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<s id="A18-0|00|01">The Mechanics of Heron of Alexandria</s>

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<s id="A18-1|00|02">First book</s>

<s id="A18-1|01|01">We want to move a known load by means of a known force through the mechanism of cogwheels.</s>

<s id="A18-1|01|02">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.</s>

<s id="A18-1|01|03">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>.</s>

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<s id="A18-1|01|13">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)</s>

<s id="A18-1|02|01">2. On the wheels.</s>

<s id="A18-1|02|02">The wheels attached to an axle always move in one direction, namely the direction in which the axle moves.</s>

<s id="A18-1|02|03">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.</s>

<s id="A18-1|02|04">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.</s>

<s id="A18-1|03|01">[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>.</s>

<s id="A18-1|03|02">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>.</s>

<s id="A18-1|03|03">Then it can occur that points move in the same direction and that they move in opposite directions.</s>

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<s id="A18-1|03|09">This our explanation has to be observed with equal circles.As for different circles, we shall demonstrate it in the following.</s>

<s id="A18-1|04|01">[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.</s>

<s id="A18-1|04|02">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.</s>

<s id="A18-1|04|03">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>.</s>

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<s id="A18-1|04|07">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.</s>

<s id="A18-1|05|01">[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.</s>

<s id="A18-1|05|02">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.</s>

<s id="A18-1|05|03">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.</s>

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<s id="A18-1|05|05">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.</s>

<s id="A18-1|06|01">[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.</s>

<s id="A18-1|06|02">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.</s>

<s id="A18-1|07|01">[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.</s>

<s id="A18-1|07|02">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'>.</s>

<s id="A18-1|07|03">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.</s>

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<s id="A18-1|07|10">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.</s>

<s id="A18-1|08|01">[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.</s>

<s id="A18-1|08|02">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.</s>

<s id="A18-1|08|03">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.</s>

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<s id="A18-1|08|07">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.</s>

<s id="A18-1|09|01">[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.</s>

<s id="A18-1|09|02">Let us first deal with the plane figures.Let us therefore assume a certain kind of line.</s>

<s id="A18-1|09|03">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.</s>

<s id="A18-1|09|04">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.</s>

<s id="A18-1|10|01">[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.</s>

<s id="A18-1|10|02">Thus let there be given a line that is in a certain ratio with another line.</s>

<s id="A18-1|10|03">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.</s>

<s id="A18-1|11|01">[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.</s>

<s id="A18-1|11|02">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.</s>

<s id="A18-1|11|03">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>.</s>

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<s id="A18-1|11|10">Thus we have constructed two mean proportionals to the two lines <ab> and <bg>, namely the lines <az> and <ge>. q.e.d.</s>

<s id="A18-1|12|01">[12] How one has to enlarge or reduce regular plane or solid figures in a certain ratio, we have now explained.</s>

<s id="A18-1|12|02">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.</s>

<s id="A18-1|12|03">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.</s>

<s id="A18-1|12|04">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.</s>

<s id="A18-1|13|01">[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.</s>

<s id="A18-1|13|02">If the line moves in a plane, the determined figures will be planar.</s>

<s id="A18-1|13|03">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.</s>

<s id="A18-1|13|04">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.</s>

<s id="A18-1|14|01">[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.</s>

<s id="A18-1|14|02">We shall prove this first for the plane.</s>

<s id="A18-1|14|03">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.</s>

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<s id="A18-1|14|06">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.</s>

<s id="A18-1|15|01">[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.</s>

<s id="A18-1|15|02">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.</s>

<s id="A18-1|15|03">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.</s>

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<s id="A18-1|15|08">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.</s>

<s id="A18-1|16|01">[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.</s>

<s id="A18-1|16|02">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.</s>

<s id="A18-1|16|03">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>.</s>

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<s id="A18-1|16|06">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.</s>

<s id="A18-1|17|01">[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.</s>

<s id="A18-1|17|02">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.</s>

<s id="A18-1|17|03">When we have done this, a solid figure forms from these lines, which is equal and similar to the one first assumed.</s>

<s id="A18-1|18|01">[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.</s>

<s id="A18-1|18|02">Let the size of the boards fit the size of the largest of the similar figures.</s>

<s id="A18-1|18|03">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.</s>

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<s id="A18-1|18|23">Thus the bodies are in this known ratio with one another.</s>

<s id="A18-1|19|01">[19] If we now want to make the back of the similar bodies, we apply the same procedure.</s>

<s id="A18-1|19|02">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.</s>

<s id="A18-1|19|03">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: </s>

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<s id="A18-1|19|22">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.</s>

<s id="A18-1|20|01">[20] Some people believe, subscribing to false views, that loads lying on the ground can only be moved by an equivalent force.</s>

<s id="A18-1|20|02">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.</s>

<s id="A18-1|20|03">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.</s>

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<s id="A18-1|20|07">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.</s>

<s id="A18-1|21|01">[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.</s>

<s id="A18-1|21|02">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.</s>

<s id="A18-1|21|03">This happens, however, to the water because its parts are not joined, but rather are easily separable.</s>

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<s id="A18-1|21|07">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.</s>

<s id="A18-1|22|01">[22] If we now want to lift a load to a higher place, we need a force equal to the load.</s>

<s id="A18-1|22|02">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.</s>

<s id="A18-1|22|03">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.</s>

<s id="A18-1|22|04">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.</s>

<s id="A18-1|23|01">[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.</s>

<s id="A18-1|23|02">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.</s>

<s id="A18-1|23|03">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.</s>

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<s id="A18-1|23|08">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.</s>

<s id="A18-1|24|01">[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.</s>

<s id="A18-1|24|02">That one in reality only speaks of gravity and inclination in bodies, nobody will deny.</s>

<s id="A18-1|24|03">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.</s>

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<s id="A18-1|24|21">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.</s>

<s id="A18-1|25|01">[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.</s>

<s id="A18-1|25|02">For Archimedes has already adopted a reliable procedure on this part in his book with the title "Book of Supports".</s>

<s id="A18-1|25|03">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.</s>

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<s id="A18-1|25|06">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.</s>

<s id="A18-1|26|01">[26] Let therefore an evenly thick and evenly dense load, <ab>, rest on pillars.</s>

<s id="A18-1|26|02">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>.</s>

<s id="A18-1|26|03">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.</s>

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<s id="A18-1|26|09">If there are even more pillars, then we learn through the same procedure, how much weight comes to each of them.</s>

<s id="A18-1|27|01">[27] If that is so, then we assume the supports <ab> and <gd> in equal positions; </s>

<s id="A18-1|27|02">let an evenly thick and heavy body rest on them, namely <ag>.</s>

<s id="A18-1|27|03">We just said that half the weight of <ag> falls to each of the two supports <ab> and <gd>.</s>

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<s id="A18-1|27|13">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.</s>

<s id="A18-1|28|01">[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.</s>

<s id="A18-1|28|02">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>.</s>

<s id="A18-1|28|03">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>.</s>

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<s id="A18-1|28|09">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.</s>

<s id="A18-1|29|01">[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.</s>

<s id="A18-1|29|02">Thus it is clear that the load can only be moved if the strength of the second man is added.</s>

<s id="A18-1|29|03">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.</s>

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<s id="A18-1|29|09">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.</s>

<s id="A18-1|30|01">[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.</s>

<s id="A18-1|30|02">We want to know how much of the load affects each of the supports.</s>

<s id="A18-1|30|03">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.</s>

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<s id="A18-1|30|08">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.</s>

<s id="A18-1|31|01">[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>.</s>

<s id="A18-1|31|02">Then it is clear that on each of the supports falls half of the load <ab>.</s>

<s id="A18-1|31|03">Let us now suspend a weight from <ab> at point <e>.</s>

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<s id="A18-1|31|10">If more weights are attached, then we learn after the same method how much weight falls to each of the two supports.</s>

<s id="A18-1|32|01">[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.</s>

<s id="A18-1|32|02">This must, however, not be said so in general, but one has to introduce a better distinction.</s>

<s id="A18-1|32|03">Let us now assume an evenly thick and heavy scale beam, namely <ab>, whose point of suspension, namely point <g>, lies in its center.</s>

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<s id="A18-1|32|07">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.</s>

<s id="A18-1|33|01">[33 ]Some have thought that inverse proportionality is not present in irregular scales.</s>

<s id="A18-1|33|02">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>.</s>

<s id="A18-1|33|03">Here, we understand with balance the rest and standstill of the scale beam, even if it is inclined to any side.</s>

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<s id="A18-1|33|09">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.</s>

<s id="A18-1|34|01">[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.</s>

<s id="A18-1|34|02">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.</s>

<s id="A18-1|34|03">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.</s>

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</chap>

<s id="A18-2|00|01">Second book</s>

<s id="A18-2|01|01">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.</s>

<s id="A18-2|01|02">Now their names are the following: the shaft with the wheel, the lever, the block and tackle, the wedge, the screw.</s>

<s id="A18-2|01|03">The shaft with the wheel is made in the following way.</s>

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<s id="A18-2|01|14">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.</s>

<s id="A18-2|02|01">2 The second power.</s>

<s id="A18-2|02|02">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.</s>

<s id="A18-2|02|03">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.</s>

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<s id="A18-2|02|07">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.</s>

<s id="A18-2|03|01">3 The third power.</s>

<s id="A18-2|03|02">The third power is the one that is called block and tackle.</s>

<s id="A18-2|03|03">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.</s>

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<s id="A18-2|03|16">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.</s>

<s id="A18-2|04|01">4 The fourth power.</s>

<s id="A18-2|04|02">The fourth power, which follows to this one, is the one called wedge.</s>

<s id="A18-2|04|03">It is used with some tools in the preparation of perfume and in order to join separate parts of carpenter's works.</s>

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<s id="A18-2|04|09">The more acute the angle of the wedge, the easier can it be worked with, as we are going to show.</s>

<s id="A18-2|05|01">5 The fifth power.</s>

<s id="A18-2|05|02">That is the one called screw.</s>

<s id="A18-2|05|03">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.</s>

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<s id="A18-2|05|21">This screw is called lentil-shaped, the other one square.</s>

<s id="A18-2|06|01">6 If the screw is used by itself, it happens in this way.</s>

<s id="A18-2|06|02">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.</s>

<s id="A18-2|06|03">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.</s>

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<s id="A18-2|06|07">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.</s>

<s id="A18-2|07|01">7 The manufacture of the five powers described earlier and their application we have just explained and elucidated.</s>

<s id="A18-2|07|02">The reason why each of these machines moves big loads with a small force, we want to explain now as follows.</s>

<s id="A18-2|07|03">Let us assume two circles around the same center, namely point <a>, whose two diameters are the lines <bg> and <de>.</s>

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<s id="A18-2|07|10">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.</s>

<s id="A18-2|08|01">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.</s>

<s id="A18-2|08|02">Incidentally, already the ancients, who were before us, have executed this introduction.</s>

<s id="A18-2|08|03">Let us now prove it for the tool called a lever.</s>

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<s id="A18-2|08|13">Thus the proof for the lever that moves loads is the same as the one put forth for the two circles.</s>

<s id="A18-2|09|01">9 Let us now assume a different lever, let it be the line <ab> and be movable around a Hypomochlion, namely <d>.</s>

<s id="A18-2|09|02">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 .</s>

<s id="A18-2|09|03">If we now press down towards the ground the end of the lever at , then we have moved the load <g>.</s>

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<s id="A18-2|09|18">That the scales can also be traced back to the circle is clear, since the circle is a scales.</s>

<s id="A18-2|10|01">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.</s>

<s id="A18-2|10|02">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.</s>

<s id="A18-2|10|03">Our predecessors have already spoken this sentence; we have put it here so our work may be complete and have a well-ordered structure.</s>

<s id="A18-2|11|01">11 Let us now discuss the basis of the tool called block and tackle.</s>

<s id="A18-2|11|02">Let us imagine an elevated wheel at point <a>, around which a rope (Hoplon), namely <bg>, is wound.</s>

<s id="A18-2|11|03">Let the load, namely <d>, be tied to the two free ends of the rope, which are also elevated above the ground.</s>

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<s id="A18-2|11|10">Thus this so-called simple pull is the one in which the rope hangs down doubled.</s>

<s id="A18-2|12|01">12 We now want to explain the double pull; it is the one in which three parts of the rope are strung.</s>

<s id="A18-2|12|02">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.</s>

<s id="A18-2|12|03">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.</s>

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<s id="A18-2|12|13">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.</s>

<s id="A18-2|13|01">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.</s>

<s id="A18-2|13|02">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>.</s>

<s id="A18-2|13|03">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.</s>

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<s id="A18-2|13|10">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.</s>

<s id="A18-2|14|01">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.</s>

<s id="A18-2|14|02">Thus it can be gathered from this that the wedge moves after the blow stops.</s>

<s id="A18-2|14|03">We also learn this in another way.</s>

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<s id="A18-2|14|07">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.</s>

<s id="A18-2|15|01">15 Now I say that every blow, even if it is only light, moves every wedge.</s>

<s id="A18-2|15|02">Let us assume some wedge, whose angle is at <a> and let its head be the line <dm>.</s>

<s id="A18-2|15|03">Let the blow <bg> move it and let its distance (from the original position) be <ad>.</s>

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<s id="A18-2|15|18">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.</s>

<s id="A18-2|16|01">16 After this it remains to explain the effective cause in the screw.</s>

<s id="A18-2|16|02">First let us start by explaining what shows in screw threads.</s>

<s id="A18-2|16|03">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.</s>

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<s id="A18-2|16|10">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.</s>

<s id="A18-2|17|01">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.</s>

<s id="A18-2|17|02">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.</s>

<s id="A18-2|18|01">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.</s>

<s id="A18-2|18|02">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.</s>

<s id="A18-2|18|03">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.</s>

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<s id="A18-2|18|08">Thus as many cogs as there are on the wheel, as many rotations the screw makes, until the wheel has made one rotation.</s>

<s id="A18-2|19|01">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.</s>

<s id="A18-2|19|02">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.</s>

<s id="A18-2|19|03">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.</s>

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<s id="A18-2|19|12">With this we have talked enough about the nature of the screw and its use.</s>

<s id="A18-2|20|01">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.</s>

<s id="A18-2|20|02">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.</s>

<s id="A18-2|20|03">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.</s>

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<s id="A18-2|20|10">Let us now consider how the hindrances that occur with these three machines can be remedied.</s>

<s id="A18-2|21|01">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.</s>

<s id="A18-2|21|02">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.</s>

<s id="A18-2|21|03">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.</s>

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<s id="A18-2|21|25">These supports have to be set up in a secure, firm place, when the load is lifted.</s>

<s id="A18-2|22|01">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.</s>

<s id="A18-2|22|02">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>.</s>

<s id="A18-2|22|03">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>.</s>

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<s id="A18-2|22|08">The same shows with multiple axles and multiple wheels and is proven in the same way.</s>

<s id="A18-2|23|01">23 Now we are supposed to move the same weight by the same force through the tool called block and tackle.</s>

<s id="A18-2|23|02">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.</s>

<s id="A18-2|23|03">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.</s>

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<s id="A18-2|23|07">then the force will counterbalance the load.</s>

<s id="A18-2|24|01">24 That the delay also occurs with this tool is clear because the process takes place in the same ratio.</s>

<s id="A18-2|24|02">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.</s>

<s id="A18-2|24|03">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.</s>

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<s id="A18-2|24|05">In this procedure the blocks and tackles lift together.</s>

<s id="A18-2|25|01">25 By means of the lever, also the same load can be moved by the same force according to the same procedure.</s>

<s id="A18-2|25|02">So let the load be at point <a> and the lever be <bg>, the Hypomochlion at point <d>.</s>

<s id="A18-2|25|03">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>.</s>

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<s id="A18-2|25|08">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.</s>

<s id="A18-2|26|01">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.</s>

<s id="A18-2|26|02">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.</s>

<s id="A18-2|26|03">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>.</s>

<s id="A18-2|26|04">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.</s>

<s id="A18-2|27|01">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.</s>

<s id="A18-2|27|02">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.</s>

<s id="A18-2|27|03">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.</s>

<s id="A18-2|28|01">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.</s>

<s id="A18-2|28|02">Thus we have proven that the ratio of wedge-angle to angle is (inverse) that of moving blow to blow.</s>

<s id="A18-2|28|03">Then the ratio of time to time is also (inverse) that of force to force.</s>

<s id="A18-2|29|01">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.</s>

<s id="A18-2|29|02">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.</s>

<s id="A18-2|29|03">Let us now prove that we can combine the four powers and by their fusion move a known load.</s>

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<s id="A18-2|29|16">When the spoke is turned, the load rises.</s>

<s id="A18-2|30|01">30 For the wedge and the screw, we apply the following procedure.</s>

<s id="A18-2|30|02">Let the angle of the wedge we want to make be <abg>, namely an acute one.</s>

<s id="A18-2|30|03">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.</s>

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<s id="A18-2|30|13">Thus it is not absolutely necessary to employ small angles in the wedge.</s>

<s id="A18-2|31|01">31 We cannot proceed in the same way with the screw.</s>

<s id="A18-2|31|02">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>.</s>

<s id="A18-2|31|03">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.</s>

<s id="A18-2|32|01">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.</s>

<s id="A18-2|32|02">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.</s>

<s id="A18-2|33|01">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.</s>

<s id="A18-2|33|02">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.</s>

<s id="A18-2|33|03">The beginning for the things that we are going to research, we derive from what is clear to us.</s>

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<s id="A18-2|33|11">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.</s>

<s id="A18-2|34|01">34 Questions. </s>

<s id="A18-2|34|03">Why do wagons with two wheels carry</s>

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<s id="A18-2|34|98">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.</s>

<s id="A18-2|35|01">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.</s>

<s id="A18-2|35|02">First now we want to show how one finds the center of gravity of an evenly thick and heavy triangle.</s>

<s id="A18-2|35|03">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>.</s>

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<s id="A18-2|35|10">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.</s>

<s id="A18-2|36|01">36 We want to find the same for the quadrangle.</s>

<s id="A18-2|36|02">Let thus the given quadrangle be <abgd>.</s>

<s id="A18-2|36|03">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>.</s>

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<s id="A18-2|36|05">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.</s>

<s id="A18-2|37|01">37 We want to prove the same for the pentagon <abgde>.</s>

<s id="A18-2|37|02">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>.</s>

<s id="A18-2|37|03">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>.</s>

<s id="A18-2|37|04">We have to imagine it the same way for all polygons.</s>

<s id="A18-2|38|01">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.</s>

<s id="A18-2|38|02">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.</s>

<s id="A18-2|38|03">Now we have to distribute it on the supports.</s>

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<s id="A18-2|38|06">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.</s>

<s id="A18-2|39|01">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.</s>

<s id="A18-2|39|02">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>. </s>

<s id="A18-2|39|03">The weight at <d> has been determined; consequently the two weights <g>, are determined.</s>

<s id="A18-2|39|04">But the weight at <a> has also been determined; consequently the weights that rest on the supports, are determined.</s>

<s id="A18-2|40|01">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.</s>

<s id="A18-2|40|02">We divide the line <ab> at point <d> so that <bd> relates to <ad> like the weight at <a> to the weight at .</s>

<s id="A18-2|40|03">Then the point for the total weight of the two loads is at point <d>.</s>

<s id="A18-2|40|04">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.</s>

<s id="A18-2|41|01">41 We want to show the same for a polygon.</s>

<s id="A18-2|41|02">Let the figure <abgde> be a polygon.</s>

<s id="A18-2|41|03">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 .</s>

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<s id="A18-2|41|06">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.</s>

<s id="A18-2|42|01">End of the second book of Heron on the lifting of heavy objects.</s>

<s id="A18-2|42|02">Übersetzung fehlt. Jutta</s>

</chap>

<s id="A18-3|00|01">Third book</s>

<s id="A18-3|01|01">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; </s>

<s id="A18-3|01|02">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.</s>

<s id="A18-3|01|03">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.</s>

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<s id="A18-3|01|12">Some people use neither boards nor rollers, but attach to the ends of the toads hard pulleys, on which they move.</s>

<s id="A18-3|02|01">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.</s>

<s id="A18-3|02|02">The one with one support has the following appearance.</s>

<s id="A18-3|02|03">We take a long beam of greater height than the distance to which we want to lift the load.</s>

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<s id="A18-3|02|12">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.</s>

<s id="A18-3|03|01">3 The machine with two supports is made in the following way.</s>

<s id="A18-3|03|02">One uses the tool called <ou)do\s> and erects the supports on it.</s>

<s id="A18-3|03|03">Let these incline towards the top a little, for about one fifth of their lower spacing.</s>

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<s id="A18-3|03|08">Then we bring the stone into the necessary position and transport the base to the other side of the building, depending on need.</s>

<s id="A18-3|04|01">4 The machine with three pillars is made in the following way.</s>

<s id="A18-3|04|02">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.</s>

<s id="A18-3|04|03">If now the ropes of the block and tackle are tightened, the load rises.</s>

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<s id="A18-3|04|05">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.</s>

<s id="A18-3|05|01">5 As now for the tool with four supports, it is used for enormous loads.</s>

<s id="A18-3|05|02">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.</s>

<s id="A18-3|05|03">Then we attach to the ends of these pillars pieces of wood that are connected with one another, we make them firm and secure.</s>

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<s id="A18-3|05|07">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.</s>

<s id="A18-3|06|01">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".</s>

<s id="A18-3|06|02">We draw on the surface of the stone, namely the plane <abgd>, a figure like the one illustrated in the drawing.</s>

<s id="A18-3|06|03">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>.</s>

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<s id="A18-3|06|10">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.</s>

<s id="A18-3|07|01">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.</s>

<s id="A18-3|07|02">On their (the supports') ends, crossbeams are put and fastened with ropes, then lifted so they lift the load.</s>

<s id="A18-3|07|03">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.</s>

<s id="A18-3|08|01">8 For the same purpose one also applies another procedure that is easier and safer than this one.</s>

<s id="A18-3|08|02">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.</s>

<s id="A18-3|08|03">Let this hollow have sharp sides, i.e., let it have on two sides a considerable bulge.</s>

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<s id="A18-3|08|15">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.</s>

<s id="A18-3|09|01">9 The ways of lifting heavy objects and of bringing them to a height are thus the ones mentioned by us.</s>

<s id="A18-3|09|02">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).</s>

<s id="A18-3|09|03">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.</s>

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<s id="A18-3|09|07">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.</s>

<s id="A18-3|10|01">10 Some also want, with this procedure, to lift big pillars and let them down on their bases in any place.</s>

<s id="A18-3|10|02">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.</s>

<s id="A18-3|10|03">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.</s>

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<s id="A18-3|10|05">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.</s>

<s id="A18-3|11|01">11 Some wanted to move big loads by sea after the following method.</s>

<s id="A18-3|11|02">Namely, one makes from wood a quadrangular raft, whose individual parts are fastened to each other by nails and bolts.</s>

<s id="A18-3|11|03">One makes for it strong walls and brings it onto the water, where one wants to load the load.</s>

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<s id="A18-3|11|07">Then one has the two boats go out to sea and they plow through it, carrying the raft.</s>

<s id="A18-3|12|01">12 Others thought of transporting big blocks of stone in the same manner, swimming on the sea.</s>

<s id="A18-3|12|02">Some applied the following method to straighten up walls that inclined in earthquakes.</s>

<s id="A18-3|12|03">One digs, on the side towards which the wall is inclined, following the length of the wall, a trench into the ground.</s>

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<s id="A18-3|12|09">Then one removes the beams and the wall stands firmly in its perpendicular state.</s>

<s id="A18-3|13|01">13 What is necessary for moving loads and what is useful for this we have now explained in a sufficient manner.</s>

<s id="A18-3|13|02">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.</s>

<s id="A18-3|13|03">The beam called Oros, which others also call press, is nothing but a lever and its Hypomochlion.</s>

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<s id="A18-3|13|07">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.</s>

<s id="A18-3|14|01">14 We now want to look at the suspending of the stone [block].</s>

<s id="A18-3|14|02">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).</s>

<s id="A18-3|14|03">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).</s>

<s id="A18-3|14|04">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.</s>

<s id="A18-3|15|01">15 We have yet another method to lower the [piece of] wood called Oros and to lift the stone [block] called Lithos.</s>

<s id="A18-3|15|02">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.</s>

<s id="A18-3|15|03">When lifting the stone [block] we also have to use long spokes, in order to rotate the shaft with them.</s>

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<s id="A18-3|15|28">This procedure is strong and sound, with a secure outcome (safe) and without much effort.</s>

<s id="A18-3|16|01">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.</s>

<s id="A18-3|16|02">This one fills with any [kind of] material, puts it under the beam called Oros and lowers the beam down on it.</s>

<s id="A18-3|16|03">Through this one gets a wide space for what one wants to press and the work is made easier.</s>

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<s id="A18-3|16|13">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.</s>

<s id="A18-3|17|01">17 As now for the other Galeagra, the connection of its walls*) with one another is established by three crossbeams on each of them.</s>

<s id="A18-3|17|02">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.</s>

<s id="A18-3|17|03">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.</s>

<s id="A18-3|18|01">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.</s>

<s id="A18-3|18|02">First we explain the difference between them and then we describe their manufacture.</s>

<s id="A18-3|18|03">So we say that the beam called Oros is nothing but a lever that is pressed down by a weight.</s>

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<s id="A18-3|18|09">That is the difference between the tools.</s>

<s id="A18-3|19|01">19 The tools whose manufacture we are going to discuss now serve for the pressing of olive oil.</s>

<s id="A18-3|19|02">They are easy to handle, can be transported and brought to any place.</s>

<s id="A18-3|19|03">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.</s>

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<s id="A18-3|19|32">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.</s>

<s id="A18-3|20|01">20 There is another tool with a screw.</s>

<s id="A18-3|20|02">It consists of attaching two posts to the table that support the crossbeam in which is the nut.</s>

<s id="A18-3|20|03">Let the nut be in the center of this [piece of] wood.</s>

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<s id="A18-3|20|06">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.</s>

<s id="A18-3|21|01">21 The nut now is manufactured in the following way.</s>

<s id="A18-3|21|02">One takes a hard piece of wood (bc), whose length is twice as great as the nut and whose thickness is equal to it.</s>

<s id="A18-3|21|03">On the one side we make a screw (d, d) on half of the [piece of] wood, according to the description given earlier.</s>

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