1. I have briefly set forth what I thought necessary about the principles of hoisting machines. In them two different things, unlike each other, work together, as elements of their motion and power, to produce these effects. One of them is the right line, which the Greeks term εὑθεια; the other is the circle, which the Greeks call κυκλωτἡ; but in point of fact, neither rectilinear without circular motion, nor revolutions, without rectilinear motion, can accomplish the raising of loads. I will explain this, so that it may be understood.

2. As centres, axles are inserted into the sheaves, and these are fastened in the blocks; a rope carried over the sheaves, drawn straight down, and fastened to a windlass, causes the load to move upward from its place as the handspikes are turned. The pivots of this windlass, lying as centres in right lines in its socket-pieces, and the handspikes inserted in its holes, make the load rise when the ends of the windlass revolve in a circle like a lathe. Just so, when an iron lever is applied to a weight which a great many hands cannot move, with the fulcrum, which the Greeks call ὑπομὁχλιον, lying as a centre in a right line under the lever, and with the tongue of the lever placed under the weight, one man's strength, bearing down upon the head of it, heaves up the weight.

3. For, as the shorter fore part of the lever goes under the weight from the fulcrum that forms the centre, the head of it, which is farther away from that centre, on being depressed, is made to describe a circular movement, and thus by pressure brings to an equilibrium the weight of a very great load by means of a few hands. Again, if the tongue of an iron lever is placed under a weight, and its head is not pushed down, but, on the contrary, is heaved up, the tongue, supported on the surface of the ground, will treat that as the weight, and the edge of the weight itself as the fulcrum. Thus, not so easily as by pushing down, but by motion in the opposite direction, the weight of the load will nevertheless be raised. If, therefore, the tongue of a lever lying on a fulcrum goes too far under the weight, and its head exerts its pressure too near the centre, it will not be able to elevate the weight, nor can it do so unless, as described above, the length of the lever is brought to equilibrium by the depression of its head.

4. This may be seen from the balances that we call steelyards. When the handle is set as a centre close to the end from which the scale hangs, and the counterpoise is moved along towards the other arm of the beam, shifting from point to point as it goes farther or even reaches the extremity, a small and inferior weight becomes equal to a very heavy object that is being weighed, on account of the equilibrium that is due to the levelling of the beam. Thus, as it withdraws from the centre, a small and comparatively light counterpoise, slowly turning the scale, makes a greater amount of weight rise gently upwards from below.

5. So, too, the pilot of the biggest merchantman, grasping the steering oar by its handle, which the Greeks call οἱαξ, and with one hand bringing it to the turning point, according to the rules of his art, by pressure about a centre, can turn the ship, although she may be laden with a very large or even enormous burden of merchandise and provisions. And when her sails are set only halfway up the mast, a ship cannot run quickly; but when the yard is hoisted to the top, she makes much quicker progress, because then the sails get the wind, not when they are too close to the heel of the mast, which represents the centre, but when they have moved farther away from it to the top.

6. As a lever thrust under a weight is harder to manage, and does not put forth its strength, if the pressure is exerted at the centre, but easily raises the weight when the extreme end of it is pushed down, so sails that are only halfway up have less effect, but when they get farther away from the centre, and are hoisted to the very top of the mast, the pressure at the top forces the ship to make greater progress, though the wind is no stronger but just the same. Again, take the case of oars, which are fastened to the tholes by loops,—when they are pushed forward and drawn back by the hand, if the ends of the blades are at some distance from the centre, the oars foam with the waves of the sea and drive the ship forward in a straight line with a mighty impulse, while her prow cuts through the rare water.

7. And when the heaviest burdens are carried on poles by four or six porters at a time, they find the centres of balance at the very middle of the poles, so that, by distributing the dead weight of the burden according to a definitely proportioned division, each labourer may have an equal share to carry on his neck. For the poles, from which the straps for the burden of the four porters hang, are marked off at their centres by nails, to prevent the straps from slipping to one side. If they shift beyond the mark at the centre, they weigh heavily upon the place to which they have come nearer, like the weight of a steelyard when it moves from the point of equilibrium towards the end of the weighing apparatus.

8. In the same way, oxen have an equal draught when their yoke is adjusted at its middle by the yokestrap to the pole. But when their strength is not the same, and the stronger outdoes the other, the strap is shifted so as to make one side of the yoke longer, which helps the weaker ox. Thus, in the case of both poles and yokes, when the straps are not fastened at the middle, but at one side, the farther the strap moves from the middle, the shorter it makes one side, and the longer the other. So, if both ends are carried round in circles, using as a centre the point to which the strap has been brought, the longer end will describe a larger, and the shorter end a smaller circle.

9. Just as smaller wheels move harder and with greater difficulty than larger ones, so, in the case of the poles and yokes, the parts where the interval from centre to end is less, bear down hard upon the neck, but where the distance from the same centre is greater, they ease the burden both for draught and carriage. As in all these cases motion is obtained by means of right lines at the centre and by circles, so also farm waggons, travelling carriages, drums, mills, screws, scorpiones, ballistae, pressbeams, and all other machines, produce the results intended, on the same principles, by turning about a rectilinear axis and by the revolution of a circle.