Herbert Jennings Rose's Greek mathematical literature
To begin with the most fundamental discipline of all, and one which in the nature of the case must receive only cursory treatment in a manual of literature, Mathematics made enormous progress between the death of Alexander and the establishment of Roman supremacy. The Pythagoreans had devoted great attention to it, and some of them, notably Archytas, were a long way past the rudiments of the subject. [Archytas's subject being for the most part mathematical technicalities, it is not to be expected that his purely literary interest should be high; one quaint passage essaying to prove that calculation alone is able to breed confidence among men and prevent strife indicates that he was able to express himself, on occasion, well enough by other means than figures and diagrams.] Plato had regarded it as one of the most important subjects in the training of a philosopher, and was himself a good mathematician, keenly interested in the development of the subject, though only on its theoretical side. Many of his scholars followed their master in this respect, and some of them, notably Eudoxos, attained eminence; Eudoxos' pupil Menaichmos is regarded as the founder of the theory of conic sections, which was continued by Aristaios, in the third century. Outside these schools, Demokritos had done more than a little to advance mathematics, apparently attaining at least a rudimentary idea of infinitesimals and more than a rudimentary idea of projection.
[Demokritos of Abdera, of the philosophers, was perhaps the greatest of all who wrote in Ionic. His fragments are numerous, but generally short, extending but to a line or two apiece. Many consist of some pithy turn of phrase attributed to him, such as "Neither excellence of speech can hide an evil act, nor a good act be harmed by abuse in words", or "Pleasures out of season being forth unpleasantness". Only here and there have we a long enough extract to see that his style was that of a practised and experienced writer, capable of expounding his views with fullness and even eloquence, and not a mere coiner of epigrams. It is a very regrettable gap in our knowledge of the history of philosophy that we have lost the works of this notable man, the father of the atomic theory. Even of his life we are but ill informed, his great age having been doubtless exaggerated and his dates therefore uncertain; he was some thirty or forty years younger than the fifth century and lived well into the fourth. More than the usual number of anecdotes were told of him and need not be considered save in a work on folklore. In addition to the anecdotes, a most portentous literature of forgeries grew up around him. He was a wise man, therefore he had travelled and met Eastern sages and Egyptian priests, the fountainheads of knowledge and wisdom; therefore there were works dealing with his travels and the wonders he had seen. So later ages seem to have reasoned, and the works were soon forthcoming. A certain Bolos, early enough for Vitruvius, in the time of Augustus, to take him for genuine Demokritos, wrote a work dealing with sympathies and antipathies, whereof a good many fragments remain, and this was fathered on the sane and scientifically minded philosopher, by those who had not read him. later still, his name was attached to alchemical writings.]
There was thus a considerable body of knowledge from which further advances might start, and of it the Alexandrians made excellent use, for enough of their work remains for us to judge with some exactness how much they knew. The most famous name is that of Euclid (not to be confused with the Megarian philosopher of that name). Of his life we know nothing, but as he used Eudoxos, and Archimedes (see below) used him, probably his activities lie towards the beginning of the third century B.C. His great work, familiar till recently in our schools, is the Elements, the mathematical handbook for the rest of antiquity (even the totally unmathematical Romans heard of it), and for most later ages. It is preserved entire; of its thirteen books, the first six deal with plane geometry, the next three with arithmetic, Book X, the longest of all, with irrational quantities, and the remainder with solid geometry. A sort of appendix, not by Euclid, is conventionally labelled Books XIV and XV. One other work of his on pure mathematics survives, the Data; but in addition we have notices of several more, the treatise On Divisions (of plane figures), whereof part is preserved in an Arabic version, that on Conic Sections, the Porismata (problems on the determination, from certain data, of some part of a figure; the simplest example was 'given a circle, to find its centre'), that on Mathematical Fallacies (one is reminded of De Morgan's Bundle of Paradoxes). The Optics is preserved in a later compilation (we may render this freely, Outlines of Astronomical Mathematics; the epithet 'little' implies a contrast with Ptolemy's Almagest), dating from the third century A.D. or thereabouts, and with it the Phaenomena, an elementary astronomical treatise. Of the 'Greek Title' we know that it dealt with conic sections, and consisted of two books. A little work on music also survives, entitled The Division of the Monochord, i.e., the arithmetical relations between the notes of the diatonic scale.
Falsely ascribed to him are the Catoptrics and the Introduction to Musical Theory.
Euclid seems to have been not so much an original discoverer as an organizer of mathematical knowledge. Of other members of the Alexandrian school we know the names of Konon the astronomer, who was the friend and correspondent of Archimedes, Eratosthenes the geographer, who invented a mathematical instrument for finding the two mean proportionals necessary for the duplicature of a cube, and, about a century after Euclid, Apollonios of Perge, author of a work on conic sections, which survives partly in the original, partly in an Arabic version, a treatise on harmonic section, whereof an Arabic translation only has come down to us, and several other works now lost. This man was trained and lived partly in Pergamon, which in science as in literature was the rival of Alexandria.
[Minor works of Kallimachos included the Lock of Berenike, which we know, not so much from the score of lines surviving, as from the fairly literal translation of Catullus. It is a pretty bit of courtly astronomy; Konon the astronomer had identified a small group of stars near Leo as a new constellation, the Lock of Hair, and declared it must be a lock dedicated by the queen which had mysteriously disappeared from its place in the temple.]
A greater mathematician than any of the above was Archimedes of Syracuse, who also was trained in Alexandria, though he owed most to his own genius. His dates are 287-212. Mathematical ability was perhaps a family characteristic, for his father Pheidias was an astronomer. The young Archimedes began with applied mathematics, his earliest known work, On Mechanical Theorems being dedicated to Eratosthenes. But he soon passed on to more abstract problems, some of which were discussed in his treatise On Sphere and Cylinder. This he had meant to dedicate to Konon, but as the latter died before the work was ready, he dedicated it to his pupil Dositheos. There followed a more important work, On conoidal and spheroidal figures, which among other things discussed the problem, already famous, of 'squaring the circle'. A little work which has not come down to us in its original form attacked the same problem from another angle, and evaluated π within fairly close limits; yet another, now lost, calculated it still more closely. This and other questions which interested him involved finding a better system of numeration than that generally in use among Greeks; from this sprung a popular essay, addressed to Gelon, son of Hieron II of Syracuse, known as The Sand-reckoner, showing that it was perfectly possible to reckon the number of grains of sand required to fill the whole universe, by employing an ingenious system of his own for expressing high figures, and therefore that the common proverb 'numberless as the sands' was wrong. For Archimedes was capable of joking, provided the joke was mathematical; he seems to have liked puzzles, since there survive fragments of a work on the theory of the stomachion ('teaser'), a sort of tangram, whose fourteen pieces can be put together into a square and also into figures of various sorts; he is also alleged to be the author of an epigram inviting the reader to calculate the number of the cattle of the Sun from data the working out of which involves handling numbers which run into many millions. In more serious mood he wrote a work On Spirals, another on equilibrium, in other words on statics, yet another, preserved complete in a Latin translation of 1543 from a manuscript now lost, partly in a palimpsest discovered by Heiberg, on hydrostatics (literally On things carried, sc., on water or other fluid, i.e., floating). Other works are preserved in fragments, or not at all.
After the great age, mathematics continued to be studied, even to make advances here and there, for several centuries. Zenodoros is of uncertain date; we have a work of his On Figures of equal Perimeter. Out of several works dealing. with spherical trigonometry we have an Arabic version of one, the Sphaerica of Menelaos of Alexandria, who lived in the first century A.D. The discovery of a manuscript at Constantinople has given us a writing of Heron, On Mensuration; he also commented on Euclid's Elements. Nikomachos, a Neo-Pythagorean of about the second century A.D., wrote an Introduction to Arithmetic which we still have, and a work on Arithmetical Theology, i.e., on the mystic meanings given by Pythagoreanism to the first ten numbers, of which we have an abstract and some fragments. A better author, though later (third century A.D.), is Diophantos of Alexandria, of whose thirteen books on arithmetic six survive; their contents include a good deal of what we now call algebra. Pappos, also of the third century, was a very respectable mathematician, widely read in the classical works of Archimedes and the rest; hence his Collection of comments on and supplements to their writings, with historical notes and other welcome information, is of much use to students of the history of the subject, besides presenting a picture of what mathematical studies were like at that date. After Pappos comes a long line of commentators, reaching to the Revival of Letters and leading up to the modern renaissance and progress of the subject.
Mathematical knowledge was required, then as now, for any serious work on Astronomy. This was ardently studied at Alexandria, and great progress made, which might have been greater if Aristarchos of Samos had succeeded in convincing the world that the sun and not the earth is the centre of our system. Unhappily for the progress of science, his theory was rejected, partly on theological grounds; his works are lost, save for a little treatise On the sizes and distances of the Sun and Moon, and his only known follower was the rather obscure Seleukos, about 150 B.C., that is to say some 130 years later. But most of the astronomical works which are known to us are observational rather than theoretical, and their increasing accuracy fills a modern student with respect for the powers of their authors, unassisted as they were by any but the rudest instruments. Eudoxos, already mentioned, had published a handbook describing the relative positions and apparent movements of the stars (Phainomena), which contained a good many misstatements, as might be expected of an early attempt (his dates are 408-355). He also interested himself in calendar reform; like Meton of Athens before, Kallippos of Knidos and Hipparchos of Nikaia after him, he proposed a cycle of lunar years which should, by carefully arranged intercalations, amount to the same total number of days as the same number of solar years. Our earliest surviving work, however, is by a later writer, Autolykos of Pitane, who lived towards the end of the fourth century and wrote, so far as we know, two little treatises, one On the Moving Sphere, the other On Risings and Settings (sc., of the fixed stars). Hypsikles, already mentioned, is the first surviving author to mention, in an essay called 'Discourse on rising', what he clearly did not invent or introduce, the division of a circle into 360 degrees. As 60 is a common Babylonian factor (e.g., the talent, which is of Babylonian origin, is divided into 60 minae), it is fairly certain that we have here Babylonian influence, be it direct or otherwise. Hence it is not surprising that the very obscure Kleostratos, said to have been the first Greek to mention the signs of the zodiac, owes something - how much is a point as disputable as his date, which is variously computed at the sixth century or the fourth - to the Babylonians also. At all events, their observations were, at least in some measure, accessible to Greek astronomers after Alexander. Hence the constellations, especially those of the zodiac, become familiar subjects at this epoch, though scattered mentions of them are to be found from Homer onwards, and the differences between the Greek and the non-Greek picture of the heavens a matter for comment. With constellations came star-myths, very rare things in pre-Alexandrian times, apart from fanciful interpretations by moderns. That Konon furnished Kallimachos with material for a new one has already been mentioned. Eratosthenes was therefore quite in accordance with the taste of his day when he wrote a work on catasterisms, i.e., miraculous transformations of mythological persons into constellations, Kallisto into the Great Bear and so forth, whereof an epitome by some unknown hand survives.
It is very significant of the relative importance of literature and science for the succeeding generations that what we have left of Alexandrian astronomy concerns itself largely with Aratos' poem. Hipparchos of Nikaia was a very notable astronomer and mathematician, making careful use of Babylonian records of eclipses and of his own amazingly exact observations for such weighty matters as determining the exact length of the solar year, drawing up a catalogue of the fixed stars, settling the exact time of the apparent risings and settings of the heavenly bodies, and other things well worth the interest of a genuine scientist; yet all we have left of him, apart from mentions in later authors, is one little work, in which he corrects the errors in Eudoxos, and consequently in Aratos, for the benefit of a friend interested in astronomical studies. His date is fixed by certain observations which he is known to have made at the second century B.C. Geminos of Rhodes is later, about the seventies of the first century B.C. He would appear to have written a commentary on the astronomical work of Poseidonios, and is the author of a short handbook, the Introduction to Astronomy, which still survives.
[Antipatros had a pupil whose name was known far outside the circle of professed philosophers, Poseidonios of Apameia, a pupil also of Panaitios. Poseidonios not only was, like his teacher, a member of the Scipionic circle, but one of the most influential of Hellenistic writers. He was an eclectic philosopher, nominally a Stoic, but a most unorthodox one, for he combined the doctrines of his own school with those of practically all the rest save the Epicureans and Skeptics, and thus produced a blend of good practical ethics combined with comforting eschatological doctrine, a leaning towards mysticism and a quasi-scientific justification of wonders of all sorts. Of his learning and many-sided industry there can be no doubt; besides treatises on epistemology, on ethics, the nature of the soul and the universe, astronomy, fate, divination, and the nature of the gods, he was the author of books on literary criticism and rhetoric, and finally of a long historical work in fifty-five books, beginning where Polybios had ended, and of dissertations on meteorology and on geographical subjects. It is therefore no wonder that he was drawn upon by writers on all these topics, the more so as his style seems to have been pleasant and lively and much of what he wrote, at all events on geography and ethnology, the fruit of his own travels and firsthand observations. It is a pardonable exaggeration to trace to him almost everything in Roman and later Greek writers which deals with any of his subjects and cannot be definitely assigned to anothe source; but it is an exaggeration nevertheless, and criticism is beginning to correct the too exuberant assumptions of enthusiasts for this highly important, but unfortunately lost writer. He probably was born about 135, or at most some eight or nine years sooner; he lived in Rhodes for some time from about 97 onwards, till Panaitios died; after that he set out on his travels, which included the western countries (Gaul and Spain) then almost unknown to Greeks. he died somewharer between 60 and 50 B.C.]
Poseidonios' own astronomical studies were probably the result of his interest in philosophy in general rather than this science in particular, but there seems little doubt that he wrote a considerable work, the 'Greek Title', besides some smaller essays; of the former we have perhaps a sort of synopsis in the compendium of astronomy by Kleomedes, a writer otherwise unknown. Theon of Smyrna was frankly writing for non-specialists when he produced a monograph whose title may be rendered Mathematics for Students of Plato.
But the most famous name in this connexion is that of Ptolemy (Claudius Ptolemaeus), a contemporary of Theon, whose works, through the fault chiefly of readers and not of himself, kept astronomy at a standstill for more than a millennium after his death. He lived in the second century A.D., about 100-178, and was evidently a man of great learning and capable of at least routine observational work. His most important writing is familiar to later ages under its Arabic title of Almagest (Tabrir al magesthi); he himself called it the System of Mathematics. It is a textbook of astronomy, with tables and diagrams suggestive of a modern author, although in some respects its contents were already antiquated when he wrote; especially, his firm upholding of the geocentric theory of the universe has caused that view to be called Ptolemaic in modem times. Its thirteen books cover 1254 Teubner pages in all, and contain, in its author's opinion, 'practically everything that should be studied in such a system, in the light of contemporary knowledge '. Hence it served as the manual of all serious students down to the great developments associated with the names of Copernicus and Galileo. Besides some minor writings, Ptolemy also composed an astrological treatise in four books, hence known as the Tetrabiblos; for astrology was flourishing in his day, and had done so since about the third century B.C.
A mass of astrological works have come down, but remain largely unpublished; a full list of all known manuscripts containing them, with extracts from those not yet printed, exists under the general title of Catalogus codicum astrologorum Graecorum, and we have two considerable treatises in verse, one by a certain Maximus, perhaps about contemporary with Geminus, another by several hands and of very different dates, all thrown together by some late copyist or editor under the name of Manetho. There exist also fragments of a more famous poet of this kind, Dorotheos, who is supposed to have lived about the same time as Ptolemy. Numerous other writers in prose and verse are known to us by their (real or assumed) names and citations from their works. The interest of all these writers is non-literary; there is room for a full treatise on this curious aberration, but it would fall wholly outside the scope of a book like this. The one poet of any merit astrology produced was the Latin Manilius.
Another development of mathematics was Mechanics, a department of knowledge concerning which the Greeks seem to have written comparatively little. Archimedes, however, gave considerable attention to it. Besides his writings he was on occasion active as an engineer, and even those who knew nothing of his science remembered, and probably exaggerated, the efficiency and ingenuity of the machines with which he defended Syracuse. The most renowned of Alexandrian authorities on this subject, Ktesibios, who may have lived in the third century B.C., has not come down to us directly, but later writers make use of him. Among these are Philon of Byzantion, of whom we have a certain amount left, partly in the original and partly in Arabic and Latin versions, and Heron (about 50 B.C.?), who was concerned less with mechanical theory than with practical, and often very ingenious, methods of constructing machines great and small, from war engines to toys of various sorts, including puppet-theatres and one or two rudimentary applications of the power of steam.
Not unconnected with mathematics is Music, and it may be mentioned here that Ptolemy wrote three books of Harmonies (i.e., musical theory, not what our musicians call harmony, for that did not exist in Greek music), which still survive, and another writer, perhaps of the third century A.D., Aristeides Quintilianus, also produced three books, On Music. Of other writers, notably Aelius Dionysios, we know something from references to them, but have no complete works.
Clearly, none of the above subjects give much scope for purely literary treatment; it may be said generally that these technical writers have plain, straightforward styles, free from misplaced attempts at eloquence. They write in the Greek of their own day, that is in the 'common dialect', for the most part; Archimedes uses his native Sicilian Doric for several works (in the case of some which are in the common dialect, it may be suspected that they were originally in Doric also). The worst, stylistically, is perhaps Ptolemy, who is prone to heavy, over-long sentences and a certain pretentious pedantry.
More nearly literary is Geography, which by this time was definitely a science, concerning itself with mathematical theories of the shape and size of the earth, the climates of different zones and so forth, and tolerably expert in calculating latitude and longitude; while at the same time such departments as ethnography were fairly well advanced. Hence it is to be regretted that we have not the three books on this subject written by the many-sided Eratosthenes, nor the more popular work of Poseidonios (see above).
[Eratosthenes, the great scholar and scientist who was born about 275, was blessed with a number of nicknames, all indicating that he was second best at everything (e.g. Beta, number 2, and Pentathlete, a good all rounder). One of these things was poetry, and we have evidence that he was fairly active in verse-making. One work of his, the Erigone, is usually considered the chief source of the pretty legend of Erigone and Ikarios as we have it from later writers; another, the Hermes, seems to have combined the mythology of that god with an episode in which the heavens were described somewhat after the manner of Artos; this was in hexameters. A few more fragments belong to poems on various themes. Another of his activities was chronology, in which area he is the first notable name; of his chronicle but few fragments remain, but it is spoken of with respect and cited as an authority by several judicious writers.]
There remain, however, two considerable writings, capable of teaching us much that is interesting, the more so as their authors were rather compilers than original researchers, and so tell us something of the older geographers as well.
Strabo of Amiseia in the Roman province of Pontos, was born about 63 B.C., and seems to have lived till about A.D. 19. He was of a good family, originally Cretan, and was thoroughly educated under some of the most distinguished teachers of the day. He became a Stoic of the same moderate sort as Panaitios and Poseidonios, lived for some time in Rome (about 29-24 B.C., and again a few years later) and travelled over a great part of the known world. Evidently Polybios influenced him; for not only did he, like most scientists of the time, keep clear of the craze for Atticism which began in his day but his work was modelled to a great extent on that of the earlier writer. He composed a long history (43 books) leading up to Polybios and then continuing him, evidently in considerable detail, down to his own day. This is unfortunately lost; but we have the greater part of the seventeen books on geography which earned him the title of 'The Geographer' in the Middle Ages, as Homer was 'The Poet'. This is a description on a generous scale of the then known world, starting in the west and continuing to the farthest east with which Graeco-Roman civilization had any acquaintance. It is such a treatise as one would expect from an admirer of Polybios who was not, like his model, a man of action and practical experience. History and geography, the former taken from his own work, go hand in hand; there is little interest shown in the purely scientific aspects, much in such things as may furnish useful moral lessons. The sources, despite Strabo's first-hand knowledge of several countries, are usually books, and these of various dates and degrees of trustworthiness, not always critically handled or their discrepancies noticed and accounted for. Probably some part of this failing is due to the work not having received its final revision when its author died; but his critical abilities were not great, and therefore he was somewhat prone to treat his authorities from the standpoint of his own, or his school's notions of probability, making little, if any, original inquiry. Thus he greatly undervalues Pytheas, for instance (see P. 312), although he has the good sense to recognize Megasthenes' fables as such.
Before and after Strabo a number of writers produced works on geography, mostly in prose but occasionally in verse: their remains are to be found in Müller's Geographi Graeci minores. Of these, several had no literary pretensions at all, but wrote lists of the towns and other conspicuous features to which a traveller along a given coast would come, with notices of the distances between them. A good example of this type of guide-book is the anonymous Voyage around the Red Sea (the name includes more than our 'Red Sea', for the regions described extend to India), written by some unknown Graeco-Egyptian merchant of about the first century A.D., a practical man who set down plainly what he knew about routes, the character of the inhabitants, the goods to be had (they include sugar) and other such matters. The manuscript of this treatise absurdly names Arrian as its author; the real Arrian wrote a description of India and a Periplus of the Euxine, the former derived from older authors, the latter from observations of his own. There is a work. apparently much interpolated by those who used it as a school-book, falsely attributed to Skylax of Karyanda, but really composed about the thirties of the fourth century B.C., the 'Greek Title', which takes the reader around the shores of the entire civilized world as known to the author, diversifying the list of names by notes on ethnology, local myths and the like. Of more pretentious writings, rather literary than either scientific or practical in intention, the considerable remains of Agatharchides' works may serve as a specimen. He was a Peripatetic, apparently a man of upright character as well as considerable learning, and a determined opponent of the Asianists. A native of Knidos, he spent a great part of his life in Alexandria, where he was tutor to the young Ptolemy VIII. His best-known work, preserved for us in extracts and an epitome by Photios, is the treatise On the Red Sea, originally in five books. It contained much that was interesting, including a famous description of the inhuman treatment of convicts in the gold-mines at the southern extremity of Egypt, but it was by no means confined to geographical facts, much less to the advancement of the theory of that science, but digressed on all manner of topics, mythological, stylistic and moral. Of the verse writers, two deserve mention; one used iambic trimeters, explaining in a sort of preface why he did so; they are the metre of Comedy, which has the great virtue of putting things clearly, briefly and at the same time pleasantly. Who he was is not known. For some time he was called Skymnos of Chios; but Skymnos, as we now know from inscriptional evidence, was living 185/4 B.C., while this man dedicates his work to Nikomedes II of Bithynia, 147-95. The real Skymnos wrote a description of Europe, Asia and Africa, whereof some fragments are preserved in the scholia on Apollonios Rhodios, but it was in prose. The other versifying geographer is Dionysios, sometimes called the Periegete to distinguish him from the numerous other persons of the same name; much ink was spilt over the question of his age and country, until G Leue had the perspicacity to notice that the poem is signed and dated; lines 113-134 (Müller) and 522-532 are acrostics, informing us respectively that the work is 'by Dionysios, one of those inside Pharos', i.e., in Alexandria, and 'of the time of Hadrian'. The Byzantine schools seized upon this work - its 1187 hexameters are not an impossible amount to learn by heart - and consequently we have a mass of scholia, also a commentary of portentous length by Eustathios, bishop of Thessalonike (Saloniki); but that it was popular in the West also is indicated by the fact that Avienus, in the fourth century, turned it into Latin verse.
More important than any of these is the geography, or rather gazetteer, of Ptolemy, in eight books, under the general title of 'Greek Title'. Book I is introductory, Book VIII gives full directions for drawing a map of the then known world, the intervening books consist of lists of place-names, 8,000 or so in all, arranged by provinces of the Roman Empire, and with the latitude and longitude of each carefully set down. The absence of literary merit, and therefore of interest to the non-specialist, the immense amount of work involved in sifting the manuscripts tradition - this is extensive and includes maps, whose exact relation to Ptolemy himself is anything but certain - and the recurrent problems of identifying doubtful names and deciding whether a given mistake is due to the author or a copyist have combined to prevent a full critical edition being published hitherto.
JOC/EFR July 2012
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