This book is the editio princeps, together with a French translation and introduction, of the oldest known Arabic manuscript of a hydrostatic treatise by Menelaus (first century CE), who was an important figure in the astronomy and mechanics of the Hellenistic Greek world. Menelaus is mentioned by Ptolemy, Pappus of Alexandria, Proclus, and others. None of Menelaus’ works survive in their original Greek, but some do survive in Arabic translations. The editor/translator of this treatise, Roshdi Rashed, is possibly the foremost contemporary scholar of the mathematics and physical sciences of medieval Islamic lands. Rashed includes alternative texts of some passages preserved in Arabic by later medieval translators, as well as a tenth century commentary on the treatise by Mohammed ibn al-Haytham. Rashed discovered the single manuscript of Menelaus’ hydrostatics and al-Haytham’s commentary in separate manuscripts. He dates the copying of the Menelaus text for his edition to 742 (1342 CE) in the Islamic West (33). Menelaus’ text on hydrostatics became available in the Arabic world before Archimedes’ On Floating Bodies.
The condition of the Greek (or other language) manuscripts from which Arabic translations were made is often an issue in approaching Arabic translations of scientific texts. Menelaus’ Hydrostatics (Book on the method by which one knows each of a number of mixed substances[1]) has two books. The first treats the determination of the relative amounts of two substances in a body. Menelaus’ method given in the text of Book I is cogent. Book II concerns the determination of the relative amounts of more than two substances in a body. The text of Book II that came to al-Khāzinī, author of Book on the Balance of Wisdom (twelfth century CE), was so confused that the medieval scientist did not attempt to interpret it. The text’s modern translator, J. Wurschmidt,[2] did not translate it for the same reason. Rashed’s newly edited text is no clearer for Book II. Rashed edited and translated Book I and the parts of Book II that could be interpreted. The corruption of the text had, nevertheless, led the tenth-century Arabic scholar, Muhammad ibn al-Haytham, to give his own account of determining the relative amounts of three substances in a mixed body, an account that Rashed deems correct if not clear. Rashed says that al-Haytham’s step by step reconstruction of how to determine the relative amounts of three mixed substances shows well the location of the lacunae in Book II of Menelaus’ text (3–4). Rashed translates Menelaus’ Book I and fragments of Book II (§I.3) and ibn Al-Haytham’s text (§II.2).
Ancient hydrostatics took its point of departure from the problem given at the beginning of Menelaus’ treatise: to determine how much of gold versus some other substance was present in a crown given to king Hieron II—in particular, the proportion of gold to silver, the silver being an adulteration of the gold crown. (The story about Archimedes and Hieron’s crown was related by Vitruvius in the first century BCE, with embellishments not present in Menelaus’ text.) The problem is to determine the amount of the two substances in the crown without dividing or breaking it. At the beginning of his treatise, Menelaus gives credit to Archimedes for solving the problem. He says he has not himself seen Archimedes’ proof, having only heard of it, but deems it not something difficult for Archimedes to achieve (34–35). He mentions a Manatius who had come up with something of the right approach in a work on rarity and density of bodies but who did not arrive at it by a universal method (bi ṭarīqi kullī). Manatius has a correct rule (bāb). His method, however, appears to have been largely empirical, requiring more than one balance and offering many occasions for inaccuracy. Menelaus’ account of Manatius’ treatment suggests that, in one form or another, the basic principle was well-known by Menelaus’ time. To solve the problem by demonstration (burhān), Menelaus says that the fundamental notion is that equal weights of the same substance on a balance of equal arms will be in equilibrium on a horizontal beam. This sounds like the simple law of balance but the key word is ‘substance’ (in a non-metaphysical sense, jawhar), because the density of different substances is key to the solution. Gold is one of the densest substances on earth, denser than either silver or lead.
If the balance with equal weights is immersed in a fluid, the equal weights of the same substance will still be in equilibrium. In addition, they will each displace the same amount of the fluid. The weight of each in fluid is their weight before immersion minus the weight of fluid displaced for each body. Menelaus says that Archimedes proved this (40). Menelaus shows that the size, body or place of the equal weights (their volumes) are in the same ratio as their weights. The ratio of these volumes is also the same ratio as the ratio of the weights of fluid displaced. From this, Menelaus shows that, given bodies of different substances whose weights (in air) are the same, when these are immersed in the same fluid their weights will be unequal, and the more dense substance (aljawhar alakthaf) will be the heavier. If A is the denser substance and B the less dense, but their weights in air are equal, then B clearly is larger in size than A. Consequently, the volume of fluid occupied by B when immersed is larger than the volume of fluid occupied by A. Accordingly, the weight of fluid displaced by the less dense body is greater than the weight of fluid displaced by the denser body: “When from equal things are substracted things unequal, the remainders are unequal, and the greater of the remainders is the one left after the lesser is removed” (42). This means the denser A is heavier in fluid than the less dense B. So, imagining the immersed balance, the end with weight A will decline and the end with weight B will rise. Although he returns to the immersed balance in later proofs, Menelaus makes the comparison and proves this initial proposition solely by means of the amount of fluid displaced. He applies his equal weights theorem in various situations, e.g., if weights of different substances are equal in one fluid, when they are immersed in a more dense fluid, the denser substance will be heavier, and if they are immersed in a rarer fluid than the first, the denser body will become lighter (46). His example involves fresh versus sea water. Returning to the problem of the crown possibly not made of pure gold, Menelaus details more than one approach to revealing the adulteration, depending on whether one starts with equal weights or equal volumes of gold and silver and the crown to be tested (51–65).
Menelaus is most famous in medieval Islam for his work on spherical geometry, which was translated into Arabic at least three times. The treatise dealt not with relations within the sphere as a solid but with properties of the surface of a sphere. He proved a theorem concerning the relative lengths of arcs of a spherical triangle—a triangle defined by the intersection of three great circles on the surface of a sphere. The subject matter of his Spherics was non-Euclidean.[3] Menelaus’ Elements of Geometry was translated into Arabic by Thābit ibn Qurra. There is likely much to be learned about Greek mathematics, including its non-Euclidean and more Archimedean branches, from the study of Menelaus’ first century texts, which were translated into Arabic beginning in at least the tenth century CE.
Rashed includes photographs of the Menelaus text he edited, which is in the Biblioteca del Escorial. There are simple line drawings in the manuscript. He provides valuable biographic and bibliographic sketches of both Menelaus and Mohammed ibn al-Haytham. Rashed has long argued that Mohammed ibn al-Haytham was a separate scientific figure of the same time period as the more famous al-Ḥasan ibn al-Haytham.[4] His section II.3 renews this claim. Besides presenting textual and archival evidence, Rashed argues that The Configuration of the Universe, often ascribed to al-Ḥasan exhibits a simplistic acceptance of Ptolemy and an addition of cosmology to the Ptolemaic universe that is inconsistent with al-Ḥasan’s expressed doubts about the mathematics of Ptolemy’s system (130). A.I. Sabra’s equally meticulous counter-argument is worth consulting, including on the disputed Configuration.[5]
Rashed supplies numbering for Menelaus’ theorems. He uses modern terminology and notation in his paraphrase of Menelaus’ argument (Introduction). Creating equations, however, even if reader-friendly, can obscure the nature of ancient reasoning by ratio and proportion. In the translation, the language of equality is used where the text speaks of proportion or accord. This will be for most readers a minor point. Rashed does not undertake a comparison of Menelaus’ hydrostatics to that of Archimedes. In general, Archimedes’ On Floating Bodies is more sophisticated and complete. For instance, Archimedes begins by establishing geometric properties of a fluid as a fluid and then considers what happens when bodies of the same or different weight as the fluid are immersed in it. Neither Archimedes’ nor Menelaus’ account is quite the same as what is called “Archimedes’ principle of bouyancy” in modern elementary physics.
Rashed has added to our knowledge of Menelaus and made a comparison of Archimedes and Menelaus more accessible to students of ancient mechanics. At the same time, he has made an important addition to the history of mechanics in medieval Islam. This erudite and meticulously executed book will hold the attention of any reader interested in a more complete retrieval of Hellenistic mathematics and mechanics.
Notes
[1] Kitāb Mānālāwus … fī al-ḥīla allatī yu‘rafu bihā kullu wāḥidin min ‘iddat ajsām mukhtaliṭa.
[2] J. Wurschmidt, ‘Die Schrift des Menelaus über die Bestimmung der Zusammensetzung von Legierungen,’ Philologus 80 (1925), 377–409.
[3] Menelaus did not make use of a parallel postulate, for example. See Roshdi Rashed and Athanase Papadopoulos, Menelaus’ Spherics: Early Translation and al-Māhānī / al-Harawī’s Version (Berlin/Boston: De Gruyter, 2017), XI–XIV.
[4] Rashdi Rashed, Les mathématiques infinitésimales du IXe–XIe siècles, vol. II, Ibn al-Haytham (London: Al-Furkān Islamic Heritage Foundation), 1993, 1–19.
[5] A.I. Sabra, ‘One Ibn al-Haytham or two?: An Exercise in Reading the Bio-Bibliographical Sources,’ Zeitschrift für Geschichte der arabischen-islamischen Wissenschaften 12 (1998), 1–51, and ’One Ibn al-Haytham or two: Conclusion,’ in volume 15 of the same journal (2002/03), 95–108. Sabra addresses the Escorial manuscript in (1998), 13–14, and the question of The Configuration in (2002/03), 102–103.