Diodorus of Alexandria
Diodorus of Alexandria (fl. first century BCE), was the author of the first treatise on sundials from which any significant amount of material has survived, and he is described in The Palatine Anthology as “famous among the gnomonists.” (The gnomon is, in this context, the rod that casts a shadow in a sundial). He is also known as one who attempted a proof of Euclid’s famous parallel postulate “with a number of different figures.”
Among the nonmathematical writings of Diodorus are treatises on physics and uranography. Although some have suggested that the authors in the three areas were different people, all three are referred to as “a mathematician” in ancient sources and, as Don R. Edwards has put it (1984, p. 153), “if little leads one to identify the three, nothing whatever prevents one.” On the basis of the identification of these three writers named Diodorus as being one and the same person, then, Diodorus of Alexandria may be dated to the first century BCE. This is largely because of his having disagreed with Posidonius (c. 135–50 BCE) on the Milky Way and his having been quoted by the philosopher Eudorus (d. c. 25 BCE). More than this the sources do not reveal.
Work on the Heavens . His writing on uranography (that branch of Greek astronomy which describes the appearance of the stars and their constellations) appears, from the surviving references, to have been a commentary on Aratus’s versified form of Eudoxus’s Phenomena. For example, a scholium on Aratus cites Diodorus as commenting on a controversy between Attalus and Hipparchus about the position of stars in the constellation Draco. It is not clear if this is the same work as one referred to in both an ancient catalog of writers and a Byzantine scholium as one of those who “wrote about the pole [of the cosmos].” (Ernst Maass takes this, quite plausibly, to refer to a treatise on celestial phenomena.)
In physics, Macrobius cites Diodorus as arguing that the Milky Way is made of fiery particles. As regards the differences between mathematics and physics, Diodorus argues that although there is a conceptual difference between physical and mathematical argument one ought, when investigating a question where both are relevant, to ignore the conceptual distinction.
Work on Sundials . It was, however, his Analemma, a treatise on sundials, which brought Diodorus a lasting reputation. His writing in this area first appears in the first century CE incorporated in the writings of the Roman surveyor Hyginus Gromaticus, who describes a method for finding the local meridian (north-south) line. Hyginus does not name his source, but from Arabic works (below) we know that Diodorus invented the method. Then Pappus of Alexandria (c. 300 CE) composed a commentary on Diodorus’s Analemma in which, as he says (in Book IV of his Collection), “we used the aforesaid line [the cochloid of Nicomedes] when we sought to trisect the angle.” Indeed, because a circular arc representing the period of daylight on the face of a sundial would have to be divided into twelve segments (corresponding to the twelve hours into which the ancients divided the period of daylight), the problem of constructing sundials immediately leads one to the problem of trisecting an arc.
The word analemma is also used in a mathematical sense by Vitruvius Pollio, who lived at the time of Augustus, in his Architecture IX, 7 to describe a geometrical construction to determine the length of the noon shadow at a locality of given latitude when the sun has a given declination. Later, Heron of Alexandria applies the word to a similar method for determining the distance between Alexandria and Rome. Most famously the word is part of the title of Claudius Ptolemy’s treatise on sundials, On the Analemma. In all these cases, the context is a solution to a problem by rotating geometrical objects—angles, circular arcs, or triangles—associated with the celestial sphere into a working plane where angles and arcs can be faithfully represented and measured. (The working plane represents a plane in the celestial sphere—often the local meridian—and the object is rotated about the line of the intersection of the working plane and the plane containing the object.) Thus, in Vitruvius’s analemma, circle ABC represents the local meridian of the celestial sphere, which provides the working plane. The horizontal line represents the local horizon, and semicircle ADB represents the eastern half of the circle traced out by the sun (during the course of a given twenty-four-hour day) rotated 90° about AB into the working plane. The perpendicular DE represents the intersection of the semicircle with the horizon and hence arc AD measures half the length of daylight on the given day.
Diodorus’s treatise used such methods to design sundials. But to orient the completed dial properly one would need to know the local meridian. And in the surviving extract from his book, Diodorus shows how to construct a geometrical diagram in the horizontal plane to determine this meridian. In his method, three triangles formed by a gnomon and its shadow in the planes of three altitude circles at different times during the day are represented by similar triangles in the plane of the horizon. And it is these triangles that allow the construction of the local north-south line.
Arabic References . Diodorus’s Analemma was translated into Arabic and quoted by the gifted mathematician Ibrāhīm ibn Sinān of Baghdad (908–946). In his Treatise on Shadow Instruments, Ibn Sinān writes, “we take the tip of the gnomon as center of the [celestial] sphere as well as of all great circles which are on the sphere, such as the horizon and others. Diodorus and others took this as a principle in their works on the instrument.”
Later in the tenth century, a little-known geometer, Abū Sa īd al-Darīr (“the blind”), in his treatise On the Determination of the Local Meridian, cites a work titled The Analemma as the source of most of his treatise. That it was Diodorus’s Analemma referred to is clear from Abū Rayhān al-Bīrūnī’s (d. 1048) Exhaustive Treatise on Shadows, where the method found in Abū Sa īd is attributed to Diodorus’s Analemma. (It is a curious turn of fate that, although Ptolemy did not mention Diodorus’s Analemma in his work of the same name, it was via Diodorus’s treatise, and not that of Ptolemy, which was unknown in medieval Islam, that the method of the analemma was introduced to medieval Islam.)
An important part of Abū Sa īd’s proof is a demonstration that two lines occurring in the procedure intersect on a certain side of a transversal. The proof of this demands, of course, a use of Euclid’s parallel postulate, and it may be significant in this regard that the tenth-century writer al-Nairīzī, following Simplicius, mentions that Diodorus proved the parallel postulate by means of many different figures.
Finally, in addition to the above-cited discussion of the parallel postulate it appears that Diodorus also wrote on Euclid’s Data. Because the Data shows that if certain features of a geometrical configuration are given, other features are given as well, it was useful in the analysis of geometrical problems. There was, however, some discussion of what, exactly, Euclid meant by “given,” and Marinus of Neapolis cites Diodorus as taking “given” to mean “known…and whatsoever comes to a certain comprehension…even if it is not rational.”
WORK BY DIODORUS OF ALEXANDRIA
Bīrūnī, Muhammad ibn Ahmad. The Exhaustive Treatise on Shadows by Abū Rayhān al-Bīrūnī [Ifrād al-maqāl fi amr alzilāl], translated by Edward S. Kennedy. 2 vols. Aleppo, Syria: University of Aleppo, 1976. Contains an English translation of the Arabic version of the surviving fragment from Diodorus’s Analemma.
Besthorn, Rasmus O., and Johann L. Heiberg, eds. Euclidis Elementa ex interpretatione al-Hadschdschadschī cum commentaries al-Nairīzī. Part 1. Copenhagen, 1907.
Blume, Friedrich, Karl Lachmann, and Adolf Rudorff, eds. Die Schriften der römischen Feldmesser. Berlin: O. Reimer, 1848. Reprinted, Hildesheim, Germany: Olms, 1967.
Edwards, Don R. Ptolemy's : An Annotated Transcription of Moerbeke’s Latin Translation and of the Surviving Greek Fragments with an English Version and Commentary. Thesis, Brown University, 1984.
Kennedy, Edward S. “Bīrūnī’s Graphical Determination of the Local Meridian.” Scripta Mathematica 24 (1959): 251–255. Reprinted in Studies in the Islamic Exact Sciences, edited by Edward S. Kennedy, et al. Beirut: American University of Beirut Press, 1983.
Maass, Ernst. “Aratea.” In Philologische Untersuchungen, vol. 12. Berlin: Weidmann, 1892.
Neugebauer, Otto N. A History of Ancient Mathematical Astronomy. 3 parts. New York: Springer-Verlag, 1975.
Schoy, Carl. “Abhandlung über die Ziehung der Mittagslinie, dem Buche über das Analemma entnommen, samt dem Beweis dazu von Abū Sa īd ad-Darīr.” Annalen der Hydrographie und maritimen Meteorologie 10 (1922): 265–271.
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