(fl. Rhodes [?], ca. 70 b.c.)
Geminus’ name is Latin, but his works and manner are patently Greek—the forms Γ’єμινος and Γ’єμινος, also found in the manuscripts, are probably false analogies based on true Greek forms such as ’Aλέζåνand ’Eτγåνos. He was author of an Introduction (Isagoge) to Astronomy (still extant) and of a work on mathematics (lost, except for quoted extracts from it). Nothing is known of the circumstances of his life, but his date and place of work may be inferred from internal evidence in the Isagqge. In chapter 8, sections 20–24 of the Manitius edition, Geminus corrects a widespread Greek view that the Egyptian festival of Isis coincided with the winter solstice; he explains that although this was so 120 years ago, in his own time there was a whole month’s difference between the two dates, since every four years the Egyptian calendar (based on a “year” of twelve months of thirty days each plus five additional days)1 became out of step with the solar year by one day. The festival of Isis took place from the seventeenth to the twentieth day of the Egyptian month Athyr, and a papyrus fragment tells us that in the calendar of Eudoxus of Cnidus the winter solstice occurred on 20 or 19 Athyr. By Julian reckoning the winter solstice in Eudoxus’ time (ca.350 b.c.) occurred on 25 December or 28 December, and the first years in which these Julian dates can coincide with 19 Athyr are 185 b.c. and 197 b.c.2 Hence, the papyrus presumably was written at this time; and subtracting the 120 years mentioned by Geminus, we obtain a date for the Isagoge of 65 or 77 b.c., which agrees with other evidence indicating a date about 70 b.c.3
We have no definite evidence about where Geminus was born or worked, but the commonly held opinion that his place of work was Rhodes may be correct. It is the “clima” of Rhodes that he uses to illustrate his account of various astronomical phenomena (e.g., 1, 12; III, 15; V, 25); in XVII, 4, he refers to Mt. Atabyrius,4 i.e., the modern Mt. Attaviros, in the center of Rhodes, without feeling it necessary to specify its location, whereas in the same passage he is careful to explain that Mr. Cyllene is in the Peloponnesus; and Rhodes had a reputation in the last two centuries b.c. as a center for those subjects (philosophy, astronomy, and mathematics) with which Geminus was concerned, Panaetius, Posidonius, and Hipparchus all having worked there.
On the other hand, the choice of Rhodes as a typical example may simply have been dictated by its position on the best-known and most central parallel of latitude, 36 °N. (Geminus himself says [XVI, 12] that all globes were constructed for this “clima,” which for practical purposes was regarded as the latitude of Greece as well),5 or Geminus may merely have used the examples he found in his sources, in this case almost certainly Hipparchus (see below), just as Ptolemy in the Almagest (II, 3 and 4), some 200 years later, was to use the same example from the same source. Similarly, Geminus may have taken the examples of Mt. Cyllene and Mt. Atabyrius straight from Dicaearchus (who is cited for their heights in XVII, 5); and even though two of his chief authorities, Hipparchus and Posidonius, worked in Rhodes, it does not necessarily follow that Geminus did.
The Isagoge is an early´ example of an elementary astronomical handbook written to popularize the main ideas in the technical treatises of the scientists; it belongs to the same tradition as the De motu ciruclars corporum caelestium of Cleomedes, and there are many similarities in style and arrangement between the two works, although, as might be expected from its later date, Cleomedes’ work is fuller and less elementary than its precursor. One interesting difference is that Geminus includes a chapter (Isagoge, II) on the astrological “aspects” of the zodiacal signs, i.e., their arrangement in pairs, triplets, quadruplets, etc., according to which the astrologers calculated the signs’ influence on human affairs; Cleomedes has no such chapter and, in fact, does not mention astrological doctrines at all. Geminus gives a simplified description of basic astronomy as known in the time of Hipparchus, omitting most of the mathematics and giving (1, 23 f.) little information about the planets apart from their zodiacal periods (thirty years for Saturn, twelve for Jupiter, two and a half for Mars, one year each for Mercury, Venus, and the sun,6 and 27⅓ days for the moon); there is no mention of epicycles, but the eccentricity of the sun’s path relative to the earth is carefully described (I, 31 ff.) and the Hipparchian values for the four astronomical seasons are given (I, 13 f.) — 94½, 92½, 88⅛, and 90⅛ days, respectively, starting from the vernal equinox.7
Chapter III describes the main constellations; chapters IV and V describe the chief circles of the celestial sphere; and chapter VI explains the variations in the lengths of day and night at different latitudes. Chapter VII (which is based largely on Aratus and, almost certainly, on Hipparchus’ commentary on Aratus’ astronomical poem entitled Phaenomena) deals with the rising of the zodiacal signs, and chapter VIII with the length of the lunar month, for which a figure of 29½ + 1/33 days is given (rounded off from the accurate Hipparchian value of 29 days, 12 hours, 44 minutes, 3 seconds);8 this last chapter is especially valuable for its account of how the Greeks developed an astronomically based calendar with a scientific system of intercalation.9 Chapter IX explains the phases of the moon, chapters X and XI solar and lunar eclipses, and chapter XII the general motions of the planets. Chapters XIII and XIV deal with the risings and settings and courses of the fixed stars, and chapters XV and XVI with the zones of the terrestrial globe, delineated according to the estimates of Eratosthenes, which Hipparchus also accepted.10
Chapter XVII discusses sensibly the principles on which the “parapegmata” (astronomical calendars containing weather prognostications connected with the risings and settings of certain stars and constellations)11were base. Chapter XVIII deals with the “exeligmus” (εξεγiιóς), or shortest period containing a whole number of synodic months, of anomalistic months, and of days (ef. Almagest, IV, 2); this chapter, far more technical than the others and out of keeping with the elementary character of the rest of the book, may well be an unrelated fragment.12 Finnally, there is an astronomical calendar, or “parapegma,” which begins with Cancer and is evidently based on figures for the astronomical seasons which are not Hipparchian;13 it seems probable that this was not part of the original treatise but represents older material.
Geminus’ exposition is usually sound and clear and includes some intelligent criticism, e.g., of the notion held “by many philosophers” that the planetary bodies do not in fact exhibit two opposite motions (the diurnal and the zodiacal) but only seem to do so because they revolve at different speeds (Isagoge, XII, 14 ff.); this idea, says Geminus, “is not in accordance with observed phenomena” (XII, 19, α σύ µ Φ ω ν ós εστι ιοīς φ αLvoµενοις). He also gives a shrewd appreciation of the limitations of parapegmata, points out that weather forecasting based on observation of the rising and setting of various stars is by no means an exact science but a rough codification of general trends, and asserts firmly that stellar risings do not cause changes in the weather but merely indicate them (ούκ αύταί παραίτιοί είσι ιών, άλλά σημέīα έκκινται XVII, 11; see the whole of this chapter, especially 6–41); this last assertion contrasts strangely with the belief underlying chapter II.
There are, of course, some errors and infelicities. The chapter which gives a rational discussion of parapegmata (XVII) also contains an absurdly exaggerated view of the restricted range of atmospheric phenomena, according to which no wind or rain is experienced at the top of a mountain under 10,000 feet high (XVII, 3–5). In chapters X and XI no mention is made that, for an eclipse to take place, the moon must be at a node: and in XVI, 13, it is wrongly stated that the magnitudes of eclipses are the same for people living on the same parallel of latitude. There are also minor errors in the description of a star’s evening rising (XIII, 13) and morning setting (XIII, 16), which may be the result of scribal errors in the transmission of the text (cf. Manitius. Anmerkung 25, 26); but Manitius’ theory (pp. 246–248) that our present text represents a Byzantine compilation from an epitome of Geminus’ original work, although it cannot be disproved, rests on very slender evidence.14
A work on mathematics (known only from extracts quoted by later writers) is also attributed to Geminus; and although it has been thought that this is not the same man as the author of the Isagoge, there can, in fact, be little doubt of the identity of the two.15 The title of the work was probably Theory of Mathematics (Θέωρία τών µαθηµáτν)16 and it must have extended to at least six books, since Eutocius quotes from the sixth. Our chief source for its contents is Proclus,17 who quotes appaear in the scholia to book I of Euclid’s Elements18 and in the collection of writings attributed to Hero of Alexandria;19 and there are isolated quotations elsewhere 20. There is also an Arabic commentary on Euclid by al-Nayrīzī21 which contains extracts from a commentary on the Elements, by Simplicius, who quotes a certain “Aghanis” on the definition of parallels; from similarities with what we know from Proclus of Geminus’ views on this subject, it has been suggested that Aghanis is actually Geminus, but there are grave objections to such an identification, which must therefore remain doubtful.22 A Latin translation of this Arabic text by Gerard of Cremona is extant. 23
The Theory of Mathematics apparently dealt with the logical subdivisions of the mathematical sciences, discussing the philosophical principles of their classification, distinguishing carefully between such terms as “hypothesis” and “theorem.” “postulate” and “axiom,” and paying particular attention to accurate definitions not only of the various branches of mathematics but also of concepts such as “line,” “surface,” “figure,” and “angle.” It seems to have been a more substantial work than the Isagoge and to have contained some very pertinent criticism of Euclid’s postulates, particularly the fifth, the so-called parallel postulate, for which Geminus believed he had found a proff.24
We learn from Simplicius that Geminus wrote an exegesis on the Meteorologica of Posidonius;25 Simplicius cites Alexander of Aphrodisias as transcribing along extract from an “epitome” of Geminus’ work (εκ της έπιτοµης τών Πoσέιδωνιoν Mετεωρoλoγικwν εξηγησέως), but an epitome of an exegesis sounds unlikely and it is tempting to excise the word έπιτOμης as an otiose gloss that has crept into the text. The extract discusses the different aims of physics and astronomy and has obvious relevance to a work on meteorology, which has affinities with both these sciences. Simplicius ends his citation with the following: “In this manner, then, Geminus also, or rather Posidonius in Geminus, expounds the difference …” oτνω μέν oνν και o τέμινoς ητoι o παρα τω τέμινω πoσέισωνoς την καιδιαΦoραν … παραδιδωσιν). From this it has been implied that Geminus did little else but reproduce the opinions of Posidonius.
Such an implication certainly overestimates Geminus debt to Posidonius. There is no doubt that both writers subscribed to Stoic views of the universe, and both were concerned to combat the attacks on the validity of the mathematical sciences, which, as we learn from Proclus and Sextus Empiricus, were mounted by both Skeptic and Epicurean philosophers;26 but Geminus shows his independence of Posidonius in several respects. In Isagoge XVI, it is Eratosthenes’ estimate of 252,000 stades that is taken as the basis of the division of the earth into zones and not Posidonius’; figure of 240,000 (Cleomedes, De Motu circulari, I, 10) or 180,000 (Strabo, Geography, 95),27 neither of which is mentioned (nor, in fact, is Posidonius named in the entire treatise). Chapters VIII and XVIII, on the calendar and the length of the lunar month, almost certainly owe nothing to Posidonius, as there is no evidence that he did any work on calendrical problems; and there are other indications that Geminus, although he must have been well acquainted with Posidonius’ opinions and was probably Proclus’ chief authority for the latter,28 did not hesitate to differ from the latter in following other sources (particularly Eratosthenes and Hipparchus for geography and astronomy) or putting forward his own views, as in his criticism of Euclid’s postulates.
1. On this, see O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed. (Providence, R.I., 1957), pp. 81, 94.
2. 25 December according to W. Kubitschek, Grundriss der antiken Zeitrechnung (Munich, 1928), p. 109; 28 December according to Böckh, cited by Manitius in his ed. of Elementa, p. 264. See also “Synchronistic Table,” in E. J. Bickerman, Chronology of the Ancient World (London-Ithaca, N.Y., 1968), p. 150.
3. For details, see Manitius ed. of Elementia, Anmerkung 16, pp. 263–266.
4. The Σαταβνρων of the MSS must be corrected to Aταβνριoν see D. R Dicks, Geographical Fragements of Hipparchus (London, 1960), p. 30.
5. See V, 48; Dicks. op. cit., pp. 123, 130, 176.
6. See D. R. Dicks, Early Greek Astronomy to Aristotle (London, 1970), notes 174, 345.
7. See Ptolemy, Almagest, III, 4.
8.Ibid., IV, 2.,
9. See Dicks, Early Greek Astronomy, pp. 86 f.
10. See Dicks, Geographical Fragments, p.148.
11. Se Dicks, Early Greek Astronomy, pp. 84 f.
12. See Manitius ed. of Elementa, p. 278.
13.Ibid., Anmerkung 34
14. See Tittle, “Geminos I,” cols. 1031–1032.
15.Ibid., cols. 1029–1030.
16. See Eutocius, Commentaria in Conica I, in Apollonii Pergaei quae graece exstant, J. L. Heiberg, ed., II (1893), 170, 25.
17.Commentarii in primumuth Euclidis elementorum librum, G. Friedlein, ed. (Leipzig, 1873.)
18.Euclidis opera omnia, J. L. Heiberg and H. Menge, eds., V (Leipzig, 1888), 81, 4; 82, 28; 107, 20.
19.Heronis Alexandri reliquiae, F. Hultsch, ed (Beriln, 1864), nos 5–14, 80–86.
20. See Tittle, op. cit., cols 1039–1040.
21.Codex Leidensis 399, I R. O. Besthorn, J. L. Heiberg, G. Junge, J. Raeder, and W. Thomson, eds., 3 pts. (Copenhagen, 1893–1932).
22. T.L. Heath, History of Greek Mathematics, II, 224, accepts it; but the same scholar, in The Thirteen Books of Euclid’s Elements, 2nd ed. (Cambridge, 1925), pp. 27–28, rejects it. Compare A. I. Sabra, “Thabit Ibn Qurra on Euclid’s Parallels Postulate,” in Journal of the Warburg and Courtauld Institutes, 31 (1968), 13.
23.Euclidis opera omnia. Supplementum: Anaritii in decent libros priores Elementorum Euclidis commentarii, M. Curtze, ed. (Leipzig, 1899).
24. For details see Heath, History of Greek Mathematics, II, 223–231.
25.Sirnplicii in Aristotelis Physicorum libros IV priores commentaria, H. Diels, ed. (Berlin, 1882), pp. 291–292; this passage is also printed by Manitius in his ed. of the Isagoge as Fragmetntum I, pp. 283–285.
26. Proclus, G. Friedlein, ed., pp. 199 f., 214 if.; Sextus Empiricus, Adversus mathematicos, H. Mutschmann-J. Mau, eds., I, 1 ff,; compare the whole of III
27. See E.H Bunbury, History of Ancient Geography, II (London, 1879), 95–96; Dicks Geographical Fragments., p.150.
28. See Title, op. cit., col. 1042.
K. Manitius edited the Gemini Elementa astronomiae (Leipzig, 1898); chs. I, III-VI, VIII-XVI have been published by E. J. Dijksterhuis as Gemini Elementorum astronomiae, no. 22 in the series Textus Minores (Leiden, 1957).
See T. L. Heath, History of Greek Mathematics, II (Oxford, 1921; repr. 1960), 222–234; and Tittel, “Geminos I,” in Real-Encyclopádie, Halbband XIII (1910), cols, 1026–1050.
D. R. Dicks
Geminus (also Known as Lambrit or Lambert), Thomas
Geminus (also Known as Lambrit or Lambert), Thomas
(b Lixhe, Belgium, ca 1510; d London, England, June 1562)
Geminus— his assumed name may indicate he was the twin of his brother Jasper Lambrit-migrated about 1540 to England, where he practiced the arts of engraving, printing, and instrument making, the last possibly with the assistance of Humphrey Cole. The unique copper-engraved anatomical figure of a woman (ca. 1540), now in the Wellcome Historical Medical Library (Catalogue of Printed Books, 1 [London, 1962], no. 29.0), was probably the work of Geminus, as were the, Cornpendiosa totius anatomie Delineatio (1545) and the unique copy of Morysse and Damashin Renewed and Encreased Very Profitable for Goldsmiths and Embroiderars (London. 1548), now in the Landesmuseum, Münster. Geminus both engraved and printed a map entitled Nova descriptio Hispaniae (London, 1555), on four sheets, and printed Brittmniae insulae.… nova descriptio, (London, 1555); the unique copies of both are now in the Bibliothèque Nationale. Paris. He also printed Leonard Digges’s Prognostication of Right Good Effect in 1555 and a second edition in 1556, and he was possibly responsible for the diagram of the signs of the zodiac on the title page; he also brought out Digges’s Boke Named Tectonicon (1562). In addition, he produced and engraved several astrolabes: one with the arms of the duke of Northumberland, Sir John Cheke, and Edward VI, dated 1552 (now in the Royal Belgian Observatory, Brussels), two about 1555 (National Maritime Museum, Greenwich, and Museo di Storia delle Scienze, Florence), and another for Queen Elizabeth (Museum of the History of Science, Oxford).
Geminus most important work was the series of handsome, copper-engraved anatomical figures, most of them plagiarized from Vesalius’ Fabrica and a few from his Epitome. Upon being presented with a set of these engravings in 1544, Henry VIII urged Geminus to issue, them in the form of a book. This was done in the following year, under the title Compendiosa totius anatomie delineatio published in London by J. Herford rather than by Gemins, who apparently had not yet established his own press. The illustrations were accompanied by the Latin text of Vesalius’ Epitome, slightly defective in its concluding chapter as the result of verbal compression, and by the Latin text of Vesalius’ descriptions (indices) of his illustrations. It appears likely that this first plagiarism of the major Vesalian texts and illustrations was intended as a treatise to accompany the anatomical lectures given at the recently (l540) organized United Company of Barber-Surgeons of London.
Although Geminus’ book introduced Vesalian anatomy to England, it appears to have been too advanced for the surgeons, who, moreover, because of guild rather than university training, were unable to comprehend the Latin text. In consequence of what was probably a disappointing sale, the Compenaious Anatomy was published again 1553 (“Impranted at London by Nycholas Hyll…for Thomas Geminus”), with an English “Treatyse” substituted for the Latin text of the Epitome and the Vesalian descriptions of the illustrations translated into English. The translations and the text were the work of Nicholas Udall, the new text being basically the same as that employed by Thomas Vicary in his Anatomie of Mans Bodie (1548); i.e.; a fourteenthcentury compilation mostly representative of Henry of Mondeville surgical anatomy, known today through the existence of a late fifteenth-century manuscript copy (Welcome MS. no. 564). To this Udall added some sections from Ludovicus Vassaeus’ Galenic In anatomen corporis humani tabula quatuor (1540) and a few fragments from Guy de Chauliac. The result therefore was essentially a medieval anatomical text factually denied by the accompanying Vesalian illustrations and their Vesalian descriptions; it could have been of little use for anatomical instruction despite the relatively low level of anatomical studies in England at that time. It was probably because of the spectacular Vesalian illustrations rather than the text that another edition of the Compendious Anatomy seems to have been called for in 1559, this time with some but very slight editorial assistance from Richard Eden (“Imprinted at London within the blacke fryars: by Thomas Gemini. Anno salutis. 1559”).
Despite the questionable value of the text, Geminus’ plagiarized illustrations were of considerable influence and were plagiarized in turn for Raynold’s Byrth of Mankynde (London, 1545, and certain succeeding editions), William Bullein’s Government of Health (London, 1558, 1559), Vicary’s Profitable Treatise of the Anatomie of Mans Body (London, 1577), and Banester’s Historie of Man Sucked From the Sappe of the Most Approved Anathomistes (London, 1579). In 1560 or shortly thereafter Geminus’ plates were acquired by the Parisian printer Andtré Wechsel and, with the editorial assistance of Jacques Grévin, were reproduced with Latin and French texts (1564, 1565, 1569) borrowed from Geminus’ first edition of 1545. In turn these Parisian editions led to still others in France, Germany and the Netherlands as late as the mid-seventeenth century.
Aside from these activities very little is known of Geminus’ life. During the reign of Henry VII he received wages as a royal surgeon; although there is no indication of his ever having had the necessary training, except perhaps for some mechanical skill and whatever he may have learned of anatomy from his plagiarism of the Vesalian illustrations and texts. Indeed, in 1555 he was penalized by the College of Physicians of London for practicing medicine without a license and undertook to print 200 copies of a proclamation against quackery for the college in place of his fine. No copy of this proclamation is known to exist today. Geminus’ printing and engraving activities; located in Blackfriars, appear to have been financially successful, as is indicated by the distribution of his possessions according to his will, dated 22 May 1562 (O.S.).
All of Geminus’ works are today very rare. For bibliographical details on those pertaining to anatomy, see Harvey Cushing. A Bio-bibliography of Andreas Vesalius, 2nd ed. (Hamden, Conn., 1962:), pp. 121-130, 131-132, 135-136. The 1553 ed. Of the Compendiosa totius anatomie delineatio Has been published in a facs, ed. (London, 1959); the intro to it by C. D. O’Malley contains the fullest account of Geminus and his activities. There is also a facs. ed. of the Illustrations, curiously entitled Andreas Vesalius (Geneva, 1964). The accompanying explanatory text unfortunately contains many errors of fact.
C. D. O’Malley
c. 130-c. 70 b.c.
Greek Philosopher, Astronomer, and Mathematician
The most important contribution of Geminus to mathematics was his classification of mathematical disciplines, and his efforts to define both mathematics and science. He also gave his own version of Euclid's parallel postulate, and wrote a survey of Greek astronomy. Much of his work as mathematician and astronomer, however, was in service to his primary career as a Stoic philosopher intent on justifying the Stoic worldview against attacks on the sciences by Epicureans and Skeptics.
Most likely Geminus came from the Greek island of Rhodes, a center of astronomical study. Though his name was Latin, he was not of Italian background; rather, the world in which he lived was one completely dominated by Rome. Geminus studied under Posidonius (c. 135-c. 51 b.c.), and though the latter was only slightly older than he, Posidonius had a reputation as, if not the wisest man of his day, certainly the most learned of the Stoics. (It is possible Geminus's dates were c. 10 b.c.-c. a.d. 60, in which case he would have been studying in Posidonius's school, but not under Posidonius. The dates of his life are based on the calendar in his Isagoge or Introduction to Astronomy, but the details of that calendar are subject to at least two possible interpretations.)
Geminus's Theory of Mathematics, now lost, presented an overview of geometry. The latter was by then a long-established discipline among the Greeks, and this gave him a certain perspective that would have been beyond the reach of his predecessors. Thus he undertook to define mathematics as a whole, and to classifying it within the context of the sciences.
At least from the time of Aristotle (384-322 b.c.), Greek schools of thought tended to view the world hierarchically; thus the Stoics viewed all of existence as a continuum from the lowest inanimate object to God, or Absolute Reason. This hierarchical conception naturally informed Geminus's schema of classification, as did the Aristotelian notion that pure science is above applied—or as Geminus called it, "mixed" or "impure"—science.
Pure science dealt with what Geminus referred to as the intelligible, whereas applied science concerned that which he described as the tangible. He wrote that "geometry and arithmetic, sciences which deal with... the eternal and unchangeable... [were] extended by later writers to cover what we call 'mixed' or applied mathematics...which has to do with sensible objects, e.g. astronomy or optics." Because they were primarily concerned with what he regarded as the intelligible, arithmetic and geometry were therefore at the apex of Geminus's system. As for the mathematics of the tangible, he divided it into six classes: mechanics, astronomy, optics, geodesy, canonics, and logistics.
This system says as much about the Greek mind—for instance, the emphasis on the ideal over that which can be experienced, a legacy of Plato (427-347 b.c.)—as it does about mathematics itself. Nonetheless, it constituted a valuable early attempt to give shape to the mathematical discipline, and to place it within the context of scientific study.
Elsewhere in Geminus's mathematics text, pieces of which survive only in quotes from others, is his discussion of Euclid's (c. 325-c. 250 b.c.) parallel postulate. Proclus (410?-485) quotes him thus: "...when the right angles are lessened, [that] the straight lines converge [as Euclid had indicated] is true and necessary; but the statement that, since they converge more and more as they are produced, they will sometime meet is plausible but not necessary...."
In his Introduction to Astronomy, a work that does survive, Geminus dealt with a "tangible" discipline. His approach to astronomy was not original, however, but rather based on the work of Hipparchus (190-126 b.c.).