|Heron Alexandrinus Mechanica 1999, tr. Jutta Miller|
We want to move a known load by means of a known force through the mechanism of cogwheels. For this purpose one builds a frame, similar to a box, in the longest parallel sides of which rest parallel axles at a space measured so that the cogs of the one mesh with the cogs of the others, as we are going to explain directly. Let this frame be a box, designated with <abgd>, in it let rest a light mobile axle, designated <ez>, on which is attached a cogwheel, the wheel <hq>. Let its diameter be, for instance, five times the diameter of the axle <ez>. But in order to explain our construction with an example let us assume the load to be pulled is one thousand talents and the moving force is five talents, that is the man or the boy who alone, without a machine, can move five talents. If we now insert the ropes fastened to the load through a hole in the side <ab> so they wind up on axle <ez>, by the rotation of cog <hq> and the winding up of the ropes the load can be moved. To make the cogwheel <hq> move, however, one needs two hundred talents of force, because the diameter of the cogwheel is five times the diameter of the axle, according to our assumption - this has been shown in the proofs of the five simple powers. We do not, however, have a force of 200 talents, since the force assumed by us is five talents; thus the cogwheel will not be moved. Let us now construct another axle, parallel to axle <ez>, namely the axle <kl>, and let a cogwheel, namely the cogwheel <mn>, be attached to it; let further the wheel <hq> also have cogs that mesh with the cogs of wheel <mn> and let another wheel be attached to the axle <kl>, namely <co>, whose diameter is five times the diameter of <mn>, so that one needs, in order to move the load through the wheel <co>, 40 talents of force, since a fifth of 200 talents is 40 talents. We further let the wheel <co> mesh with another wheel, namely the wheel <px>, which is attached to another axle, namely the axle <fi>, further let another cogwheel be attached to this axle, whose diameter is five times the diameter of <px>, namely the wheel <ss>, then the force that moves the load at the sign <ss> will be 8 talents; the force assumed by us is, however, only five talents. Let us therefore put in another cogwheel, namely the cogwheel <tt'>, whose diameter is double the diameter of wheel <ss>, and let it be attached to another axle, the axle <h'd'>, so that the wheel <tt'> needs four talents of force, so there is in this force a surplus of one talent, which one uses to overcome the resistance of the wheels that may occur. Our explanation illuminates: When the mover sets the wheel <tt'> in motion, the axle <h'd'> rotates and through its rotation the wheel <ss> rotates; therefore the axle <fi> rotates and the wheel <px> rotates; at the same time the wheel <co> and the axle <kl> rotate; therefore, the wheel <mn> rotates and the wheel <mn> sets the wheel <hq> in rotation, because of which also the axle <ez> rotates, the ropes wind up around the axle and the load is lifted. Thus we have, through a force of five talents, lifted a load in the amount of 1000 talents, by means of the mechanism just described. q.e.d. Note: It is necessary that the axis IO goes out to I, and on it, the perpendicular I, is erected equal to the semi-diameter of wheel IP or more than it, and god knows better. TB (note on the margin, not translated by Nix/Schmidt)
2. On the wheels. The wheels attached to an axle always move in one direction, namely the direction in which the axle moves. The wheels that are resting on two axles and whose cogs mesh with each others', move in two different directions, so that the one goes to the right, the other to the left. If both wheels are equal, the rotation of the one to the right entirely corresponds to the rotation of the other to the left; if they are, however, unequal, so that one is larger than the other, the smaller one rotates more often, until the larger one rotates once, according to their sizes.
 After this has been made clear in this introduction, let us rotate two equal circles, namely <hekd> and <zgqe>, around their centers <a>, <b>, while they touch at point <e>. If they now move from point <e> for the same time for half their extant, point <e> in this time runs through the arc <ehd> and reaches the point <d> by moving like the point <g> on the arc <gqe>. Then it can occur that points move in the same direction and that they move in opposite directions. The ones positioned on the same side move in opposite directions, the ones opposed to each other move in the same direction. It may occur however, that points that are described as being in opposite motion go in the same direction (both upward or both downward).For, when points move and their motion starts from one point, namely the point <e>, and we imagine two lines <zaq> and <hbk> perpendicular to the line <gd>, then the motion on the arc <ez> is the opposite of the motion on the arc <eh>, since the one goes to the right, the other to the left side. The motion can also occur in the same direction, if we imagine the distance of the points staying the same from <zh> (text <zk>).Likewise when the motion on the arc <zg> and <hd> towards <g> and <d> is balanced.We also have to assume the same for the arcs <gq>, <dk> and for the arcs <qe> and <ke>. We further say that they can move in the same direction.For we say that the points <de> move in the same direction (this time to the left), when point <e> moves on the arc <ezg> and point <d> on the arc <dke>, and their distance from points <z>, <k> as well as their approach to them remains the same, so the motion is still called opposite (because <e> moves up, then down, <d> down, then up). Therefore the same and the opposite are just complementary and in any motion one has to distinguish between the same and the opposite. This our explanation has to be observed with equal circles.As for different circles, we shall demonstrate it in the following.