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 Roman 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:[3]

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[4] 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
(ref)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]By Frova, via the Vardaman article.↩
  4. [4]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.↩

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.”↩

“Burned without pity” – the fake quotation taken back to 1930!

Posted on by Roger Pearse

A few weeks ago, I discussed a fake quotation attributed to Pope Innocent III:

Anyone who attempts to construe a personal view of God which conflicts with church dogma must be burned without pity.

These kinds of “quotes” are often derived from opinions by modern writers, which someone has then turned into a quote by the object of the opinion.  And so it proved; it was a quote from Peter Tompkins, The Magic of Obelisks.

But a correspondent then pointed out that the phrase appeared earlier, in the English translation of a book by Frenchman Maurice Magre, in 1931.  The US title was “Magicians, Seers and Mystics”; the UK title “Return of the Magi”, London, 1930.

This I have now obtained, and as it is public domain, I have uploaded it to Archive.org here.  And indeed this is correct – on p.60, our “quote” appears.

Three terrible figures dominate the great Albigensian massacre. For the massacre to be possible, it was necessary that an extraordinary genius for violence, for organisation, and for hypocrisy, should take shape in three men, who were all equally devoid of pity and, possibly, equally sincere in their hatred of heresy and love of the Church.

It was Pope Innocent III who, with obstinate determination, desired and decided on the crusade. The murder of the papal legate Pierre de Castelnau was only a pretext. Historians are unanimous in gloryifing this pope. To them the great men of history are men who do something, who have a powerful will and exert it to attain an aim. It makes no difference whether the aim is sublime or abominable; it is success in attaining the aim which gives the measure of genius.

As soon as he was elected pope, in all his public utterances Innocent III began to talk of ” exterminating the impious.” It was the dominating idea of his life, and he realised it wholeheartedly. He had a deep-rooted conviction that any man who attempted to build up a personal view of God which conflicted with the dogma of the Church must be burned without pity at the stake.

Italics mine.

No reference or source is given for the claim.

The author appears to be an occultist.  A chapter is devoted to the supposed origins of Rosicrucianism, recited uncritically; another to a biography of Apollonius of Tyana, equally uncritically given.  The prose style of the author is that of a historian; the content is nonsense.

With luck, this is the final origin of this striking phrase.

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Life of the Coptic Patriarch Isaac (686-689 AD) by Anthony Alcock

Posted on by Roger Pearse

Anthony Alcock has kindly translated for us all a Bohairic Coptic account of the life of the Coptic patriarch Isaac (686-689 AD), which he has sent to me for publication.  The PDF is here:

Isaac does appear in the History of the Patriarchs of the Coptic Church, but only briefly – this Life is much longer, but also hagiographical.  It is translated from the text in the Patrologia Orientalis 11 (1914).

Our thanks to Dr. Alcock for making this accessible!

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From my diary

Posted on by Roger Pearse

I am now recovered from the virus that struck me down last week.  Thank you everyone who prayed for me.  I’ve spent the last few days preparing for a job interview with a new client.  This required quite a bit of revision of my skills, for the inevitable technical test.  But I was successful. Evidently the “other guy” bombed the technical test, for they got back to me in a couple of hours, before I had even got home.  So gainful work beckons in a week’s time.  Sadly I shall have to spend much of that week haggling over contractual terms, but such is the nature of the business.  It is my dream to find a client one day who does not attempt to impose an unreasonable contract.  But in twenty years this has yet to happen.  It will be back to dwelling in a hotel for four nights a week.

In the limited time available, I’ve been OCRing the 1845 French translation of Serenus Sammonicus.  It looks to me as if Google Translate would process this into English quite easily.  Indeed the Latin itself is not difficult.  So I will carry on with this.

An email today asked me if I knew where the “pine cone” in the Vatican courtyard came from.  It was previously in the atrium before Old St Peter’s.  Before that … I can’t say that I know.  Something to think about one day!

I have a little list of blog post topics that has built up over the past few weeks.  I shall get to them all one day!

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From my diary

Posted on by Roger Pearse

A week of illness, as I have had yet another bout of fever and stomach troubles, assisted by a dental problem.  It seems odd to get two episodes of the fever within two weeks, but so it is.  I’ve been unable to respond much to emails.

Most days I have taken a ten minute drive down to the sea front, and walked a little on the promenade.  The section that I go to is far from the pier, which means that it is quieter, and fortunately the street parking is free, which is a blessing.  I can sit there and gaze at the waters for a bit, without being charged.  Two very sunny days presage the coming of summer; but today we’re back to cold March weather.

The rest of my days has been spent lying on the sofa, and reading websites.  Many of these are political, and of course these are always full of the latest crisis; which is not really good reading for an invalid.  My twitter feed is now entirely without political content.  In fact I’ve been reading lots of posts about the Sudan and its monuments.  I’ve even followed the Khartoum weather centre tweets – somehow hearing that it is 35ºC out there, under a cobalt sky, is cheering.

At night I’ve been reading from Paupers and Pig-Killers: The Diary of William Holland, a Somerset Parson, 1799-1818 (Amazon).

This is what it sounds like: the daily journal of a clergyman, complete with sharp observations of his neighbours.  Each entry is short, and it is just as gently readable as James Woodforde, The Diary of a Country Parson (1758-1802), Amazon, who lived in Norfolk.  Sadly a lot of the volumes of Holland’s diary are lost, including the year of Waterloo.

Almost any account of daily life, it seems, is of interest once enough time has passed.

I’m a little sick of reading Twitter, so I have pulled down the white hardback of Kent Weeks’ The Lost Tomb.

Weeks is an archaeologist working in the Valley of the Kings in Egypt.  Reliant on whatever funding he can obtain – often, clearly, meagre – he has done what has never been done, and actually mapped the valley in a professional way.  It is extraordinary, considering a century of digging, that no proper surveys were ever taken.  In the process he relocated the sprawling tomb of the sons of Ramasses II, KV 5, which was lost; and excavated it, finding many galleries and stairways never seen before.  There used to be an excellent website, www.kv5.com, but this now redirects to his main website.  There are still details there, and a zoomable map, of which this is a screen grab.  You go down steps into room 1 (marked by my red arrow), into the main chamber, and doors go back and down steps behind you.


Note the unexplored corridor leaving the main chamber at the SW (top left).  I wonder why it has been left?  The corridor going down stairs at the SE (bottom left) itself has not been fully cleared, it seems.

I can’t say that I have ever read Weeks 1998 volume since I obtained it.  It is a big white hardback, and unfortunately Weeks is not an engaging writer.

I have still been throwing out books that I have not reread, or which I feel I might not read again.  Being ill though leads me to doubt – won’t I want some of these?  It is an odd feeling, to see gaps in my shelves, after all these years.

I’ve also been looking for a new contract, as I work freelance.  There ought to be more than there is, I feel.  No luck yet.  It usually happens suddenly.  All the same, the government here is trying to tax the heck out of my sector, and passing endless new laws.  This, together with the political uncertainty of Brexit and Trump, is causing employers to hesitate.  Your prayers for a contract reasonably close to home that pays well would be appreciated.

I shall now return to my sofa!

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Nice big image of the Nuremberg drawing of Rome (with Meta Romuli)

Posted on by Roger Pearse

Again seen on twitter, a link to a wonderfully large image of the page in the Nuremberg Chronicle (1493) with a picture of Rome.  It may be found here, but I reproduce it below because images vanish from the web like butterflies.  The basilica of Old St Peter’s may easily be seen; but also the vanished pyramidal monument, the Meta Romuli, between St Peter’s and the Castel S. Angelo.

Click to enlarge.

UPDATE: A correspondent writes to say that the Nuremberg Chronicle 1497 (767 pages) is available at the following web sites:

  • University of Adelaide (Australia) ebooks here .
  • Bayerische Staatsbibliothek here (downloadable as a PDF file (349MB) and also as JPEG files (300DPI).

Always useful to know!

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The Meta Sudans in 1849 in Pierre Monami

Posted on by Roger Pearse

A twitter post alerted me to the existence of an oil-painting from 1849 by Pierre Monami, depicting the Roman forum with the Arch of Constantine, the Meta Sudans, the Temple of Venus and Rome, and the Via Sacra leading to the Arch of Titus.  The painting was sold recently at Bonhams, who have a viewer on it here.

Zooming in we get this:

P. Monami, The Meta Sudans, 1849 (excerpt)

The most notable feature is that the Meta Sudans looks pretty much exactly as it does in 1930. The demolition of the top section took place earlier, it seems.

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The Annals of Eutychius of Alexandria (10th c. AD) – chapter 19b – Abbasids part 2

Posted on by Roger Pearse

We continue with a couple more caliphs.

CALIPHATE OF HARUN AR-RASHID (170-193 / 786-809).

1. The bay’ah was given to Harun ar-Rashid b. al-Mahdi – his mother was al-al-Khayzuran – in the same night that Musa al-Hadi died, the night of Friday 14 Rabi al-awwal in the year 170.  That night his son al-Ma’mun was born.  He entrusted the management of his business in Yahya b. Khalid b. Barmak.  During his caliphate he made the pilgrimage to Mecca  nine times and he invaded the territories of Rum eight times.  He removed his favour from the Barmakees in the month of Safar of the year 187 of the Hegira. His caliphate lasted twenty-three years, two months and sixteen days.

2. Leo (IV), son of Constantine, son of Leo, king of Rum, died.  After him there was made king of Rum Nicephorus (I), son of Istirāq[1], who asked for a truce from [Harun] ar-Rashid.  Ar-Rashid gave him a respite of three years.  There ruled in Egypt, in the name of ar-Rashid, Musa b. Isa al-Hashimi, who extended the Great Mosque of Misr at the rear of the building which may still be seen.  Ar-Rashid then deposed Musa ibn Isa and entrusted the government of Egypt to Abd Allah ibn al-Mahdi.  Abd Allah sent as a gift to ar-Rashid a young girl of his choice from among the Yemenis who lived in the south of Egypt.  She was very beautiful and ar-Rashid fell intensely in love.  The young girl was then hit by a serious disease.  The doctors cared for her but no medicine was effective.  They said to ar-Rashid: “Send word to your governor in Egypt, Abd Allah, to send you an Egyptian doctor.  The Egyptian doctors are more able than those of Irāq to cure this young girl.”  Ar-Rashid sent word to Abd Allah ibn al-Mahdi to choose the most skillful Egyptian doctor and send him to him, telling him about the young girl and of what had happened.  Abd Allah sent for Politianus, the Melkite Patriarch of Alexandria, expert in medicine, made him aware of the young girl and the disease that had struck her, and sent him to ar-Rashid.  [Politianus] brought with him some Egyptian durum “ka’k”[2], and some pilchards.  When he arrived in Baghdad and presented himself to the young girl, he gave her some rustic ka’k and pilchards to eat.  The young girl recovered her health at once, and the pain disappeared.  After that [ar-Rashid] began to order from Egypt, for the sultan’s use, durum ka’k and pilchards.  Ar-Rashid gave lots of money to the patriarch Politianus and gave him in writing an order which provided that all the churches that the Jacobites had taken away from the Melkites and of which they had taken possesion, should be returned.  The patriarch Politianus returned to Egypt and got back his churches.  The patriarch Politianus died after having held the patriarchal seat for forty six years.  After him there was made patriarch of Alexandria Eustathius[3], in the sixteenth year of the Caliphate of ar-Rashid.  Eustace was a linen-maker and had found a treasure in the house in which he used to prepare linen.  He had embraced the monastic life at “Dayr al Qusayr”, later becoming the superior.  He built at “Dayr al-Qusayr” the church of the Apostles, and a residence for the bishops.  Later he was made patriarch of Alexandria, held the office for four years and died.  After him there was made patriarch of Alexandria Christopher[4] in the twentieth year of the Caliphate of ar-Rashid.  The patriarch Christopher was hit by hemiplegia and could only move if supported.  There was therefore appointed a bishop named Peter after a vote whom the bishops put in place of the patriarch.  Christopher held the office for thirty-two years and died.  In the eighth year of the Caliphate of ar-Rashid there was made patriarch of Antioch Theodoret.  He held the office for seventeen years and died.  During the caliphate of ar-Rashid there was, after the afternoon prayer, an eclipse of the sun so intense that you could see the stars, and people stood screaming at the sky imploring God – may His name be glorious!  In Khurasan there rebelled against ar-Rashid, Rafi ibn al-Layth and occupied it.  Ar-Rashid invaded Khurasan, but at Gurgan he became ill and stopped at Tus, sending al-Ma’mun to Merv to the head of a large army.

3. Ar-Rashid died in the month of Jumādà al-Akhar in the year 193 [of the Hegira], at the age of forty-six.  He was buried in Tus, in the city of an-Nirāt [5].  The sons who were with him, those of his family and his commanders gave the bay’ah to his son Muhammad ibn Zubaydah.  Al-Fadl ibn ar-Rabi returned with his men to Baghdād.  Ar-Rashid was of perfect stature, handsome of face, with a black and flowing beard which he used to cut when he went on pilgrimage.  The leaders of his bodyguard were al-Qasim ibn Nasr b. Malik first, then Hamza ibn Hazim b. Obayd Allah b. Malik, then Hafs ibn Umar b. ash-Shugayr.  His hāgib was Bishr ibn Maymun b. Muhammad b. Khalid b. Barmak.  Then al-Fadl ibn Rabi regained this position.

CALIPHATE OF MUHAMMAD AL-AMIN (193-198 / 809-814).

1. The news of the death of ar-Rashid arrived in Baghdad on Wednesday, twelve days before the end of Jumāda al-Akhar.  The crowds gathered, his son Muhammad went out in the pulpit, and invited them to mourn his death.  The people gave him the bay’ah on that day.  Then there appeared strong differences between him and his brother al-Ma’mun.  The mother of Muhammad al-Amin was called Umm Jaffar[6], and was the daughter of Abu Jaffar al-Mansur.  Muhammad al-Amin sent Ali ibn Isa b. Mahan to Khurasan to fight against al-Ma’mun, who sent against him, from Merv, Zahir ibn al-Husayn b. Sa’b al-Būsagi.  Zahir killed Ali ibn Isa, put to flight the armies of Muhammad al-Amin and came to Baghdad, where he was joined by Hartama ibn A’yan and Humayd ibn Abd al-Hamid at-Tusi. Al-Ma’mun was hailed as caliph in Khurasan in the year 196.  The civil war then moved to Baghdād.

2. Muhammad al-Amin was killed in Baghdad on Saturday, five days before the end of the month of Muharram of the year 198 [of the Hegira].  His caliphate, until the day of his murder, had lasted four years, eight months and six days. He was killed at the age of twenty-eight years.

3. Nicephorus, son of Istabraq, king of Rum, died.  After him there reigned over Rum  Istabraq[7], son of Nicephorus, son of Istabraq.

4. In the third year of the caliphate of Muhammad al-Amin there was made patriarch of Jerusalem Thomas, nicknamed Tamriq[8].  He held the office for ten years.

5. Muhammad al-Amin was handsome, with a perfect constitution, white-skinned, fat, strongly built, with thin fingers.  His body was buried at Baghdād and his head brought to Khurasan.  The leader of his bodyguard was Ali ibn Isa b. Mahan and his hāgib al-Fadl ibn ar-Rabi, who was also his confidential adviser.

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  1. [1]Elsewhere “Istabrāq”; i.e. Stauracius.↩
  2. [2]A collective term for various pastries and pretzels.↩
  3. [3]813-817 AD↩
  4. [4]817-848.↩
  5. [5]Possibly means “Iran”?↩
  6. [6]I.e. Zubaydah, the wife of Harun ar-Rashid.↩
  7. [7]Stauracius, emperor of the East from 26 July 811 to 2 October 811.↩
  8. [8]807-821.↩