The Mechanics of Heron of Alexandria First book

The Mechanics of Heron of Alexandria

First book

Heron Alexandrinus Mechanica 1999, tr. Jutta Miller
author index 2 / 34

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.

[3] After this has been made clear in this introduction, let us rotate two equal circles, namely <hekd> and <zgqe>, around their centers <a>, , 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.

[4] On the different circles.Now let the circles be unequal; and let their centers lie on the two points <a> and , further let the larger of the two circles be the one whose center lies on point <a>, then the order with these circles will not be perfect as with equal circles. Let us now assume two points that we let rotate from point <e> and let us make, to pose an example, the diameter <ge> twice the size of the diameter <ed>, then the arc <ezg> will be twice the arc <ehd>, for Archimedes has already proven this. Then in the same time that point <e> in its motion towards <g> runs along the arc <ez>, point <e> will run in the opposite direction along the arc <ehd>. Further in the same time that point <e>, starting at <z>, runs along the arc <zg>, point <e> will, starting at <d>, run along the arc <dke> and reach point <e>. Thus the point that runs along the arc <ehdke> will one time make the opposite motion to the point that runs along the arc <ezg>, the other time be equal to it. Further, in the same time that point <g> runs through the arc <gqe>, point <e> runs through the arc <ehdke>, partially in the same, partially in the opposite direction as <g>. If now the one arc is three times as large as the other one, or in any other relation to it, then we shall show that the moving points are moving partially in the same, partially in the opposite direction.

[5] If we imagine a third constructed circle which touches the circle with the center , so we shall prove for the third circle what we mentioned about the first one.For if the first circle is moving in the direction opposite from the second one, the second one however moves opposite to the third, then the motion of the first circle is the same as that of the third. For if something is moving in the same manner as something else, this however moves in the opposite direction of a third thing, so is the first thing moving in the direction opposite to the third. If further a fourth circle is present, we proceed after the same method.In general, what ensues from the three circles will occur with all circles whose number is odd and what ensues from the two circles takes place with all circles whose number is even. But one not only sees with two and more circles that the motion is now equal, now opposite, but in one circle one sees the same point move now in one direction, now in its opposite. For when the moving point starts moving at any point, it does not stop moving in the same direction until it has run through a semicircle; when it now runs through the second semicircle it moves in the direction opposite to it.

[6] Further, the larger circles do not aloways move faster than the small ones, but sometimes the smaller ones are faster than the larger ones.For when the circles are attached to one axle the larger ones move faster than the smaller ones. When, however, the circles are distant from one another, but on the same body, namely not on the same axle, as occurs with a wagon with many wheels, the smaller circles move faster than the large ones, because their locomotion is one and the same and each of them in the same time moves (the same distance); therefore the smaller circle has to make several rotations until the large one makes one, so therefore the smaller one is in a faster motion.

[7] Sometimes however the motion of the smaller and the larger circles can be equally fast, even when the circles are attached to the same center and rotate around it. Let us assume two circles attached to the same center <a> and let a tangent to the larger circle be given, namely the line <bb'>. If we further connect the points <a>, , then line <ab> is perpendicular to line <bb'> and line <bb'> is parallel to line <gg'>; then line <gg'> is a tangent of the smaller circle. If we further draw through point <a> a line that is parallel to these lines, namely line <aa'>, then if we imagine the larger circle rolling on line <bb'>, the smaller circle will roll by running through the line <gg'>. When now the larger circle has made one rotation, we see that the smaller one also has made one rotation, so that the position of the circles is the position of those circles whose center is at <a'> and the position of line <ab> is that which is taken by line <a'b'>.Therefore line <bb'> equals line <gg'>. Line <bb'> however is the line on which the larger circle rolls when it makes one rotation and line <gg'> is the line on which the smaller circle rolls when it makes one rotation; thus the motion of the smaller circle is equally fast as that of the larger one, because line <bb'> equals line <gg'>. Things that run through the same distance in the same time, however, have equal speed and equal motion. One might think this sentence is absurd, since it is impossible that the circumference of the larger circle should equal the circumference of the smaller one. We now say that not only the circumference of the smaller circle has rolled on line <gg'>, but that the smaller circle also runs through the path of the larger one, thus we see that the smaller circle through two motions reaches the same speed as the larger one; then, if we imagine the larger circle rolling, the smaller one, however, not rolling, but only attached to the point <g>, then it will in the same time cover line <gg'>; then the center <a> covers in this time line <aa'>. This however equals the lines <bb'> and <gg'>; thus the continuous rolling of the smaller circle does not make any difference in the motion and as a consequence the length of the distance of the larger circle is the same as that covered by the small circle; for we see that the center, without rolling, covers the same distance, due to the motion the large circle is in.

[8] As for the case that a point that is moving in two motions of constant speed can follow unequal lines, we are going to prove that now. Let us assume a rectangle, namely <abgd>, and let the line <ad> be a diagonal; further let the point <a> run in steady motion on the line <ab> and let the line move in steady motion on the two lines <ag>, <bd>, so that it is always parallel to line <gd>; also let the time in which point <a> runs to be equal to the time in which line <ab> reaches <gd>; thus I say that point <a> in a certain time covers two unequal lines. Proof for this: If line <ab> moves for a certain time and is placed on line <ez>, then the point moving on line <ab> in the same time comes to rest on line <ez> and a constant relation sets in. The relation between line <ag> and line <ab>, i.e., with line <gd>, is, namely, equal to the relation of line <ae> with the line that lies between point <e> and the point moving on it. Line <ag>, however, is to line <gd> as <ae> is to <eh>.Thus the point moving on line <ab> after <h> falls on the line <ad>, which is the diagonal. Similarly we prove that the point running on line <ab> constantly moves forward on line <ad> and is at the same time moving on lines <ad> and <ab>.The two lines <ad> and <ab> are, however, different; thus, the point moving forward in steady motion covers unequal lines in the same time. But, as I said, the motion of the point on line <ab> is simple, its motion on the diagonal <ad> however is derived from the motion of <ab> on the two lines <ag> and <bd> and the motion of <a> on line <ab>.Thus the one point <a> covers in steady motion two unequal lines. q.e.d.

[9] How one enlarges or reduces plane or solid figures in a certain ratio, we want to explain now, in order to be able to enlarge, for example, one cubit of a solid or plane figure in the same ratio. Let us first deal with the plane figures.Let us therefore assume a certain kind of line. Now we want to find such a line that the similar figures described above the two lines are in a ratio to one another that equals the known ratio. Let the known line be in a known ratio to another one and let us assume the mean proportional* between the two known lines, then this is the line sought; for if the lines are proportional among each other, then the ratio of the first to the third equals the ratio of the similar figures that are described above the first and the second according to the similarity.

[10] Now, however, we want to find a line, such that the similar solid figures, described according to similarity, above the two lines are in a certain ratio with one another. Thus let there be given a line that is in a certain ratio with another line. Let us now assume between the two lines two other lines with continuous proportion, the ratio of the first to the fourth equals the ratio of each of the solid figures constructed above the first to the similar solid object/shape described above the second according to similarity.

[11] How to find two mean proportionals to two given lines, however, we want to explain now with the aid of an instrument, whereby we do not need solid figures; and we want to show the easiest method for this. Let the two given lines be the lines <ab> and <bg>; one is perpendicular to the other one and let the two be the lines for which we want to find the two mean proportionals. Let us now complete the rectangle <abgd> by drawing the two lines <dg> and <da>.Let us further connect with <d> and <g> with <a> and put a ruler to point which intersects the lines <de> and <az>, turn it until the line starting at point <h> towards the intersection of <ge> equals the line starting at point <h> towards the intersection of <az>. Let the position of the ruler be <ebz> and the two lines <eh> and <hz> be equal; then I say that the two lines <az> and <ge> are the two mean proportionals to <ab> and <bg>, <ab> being the first, <za> the second, <ge> the third and <bg> the fourth proportional. Proof: because the quadrangle <abgd> is a rectangle, the four lines <dh>, <ha>, <hb> and <hg> are equal; and because line <hd> equals line <ha>, moreover the line <hz> is drawn, thus is<dz> x <za> + <ah>2 = <hz>2.Also<de> x <eg> + <gh>2 = <he>2. The lines <eh> and <hz> are, however, equal; accordingly is<dz> x <za> + <ah>2 = <de> x <eg> + <hg>2. It is, however,<ah>2 = <hg>2Therefore by subtraction:<dz> x <za> = <de> x <eg> Then the line <ed> relates to <dz> as the line <za> to <ge>. The line <ed>, however, relates to <dz> as the line <ba> to <az> and as the line <eg> to <gb>.Then the line <za> relates to <ge> and the line <ge> and <gb> as the line <ab> to <az> (i.e. <ab>/<az> = <az>/<ge> = <ge>/<gb>). Thus we have constructed two mean proportionals to the two lines <ab> and <bg>, namely the lines <az> and <ge>. q.e.d.

[12] How one has to enlarge or reduce regular plane or solid figures in a certain ratio, we have now explained. Now it is, however, also very necessary to devise a method for the irregular plane and solid figures, by means of which the same procedure is possible for us. But first we want to say in advance some things that are suited to facilitate its understanding; we shall then let the proof of that follow. It is said that plane and solid figures, be they regular or irregular, are congruent if one can describe on one of them such a figure of straight lines, that it is equal and similar to the one that is described on the other one; and it is said that figures are similar to each other if one can describe in one of them figures of straight lines in a manner that one can describe in the other one [figures] similar to them.

[13] If a line moves around a point and one assumes on this line two points that, starting at the fixed point, divide the line according to a given ratio, then the two points that are moving with this line will determine similar figures. If the line moves in a plane, the determined figures will be planar. If the line, however, does not move in a plane but in a body, then the determined figures are solid, if we assume that the points in their close proximity to one another describe the surfaces of the figures.For nothing prevents this sentence to be assumed among the things that are perceptible by the senses; among those that are only imagined it is even more true and correct. From another point of view figures are called similar when one draws the one inside the other and assumes one point so that the lines, drawn from that point towards the borders of the figures, be they lines or planes, are intersected by the borders of the figures in that ratio.

[14] After having said this in advance, we are going to prove that we can find, for any given figure, a similar one that is in a given proportionality to it. We shall prove this first for the plane. Let us assume any line, namely the line <ab>, that is fixed at point <a> and moves in a plane.Let there be two points on it, namely the points , <h>, that move with the line. Point describes in the plane the (circle-)line <bgdez> and point <h> the (circle-)line <hqklm>, so we say that the two (circle-)figures <bgdez> and <hqklm> are similar to one another. Proof: Let us draw into <bgdez> a figure of straight lines, namely the figure <bgdez>; let us further draw the figure <hqklm> by drawing lines from point <a> to the points <bgdez>, namely the lines that we have already drawn; let us further connect the points <hqklm>, then, because the lines <ba>, <ga>, <da>, <ea>, <za> are, according to our assumption, similarly divided at the points <hqklm>, the one figure is of straight lines, namely <bgdez>, similar to the other figure of straight lines, namely <hqklm>. In a similar way we prove that we can draw inside the figure <hqklm> a figure of straight lines that is similar to any (arbitrary) figure of straight lines drawn inside <bgdez>, because the figures described by the two points are similar.

[15] Let us now prove how, with the aid of an instrument, to find for a given plane figure a similar one that is in a given ratio to it. Let us make two round discs (ac, ab), that are cogged regularly, around the same center (a), that are attached to it and are moving around the same axle in the same plane that the figure, for which we want to construct a similar one, lies in.Let the ratio of the discs be that known ratio. Let there be a ruler (pr, lo) at each of the two discs, with cogs towards that direction (a) and their cogs are to mesh with the cogs of the discs. Let these rulers run in the groove of another ruler (ahk), that can be moved on the axle by means of a round hole. Let there be markings (m, n) for the line of the similar figures on the edges of the cogged rulers and these markings are to run on a straight line (amn) that goes through the center of the discs. In order, however, to have both of them always move so that the motion takes place on a straight line that goes through the center, and so that the three points always do the same and always remain on the same straight line, we have to put the markings on the cogged rulers at the same distance from the center of the discs, as the shortest distance of the center of both discs from the edges of the rulers. Then we shift those so that they meet the plane that we want to draw the similar figures on. If one now moves one of the markings so that it comes to rest on the perimeter of that figure and the other one so far from it that the distance between the first one and the center of the discs relates to the distance between this and the other marking like the diameters of the cogged discs to one another (let, however, the ruler that has the groove be a little bent, so the marking that lies on the line mentioned by us runs on this line), then the other marking describes the figure that is similar to the first one and describes it in the given ratio, because the cogged discs are in this ratio.

[16] The figure that is similar to the known one and is in a given ratio to it we have designed in the place where it is itself and where we want to construct the one similar to it. If one is, however, supposed to draw the figure that is to be found, not in that place but in another one, wherever its constructor wants to have it, then one does the following. Let the figure similar to the known one be the figure <abgdez> and let the place to which we want to transfer it be the vicinity of the point <h>. Let us assume inside of figure <abgdez> any point, the point <q>, and let us describe around the two points <h> and <q> two equal circles in the plane and divide them at the points <klmnco> and <pxsstt'> into equal parts, let us connect the dividing points with the centers and make the lines starting from point <h> equal to those in figure <abgdez>; let the line <ak> be equal to line <xd'>, line <lb> equal to line <sf'>, <mg> equal to line <sy>, <nd> equal to line <tf>, <ce> equal to line <t'q'> and <oz> to line <ph'>. If we further draw lines through the points <h'd'f'yfq'> and the points similar to them, then, when we divide the same circles around the centers <h> and <q> into more parts, the line drawn will be the more correct and certain, since the points lie closer to one another. If we now draw the line <h'd'f'yfq'>, then this line and line <abgdez> will be congruent, because the congruent figures correspond to each other.

[17] Also with the solid figures, the regular as well as the irregular, we have to imagine the transfer in a similar way - only with a sphere taking the place of the circle, inside or outside of which we construct the congruent figures. Thus we assume similarly situated points on the sphere and draw, starting from them towards other points situated inside the figure, lines and extend them. When we have done this, a solid figure forms from these lines, which is equal and similar to the one first assumed.

[18] In order to construct similar solid figures, we proceed in the following way.We take two plane boards of wood that can be moved around a common line, so that the line remains one and the same line in any motion.We achieve this when the centers of the hinges, around which the boards move, fall on this common line. Let the size of the boards fit the size of the largest of the similar figures. The manufacture and use of the tool we shall now explain.Let us take two frames of iron that resemble the letter called upsilon and let the parts of each of them spread out be similar to one another. Let us now bend their ends so that the bend has a point and the bending of both of them may result in the figure of a triangle. Let further the known ratio of the one of the similar figures to the other one be equal to the tripled (i.e. cubic) ratio of the proportional sides of the two triangles and let us assume this now for the lines <ab>, <ag> and <ad>, while the lines that were bent are <ge>, <bz> and <dh>; let the other frame consist of the lines <qk>, <ql> and <qm>, and the bent lines be <kn>, <lc> and <mo>; let the two similar triangles be <hez> and <noc>. Let us now draw above the line (cb), that is common to the movable boards, on one (ab) of the boards, a figure (<hez>) congruent to the iron frame and let us further draw through one of the sides of the triangle a line (<oc>) that is parallel to the baseline (<ez>) of the triangle and which cuts off another triangle (<noc>), that is equal (congruent) to the iron triangle which resembles the letter upsilon. On each of the upsilon-frames let a tin rod (S<a> and s<q>), the end of which is very pointed, be attached so that when it is bent and then released, it is firm, i.e. does not tremble, like the tin rods that are used for the human pictures(?). Let the form of this letter called upsilon (after bending) be similar to the tool called Galeagra.Let the motion of the mentioned boards against one another be such that, when the motion stops, they stand firm and cannot be shaken, like the "crabs". This is the manufacture of the tool; we want to explain its use directly. If we now want to make a solid figure similar to another one, which is in a known ratio to it, we bring the surface of the solid figure close to the upsilon-frame, so that the markings on all sides touch the plane and we also bring the other upsilon-frame close to the body to be constructed. If we now want to make it larger than the existing body, we bring the larger body to the larger triangle, the other one to the second. Let us assume we want to make a similar body from stone or wood or any other matter and bring the markings to each body. Let us locate the assumed markings in similar positions on the bodies and we construct the remaining parts on the basis of this procedure. In order to make our explanation clearer, let us assume that we want to attach an eye to the picture of a human or the picture of something else. Let us therefore put the markings of the upsilons on the existing, I mean the given [object], for which we want to make a similar figure, and let us bend the tip (S) of the tin rod that is on the upsilon until the tip touches the eye concerned; then we take the upsilon and put it on the triangle (<hez>), which is drawn on the board (ab); then we lower or raise the other board (cd), on which nothing is drawn, until in its rising or sinking it meets the tip of the rod. Then we remove the upsilon and draw, starting from the point (m) that the tin rod has made on the board (cd), two lines (m<h>, m<z>) towards the end points of the side of the triangle that are lying on the line common to both boards, and make sure that the boards do not move against one another, draw through the other point (<c>), which is located on the line common to both boards, a line (n<c>) parallel to m<z> (text: to the largest lines that are near the lines parallel to the baseline), until it intersects the other drawn line (<h>m). Then we take the other upsilon, put the pointed tips of the teeth that were bent on the triangle (<noc>) that is on the board (ab) and is equal (congruent) to the triangle formed by the ends of those parts (<kn>, <mo>, <lc>), bend the tin rod until it meets the point (n) that was determined by the parallel line (n<c>) on the other board (cd), remove the upsilon and put it on the given points of the body not used yet. The point where the end of the rod meets on the body is the point determined on the picture for the place of the eye, which has a similar position as the one on which we bent the first rod. In the same manner, we bend the rod towards the other parts of the picture and mark the similarly situated points on the stone; then we construct the plane according to the assumed points, which are the points that make the figure similar to the one given first and that has a ratio to it as the one mentioned. Now concerning the mentioned parallel line, it is lightly drawn on the other board, when we draw on the board any parallel to the common line (?). That the figures gained in this manner are similar becomes evident from the fact that they originate from similar, similarly situated pyramids whose bases are the triangles (<hez>, <noc>) defined by the upsilons on the bodies, and whose tips are the points (m, n) marked by the ends of the rods on each of the bodies. That they are in a known ratio is clear, because the ratio of the pyramids that the bodies were made from is the tripled (i.e. cubic) ratio of the proportional sides, for the sides of the similar triangles (<hez>, <noc>) were so assumed. Thus the bodies are in this known ratio with one another.

[19] If we now want to make the back of the similar bodies, we apply the same procedure. We assume on the backs of each of the two figures three points that have a similar position, and determine through the lines connecting them two triangles which are equal (congruent) to the triangles constructed through the letter upsilon, namely the ones drawn on one of the boards; then we put the two upsilons on the back and assume in succession points through which we construct the mentioned parts of the body. If, however, we want to make pictures, one of which is the counterpart of the other, so that when one puts forward its right foot, the other puts forward its left in a step that is similar to that of the right foot of the other - and so forth with the remaining limbs -, then we proceed as follows: We transfer the point given on the second board (<e> = m) to the other side, so that it assumes a similar position, i.e. that the perpendicular (<ez>) drawn from the point (<e>) mentioned towards the common line (<ab>) has the same distance from the end point as the other perpendicular (<qh>) from the other (end) point (<gz> = <dh>) situated on the other side, and that it is equal to the other perpendicular (<ez> = <qh>). In other words: let the line common to both boards be the line <ab> and let the end points of the side of the triangle be the points <g>, <d>, the given point the point <e>; we now draw a perpendicular to line <gd>, namely the perpendicular <ez> and make line <dh> equal to line <gz>; let the line <hq>, which is equal to <ez>, be the perpendicular to it (on <dh>). Now we do not bend the tip of the rod in the direction of point <e>, but in [the direction] of point <q>. We proceed in the same manner by always transferring it (the point concerned) to the other side and making the limbs counterparts. How one creates on a disc a certain number of cogs that mesh with a known screw, we want to explain now, because it is of great benefit for what we want to explain later. Let the screw be situated at <ab> and let the screw thread not be lentil-shaped.Let further the spaces of the screw grooves be the amount of <gd>, <de>, <ez> and let these three lines be equal, so that we want to find a disc with twenty cogs that mesh with the thread of the screw. Let us assume any circle of arbitrary size, namely the circle <hqk> and let the center of it be at point <l>.Let us now divide the circumference of the circle into twenty equal parts and let one of these parts be the arc <hq>. Let us connect the points <hq>, <lq>, <lh> and let us assume the line <hm> is equal to one of the lines <gd>, <de>, <ez>, let us draw through point <l> a parallel to <hq>, namely <ln> and let this be equal to line <hm>.If we connect points <m> and <n> by the line <mn>, then it will intersect line <lq>.Let the point of intersection fall on point <s>. Let us now draw around the center <l> at a distance <ls> a circle, namely the circle <sop>, then we see, that the arc <so> is one of the twenty parts of the circle <sop>, because arc <hq> is one twentieth of the circumference of <hqk>. The circle <sop> is, however, the inner circle.It is thus the circle to be determined, if we extend line <ls> by a line to the amount (<sf>) of the depth of the screw grooves and draw with this complete line (<lf>) a circle around the center <l>. One has to know that the parts situated outside the circle have to mesh with the depth of the screw, because <so> equals <gd>. In reality they do not mesh, however, because the space of the outer part of the screw threads is equal to the inner spaces of the screw grooves; for the cogs, however, the space between their outer points is greater than between the deeper lying inner ones.Since the difference here is not noticeable, however, it does not cause a hindrance for the work. Further, one must not make the pieces cut out on the surface of the front side of the wheel perpendicular, as we teach it for the cog wheels whose cogs we want to mesh with one another, but we make them oblique, so the cogs always mesh with the entire position of the screw groove. This ensues if we divide a circle at the rim of the wheel into twenty parts equal to each other and draw from a dividing point a line under the same inclination as the inclination of the screw groove and divide the other side of the wheel into parts corresponding to the first ones.If we now connect these points by lines on the surface of the rim of the wheel and cut out the cogs, then the screw grooves fit with them and the cogs of the wheel mesh with them. We want to explain now, how the inclination on the front side of the wheel has to be for rotation - for we make the inclination of the cogs on the front side of the wheel so that they mesh with the hollow of the screw threads. Let us assume a wheel and let the distance of one of the cogs be the line <ab> and let the screw groove on the screw be the line <ge> between two lines parallel to the base of the cylinder, namely <gz> and <ed>. Let us now assume two lines, one of which is perpendicular to the other, namely <hq> and <qk>, and let <ed> be equal to line <hq> and <ge> be equal to line <qk>. If we now connect the two points <h> and <k> and draw, starting at point <a>, a line that is perpendicular to the wheel, on the thickness of the wheel, namely <al>, then <al> will be the thickness of the wheel. Let now line <qm> be equal to line <al> and let us draw line <mn> parallel to line <hk>, let further line <ls> be equal to line <qn> on the other circle of the wheel, and let us connect the two points <s> and <a> and divide circle <ls>, starting from point <s>, according to the number of the amount of the cogs, and let <so> be such a part.If we now draw <ob>, then the hollow of the cog is determined by the two lines <ob> and <as>.Let the same be done with the other cogs.

[20] Some people believe, subscribing to false views, that loads lying on the ground can only be moved by an equivalent force. Let us therefore prove that loads, lying in the way described, can be moved by a force less than any known one, and let us explain the reason why this phenomenon is not indeed obvious. Let us therefore assume a load lying on the ground, let it be regular, smooth and joined in its parts; let the plane that the load is lying on be inclinable to both sides, namely to the right and to the left. Let it first incline to the right; then we see that the given load inclines to the right, because it is the natural tendency of the loads to move downward, if they are not supported by something and prevented from motion.If further the inclined side rises back to the horizontal plane and (the entire plane considered in and of itself) comes into balance, then the load will remain in this situation. If it now inclines to the other, i.e., to the left side, then the load too will incline towards the lowered side, even if the incline is very small; thus the load does not need a force that moves it, but a force that supports it so it does not move. If now the load also returns to balance and does not incline to any side, then it remains like that, without the presence of a force that supports it and remains in rest, until the plane inclines to any side and then it inclines towards that direction. Doesn't therefore the load that is positioned to turn to any direction need only a small force to move, that is, in the amount of the force that makes it incline?Therefore, the load can be moved by any small force.

[21] The stretches of water, now, that are situated on not inclined planes, do not flow, but they are still, without inclining towards any side. If they experience, however, even the slightest incline, then they all flow towards that side, so that not even the smallest part of the water remains on it, unless there might be depressions in the plane, so that small parts would remain in the hollow of these depressions, as sometimes occurs with vessels. This happens, however, to the water because its parts are not joined, but rather are easily separable. Since, however, the joined bodies are by nature not smooth on their surfaces and cannot easily be made even, the roughness of the bodies causes them to support one another and this again causes them to lean against one another like cogwheel mechanisms, so that they are prevented from it; for when they are numerous and tightly joined to each other through a mutual fusion, a united large force is necessary. From experience one has now drawn the lesson; for one started to put under the "turtles" pieces of wood, whose surface is formed cylindrically, touching only a small part of the plane, which is why only the smallest friction occurs.Now one uses stakes, so the load can easily be moved on them, under the condition that the load is increased by the weight of the tool. Others attach planed boards to the bottom, because of their smoothness, and smear them with fat, in order to smooth out the roughness on them, and then move the load with very little force. As for the cylinders, they can, if they are heavy and lie on the ground so that only a single line touches the ground, be moved with ease, and as well the spheres that we already talked about.

[22] If we now want to lift a load to a higher place, we need a force equal to the load. Let us assume a mobile pulley, fixed in height, perpendicular to the plane, which can easily be moved around the centers on an axle.Let there be lying around its rim a rope, one of whose ends is fastened to the load; let the other one be with the pulling force.Now I say that this load can be moved by a force equal to it. Let there be on the other end of the rope not a force, but another weight attached, so it will show that the pulley, if the weights are equal, does not move towards any side and that the first weight is not strong enough for the second fastened one, nor the weight for the load, because the second fastened weight is equal to the first load. If however a small amount is added to the weight, then the other weight is pulled upward.If therefore the force moving the load is larger than the load, then it is strong enough for it and moves it except if friction occurs in the turning of the pulley or stiffness of the ropes, so that it would cause a hindrance for the motion.

[23] Now as for the loads situated on inclined planes, they have the natural tendency also to move downward, as is the motion of all bodies.If that is not as mentioned, then we have also here to think of the reason mentioned before. So let us assume we want to move a load upward on an inclined plane.Let its bottom be smooth and even, the same also the part of the load it supports. For this purpose we have to attach a force or a weight to the other side so that it may be equal to the load, i.e., keep its balance equilibrium so that the surplus of force over it may be strong enough for the load, and lift it upward. In order that our claim is proven correct, we want to prove it with a given cylinder.Since a large part of the cylinder does not touch the ground, it has the natural tendency to roll downward. If we now imagine a plane that goes through the line that touches the ground and is perpendicular to this ground, then it turns out that this plane goes through the axis of the cylinder and divides it into two halves; for if a line touches a circle and if one erects a perpendicular at the point of contact, then this goes through the center of the circle. If we further put through the same line, namely the line of the cylinder, a plane perpendicular to the horizon, then it will not be the first positioned plane and will divide the cylinder into two different parts, the smaller of which lies to the top, the larger to the bottom.So the larger one has the superior weight over the smaller one, since it is larger, and the cylinder rolls. If we now imagine on the other side of the plane perpendicular to the horizon the amount of superior weight over the smaller [part] taken away from the larger part, then both parts are in balance and the weight of both of them remains on the line touching the ground, without inclining to any side, namely neither upward nor downward. Thus we need a force equivalent to this difference that withstands it.If however a small surplus is added to this force, then it gains superior weight over the load.

[24] Now I am of the opinion that it is necessary to enlighten those who cultivate mechanics about what is gravity and what the center of gravity, be it in a body or in a non-body. That one in reality only speaks of gravity and inclination in bodies, nobody will deny. When we say, however, concerning geometrical figures, solid and plane ones, that the point of inclination or the center of gravity is a certain point, so has this been sufficiently explained by Archimedes. Thus it has to be understood on the basis of what we are now explaining about it.Posidonius, a Stoic, has determined the center of gravity and the point of inclination in a natural (physical?) definition; he said: the center of gravity or point of inclination is such a point that, when a load is suspended at it, it is divided into two equal parts. Therefore Archimedes and his followers in mechanics have particularized this theorem and distinguished between the point of suspension and the center of gravity. As for the point of suspension, it is such a point on the body or non-body that, when the object to be suspended is suspended at it, its parts are in balance, and with this I mean that it does not sway and does not incline. For balance occurs when an object is equal in weight to another one, as is the case in wagons when they sway parallel to the plane of the horizon or to a plane parallel to it.Thus Archimedes says: loads do not incline on a line and on a point. On a line, when a load rests on two points of that line, so that the line does not incline, and the plane put through that line perpendicular to the horizon, how ever the line may be moved, remains perpendicular, the load will not incline at all. If we say: the load inclines, we only mean by that its drop downward, i.e., its motion towards the ground. As for the balance at a point, however, it occurs when a load is suspended from the same and the parts of the body, in every motion it makes, remain in equilibrium with each other. One load keeps the balance of another one, when they are suspended from two points of a line divided into two halves and at the dividing point of this line, and this line is parallel to the horizon, after the amounts of the loads are in the [same] ratio to each other as the amounts of their inverse proportional distances from their points of suspension. That loads suspended in this manner remain balanced with one another in the inclination, Archimedes has proven in his works on balance in figures on which levers are used. Suspensions and supports show the same phenomena, because the suspension and the support of a force (or: and the support according to the force imposed?) are the same, for the supports that a load is put on are the ones that support the load. Such supports can be very numerous, even unlimited in number.As for the center of inclination, it is a single point in each of the bodies, towards which the lines that are perpendicular (to the horizon) from the points of suspension incline.Sometimes the centers of inclination also lie outside of the substance of an individual body, as is the case for wheels and rings. Now, that the lines of suspension meet at a common point will become clear to us if we imagine a plane standing perpendicular on the horizon and the same intersecting any body according to the balance.For we see that the body is divided by the plane into two halves; thus it goes through the body. If we now imagine another plane that intersects the body like that plane, then it penetrates it like that plane and the two planes intersect in one line.For if the line of intersection did not go through the point of suspension, then it would show that the same bodies would be in balance and not in balance. Let us now apply this conclusion to the supports.Let us imagine a body resting on a line that is perpendicular to a plane and let the body rest in balance of its parts on this line.If now this line is extended, then it goes through the body. For if the extended line falls outside of the body, then also the plane put through it falls outside of the same; that is, as we have just seen, impossible.Thus the line goes through the body and divides it into two parts that remain balanced. If we now assume as point of balance one that is different from this point, then we see here the same as with the first one, namely, that the line drawn through that point goes through the center of the body, so that the two lines are distant from one another.If now two planes are put through them, then those do not intersect; for one can put through two lines two planes that do not intersect.Thus here occurs the same as in the first case; thus it is not possible. From that you see that the planes intersect and the lines meet, so that they fall in one plane.If now this plane is pulled towards the surface of the body, then it makes, according to the points of intersection, a line.Then there is a third point falling outside of this line. If we now assume this point also to be a point of balance, on which the body remains balanced, and draw a supporting line through this point, then this line, according to what we have said already, when it is drawn, will meet those two lines, through which the plane has been put, but not another point, except their point of intersection.For if any line meets two intersecting lines, but it lies on a different plane, then it meets them at their point of intersection.If however its meeting with the two does not occur at their point of intersection, then necessarily one part of the line lies on one plane and the rest on another one.Thus all lines that serve suspension unite at one point, namely the one that is called the point of inclination or center of gravity.

[25] It is now urgently needed to give some explanations concerning pressure, transport and support with regard to quantity, as are suitable for an introduction. For Archimedes has already adopted a reliable procedure on this part in his book with the title "Book of Supports". We want to pass over of it what we need for other things and use of it now what refers to the amount of quantity, as is suitable for students. The general point of view here is this: If one has any number of pillars and crossbeams or a wall rests on them; further [if it rests] in the same or in a different position on the two outermost ones of them (the pillars), so that they extend beyond one of them or both at the same time, and if the distance between the pillars is equal or different, then we want to learn how much of the load affects each of the pillars. An example for this is the following: If one has a long beam of even weight that is carried by men evenly distributed to the length and the ends of the beam, and one or both of the ends jut out, then we want to learn from each man, how much of the load comes to him; for the question is the same in both cases.

[26] Let therefore an evenly thick and evenly dense load, <ab>, rest on pillars. Let it rest on two pillars, namely <ag> and <bd>; then each of the two pillars <ag>, <bd>, is affected by half the load <ab>. Let now third pillar <ez> be present and let it divide the distance <ab> arbitrarily; then we want to learn about each of the pillars <ag>, <ez>, <bd>, how much of the load comes to it. Let us now imagine the load <ab> to be divided at point <e> following a perpendicular to the pillar, then we see that part <ae> affects each of the two pillars <ag>, <ez> with half its weight and part <eb> each of the two pillars <eb>, <bd> with half its weight, because there is no difference, as far as the pillars are concerned, whether the object put on them is joined or broken; for may it be joined or broken, it rests entirely on the pillar. Thus to pillar <ez> comes half of the weight of <eb> and half of the weight of <ae>, i.e. half of the entire weight of <ab>; and to pillar <ag> comes half of the weight of <ae>, to <bd> half of <eb>. If we now divide half of <ab> in the ratio of the space <ae> to space <eb>, then the weight of the part proportional to <ae> falls to <ag> and the weight corresponding to the distance <eb> to <bd>. If we now erect another pillar, <hq>, then the result is that half of <ae> falls to <ag>, half of <hb> to<bd>, half of <ah> to<ez> and half of <eb> to<hq>. Half of <ae> plus half of <hb> plus half of <ah> plus half of <eb> is however all of <ab> and that is what rests on all the pillars together. If there are even more pillars, then we learn through the same procedure, how much weight comes to each of them.

[27] If that is so, then we assume the supports <ab> and <gd> in equal positions; let an evenly thick and heavy body rest on them, namely <ag>. We just said that half the weight of <ag> falls to each of the two supports <ab> and <gd>. If we now move support <gd> and bring it closer to <ab>, namely to position <ez>, then we want to know which of the weight falls to <ab> and <ez>. Now we say that distance <ae> is either equal to distance <eg> or smaller or greater than it. Let it first be equal to it, then we see that the weight of <ae> keeps the balance of the weight of <eg>. Thus if we remove the support <ab>, the weight <ab> remains steadily in its position and we see that none of the weight fell to support <ab>, but the weight <ag> was only on <ez>. If now distance <ge> is greater than distance <ea>, then the load inclines towards <g>. Let now the space <ge> be smaller than space <ea> and let <ge> equal <eh>, then <gh> remains balanced on <ez> alone. If we now put in a pillar at <h>, then, if we imagine the entire load cut at point <h>, <hg> rests on <ez> alone and half of <ah> rests on each of the two supports <ag> and <hq>. If we now remove support <hq>, point <h> obtains its force, if the body is joined, and to <ab> falls half the weight of <ha>, to <ez> the rest, namely <gh> and half of <ah>; if we imagine <ag> bisected at point <k>, then <ke> is half of <ah>. If now the support that was first at <e> moves under point <k>, then it is affected by the entire weight of <ag>. And the further the support moves away from the point of intersection which divides the load in half, the more of the load goes to <ab>, while the rest of it rests on the other support.

[28] If this is so, then we want to assume two supports, namely <ab> and <ez> in the position mentioned before and let the load <eg> be jutting out. If we now divide the load in two halves at point <k>, then we have proven, that the weight <ke> falls to <ab> and the rest of the load <ag> to <ez>. If we now assume a support under point <g>, namely support <gd>, then it is also proven, that the support <ab> is affected by half the weight of <ae> and support <gd> by half the weight of <ge>, finally the support <ez> by half the weight of <ag>. Before we put in support <gd>, we showed, how much weight falls to each of supports <ab> and <ez>. It is also clear that, after the support <gd> came under the load, more of the load comes to support <ab> than before, in fact, half of <eh> = <eg> more, to <ez>, however, less by the amount of <eg>. Consequently, to <gd> comes half of <eg>, because the support added below the load removed, from what affects <ez>, an amount equal to <eg>, and it added to <ab> an amount equal to half of <eg>; thus <gd> is affected by the other half of <eg>. That much also affected it after the other procedure. It is therefore evident that, if a load rests on supports that support it, and if one adds to these supports another one, the first of the former supports is affected by more of the load than before the addition, and the other one by less than affected it before the addition. Since now, when <ab>, <ez> and <gd> were the supports, the part falling to <ab> was half of <ae>, after however <gd> was removed, the part falling to <ab> was half of the weight of <ah>, it demonstrates that <eg>, by floating, worked as a lever and took over part of the weight resting on <ab>; however, it shifted a larger weight to <ez> than had rested on it before, while the load <ag> kept its position.

[29] That small forces cannot move big loads without the use of a machine is proven by clear events; for two men move with ease a load, that one, even summoning up all his strength, does not move. Thus it is clear that the load can only be moved if the strength of the second man is added. That the second man alone does not move the load, is clear; for if the first man rests and leaves it to the second one, then he does not move it. If however the load is divided into two halves, then the first man moves half the load and the other half remains still. Thus it demonstrates that the half, that one man moves, was pulled by the other half, before it was separated from it. Also, if many forces move a load and one of these forces is removed, all forces, after the removal of that one, no longer move the load. If, however, the reunited force starts lifting the load, then it moves the load easily through the addition of that remaining given force. The same we see in striking, because an object that has been worn down brakes into pieces by the addition of a single blow, not only by combining them, but also through this one alone. This also shows in perceptions; for if we lift a load whose weight we manage, even if it is with strain and effort, then obviously our strength is equal to that load.

[30] Let us now assume the supports <ab> and <gd> and let rest on them an evenly heavy and thick body, namely <ez>, which juts out beyond each of the supports. We want to know how much of the load affects each of the supports. Since we have proven that, when the load <az> rests on <gd> and <ab>, <gd> is affected by twofold more [of the share] of <gz> than <ab>; and if <ge> rests on <gd> and <ab>, <ab> [is affected] by twice [the amount] of <ae> more of the load, then the result is that that much more of the load falls on <gd> than on <ab>, as the surplus of the double of <gz> over the double of <ae> amounts to. If now <gz> and <ae> are equal, then the weight falling on <gd> and <ab> is equal. The greater, however, the distance becomes, the more of the surplus of the load falls to that support. The above stated makes evident that when a crossbeam or wall that is evenly heavy and thick rests on pillars or supports and the distances between them are randomly different, we can learn on which supports falls a greater weight and how large the excess is. If a crossbeam or anything else rests on the supports, we employ the same method. If, further, people carry a beam on their shoulders or in a loop, some in the middle, some at its ends, and if the load juts out at one or both sides, then it will become in the same way evident to us, how much of the load comes to each of the bearers.

[31] Let now another, also even and evenly heavy, load be given, namely <ab>, which rests on supports in the same position, namely <ag> and <bd>. Then it is clear that on each of the supports falls half of the load <ab>. Let us now suspend a weight from <ab> at point <e>. If point <e> divides <ab> equally, then it is clear that to each of the supports falls half of the load <ab> and half of the weight suspended or put on at point <e>. If point <e> does not divide the load in half, however, and if one divides the load in the ratio of <be> and <ea>, then the weight of the part proportional to <eb> falls to <ag> and the weight of the part proportional to <ea> [falls] to <bd>, furthermore each of the supports bears half of <ab>. If we now suspend another weight at point <z> and divide it in the ratio of <az> and <zb>, then on <db> falls the weight of the part proportional to <az> and on <ag> the weight [of the part proportional] to <zb>; and each of the supports is affected by half of <ab>. The relation of <zb> to <ag> has just been mentioned. The loads that affected them, before the weights attached at the points <e>, <z> were suspended, have also already been mentioned; thus everything has been given that falls to the two supports <ag> and <bd>. If more weights are attached, then we learn after the same method how much weight falls to each of the two supports.

[32] Some people believe that, when in scales the weights are in balance with the weights, the weights are in the inverse proportional ratio to the distances. This must, however, not be said so in general, but one has to introduce a better distinction. Let us now assume an evenly thick and heavy scale beam, namely <ab>, whose point of suspension, namely point <g>, lies in its center. Let now be suspended at random points, for instance points <de>, ropes, namely the two ropes <dz> and <eh>, to which are fastened two weights and let scales after the suspension of the weights be in balance. If we imagine the two ropes going through points <q> and <k>, then in the equilibrium of the scales the distance <qg> will relate to the distance <gk> like weight <h> to weight <z>. Archimedes has proven this in his works with the title: Writings on Levers. If we now cut off the scale beam the parts on each side, namely <qa> and <kb>, then the scales will no longer be in balance.

[33 ]Some have thought that inverse proportionality is not present in irregular scales. Let us therefore also imagine a differently heavy and dense scale beam of any material that is in balance when it is suspended at point <g>. Here, we understand with balance the rest and standstill of the scale beam, even if it is inclined to any side. Then we suspend weights at random points, namely <d> and <e>, and we let the beam again be in balance after their suspension. Now Archimedes has proven that also in this case weight to weight is inverse to distance to distance. As now for the irregular bodies, in which the space is inclined, there we have to imagine the following: Let the suspension string situated at point <g> be extended towards <z>. Let us now draw a line and imagine it going through point <z> and equal to line <zhq>; let it be "solid", i.e. perpendicular to the string. Since now the two strings situated at points <d> and <e>, namely <dh> and <eq>, are like this, then the distance that exists between line <gz> and the weight suspended at point <e> is <zq> and with the scales in rest, just as <zh> is to <zq>, so the load suspended at point <e> is to the one suspended at point <d>, which has been proven in the preceding.

[34] Let a circular disc or a pulley be mobile on an axle around the center <a>; let its diameter, the line <bg>, be parallel to the horizon. Let us now suspend at points and <g> two strings, namely <bd> and <ge>, on which two equal weights are hanging, then we see that the pulley does not incline to any side, because the two weights are equal and the two spaces from the point of suspension <a> are equal. Let now the weight at point <d> be greater than the one at <e>, then we see that the pulley inclines towards and point drops together with the weight. Now we have to learn, in what place the greater weight <d>, when it drops, comes to rest. Let us therefore lower point and let it come to <z> and let then the string <bd> be with string <zh>, so the weight comes to a stand still. We now see that string <ge> winds itself around the rim of the pulley and that it hangs on the weight from point <g>, because its wound-up part no longer hangs down. If we now extend <zh> to point <q>, then, because the two weights are in balance, the ratio of the one weight to the other is in the same (inverse) ratio of the distance of point <a> from the strings and it is like <ag> is to <aq>, so the load at <h> is to the load at <e>. If we make the ratio of <ga> to <aq> equal to the (inverse) ratio of load to load and shift points , <g> towards <zq> at a right angle, then we see that the pulley has moved from point to point <z> and is at rest.The same observation is also true for other weights. Thus according to this point of view every load can keep the balance of a load that is smaller than itself. This may be enough for the first book of the introduction to mechanics. In the following we are going to deal with the five powers by means of which loads are moved, explain what they are based on and how the natural effect in them occurs. Furthermore we shall speak of other things that are of great benefit in the lifting and carrying of the loads. End of the first book of Heron's writings on the lifting of heavy objects.

Second book

1 Since the powers through which one moves a known load by a known force are five, we necessarily have to explain their forms, their application and their names, because these powers go back to one natural principle while they are quite different in form. Now their names are the following: the shaft with the wheel, the lever, the block and tackle, the wedge, the screw. The shaft with the wheel is made in the following way. One takes a hard, square piece of wood in the form of a beam; make its ends round by planing them and fasten to them fitting rings of copper, so the roughness of the axle has no effect, so they rotate easily when they are put in round bronze-clad holes in a firm, immovable support. This [piece of] wood, made according to the description just given, is called an axle. Then one attaches to the center of the axle a wheel that is fitted with a square hole according to the center of the axle and fitting the measurements of the axle, so the axle and the wheel, when the latter is attached to the former, rotate together. This wheel is called Peritrochion, the translation of which is "the surrounding". When we have done this, then we make on both sides of the wheel a curl-like groove, so the same may be a winch, on which the ropes wind up. Then we make on the front side of the wheel, i.e. on its perimeter, holes, whose number suits necessity, that are made regularly so that when spokes are fastened in them, the wheel and the pulley are set in motion by these spokes. We have just explained how one has to construct the axle; how one works with it we are going to explain now. If one wants to move a big load with a small force one fastens the ropes tied to the load to the place grooved on both sides of the wheel. Then one puts spokes into the holes bored into the wheel and presses the spokes downward so the wheel rotates and the load is moved by the small force and the ropes wind up around the axle or we stack them on one another so they do not wind up on the whole axle. The size of this machine has to be set up according to the size of the load that one wants to move with it. Its calculation has to take place according to the ratio of the load one wants to move to the force that is meant to move it, as we are going to explain in the following.

2 The second power. The second power is the one that is called lever and this power is perhaps the first thing one thought of for the moving of excessively heavy bodies. For, since the first thing that one needed if one wanted to move a body of excessive weight was to lift it off the ground in its motion, but one did not have any hold on it, to grip it, since all parts of its base were lying on the ground, so one necessarily thought of this procedure, made below the body a small pit in the ground, took a long [piece of] wood, put one end of it into that pit and pressed the other one downward; so the load rose. Then one put under that [piece of] wood a stone [block], which one called Hypomochlion, i.e., that which is put under the lever, and pressed it downward again, so the load rose even more. When this power became known, one understood that it was possible to move big loads in this way. This [piece of] wood is called a lever, may it be round or square. The closer one brings the stone [block], that one puts under it, to the load, the more comfortable it is for the motion, as we are going to show in the following.

3 The third power. The third power is the one that is called block and tackle. For if we want to lift any load, we tie ropes to this load and want to pull the ropes tight, until we lift it. Therefore we need a force equal to the load to be lifted. But if we remove the rope from the load, tie one of its ends to a firm crossbeam, put the other one over a pulley that is attached to the middle of the load, and pull the rope, then it is easier to move that load. If we now attach another pulley to the firm crossbeam, put the end of the rope over it and pull it, then the moving of the load is even easier for us. If we further fasten a second pulley to that load and pull the end of the rope over it, then this increases for us the ease in moving the load. In this manner we continue to attach pulleys to the firm crossbeam and to the load that we want to lift and to put the one end of the rope over the pulley tied to the firm crossbeam and the load and let the rope run back to it again and again, then through this the ease to lift that load increases for us. The more numerous the pulleys are that the rope runs on, the easier can that load be lifted. The one end of the rope has to be firmly tied to the firm crossbeam and the rope has to run from this towards the load. The pulleys on the firm crossbeam have to be attached to another [piece of] wood and rotate around the same axle. This axle is called Manganon. The [piece of] wood is fastened to the firm crossbeam by means of other ropes. The pulleys on the load sit on another axle, equal to that first one and fastened to the load. The pulleys have to be attached to the axle in such a manner that they cannot come in contact with one another; for if they can come in contact the rotation is impeded. Why now the ease in lifting is increased for us, when the number of pulleys is increased and why the one end of the rope is tied to the crossbeam, we are going to explain later.

4 The fourth power. The fourth power, which follows to this one, is the one called wedge. It is used with some tools in the preparation of perfume and in order to join separate parts of carpenter's works. Their uses are manifold; we use them the most frequently, however, when we intend to split the lowermost part of stone [blocks] we want to cut, after having already detached the side parts off the mountain from which we want to break them off. In this none of the other powers works, even if they were all united. The wedge however works in this all by itself. Its effect is based on the blow that hits it, whatever may be the nature of the blow, and its effect does not stop after the end of the blow. That is clear; for often he causes a noise, without being hit and a cracking of what it splits with its force. The more acute the angle of the wedge, the easier can it be worked with, as we are going to show.

5 The fifth power. That is the one called screw. The principles of the tools mentioned until now are clear and perfect in themselves.In the effect and application of the screw, however, exists a difficulty, whether it works by itself or another power together with it. It is however just a winding wedge, which, however, one cannot hit, but [it] is moved by means of the lever. That will become clear from what is going to be mentioned. Now we say that the nature of the line described around it is the following: if one assumes any side of a cylinder moving on a plane and at the end of this side a point that moves on it and runs completely through it, at the same time in which that side circles the surface of the cylinder once and returns to the place where it started its motion, then the line, which that point describes on the surface of the cylinder, is a screw thread, which one calls a screw. If we want to describe this line on the surface of a cylinder, we proceed like this: if we assume in a plane two lines, one of which is perpendicular to the other one and one of which is equal to the side of the cylinder, the other one equal to the circle of the cylinder, i.e. equal to the circle of its base, if we further connect the end points of the line surrounding the right angle, then put the line equal to the side of the cylinder on the side of the cylinder and the line equal to the circle of the cylinder's base on that, then the side opposite the right angle winds up on the surface of the cylinder and a screw thread is formed on it. We can also divide the side of the cylinder into any number of equal parts and describe a screw thread on each of them, so that many threads are formed on the cylinder and the cylinder becomes a screw. The cylinder, on which a hypotenuse was wound up, is called a screw with a thread if, namely, the side of the cylinder contains only one line that starts at one of its ends and goes to the other one. If we now want to use the screw, we cut, following this line drawn on the cylinder, a deep groove, which penetrates into the surface of the cylinder so far, that we can insert the [piece of] wood called Tylos. Then we use the screw in the following way. We fit its two ends with a smooth curve and insert them in round holes in firm supports, so it can be easily rotated in these holes. Then one fastens the [piece of] wood called Kanon perpendicular and parallel to the wood of the screw. Let there be in this Kanon a deep canal-like groove, that shows on the side of the wood facing the screw. Then we put the one side of the [piece of] wood called Tylos into the groove of the screw, the other one into the groove of the Kanon. If we now want to lift a heavy load with this tool, we take one of the ropes called Hoplon, tie one of its ends to the load we want to move, the other one to the [piece of] wood called Tylos, after having pierced the end of the screw with holes opposite to each other. Now we insert spokes into these holes and with them turn the screw, then the Tylos rises because of its motion in the groove, that is in the screw, and at the same time the rope rises, lifting the load that is fastened to it. Instead of spokes we can attach to the end of the screw that is outside of the firm support a quadrangle fitted with holds, with the aid of which we turn the screw and the load rises. The screw groove that is on the cylinder is sometimes square and sometimes lentil-shaped. The square one is the one with perpendicular cuts whose groove ends in two lines, the lentil-shaped one the one whose cuts are inclined and end in only one line. This screw is called lentil-shaped, the other one square.

6 If the screw is used by itself, it happens in this way. If one uses it however differently, in connection with another power, namely the one effective through the shaft with the fitted wheel, it happens in the following way. Let us assume cogs on the wheel on the shaft, while a screw stands opposite the wheel, either perpendicular to the ground or parallel to the plane of the ground. The cogs are to mesh with the screw thread and the ends of the screw lie in two round holes in two firm supports, as described before. Let there be at the one end of the screw a protrusion jutting out from the firm support, to which is attached a quadrangle with holds, or we drill holes into this jutting protrusion in order to fasten spokes in them, with which we turn the screw. If we want to lift any load with this tool, we tie the ropes that are fastened to the load to the shaft on both sides of the wheel. Then we turn the screw that we had the cogs of the wheel mesh with, then the wheel will rotate with the shaft and that load will rise.

7 The manufacture of the five powers described earlier and their application we have just explained and elucidated. The reason why each of these machines moves big loads with a small force, we want to explain now as follows. Let us assume two circles around the same center, namely point <a>, whose two diameters are the lines <bg> and <de>. Let the two circles be mobile around the point <a>, their center, and let them be perpendicular to the horizon. If we now suspend at the two points and <g> equal weights, namely <h> and <z>, then it is clear that the circles do not incline to any side, since the weights <z> and <h> are equal and the spaces <ba> and <ag> are also equal, so <bg> is a scale beam that can be set in motion around the point of suspension, namely point <a>. If we now shift the weight at <g> and suspend it at <e>, then the load <z> will sink and set the circles in rotation. If we however increase weight <q>, then it will again keep the balance of weight <z> and load <q> then relates to load <z> like the distance <ba> to the distance <ae> and we thus imagine the line <be> as a balance that can be set in motion around the point of suspension, namely point <a>. Archimedes has proven this in his work on the balancing of inclination. From that it is evident that it is possible to move an immense bulk with a small force. For if one has two circles around the same center and the bigger load is on any arc of the smaller one, the smaller on any arc of the larger one, if however the ratio of the line starting from the center of the larger circle to the one starting from the center of the smaller one is greater than the ratio of the big load to the small force that moves it, then the small force counterbalances the big load.

8 Since we have now found this to be correct in our example with the circle, we now want to show the same for the five powers and, having done that, the proof will also have been provided for those. Incidentally, already the ancients, who were before us, have executed this introduction. Let us now prove it for the tool called a lever. The lever moves heavy objects in two ways: either being in a position parallel to the ground or by rising off the ground and standing inclined towards it. One applies it in pressing its end that is above the ground downward toward the ground. Let us first assume it to be parallel to the ground. Let the lever be the line <ab> and the load to be moved by it, namely <g>, at point <a>, the moving force at point , the stone [block] under the lever, on which it moves, at point <d> and let <bd> be greater than the line <da>. If we now lift the end of the lever that is at , so that the lever arm rises above the stone [block], around which the lever rotates, then the load <g> moves towards the other side. Then point describes a circle around the center <d> and the point <a> around the same center a smaller circle than the one described by point . If now line <bd> relates to <da> like the load <g> to the force at , then the load <g> keeps the balance to the force . If the ratio <bd> : <da> is greater than that of the load to the force, then the force has the superior weight over the load, because two circles exist around the same center, the load is at the arc of the smaller circle, and the moving force is at the arc of the larger one. Thus it is clear that the same phenomenon occurs with the lever as with the two circles around the same center. Thus the proof for the lever that moves loads is the same as the one put forth for the two circles.

9 Let us now assume a different lever, let it be the line <ab> and be movable around a Hypomochlion, namely <d>. Let the one end of the lever, namely point <a>, be under the load <g>, the other one raised above the ground, namely at point . If we now press down towards the ground the end of the lever at , then we have moved the load <g>. Now I say that in this procedure it does not move as in the first one. For in this procedure part of the load moves and the other one remains lying on the ground. Let us now imagine a plane going perpendicularly through the point <e>, forming a jutting part of the load, namely <ezq>. Let this load <ezq> be in balance with another part, <ehz>. If we now imagine this whole weight <ehq> separated from the load and put up in the place where it is located, it will not incline to any side, neither towards <q>, nor towards <h>, because the two weights <eqz> and <ehz> remain in balance with each other. Thus the part <ehq> does not need any force at all, and, consequently, it is only the part <ehkl> of the load that the lever moves. If the lever <ab> were to move all of the load <eqkl>, then <bd> would relate to <da> like the load <eqkl>to force ; it does not move all of it, however, because part of the load is held up by the imagined (perpendicular) plane, so let this part be, for instance, half [of it]. For if we do not imagine that (perpendicular) plane, so that the whole appears as scales around point <d>, and we add to the moving force an amount equal to that surplus (half of the load as mentioned above as an example), then the force will be pressed down and the end of the lever at <a> rises, because the loads spread out evenly over the moving forces (not proportionally, through which equilibrium would occur). Übersetzung fehlt (jutta) If now the force at keeps the balance of the load <ehkl>, then consequently <bd> relates to <da> like the load <ehkl> to force ; and a correspondingly smaller force is necessary as the load rises off the ground. This load finally assumes such a position that it does not require any force, if, namely, the perpendicular plane going through point <e> divides the load into two halves. This procedure with the lever too can be traced back to the circle; but it is not as in the first procedure. That the scales can also be traced back to the circle is clear, since the circle is a scales.

10 As for the shaft with the wheel, it is nothing else but two circles around the same center, one of which, namely the circle of the shaft, is small, the other, namely that of the wheel, is larger. Therefore, the suspending of the load rightly occurs at the shaft and the moving force is at the wheel, because in this procedure the small force counterbalances a big load. Our predecessors have already spoken this sentence; we have put it here so our work may be complete and have a well-ordered structure.

11 Let us now discuss the basis of the tool called block and tackle. Let us imagine an elevated wheel at point <a>, around which a rope (Hoplon), namely <bg>, is wound. Let the load, namely <d>, be tied to the two free ends of the rope, which are also elevated above the ground. Then it is clear that the two ends of the rope hanging down hang down equally far and each of them carries half of the load <d>; for of the suspended parts, if the piece hanging down were not equal, the higher one would pull up the one hanging farther down. We do not notice any of this, however, because each of the two ends of the rope hanging down remains still. If we now divide the load into two halves, i.e. into two equal parts, then we see that the two parts of the rope hanging down remain at rest, because the force tightening them is the same, namely the one that tightened them at first. Thus, half the load keeps the balance of the load equal to it. The two parts of the rope hanging down are equal also from another point of view, because equal weights are suspended from equal lines; for the tightened rope touches two points of the wheel opposite each other and whose distance from the center is the same; thus it is as if the two weights were suspended at these two points. In this procedure and in this way a heavy load or a big weight does not keep the balance of a small force, and, therefore, this kind of the tool called block and tackle is called simple pull. Thus this so-called simple pull is the one in which the rope hangs down doubled.

12 We now want to explain the double pull; it is the one in which three parts of the rope are strung. In the same manner, the more often one strings a rope back and forth, according to the number of repetitions the tool is named as of so and so many pulls, after one has subtracted one from the number of repetitions of the tightenings, so that the name gives the number that is by one smaller than that number, namely the number of repetitions of the rope. Let us now imagine the end of the rope situated at <d> running over a pulley and going to a firm support that is connected to the pulley <a>, namely to the point <h>, then the tightening of the strings is equal, for the reason given by us, because each of them pulls one third of the load. If now the load <z> is divided into three equal parts, so that the part falling to <qb> is twice that of <g>, then the load is at rest and nothing of it inclines towards any side, so that the weight suspended from string <g> keeps the balance of the one suspended from string <d>, while the latter is twice the other side. If we now attach instead of <g>, i.e., one third of the load, a force corresponding to the weight which holds the rope, then the remaining load will not counterbalance it, although it is smaller than that. The same occurs if we have the end of the rope at <h> run over a pulley fixed at <h> and pull it until its end is tied to the load <z> at point <k>; for each of the ropes carries one forth of the load's weight. If one divides the load again, so that the piece of it that falls to the points <qbg> is thrice that of the one falling to point <k>, then the weight at <k> keeps the balance of the remainder of the load and the number of tightened ropes that lift the load relate to the rope that pulls like the load to the (counter-)load. In general in all these loads the ratio of the known load to the force moving it has to be the same as the ratio of the strung ropes that lift the load to the ropes that are moved by the moving force. If, for instance, the load is fifty talents and the moving force five, then the tightened ropes that carry the load have to be ten times the ropes that are pulled by the force of five talents, so that the tightened ropes that carry the load are ten, while the rope at the moving force is one. If however the ropes that carry the load are twenty, then the ropes at the moving force are two. Under these conditions the force keeps the balance of the load. If however the force is supposed to counterbalance the load, then we increase either the force or the ropes that carry the load. Thus the proof for the pulleys called block and tackle is provided and we see from it that one can move a known load by a known force.

13 It happens that, in an operation, the rope that is folded and strung into only two strings is called now simple, now double pull, depending on the force that we apply. Let us, for instance, assume a pulley at point <a>, over which goes a rope, and let the two parts of the rope hanging down be at points and <g> and let and <g> be tied to any load, namely load <e>. If we now divide this load into two halves, then the two parts will on both sides keep the balance; this pulley is called simple pull, because the force here keeps the balance of the weight equal to it. Let us now imagine a different load at <z> and fasten to it the pulley <h>, pull over this pulley a rope and tie its two ends to a firm crossbeam, so that the load <z> floats, then each of the two tightened parts of the rope lifts the weight of half the load. If somebody now unties the one end of the rope tied at <k> and stands there himself and holds the rope, then he carries half the load and the whole load is twice the force that holds it. From that it is evident that from the firm support at the tied end of the rope another force, which is equivalent to the one holding the other end of the rope, also pulls the load. Therefore this pulley is rightly called a double pull. Consequently one can call the rope that is folded and divided into two parts and tightened simple and double pull. Therefore it is clear now that the other end of the rope has to be fastened to a firm crossbeam and not to the load given for lifting, because a certain force from that firm support keeps the balance of the moving force and helps it with moving the load. Thus it turns out that the load keeps the balance of a force equal to it if one end of the rope is tied to the load; if however the other end is fastened to a firm crossbeam, then the force keeps the balance of a load twice as big and the load can be moved by a lesser force than the first time.

14 As now for the wedge, the blow moves it in a certain time, because there is no motion without time; this blow only has an effect through the contact, which does not adhere to the wedge, not even for the shortest time. Thus it can be gathered from this that the wedge moves after the blow stops. We also learn this in another way. For a certain time after the blow there are noises and crackings coming from the wedge, from the cracking at its tip. That the blow, however, even if it does not, not even for the shortest time, adhere to the wedge, has an effect on it, can be gathered from rocks that one throws, or arrows, whether they are hurled by hand alone or by any other tool. For when the rock has left the hand, one sees it flying towards a place with might, without the hand pushing it any longer. We gather from this that the blow does not remain on the wedge for even the shortest time, that the wedge, however, starts moving after the blow.

15 Now I say that every blow, even if it is only light, moves every wedge. Let us assume some wedge, whose angle is at <a> and let its head be the line <dm>. Let the blow <bg> move it and let its distance (from the original position) be <ad>. Let it now be possible to move it with a light blow. Let us take from the blow <bg> a blow, for instance <be>, which is the smallest of all known blows, then I say that the blow <be> by itself moves part of the wedge. Proof: Since the blow <bg> causes the motion of the distance <ad>, <eg> causes a smaller motion than the distance <ad>; let it cause the distance <az>. If further the blow <be> is added and if the distance <ad> is caused by blow <bg>, so blow <be> by itself causes the distance <dz>. Let us now imagine the blow <bg> divided into blows equal to <be>, namely into <be>, <eh>, <hq>, <qg>, then the distance <ad> is divided into <dz> equal parts, namely <ak>, <kl>, <lz>, <zd> and so each of the blows <be>, <eh>, <hq>, <qg> causes one of the distances <dz>, <zl>, <lk>, <ka> respectively. Let us now imagine lines parallel to the head of the wedge, the line <dm>, namely <zn>, <ls>, <ko>; also parallel to line <ad>, namely <pn>, <xs>, <ro>, then the lines <dr>, <rx>, <xp>, <pm> will be equal to one another. If we now connect the points , <x>, <r> with point <a>, then four wedges are formed whose points are at point <a> and whose heads are the lines <mp>, <px>, <xp> and <rd>. Each of them is moved by a blow equal to blow <be> for a distance equal to line <ad>, and thus it is the same whether one says the blow <be> makes the whole wedge penetrate for the distance <dz> = <ka> or the blow <be> makes the wedge whose head is the line <rd> penetrate for the line <ad>, because in the penetration of the whole wedge the line <ok> shifts for <ak> and in the penetration of the wedge, whose head is <dr>, a distance equal to line <ko> is covered, namely the space <dr> for the distance <ad>; consequently <rd> moves through the blow <be> for <ad>. From this it is evident that the amount of the blow <be> of <bg> corresponds to the wedge, whose head is <dr>, of the whole wedge. The same holds for the time within which the wedge, whose head is <dr>, moves and for the amount of the distance that the whole wedge covers through the blow <bg> and this ratio is the same as that of the blow <be> to the whole blow. Also, from another point of view we do not find any difference between the motion of blow <bg> to <dm>, i.e., to the whole wedge and between the motion of each of the individual blows <be>, <eh>, <hq>, <qg> to each of the individual wedges, whose heads are <mp>, <px>, <xr>, <rd>, because the partial blows equal the whole blow. Thus the blow <be> drives in the wedge with the head <mp> as far as the whole blow drives in the whole wedge and each individual of the remaining blows each individual of the remaining wedges. If the wedge to be driven in is one of the small wedges and is driven in through a fierce blow, then it is driven in for as much as the whole wedge through the totality of the blows. This occurs through the corresponding amount of blows, namely through the amount of the blows <be>, <eh>, <hq>, <qg>; accordingly the ratio of time to time is like that of blow to blow and that of the whole wedge head to the head of the individual small wedge. The smaller now the angle of the wedge becomes, the smaller can also be the force in relation to the force that drives in the whole wedge.

16 After this it remains to explain the effective cause in the screw. First let us start by explaining what shows in screw threads. Thus we say: if we want to construct a screw, we take a strong hard [piece of] wood of the length corresponding to our purposes; let the part we want to make into a screw be turned and its density be even in all parts, so that its surface is a cylinder, and let us draw on its surface one side of the cylinder. Let us now divide this side into equal parts, corresponding to the height of the screw groove and let us assume in a plane two straight lines, one of which is perpendicular to the other one, and make one of them equal to the circumference of the cylinder, the other one according to the height of the screw groove, connect the two end points of the two lines through a line opposite to the right angle and make from thin brass a triangle equal (congruent) to this triangle, of such thinness that we can bend it as we want. After doing this, we put the side that is equal to the height of the screw groove on the first of the equal spaces that we have divided up on the side of the cylinder, then we wind the thin brass triangle around the cylindrical [piece of] wood and let the remaining acute angle of the triangle reach the right angle of the brass figure, because the baseline of the triangle equals the circumference of the cylinder. Then we tack together the two angles and draw the screw thread according to the line opposite the right angle. After that we turn the triangle to the second space and put the height of the triangle on the second space. In the same manner as before we draw also the second screw thread in immediate connection with the first one. We proceed in the same manner until we have drawn all spaces of the cylindrical [piece of] wood. But since we needed, in using the screw, to put the [piece of] wood called Tylos into the first depression of the screw thread, and it is the one that lifts the load, thus this [piece of] wood rises at the rotation of the screw and the load rises with it.

17 We have however to imagine the screw just as a twisted wedge, because the triangle which determines the screw thread has the form of a wedge; its head is the side that represents the height of the screw thread and the acute angle of the wedge is the remaining angle of the triangle, in which is the [piece of] wood called Tylos. Therefore the screw is a twisted, wound up wedge that does not have an effect by striking, but by its rotation. Turning here takes the place of the striking of a wedge, so it lifts the load. By lifting the load it has the opposite effect to the wedge, because the wedge only has an effect by penetrating into the inside and so moving the load, while the load remains in its place; the screw however is a twisted wedge that lifts the load towards itself while remaining in its place. As it has been proven for the wedge that the one with the smaller angle moves the load by means of a lesser force than the one that moves the load by means of a wedge with a larger angle, we have to say as well for the screw in which the spaces between the screw threads are smaller that it moves the load easier than the screw in which the spaces between the threads are larger, because the smaller space causes a smaller angle. Therefore the screw whose threads are steeper moves the load by means of a bigger force, while the shallow screw moves the load by means of a smaller force.

18 If now a wheel with cogs meshes with the groove of a screw, then the screw moves, with each rotation it makes, one cog of the wheel further. This we are going to demonstrate in the following way. Let us imagine a screw, let it be the screw <ab> and let its screw grooves be <aq>, <de>, <zg> and let each single one of these threads be single. Let us now imagine a wheel with cogs put to it, namely <hgeq> and let its teeth <hd>, <ge>, <eq> be fitting to mesh with the screw grooves. Let the cog <ge> mesh completely with a screw groove, then the remaining cogs will not mesh with the other screw grooves. If we now turn the screw until the point <e> is brought to the position of <g>, then <e> falls on <g>. If thus the screw makes one rotation and the cog <ge> comes to the position of cog <gh>, the cog <eq> to the position of cog <ge>, and the cog <eq> now takes the position of <ge>, then, in the rotation that the screw makes, the entire space of the cog shifts. In the same manner we have to imagine the procedure with the other cogs. Thus as many cogs as there are on the wheel, as many rotations the screw makes, until the wheel has made one rotation.

19 When the screw rotates, it moves the [piece of] wood, according to what was said before, and lifts the load in a straight direction. When the screw is not moving, this Tylos has to remain still and fixed in its position by some force having an effect on it, so the load, when the screw stops rotating, does not gain superior weight over it, i.e. when this [piece of] wood meshes with the screw thread and is like a support for it, it must not slide out of the screw thread, because, when it slides out, the whole load drops towards the position it had been lifted from. This [piece of] wood does not slide out of the screw thread, if its end fits right into the groove and the groove is like a boot for it. Therefore we have to place the screw grooves close to each other so they become almost parallel to the base of the cylinder that the screw is constructed on. If the threads are laid out like that, then they are like a boot for the [piece of] wood that lifts the load. If however the screw grooves in the screw threads are very steep, so that they are almost parallel to the side of the cylinder, then the [piece of] wood called Tylos holds back the rotation of the screw when a heavy load is suspended from it or a great force presses it, and causes a rotation opposite to the first one. From this it is evident that the screw can set the [piece of] wood called Tylos in motion as well as it can be set in motion by this [piece of] wood; it will move the [piece of] wood when the screw thread consists of threads that are close to each other; when the screw stops rotating the same remains in its position and the load remains suspended from it. If, however, the screw threads are very steep and the Tylos does not stand still when the rotation of the screw stops, then it is this [piece of] wood which moves the screw, because, if a rope is attached to the place of the screw that does not have a thread and a weight is tied to this rope and the screw thread is very steep, we lift the weight too when we lift the [piece of] wood called Tylos; when we stop lifting the [piece of] wood, however, the weight is still and remains hanging; for this [piece of] wood resists the screw thread, if this thread is almost parallel to the side of the cylinder. If now there is no screw thread on the cylinder, but along one side of the cylinder a canal is made, then this [piece of] wood called Tylos is a particular resistance for this canal. If however the screw threads are very close together and we lift the [piece of] wood called Tylos, then we shall not lift the weight if a great force does not make the Tylos rise. If now the load is suspended from the Tylos and we turn the screw while the screw threads are very close to each other, then the load rises, and if we stop turning the screw, then the load is still and remains balanced. If however the screw threads are steep, then we do not move the load except if we have a great force that overcomes the screw. With this we have talked enough about the nature of the screw and its use.

20 That the five powers that move a load are similar to circles around one center is proven by the figures that we have designed in the preceding; but it appears to me that they look more similar to the balance than to the circles, because in the preceding the bases of the proof for the circles resulted from the balance. For it was proven that the load suspended from the smaller side relates to the one suspended from the larger side like the larger scale beam to the smaller one. For all of these five powers there is in practice one hindrance if we want to move with them big loads by means of a small force. The first three require that we increase their size according to the size of the load we want to move, namely the wheel on the shaft, the lever and the block and tackle; the two remaining ones, namely those that are formed through the wedge and the screw, require that we decrease their size in the same ratio. If, for instance, we want to move a load of one thousand talents by a force that corresponds to five talents and use for this motion the wheel on the shaft, then the line going from the center of the wheel to its circumference has to be a little more than two hundred times the size of the one going from the center of the shaft to its circumference. If we use, however, the lever in this, then its larger arm, which goes to the force moving the load, has to be according to this ratio or a little larger. A procedure with these tools is difficult and almost impossible; for if we make the diameter of the shaft one half of a cubit, so that it is strong enough to suspend the load from it, then we have to make the diameter of the wheel one hundred cubits or a little larger. Doing this is difficult, however. The same holds for the lever and the block and tackle, because we cannot set up the division of the lever in this manner and not set up the number of pulleys according to this amount. Let us now consider how the hindrances that occur with these three machines can be remedied.

21 Now we say that the circle possesses among all figures the greatest and easiest mobility, whether the circle moves around a center or on a plane on which it stands perpendicular. The same holds for the figures related to it, the spheres and the cylinders; for their motion is a rotating one as we have proven in the preceding book. Let us now assume that we first wanted to move a big load by means of the wheel on the shaft by a small force, without that hindrance occurring. Let the load, that we want to move, be one thousand talents and the force, with which we want to move it, five talents. Now we first have to bring the force into balance with the load because, when that occurs, we are able to have that force gain the superior weight of that load, by attaching a little excess to the tool. Let us now attach the shaft that the rope fastened to the load winds up on, at point <a> and let the wheel fitted to it be at . In order to make the manufacture of the tool easier for us, we make the diameter of the wheel five times the size of the diameter of the shaft. Doing this it is necessary that the force moving the wheel , which keeps the balance of the load of one thousand talents, be two hundred talents; the assumed force that we have is, however, only five talents. Thus we cannot move the assumed load by this force with the wheel . Let us therefore make a cogged shaft, namely shaft <g>, which meshes with the cogs of wheel , so that, when shaft <g> moves, through its motion the cogwheel starts moving together with the first assumed shaft, so that the load moves when shaft <g> rotates. This shaft can be set in motion by the force that moves cogwheel , because we have proven that all objects moving around particular centers can be moved by a small force. Thus there is no difference whether the load is moved by the cogwheel or by the shaft <g>. Let there further be on shaft <g> a wheel attached to it, namely <d>, whose diameter is, for example, five times that of the shaft <g>, then the force that in wheel <d> keeps the balance of the load has to be forty talents. Let us further assume another shaft, namely <e>, that meshes with this cogwheel, then the moving force in <e> will also be forty talents. Let there now be another cogwheel, which is attached to the axle <e>, namely the wheel <z>, and let its diameter be eight times that of the diameter of axle <e>, because the force of forty talents is eight times the force of five talents, then the force in <z>, which keeps the balance of the load of one thousand talents, will be five talents, as was given. For the force to gain the superior weight of the load, however, we have to make the wheel <z> a little larger or the axle <e> a little thinner. If we do this, the force counterbalances the load. If we want to use in this procedure even more wheels and axles, then we have to apply the same ratio, because all ratios have to correspond to the load, if we want to make the force balance the load. If we want it to counterbalance the load, however, we have to give to all the ratios together an excess over the equilibrium of the load. Thus through the axle that goes through a wheel a load can be moved in this manner. If we do not want to make the wheels cogged, however, then we wind ropes around the axles and wheels and the same work can be done, because through the wheel that moves last the first axle, which pulls the load, is moved. Using this kind of wheels and axles has to happen on firm supports that have holes that the ends of the axles go into. These supports have to be set up in a secure, firm place, when the load is lifted.

22 A delay occurs however with this tool and those similar to it of great power, because the smaller the moving force is in relation to the load to be moved, the more time we need, so that force to force and time to time are in the same (inverse) ratio. An example for this is the following: Since the force in wheel was two hundred talents and it moved the load, one requires one rotation for the rope wound around <a> to wind up, so that the load through the motion of wheel moves the amount of the circumference of <a>. If it is moved, however, through the motion of cogwheel <d>, the wheel on <g> has to move five times for the axle <a> to move once, because the diameter of is five times the diameter of the axle <g>. Thus five times the amount of <g> is equal to a single , if we make the respective axles and wheels equal to one another. But if not, then we find a proportionality similar to this one. The cogwheel <d> moves at and the five revolutions of <d> take fivefold the time of a single one (of ), and the two hundred talents are five times the amount of forty talents. Thus the ratio of the moving force to the time is inverse. The same shows with multiple axles and multiple wheels and is proven in the same way.

23 Now we are supposed to move the same weight by the same force through the tool called block and tackle. Let the weight be called <a>, the place that it is pulled away from and the place opposite to it <g>, which is the firm point of support to which we want to lift the weight. Let the block and tackle have for instance five pulleys and let the pulley, from where the load is pulled, be at point <d>, then the force at <d>, which keeps the balance of the one thousand talents, has to be two hundred talents; the force given to us is, however, only five talents. Let us therefore pull a rope from the pulley <d> to a block and tackle at point <e>, and let opposite to it be a firm point of support at <z>; let there be five pulleys at this firm point of support and in its vicinity at point <e> and let the pulley be at <h>, then the force at <h> has to be a force of forty talents. Let us pull again the end of the rope at <h> to another block and tackle at <q> and let the firm point of support be at <k> and let [it] be pulled at <k>, then, because forty talents are eight times the amount of five talents, the block and tackle will have to have eight pulleys, so that the force at <k>, which keeps the balance of the one thousand talents, is five talents. For the force at <k> to gain the superior weight of the load however, the pulleys have to be more than eight in number; then the force will counterbalance the load.

24 That the delay also occurs with this tool is clear because the process takes place in the same ratio. For if the force at <d>, which is two hundred talents, lifts the load from to <g>, then it wants to wind up the five ropes strung around the five pulleys the amount of the distance between the points and <g>, while the force at <h> has to wind up the five ropes five times. If we now make the distances <bg> and <ez> equal to one another, then, while winding up one of the ropes of the distance <bg>, it winds up five of the ropes of the distance <ez>, because, if the load moves the distance between and <g>, five ropes have to be wound up the amount of the distance <bg>, so that time relates to time (inversely) like moving force to moving force. So that the ropes do not become too numerous, the distance <ez> has to be five times the amount of the distance <bg>, and <qk> eight times the amount of <ez>. In this procedure the blocks and tackles lift together.

25 By means of the lever, also the same load can be moved by the same force according to the same procedure. So let the load be at point <a> and the lever be <bg>, the Hypomochlion at point <d>. We move the load by means of the lever, which is parallel to the ground, and we let <gd> be five times the amount of <db>. Thus the force at <g> which keeps the balance of one thousand talents will be two hundred talents. Let there now be another lever, namely <ez>, and let the point <e>, the head of the lever, butt against the point <g>, so that with the motion of <e> <g> moves too; let the Hypomochlion be at point <h> and let (the lever arm <e>) move towards <d>; let further <zh> be five times the amount of <he>, then the force at <z> is forty talents. Let now another lever be present, namely <qk> and let us connect the point <q> with the point <z> and let this move in the opposite direction of <e>; let further the Hypomochlion be at point <l> and <kl> be eight times the amount of <lq> and let this move in the direction that <e> does not move in, then the force at <k> is five talents and keeps the balance of the load. However, if we want the force to counterbalance the load, then we have to make <kl> greater than eight times the amount of <lq>. Thus if <kl> is eight times the amount of <lq>, <zh> five times the amount of <he>, and <gd> greater than five times the amount of <db>, then the force will counterbalance the load.

26 Here, too, the delay shows in the same ratio, because there is no difference between these levers and the shafts that go through wheels and move around centers. For the levers are like the shafts, by moving around points <d>, <h>, <l>, namely around the stone [block]s around which the levers rotate. The axle circles are the circles described by points , <e>, <q> and the wheels are those circles that are described by points <g>, <z>, <k>. Just as we have proven for those axles that the ratio of force to force is (inverse) that of time to time, in the same way we prove it here.

27 For the wedge and the screw we cannot put forward that claim, however, because, as we have proven in the preceding, with these no hindrance occurs, but the contrary of that, the greater the force becomes with the two of them, the smaller each of them becomes. Our purpose was, however, to reflect on the machines that become larger with the increase of the load in order to be able to work on them with small machines and so it might become easier. Thus, for the screw and the wedge we do not have to think about their reduction to be able to work more easily with them.

28 That the delay also occurs with these two is clear, because many blows take more time than a single one and the frequent rotations of a screw takes more time than one rotation. Thus we have proven that the ratio of wedge-angle to angle is (inverse) that of moving blow to blow. Then the ratio of time to time is also (inverse) that of force to force.

29 In the preceding, we have moved the known load by means of many shafts with wheels, many combined levers and many block and tackles. We can, however, also move the known load by a fusion of those and by a combination of individual ones, except for the wedge, because this is only moved by blows. Let us now prove that we can combine the four powers and by their fusion move a known load. Let the known load be at point <a> and let at points <bg> be a lever; let the point be the one underneath the load and the point <g> be raised; let the Hypomochlion be the point <d> and <gd> be five times the amount of <db>; then the force at <g> is two hundred talents, so it keeps the balance of load <a>. Let us attach to the end of the lever at point <g> a block and tackle that is at <e>, and let the other part of the tool be parallel to it on a firm support, namely at point <z>. Let the point of attack of this tool be at point <g> and let this have five pulleys; then the pulling force is forty talents. Let now also a shaft with a wheel be present, namely <qk>. and let the shaft be designated <q>, the wheel be designated <k>, and let the rope that runs over the pulleys be wound around the axle. Let the wheel be cogged and let it stand perpendicular on the given plane. Let a screw mesh with its cogs, namely the screw <l>, with a handle that sets it in rotation, and let the cogs mesh with the screw thread. If we now set the handle <m> in rotation, the screw <l> rotates, and at the same time as the screw the wheel <k> rotates; through its rotation, the axle rotates, and the rope of the pulleys winds up on it, presses down the end of the lever at <g>, and the load rises. Let now the diameter of the wheel <k> be four times the measure of the diameter of the axle <q>, so that the force in <k> is ten talents, and let the spoke be twice the amount of the diameter of the screw cylinder, then the force at <m>, which keeps the balance of the one thousand talents, is five talents. If we extend the spoke at point <m> a little, however, then the force of five talents predominates. Let the wheel with the shaft and the screw be fastened in a frame of a kind of a box, so that the ends of the axle rest in the perpendicular walls of the frame, the lower end of the screw rotates in the bottom of the frame and its upper end in the center of the upper plane. One makes this end square and attaches to it a disc into which the spoke is fitted. Let this firm box-like frame be in a solid, well founded place of strong firmness. When the spoke is turned, the load rises.

30 For the wedge and the screw, we apply the following procedure. Let the angle of the wedge we want to make be <abg>, namely an acute one. Then I say that the wedges, whose angles are more acute, move the load through lesser blows, i.e., by means of a smaller force, and they may attain such smallness that they cannot be used because of their tip. Let us draw a line perpendicular to <bg>, namely <bd>, so that the wedge may become effective. Further, a parallel to <bg>, namely <de>; let us now draw through the point <e> a line under a right angle, namely <eg>, and one makes a wedge, like the one just determined, namely <abde>. Let us drive in its side <bd> so that a small part of it comes underneath the load, and let its head be <ae>, then we see that, when the wedge <abg> is driven in, it also drives in <abde>. Proof. If we extend the two lines <ab> and <de> towards <z>, then the angle <aze> becomes equal to the angle <abg>; thus, <aze> is also a wedge that can be moved by the same force. Let us now imagine the part of it that is situated at <bzd> underneath the load, then the wedge is driven in. Thus, this is the proof for the wedge. However, it is not absolutely necessary to employ acute angles in the wedge, because we have just proven that every slight blow can move every wedge, if the blows fall in a great number. We do, however, use the acute angle because of the slight blows. Thus it is not absolutely necessary to employ small angles in the wedge.

31 We cannot proceed in the same way with the screw. Therefore we have to attach to the angle of the screw groove, namely <abg>, a perpendicular <ag> on <bg>, equal to the thickness of the Tylos that we want to have mesh with the screw thread, and make a cylinder, whose circumference equals line <bg>. Let us now construct from these lines a screw thread of the height <ag> and hollow out the screw groove, whose space is equal to line <ag>, then we can, according to this procedure, fit that [piece of] wood into the screw thread.

32 Since we have just proven for each one of the individual powers that by a given force a given load can be moved, we have to add that, if all machines to be constructed could be turned with a file, even in weight, evenness and smoothness, one could apply for each individual one of them the procedures mentioned, according to those ratios. Since it is however not possible for humans to make them in perfect smoothness and evenness, one has to increase the forces because of the friction of the machines that occurs, and enlarge them, by building them in larger scale than according to those ratios that we have mentioned, so that no hindrance occurs, while our observation of the use of the tools declares incorrect that for which the proof has just been found correct.

33 It is now absolutely necessary for those who occupy themselves with the science of mechanics to know the causes that are in effect in the use of each motion, as we have explained for the lifting of heavy objects with natural proofs, and set out everything that occurs with each individual of the powers mentioned, so that nothing unproven for them happens about which they are in doubt, but the truth of it for each single one we mentioned appears for them, when they look intently at each of their tasks. Now we want to talk of things that the ancients already stated, because of the usefulness they have in this chapter, and we are going to be amazed at the things that, when we have proven them, will be the contrary to what we had knowledge of before. The beginning for the things that we are going to research, we derive from what is clear to us. The things of whose causes we can only talk after the most clear objects will, however, even increase our amazement when we see that the things that we apply are the contrary to what we have gotten used to and what was certain for us. It is now clear that anyone who wants to find the causes thoroughly, necessarily has to apply natural principles, either one or more, and has to link everything that he researches with them and that the solution of every single question is given fundamentally, if its cause has been found and this is something that we have already understood. Let the principle now hold for us that the light [object] is easy to move, the heavy one is hard to move and that the same load can more easily be moved by a greater force than by a smaller one; for the one results from the other and is clear and obvious to us. We have to know, however, that there is in every question something dark, not obvious, because there is almost never a question about anything whose cause is clear and plain. Furthermore, it has to be taken into consideration that all questions that arise in mechanics, and in which there is a darkness concerning the cause, result from the fact that we cannot see how heavy bodies are distributed among the forces moving them; this reason becomes obvious through many circumstances, in particular, however, through the motion of these bodies. For a body that one man does not move or which, if one man moves it, becomes too heavy for him, is moved by a number of men and it is easy for them to move it. If it would be the case that all of the load to be moved were lying on each individual of those moving [it], then there would be no difference in the motion, between the motion of the individual and the motion of the totality. We see, however, that the motion is easier for the totality. And since some of the load falls to each individual of the totality and motion becomes easy for them, it is clear that the load is distributed among those that carry it.

34 Questions. a. Why do wagons with two wheels carry the loads more easily than wagons with four wheels? Because the load on wagons with two wheels is distributed in two equal parts to both sides of the axle. That won't do for wagons with four wheels; the load cannot be distributed in a way that the two parts of it are equal on both sides, but the entire load lies in front of the rear wheels and behind the front wheels, and the difference in the position determines the speed of the motion of the wheels; for the wheel only has a fast motion, since its load rests equally on all of its parts. b. Why is the pulling of a wagon in sand hard for draught animals? Because part of the curvature of the wheels is in the groove of the sand and, when the wagon is pulled, the sand, that is in front of the wheel, supports it. Furthermore it is difficult because the feet of the animals go into the sand and it is hard to pull them out. This, however, does not happen on hard ground. c. Why does the same weight cause different inclination in balances that are in equilibrium, in a way that there is a greater inclination with a lesser load? If one has, for example, two scales with three mines in each, and we put another half mine into one of the two scales, then this scale inclines very strongly. If there are, however, ten mines in each of the scales, and we add half a mine to one of the scales, then the inclination of the beam is very small. Because the first case shows that the load is moved by a great force, the three mines being moved by the same plus a sixth of it; the ten mines, however, are moved by the same plus a twentieth of it. For one half mine is the twentieth of ten, but a sixth of three mines, and the load that is moved by the greater force is easier to be moved. d. Why do big loads fall to the ground in a shorter time than lighter ones? Because, as it holds for them that they can be more easiy moved if the force moving them from outside is greater, they also move more quickly, if the force within them is greater. The force and the attraction are, however, greater for the bigger load in natural motions than for the smaller load. e. Why does the same weight, if it is wide, fall more slowly to the ground than if it is spherical? Not because, as some believe, the wide one in its breadth meets with a lot of air, the spherical however, because its parts meld into each other, only meets with little air, but because the load that sinks wide has many parts, to each of which, according to its breadth, comes a part of the force, so that in the motion of this load each of its parts receives some of the moving force according to its weight, but one force does not meet it as a whole. f. Why does a shot from the center of the string propel the arrow into a great distance? Because the tautness there is the strongest and the propelling force the greatest. That is why one makes the bows from horns, because here bending is possible. When they are strongly bent, the string with the arrow is more strongly drawn, so that a greater force gets into it and it therefore covers a longer distance. That is why hard bows whose ends cannot be bent only propel the arrow a short distance. g. Why can wood be broken more quickly if one's knee is brought into its center? Because, if one's knee is brought into a smaller distance (form one end) than the center, so that one of the two parts is shorter than the other one, then it is a balance divided into two unequal parts, which is why the hand that is further from the knee has the superior weight over the one closer to it. The one reaches, however, the force of the other one only when both are at the end of the [piece of] wood (at an equal distance from the center). h. Why is a piece of wood, the longer it is, the weaker and why does its bending increase when it is raised at one of its two ends? Because in the long [piece of] wood great force is distributed over its parts so that the whole has the superior weight over its solid part, on which it rises. Therefore, here the same phenomenon occurs as in a short [piece of] wood, when from its ends something is suspended that presses it down. Thus the increase in length of the [piece of] wood corresponds to the weight that pulls down the shorter [piece of] wood. Therefore the same [thing] happens to the long [piece of] wood, because of its length, as [happens] to the short [piece of] wood, when something heavy is tied to its end. i. Why does one use pliers when pulling teeth, and not the hand? Because we cannot grip the tooth with the entire hand, but only with part of it; and just as it is harder for us to lift a weight with only two fingers, than with the entire hand, so it is also harder for us to grip the tooth with two fingers and to press, than with the entire hand. In both cases the force is the same but the division of the pliers in its nail causes the hand to have the greater strength than the tooth; for it is a lever, at the greater part of which is the hand, and the space of the pliers makes the moving of the tooth easier. For the root of the tooth is that around which the lever moves. But since the space of the pliers is greater than the root of the tooth, around which something big moves, the hand outweighs the force resting in the root of the tooth. For there is no difference between the moving of a weight and the moving of a force which equals that weight. For if we close the hand after it was stretched out, then a resistance occurs, not because of the weight of the hand, but because of the force with which the muscles adhere to one another. k. Why can balances, be they loaded or not, be moved faster if one turns them (horizontally) than they move to one side, towards which one inclines them? Because, when one turns them, their weight remains positioned similarly and evenly in all directions, so that it thus rotates around a center, namely the point of suspension. But if we pull the balance to one of the two sides, then we lift a load, because the drop of the one scale pushes the other one up, so that its motion is not natural, I mean the motion upward of a load; for the natural motion is easy, namely the attraction downward of a weight. Therefore it is easier to pull weights downward than to lift them upward. l. Why is it easy to move suspended weights? Because the entire force of the weight is outweighed by the force through which it was suspended. Thus, because they do not have great force left, it is easy to push them. The same we also see in scales; when it is suspended and we pull it, it moves very easily. m. Why are the stone [block]s of considerable size that are found at the shore of the sea mostly round? Because they were first sharp-edged, but through the motion of the sea one bumps into the other, so that the edges break against one another because of their weakness. n. Why is it harder to move suspended loads that one wants to move, the further one removes the hand from them, until it reaches the firm support that they are suspended from, or comes close to it? Because, if we want to move them in the firm place that they are suspended from, this is very hard for us, and completely impossible. But if the hand moves away from the firm support, then the weight moves, though with difficulty, namely because it is still too close to the point where motion stops completely. Yet the further the mover goes away from the firm support, the easier the moving becomes to him. Let us, for instance, imagine the firm support that the load is suspended from at point <a> and let the rope be the line <ab>. Let us now draw the line <ag> perpendicular to <ab> and let us assume on line <ab> two randomly positioned points, <d> and <e>, and pull the rope at point <d>, then we brake it, until it takes the form of <azh>. Then the load is at <h>. Now I say that <h> lies higher than . Proof. If we extend line <hz> towards <g>, then, since <azh> is greater than <gzh>, point <h> is higher than point . Let further the rope to be tightened at point <e> have a position perpendicular to <ag> so that the load is again in the same place, namely the same as <ab>. Since now <ae> is greater than <az>, <e> will come to rest lower than <z>, at approximately <q>. If we now pull <aq> then <ab> is broken towards <aqh>. Now I say that the suspended weight comes lower than <h>. Proof. Since <az> plus <zq> is greater than <aq>, then, if <hq> is added to both sides, <az> plus <zh>, i.e., <ab>, is greater than (<aq> + <qh>). Let now (<aq> + <qk>) equal <ab>, then the load comes to <k>, and <k> lies lower than <h>. Thus if we pull the load from <e>, it comes to <k>; if we pull it, however, from point <d>, then it reaches <h>, so that the load is lifted higher from point <d> than from point <e>. The load, however, that is lifted to a higher point, strains the force more than the one lifted to a lower point, because the one lifted to a higher point takes more time. o. Why do objects that are floating on water have a greater speed if they are lying on only one side? Because the part above water is very light, so that the water supporting it is also very little and the wind that hits it, has greater strength than the water that resists its motion. p. Why does the rudder, although it is very small, deflect big ships? Because a man who runs and whom someone pulls to any side, quickly turns towards that side. The rudder however is supported by the water, so that it has greater strength than the ship. q. Why do arrows penetrate into coats of mail and armor but not into spread-out canvas? Because the weapon, when it hits an object that gives way to it and does not offer resistance to it, does not have a great effect, since the speed and the size of the force splits when hitting a yielding and non-resistant object. If, however, something hard hits something equally hard and gives it a blow, then the hard object does not give way and offers resistance, so that nothing of the force splits, but the impact is a strong one. For the same reason there is no damage to those who fling themselves into the water from a great distance. r. Why do liquids that are heavy in nature move quickly with ease? For we see that a single man moves one thousand Kist of water at the same time. Because water is a homogenous object, whose parts can, however, quickly be separated. Therefore it does not have any firmness in itself but it flows downward. That's the reason why we move only a small part of it and the remaining parts incline towards the place where the small part of it has been brought.

35 Now we still have to explain some things that we require for pull and pressure, but not of the kind mentioned in the last book, rather, of greater importance than those, things that Archimedes and others have already clarified. First now we want to show how one finds the center of gravity of an evenly thick and heavy triangle. Let the known triangle be the triangle <abg> and let us divide the line <bg> at point <d> into two halves and connect the two points <a>, <d>. If we now put the triangle onto the line <ad>, then it does not incline to any side, because the triangles <abd> and <adg> are equal. If we further divide the line <ag> at point <e>, and connect the two points , <e>, then put the triangle onto line <be>, then it does not incline to any side. Since now the triangle, when put onto each of the two lines <ad> and <be>, is in balance in all its parts and does not incline to any side, then the common point of intersection is the center of this weight, namely the point <z>. We have, however, to imagine the point <z> in the middle of the thickness of the triangle <abg>. Now it turns out that if we connect the two points <a>, <d> and divide the line <ad> at point <z> into two parts in a manner that one of them, namely <az>, is twice the amount of <dz>, that point <z> is the center of gravity; for if we connect the two points <d>, <e>, then the line <ab> is parallel to line <de>, since the two lines <ag> and <bg> were bisected at the points <d> and <e>. Then <ag> relates to <ge> like <ab> to <ed>; <ag> is, however, twice the amount of <ge>; consequently, <ab> is twice the amount of <ed>. Furthermore, <ab> relates to <ed> like <az> to <dz>; consequently, <az> is twice the amount of <zd>, because the two figures <abz> and <dze> equal one another in their angles.

36 We want to find the same for the quadrangle. Let thus the given quadrangle be <abgd>. Let us draw <bd> and bisect it at point <e>, connect the two points <a>, <e> and <e>, <g> respectively and divide the connecting lines at points <z>, <h>, so that <az> is twice the amount of <ze> and <gh> twice the amount of <he>, then the center of gravity of the triangle <abd> is at <z> and the center of gravity of the triangle <bdg> at point <h> and we do not find any difference, if we imagine the entire weight of the triangle <abd> at point <z> and as well the weight of the triangle <bgd> at point <h>. Thus the line <zh> is a balance at the ends of which are these two quantities. If we now divide the line <zh> at point <q> so that <qh> relates to <zq> like the load <z>, i.e., the weight of the triangle <abd>, to the load <h>, i.e. the weight of the triangle <bdg>, then the point <q>, at which the two loads are keeping the balance, is the center of gravity of this quadrangle.

37 We want to prove the same for the pentagon <abgde>. Let us draw <be> and determine the center of gravity of the triangle <abe>; let it fall on point <z>; let the center of gravity of the quadrangle <bgde> be at point <h>. Let us connect the two points <z> and <h> and divide the line <zh> in two parts so that <hq> relates to <qz> like the weight of the triangle <abe> to the weight of the quadrangle <bgde>, then the point <q> is the center of gravity of the figure <abgde>. We have to imagine it the same way for all polygons.

38 If <abg> is an evenly thick and heavy triangle and under the points <abg> there are supports in the same position, we want to show how to find the amount of the weight of the triangle <abg> that each of them bears. Let us bisect <bg> at point <d> and connect the two points <a> and <d>, divide the line <ad> at point <e> so that the part <ae> is twice the amount of <ed>, then the point <e> is the point of the entire weight of the triangle. Now we have to distribute it on the supports. But if we imagine the line <ad> in equilibrium when it is suspended at point <e>, then the load at <d> is twice the amount of that at <a>, because the line <ae> is twice the amount of line <de>. And if we imagine the weight at <d> distributed to the two points , <g> and the line <bg> is in equilibrium, then at each of the two points , <g> rests half of the weight that is at <d>, because the two lines <bd> and <dg> are equal to one another. The weight at <d> was, however, twice the amount of the weight at <a>; consequently, the loads at points <a>, , <g> are equal to one another and thus the supports bear equal weights.

39 Let further the triangle <abg> be evenly heavy and thick, on supports in the same position, and let any weight be put on or suspended at point <e>, in fact, let point <e> have any random position, then we want to find out, how much of the weight at <e> each of the supports bears. Let us draw <ae> and extend it towards <d>, divide the weight at <e> so that, if the triangle lies on line <ab> in equilibrium, the load at <d> relates to the load at <a> like the line <ae> to the line <ed>. Let us further divide the weight at <d> so that <bg>, if it is suspended, is in equilibrium, then the weight of <g> relates to the weight of like the line <bd> to the line <gd>. The weight at <d> has been determined; consequently the two weights <g>, are determined. But the weight at <a> has also been determined; consequently the weights that rest on the supports, are determined.

40 If a triangle <abg> is given and known weights are suspended at points <a>, , <g>, we want to find in the interior of the triangle such a point that the triangle, if it is suspended at it, is in equilibrium. We divide the line <ab> at point <d> so that <bd> relates to <ad> like the weight at <a> to the weight at . Then the point for the total weight of the two loads is at point <d>. If we now connect the two points <d> and <g> by the line <dg> and divide it at point <e> so that <ge>relates to <ed> like the weight of <d> to the weight of <g>, then the point <e> is the point for the total weight of all and therefore the point of suspension.

41 We want to show the same for a polygon. Let the figure <abgde> be a polygon. Let us suspend known weights at the points <abgde> and divide the line <ab> at point <z> so that the line <bz> relates to <za> like the weight <a> to weight , then the point <z> is the center of gravity of the two weights at <a> and . Let us also divide the line <de> at point <h> so that the distance <dh> relates to <he> like the load <e> to the load <d>, then the point <h> is the point for the total weight of the two points <e>, <d>. Let us now draw <zh> and divide it at point <q> so that (<a> + ) relates to (<d> + <e>) like <hq> to <qz>, then the point <q> is the point for the total weight of <abde>. Let us yet connect the two points <g>, <q> by the line <gq> and divide it at point <k> so that <gk> relates to <kq> like the total weight of <abde> to the weight of <g>, then the point <k> is the point for the weight combined from all of them.

End of the second book of Heron on the lifting of heavy objects. Übersetzung fehlt. Jutta

Third book

1 In the preceding book we have talked about the five powers and have explained the causes by which big loads can be moved by means of small forces, and have dealt with it, in our opinion, in greater detail than our predecessors; we also have explained the reason why there is a delay in tools of great force and have treated other things clearly, which are for students, where inclination and pressure are concerned, of great benefit, things that the students can be contented with. In this book we are going to describe machines that are useful to facilitate that of which the existence and the application has already been shown and that are also beneficial in the moving of heavy bodies. Furthermore we are going to construct tools through which we have benefits in pressing, because these too require a great force in their use. Loads that are pulled on the ground are done so on "toads". The "toad" is a solid body that is made from a square [piece of] wood whose ends are rounded off. Onto these toads one puts the loads and attaches to their ends ropes or anything else that serves for pulling, with which one moves the toads. These ropes are either pulled with the hands or with other tools. If now the ropes are tightened, the toads glide on the ground.Underneath the toads one attaches thin [pieces of] wood or boards so the toads move on them. If the load is light one has to use round [pieces of] wood, if it is, however, heavy, boards, because these cannot be moved quickly. For if the rollers roll under the load, they will be crushed under it, because of the strong speed of their motion. Some people use neither boards nor rollers, but attach to the ends of the toads hard pulleys, on which they move.

2 In order to lift heavy objects one needs machines; some of these have one support, others two, others again three and some have four supports. The one with one support has the following appearance. We take a long beam of greater height than the distance to which we want to lift the load. Even though this beam is solid in itself, we yet take a rope, tie it to it and wind it in equal spaces around it; let the perpendicular line measured between the individual coils be four hands. This way the force of the wood is increased and the coils of rope around it are like a flight of steps for someone who has to do something at the top of the beam, by which the work is made easier. If, however, that beam is not strong enough in itself, we have to consider the amount of the load that we want to lift, so that the force of the load does not become greater than the force of that support. Thus we put the support perpendicular on a [piece of] wood in which it can move, and tie to the top of this support three or four ropes, pull them towards solid, firm supports and fasten them to them. Then we attach to the end of the support pulleys that are tied to it with ropes, and fasten the ropes of the pulleys to the load that we want to lift. Hereupon we pull the ropes tight, either with our hands or with some other tool, and then the load rises. If one now wants to bring a stone onto a wall or to any place, one unties the rope from one of the firm supports that hold the supporting beam that the pulleys are attached to, indeed on the side opposite to the one to which one wants to bring the stone, and the beam inclines towards that side; then one slowly lowers the rope on the pulley to the place, where one wants to put the stone. If, however, one cannot incline the supporting beam that the pulleys are attached to so much as to let the lifted load reach the intended place, then we attach rollers underneath it, on which we let it run, or we drive it with levers so far, until we bring it to the intended place. When that has happened, we bring the beam back into its position, towards the side facing us, fasten it again and proceed with it as was done earlier.

3 The machine with two supports is made in the following way. One uses the tool called <ou)do\s> and erects the supports on it. Let these incline towards the top a little, for about one fifth of their lower spacing. Then one fastens the two supports to the base, so that their two (lower) ends are connected to each other, and one attaches to the (upper) ends of the supports a crossbeam, to which is fastened a block and tackle. Let another block and tackle be at the stone. Then one tightens the rope, as was done earlier, either with the hands or by a draught animal, and so the load rises. For these supporting beams to remain upright, they have to be tied with ropes, as described before. Then we bring the stone into the necessary position and transport the base to the other side of the building, depending on need.

4 The machine with three pillars is made in the following way. We make pillars inclined towards one another, whose tips meet at one point, and attach at this point, where the three beams meet, a block and tackle, whose other part is fastened to the load. If now the ropes of the block and tackle are tightened, the load rises. The base of this tool is firmer and more secure than others, but it cannot be used well in every place, rather, only in places where we want to lift the load in the center of the tool. Thus if we have to bring the load to a place, around which we can put up this tool, then we use it for that.

5 As now for the tool with four supports, it is used for enormous loads. It consists in erecting four pillars of wood in the form of a square enclosure with parallel sides, so far apart that the stone can easily be moved and lifted in it. Then we attach to the ends of these pillars pieces of wood that are connected with one another, we make them firm and secure. Onto these pieces of wood we put in opposite order (i.e. diagonally) others again, so that all supports are connected with one another. Hereupon we fasten the block and tackle in the center of these [pieces of] wood, at the point, at which the [pieces of] wood meet one another. Now we fasten the ropes of the block and tackle to the stone, tighten them, and the load rises. One has to take care, however, not to use nails or pegs with the mechanical tools, indeed, with any load, in particular, however, with big loads; on the other hand, we use ropes and lines and tie together with them what we want to, instead of wanting to nail something.

6 Since it happens sometimes with the tool that looks like a catapult, with which one lifts stones, that it is awkward to put the stone where one has to put it, we use the instrument that is called "loop". We draw on the surface of the stone, namely the plane <abgd>, a figure like the one illustrated in the drawing. Each of the planes <ezhq> and <klmn> is, namely a rectangle; let <ezhq> be wider than <klmn>.But let them be equal in length, i.e., let the line <kl> be equal to line <eh>. Then we carve this figure deeply into the stone, let the depth of the hollow correspond to the weight of the stone. Let the hollow of the plane <ezhq> be evenly perpendicular, that of plane <klmn>, however, inclined, i.e., let the lower part be wider than the upper one, so that a hollow like a wooden lock is formed. Let the narrow (part) be equal to <klmn>, the wide one equal to <ezhq>. Then we make a body of iron that also looks like a wooden bolt, that fits into this hollow, to the upper part of which a ring is attached and that goes into the hollow <ezhq>, so that it is entirely inside; then one pushes and shoves it, until it goes into the parent hollow (<klmn>), without shifting. Now one puts into the hollow <ezhq> a [piece of] wood so that the iron does not slip out. Then one attaches to the ring that is on the iron peg the ropes that carried the catapult, in which the stone was lying, and lifts it in this way, until it reaches the intended place, without being hindered by anything. When the stone is put in its place, the wooden peg is removed, the iron pulled out, to be inserted into another stone that is also lifted.

7 Stones can also be lifted by means of the tools called "crabs", if they have three or four supports and their ends are bent so they look like fish-hooks and these hooks are brought into the sides of the load. On their (the supports') ends, crossbeams are put and fastened with ropes, then lifted so they lift the load. We have to attach the crossbeams to the ends of these supports so that they come together with their ends outside the stone, so the stone, when it is suspended from them and rises, does not fall down, but these crossbeams have to be tied together and the ropes have to be connected with the pulleys on their outside; when they are tightened they lift the stone.

8 For the same purpose one also applies another procedure that is easier and safer than this one. Let the surface of the stone be designated <abgd> and let us carve into it a figure similar to a rectangle, namely <ezhq>, of even depth. Let this hollow have sharp sides, i.e., let it have on two sides a considerable bulge. Above this bulge let it be very strong so it can carry the stone that is suspended from it. We now use two iron pegs, let their sides be inclined, similar to the letter Gamma. Let there be a ring or a hole at their top; then we put each of the two pegs into one side of the hollow and bring their inclined part into the inclined bulge, make a third peg of iron that we insert between the first two so it keeps them from shifting. Let the third peg also be drilled through at its upper end with a hole corresponding to the hole of the first two pegs. Then we insert into the three holes an axle, one of whose ends is thicker. Now the three pegs fill the hollow <ezhq> and the inclined (parts) of two pegs are in the bulge that is on both sides of the bed of the hollow, while the third fills the space between the first two; thus the three pegs look like a single body. Hereupon we fasten the ropes leading to a block and tackle to the axle that goes through the three pegs. Let there be on the upper part of the machine, by means of which we lift the load, a block and tackle parallel to the one attached to the stone; if we lead the ropes through it and tighten [them], the stone rises, because the middle peg does not let go of the two other ones whose inclined parts are stuck in the stone. Then one lifts it until it is opposite the place where we want to insert it, and lets it settle in this place. When the stone rests in its place the axle is taken out and the middle peg and then the two pegs with inclined sides are removed; hereupon we prepare another stone and proceed with it as before. In this procedure one has to take care not to use too hard iron, so it does not break; but one also has to beware of too soft [iron], so it does not curve and bend because of the weight of the stone, but one uses medium [iron], not too hard and not too soft. One should also watch for a bend or fold in the iron, or a crack that could befall it during the work, for a fault in it is very dangerous, not only because the stone might fall, but also because it hits the workers when it falls.

9 The ways of lifting heavy objects and of bringing them to a height are thus the ones mentioned by us. We have, however, to take into consideration place, time and other requirements and to explain how we proceed according to each single one of (these circumstances). For transporting big blocks off the peaks of high mountains one employs equipment to keep the stone [block] from rolling because of the slope of the mountain through its own downward motion, and from falling onto and destroying the draught animals that pull it and the wagon. Therefore one uses on the mountain, in the place where one wants to transport the stone [block] downhill, two tracks that one possibly levels off, and one takes two four-wheeled wagons, one of which one puts in the highest position of the track that one wants to transport the stone [block] on, the other one in the lowest position of the second track. Then one attaches pulleys to a firm post between the two tracks, leads from the wagon that carries the stone [block] ropes over the pulleys and makes them go to the lower wagon. This lower wagon one loads with small rocks that result from the cutting of the big block, until it is loaded with a weight (a little) less than that of the stone [block] that is to be transported down. Hereupon one hitches up to this wagon draught animals that pull it upward and through the gradual climbing of this wagon the big stone [block] also moves easily and gradually downward.

10 Some also want, with this procedure, to lift big pillars and let them down on their bases in any place. According to this method, one ties ropes to the upper part of the pillar that one wants to lift, leads them through pulleys that are tied to a firm support and pulls them through until they come out on the other side of the pulley. Then one attaches to the ends of those that have been pulled through containers into which one can put rocks and heavy objects, I mean boxes or such. Then one fills the containers with rather heavy rocks and weights, until they counterbalance the weight of the pillar and gain the superior weight over it; for so they lift it and it remains upright on its base. The lower part of the pillar has to be tied to the base so that it does not, when the pillar is lifted, leave the base or move away from it; or one winds ropes around the base of the pillar that surround it like a wreath, so that the lower part of the pillar, when it (the pillar) is lifted, rests firmly in those ropes that were put around it.

11 Some wanted to move big loads by sea after the following method. Namely, one makes from wood a quadrangular raft, whose individual parts are fastened to each other by nails and bolts. One makes for it strong walls and brings it onto the water, where one wants to load the load. Underneath the raft one puts sacks filled with sand, with the openings tied shut, and puts the raft onto the sacks. Then one takes two boats, ties them with ropes to the walls on both sides of the raft. Then one brings the load onto the raft, opens the sacks and lets the sand run out. Then one has the two boats go out to sea and they plow through it, carrying the raft.

12 Others thought of transporting big blocks of stone in the same manner, swimming on the sea. Some applied the following method to straighten up walls that inclined in earthquakes. One digs, on the side towards which the wall is inclined, following the length of the wall, a trench into the ground. At a short distance from the wall one puts into it a square beam and erects between the wall and the square beam lying in the trench other beams (that are connected by a crossbeam). To the ends of the perpendicular beams one attaches pulleys and leads over them ropes to the tool called a "winch". Then one turns this tool until the ropes tighten and the crossbeam is pulled up. By this the perpendicular beams are pulled up and cause the wall to incline until they return it to its former place. When it has been returned to its place one leaves it standing for some time, supported by these beams, so that the stones settle into one another. Then one removes the beams and the wall stands firmly in its perpendicular state.

13 What is necessary for moving loads and what is useful for this we have now explained in a sufficient manner. Now agricultural tools, namely those with which one presses wine and oil, are not far removed from the use of the levers that we have mentioned; for it is necessary to explain this and to clarify as much of it as one needs to know. The beam called Oros, which others also call press, is nothing but a lever and its Hypomochlion. The latter is here the wall of the press, to which the end of the pressing beam is attached. The load is the net that is wound around the grapes to be pressed, and the moving force is the stone [block] that is suspended from the end of the pressing beam, called Lithos. In big pressing beams we see that also the weight of the stone [block] is great, so that it is strong enough for pressing. The long pressing beams are sometimes up to twenty-five cubits long and the stone [block] suspended from them, called Lithos, has a weight of twenty talents.

14 We now want to look at the suspending of the stone [block]. We proceed like this: We take a block and tackle and attach one pulley to the end of the Oros, the other one to the stone [block] (and lead a rope over the pulleys). To the stone [block] we furthermore attach a crossbeam above the pulley, which is attached to the [piece of] wood called Oros (in order to suspend the stone [block] from the pressing beam, when it is lifted by means of the block and tackle). Then we lead that rope to a shaft with the wheel, rotate the wheel, so that the rope winds up around the shaft and the stone [block] rises.

15 We have yet another method to lower the [piece of] wood called Oros and to lift the stone [block] called Lithos. For the stiffness of the ropes causes a hindrance for the lowering of the [piece of] wood and the lifting of the stone [block], because the rope, if it is stiff, does not run over the pulleys, upward when lifting the stone [block] and downward when lowering the beam. When lifting the stone [block] we also have to use long spokes, in order to rotate the shaft with them. We are, however, when the grapes to be pressed that are lying under the beam, are very many, or the people who turn the shaft that the rope is on are of greater number, not safe from breaking individual spokes, so that (the stone [block]) falls down and so an accident happens to them or so that they (the spokes) slip out of the hole, so that (the stone [block]) also falls and the same accident would happen to them. So one has found a different method for which no rope is necessary, which is easier and safer. Its description is the following. We use a square body of wood that looks like a brick* and attach it underneath the beam called Oros in the place where the rope was. The one of its parts that is directed upward we make curved and on both sides of the firm support we attach impediments (?) that are fastened to the beam called Oros, so that this brick does not run further than is necessary, but still can move to both sides. Then we lift the pressing beam as high as possible, in order to put the grapes under it, measure the distance between the brick and the stone [block], take half of that or a little more and make according to this length an evenly thick, lentil-shaped screw. But let the screw thread on the one side not go to the end of the screw wood; but on the other side the thread has to reach the end of the screw wood. The protruding part of the wood we make square and cut into this square a hollow, called Tormos. That is a circular hole, which is drilled into the end of the [piece of] wood, so that this [piece of] wood can be joined with the [piece of] wood (the brick), with which it is to be connected. Then we add this Tormos to one of the sides of the brick that are lying under the pressing beam, take iron cross nails, insert their ends into this hole and nail the rest of them to the brick. Now we use an iron axle, which we insert into this Tormos, let it go to the brick and fasten it to it, so that it increases the strength of the bond with the brick. Hereupon we take another square piece of strong hard wood, of the same length as the screw; let its width, which is determined by one of the sides of the square of its base, be as much larger as the diameter of the screw cylinder, so that we can insert this cylinder into the interior of this square [piece of] wood. Then we split it lengthwise, make a round, canal-like groove into each of the two parts, in order to make from it a nut, cut the screw thread, so that the bolt can be inserted. Then we (re)connect the two parts, so that they become one piece. The thread has also in the nut wood to go on one side to the end of the [piece of] wood; on the other end one leaves it undrilled, solid. If we now insert the end of the screw into the end of the hollowed [piece of] wood, whose thread goes to its outermost (edge), and the screw is turned, then it goes completely into the hollowed [piece of] wood, until it disappears entirely. When we have done this, we cut at the end of this [piece of] wood that is hollowed on the inside on its neck at a short distance from the end a circle (a), and put an iron ring (c) on it, as one does with wagon axles. Then we carve into the stone [block] a hole (c), so wide, that the end of this [piece of] wood (a, d) fits inside and we can easily rotate it inside. Now we insert the end of the [piece of] wood into this hole and attach iron hooks (f, f) to it (the stone [block]), which prevent the [piece of] wood from slipping out of the hollow in the stone [block]. To the circle that is carved into the end of the [piece of] wood we also attach an iron ring (d), so that it can be rotated more easily. Above this end that is set into the stone [block] we make two holes opposite to each other, from which protrude the four ends of two spokes. When we have done this and want to use the beam called Oros, we approach the two ends of the screw and the [piece of] wood that is hollowed inside to one another. Then the four spokes are turned, until the screw penetrates into the hollow, the beam is pushed down, the stone [block] rises and so everything under the beam is pressed out. When the stone [block] has been lowered so far that it sits on the ground, we turn in the opposite (direction), so the beam rises, while the stone [block] remains lying. This procedure is strong and sound, with a secure outcome (safe) and without much effort.

16 Some have thought of inventing different kinds of pressing tools; instead of the net that was wound around the grapes to be pressed and the baskets, into which one puts the olives after an incision (?) has been made on them, and that one brings under the Oros, they have made an instrument from wood, that is called Galeagra. This one fills with any [kind of] material, puts it under the beam called Oros and lowers the beam down on it. Through this one gets a wide space for what one wants to press and the work is made easier. The manufacture of the Galeagra occurs in two ways. One of them is assembled and completed according to the following procedure. One takes wood of strong and hard consistency and makes from it slats in the length of the instrument one wants to make. Let their width measure two spans and their thickness six fingers. Then we carve on both ends of each slat, on both sides, after leaving six fingers wide free, a notch in the upper part and let it penetrate into the depth of the slat for a quarter of its thickness. The same we do on the lower side, so that the rest of the thickness of the wood is half. The notches in the slats have to be regular, so that one fits into the other. Then we assemble the slats so that from the assembling of all an equilateral, square, box-like frame is formed. The inner joins between the slats have to be wide, so that the fluids drain quickly. In this tool the [piece of] wood that is lying on top of the grapes and the boards that are stacked above it, do not have to be very thick, because, when the grapes are pressed, (by putting on new boards), depending on the amount of what has already been pressed out, they jut out beyond the slats so that those do not become a hindrance.

17 As now for the other Galeagra, the connection of its walls*) with one another is established by three crossbeams on each of them. On the sides of these three crossbeams there has to be a protrusion that is fitted with a notch extending to the middle of its thickness, so that the four walls are firmly joined, when they are assembled. Also in this tool the joins have to be wide and on top of the uppermost board a piece of wood has to be put, which, according to what was just said, protrudes on top, so that the pressing beam does not touch part of the grapes, but the wood block drops to the bottom of the Galeagra.

18 Now we want to discuss the manufacture of the presses that press strongly and powerfully, and we want to state the difference that exists between the tools already mentioned and the following ones, which are among the strongest and most perfect that exist. First we explain the difference between them and then we describe their manufacture. So we say that the beam called Oros is nothing but a lever that is pressed down by a weight. The weight that presses on it is located on the one of its ends that is raised off the ground. When it exerts pressure, the fluids flow continuously, until the weight rests on the bottom. The tools, that we want to describe now, while they are very powerful, still their pressure is not continuous and of even strength. Therefore one has to repeat the turning and pressing from time to time. With the beam called Oros, however, the stone [block] alone, when it is suspended and then released, exerts the pressure, and a constant repetition of the pressing is not necessary. That is the difference between the tools.

19 The tools whose manufacture we are going to discuss now serve for the pressing of olive oil. They are easy to handle, can be transported and brought to any place. One does not need for them long, even beams of hard nature, nor a big, heavy stone [block], nor strong ropes, we also do not meet with any hindrance because of the hardness of the ropes; but they are free of all that, exert strong pressure and entirely press out all fluids. Their manufacture is done in the way that we are going to explain now. We take a square [piece of] wood of six spans of length; let its width be no less than two feet, and its thickness no less than one foot. Let this [piece of] wood be of a strong kind, not too soft and not too dry, but of medium (quality). We call it a "table". We now put the table flat down and drill on its two edges, in equal distance, two deep, round holes. Into each hole we put two locking wooden bars (b, b) that protrude into the depth of the table. Let their two ends be arcs that meet, so that a small circle is formed, which is smaller than the drilled circular holes. Let these two locking wooden bars be cut obliquely, so that, when they are inserted, they stay firm and do not give way. Then we take two hard, square, ruler-straight [pieces of] wood, of equal thickness and width; one of their heads we leave, for an adequate space, square. We take the edges of the remaining part of the two [pieces of] wood and curve them with a file and construct on them a screw of even thickness. To the end of the screw wood that we have left square we attach a disc with four holes, into which we insert four spokes. Let run around the other end of the two [pieces of] wood a rough circular cut that is removed from the end as far as the round hole, that we have drilled into the table (d), is deep. Let the diameter of this circle equal half the diameter of the circle of the base of the screw. When this has happened, we insert the end (a) of the screw, in which is the carved circle, into the round hole in the table. Then we drive the blocking woods (b, b), that we have made [with wedges (c, c)], until they penetrate into the carved circle and are stuck in it and thus do not let the screw slip out. In the same manner we proceed with the screw that goes to the other end of the table. Now we take a long, square [piece of] wood of the length of the lower [piece of] wood, into which the screws are inserted. In this [piece of] wood there are two circles that go into the wood and through to the other side, in the same position as the two circular hollows, in which the ends of the screw are sitting. Let there be a screw thread on the interior of these two circular openings, so that they form the nuts, so that, when the two screws are turned, the [piece of] wood drops and rises in the same way when the screws are turned to the other side. How one manufactures the nut, however, we are going to explain in the following. The length and thickness of this [piece of] wood has, as I said, to have the measurements of length and thickness of the table; its width has to be one quarter less [than the width of the table]. Hereupon we make a square, right-angled foot for the table, whose lower part looks like a step and whose length is a little greater than the width of the table, so that the entire tool firmly stands on it. We have to give the middle of this foot an adequate groove and the middle of the table with a tenon corresponding to the groove and insert it into that, so that it sits very firmly. Then we set up on the table, between the two screws, four interconnected walls of thin boards, that are less than one finger thick; let the length and width of the square between these boards be such that, when the Galeagra comes into its center, there remains a free space between the two, surrounding the Galeagra, in which the fluid flows. In the center of the table we have to make a pit, as wide as the base of the Galeagra that touches the table, i.e., it has to fit into it and we insert the Galeagra into this pit. Then we put a thick board on top that fills it (width-wise) and on top of it a piece of wood of lesser length and width than the board, but so thick that it fills the Galeagra (depth-wise). Then we turn the screws by means of the spokes that are on the discs, until the [piece of] wood, in which are the nuts, drops to the piece of wood. Then the piece of wood is pressed and the piece of wood presses the board inside the Galeagra and presses out the body in the Galeagra and the fluid drains. Then one turns the screw towards the other side, so that the [piece of] wood rises, the piece of wood is removed and the object to be pressed is exchanged, until all of its fluid is out.

20 There is another tool with a screw. It consists of attaching two posts to the table that support the crossbeam in which is the nut. Let the nut be in the center of this [piece of] wood. Then we insert the screw into this hole and turn it by means of the spokes that are in the disc, until the screw drops to the board that has been laid on top of the Galeagra, presses it, and the fluid drains. One has to repeat the pressing a number of times, until there is no fluid left in the body to be pressed. There still are many other kinds of presses, describing which does not appear good to us, however, because their use is frequent and usual among the people, although in effect they are inferior to the ones mentioned by us.

21 The nut now is manufactured in the following way. One takes a hard piece of wood (bc), whose length is twice as great as the nut and whose thickness is equal to it. On the one side we make a screw (d, d) on half of the [piece of] wood, according to the description given earlier. Let the depth of its threads be as great as that of the threads on the screw that we want to screw into the nut. On the other side we remove by turning the amount of the thickness of the screw grooves, so that the [piece of] wood becomes like an evenly thick peg (e, e). Then we draw the diameter of the base of the [piece of] wood and divide it into three equal parts. In one of the two dividing points we erect a perpendicular on the diameter. Then we draw from the end (points) of this line that is perpendicular to the diameter, on the entire length of the peg, two straight lines (ty, uz). We achieve this by putting the peg on a straight board and furrowing it with pliers, until we reach the thread. Then we carefully use a fine saw and saw it through to the thread. Then we detach the marked third (utzy) from the peg and carve into the center of the two remaining parts a canal-like groove (fghi) on the entire length, half as deep as the remaining thickness. Then we take an iron rod (lknm) and turn it according to the screw threads. Hereupon we attach it to the peg (ee), in which is the groove, and let its (blunt) end approach the thread (of wood ending in u), after having firmly connected the two pieces (the sawed off one and the hollowed one), so that one adheres to the other and there remains absolutely no space between them. Then we take a small wedge (opqrvw), insert it into the canal-like groove and hit it, until it drives out the iron rod and falls between the two parts. Having done this, we insert the screw into a drilled [piece of] wood (<z>, <z>), in which is a completely straight hole measuring the same as the thickness of the screw (dd). Then we drill into the walls of this wide hole small holes, placed side by side, insert into them small, oblique plugs (<h>) and let these penetrate so far until they mesh with the thread of the screw.*) Then we take the [piece of] wood (T), into which we want to drill the nut thread, drill a hole (<ee>) corresponding to the screw peg (ee) into it and connect this [piece of] wood with the one into which we have inserted the screw, by two posts that we fasten completely. Then we insert the peg (<g>), in which is the wedge, into the hole (<ee>), which is in the [piece of] wood (T) that is to be drilled to [form] a nut thread, drill into the upper end () of the screw holes, into which we insert spokes, and turn it, until it (the peg <g>) penetrates into the wood. We do not stop turning it up and down and driving the wedge again and again,**) until the nut thread is drilled as we intended. Then we have drilled the nut threads. and this is the shape, and with its completion the book is completed (TB)