Extracts from Hero of Alexandria’s “Mechanics”

Earlier this week I saw a reference online to a work by Hero of Alexandria, the ancient constructor of machines who lived at an uncertain time, possibly even in the late 1st century AD.^[1]  The reference was to his Mechanics.

In the Mechanica, I am told, Hero explored the parallelograms of velocities, determined certain simple centers of gravity, analyzed the intricate mechanical powers by which small forces are used to move large weights, discussed the problems of the two mean proportions, and estimated the forces of motion on an inclined plane.

The original Greek of the Mechanica is lost.  Fragments are quoted by the 3rd century AD author Pappus in his Mathematical Collection; but the complete work is preserved in an Arabic translation by Qusta Ibn Luqa (9th century).  (Remarkably, I found a manuscript of the Arabic online here, and a discussion of Greek mechanical texts here; an interesting page with bibliography on Hero here.).^[2]  The work was edited with a German translation by L. Nix in Heronis Alexandrini Opera quae supersunt, Leipzig: Teubner (1899-1914), vol. II.2 (1900), which may be found online at Wilbour Hall, here.  There are extracts in English in A.G. Drachmann, The Mechanical Technology of Greek and Roman Antiquity, Copenhagen: Munksgaard, 1963, some of which are quoted by Papadopoulos at this article preview here.  I wish I had access to Drachmann.  UPDATE:  I had forgotten the complete French translation by Barob Carra de Vaux.^[3]

The reference that I saw online was in the form of a blog post. The blogger posted a couple of pages of the Arabic (p.177 and 179 of the Nix edition, book II chapter 34, questions  f-i) – which relate to topics like why breaking a stick is easier if you put your knee in the middle, bending of planks, and so on- ^[4] and asked if someone would make a translation for him; someone familiar with medieval Arabic, and the relevant technical terms, which is quite a  request. He also said that he was familiar with the standard English translation, which annoyingly he did not name, but he wanted specialist interpretation of specific words, with a view to scholarly publication.  I found that, oddly, comments on the blog are only permitted from those who sent him money.^[5]

Anyway, all this made me search for whatever English translation that I could find.  The work has not been translated as a whole, as far as I could tell.

However I found that extracts have been turned into English in Morris R. Cohen and Israel E. Drabkin, A Source Book in Greek Science (1958), which is itself a rather remarkable and useful volume.  Few will have access to it, however, so I thought that I would give a couple of these extracts here.  The footnotes are those of Cohen and Drabkin.

Book I, chapters 20-23:^[6]

20. Many people have the erroneous belief that weights placed on the ground may be moved only by forces equivalent to these weights. Let us demonstrate that weights placed as described may [theoretically] be moved by a force less than any given force, and let us explain the reason why this is not the case in practice.^[7] Suppose that a weight, symmetrical, smooth, and quite solid, rests on a plane surface, and that this plane is capable of inclining toward both sides, that is, toward the right and the left. Suppose it inclines first toward the right. In that case we see that the given weight moves down toward the right, since it is in the nature of weights to move downward unless something supports them and hinders their motion. If, now, the side sloping downward is again lifted to the horizontal plane and restored to equilibrium, the weight will remain fixed in this position.

Again, if the plane is inclined toward the other side, that is, toward the left, the weight, too, will tend toward the lowered side, even if the slope is extremely small. The weight, in this case, does not require a force to set it m motion but rather a force to keep it from moving. Now when the weight again returns to equilibrium and does not tend in either direction, it remains in position without any force to support it. It continues to be at rest until the plane is made to slope towards either side, in which case the weight, too, tends in that direction Thus it follows that the weight, which is prone to move in any desired direction, requires, for its motion, only a very small force equal to the force which inclines it.^[8] Therefore the weight will be moved by any small force.

21. Pools of water that lie on non-sloping planes do not flow but remain still, not tending toward either side. But if the slightest inclination is imparted to them they flow completely toward that side, until not the least particle of water remains in its original position (unless there are declivities in the plane in the recesses of which small parts of water remain,as sometimes happens in the case of vessels).

Now this is the case with water because its parts are not strongly cohesive but are easily separable. Since, however, bodies that cohere strongly do not, naturally, have smooth surfaces and are not easily smoothed down, the result is that because of their roughness they support one another. That is, they are engaged like cogged wheels in a machine, and are consequently prevented [from rolling].^[9]

For when the parts are numerous and closely bound to one another by reason of mutual cohesion, a large coordinated force is required [to produce motion of one body made up of such parts over another]. Experience has taught men to lay logs with cylindrical surfaces under tortoises,^[10] so that these logs touch only a small part of the plane, whence only the smallest amount of friction results. Logs are thus used to move weights easily, but the weight of the moving apparatus must exceed that of the load to be moved. Others plane down boards to render them smooth, fasten them together on the ground, and coat them with grease, so that whatever roughness there is may be smoothed out. Thus they move the load with little force. Columns [cylinders], even if they are heavy, may be moved easily if they lie upon the ground in such a way that only one line is in contact with the ground. This is true also of the sphere,^[11] which we have already discussed.

22. Now if it is desired to raise a weight to a higher place, a force equal to the weight is needed. Consider a rotating pulley suspended perpendicular to the plane and turning about an axis at its midpoint. Let a cord be passed around the pulley and let one end be fastened to the weight and the other be operated by the moving force. I say that this weight may be moved by a force equal to it. For suppose that, instead of a force, there is, at the other end of the cord, a second weight. It will be seen that if the two weights are equal the pulley will not turn toward either side. The first weight is not strong enough to overbalance the second, and the second is not strong enough to overbalance the first, since both are equal. But if a slight addition is made to one weight, the other will be drawn up. Therefore, if the force that is to move the load is greater than the load, it will be strong enough to move the latter, unless friction in the turning of the pulley or the stiffness of the cords interferes with the motion.

23. Weights on an inclined plane have a tendency to move downward, as is the case with all bodies. If such movement does not take place we must invoke the explanation given above.^[12]Suppose we wish to draw a weight up an inclined plane the surface of which is smooth and even, as is also the surface of that part of the weight which rests on the plane. For our purpose we must have a force or weight operating on the other side and just balancing the given weight, that is, conserving the equilibrium so that any addition of force will be sufficient to move the weight up the plane.

To prove our contention, let us demonstrate it in the case of a given cylinder. The cylinder has a natural tendency to roll downward because no large part of it touches the surface of the plane. Consider a plane perpendicular to the inclined plane and passing through the line of tangency between the cylinder and the inclined plane.

Clearly, the new plane will pass through the axis of the cylinder and divide the cylinder into two halves. For, given a circle and a tangent, a line drawn from the point of contact at right angles to the tangent will pass through the centre of the circle. Now pass a second plane through the same line (i.e., the line at which the cylinder touches the inclined plane) perpendicular to the horizon. This plane will not coincide with the plane previously constructed, but will divide the cylinder into two unequal parts, of which the smaller lies above and the larger below. The larger part, because it is larger, will outweigh the smaller, and the cylinder will roll down. If, now, we suppose that from the larger [of the two parts into which this plane perpendicular to the horizon divides the cylinder] that amount be removed by which the larger exceeds the smaller portion, the two parts will then be in equilibrium and their joint weight will remain unmoved on the line of tangency to the inclined plane, tending neither upward nor downward. We need, therefore, a force equivalent to this difference to preserve equilibrium.^[13] But if the slightest addition be made to this force, it will overbalance the weight.

The next fragment is from Book II, chapter 34d:

d. Why do heavier bodies fall to the ground in shorter time than lighter bodies?

The reason is that, just as heavy bodies move more readily the larger is the externaI force by which they are set in motion, so they move more swiftly the larger is the internal force within themselves. And in natural motion this internal force and downward tendency are greater in the case of heavier bodies than in the case of lighter.

These kinds of works all need reliable translation.  It is telling that Cohen and Drabkin plainly just translated the German translation of Nix.

It would be interesting to find if there is a real translation of Mechanics II, 34!

1. ^[1]Evangelos Papadopoulos, here: “The chronology of Heron’s works is disputed and not absolutely certain to date. Many contradictory references on Heron exist, partly because the name was quite common. However, historians cite that he came after Apollonius, whom he quotes, and before Pappos, who cites him. This suggests that he must have lived between 150 BC and 250 AD (Thomas, 2005). In 1938, Neugebauer, based on a reference in Heron’s Dipotra book of a moon eclipse, he found that this must have happened on March 13, 62 AD. (Neugebauer. 1938). Since the reference was made to readers who could easily remember the eclipse, this suggests that Heron flourished in the late first century AD. According to Lewis (2001), and assuming that Cheirobalistra, a powerful catapult, is genuinely his, Heron should have been alive at least till 84 AD, the year in which the Cheirobalistra, was introduced.”↩
2. ^[2]The shelfmark is British Library Additional Ms. 23390, fol. 3r-50r; 17th century.↩
3. ^[3]Heron d’Alexandrie, Les Mechaniques ou l’elevateur de Heron d’Alexandrie, 1894 (Google books).  In: Journal asiatique, IXe serie, tome II, 1893, 152-289 and 420-514.  Online at Remacle.org here: http://remacle.org/bloodwolf/erudits/heron/table.htm↩
4. ^[4]The German translation is:

   “f. Warum treibt ein Schufs von der Mitte der Sehne den Pfeil auf eine grolse Entfernung hinaus?

   Weil die Spannung daselbst am stärksten und die treibende Kraft am gröfsten ist. Deshalb macht man auch die Bogen aus Hörnern, weil hierbei das Biegen möglich ist. Wenn sie stark gebogen sind, ist auch die Sehne mit dem Pfeil stärker gespannt, so dais eine gröfsere Kraft in ihn kommt und er deshalb eine weitere Strecke durchdringt. Deshalb treiben harte Bogen, deren Enden sich nicht biegen lassen, den Pfeil nur auf kurze Strecken.

   g. Warum läfst sich Holz schneller brechen, wenn man das Knie bei demselben in die Mitte bringt?

   Weil, wenn man das Knie dabei in geringere Entfernung (vom einen Ende) als die Mitte bringt, so dafs der eine der beiden Teile kürzer ist als der andre, es eine in zwei ungleiche Teile geteilte Wage ist, weshalb die von dem Knie entferntere Hand das Übergewicht über die ihm nähere hat. Die eine erreicht aber die Kraft der anderen nur, wenn beide an dem Ende des Holzes (gleichweit von der Mitte) sind.

   h. Warum ist ein Stück Holz, je länger es ist, desto schwächer und warum nimmt seine Biegung zu, wenn es in einem seiner beiden Enden aufgerichtet wird?

   Weil im langen Holze grofse Kraft auf seine Teile verteilt ist, so dafs das Ganze das Übergewicht hat über den festen Teil desselben, auf welchem es sich erhebt. Daher tritt hierbei dieselbe Erscheinung ein wie hei kurzem Holz, wenn an dessen Enden etwas hängt, das es niederdrückt. Der Zuwachs an Länge des Holzes entspricht also dem Gewichte, welches das kürzere Holz herabzieht. Deshalb begegnet dem langen Holze durch sich selbst wegen seiner Länge dasselbe, wie dem kurzen Holz, wenn an seinem Ende etwas Schweres angebunden wird.

   i. Warum benutzt man beim Zahnausziehen Zangen und nicht die Hand?

   Weil wir den Zahn mit der ganzen Hand nicht packen können, sondern nur mit einem Teil derselben; und wie es uns schwerer fällt, ein Gewicht mit nur zwei Fingern zu heben, als mit der ganzen Hand, so ist es auch schwerer für uns, den Zahn mit zwei Fingern zu packen und zu drücken, als mit der ganzen Hand. In beiden Fällen ist die Kraft dieselbe, aber die Teilung der Zange bei ihrem Nagel bewirkt dazu, dafs die Hand die Übermacht über den Zahn hat; denn es ist ein Hebel, an dessen gröfserem Teil die Hand ist, und der Abstand der Zange erleichtert das Bewegen des Zahnes. Denn die Zahnwurzel ist das, um was sich der Hebel bewegt. Weil aber der Abstand der Zange gröfser ist als die Zahnwurzel, um die sich etwas Grofses bewegt, so überwiegt die Hand über die in der Zahnwurzel liegende Kraft. Es ist nämlich kein Unterschied zwischen dem Bewegen eines Gewichtes und dem Bewegen einer Kraft, die jenem Gewichte gleichkommt. Denn wenn wir die Hand schliefsen, nachdem sie ausgebreitet war, so entsteht ein Widerstand, nicht wegen…”.  This Google Translate renders as:

   f. Why does a creature from the middle of the tendon push the arrow out to a great distance?

   Because the tension is greatest there, and the driving force is greatest. That is why the bow is made of horns, because bending is possible. When they are strongly bent, the tendon is tightened with the arrow, so that a greater force comes into it and therefore it penetrates a further distance. Therefore, hard bows, whose ends can not be bent, drive the arrow only for short distances.

   g. Why can wood break faster when the knee is brought into the middle?

   If the knee is placed at a distance (from the one end) as the middle, so that one of the two parts is shorter than the other, it is a divided into two unequal parts, which is why the hand distant from the knee The overweight has got closer to him. But the one reaches the power of the other only when both are at the end of the wood (equidistant 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 in one of its two ends?

   Because great force is distributed over its parts in the long wood, so that the whole has the preponderance over the fixed part on which it rises. Hence the appearance of the wood is like that of a short wood, when there is something hanging on the ends of it, which it presses down. The increase in the length of the wood therefore corresponds to the weight which the shorter wood draws. Therefore the long wood meets with itself by its length, as the short wood, when at its end something heavy is tied.

   i. Why use forceps when pulling teeth and not the hand?

   Because we can not grasp the tooth with the whole hand, but only with a part of it; And as it is harder for us to lift a weight with only two fingers than with the whole hand, it is also harder for us to grasp and press the tooth with two fingers than with the whole hand. In both cases the force is the same, but the division of the forceps with its nail causes the hand to have over the tooth; For it is a lever, on the greater part of which the hand is, and the distance of the forceps facilitates the movement of the tooth. Because the tooth root is what the lever moves. But because the distance between the tongs is greater than that of the tooth-root, around which a great extent moves, the hand overcomes the force in the tooth-tooth. Indeed, there is no difference between moving a weight and moving a force equal to that weight. For when we close the hand after it has spread, there arises a resistance, not because of …”.  This gives the idea at least.↩

5. ^[5]The appeal may be found here.↩
6. ^[6]p.197, n.1. “The Mechanics is extant in an Arabic version in three books. Of the original Greek only fragments are extant, the longest of which is a passage, which Pappus quotes (p 224, below), on the five simple machines. The first book deals with various problems in statics, dynamics, and kinematics, the second discusses the simple machines and includes a collection of problems similar to that in the Aristotelian Mechanics, while the third deals with the construction and operation of machines, especially those for lifting large weights For the selections given here the German translation of L. Nix has been used as a basis. Professor A S. Halkin has helped interpret the Arabic text.”↩
7. ^[7]p.197 n.2. “The general point of this and the following section (I.21) is clear. The passage constitutes an important step in the development of the principle of inertia. Particularly to be noted is the treatment of friction and the means of reducing it. There is some difficulty in the precise interpretation of certain points; see the following note.”↩
8. ^[8]p.198 n.1 “The idea seems to be that the force required to set the weight in motion on the horizontal plane is equal to the resistance that enables the weight to continue at rest while the plane is inclined until the critical angle is reached beyond which the weight moves down the plane. If a plane be inclined at this critical angle, a, we should say that the force required to draw weight W up the plane is W sin a + R, where R is the component due to friction. Hero’s point seems to be, but it is not put very clearly, that the same force would be necessary to set the weight in motion on the horizontal plane. Of course, where R approached zero, a would approach zero, as would also the force necessary to set the weight in motion on the horizontal plane.”↩
9. ^[9]p.198 n.2. “I.e., even in the case where the surface is somewhat inclined. The text is not clear but the reference may be to the interaction of a body and the surface on which it rests.”↩
10. ^[10]p.199 n.1: “The reference is to sheds moved by rollers and used in military operations.”↩
11. ^[11]p.199 n.2: “Here the tangency approaches a single point.”↩
12. ^[12]p199 n.3 “I.e., the effect of friction.”↩
13. ^[13]p.200 n.1 “The method of Hero amounts to holding that (in the case of a cylinder of radius r, height A, and density d, rolling on a plane inclined to the horizontal at angle a) the force (F) required to draw the weight (W) up the plane is r = W[2(a + cos a sin a) /r] — 2r²hd (a + cos a sin a). The modern formulation is F = W sin a. Hero’s formula is obtained by noting that he takes the part of the cylinder standing over LMN as the part to be balanced by force F, so that F/W = area LMN/(πr²)\ and area LMN = sector MON + 2 ΔNOL = r² (2a + sin 2a). Compare the problem of the inclined plane as formulated and treated by Pappus (p. 194). The modern solution was not discovered until the days of Stevinus and Galileo.”↩