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 form
s 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.
The Complete Works of Aristotle Page 227
