The Politics of Aristotle
Page 228
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 dir
ection 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 men
tioned 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.