Heron Alexandrinus, Mechanica, 1999

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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)

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