Our wonder is excited, firstly, by phenomena which occur in accordance with [847a10] nature but of which we do not know the cause, and secondly by those which are produced by art despite nature for the benefit of mankind. Nature often operates contrary to human interest; for she always follows the same course without [15] deviation, whereas human interest is always changing. When, therefore, we have to do something contrary to nature, the difficulty of it causes us perplexity and art has to be called to our aid. The kind of art which helps us in such perplexities we call Mechanical Skill. The words of the poet Antiphon are quite true: [20]
Mastered by Nature, we o’ercome by Art.
Instances of this are those cases in which the less prevails over the greater, and where forces of small motive power move great weights—in fact, practically all those problems which we call Mechanical Problems. They are not quite identical [25] nor yet entirely unconnected with Natural Problems. They have something in common both with Mathematical and with Natural Speculations; for while Mathematics demonstrates how phenomena come to pass, Natural Science demonstrates in what medium they occur.
Among questions of a mechanical kind are included those which are connected [847b10] with the lever. It seems strange that a great weight can be moved with but little force, and that when the addition of more weight is involved; for the very same weight, which one cannot move at all without a lever, one can move quite easily with it, in spite of the additional weight of the lever. [15]
The original cause of all such phenomena is the circle. It is quite natural that this should be so; for there is nothing strange in a lesser marvel being caused by a greater marvel, and it is a very great marvel that contraries should be present together, and the circle is made up of contraries. For to begin with, it is formed by [20] motion and rest, things which are by nature opposed to one another. Hence in examining the circle we need not be much astonished at the contradictions which occur in connexion with it. Firstly, in the line which encloses the circle, being without breadth, two contraries somehow appear, namely, the concave and the [25] convex. These are as much opposed to one another as the great is to the small; the mean being in the latter case the equal, in the former the straight. Therefore just as, if they are to change into one another, the greater and smaller must become equal [848a1] before they can pass into the other extreme; so a line must become straight in passing from convex into concave, or on the other hand from concave into convex and curved. This, then, is one peculiarity of the circle.
Another peculiarity of the circle is that it moves in two contrary directions at [5] the same time; for it moves simultaneously to a forward and a backward position. Such, too, is the nature of the radius which describes a circle. For its extremity comes back again to the same position from which it starts; for, when it moves continuously, its last position is a return to its original position, in such a way that it [10] has clearly undergone a change from that position.
Therefore, as has already been remarked, there is nothing strange in the circle being the origin of any and every marvel. The phenomena observed in the balance can be referred to the circle, and those observed in the lever to the balance; while [15] practically all the other phenomena of mechanical motion are connected with the lever. Furthermore, since no two points on one and the same radius travel with the same rapidity, but of two points that which is further from the fixed centre travels more quickly, many marvellous phenomena occur in the motions of circles, which will be demonstrated in the following problems.
[20] Because a circle moves in two contrary forms of motion at the same time, and because one extremity of the diameter, A, moves forwards and the other, B, moves backwards, some people contrive so that as the result of a single movement a number of circles move simultaneously in contrary directions, like the wheels of [25] brass and iron which they make and dedicate in the temples. Let AB be a circle and CD another circle in contact with it; then if the diameter of the circle AB moves forward, the diameter CD will move in a backward direction as compared with the circle AB, as long as the diameter moves round the same point. The circle CD [30] therefore will move in the opposite direction to the circle AB. Again, the circle CD will itself make the adjoining circle EF move in an opposite direction to itself for the same reason. The same thing will happen in the case of a larger number of circles, [35] only one of them being set in motion. Mechanicians seizing on this inherent peculiarity of the circle, and hiding the principle, construct an instrument so as to exhibit the marvellous character of the device, while they obscure the cause of it.
[848b1]1 · First then, a question arises as to what takes place in the case of the balance. Why are larger balances more accurate than smaller? And the fundamental principle of this is, why is it that the radius which extends further from the centre is displaced quicker than the smaller radius, when the near radius is moved [5] by the same force? Now we use the word ‘quicker’ in two senses; if an object traverses an equal distance in less time, we call it quicker, and also if it traverses a greater distance in equal time. Now the greater radius describes a greater circle in equal time; for the outer circumference is greater than the inner.
[10] The reason of this is that the radius undergoes two displacements. Now if the two displacements of a body are in any fixed proportion, the resulting displacement must necessarily be a straight line, and this line is the diagonal of the figure, made by the lines drawn in this proportion.
Let the proportion of the two displacements be as AB to AC, and let A1 be brought to B, and the line AB brought down to GC. Again, let A be brought to D [15] and the line AB to E; then if the proportion of the two displacements be maintained, AD must necessarily have the same proportion to AE as AB to AC. Therefore the small parallelogram is similar to the greater, and their diagonal is the same, so that [20] A will be at F. In the same way it can be shown, at whatever points the displacement be arrested, that the point A will in all cases be on the diagonal.
Thus it is plain that, if a point be moved along the diagonal by two displacements, it is necessarily moved according to the proportion of the sides of the parallelogram; for otherwise it will not be moved along the diagonal. If it be moved [25] in two displacements in no fixed ratio for any time, its displacement cannot be in a straight line. For let it be a straight line. This then being drawn as a diagonal, and the sides of the parallelogram filled in, the point must necessarily be moved according to the proportion of the sides; for this has already been proved. Therefore, [30] if the same proportion be not maintained during any interval of time, the point will not describe a straight line; for, if the proportion were maintained during any interval, the point must necessarily describe a straight line, by the reasoning above. So that, if the two displacements do not maintain any proportion during any interval, a curve is produced.
Now that the radius of a circle has two simultaneous displacements is plain from these considerations, and because the point from being vertically above the [849]a1 centre comes back to the perpendicular,2 so as to be again perpendicularly above the centre.
Let ABC be a circle, and let the point B at the summit be displaced to D, and come eventually to C. If then it were moved in the proportion of BD to DC, it would [5] move along the diagonal BC. But in the present case, as it is moved in no such proportion, it moves along the curve BEC. And, if one of two displacements caused by the same forces is more interfered with and the other less, it is reasonable to suppose that the motion more interfered with will be slower than the motion less interfered with; which seems to happen in the case of the greater and less of the radii [10] of circles. For on account of the extremity of the lesser radius being nearer the stationary centre than that of the greater, being as it were pulled in a contrary direction, towards the middle,3 the extremity of the lesser moves more slowly. This is the case with every radius, and it moves in a curve, naturally along the tangent, [15] and unnaturally towards the centre. And the lesser radius is always moved more in respect of its unnatural motion; for being nearer to the retarding centre it is more constrained. And that the less of two radii having the same centre is moved more [20] than the greater in respect of the unnatural motion is plain from what follows.
Let BCED be a circle, and XNMO another smaller circle within it, both having the same centre A, and let the diameters be drawn, CD and BE in the large [25] circle, and MX and NO in the small; and let the rectangle DYRC be completed. If the radius AB comes back to the same position from which it started, i.e. to AB, it is plain that it moved towards itself; and likewise AX will come to AX. But AX moves [30] more slowly than AB, as has been stated, because the interference is greater and AX is more retarded.
Now let AHG be drawn, and from H a perpendicular upon AB within the circle, HF; and, further, from H let HZ be drawn parallel to AB, and ZU and GK [35] perpendiculars on AB; then ZU and HF are equal. Therefore BU is less than XF; for in unequal circles equal straight lines drawn perpendicular to the diameter cut off smaller portions of the diameter in the greater circles; ZU and HF being equal.
[849b1] Now the radius AH describes the arc XH in the same time as the extremity of the radius BA has described an arc greater than BZ in the greater circle; for the natural displacement is equal and the unnatural less, BU being less than XF [5] Whereas they ought to be in proportion, the two natural motions in the same ratio to each other as the two unnatural motions.
Now the radius AB has described an arc GB greater than ZB. It must necessarily have described GB in this time; for that will be its position when in the two circles the proportion between the unnatural and natural movements holds [10] good. If, then, the natural movement is greater in the greater circle, the unnatural movement, too, would agree in being proportionally greater in that case only, where B is moved along GB while X is moved along XH. For in that case the point B comes by its natural movement to G, and by its unnatural movement to K, GK being [15] perpendicular from G. And as GK to BK, so is HF to XF. Which will be plain, if B and X be joined to G and H. But, if the arc described by B be less or greater than GB, the result will not be the same, nor will the natural movement be proportional to the unnatural in the two circles.
[20] So that the reason why the point further from the centre is moved quicker by the same force, and the greater radius describes the greater circle, is plain from what has been said; and hence the reason is also clear why larger balances are more accurate than smaller. For the cord by which a balance is suspended acts as the centre, for it is at rest, and the parts of the balance on either side form the radii. [25] Therefore by the same weight the end of the balance must necessarily be moved quicker in proportion as it is more distant from the cord, and some weight must be imperceptible to the senses in small balances, but perceptible in large balances; for there is nothing to prevent the movement being so small as to be invisible to the eye. [30] Whereas in the large balance the same load makes the movement visible. In some cases the effect is clearly seen in both balances, but much more in the larger on account of the amplitude of the displacement caused by the same load being much greater in the larger balance. And thus dealers in purple, in weighing it, use [35] contrivances with intent to deceive, putting the cord out of centre and pouring lead into one arm of the balance, or using the wood towards the root of a tree for the end [850a1] towards which they want it to incline, or a knot, if there be one in the wood; for the part of the wood where the root is is heavier, and a knot is a kind of root.
2 · How is it that if the cord is attached to the upper surface of the beam of a balance, if one takes away the weight when the balance is depressed on one side, the beam rises again; whereas, if the cord is attached to the lower surface of the beam, it does not rise but remains in the same position? Is it because, when the cord is [5] attached above, there is more of the beam on one side of the perpendicular than on the other, the cord being the perpendicular? In that case the side on which the greater part of the beam is must necessarily sink until the line which divides the beam into two equal parts reaches the actual perpendicular, since the weight now [10] presses on the side of the beam which is elevated.
Let BC be a straight beam, and AD a cord. If AD be produced it will form the perpendicular ADM. If the portion of the beam towards B be depressed, B will be displaced to E and C to F; and so the line dividing the beam into two halves, which was originally DM, part of the perpendicular, will become DH when the beam is [15] depressed; so that the part of the beam EF which is outside the perpendicular AM will be greater by HP than half the beam. If therefore the weight at E be taken away, F must sink, because the side towards E is shorter. It has been proved then that when the cord is attached above, if the weight be removed the beam rises [20] again.
But if the support be from below, the contrary takes place. For then the part which is depressed is more than half of the beam, or in other words, more than the part marked off by the original perpendicular; it does not therefore rise, when the weight is removed, for the part that is elevated is lighter. Let NO be the beam when horizontal, and KLM the perpendicular dividing NO into two halves. When the [25] weight is placed at N, N will be displaced to S and O to R, and KL to LH, so that KS is greater than LR by HLK. If the weight, therefore, is removed the beam must necessarily remain in the same position; for the excess of the part in which SK is over half the beam acts as a weight and remains depressed.
3 · Why is it that, as has been remarked at the beginning of this treatise, the [30] exercise of little force raises great weights with the help of a lever, in spite of the added weight of the lever; whereas the less heavy a weight is, the easier it is to move, and the weight is less without the lever? Does the reason lie in the fact that the lever acts like the beam of a balance with the cord attached below and divided into two [35] unequal parts? The fulcrum, then, takes the place of the cord, for both remain at rest and act as the centre. Now since a longer radius moves more quickly than a shorter one under pressure of an equal weight; and since the lever requires three elements, viz. the fulcrum—corresponding to the cord of a balance and forming the centre—and two weights, that exerted by the person using the lever and the weight which is to be moved; this being so, as the weight moved is to the weight moving it, [850b1] so, inversely, is the length of the arm bearing the weight to the length of the arm nearer to the power. The further one is from the fulcrum, the more easily will one raise the weight; the reason being that which has already been stated, namely, that a longer radius describes a larger circle. So with the exertion of the same force the [5] motive weight will change its position more than the weight which it moves, because it is further from the fulcrum.
Let AB be a lever, C the weight to be lifted, D the motive weight, and E the fulcrum; the position of D after it has raised the weight will be G, and that of C, the weight raised, will be K.
[10]4 · Why is it that those rowers who are amidships move the ship most? Is it because the oar acts as a lever? The fulcrum then is the thole-pin (for it remains in the same place); and the weight is the sea which the oar displaces; and the power that moves the lever is the rower. The further he who moves a weight is from the [15] fulcrum, the greater is the weight which he moves; for then the radius becomes greater, and the thole-pin acting as the fulcrum is the centre. Now amidships there is more of the oar inside the ship than elsewhere; for there the ship is widest, so that on both sides a longer portion of the oar can be inside the two walls of the vessel. The [20] ship then moves because, as the blade presses against the sea, the handle of the oar, which is inside the ship, advances forward, and the ship, being firmly attached to the thole-pin, advances with it in the same direction as the handle of the oar. For where the blade displaces most water, there necessarily must the ship be propelled [25] most; and it displaces most water where the handle is furthest from the thole-pin. This is why the rowers who are amidships move the ship most; for it is in the middle of the ship that the length of the oar from the thole-pin inside the ship is greatest.
5 · Why is it that the rudder, being small and at the extreme end of the ship, has such power that vessels of great burden can be moved by a small tiller and the [30] strength of one man only gently exerted? Is it because the rudder, too, is a lever and the steersman works it? The fulcrum then is the point at which the rudder is attached to the ship, and the whole rudder is the lever, and the sea is the weight, and [35] the steersman the moving force. The rudder does not take the sea squarely, as the oar does; for it does not move the ship forward, but diverts it as it moves, taking the sea obliquely. For since, as we saw, the sea is the weight, the rudder pressing in a contrary direction diverts the ship. For the fulcrum turns in a contrary direction to [851a1] the sea; when the sea turns inwards, the fulcrum turns outwards; and the ship follows it because it is attached to it. The oar pushing the weight squarely, and being itself thrust in turn by it, impels the ship straight forward; but the rudder, as it has [5] an oblique position, causes an oblique motion one way or the other. It is placed at the stern and not amidships, because it is easiest to move a mass which has to be moved, if it is moved from one extremity. For the fore part travels quickest, because, just as in objects that are travelling along, the movement ceases at the end; so too, in [10] any object which is continuous the movement is weakest towards the end, and if it is weakest in that part it is easy to check it. For this reason, then, the rudder is placed at the stern, and also because, as there is little motion there, the displacement is much greater at the extremity, since the equal angle stands on a longer base in [15] proportion as the enclosing lines are longer. From this it is also plain why the ship advances in the opposite direction more than does the oar-blade; for the same bulk moved by the same force progresses more in air than in water. For let AB be the oar [20] and C the thole-pin, and A the end of the oar inside the ship, and B, that in the sea. Then if A be moved to D, B will not be at E: for BE is equal to AD, and so B, if it were at E, would have changed its position as much as A, whereas it has really, as we saw, traversed a shorter distance. B will therefore be at F. H then cuts AB not at C but below it. For BF is less than AD, so that HF is less than DH, for the triangles are similar. The centre C will also have been displaced; for it moves in a contrary [25] direction to B, the end of the oar in the sea, and in the same direction as A, the end in the ship, and A changes its position to D. So the ship will also change its position, and it advances in the same direction as the handle of the oar. The rudder also acts in the same way, except that, as we saw above, it contributes nothing to the forward [30] motion of the ship, but merely thrusts the stern sideways one way or the other; for then the bow inclines in the contrary direction. The point where the rudder is attached must be considered, as it were, the centre of the mass which is moved, corresponding to the thole-pin in the case of the oar; but the middle of the ship moves in the direction to which the tiller is put over. If the steersman puts it [35] inwards, the stern alters its position in that direction, but the bow inclines in the contrary direction; for while the bow remains in the same place, the position of the ship as a whole is altered.
6 · Why is it that the higher the yard-arm is raised, the quicker does a vessel travel with the same sail and in the same breeze? Is it because the mast is a lever, and the socket in which it is fixed, the fulcrum, and the weight which it has to move [851b1] is the boat, and the motive power is the wind in the sail? If the same power moves the same weight more easily and quickly the further away the fulcrum is, then the yard-arm, being raised higher, brings the sail also further away from the mast-socket, which is the fulcrum. [5]
7 · Why is it that, when sailors wish to keep their course in an unfavourable wind, they draw in the part of the sail which is nearer to the steersman, and, working the sheet, let out the part towards the bows? Is it because the rudder cannot counteract the wind when it is strong, but can do so when there is only a little wind, [10] and so4 they draw in sail? The wind then bears the ship along, while the rudder turns the wind into a favouring breeze, counteracting it and serving as a lever against the sea. The sailors also at the same time contend with the wind by leaning their weight in the opposite direction.
8 · Why is it that spherical and circular forms are easier to move? A circle [15] can revolve in three different ways: either along its circumference, the centre correspondingly changing its position, as a carriage wheel revolves; or round the centre only, as pulleys move, the centre being at rest; or it can turn, as does the [20] potter’s wheel, parallel to the ground, the centre being at rest. Do not circular forms move quickest, firstly because they have a very slight contact with the ground (like a circle in contact at a single point), and secondly, because there is no friction, for the angle is well away from the ground? Further, if they come into collision with [25] another body, they only are in contact with it again to a very small extent. (If it were a question of a rectilinear body, owing to its sides being straight, it would have a considerable contact with the ground.) Further, he who moves circular objects moves them in a direction to which they have an inclination as regards weight. For when the diameter of the circle is perpendicular to the ground, the circle being in [30] contact with the ground only at one point, the diameter divides the weight equally on either side of it; but as soon as it is set in motion, there is more weight on the side to which it is moved, as though it had an inclination in that direction. Hence, it is easier for one who pushes it forward to move it; for it is easier to move any body in a direction to which it inclines, just as it is difficult to move it contrary to its [35] inclination. Some people further assert that the circumference of a circle keeps up a continual motion, just as bodies which are at rest remain so owing to their resistance. This can be illustrated by a comparison of larger with smaller circles; larger circles can be moved more readily with an exertion of the same amount of force and move other weights with them, because the angle of the larger circle as compared with that of the smaller has an inclination which is in the same proportion [852a1] as the diameter of the one is to the diameter of the other. Now if any circle be taken, there is always a lesser circle than which it is greater; for the lesser circles which can be described are infinite in number.
Now if it is the case that one circle has a greater inclination as compared with another circle, and is correspondingly easy to move, then it is also the case that if a [5] circle does not touch the ground with its circumference, but moves either parallel to the ground or with the motion of a pulley, the circle and the bodies moved by the circle will have a further cause of inclination; for circular objects of this kind move most easily and move weights with them. Can it be that this is due to a reason other than that they have only a very slight contact with the ground, and consequently encounter little friction? This reason is that which we have already mentioned, namely, that the circle is made up of two forms of motion—and so one of them [10] always has an inclination—and those who move a circle move it when it has, as it were, a motion of its own, when they move it at any point on its circumference. They are moving the circumference when it is already in motion; for the motive force pushes it in a tangential direction, while the circle itself moves in the motion which takes place along the diameter.
[15] 9 · How is it that we can move objects more easily and quickly when they are lifted or drawn along by circles of large circumference? Why, for example, are large pulleys more effective than small, and similarly large rollers? Is it because the longer the radius is the further the object is moved in the same time, and so it will do [20] the same also with an equal weight upon it? Just as we said that large balances are more accurate than small; for the cord is the centre and the parts of the beam on either side of the cord are the radii.
10 · Why is it that a balance moves more easily without a weight upon it [25] than with one? So too with a wheel or anything of that nature, the smaller and lighter is easier to move than the heavier and larger. Is it because that which is heavy is difficult to move not only vertically, but also horizontally? For one can move a weight with difficulty contrary to its inclination, but easily in the direction of its inclination; and it does not incline in a horizontal direction.
11 · Why is it that it is easier to convey heavy weights on rollers than on [30] carts, though the latter have large wheels and the former a small circumference? Is it because a weight placed upon rollers encounters no friction, whereas when placed upon a cart it has the axle at which it encounters friction? For it presses on the axle from above in addition to the horizontal pressure. But an object on rollers is moved [35] at two points on them, where the ground supports them below and where the weight is imposed above; the circle revolves at both these points and is thrust along as it moves.
12 · Why is it that a missile travels further from a sling than from the hand, although he who casts it has more control over the missile in his hand than when he [852b1] holds the weight suspended? Further, in the latter case he moves two weights, that of the sling and the missile, while in the former case he moves only the missile. Is it because he who casts the missile does so when it is already in motion in the sling (for he swings it round many times before he lets it go), whereas when cast from the [5] hand it starts from a state of rest? Now any object is easier to move when it is already in motion than when it is at rest. Or, while this is one reason, is there a further reason, namely, that in using a sling the hand becomes the centre and the sling the radius, and the longer the radius is the more quickly it moves, and so a cast from the hand is short as compared with a cast from a sling? [10]
13 · Why is it that longer bars are moved more easily than shorter ones round the same capstan, and similarly lighter windlasses are moved more easily by the same force than stouter windlasses? Is it because the windlass and the capstan form a centre and the outer masses the radii? For the radii of greater circles are moved more readily and further by the same force than those of lesser circles; for [15] the extremity further from the centre is moved more readily by the same force. Therefore in the case of the capstan they use the bars as a means whereby they turn it more easily; and in the case of the lighter windlasses the part outside the central cylinder is more extended, and this portion forms the radius of the circle. [20]
14 · Why is it that a piece of wood of the same size is more easily broken against the knee, if one breaks it holding the ends at equal distance from the knee, than if it is held close to the knee? And if one leans a piece of wood upon the ground and places one’s foot on it, why does one break it more easily if one grasps it at a [25] distance from the foot rather than near it? Is it because in the former case the knee, and in the latter the foot is the centre, and the further an object is from the centre the more easily is it always moved, and that which is to be broken must be moved?
15 · Why is it that the so-called pebbles found on beaches are round, though they are originally formed from stones and shells which are elongated in shape? Is it [30] because objects whose outer surfaces are far removed from their middle point are borne along more quickly by the movements to which they are subjected? The middle of such objects acts as the centre and the distance from there to the exterior becomes the radius, and a longer radius always describes a greater circle than a shorter radius when the force which moves them is equal. An object which traverses [35] a greater space in the same time travels more quickly, and objects which travel more quickly from an equal distance strike harder against other objects, and the more they strike the more they are themselves struck. It follows, therefore, that objects in which the distance from the middle to the exterior is greater always become broken, and in this process they must necessarily become round. So in the case of pebbles, [853a1] because the sea moves and they move with it, the result is that they are always in motion, and, as they roll about, they come into collision with other objects; and it is their extremities which are necessarily most affected.
[5] 16 · Why is it that the longer a plank of wood is, the weaker it is, and the more it bends when lifted up? Why, for example, does a short thin plank about two cubits long bend less than a thick plank a hundred cubits long? Is it because the [10] length of the plank when it is lifted forms a lever, a weight, and a fulcrum? The first part of it, then, which the hand raises becomes, as it were, a fulcrum, and the part towards the end becomes the weight; and so the longer the space is from the fulcrum to the end, the more the plank must bend; for it must necessarily bend more the [15] further away it is from the fulcrum. Therefore the ends of the lever must be subject to pressure. If, then, the lever is bent, it must bend more when it is lifted up. This is exactly what happens in the case of long planks of wood; whereas in the case of shorter planks, the extremity is near the fulcrum which is at rest.
[20]17 · How is it that great weights and masses can be split and violent pressure be exerted with a wedge, which is a small thing? Is it because the wedge forms two levers working in opposite directions, and each has a weight and fulcrum which presses upwards or downwards? Further, the impetus of the blow causes the weight which strikes the wedge and moves it to be very considerable; and it has all the more [25] force because by reason of its speed it is moving what is already moving. Although the lever is short, great force accompanies it, and so it causes a much more violent movement than we should expect from an estimate of its size. Let ABC be the wedge, and DEGF the object which is acted upon by it; then AB is a lever and the weight is below at B, and the fulcrum is FD. On the opposite side is the lever BC. [30] When AC is struck it brings both of these into use as levers; for it presses upwards at the point B.
18 · Why is it that if one puts two pulleys on two blocks which are in [35] opposite positions, and places round them a cord with one end attached to one of the blocks and the other supported by or passed over the pulleys, if one pulls at the end of the cord, one can move great weights, even if the force which draws them is small? Is it because the same weight is raised by less force, if a lever is employed, [853b1] than by the hand, and the pulley acts in the same way as a lever, so that a single pulley will draw more easily and draw a far heavier weight with a slight pull than the hand alone can? Two pulleys raise this weight with more than double the velocity; for the second pulley draws a still less weight than if it drew alone by itself, [5] when the rope is passed on to it from the other pulley; for the other pulley makes the weight still less. Thus if the cord is passed through a greater number, the difference is great, even when there are only a few pulleys, so that, if the load under the first weighs four minae, much less is drawn by the last. In building operations they easily [10] move great weights; for they transfer them from one pulley to another and thence again to windlasses and levers, and this is equivalent to constructing a number of pulleys.
19 · How is it that, if you place a heavy axe on a piece of wood and put a heavy weight on the top of it, it does not cleave the wood to any considerable extent, [15] whereas, if you lift the axe and strike the wood with it, it does split it, although the axe when it strikes the blow has much less weight upon it than when it is placed on the wood and pressing on it? Is it because the effect is produced entirely by movement, and that which is heavy gets more movement from its weight when it is in motion than when it is at rest? So when it is merely placed on the wood, it does [20] not move with the movement derived from its weight; but when it is put into motion, it moves with the movement derived from its weight and also with that imparted by the striker. Furthermore, the axe works like a wedge; and a wedge, though small, can split large masses because it is made up of two levers working in opposite directions.
20 · Why is it that steelyards weigh great weights of meat with a small [25] counterpoise, the whole forming only a half balance? For a pan is fixed only at the end where the object weighed is placed, and at the other end there is nothing but the steelyard. Is it because the steelyard is at once a beam and a lever? For it is a beam, [30] inasmuch as each position of the cord becomes the centre of the steelyard. Now at one end it has a pan, and at the other instead of a pan the counterpoise which is fixed in the beam, just as if one were to place the other pan with the counterpoise in it at the end of the steelyard; for it is clear that it draws the same weight when it lies in [35] this second pan. But in order that the single beam may act as many beams, many such positions for the cord are situated along a beam of this kind, in each of which the part on the side of the counterpoise forms half the steelyard and acts as the weight,5 the positions of the cord being moved through equal intervals, so that one can calculate how much weight is drawn by what lies in the pan, and thus know, [854a1] when the steelyard is horizontal, how much weight the pan holds for each of the several positions of the cord, as has been explained. In short, this may be regarded as a balance, having one pan in which the object weighed is placed, and the other in which is the weight of the steelyard, and so the steelyard at the other end is the [5] counterpoise. Hence it acts as an adjustable balance beam, with as many forms as there are positions of the cord. And in all cases, when the cord is nearer the pan and the weight upon it, it draws a greater weight, on account of the whole steelyard [10] being an inverted lever (for the cord in each position is a fulcrum, although it is above, and the weight is what is in the pan), and the greater the length of the lever from the fulcrum, the more easily it produces motion in the case of the lever, and in the case of the balance causes equilibrium and counterbalances the weight of the [15] steelyard near the counterpoise.
21 · How is it that doctors extract teeth more easily by applying the additional weight of a tooth-extractor than with the bare hand only? Is it because [20] the tooth is more inclined to slip in the fingers than from the tooth-extractor? or does not the iron slip more than the hand and fail to grasp the tooth all round, since the flesh of the fingers being soft both adheres to and fits round the tooth better? The truth is that the tooth-extractor consists of two levers opposed to one another, with the same fulcrum at the point where the pincers join; so they use the [25] instrument to draw teeth, in order to move them more easily.
Let A be one extremity of the tooth-extractor and B the other extremity which draws the tooth, and ADF one lever and BCE the other, and CHD the fulcrum, and let the tooth, which is the weight to be lifted, be at the point I, where the two levers [30] meet. The doctor holds and moves the tooth at the same time with B and F; and when he has moved it, he can take it out more easily with his fingers than with the instrument.
22 · Why is it that men easily crack nuts, without striking a blow upon them, in the instruments made for this purpose? For with nut-crackers much power is lost, namely, that of motion and violent impetus. Further, if one crushes them with a [35] hard and heavy instrument, one can crack them much more quickly than with a light wooden instrument. Is it because the nut is crushed on two of its sides by two levers, and weights can easily be divided with a lever? For the nut-cracker consists [854b1] of two levers, with the same fulcrum, namely, A, their point of connexion. As, therefore, E and F would have been easily pushed apart, so they are easily brought together by a small force,6 the levers being moved at the points D and C. So EC and [5] FD being levers exert the same or even greater force than that which the weight exerted when the nut was cracked by a blow; for when weight is put upon the levers they move in opposite directions and compress and break the object at K. For this very reason, too, the nearer K is to A, the sooner it is subjected to pressure; for the further the lever extends from the fulcrum, the more easily and more powerfully does it move an object with the exercise of the same force. A, then, is the fulcrum, [10] and DAF and CAE are the levers. The nearer, therefore, K is to the angle at A, the nearer it is to the point where the levers are connected, and this is the fulcrum. So with the same force bringing them together, F and E must be subjected to more weight; and so, when weight is exerted from two contrary directions, more compression must take place, and the more an object is compressed, the sooner it [15] breaks.
23 · Why is it that in a rhombus, when the points at the extremities are moved in two movements, they do not describe equal straight lines, but one of them a much longer line than the other? Further (and this is the same question), why does the point moving along the side describe a resultant line less than the side? For the point describes the diagonal, the shorter distance, and the line moves along the side, [20] the longer distance; and yet the line has but one movement, and the point two movements.
For let A move along AB to B, and B to A with the same velocity; and let the line AB move along AC parallel to CD with the same velocity. Then the point A must move along the diagonal AD, and B along BC; and both must describe these [25] diagonals simultaneously, while AB moves along the side AC.
For let A be moved the distance AE, and the line AB the distance AF, and let FG be drawn parallel to AB, and a line drawn from E to complete the parallelogram. The small parallelogram then thus formed is similar to the whole parallelogram. Thus AF equals AE, so that A has been moved along the side AE, while the [30] line AB would be moved the distance AF. Thus A will be on the diagonal at H, and so must always move along the diagonal; and the side AB will describe the side AC, and the point A the diagonal AD simultaneously. In the same way it may be proved [35] that B moves along the diagonal BC, BE being equal to BG. For, if the parallelogram be completed by drawing a line from G, the interior parallelogram will be similar to the whole parallelogram; and B will be on the diagonal at the point where the sides meet; and the side will describe the side; and the point B describes [855a1] the diagonal BC.
At the same time then B will describe a line which is much longer than AB, and the side will pass along the side which is shorter, though the velocity is the same, in the same time (and the side has moved further than A, though it is moved by only [5] one movement). For as the rhombus becomes more acute, AD becomes the lesser diagonal and BC greater, and the side less than BC. For it is strange, as has been remarked, that in some cases a point moved by two movements travels more slowly than a point moved by one, and that, while both the given points have equal velocity, either one of them describes a greater line. [10]
The reason is that, when a point moves from an obtuse angle, the sides are in almost opposite directions, namely, that in which the point itself is moved and that in which it is moved down by the side; but when it moves from an acute angle, it moves, as it were, in actual fact towards the same position. For the angle of the sides contributes to increase the speed of the diagonal; and in proportion as one makes the one angle more acute and the other more obtuse, the movement is slower or quicker. [15] For the sides are brought into more opposite direction by the angle becoming more obtuse; but they are brought into the same direction by the sides being brought nearer together. For B moves in practically the same direction in virtue of both its movements; thus one contributes to assist the other, and more so, the more acute the [20] angle becomes. And the reverse is the case with A; for it itself moves towards B, while the movement of the side brings it down to D; and the more obtuse the angle is, the more opposite will the movements be; for the two sides become more like a [25] straight line. If they became actually a straight line, the components would be absolutely in opposite directions. But the side, being moved in one direction only, is interfered with by nothing. In that case it naturally moves through a longer distance.
24 · There is a question why a large circle traces out a path equal to that of a [30] smaller circle, when they are placed about the same centre, but when they are rolled separately, their paths are to one another in the proportion of their dimensions. And, further, the centre of both being one and the same, at one time the path which they trace is of the same length as the smaller traces out alone, and at another time [35] of the length which the larger circle traces. Now it is manifest that the larger circle traces out the longer path. For by mere observation it is plain that the angle which the circumference of each makes with its own diameter is greater in the case of the larger circle than in the smaller; so that, by observation, the paths along which they [855b1] roll will have this same proportion to one another. But, in fact, it is manifest that, when they are situated about the same centre, this is not so, but they trace out an equal path; so that it comes to this, that in the one case the path is equal to that [5] traced by the larger circle, in the other to that traced by the smaller.
Let DFC be the greater circle, EGB the lesser, A the common centre, FI the path along which the greater circle moves by its own motion, and GK the path of the smaller circle by its own motion, equal to FL.
[10] When, then, I move the smaller circle, I move the same centre A; and now let the large circle be fixed to it. Whenever, therefore, AB becomes perpendicular to GK, AC at the same time becomes perpendicular to FL; so that they will always have traversed an equal distance, GK representing the arc GB, and FL representing [15] the arc FC. And if one quadrant traces an equal path, it is plain that the whole circle will trace out a path equal to that of the other whole circle; so that whenever the line GB comes to K, the arc FC will move along FL; and the same is the case with the whole circle after one revolution.
In like manner if I roll the large circle, fastening the smaller circle to it, about [20] the same centre, AB will be perpendicular and vertical at the same time as AC, the latter to FI, the former to GH. So that, whenever the one shall have traversed a distance equal to GH and the other a distance equal to FI, and FA again becomes perpendicular to FL and AG to GK, they will be in their original position at the points H and I. And, since there is no halting of the greater for the lesser, so as to be [25] at rest during an interval at the same point (for in both cases both are moved continuously), nor does the lesser skip any point, it is strange that in one case the greater should traverse a distance equal to that traversed by the lesser, and in the other case the lesser a distance equal to that traversed by the greater. And, further, [30] it is wonderful that, though there is always only one movement, the centre that is moved should be rolled forward in one case a great and in another a less distance. For the same thing moved at the same velocity naturally traverses an equal distance; and to move a thing at the same velocity is to move it an equal distance in both cases.
As to the reason, this may be taken as a principle, that the same, or an equal force, moves one mass more slowly and the other more quickly.
Suppose that there is a body which is not naturally in motion of itself; if [35] another body which is naturally in motion move it and itself as well, it will be moved more slowly than if it were being moved by its own motion alone; and if it be naturally in motion and nothing is moved with it, the same is the case. So it is quite impossible for any body to be moved more than that which moves it; for it is not moved according to any rate of motion of its own, but at the rate of that which moves it. [856a1]
Let there be two circles, a greater A and a lesser B. If the lesser were to push along the greater, when the greater is not rolling alone, it is plain that the greater will traverse so much distance as it has been pushed by the lesser. And it has been [5] pushed the same distance as the small circle has moved; so that they have both traversed an equal straight line. Necessarily, therefore, if the lesser be rolling while it pushes the greater, the latter will be rolled, as well as pushed, just so far as the lesser has been rolled, if the greater have no motion of its own; for in the same way and so far as the moving body moves it, so far must the body which is moved be moved thereby. So, indeed, the lesser circle has moved the greater so far and in the [10] same way, viz., in a circle and for the distance of one foot (for let that be the extent of the movement); and consequently the larger circle has moved that distance.
So too, if the large circle move the lesser, the lesser circle will have been moved just as far as the large circle, in whatever way7 the latter be moved, whether quickly [15] or slowly, by its own motion; and the lesser circle will trace out a line at the same velocity and of the same length as the greater traced out by its natural movement. And this is just what causes the difficulty, that they do not act any longer when they are joined together in the same way as they acted when they were not connected; that is to say, when one is moved by the other not according to its natural motion, nor according to its own motion. For it makes no difference whether one is fixed [20] round the other or fitted inside it, or placed in contact with it; for in all these cases, when one moves and the other is moved by it, the one will be moved just so far as the other moves it.
Now when one moves a circle by means of another circle in contact with it, or suspended from it, one does not revolve it continuously; but if one places them about [25] the same centre, the one must be continuously revolved by the other. But nevertheless, the former is not moved in accordance with its own motion, but just as if it had no proper motion; and if it has a proper motion, but does not make use of it, it comes to the same thing.
Whenever, therefore, the large circle moves the small circle affixed to it, the small circle moves the same distance as the large, and vice versa. But when they are [30] separate each has its own motion.
If any one raises the difficulty that, when the centre is the same and is moving the two circles with equal velocity, they trace out unequal paths, he is reasoning falsely and sophistically. For the centre is, indeed, the same for both, but only [35] accidentally, just as the same thing may chance to be musical and white; for to be the centre of each of the circles is not the same for it in the two cases.
In conclusion, when it is the smaller circle that moves the greater, the centre and source of motion is to be regarded as belonging to the smaller circle; but when the greater circle moves the lesser, it is to be regarded as belonging to the greater circle. Thus the source of motion is not the same absolutely, though it is in a sense the same.
25 · Why do they construct beds so that one dimension is double the other, [856b1] one side being six feet long or a little more, the other three feet? And why do they not stretch bed-ropes diagonally? Do they make them of this size so as to fit the [5] body? Thus they have one side twice the length of the other, being four cubits long and two cubits wide.
The ropes are not stretched diagonally but from side to side, so that the wooden frame may be less likely to break; for wood can be cleft most easily if split thus in the natural way, and when there is a pull upon it, it is subject to a considerable strain. Further, since the ropes have to be able to bear a weight, there will be less of [10] a strain when the weight is put upon them if they are strung crosswise rather than diagonally. Again, less rope is used up by this method.
Let AFGI be a bed, and let FG be divided into two equal parts at B. There is an equal number of holes in FB and FA; for the sides are equal, each to each, for the [15] whole side FG is double the side FA. They stretch the rope on the method already mentioned from A to B, then to C, D, H, and E, and so on until they turn back and reach another angle; for the two ends of the rope come at two different angles.
Now the parts of the rope which form the bends are equal, e.g. AB, BC are [20] equal to CD, DH—and so with other similar pairs of sides, for the same demonstration holds good in all cases. For AB is equal to EH; for the opposite sides of the parallelogram BGKA are equal, and the holes are an equal distance apart from one another. And BG is equal to KA; for the angle at B is equal to the angle at G (for the exterior angle of a parallelogram is equal to the interior opposite angle); [25] and the angle at B is half a right angle, for FB is equal to FA, and the angle at F is a right angle. And the angle at B is equal to the angle at G; for the angle at F is a right angle, since the bed is a rectangular figure, one side of which is double the other, and divided into two equal parts; so that BC is equal to EG, as also is KH; for it is [30] parallel. So that BC is equal to KH, and CE to DH. In like manner it can be demonstrated that all the other pairs of sides which form the bends of the rope are equal to one another. So that clearly there are four such lengths of rope as AB in the bed; and there is half the number of holes in the half FB that there is in the whole [35] FG. So that in the half of the bed there are lengths of rope, such as AB, and they are of the same number as there are holes in BG, or, what comes to the same thing, in AF, FB together. But if the rope be strung diagonally, as in the bed ABCD, the [857a1] halves are not of the same length as the sides of both, AF and FG; but they are of the same number as the holes in FB, FA. But AF, FB, being two, are greater than AB, so that the rope is longer by the amount by which the two sides taken together are greater than the diagonal.
26 · Why is it more difficult to carry a long plank of wood on the shoulder if [5] one holds it at the end than if it is held in the middle, though the weight is the same? Is it because, as the plank vibrates, the end prevents one from carrying it, because it tends to interrupt one’s progress by its vibration? No, for if it does not bend at all [10] and is not very long, it is nevertheless more difficult to carry if it is held at the end. It is easier to carry if one holds it in the middle rather than at the end, for the same reason for which it is easier to lift in that way. The reason is that, if one lifts it in the middle, the two ends always lighten one another, and one side lifts the other side up. For the middle, where the lifter or carrier holds it, forms, as it were, the centre, and [15] each of the two ends inclining downwards raises up and lightens the other end; whereas if it is lifted or carried from one end, this effect is not produced, but all the weight inclines in one direction. Let A be the middle of a plank which is raised or carried, and let B and C be the extremities. When the plank is lifted or carried at the point A, B inclines downwards and raises C up, and C inclines downwards and [20] raises B up; the effect is produced by their being raised up at the same moment.
27 · Why is a very long object more difficult to carry on the shoulder, even if one carries it in the middle, than a shorter object of the same weight? In the last case we said that the vibration was not the reason; in this case it is the reason. For [25] the longer an object is, the more its extremities vibrate, and so it would be more difficult for the man to carry it. The reason of the increased vibration is that, though the movement is the same, the extremities change their position more the longer the piece of wood is. Let the shoulder, which is the centre (for it is at rest), be at A, and [30] let AB and AC be the radii; then the longer the radius AB or AC is, the greater is the amplitude of movement. This point has already been demonstrated.
28 · Why do they construct swing-beams by the side of wells by attaching the lead as a weight at the end of the bar, the bucket being itself a weight, whether it [35] is empty or full? Is the reason that, the drawing of water being divided into two operations distinct in time (for the bucket has to be dipped and then drawn up), it is an easy task to let it down when it is empty, but difficult to raise it when it is full? It [857b1] is therefore of advantage to lower it rather more slowly with a view to lightening the weight considerably when it is drawn up again. This effect is produced by the lead or stone attached to the end of the swing-beam. In letting it down there is a heavier [5] weight to lift than if one has merely to lower the empty bucket; but when it is full, the lead, or whatever the weight attached is, helps to draw it up; and so the two operations taken together are easier than on the other method.
29 · Why is it that when two men are carrying an equal weight on a piece of wood or something of the kind, the pressure on them is not equal unless the weight is [10] in the middle, but it presses more on the person carrying it to whom it is nearest? is it because the wood, when they hold it in this way, becomes a lever, and the load [15] forms the fulcrum, and the carrier nearer to the load becomes the weight which is to be moved, while the other carrier becomes the mover of the weight? The further the latter is from the weight, the more easily he moves it, and the more he presses down the other man, since the load placed on the wood and acting as a fulcrum, as it were, offers resistance. But if the load is placed in the middle, one carrier does not act as a weight on the other any more than the other on him, or exercise any motive force [20] upon him, but each is equally a weight upon the other.
30 · Why is it that when people rise from a sitting position, they always do so by making an acute angle between the thigh and the lower leg and between the chest and the thigh, otherwise they cannot rise? Is it because equality is always a cause of rest, and a right angle causes an equality and so causes equilibrium? So in [25] rising a man moves towards a position at equal angles to the earth’s circumference; for it is not the case that he will actually be at right angles to the ground. Or is it because when a man rises he tends to become upright, and a man who is standing must be perpendicular to the ground? If, then, he is to be at right angles to the [30] ground, that means that he must have his head in the same line as his feet, and this occurs when he is rising. As long, then, as he is sitting, he keeps his feet and head parallel to one another and not in the same straight line. Let A be the head, AB the line of the chest, BC the thigh, and CD the lower leg. Then AB, the line of the chest, [35] is at right angles to the thigh, and the thigh at right angles to the lower leg, when a man is seated in this way. In this position, then, a man cannot rise; but to do so he must bend the leg and place the feet at a point under the head. This will be the case if CD be moved to CF, and the result will be that he can rise immediately, and he [858a1] will have his head and his feet in the same straight line;8 and CF will form an acute angle with BC.
31 · Why is it that a body which is already in motion is easier to move than one which is at rest? For example, a wagon which is in motion can be propelled more [5] quickly than one which has to be started. Is it because, in the first place, it is very difficult to move in one direction a weight which is already moving in the opposite direction? For though the motive force may be much quicker, yet some of it is lost; for the propulsion exerted by that which is being pushed in the opposite direction must necessarily become slower. And so, secondly, the propulsion must be slower if the body is at rest; for even that which is at rest offers resistance. When a body is [10] moving in the same direction as that which pushes it, the effect is just as if one increased the force and speed of the motive power; for by moving forward it produces of itself exactly the effect which that power would have upon it.
32 · Why is it that an object which is thrown eventually comes to a [15] standstill? Does it stop when the force which started it fails, or because the object is drawn in a contrary direction, or is it due to its downward tendency, which is stronger than the force which threw it? Or is it absurd to discuss such questions, while the principle escapes us?
33 · How is it that a body is carried along by a motion not its own, if that which started it does not keep following and pushing it along? Is it not clear that in the beginning the impelling force so acted as to push one thing along, and this in its turn pushes along something else? The moving body comes to a standstill when the force which pushes it along can no longer so act as to push it, and when the weight of [20] the moving object has a stronger inclination downwards than the forward force of that which pushes it.
34 · Why is it that neither small nor large bodies travel far when thrown, but they must have due relation to the person who throws them? Is it because that [25] which is thrown or pushed must offer resistance to that from which it is pushed, and whatever does not yield owing to its mass, or does not resist owing to its weakness, does not admit of being thrown or pushed? A body, then, which is far beyond the force which tries to push it, does not yield at all; while that which is far weaker offers no resistance. Or is it because that which travels along does so only as far as it [30] moves the air to its depths, and that which is not moved cannot itself move anything either? Both these things are the case here; that which is very large and that which [858b1] is very small must be looked upon as not moving at all; for the latter does not move anything, while the former is not itself at all moved.
35 · Why is it that an object which is carried round in whirling water is [5] always eventually carried into the middle? Is it because the object has magnitude, so that it has position in two circles, one of its extremities revolving in a greater and the other in a lesser circle? The greater circle, then, on account of its greater velocity, draws it round and thrusts it sideways into the lesser circle; but since the object has breadth, the lesser circle in its turn does the same thing and thrusts it into [10] the next interior circle, until it reaches the centre. Here the object remains because it stands in the same relation to all the circles, being in the middle; for the middle is equidistant from the circumference in the case of each of the circles. Or is it because an object which, owing to its magnitude, the motion of the whirling water cannot [15] overcome, but which by its weight prevails over the velocity of the revolving circle, must necessarily be left behind and travel along more slowly? Now the lesser circle travels more slowly—for the greater and the lesser circle do not9 revolve over the same space in an equal time when they move round the same centre—and so the object must be left revolving in a lesser and lesser circle until it reaches the middle. [20] If the force of the whirling water prevails at first, it will go on doing so to the end; for one circle must prevail and then the next over the weight of the object owing to their velocity, so that the whole object is continually being left behind in the next circle towards the centre. For an object over which the water does not prevail must be carried either inwards or outwards. Such an object cannot then be carried along in [25] its original position; still less can it be carried along in the outer circle, for the velocity of the outer circle is greater. The only alternative is that the object over which the water does not prevail is transferred to the inner circle. Now every object [30] has a tendency to resist force; but since the arrival at the middle puts an end to motion, and the centre alone is at rest, all objects must necessarily collect there.