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Return to Vignettes of Ancient Mathematics
A note on the translation
Let a liquid be hypothesized as having such a nature that with the parts of it lying equally and being continuous, the smaller in being squeezed is pushed out by what is squeezed more, and each of the parts of it are squeezed by the liquid above it along a perpendicular, unless the liquid has been confined in something and is being squeezed by something else.
1. If some surface is being cut by a plane through some point, always the same one, makes its section a circular-arc of a circle, having as center the point through which it is cut by the plane, the surface will be of a sphere.
2. For every liquid that’s come-to-a-state so that it remains motionless, it will have its surface as a figure of a sphere having the same center as the earth.
3. Among solid magnitudes, those having equal weight with the liquid, when let go into the liquid go down so that nothing protrudes out of the surface of the liquid and will no longer move downwards.
4. Among solid magnitudes that which is lighter than the liquid, when let go into the liquid will not wholly sink down, but some [part[ of it will be outside the surface of the liquid.
5. Among solid magnitudes what’s lighter than the liquid, when let go into the liquid, will sink down to the extent that as large a volume of the liquid as is the volume of the sunken magnitude has a weight equal to the magnitude as a whole.
6. Solids lighter than the liquid on being forced into the liquid will go up by as much force as is the weight by which the liquid having a volume equal to the magnitude is heavier than the magnitude.
7. [Bodies] heavier than the liquid on being let go into the liquid will move down as much might sink and will be lighter in the liquid so much as the weight of the liquid having so much volume as is the volume of the solid magnitude.
Let it be supposed that each of the [bodies] moving upwards in the liquid moves upward along the perpendicular drawn through the center of its weight.
8. If some solid magnitude is lighter than the liquid, having the shape of a segment of a sphere, is let go in the liquid in a way so that the base of the segment does not touch the liquid, the figure will come to rest upright in a way so that the axis of the segment is along the perpendicular. And if the figure is dragged by something in a way so that the base of the segment touches the liquid, it will not remain tilting if it is let go, but will return to rest upright.
9. And moreover, if some solid magnitude that is lighter than the liquid is let go into the liquid in a way so that the base of it, as a whole, is in the liquid, the figure will-come-to-stand upright in a way so that the axis of it is on a perpendicular.
1. If some magnitude that’s lighter than the liquid is let go into the liquid, it will have the ratio in weight to the liquid which the sunken magnitude has to the whole magnitude.
4. The upright segment of the right-angled conoid, whenever it is lighter than the liquid and has the axis more than half-again the straight-line up to the axis, when it has in weight a ratio to the equivoluminous liquid no smaller than that which the square from the excess by which the axis is larger than half-again the straight-line up to the axis to the square from the axis, when let go into the liquid in the way so that its base does not touch the liquid, when placed inclining it will not remain inclining, but will come-to-a-stand at an upright position.
Note on the translation: Heiberg's text is mostly on two manuscripts: the translation of William of Moerbeke (Vat. Ottob. lat. 1850, which he labels B) and the Archimedes Palimpsest, Metochii Constantinopolitani S. Sepulchri monasterii Hierosolymitani 355, (C), as well as discussions of two Arabic manuscripts providing a list of propositions, without proofs, of FB I 1-8, II 1 by M.H. Zotenberg (Journal Asiatique 12 (1879): 509-515) and Eilhard Weidemann (Physik.-Med. Societät zu Erlangen 38 (1906), 152-162). The publication of the palimpsest by R. Netz, W. Noel, N. Tchernetska, and N. Wilson (Cambridge, 2011) provides a full text of C for Book I and more of Book II than Heiberg had available. For Moerbeke's translation, we also now have Marshall Clagett, Archimedes in the Middle Ages, vol. 2 (Philadelphia: Am. Phil Assoc., 1976), 358-364, with variant readings, 423-5, and notes, 574-8, and diagrams, 631-3. The basis for any edition is Vat. Ottob. 1850, thought to have been prepared by Moerbeke himself. Heiberg labels the manuscript B, while Clagett uses'B' for the lost Greek manuscript used by Moerbeke, while he refers to Vat. Ottob. lat. 1850 as O. For difficulties in Moerbeke's translation, as well as the poor quality of the Greek manuscript he used, of FB I, see 52-3 and vol. 3 (Phil, 1978), 625-8. When he was unsure of how to translate, Moerbeke occasionally provided in the margins copies of the Greek original.
Since the exemplar of Vat. Ottob. 1850 can only be reconstructed from Moerbeke, whether by back translating (often easy, but never certain) or rarely from the marginal quotations, I shall retain Heiberg's label B.
Even with the proviso that, as Clagett observes, Moerbeke did not always understand the text being translated, B is frequently better than C. So even where both manuscripts are available, one should not follow C. Ideally, one should reconstruct the text out of the new edition of the Palimpsest and Clagett's edition of Moerbeke, with an eye to the two basic manuscripts, which is more than I shall do here.
Furthermore, the availability of newer editions of B and C does not eliminate the value of Heiberg's text. Especially given the difficulty of the text, there has been a robust tradition of corrections in B, of matheamtically improved editions, notably by Commandino, of which Heiberg is a fine example.
Heiberg's text is defective in one important way, in his propensity, long ago noticed by Wilbur Knorr, to clean Doric texts of koinê dialect. Knorr could not have noticed this about Floating Bodies, but it is evident here too, as Reviel Netz points out. There is one odd exception, where Heiberg prints σχῆμα (shape or figure) or σχᾶμα in C. Fortunately, this has no effect on the translation, which does not attempt to mix dialects of English.
All of this said, where C was available to Heiberg, I have followed Heiberg on the principle that he was often great at finding the right fix for a bad text (but with an eye to Netz et al. and Clagett), and C has now become available I have followed Netz et al. Where C is not available, I have followed Clagett. Issues are usually marked in footnotes.
A note on diagrams is also in order. The diagrams for FB I 3, 5 are clearly wrong in C, as they have liquids protruding from the surface and these liquid is supposed to equal the volume of the sunken part of the solid. Moerbeke's Vat. Ottob. 1850, as reported by Clagett, is correct for these propostions. Again, the lettering in the diagram for FB I 7 is also wrong in that it treats a line as defined by endpoints BG, whereas B and G are actually two segments of the line. Again, Vat. Ottob. 1850 gets it right. Neither diagram for FB I 8 is more right than the other, but Vat. Ottob. 1850 is less exaggerated, for what its worth. This sets up a rivalry between the two manuscripts. The 'general diagram' for each theorem is based on one or the other (for Prop. 8, I provide both diagrams based on both). However, I do not pretend to reproduce the diagrams of either, e.g. in that I fully letter and color them. The diagrams within the proofs follow my normal practice.
On the phrase, ἐν ᾦ τὸ A (on-which-A-is), used in FB I 4 (once), 6 (frequently), 7 (frequently), as well asFB II 3, see FB I 6.