Aristotle · a new plain-English translation from the original language
1| [section 1] Wonder attaches to those things that happen in accordance with nature, whenever the cause is unknown, and to those that happen contrary to nature, when they come about through craft for man's advantage. For in many cases nature acts in a way opposed to what is useful to us; for nature always keeps to the same way and acts simply, while what is useful changes in many ways. [section 2] So whenever something must be done contrary to nature, the difficulty of it raises a puzzle and calls for craft. That is why we call the part of craft that comes to the aid of such puzzles a device. For as the poet Antiphon put it, so it is: by craft we master things by which we are overcome by nature. [section 3] Such are the cases in which the lesser masters the greater, and things having a small inclination move great weights, and pretty much everything we call mechanical problems. These are neither entirely the same as natural problems, nor altogether separate from them, but are common to both mathematical and natural inquiry: the how of it is made clear through mathematics, but the matter it concerns is made clear through natural science. [section 4] Among the puzzles contained in this class are those concerning the lever. For it seems strange that a great weight should be moved by a small force, and that too along with a greater weight besides; for the very weight that one cannot move without a lever, that same weight, once one has further taken on the weight of the lever as well, one moves more quickly.
2| [section 5] Of all such things, the circle holds the starting point of the cause. And this has come about reasonably; for it is nothing strange that something wonderful should come about from something more wonderful still, but the most wonderful thing of all is that opposites should come to be together. Now the circle is composed of just such things: for straightaway it is generated both from something moving and something remaining at rest, and the natures of these are opposite to one another. [section 6] So that here too, on inspection, one may wonder the less at the opposite properties that occur in connection with it. First, in the line that encloses the circle, though it has no breadth, opposites are in a way displayed together: the concave and the convex. These stand apart from one another in the way that the great and the small do; for the mean between the latter pair is the equal, and between the former, the straight. [section 7] Hence, when they change into one another, Hence, when they change into one another, the great and the small must first become equal before passing to either extreme, and the line must become straight, when it changes from convex to concave, or again from this back to convex and curved. This, then, is one of the strange things about the circle; a second is that it moves with opposite motions at the same time: for it moves at once into the place in front and into the place behind. [section 8] The line that traces out the circle is in the same condition: for from the place where it begins, its endpoint comes again to this same place; for as it moves continuously, its last point in turn arrives first, so that it is also clear that it has moved on from there.
3| [section 9] Hence, as has been said before, it is nothing strange that the circle should be the starting point of all these wonders. Now the phenomena concerning the balance are traced back to the circle, those concerning the lever to the balance, and pretty much all the rest of the phenomena concerning mechanical motions to the lever. [section 10] Further, because, there being a single line from the center, none of the points on it travels at the same speed as any other, but the point further from the fixed endpoint always travels faster, many of the wonders concerning the motions of circles come about; about these it will become clear in the problems that follow. [section 11] And because the circle moves with opposite motions at the same time — one endpoint of the diameter, the point A, moving forward, and the other, the point B, moving backward — some construct devices so that, from a single motion, many opposite circles move at once, like the little wheels of bronze and iron that people have made and dedicated in temples. [section 12]
4| For if there were, touching the circle on which A B lies, another circle on which C D lies, then, as the diameter of the circle on which A B lies moves forward, C D will move backward on the circle on which A lies, since the diameter is moving about the same point. The circle on which C D lies will therefore move in the opposite direction to the one on which A B lies, and this circle in turn will move the next one, on which E F lies, in the opposite direction to itself, for the same cause. [section 13] And in the same way, even if there are more circles than this, they will do the same though only one is set in motion. Taking hold, then, of this nature that belongs to the circle, craftsmen construct an instrument that hides its starting principle, so that only the wonder of the device is apparent, while the cause remains unclear. [section 1] First, then, the phenomena concerning the balance raise a puzzle: for what cause are the larger balances more accurate than the smaller ones? The starting point of this is: why, in a circle, does the line further from the center, moving with the same force as one nearer, travel faster than the smaller one? For 'faster' is said in two ways: for we say a thing is faster both if it traverses an equal distance in less time, and if it traverses a greater distance in equal time.
5| [section 2] Now the greater line, in equal time, describes a greater circle; for the outer is greater than the inner. The cause of this is that the point tracing out the circle moves with two motions at once. Now whenever something moves in some fixed ratio, what is moving must move along a straight line, and this line becomes the diagonal of the figure formed by the lines composed in that ratio. [section 3] For let the ratio in which the moving thing moves be that which A B has to A C; and let A C be carried toward B, and let A B be carried down toward G C; and let A have been carried to D, and the line A B to E. If, then, in the motion the ratio was that which A B has to A C, then A D must also have this same ratio to A E. [section 4] The small quadrilateral is therefore similar, in point of ratio, to the larger one, so that they also have the same diagonal, and A will fall on F. In just the same way it will be shown wherever the motion is divided; for the point will always be on the diagonal. It is clear, then, that what moves along the diagonal under two motions must move in the ratio of the sides.
6| [section 5] For if it moved in some other ratio, it would not travel along the diagonal. And if it moves with two motions in no fixed ratio at all, for any length of time, it is impossible for its motion to be a straight line. For suppose it is straight. Then, with this line set as diagonal, and the sides filled in to complete the figure, what is moving must be carried in the ratio of the sides; for this has been shown before. [section 6] Hence what moves in no fixed ratio, for no length of time, will not produce a straight line. For if it is carried in some ratio for some time, then for that time its motion must be a straight line, because of what has been said. So it becomes circular, given that it moves with two motions in no fixed ratio for any length of time. [section 7] That the point tracing out the circle, then, moves with two motions at once is clear from this, and also from the fact that the point moving in a straight line arrives at the perpendicular, so that this line, drawn from the center, is again a perpendicular. Let there be a circle A B C, and let the endpoint B be carried toward D; at some point it arrives at C. Now if it were moving in the ratio which B D has to D C, it would travel along the diameter B C. [section 8] But as it is, since it moves in no fixed ratio, it travels along the arc B E C. And if, of two things moving under the same force, one is deflected more and the other less, it is reasonable that the one deflected more should move more slowly than the one deflected less; and this is what appears to happen with the greater and the lesser of the lines drawn from the center that trace out the circles.
7| [section 9] Because the endpoint of the shorter radius is closer to the point that stays fixed than the endpoint of the longer one, and is, as it were, dragged back in the opposite direction, the endpoint of the shorter radius moves more slowly toward the middle. Now this happens with every line describing a circle: it moves in accordance with nature along the circumference, but contrary to nature toward the side and toward the center. [section 10] And the shorter radius always moves more contrary to nature; for because it is closer to the center that drags it back, it is more overpowered. That the shorter of the lines drawn from the center that describe the circles moves more contrary to nature than the longer one is clear from the following. [section 11] Let there be a circle on which lie Β, Γ, Δ, Ε, and within it another, smaller circle on which lie Χ, Ν, Μ, Ξ, about the same center Α; and let the diameters be drawn out — in the larger circle, ΓΔ and ΒΕ, and in the smaller, ΧΝΞ — and let the oblong figure ΔΨΡΓ be completed. [section 12] If, then, the line ΑΒ, in describing a circle, comes back to the same point from which it set out, namely to ΑΕ, clearly it moves back toward itself. Likewise ΑΧ will also come back to ΑΧ. But ΑΧ moves more slowly than ΑΒ, as has been said, because the deflection becomes greater and ΑΧ is dragged back more.
8| [section 13] Let ΑΘΗ be drawn, and from Θ let ΘΖ be drawn perpendicular to ΑΒ within the circle; and again from Θ let ΘΩ be drawn parallel to ΑΒ, and ΩΥ perpendicular to ΑΒ, and also ΗΚ. Now the lines ΩΥ and ΘΖ are equal. Therefore ΒΥ is less than ΧΖ; for equal straight lines set at right angles to the diameter in unequal circles cut off a smaller segment of the diameter in the larger circles, and ΩΥ is equal to ΘΖ. [section 14] In the time in which ΑΘ has traveled ΧΘ, in that same time, in the larger circle, the endpoint of ΒΑ has traveled more than ΒΩ. For the motion in accordance with nature is equal, but the motion contrary to nature is less — that is, ΒΥ compared with ΖΧ. And it must be proportional: as the natural motion is to the natural motion, so the motion contrary to nature is to the motion contrary to nature. [section 15] Therefore it has traversed a greater arc, ΗΒ, than ΩΒ. And it is necessary that it has traversed ΗΒ in this time; for that is where it will be, when the ratio of the contrary-to-nature motion to the natural motion comes out proportional in both cases. If, then, the natural motion is greater in the larger circle, the contrary-to-nature motion would also more likely coincide there in only this one way, so that Β would have traveled ΒΗ to the point marked Χ.
9| [section 16] For here the center comes to coincide with the point Β in accordance with nature (since this is the perpendicular from ΗΚ), while contrary to nature it moves toward ΚΒ. And ΗΚ is to ΚΒ as ΘΖ is to ΖΧ. This is clear if ΒΧ is joined to ΗΘ. But if, when Β had been carried, ΗΒ turns out to be less or greater, then the ratio of the natural motion to the contrary-to-nature motion will not be the same or proportional in both cases. [section 17] The reason, then, why under the same force the point farther from the center moves faster is clear from what has been said; and why the larger scales are more accurate than the smaller ones is also clear from this. For the cord becomes the center (since it stays fixed), while the lines from the center run to each side of the balance-pan. [section 18] From the same weight, then, the end of the balance-pan must necessarily move faster the farther it is from the cord; and in small scales the weights placed on them are sometimes not perceptible to the senses, while in large scales they are perceptible — for nothing prevents a smaller amount of movement in the one case than is visible to sight.
10| [section 19] On the large balance-pan the same weight produces a visible amount of movement. Some things are visible in both cases, but much more so in the larger, because the amount of the tilt produced by the same weight is much greater in the larger scales. [section 20] And this is why dealers in purple dye cheat when they set up their scales — by not placing the cord in the middle, and by pouring lead into one side of the beam, or by making the part of the wood near the root lean toward whichever side they wish, or, if the wood has a knot, using that; for the part of the wood where the root is — or where a knot is, since a knot is a kind of root — is heavier. [section 1] Why is it that if the cord is above, then when the weight is removed after the balance has tilted downward, the beam rises again, but if the support is from below, it does not rise again but stays where it is? Is it because, when the cord is above, the part of the beam beyond the perpendicular becomes more than half? For the cord is a perpendicular. [section 2] So the greater part must necessarily tilt downward, until the line that bisects the beam comes to coincide with the perpendicular itself, once the weight is placed on the part of the beam that has been drawn up. Let ΒΓ be a straight beam, and ΑΔ the cord. If this is extended downward, the perpendicular will be ΑΔΜ.
11| [section 3] If, then, the weight is placed at Β, Β will be at the point Ε and Γ at the point Ζ, so that the line bisecting the beam, which was at first ΔΜ, the perpendicular itself, will, once the weight is placed on it, become ΔΘ; so that, of the beam ΕΖ, the part outside the perpendicular ΑΒ, namely ΦΠ, is more than half. [section 4] If, then, the weight is removed from Ε, Ζ must necessarily move downward, for it is less than Ε. Now if the cord is above, the beam rises again for this reason. But if the support is from below, it does the opposite; for the lower part of the beam becomes more than half of what the perpendicular divides off, so that it does not rise again — for the part that is raised up is lighter. [section 5] Let ΝΞ be the straight beam, and ΚΛΜ the perpendicular. ΝΞ is then bisected. When a weight is placed at Ν, Ν will be at the point Ο, Ξ at the point Ρ, and ΚΛ at the point ΑΘ, so that ΚΟ is greater than ΛΡ by ΘΚΛ. And so, once the weight is removed, it must necessarily stay where it is; for the excess of the half at Κ lies on it like a weight.
12| [section 1] Why is it that small forces move great weights by means of the lever — as was said also at the beginning — once the weight of the lever itself is added in? It is easier to move a smaller weight, and the weight is smaller without the lever. Or is it because the cause is the lever, [being] a kind of beam that has its cord below and is divided into unequal parts? For the fulcrum takes the place of the cord; for both of these stay fixed, like the center. [section 2] Since the longer of the lines from the center is moved faster by an equal weight, and there are three things involved in the lever — the fulcrum (the cord and center), and two weights, the one moving and the one moved — then as the weight moved is to the weight moving, so, inversely, is the length to the length. And always, the farther it is from the fulcrum, the more easily it will move the weight. [section 3] The cause is the one stated before, that the point farther from the center describes a greater circle. So from the same force they move it more; the point applying the motion will be positioned farther from the fulcrum. Let ΑΒ be the lever, Γ the weight, Δ the point applying the motion, Ε the fulcrum, Η the point to which Δ moves in applying the motion, Γ the point moved, and Κ the weight.
13| [section 1] Why do the oarsmen amidships move the ship most? Is it because the oar is a lever? For the thole-pin becomes the fulcrum (since it stays fixed), the weight is the sea, which the oar pushes against, and the one moving the lever is the sailor. [section 2] Now a mover always moves a greater weight the further he is from the fulcrum; for in this way the distance from the center becomes greater, and the thole-pin, being the fulcrum, is the center. In the middle of the ship the greater part of the oar is inside; for the ship is widest there, so that a greater portion of the oar on both sides can be inside the ship at each rail. [section 3] The ship is moved, then, because when the oar is braced against the sea, the inner end of the oar moves forward, and the ship, being fastened to the thole-pin, moves forward together with it, at the point where the end of the oar is. [section 4] For where the oar cuts the most sea, there it is necessarily pushed forward the most; and it cuts the most sea where the greatest portion of the oar is from the thole-pin. This is why the oarsmen amidships move the ship most; for amidships the portion of the oar from the thole-pin to the inner end is greatest.
14| [section 1] Why does the rudder, though small and set at the very end of the ship, have such power that great masses of ships are moved by a small tiller and the strength of one man, and that a gentle strength? Is it because the rudder too is a lever, and the helmsman works it as a lever? Where it is fitted to the ship, it becomes the fulcrum, and the whole rudder is the lever, the sea is the weight, and the helmsman is the mover. [section 2] But the rudder does not take hold of the sea broadside, as the oar does. For it does not move the ship forward, but tilts it as it moves, taking the sea obliquely. For since the sea is the weight, the rudder, bracing against it, tilts the ship in the opposite direction. For the fulcrum turns to the opposite side, and the sea is on the inside; the fulcrum turns to the outside. [section 3] The ship follows this because it is bound to it. Now the oar, pushing the weight broadside and being pushed back by it in turn, drives the ship straight ahead; but the rudder, since it sits obliquely, produces a motion to the side, this way or that. [section 4] It is placed at the end and not in the middle because it is easiest to move a moving thing by moving it from its end. For the foremost part is carried fastest, because, just as in things carried along the motion slackens at the end, so too in a continuous body the motion is weakest at the end.
15| [section 5] And if it is weakest, it is easy to deflect. For these reasons, then, the rudder is at the stern, and also because there, when a small motion occurs, a much greater interval results at the extremity, because the same angle, when it stands upon a greater base, subtends more, the greater the sides containing it are. [section 6] From this it is also clear for what cause the ship moves forward more than the blade of the oar does; for the same magnitude, moved by the same force, advances further in air than in water. Let AB be the oar, let C be the thole-pin, let A be the part on the ship, the beginning of the oar, and let B be the part in the sea. [section 7] If now A has shifted to where D is, B will not be where E is; for BE is equal to AD. It would then have shifted an equal amount. But it was less. It will then be where Z or where H is. Therefore it is along AB, and not merely at C, and from below. For BZ is less than AD, so that ΘZ is also less than DΘ; for the triangles are similar.
16| [section 8] The middle point too, the point C, will have shifted; for it moves to the opposite side from that of the extremity B in the sea, in the direction that the extremity A on the ship did not move, where D is. So the ship will be shifted, and to the place to which the beginning of the oar is carried. [section 9] The rudder does the same thing, except that it contributes nothing to forward motion for the ship, as was said above, but only pushes the stern to one side or the other; for in this way the prow turns to the opposite side. [section 10] Now where the rudder is attached, one must conceive of something like a midpoint of the moving body, just as the thole-pin is for the oar; and the midpoint gives way in the direction the tiller is shifted. If one draws it inward, the stern too shifts that way; but the prow turns to the opposite side; for while the prow remains in the same place, the whole ship has shifted. [section 1] Why do ships, the higher the yard-arm is, sail faster with the same sail and the same wind? Is it because the mast becomes a lever, the mast-step in which it is fixed becomes the fulcrum, that which the weight must move is the ship, and that which moves it is the wind in the sail? And if, the farther off the fulcrum is, the more easily and quickly the same force moves the same weight, then the yard-arm, being raised higher, makes the sail farther from the mast-step, which is the fulcrum.
17| [section 1] Why, when they wish to run before a wind that is not fully favorable, do they furl the part of the sail toward the helmsman, and, making a foot's width of the part toward the prow, let that stand? Is it because the rudder cannot pull against a great deal of wind, but can against a little, which is the part they take in? [section 2] Now the wind drives the ship forward, but the rudder brings it into a favorable course, pulling against the sea and working it as a lever. At the same time the sailors too fight against the wind, for they lean their bodies to the opposite side. Why are round and circular shapes more easily moved? A circle can be made to roll in three ways: either about its rim, with the center shifting along with it, as the wheel of a wagon rolls; or about the center alone, the center remaining fixed, as pulley-wheels do; or along a plane, the center remaining fixed, as the potter's wheel rolls. [section 2] Such things move fastest, both because they touch the plane at only a small point, as the circle does at a single point, and because they do not strike against anything; for the angle stands clear of the ground. And further, whatever body it meets, it again touches this only at a small point. But if it were rectilinear, it would touch the plane over a length with its straight side.
18| [section 3] Further, in whatever direction it inclines toward the weight, in that direction the mover moves it. For when the diameter of the circle is perpendicular to the plane, the circle touching the plane at a point, the diameter divides the weight equally on both sides; but when it moves, at once more weight falls on the side toward which it moves, as if inclining that way. [section 4] Hence it is more easily moved by whatever pushes it forward; for whatever a thing inclines toward, it is easily moved in that direction, just as it is hard to move in the direction opposite to its inclination. Further, some say that even the line of the circle is always in motion, like things that remain at rest, because of mutual resistance, as is the case also with larger circles relative to smaller ones. [section 5] For larger circles are moved faster by an equal force, and move weights faster, because the angle of the larger circle has some inclination relative to that of the smaller, which is as diameter to diameter. But indeed every circle is larger relative to some other, smaller one; for the smaller circles are unlimited in number. [section 6] And if a circle also has an inclination relative to another, and is equally easily moved, then a circle, and the things moved by a circle, would also have another inclination, even if it does not touch the plane at its rim but along the plane, or as pulley-wheels do; for in this condition too they are moved most easily and move the weight. Or it is not by touching and striking little by little, but for some other cause.
19| [section 7] This is the point made earlier, that the circle is composed of two motions, so that one of them always has a tendency present in it, and those who move it always move it as something already in motion, whenever they move it along the circumference in any way at all. For they move it while it is already in motion: the mover pushes its motion sideways, while the circle itself moves along the diameter. [section 1] Why do we move things carried and hauled by means of larger circles more easily and more quickly? For example, larger pulley-wheels move loads more easily than smaller ones, and rollers likewise. Is it because, the larger the line from the center is, the more space it traverses in an equal time, so that, given an equal weight bearing on it, it will produce the same effect — just as we said that larger scale-beams are more accurate than smaller ones? For the cord is the center, and the parts of the beam on either side of the cord are the lines from the center. [section 1] Why does the scale-beam move more easily when it carries no weight than when it has weight; and likewise a wheel or anything else of the sort, the heavier and larger one more easily than the smaller and lighter? Is it because what is heavy is hard to move not only in the contrary direction, but also sideways? For it is difficult to move it against its tendency, but easy in the direction toward which it tends; and toward the side it has no tendency at all.
20| [section 1] Why are loads carried more easily on rollers than on wagons, though some wagons have large wheels and others small? Is it because on rollers the load meets no obstruction, whereas on wagons it meets the axle and is obstructed by it? For it presses on the axle both from above and from the sides. [section 2] What is on rollers moves by virtue of two things, the ground lying beneath and the weight lying on top; for at both these points the circle rolls and, as it moves, is pushed forward. [section 1] Why are missiles carried farther when thrown from a sling than from the hand? And yet the thrower has more control with the hand than when he lets the weight hang loose in a sling. And further, thrown this way he sets two weights in motion, that of the sling and that of the missile, but the other way only the missile. [section 2] Is it because in the sling the thrower throws the missile while it is already in motion — for he swings it round in a circle many times before he lets it go — whereas from the hand the starting point is from rest? And everything is easier to move when already in motion than when at rest. [section 3]
21| Is it for this reason, and also because in slinging the hand becomes the center, and the sling is the line from the center; and the greater the line from the center is, the faster the motion? Whereas the throw from the hand alone, compared with the sling, covers a short line. [section 1] Why do the larger pegs turn more easily around the same crossbar than the smaller ones, and why do the same capstans, the thinner ones, turn more easily under the same force than the thicker? Is it because the capstan and the crossbar are the center, and the parts standing out from them are the lines from the center? [section 2] The circumferences of larger circles move faster and cover more ground under the same force than those of smaller ones; for under the same force the point farther from the center shifts more quickly. This is why, for the crossbar, they make instruments for the pegs, with which they turn them more easily; while in slender capstans the part sticking out beyond the wood becomes greater, and this becomes the line from the center. [section 1] Why is a piece of wood of the same size broken more easily over the knee if one grips it holding the ends at equal distances and breaks it, than if it is held close by the knee; and why, if one has braced it against the ground and stepped on it with the foot, is it broken more easily when the hand works far off than close by? Is it because in the one case the knee is the center, in the other the foot? And the farther a thing is from the center, the more easily the whole of it moves; and it must move in order to be broken.
22| [section 1] Why are the stones on beaches, called pebbles, round, when the stones and shells they were originally made of are long? Is it because the parts farther from the middle move faster in their motions? For the middle becomes the center, and the distance from it becomes the line from the center. And a greater line from the center, moved through an equal motion, always traces a greater circle. [section 2] And what covers a greater distance in an equal time moves faster. And things moving faster strike more violently from an equal distance. And things that strike harder are themselves struck harder in return. So the parts farther from the middle must always be broken off more; and by undergoing this they must become round. [section 3] And for pebbles, because of the motion of the sea, since they move along with the sea, it happens that they are always in motion and, as they roll, keep striking against things. And this must happen most of all to their projecting ends. [section 1] Why do wooden beams become weaker the longer they are, and bend more when lifted, even if the short one — say two cubits — is thin, while the hundred-cubit one is thick? Is it because, in the lifting of the beam, its length turns into a lever, having both a weight and a fulcrum? For the first part of it, which the hand lifts, becomes a kind of fulcrum, and the part at the far end becomes the weight.
23| [section 2] So the longer the part beyond the fulcrum is, the more it must bend; for the farther a part is from the fulcrum, the more it must bend. The ends of the lever must therefore be lifted. If, then, the lever is one that bends, it must bend more as it is lifted. This is what happens with long beams; but in short ones the far end lies close to the fulcrum that stays at rest. [section 1] Why is it that the wedge, though small, splits apart great weights and masses of bodies, and produces powerful pressure? Is it because the wedge is two levers set opposite each other, each having both a weight and a fulcrum, which it either draws up or presses down? Further, the motion of the blow makes great the weight that strikes and sets it moving; and because what is in motion sets something else in motion, it has still more force by reason of its speed. [section 2] And though it is small, great forces attend it, and so it goes unnoticed that it produces motion out of proportion to its size. Let the wedge be ABΓ, and let the thing being wedged apart be ΔΕΗΖ. Then AB becomes a lever, the weight being that below Β, and the fulcrum ΖΔ. Opposite to this is the lever ΒΓ. And ΑΓ, as it cuts, makes use of each of these as a lever; for it draws Β up.
24| [section 1] Why is it that, if someone makes two pulley-blocks on two pieces of wood set facing each other, and passes a rope around them in a circle, with the tie fastened to one of the pieces of wood while the other is fixed or attached by way of the pulleys, then if one pulls at the free end of the rope, great weights are drawn along, even though the pulling force is small? [section 2] Is it because the same weight is raised by a smaller force if it is levered up than if it is lifted by hand? And the pulley does the same thing as the lever, so that a single one will pull more easily, and from a single pull by hand it will draw up something much heavier than the hand alone would. And this is what the two pulley-wheels do, lifting with more than double the effect. [section 3] For the second pulley draws even less than it would if it drew by itself alone, when the rope is passed onto it from the other; for the other has already made the weight even smaller for it. And so, if the rope is passed over still more pulleys, even with a few pulley-blocks a great difference results, so that if the first draws four minas of the weight, by the last a much smaller amount is drawn. [section 4] And in building works they move great weights easily this way; for they transfer the pull from one pulley to another, and again from that one to capstans and levers; and this has the same effect as using many pulleys.
25| [section 1] Why is it that if someone places a large axe on the wood, with a heavy load on top of it, it does not split the wood to any degree worth mentioning; but if someone lifts the axe and strikes with it, it splits the wood, even though the striking implement has much less weight than the one lying on top and pressing down? Is it because everything produces its effect by motion, and the heavy thing gets the motion proper to its weight more when it is moving than when it is at rest? [section 2] Now when it lies on top, it does not move with the motion of its weight; but when it is carried, it has both this motion and the motion of the striker. Further, the axe becomes a wedge; and a wedge, though small, forces great things apart, because it is composed of two levers arranged opposite each other in contrary fashion. [section 1] Why do steelyards lift great weights from a small suspension-point, though the whole beam is unevenly balanced? For on the side where the weight is placed, only the pan hangs, while on the other side there is only the beam. Is it because the beam happens to be at once a balance and a lever? For it is a balance insofar as each of the cords becomes the fulcrum-point of the beam. [section 2] So on one side it has a pan, while on the other, in place of a pan, it has the sliding weight, which rests on the beam — just as if someone were to take the other pan together with its counterweight and place it at the end of the first pan; for it is clear that it draws down as much weight as is placed in the other pan.
26| [section 3] And so that the single beam may serve as many balances, many such cords are set along such a beam, of which each one, on the side toward the sliding weight, marks the midpoint of the beam; and the counterweight moves an equal distance from each of the cords in succession, so that one can measure how much weight the thing placed in the pan draws — so that one can know, when the beam stands level, from which cord the pan carries how much weight, as has been said. [section 4] In general this is a balance, having one pan in which the weight stands, and another, in which stands the counterweight on the beam. That is why the beam is, on the other side, a sliding weight. Being of this kind, it is many balances, as many as there are cords. [section 5] The cord nearer to the pan and to the weight standing in it always draws a greater weight, because the whole beam becomes an inverted lever — for each cord, being above, is the fulcrum, and the weight is that which is in the pan — and the longer the length of the lever from the fulcrum, the more easily it moves there, but here it produces a hanging load, and holds up the weight of the beam against the sliding weight.
27| [section 1] Why do doctors extract teeth more easily by adding the weight of the tooth-forceps than with the bare hand alone? Is it because the tooth slips more easily through the hand than out of the forceps? Or does the iron slip more than the hand, and fail to grip the tooth all the way round, since the flesh of the fingers, being soft, both clings better and fits round it more closely? [section 2] But rather it is because the tooth-forceps are two opposed levers, having a single fulcrum at the joint of the pincers; they use the instrument, then, because it is easier to move the tooth for extraction. [section 3] For let one end of the forceps, at which is A, and the other, at which is B, be that which does the extracting; let the one lever be AΔZ, and the other lever BΓE; let the fulcrum be ΓΘΔ; let the tooth be at the point of junction; and let this be the weight. Taking hold, then, at each of BZ together, one moves it. And once it has been moved, one draws it out more easily than with the hand. [section 1] Why do people crack nuts easily without a blow, using instruments made for cracking them? For much of the force of impact and violence is removed there. Moreover, crushing with something hard and heavy would break it faster than with a wooden and light instrument.
28| [section 2] Is it rather because in this way the nut is squeezed on both sides at once by two levers, and heavy things are easily split by the lever? For the instrument is composed of two levers, having the same fulcrum, the joint at which is A. [section 3] It is then as if two levers had been shot out from this point, by whose motion, toward the ends at Γ and Δ, EZ are brought together easily from a small force; so whatever the weight would have done in a blow, the greater force does — namely EΓ and ZΔ, being levers; for by the raising they are lifted in opposite directions, and by pressing they crack what is at K. [section 4] For this very reason, too, the nearer K is to A, the faster it is crushed; for the further the lever is from the fulcrum, the more easily it moves, and moves more, for the same amount of force. A, then, is the fulcrum, ΔAZ is a lever, and so is ΓAE. [section 5] The nearer K is, then, to the angle at A, the nearer it comes to the junction at A — and this is the fulcrum. It is necessary, then, that, from the same force bringing Z and E together, they be raised more. So that, since the raising is in opposite directions, the pressing must be greater; and what is pressed more is broken faster.
29| [section 1] Why is it that, when two points at both ends of a rhombus undergo two motions, neither of them traverses an equal straight line, but one traverses a line many times the other? The same reasoning also explains why the point moving along the side traverses a distance less than the side. For the one point traverses the shorter diagonal, the other the longer side, and the one moves with a single motion, the other with two motions. [section 2] For let A be carried along AB toward B, and B toward Δ, at the same speed; and let AB itself also be carried along AΓ, parallel to ΓΔ, at the same speed as these. Then A must be carried along the diagonal AΔ, and B along BΓ, and each must have traversed its line in the same time that AB traverses the side Γ. [section 3] For let A have traversed AE, and AB have traversed AZ; and let ZH be drawn out parallel to AB, and let the figure be completed from E. The completed figure, then, is similar to the whole. AZ is therefore equal to AE, so that A has been carried along the side AE. And AB will have traversed AZ. It will therefore be on the diagonal, at the point Θ.
30| [section 4] And it is always necessary that it be carried along the diagonal. And at the same time the side AB traverses the side AΓ, and A traverses the diagonal AΔ. In the same way it will be shown that B too is carried along the diagonal AΓ. For BE is equal to BH. When the figure is completed from H, then, the interior is similar to the whole. [section 5] And B will be on the diagonal at the junction of the sides, and at the same time the side traverses the side, and B traverses the diagonal BΓ. At the same time, then, B traverses a distance many times that of AB, and the side traverses the shorter side, though moving at the same speed, and the side has traversed more than A, though moving with a single motion. [section 6] For the more acute the rhombus becomes, the shorter the diagonal becomes, and BΓ becomes greater, and the side becomes less than BΓ. For it would be absurd, as was said, that what moves with two motions should sometimes move more slowly than what moves with one, and that, given two points of equal speed, one should traverse a greater distance than the other. [section 7] The reason is that, for the point moving from the obtuse angle, the two motions — the one it moves with itself, and the one by which it is carried along by the side — turn out to be nearly opposite; while for the point moving from the acute angle, it happens that they are carried in the same direction. For the motion of the side runs together with the motion along the diagonal; and the more one makes the one angle more acute and the other more obtuse, the motion from the obtuse angle will be slower, and the motion from the acute angle faster.
31| [section 8] For some motions become more opposite because the angle becomes more obtuse, while others become more in the same direction because the lines converge. For B is carried nearly in the same direction under both motions; so the one motion is borne along together with the other, and the more acute the angle becomes, the more this is so. [section 9] But with A it is the opposite: A itself is carried toward B, while the side carries it along toward D. And the more obtuse the angle is, the more opposite the motions become; for the line becomes straighter. And if it became entirely straight, the motions would be completely opposite. But the side, moving with a single motion, is hindered by nothing. It is therefore reasonable that it traverses the greater distance. [section 1] The question is raised why the greater circle unrolls a line equal to that of the lesser circle, when they are set about the same center. But when they roll separately, the lines they trace stand to one another as their sizes stand to one another. [section 2] Further, though both have one and the same center, sometimes the line along which they roll turns out as great as the one the lesser circle by itself rolls out, and sometimes as great as that of the greater. That the greater circle, moving by itself, rolls out a greater line is clear.
32| [section 3] For the arc belonging to each circle's own diameter seems, to perception, to be an angle — that of the greater circle a greater one, that of the lesser a lesser one — so that the lines along which the circles were rolled will, to perception, stand in this same ratio to one another. [section 4] But it is also clear that they roll out an equal line when they are set about the same center; and thus it turns out sometimes equal to the line the greater circle rolls out, sometimes less. [section 5] Let the greater circle be the one on DZG, the lesser the one on EHB, and let A be the center of both; and let the line the greater unrolls by itself be ZI, and the line the lesser unrolls by itself be HK, equal to AZ. [section 6] If, then, I move the lesser circle, I move the same center, A, and let the greater be fitted onto it. So when AB becomes perpendicular to HK, at the same time AG also becomes perpendicular to ZL, so that HK will always have traversed a line equal to the arc HB, and ZL a line equal to the arc ZG.
33| [section 7] If the fourth part unrolls an equal line, it is clear that the whole circle too will unroll a line equal to that of the whole circle, so that when the line BH reaches K, the arc ZG will also be at ZA, and the whole circle will have been unrolled. Likewise too if I move the greater circle, having fitted the lesser onto it, with the same center, then at the same moment as AG, AB will be perpendicular and at right angles, the one to ZI, the other to HTh. [section 8] So that when the one has traversed a line equal to HTh, and the other one equal to ZI, and ZA again becomes perpendicular to ZL, and AG again perpendicular, they will be, as at the start, at Th and I. And this happens without any halt occurring for the greater relative to the lesser, so that it should remain for some time at the same point; for both move continuously in both cases. [section 9] Since the lesser circle skips no point, it is absurd that the greater should traverse a line equal to that of the lesser, and the lesser one equal to that of the greater. Further, since the moving center always has a single motion, it is astonishing that it should sometimes unroll the greater line, sometimes the lesser.
34| [section 10] For the same thing, moving at the same speed, is naturally suited to traverse an equal distance; and it moves at the same speed in both cases. The following must be taken as a starting point concerning their cause: that the same and equal power moves one magnitude more slowly, and another magnitude more quickly. [section 11] If, then, there were something not naturally suited to move by itself, and if this at the same time also moves the thing that is naturally suited to move, it will be moved more slowly than if it were moved by itself alone. And if it is naturally suited to move, but nothing is moved along with it, it will be in the same condition. And indeed it is impossible for it to be moved more than the thing moving it; for it is not moved with its own motion, but with that of the mover. [section 12] Let there be a circle, the greater being A, the lesser being B. If the lesser pushes the greater, without the greater itself rolling, it is clear that the greater traverses as much of the straight line as it was pushed by the lesser. And it was pushed exactly as much as the lesser was moved. They have therefore traversed an equal amount of the straight line. [section 13] It is necessary, then, that even if the lesser, in rolling, pushes the greater, the greater is rolled together with the push exactly as much as the lesser was rolled, provided that it itself is not moved at all by its own motion. For the thing moved by another must have been moved exactly as much as that other moved it. But the circle moved the same thing by this much, both around a circle and by a foot's length (let this be the amount it was moved), and so the greater circle too was moved by this much.
35| [section 14] Likewise, too, if the greater moves the lesser, the lesser will have been moved just as much as the greater. Now each of the two, moved by itself, moves either fast or slowly, as it naturally does; but at the same speed it at once unrolls the line the greater is naturally suited to unroll. This is exactly what creates the puzzle: that they no longer behave the same way once they are fitted together. [section 15] But the solution is this: if the one is moved by the other, it is not moved with the motion natural to it, nor with its own motion. For it makes no difference whether one puts one circle around the other and fits them together, or attaches either one to the other; for likewise, when the one moves and the other is moved by it, however much the one moves it, by just that much will the other be moved. [section 16] So when one moves a thing merely attached to it or hung from it, it does not always make it roll; but when the two are set about the same center, the one is of necessity always rolled by the other. But nonetheless the one is not moved with its own motion, but just as if it had no motion of its own. And even if it does have one but does not use it, the same result follows. [section 17] So when the greater moves the lesser, bound to it, the lesser is moved exactly as far as the greater; and when the lesser moves, again the greater is moved exactly as far as the lesser. But when separated, each moves itself, on its own. And as for the fact that, though the center is the same and moving at the same speed, they turn out to traverse unequal lines, the one who raises the puzzle reasons fallaciously, in a sophistical manner.
36| [section 18] For the center is indeed the same for both, but only accidentally, as one thing can be both musical and pale; for what it is to be the center is not the same for each of the two circles. So when the mover is the lesser circle, it is that circle's center and starting point; when it is the greater, it is the greater's. It is therefore not, without qualification, the same thing that moves; it is so only in a qualified sense. [section 1] Why do they make beds with sides in a two-to-one ratio, one side six feet and a little more, the other three? And why do they string them not along the diagonal? Is it that they make them this size so that they are proportioned to bodies? For made this way the sides come out in a two-to-one ratio, four cubits in length and two cubits in width. [section 2] They string it not along the diagonal but from opposite sides, so that the wood is torn apart less; for wood splits most readily by nature when divided that way, and suffers most when pulled in that direction. Further, since the cords must be able to bear the weight, they will suffer less when the weight is placed on cords running obliquely than on ones running crosswise. And further, less cord is used up this way. [section 3] Let AZHI be the bed, and let ZH be bisected at B. Then the holes in ZB and in ZA are equal in number. For the sides too are equal; for the whole ZH is double ZA. They string it as has been described: from A to B, then to G, then to D, then to Th, then to E.
37| [section 4] And so it continues always, until they come round to another angle; for two angles hold the ends of the cord. The cords are equal at the bends, AB and BΓ equal to ΓΔ and ΔΘ. And the other such cases are the same, since the same proof holds: for AB is equal to EΘ, since the sides of the figure BHKA are equal, and the holes are spaced equally apart. [section 5] And BH is equal to KA; for angle B is equal to angle H. For in equal figures, one is exterior, the other interior; and B is half a right angle, since ZB is equal to ZA, and the angle at Z is right. And angle B is equal to the angle at H, since the angle at Z is right, because the oblong is a double-length figure and is folded at its middle. [section 6] So that AΓ is equal to EH. And KΘ is equal to this, for it is parallel. So that BΓ is equal to KΘ. And ΓE is equal to ΔΘ. In the same way the other cords are shown to be such that the two at the bends are equal to the two others. So it is clear that of cords the size of AB there are four such in the bed frame; and however many holes there are in the side ZH, there are half as many in the half ZB.
38| [section 7] So in the half of the bed there are as many lengths of cord as there are in BA, and as many in number as there are holes in BH. This is no different from saying as many as are in AZ and BZ together. But if the cords are stretched along the diagonal, as they are in the bed ABΓΔ, the halves are not as many as the two sides together, AZ and ZH, but equal to as many as the holes there are in ZB and ZA. But AZ and BZ, being two, are greater than AB. So the cord too is greater by as much as the two sides together are greater than the diagonal. [section 1] Why is it harder to carry long pieces of wood on the shoulder from the end than from the middle, when the weight is the same? Is it because, when the wood sways, the end hinders the carrying, pulling back against the motion the more as it sways? Or is it that, even if it bends not at all and is not of great length, it is still harder to carry from the end? But is it rather because it is also easier to lift from the end than from the middle, and for the same reason it is also easy to carry it that way? [section 2]
39| The cause is that when it is lifted from the middle, the two ends are always lightening each other, and the one part is helping to lift the other well. For the middle becomes like a pivot-point, at which the one lifting or carrying holds it. Each of the two ends is therefore raised upward as the other inclines downward. [section 3] But when it is lifted or carried from the end, this does not happen; rather the whole weight inclines toward one point in the middle, the point toward which it is lifted or carried. Let the middle be at A, the ends B and Γ. When it is lifted or carried at A, then B, inclining downward, lifts Γ upward, and Γ, inclining downward, lifts B upward; and they do this while being lifted upward together. [section 1] Why, when the weight is the same, is it harder to carry a piece that is very long on the shoulder, even if one carries it at the middle, than if it were shorter? It was said earlier that the swaying is not the cause; but now the swaying is the cause. For when it is longer, the ends sway more, so that it would necessarily be harder for the one carrying it as well. [section 2]
40| The cause of its swaying more is that, the motion being the same, the ends shift their position more, the longer the piece of wood is. For the shoulder is the center-point, at which A is (since this stays fixed), and AB and AΓ are the lines drawn from the center. The greater the line from the center — whether AB or AΓ — the greater the distance it shifts. This has been shown earlier. [section 1] Why do they build well-sweeps at wells in this way, adding weight to the pole in the form of lead, given that the bucket itself has weight, both when it is empty and when it is full? Is it because, the work being divided into two stages (for one must dip it down and then draw it up), it turns out that lowering it empty is easy, but raising it full is hard? [section 2] It therefore pays to have the lowering be a little slower in exchange for greatly lightening the weight for the one drawing it up. This, then, is what the lead or the stone fixed at the end of the sweep-pole does. For to the one lowering it, the weight becomes greater than if he only had to lower the bucket empty; but when it is full, the lead draws it up, or whatever weight has been fixed there. So both tasks become easier for him than they would be for someone without it.
41| [section 1] Why, when two men carry an equal weight on a piece of wood or something of the kind, are they not pressed on equally unless the weight is at the middle, but rather more, the nearer it is to one of the carriers? Is it because, things being so, the wood becomes a lever, the weight becomes the fulcrum, the carrier nearer to the weight becomes the thing that is moved, and the other carrier becomes the one who moves it? [section 2] For the further one is from the weight, the more easily one moves it, and the more one presses the other one downward, as though the weight resting on the wood were pushing back and had become a fulcrum. But when the weight is set in the middle, neither carrier becomes any more a weight to the other, nor moves the other, but each becomes an equal weight to the other in the same way. [section 1] Why do all those who stand up make the shin form an acute angle with the thigh as they rise, and the torso form an acute angle with the thigh? If they do not, they cannot stand up. Is it because equality is everywhere the cause of rest, and the right angle belongs to equality and produces a stable stance? For this reason too, bodies move toward angles similar to the curvature of the earth.
42| [section 2] Not that the body will also be at right angles to the plane. Or is it rather because the one standing up becomes upright, and it is necessary for the one who stands to be perpendicular to the ground? If, then, he is going to be at a right angle to it, and this means having the head directly over the feet, then this must come about precisely at the moment he stands up. Now when he is seated, he has his head and his feet parallel, and not on one straight line. [section 3] Let the head be A, the torso AB, the thigh BΓ, the shin ΓΔ. For one seated in this position, the torso [on which AB lies] is at right angles to the thigh, and the thigh is at right angles to the shin. So that, remaining in this position, it is impossible to stand up. It is necessary to incline the shin and bring the feet under the head. [section 4] This will happen if ΓΔ comes to lie along ΓZ, and at the same time it will turn out that he stands up and has his head and his feet on one and the same straight line. And ΓZ makes an acute angle with BΓ. [section 1] Why is what is already moving moved more easily than what is at rest — for instance, people draw wagons that are already moving faster than they get them started? Is it because it is hardest of all to move a weight that is moving in the opposite direction? For this takes something away from the power of the one who moves it, however much faster he may be; for the thrust of the mover, when thrust against, necessarily becomes slower.
43| [section 2] Second, this also holds if it is at rest; for what is at rest resists as well. And a thing moving in the same direction as what pushes it produces an effect just as if someone increased the force and speed of the mover; for whatever it would have suffered at the hands of that mover, it now does itself, moving forward along its path. [section 1] Why do things that are thrown stop being carried along? Is it when the force that released them gives out, or because of being pulled back, or because of their downward inclination, if this proves stronger than the force that threw them? Or is it absurd to raise these puzzles at all, having let go of the starting principle? [section 1] Why does something travel with a motion that is not its own, when the one who released it neither follows it nor goes on pushing it? Or is it clear that the first mover made a second thing such as to push in turn, and this a third; and it stops when the thing pushing the moving object can no longer act so as to push it, and when the weight of the moving thing inclines downward more than the forward force of the pusher. [section 1] Why do things thrown travel far neither when they are too small nor when they are too large, but must bear a certain proportion to the thrower? Is it because what is thrown and pushed must press back against the point from which it is pushed? And what yields not at all, because of its size, or presses back not at all, because of its weakness, produces neither a throw nor a push.
44| [section 2] So then, what greatly exceeds the force of the pusher yields not at all, and what is far weaker presses back not at all. Or is it rather because the thing carried travels only so far as it stirs the air to a certain depth? And what is not moved at all could not move anything either. Now it turns out that both cases share this in common: what is extremely large and what is extremely small are, in effect, both as if unmoved; for the one moves the air only as a single mass, while the other is not moved at all. [section 1] Why do all things carried along in swirling water end up being carried to the center? Is it because the thing carried has magnitude, so that it lies within two circles, a smaller one and a larger one, each of its extremities belonging to one of them? So the larger circle, because it travels faster, spins the object around and pushes it sideways into the smaller one. [section 2] And since the thing carried also has breadth, this smaller circle in turn does the same thing, and pushes it further inward, until it arrives at the center. And there it stays, because the thing carried stands in the same relation to all the circles on account of the center; for the center is equally distant from each of the circles.
45| [section 3] Or is it because, in the case of things whose motion the swirling of the water does not overpower on account of their size, but which exceed it in weight relative to the speed of the circle, these must lag behind and be carried more slowly? And the smaller circle is carried more slowly; for the large circle completes the same turn in an equal time as the small one, when they are about the same center. [section 4] So the object must necessarily lag behind into the smaller circle, until it arrives at the center. And as for the things whose motion the current overpowers at first, when that force slackens it will produce the same effect. For the speed must overpower the weight directly in the one case and differently in the other, so that everything is continually left behind toward the inner circle; for what is not overpowered must move either inward or outward. [section 5] Now then, it is impossible for what is not overpowered to be carried along within the very circle in which it is. And it is even less possible in the outer circle, for the motion of the outer circle is faster. It remains, then, that what is not overpowered shifts into the inner circle. And each thing continually advances further toward not being overpowered. And since arriving at the center puts an end to being moved, and only the center remains at rest, everything must therefore collect together at this point.