Month: April 2017

The date of Hero of Alexandria, and another translation of some extracts of the “Mechanics”

Posted on by Roger Pearse

When did Hero of Alexandria live?  The truth is that we know little other than what can be inferred from his works.

Karin Tybjerg[1] tells us that Hero quotes Archimedes, who lived ca. 287-211 or 212 BC, and is quoted by Pappus who flourished around 320 AD.  But it seems that in his Dioptra Hero refers to a lunar eclipse visible at Alexandria and Rome.  The only one that fits these criteria happened in 62 AD, around the time that St. Paul was released from house arrest in Rome, and also around the time that Mark’s gospel was being written.  Ptolemy (fl. ca. 127-158 AD) does not make use of Hero, which is perhaps an upper limit.

It has been speculated that a new model of water-organ, demonstrated to Nero in 68 AD, was probably Hero’s invention; which means that he might even have been in Rome at the same time as the apostles.  At his lavish new palace, the Domus Aurea, a new and ingenious technology entertains an emperor and his court.  Meanwhile, across the city the apostle Peter is addressing a humble congregation.  It is a reminder that the most important events of an era are not always the most heralded.

Hero’s Mechanics is about how to lift heavy weights.  The first two books go through a number of principles for doing so with limited power, while the third book describes designs for weights and presses.

I’ve come across another handbook which contains an English translation of some extracts.  This is G. Irby-Massie & P.T. Keyser, Greek Science of the Hellenistic Era : A Sourcebook, 2002, starting on p.168.  They are taken from Drachmann’s handbook which I have yet to see.[2]  Here they are:

6.11. Heron of Alexandria

Mechanics 1.20–21: weights; 2.1.1: simple machines; 2.3: the pulley; 3.2.1–2: the crane

There are many who think that weights lying on the ground are only moved by an equal force [contrast Aristotle, Physics 7.5 (250a11–19)], wherein they hold wrong opinions. So let us prove that weights placed in the way described are moved by an arbitrarily small force, and let us make clear the reason why this is not evident in fact. Let us imagine a weight lying on the ground, and let it be regular, smooth and with its parts coherent with each other. And let the surface on which the weight lies be flat, smooth and completely joined, and able to be inclined to both sides, i.e., to the right and the left. And let it be inclined first towards the right. It is then evident to us that the given weight must incline towards the right side, because the nature of weights is to move downwards, if nothing holds them and hinders them from movement; and again if the inclined side is lifted to a horizontal position and will be level [i.e., in equilibrium], the weight will come to rest in this position. And if it is inclined to the other side, i.e., to the left side, the weight will again sink towards the inclined side, even if the inclination is very small, and so the weight will need no force to move it, but will need a force to hold it so that it does not move. And if the weight again becomes level without inclination to either side, then it will stay there without a force holding it, and it will not cease being at rest until the surface inclines to one side or another, and then it will incline towards that side. Thus the weight that is ready to incline to whichever side, does it not require only a small force to move it, namely as much force as causes the inclination? And so isn’t the weight moved by any small force?

21.  Now, water on a surface that is not inclined will not flow, but remains without inclining to either side. But if the slightest inclination occurs, then all of it will flow towards that side, till not the smallest part of the water remains thereon, unless there are hollows in the surface, and small amounts stay in the bottom of the hollows, as happens often in vessels. Now water inclines like this because its parts lack cohesion and are very soluble.
As for the bodies that are coherent, since by their nature they are not smooth on their surfaces and not easily made smooth, it happens through the roughness of the bodies that they strengthen each other, and it happens that they lean upon each other like teeth, and they are strengthened thus, for if the teeth are numerous and closely joined, they require a strong and coherent force [to separate them]. And so from experiment people gained understanding: under tortoises [war machines: see Athenaios above, Section 6.9] they placed pieces of wood whose surfaces were cylindrical and so did not touch more than a small part of the surface, and so only very little rubbing occurred. And they use poles to move the weight on them easily, even though the weight is increased by the weight of the tools. And some people put on the ground cut boards (because of their smoothness) and smear them with grease, because thus their surface roughness is made smooth, and so they move the weight with smaller force. As for the cylinders, if they are heavy and lie on the ground, so that the ground does not touch more than one line of them, then they are moved easily, and so also balls; and we have already talked about that [1.2–7].

2.1.1 [simple machines (surviving in Greek)]

Since the powers by which a given weight is moved by a given force are five, it is necessary to present their form and their use and their names, because these powers are all derived from one natural principle, though they are very different in form. Their names are as follows: the axle-in-wheel [windlass], the lever [mochlos], the pulley [trochilos], the wedge [sphên], and what is called the “endless” screw [kochlia].

[construction of the axle-in-wheel]

[2.2 the lever]

2.3 [pulley (surviving in Greek)]

The third power [pulley] is also called the ‘‘multi-lifter” [poluspaston].

Whenever we want to move some weight, if we tie a rope to this weight we pull with as much force as is equal to the burden. But if we untie the rope from the weight, and tie one of its ends to a stationary point and pass its other end over a pulley fastened to the burden and draw on the rope, we will more easily move the weight. And again if we fasten on the stationary point another pulley and run the end of the rope through that and pull it, we will still more easily move the weight. And again if we fasten on this weight another pulley and run the end of the cord over it, we will much more easily move the weight. And in this way, each time we add pulleys to the stationary point and to the burden, and run one end of the rope through the pulleys in turn, we will more easily move the weight. And every time the number of pulleys through which the rope runs is increased, it will be easier to lift that weight. The more “limbs” [kôla] the rope is bent into, the easier the weight will be moved.

And one end of the rope must be securely tied to the stationary point, and the rope must go from there to the weight (Figure 6.8). As for the pulleys that are on the stationary point they must be fastened to one piece of wood, turning on an axle, and this axle is called manganon; and it is tied to the stationary point with another rope. And as for the pulleys that are fastened to the burden, they are on another manganon like the first, tied to the burden. The pulleys should be so arranged on the axles that the “limbs” do not get entangled and unwieldy. And why the ease of lifting follows from the number of “limbs,” and why the end of the rope is tied to the stationary point, we shall explain in the following.

3.2.1–2 [the crane (surviving in Greek): compare Vitruvius 10.2.8]

1. For the lifting of burdens upwards there are certain machines: some have one mast, and some have two masts, and some have three, and some have four masts. As for the one that has a single mast it is made in this way. We take a long piece of wood, longer than the distance to which we want to raise the burden, and even if this pole is strong in itself, we take a rope and coil it round, winding it equally spaced, and draw it tight. The space between the single windings should not be greater than four palms [ca. 30 cm], and the windings of the rope are like steps for the workers and they are useful for anyone wanting to work on the upper section. And if the pole is not elastic, we must estimate the burdens to be lifted, lest the mast be too weak.

2 This mast is erected upright on a piece of wood, and three or four ropes are  fastened to its top, stretched and tied to fixed points, so that the beam, however it is forced, will not give way (being held by the ropes). Then they attach to its top pulleys tied to the burden. Then they pull on the rope either by hand or with another engine, until the burden is raised [Figure 6.9].

[3.3–5: two-, three-, and four-mast cranes]

 

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  1. [1]Karin Tybjerg, “Hero of Alexandria’s Mechanical Treatises”, in: Astrid Schurmann (ed), Physik / Mechanik, Stuttgart (2005), 204-226.  Preview here.↩
  2. [2]Drachmann [1963] 46–47, 50, 53–55, 98–99; Drachmann, A.G. (1963) Mechanical Technology of Greek and Roman Antiquity, Copenhagen: Munskgaard.↩

Cleomedes: how big is the earth?

Posted on by Roger Pearse

Some time between Posidonius and Ptolemy, i.e. between the 1st century BC and the 2nd century AD, a Greek named Cleomedes wrote a 2 book basic treatise on astronomy, De motu circulari corporum caelestium.[1]  This was based mainly on the lost work of Posidonius, but also on others.

Cleomedes is our primary source for the calculations of Erastothenes, who measured the earth in the 3rd century BC.  I had never seen this, and I happened across a translation of the relevant portion, so I thought that it might be interesting to reproduce the passage here.

The text was edited in 1891 in the Teubner series by Herman Ziegler, who based his edition upon three manuscripts which he describes in a vague way.  I have supplemented his atrocious list from Pinakes:

  •  Florence, Mediceo-Laurentianus Plutei LXIX, 13.  12th century, fol.137v-164, where Cleomedes follows Diogenes Laertius, Lives of the Philosophers.
  • Leipzig Universitäts-Bibliothek gr. 16 (once 361; 250), foll. 286-298.
  • Nuremberg, Stadtbibliothek, fonds principal, Cent. V. App. 37, 15th century.

He casually informs us that others exist at the Marciana library in Venice (no. CCXIV and CCCVIII, supposedly 11-12th century), and six others which he does not trouble to name.  Fortunately we have the Pinakes list of manuscripts here.

As he gives a shelfmark only for the Florence ms., we may reasonably infer that Ziegler followed the Florence manuscript, and borrowed material from others as he found it useful.  But we must remember that the task of editing a technical astronomical work may well have been considerable.

Edit (3rd Jan. 2019): A correspondent tells me that a new Teubner edition by R. B. Todd came out in 1990, with a proper account of the manuscripts.  This I have not seen, however.

Cleomedes has recently been translated into English, I find, although again I have not seen this; Alan C. Bowen, Robert B. Todd, Cleomedes’ Lectures on Astronomy. A Translation of The Heavens with an Introduction and Commentary. University of California Press, 2004.  A French translation by “R. Goulart” in 1980[2] is described as “faulty” in the Oxford Classical Dictionary.[3]

Here’s the relevant passage, from Heath’s translation of Cleomedes, On the orbits of the heavenly bodies, I, 10.[4]

About the size of the earth the physicists, or natural philosophers,  have held different views, but those of Posidonius and Eratosthenes are preferable to the rest. The latter shows the size of the earth by a geometrical method; the method of Posidonius is simpler. Both lay down certain hypotheses, and, by successive inferences from the hypotheses, arrive at their demonstrations.

Posidonius says that Rhodes and Alexandria lie under the same meridian. Now meridian circles are circles which are drawn through the poles of the universe and through the point which is above the head of any individual standing on the earth. The poles are the same for all these circles, but the vertical point is different for different persons. Hence we can draw an infinite number of meridian circles. Now Rhodes and Alexandria lie under the same meridian circle, and the distance between the cities is reputed to be 5,000 stades.[5] Suppose this to be the case.

All the meridian circles are among the great circles in the universe, dividing it into two equal parts and being drawn through the poles. With these hypotheses, Posidonius proceeds to divide the zodiac circle, which is equal to the mendian circles, because it also divides the universe into two equal parts, into forty-eight parts, thereby cutting each of the twelfth parts of it (i.e., signs) into four. If, then, the meridian circle through Rhodes and Alexandria is divided into the same number of parts, forty-eight, as the zodiac circle, the segments of it are equal to the aforesaid segments of the zodiac. For, when equal magnitudes are divided into (the same number of) equal parts, the parts of the divided magnitudes must be respectively equal to the parts. This being so, Posidonius goes on to say that the very bright star called Canopus lies to the south, practically on the Rudder of Argo. The said star is not seen at all in Greece; hence Aratus does not even mention it in his Phaenomena. But, as you go from north to south, it begins to be visible at Rhodes and, when seen on the horizon there, it sets again immediately as the universe revolves.1 But when we have sailed the 5,000 stades and are at Alexandria, this star, when it is exactly in the middle of the heaven, is found to be at a height above the honzon of one-fourth of a sign, that is,one forty-eighth part of the zodiac circle.’ It follows, therefore, that the segment of the same mendian circle which lies above the distance between Rhodes and Alexandria is one forty-eighth part of the said circle, because the honzon of the Rhodians is distant from that of the Alexandrians by one forty-eighth of the zodiac circle. Since, then, the part of the earth under this segment is reputed to be 5,000 stades, the parts (of the earth) under the other (equal) segments (of the meridian circle) also measure 5,000 stades; and thus the great circle of the earth is found to measure 240,000 stades, assuming that from Rhodes to Alexandria is 5,000 stades; but, if not, it is in (the same) ratio to the distance. Such then is Posidonius’ way of dealing with the size of the earth.

The method of Eratosthenes1 depends on a geometrical argument and gives the impression of being slightly more difficult to follow. But his statement will be made clear if we premise the following. Let us suppose, in this case too, first, that Syene and Alexandria he under the same meridian circle, secondly, that the distance between the two cities is 5,000 stades;1 and thirdly, that the rays sent down from different parts of the sun on different parts of the earth are parallel; for this is the hypothesis on which geometers proceed Fourthly, let us assume that, as proved by the geometers, straight lines falling on parallel straight lines make the alternate angles equal, and fifthly, that the arcs standing on (i e., subtended by) equal angles are similar, that is, have the same proportion and the same ratio to their proper circles—this, too, being a fact proved by the geometers. Whenever, therefore, arcs of circles stand on equal angles, if any one of these is (say) one-tenth of its proper circle, all the other arcs will be tenth parts of their proper circles.

Any one who has grasped these facts will have no difficulty in understanding the method of Eratosthenes, which is this Syene and Alexandria lie, he says, under the same mendian circle. Since meridian circles are great circles in the universe, the circles of the earth which lie under them are necessarily also great circles. Thus, of whatever size this method shows the circle on the earth passing through Syene and Alexandria to be, this will be the size of the great circle of the earth Now Eratosthenes asserts, and it is the fact, that Syene lies under the summer tropic. Whenever, therefore, the sun, being m the Crab at the summer solstice, is exactly in the middle of the heaven, the gnomons (pointers) of sundials necessarily throw no shadows, the position of the sun above them being exactly vertical; and it is said that this is true throughout a space three hundred stades in diameter.1 But in Alexandria, at the same hour, the pointers of sundials throw shadows, because Alexandria lies further to the north than Syene. The two cities lying under the 9ame meridian great circle, if we draw an arc from the extremity of the shadow to the base of the pointer of the sundial in Alexandria, the arc will be a segment of a great circle in the (hemispherical) bowl of the sundial, since the bowl of the sundial lies under the great circle (of the meridian). If now we conceive straight lines produced from each of the pointers through the earth, they will meet at the centre of the earth. Since then the sundial at Syene is vertically under the sun, if we conceive a straight line coming from the sun to the top of the pointer of the sundial, the line reaching from the sun to the centre of the earth will be one straight line. If now we conceive another straight line drawn upwards from the extremity of the shadow of the pointer of the sundial in Alexandria, through the top of the pointer to the sun, this straight line and the aforesaid straight line will be parallel, since they are straight lines coming through from different parts of the sun to different parts of the earth. On these straight lines, therefore, which are parallel, there falls the straight line drawn from the centre of the earth to the pointer at Alexandria, so that the alternate angles which it makes arc equal. One of these angles is that formed at the centre of the earth, at the intersection of the straight lines which were drawn from the sundials to the centre of the earth; the other is at the point of intersection of the top of the pointer at Alexandria and the straight line drawn from the extremity of its shadow to the sun through the point (the top) where it meets the pointer * Now on this latter angle stands the arc carried round from the extremity of the shadow of the pointer to its base, while on the angle at the centre of the earth stands the arc reaching from Syene to Alexandria. But the arcs are similar, since they stand on equal angles. Whatever ratio, therefore, the arc in the bowl of the sundial has to its proper circle, the arc reaching from Syene to Alexandria has that ratio to its proper circle. But the arc in the bowl is found to be one-fiftieth of its proper circle.’ Therefore the distance from Syene to Alexandria must necessarily be one-fiftieth part of the great circle of the earth. And the said distance is 5,000 stades; therefore the complete great circle measures 250,000 stades. Such is Eratosthenes’ method.

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  1. [1]H. Ziegler, Cleomediis de Motu Circulari Corporum Caelestium Libri, Leipzig: Teubner, 1891, at Google Books here.  Includes Latin translation.  There is an updated Teubner by R.B. Todd, 1990.↩
  2. [2]Probably R. Goulet, Cléomède: Théorie Élémentaire. Texte présenté, traduit et commenté, Paris, 1980.↩
  3. [3]Simon Hornblower etc, The Oxford Classical Dictionary, OUP, 2012. Article preview here on p.331.  This dates the work at ca. 360 AD, and says that Cleomedes’ account of the work of Erastothenes is “mostly fictitious”.↩
  4. [4]Translated by T. L. Heath in Greek Astronomy, London (1932); via Cohen & Drabkin, A source book in Greek Science, 1948, p.149-153.  This contains diagrams and detailed but rather unclear comment.↩
  5. [5]Which is actually too much↩

Good Friday – the Pilate Stone

Posted on by Roger Pearse

It is Good Friday today.  By chance I found myself looking on Twitter at a picture of the so-called “Pilate stone”.  This is the inscription which mentions Pontius Pilate.  Most of us will be familiar with its existence, but it seems appropriate to gather some of the information about it.

In 1961 an Italian expedition was conducting the third season of excavations at Caesarea.  They found an inscribed stone in situ in the remains of the Roman theatre, where it was being used as the landing in a flight of steps which led up to the seating.  The stone was placed there during rebuilding in the 4th century AD.  In the process of reuse, the left-hand third of the inscription had been chiselled away.[1]  The stone is 82 cm high, 68 cm wide, and 20 cm thick.  The letters are 6-7 cm high, and the spaces between the lines 3-4 cm.[2]

The inscription was published by A. Frova, L’Iscrizione di Ponzio Pilato a Cesarea, Rend. Istituto Lombardo, accademia di scienze e lettere, classe di lettere 95, Milan, 1961 (Pp. 419-34, 1 map and 2 plates), which I have not seen.  It appears, I am told, in L’Annee Epigraphique in 1963 as entry 104 (ref: AE 1963 no. 104).

Three lines of the inscription are legible, and there is an acute accent from a fourth line.  Here is a transcription:

A useful picture from the web shows this, with the possible missing text.

Frova suggested that the starting “S” is perhaps the end of “Caesariensibus”.  Also there is an acute accent – an “apice” -, just like the one over the E of Tiberieum, on the fourth line.  This, it is speculated, belongs to an E, which perhaps was part of DEDIT, I.e. “he has given”.

If we accept this, we would get us something like “To the Caesareans, the Tiberium Pontius Pilate, Prefect of Judaea, ?? has given ??”, I.e. Pontius Pilate, Prefect of Judaea, has given this Temple of Tiberius to the people of Caesarea.

Sherwin-White remarked that this confirmed his own hypothesis as to the title that Pilate held.  The title of Procurator was introduced by Claudius, and its use for Pilate by Tacitus and Josephus is perhaps simply a case of those authors using the contemporary title for a provincial governor, rather than one that had dropped out of use.

Not everyone agrees with Forva’s reconstruction, or the interpretation of the Tiberium as a temple.  An alternative proposed by Géza Alföldy in 2012[3] would see it as a lighthouse, one of a pair built by Herod, now restored by Pilate for the benefit of the sailors.  He would thus read:

Nautis Tiberieum
– Pontius Pilatus
[praef]ectus Iudae[a]e
[4]e[cit]

Josephus tells us (Jewish War I, 412; Antiquities XV, 336) that Herod built colossal lighthouses at Caesarea, the largest of which stood on the western entrance to the port was named after Augustus’ step-son Drusus, Tiberius’ brother.  This then was the “Drusion”.  Alfoldy surmises that the “Tiberion” was therefore another lighthouse, perhaps on the eastern entrance of the double port.

The original stone is now in the Israel Museum in Jerusalem:

A reproduction is at Caesarea.

On which note, may I wish everyone a Happy Easter!

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  1. [1]A.N. Sherwin-White, Review of L’Iscrizione di Ponzio Pilato a Cesarea by A. Frova, JRS 54 (1964), 258-9.  JSTOR.↩
  2. [2]J. Vardaman, ‘A New Inscription Which Mentions Pilate as “Prefect”‘, Journal of Biblical Literature 81 (1962), 70-1.  JSTOR.↩
  3. [3]Géza Alföldy, “L’iscrizione di Ponzio Pilato: una discussione senza fine?” In: Gianpaolo Urso (ed), Iudaea socia – Iudaea capta, (= I Convegni della Fondazione Niccolò Canussio. Band 11). Edizioni ETS, Pisa (2012), p. 137-150. Online here.↩
  4. [4]↩

Extracts from Hero of Alexandria’s “Mechanics”

Posted on by Roger Pearse

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!

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  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.”↩