turns round its own axis or center, 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 center, 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 shews 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 Center. 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 center, whether it keeps in one and the same place, or has a progressive motion in any orbit or direction what-ever. This is the case with the card of the compass in a ship, which still keeps one of its points toward the magnetic north, let the ship be at rest, or sail round a circle of many miles diameter.

Both these cases may be exemplified either by a cube or a globe, having a pin fixed into either of its sides to hold it by: we shall suppose a cube, because its sides are flat. Sit down at 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 fixed 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 shewn 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.2. 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. though 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 him-self; 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. 10 for the conveniency of looking at both the figures, in reading the description of them. W S E N is the round immoveable plate: D the immoveable wheel on the fixed axis in the center of that plate: E the thick moveable wheel whose 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 abcd; and if it be turned round it can give no motion to the wheel D, because that wheel is fixed on an axis which is fixed into the great 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 S E N (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 fixed central wheel D, as the Moon still keeps the same side toward 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 center g (like the man on the ship whilst he stood without moving his feet on the deck) for the cross mark on its side next S will be progressively turned toward all the sides of the room.But, if we would keep the wheel F from turning round its own center, 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 round 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 center, 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, whilst the frame was turned once round, suppose in direction of the letters WSEN on the immoveable 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 in number than D has, the effect of the motion of the frame (which is to turn 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 center, 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 center of the wheel F, and if the wheel F has the same number of teeth that the fixed 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 fixed 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 of 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 fixed wheel D, without any regard to the number of teeth in the moveable wheel E.Having solved the paradox, and described the cause of the different effects which are produced upon the three wheels F, G, and H, we shall now proceed to shew some uses that may be made of the machine.

This machine is so much of an ORRERY, as is sufficient to shew the different lengths of days and nights, the vicissitudes of the seasons, the retrograde motion of the nodes of the Moon's orbit, the direct motion of the apogeal point of her orbit, and the months in which the Sun and Moon must be eclipsed.

On the great immoveable plate A (see fig. 1) are the months and days of the year, and the signs and degrees of the zodiac so placed that when the annual index b is brought to any given day of the year, it will point to the degree of the sign in which the sun is on that day. This index is fixed to the moveable frame B C, and is carried round the immoveable plate with it, by means of the knob n. The carrying this frame and index round the immoveable plate, answers to the Earth's annual motion round the Sun, and to the Sun's apparent motion round the ecliptic in a year.

The central wheel D (being fixed on the axis a, which is fixed in the center of the immoveable plate) turns the thick wheel F round its own axis by the motion of the frame; and the teeth of the wheel E take into the teeth of the three wheels F,G,H, whose axes turn within one another, like the axes of the hour, minute, and second hands of a clock or watch, where the Seconds are shewn from the center of the dial-plate.

On the upper ends of these axes are the round plates J, K, L; the plate J being on the axis of the wheel F, K on the axis of G, and L on the axis of H. So that, whichever way these wheels are affected, their respective plates, and what they support, must be affected in the same manner; each wheel and plate being independent of the others.

The two upright wires M and N are fixed in-to the plate I; and they support the small ecliptic O P, on which, in the machine, the signs and degrees of the ecliptic are marked. This plate also supports the small terrestrial globe e on its inclining axis f; which is fixed into the plate near the foot of the wire N. This axis inclines 23½ degrees from a right line, supposed to be perpendicular to the surface of the plate J, and 66 ½ to the plane of the small ecliptic O P which is parallel to that plate.

On the Earth e, is the crescent g, which goes more than half way round the Earth, and stands perpendicular to the plane of the small ecliptic O P, directly facing the Sun Z : its use is to divide the enlightened half of the Earth next the Sun from the other half which is then in the dark; so that it represents the boundary of light and darkness, and therefore, ought to go quite round the Earth; but cannot, in a machine, because in some positions;. the Earth's axis, would fall upon it. The Earth may be freely turned round on its axis by hand. within the crescent which supported by the crooked wire w, fixed to it, and into the upper plate of the moveable frame BC.

In the plate K are fixed the two upright wires Q and R; they support the Moon's inclined orbit ST in its nodes, which are the two opposite points of the Moon's orbit where it intersects the ecliptic OP. The ascending node is marked W . to which the descending node is opposite, below c, but hid from view by the globe c

. The half, W Tc of this orbit is on the north side of the ecliptic OP, and the other half ,cSW is on the South side of the ecliptic. The Moon is not in this machine: but when she is in either of the nodes of her orbit in the heavens, she is then in the plane of the ecliptic: when she is at T in her orbit, she is in her greatest north latitude; and when she is at S, she is in her greatest south latitude.

In the plate L is fixed the crooked wire UU, which points downward to the Small ecliptic OP, and shews the motion of the Moon's apogee therein, and its place at any given time.

The ball Z represents the Sun, which is supported by the crooked wire XY, fixed into the upper plate of the frame at X. A straight wire W proceeds from the Sun Z, and points always toward the center of the Earth e; but toward different points on its surface at different times of the year, on account of the obliquity of its axis, which keeps its parallelism during the Earth's annual course round the Sun Z; and therefore must incline sometimes toward the Sun, at other time from him, and twice in the year neither toward nor from the sun, but sidewise to him. The wire W is called the solar ray.

As the annual index b shews the Sun's place in the ecliptic for every day of the year, by turning the frame round the axis of the immoveable plate A according to the order of the months and signs, the solar ray does the same in the small ecliptic OP for, as this ecliptic has no motion on its axis, its signs and degrees still keep parallel to those on the immoveable plate. At the same time, the nodes of the Moon's orbit ST (or points where it intersects the ecliptic O P) are moved backward, or contrary to the order of signs, at the rate of 19 ½ degrees every Julian year; and the Moon's apogeal wire UU is moved forward, or according to the signs of the ecliptic, nearly at the rate of 41 degrees every Julian year; the year being denoted by a revolution of the Earth e round the Sun Z; in which time the annual index b goes round the circle of months and signs on the immoveable plate A.

Take hold of the knob n, and turn the frame round thereby; and in doing this, you will perceive that the north pole of the Earth e is constantly before the crescent g, in the enlightened part of the Earth toward the Sun, from the 20th of March to the 23rd of September; and the south pole all that time behind the crescent in the dark; and, from the 23rd. of September to the 20th of March, the north pole is constantly in the dark, behind the crescent, and the south pole in the light before it which shews that there is but one day and one night at each pole, in the whole year; and that, when it is day at either pole, it is night at the other.

From 20th of March to the 23rd of September, the days are longer than the nights, in those places of the northern hemisphere of the Earth which revolve through the light and dark, and shorter in those of the southern hemisphere. From the 23rd of September to the 20th of March, the reverse.There are 24 meridian semicircles drawn on the globe, all meeting in its poles; and as one rotation or turn of the Earth on its axis, is performed in 24 hour, each of these meridians is an hour distant from the other, in every parallel of latitude. Therefore, if you bring the annual index b to any given day of the year, on the immoveable plate, you may see how long the day then is at any place of the Earth, by counting how many of these meridians are in the light, or before the crescent, in the parallel of latitude of that place; and this number being subtracted from 24 hours, will leave remaining, the length of the night And if you turn the Earth round its axis, all those places will pass directly under the point of the solar ray, which the Sun passes vertically over on that day, because they are just as many degrees north or south of the equator, as the Sun's declination is from the equinoctial.At the two equinoxes, viz. on the 20th of March and 23rd of September, the Sun is in the equinoctial, and consequently has no declination. On these days, the solar ray points directly toward the equator, the Earth's poles lie under the inner edge of the crescent, or boundary of light and darkness; and, in every parallel of latitude, there are twelve of the meridians, or hour-circles, before the crescent, and twelve behind it; which shews that the days and nights then are each twelve hours long at all places of the Earth. And, if the Earth be turned round its axis, you will see that all places on it go equally through the light and the dark hemispheres.

On the 21st of June, the whole space within the north polar circle is enlightened, which is 23½ degrees from the pole, all around; because the Earth's axis then inclines 23½ degrees toward the Sun; but the whole space within the south polar circle is in the dark; and the solar ray points toward the tropic of Cancer on the Earth, which is 23½ degrees north from the equator. On the 20th of December the reverse happens, and the solar ray points toward the tropic of Capricorn, which is 23½ degrees South from the equator.If you bring the annual index b to the beginning of January and turn the Moon's orbit ST by its supporting wires Q and R till the ascending node (marked W ) comes to its place in the ecliptic OP, as found by an Ephemeris or by Astronomical Tables, for the beginning of any given year; and then move the annual index by means of the knob n, till the index comes to any given day of the year afterward, the nodes will stand against their places in the ecliptic on that day. And if you move the index onward, till either of the nodes comes directly against the point of the Solar ray, the index will then be at the day of the year on which the Sun is in conjunction with that node. At the times of those new Moons which happen within seventeen days of the conjunction of the Sun with either of the nodes, the Sun will be eclipsed: and at the times of those full Moons, which happen within twelve days of either of these conjunctions, the Moon will be eclipsed. Without these limits there can be no eclipse either of the Sun or Moon; because in nature, the Moon's latitude, or declination from the ecliptic, is too great for the Moon's shadow to fall on any part of the Earth, or for the Earth's shadow to touch the Moon.

Bring the annual index to the beginning of January, and set the Moon's apogeal wire UU to its place in the ecliptic for that time, as found by Astronomical Tables; then move the index forward to any given day of the year, and the wire will point on the small ecliptic to the place of the Moon's apogee for that time.

The Earth's axis f inclines always toward the beginning of the sign Cancer on the small ecliptic OP. And, if you set either of the Moon's nodes, and her apogeal wire, to the beginning of that sign, and turn the plate A about, until the Earth's axis inclines toward any side of the room (suppose the north side) and then move the annual index round and round the immoveable plate A, according to the order of the months and signs upon it, you will see that the Earth's axis and beginning of Cancer will still keep toward the same side of the room, without the least deviation from it ; but the nodes of the Moon's orbit ST will turn progressively towards all the sides of the room, contrary to the order of signs in the Small ecliptic OP, or from, east, by south, to west, and so on: and the apogeal wire UU will move the contrary way to the motion of the nodes, or according to the order of the signs in the small ecliptic, from west, by south, to east, and so on quite round. A clear proof that the wheel F, which governs the Earth's axis and the small ecliptic, does not turn any way round its own center; that the wheel G, which governs the Moon's orbit O P, turns round its own center backward, or contrary both to the motion of the frame B C and thick wheel E; and that the wheel H, which governs the Moon's apogeal wire UU, turns years and four-fifths of a year, and the nodes in eighteen years and a half. Notwithstanding the difference of the numbers of teeth in the wheels F, G, and H, and their being all of equal diameters. they take tolerably well into the teeth of the thick wheel E, because they are made of soft wood. But, if they were made of metal, the wheel E in Fig.1. ought to be made of the shape of E (seen edgewise) in Fig. 3. with very deep teeth, and the wheels F, G, and H, in Fig I. of diameters proportioned to their respective numbers of teeth, as F, G, and H, in Fig 3. And then the teeth of these three wheels would be of equal size, with those of the wheel E wherein they work; and the motions would be free and easy, without any pinching or shake in the teeth.