Diocles of Carystus
DIOCLES OF CARYSTUS
(b. Carystus [?]; Greece; fourth century BCE), medicine, anatomy. For the original article on Diocles see DSB, vol. 4.
Diocles of Carystus (a major town on the south end of the Greek island of Euboea, facing the East Coast of Attica) is enjoying renewed attention from historians of medicine, science, and philosophy. His significance as a major thinker, practitioner, and writer of the fourth century BCE had always been recognised from antiquity onwards, but the details of his role in the history of medicine and related areas like botany—and possibly even meteorology—are now more accurately being grasped (see van der Eijk, 2000–2001; Hankinson, 2002; Vivian Nutton, 2004).
This renewed interest comes after several decades of considerable neglect. Early and mid-twentieth century attempts to date and relate Diocles’s medical ideas to other intellectual traditions (such as Max Wellmann’s claim  that Diocles belonged to the Sicilian school of medicine, or Werner Jaeger’s thesis [1938; 1940; 1951; 1952] that Diocles was a pupil of Aristotle's) have proven too speculative and failed to find universal agreement (for surveys of earlier scholarship see von Staden, 1992, and Longrigg, 1993). A period of scepticism followed, and scholarship was aporetically stuck in “Probleme um Diokles von Karystos” (Kudlien, 1963). The chief difficulties were the fragmentary nature of the evidence (none of Diocles’s works have survived in their entirety) and the bias of the sources reporting or quoting his views. The absence of a reliable collection, interpretation, and evaluation of the surviving evidence (Wellmann’s 1901 collection widely being regarded as obsolete) presented a serious obstacle to a fresh and comprehensive investigation of his views.
New Historical Methods . Yet two related trends in classical scholarship addressed this situation and helped to restore Diocles’s prominent position in the history of thought. First, there has been a renewed and systematic examination of medical and philosophical doxography and, more generally, of ancient authors’ methods and strategies for quoting, reporting, and representing the views of earlier authorities. (Mansfeld and Runia, 1996; van der Eijk, 1999b). Scholars’ knowledge of Diocles’s ideas depends entirely on what later authors tell about him—authors such as Galen, Celsus, Pliny the Elder, Oribasius, Soranus, Caelius Aurelianus, the so-called ‘Anonymous of Paris’ (probably first century CE), Aëtius the doxographer, and Athenaeus of Naucratis’s Sophists at Dinner (a voluminous work from the second century CE full of quotations from earlier authors). It is therefore important to determine how well informed they were (e.g. whether they had direct access to Diocles’s writings or relied on intermediary sources), how they viewed his role in the history of medicine and his relationship to other medical authors, for what reasons they were interested in him, for what reasons they were citing or quoting him, what the peculiarities were of their methods of reporting, and how selective they may have been in directing their attention to specific areas of Diocles’s output. Such determination is important not only in order to estimate the extent to which all these factors may have colored their representation of Diocles’s views and scientific activity, but also to evaluate the information these source authors provide and to have some idea as to what one can expect them to say.
Secondly, and on the basis of this development, there has been a new approach to fragment collecting in classical scholarship that takes due account of the context in which a fragment is embedded and of the role of the reporting author in the representation of a thinker’s views (Burkert, 1998; van der Eijk, 1999a; Hanson, 1997). This then feeds into the reconstruction of the thinker’s views in that, during the process of piecing together the surviving evidence, the material is differentially weighted in accordance with its relative evidential value.
In the case of Diocles, these scholarly developments led to a new collection, English translation, and comprehensive interpretation of the fragments (van der Eijk 2000–2001), which provides a basis for renewed study. The net result is a substantial collection of 234 fragments (more than fifty more than in Wellmann’s collection), surviving in Greek, Latin, and Arabic; of these, nearly forty fragments lay claim to being direct verbatim quotations from his works (although in some cases their reliability is somewhat dubious), ranging in size from a few words to ten pages of text (fr. 182); the rest are reports in indirect speech, sometimes paraphrase, sometimes openly polemical in nature (for example, Caelius Aurelianus, our major source for Diocles’s therapeutic views, heavily criticised him), or associating Diocles’s views with those of other medical writers (such as many testimonies in Galen, who on the whole tends to downplay Diocles’s originality in favor of his own achievements). The evidence adds up to an overall picture of a very self-conscious scientific thinker/practitioner with wide-ranging interests and a substantial output.
The titles surviving from his works are Anatomy(according to Galen the first handbook of its kind), Affection, Cause, Treatment (Diocles’s major work on pathology), On Treatments (a detailed work in at least four books on the treatment of a wide range of different diseases), On Prognosis, On Fevers, On Digestion, On Catarrhs, On Matters related to Women (an extensive work in at least three books), On Matters of Health to Pleistarchus (Diocles’s major work on regimen in health), Archidamus (on the medicinal usage of olive oil), On Evacuations, On Bandages, On External Remedies, On Lethal Drugs, On Vegetables, On Rootcutting, In the Surgery, On Sexual Activity, Letter on the Preservation of Health to Antigonus, and On Cookery (the evidence for the latter three works is, however, not entirely secure). It is reasonably certain that at least some of these works were widely available in the Hellenistic and early Imperial age, and some were subjected to close textual and medical analysis: Thus the first century BCE writer Apollonius of Citium quotes from Diocles’s work on surgery (fr. 163), Galen reports textual variants in different copies of Diocles’s Matters of Health (fr. 188), and Oribasius in the fourth century preserved some extensive excerpts from Diocles’s dietetic and therapeutic works. In the Byzantine era, however, direct access to Diocles’s works seems to have become rare, and Arabic authors citing him seem to have been familiar with his views through Galen and other intermediaries only.
Research Approach . It is clear that, in antiquity, Diocles played a key role in the development of dissection and comparative anatomical research (fr. 24b refers to repeated animal dissection to prove a point in human anatomy), in systematic pathology (carefully distinguishing causes, symptoms, and therapies), in the further differentiation and refinement of therapeutics and surgery (where he earned fame for his spoon for the removal of arrow heads [fr. 167] and for his bowl, a particular type of bandage [fr. 166]), in gynaecology and especially dietetics and regimen in health—a field in which he acquired the greatest reputation (although again our view may be somewhat distorted by bias on the part of the sources and in the subsequent selective transmission of his ideas). He collected and systematised a large number of foods and drinks, herbs, and poisons. He went into great detail specifying their qualities and powers (dunameis) and differentiating according to mode of preparation, environmental factors such as season and climate, and according to age, living pattern, and constitution of the patient. Furthermore, his views on the role of pneuma in the psychophysiology of human cognition (in which both the heart and the brain are involved), emotion, movement, and action, as well as on blockage of the flow of pneuma as the major cause of a number of diseases, clearly paved the way for later developments in Hellenistic medicine and Stoic philosophy; and his account of hypochondriac melancholy (fr. 109) continued to be cited by later authors (Greek as well as Arabic) as the authoritative treatment of the subject.
Apart from this, Diocles is also being appreciated by historians of ancient philosophy for his methodological awareness and his theoretical views on causal explanation, inference from signs, for his careful balancing of reason and experience, and for the overall consistency and coherence of his views (Frede, 1987; Hankinson, 1998 and 2002). Diocles was not an armchair physician, and he clearly had a keen interest in the phenomena and in the practical aspects of medical care. Yet at the same time, he displayed a strong theoretical outlook, a desire to build his medical views on a general theory of nature, and a belief that the treatment of specific bodily parts has to be based on a consideration of the patient’s body (and mind) as a whole—characteristics that prompted later Greek medical writers reflecting on the history of their own subject to speak of him as a member of the Rationalist (logikos) or Dogmatist (dogmatikos) sect of medicine. It is true that Diocles sometimes showed himself eager to back up physiological reasoning by empirical evidence (e.g., in frs. 109 and 176), and he insisted that causal explanations must be empirically verifiable and relevant to the situation at hand (fr. 176). At the same time, he did not shy away from referring to hidden causes (fr. 178) and other invisible entities like pneuma; he seems to have adopted rather uncritically the principle that healing takes place by means of opposite qualities, and he shared several more speculative interests of some of the Hippocratic writers, such as the notion of critical days and the belief in the determining role of the number seven in areas like embryological development.
It is further clear that Diocles was a prolific writer and medical communicator using a variety of literary forms (including doxographical discourse and possibly letters and dialogues) and an elegance of style that would have contributed to the dissemination of medical ideas among wider audiences. He was part of a movement aiming for expansion of the area of expertise commanded by the medical profession of his time, comprising areas like wine-tasting, cookery, gymnastics, travel, and other life-style features, even the raising of children—all of which belonged to the matters of health (hugieina) covered by the health expert (hugieinos).
In all this, Diocles was most likely aware of a considerable number of other medical, scientific, and philosophical views of his time. He is seen on a number of occasions taking account of existing ideas, including some of the views found in the so-called Hippocratic writings, with which he sometimes took issue, but claims that he possessed, or even created, a Hippocratic Corpus go beyond the evidence (and there is no certainty that he took these writings to be by Hippocrates). On other occasions, he quoted or reported the works of Aristotle (fr. 40) and of a physician named Archidamus (fr. 185)—perhaps his father, although again this is not certain—and he displayed a more general concern with medical language and nomenclature. Considering his high reputation in Athens, it is plausible to assume that he was in touch with the main currents and centers of scientific thinking, such as Plato’s Academy and Aristotle’s Lyceum, and probably also the Athenian physician Mnesitheus and other dietetic writers, and although it is difficult to prove influence, similarities with Aristotelian and Peripatetic ideas and styles of reasoning and arguing are unmistakable. Yet there is no evidence that Diocles was a member or pupil at Aristotle’s school, and there is good reason to believe that any intellectual exchange there may have been in both directions (the oldest reference to Diocles being found in Theophrastus’s work On Stones).
On the negative side, associations of Diocles with Sicilian medicine (as represented by Empedocles, Philistion, and Plato) must be considered doubtful, and something similar applies to his putative connections with empiricism or scepticism: These are constructs of twentieth-century scholarship that fail to find corroboration by the evidence. Likewise, the hotly disputed question of Diocles’s date must be regarded as insoluble for lack of secure independent evidence. All that can be said with some degree of certainty is that Diocles lived somewhat later than Hippocrates and somewhat earlier than Erasistratus and Herophilus. Considering the difficulties involved in dating these medical writers, the question must remain open, and any reasonable pair of dates within the broad time-frame of the fourth century must be deemed possible.
WORKS BY DIOCLES OF CARYSTUS
Eijk, Philip J. van der. Diocles of Carystus. A Collection of the Fragments with Translation and Commentary. Vol. 1: Text and Translation. Leiden, Germany: Brill (Studies in Ancient Medicine 22), 2000; Vol. 2: Commentary. Leiden, Germany: Brill (Studies in Ancient Medicine 23), 2001. This replaces the older editions by Max Wellmann Die Fragmente der sikelischen Ärzte Akron, Philistion und des Diokles von Karystos(Fragmentsammlung der griechischen Ärzte, Band 1), Berlin: Weidmann, 1901, and Moritz Fraenkel, Dioclis Carystii fragmenta quae supersunt, Berlin: Typus Haynianis, 1840. Collection of the fragments of Diocles.
Burkert, Walter, ed. Fragmentsammlungen philosophischer Texte der Antike—Le raccolte dei frammenti di filosofi antichi. Göttingen, Germany: Van den Hoeck & Rupprecht, 1998.
Eijk, Philip J. van der. “Some methodological issues in collecting the fragments of Diocles of Carystus.” In I testi medici greci III. Tradizione e ecdotica, edited by Antonio Garzya and Jacques Jouanna. Naples: M. d’Auria, 1999a.
——, ed. Ancient Histories of Medicine. Essays in Medical Historiography and Doxography in Classical Antiquity. Leiden, Germany: Brill, 1999b.
——. “Diocles and the Hippocratic writings on the method of dietetics and the limits of causal explanation.” In Medicine and Philosophy in Classical Antiquity. Doctors and Philosophers on Nature, Soul, Health and Disease, by Philip J. van der Eijk. Cambridge, U.K.: Cambridge University Press, 2005.
——. “The Heart, the Brain, the Blood and the Pneuma: Hippocrates, Diocles and Aristotle on the Location of Cognitive Processes.” In Medicine and Philosophy in Classical Antiquity. Doctors and Philosophers on Nature, Soul, Health and Disease, by Philip J. van der Eijk. Cambridge, U.K.: Cambridge University Press, 2005.
——. “To Help, or to Do no Harm. Principles and Practices of Therapeutics in the Hippocratic Corpus and in the Work of Diocles of Carystus.” In Medicine and Philosophy in Classical Antiquity. Doctors and Philosophers on Nature, Soul, Health and Disease, by Philip J. van der Eijk. Cambridge, U.K.: Cambridge University Press, 2005.
Frede, Michael. “The Original Notion of Cause.” In Essays in Ancient Philosophy, edited by Michael Frede. Oxford: Clarendon Press 1987.
Hankinson, R James. Cause and Explanation in Ancient Greek Thought. Oxford: Clarendon Press, 1998.
——. “Doctoring history: ancient medical historiography and Diocles of Carystus,” Apeiron 35 (2002): 65–81.
Hanson, Ann Ellis. “Fragmentation and the Greek medical writers.” In Collecting Fragments—Fragmente sammeln, edited by Glenn W. Most. Göttingen, Germany: Van den Hoeck & Rupprecht, 1997.
Harig, Gerhard, and Kollesch, Jutta. “Diokles von Karystos und die zoologische Systematik.” NTM Schriftenreihe zur Geschichte der Naturwissenschaften, Technik und Medizin 11 (1974): 24–31.
Jaeger, Werner Wilhelm. Diokles von Karystos. Die griechische Medizin und die Schule des Aristoteles, Berlin: Weidmann, 1938a.
——. “Vergessene Fragmente des Peripatetikers Diokles von Karystos. Nebst zwei Abhandlungen zur Chronologie der dogmatischen Ärzteschule.” Abhandlungen der preussischen Akademie der Wissenschaften, Philosophisch–historische Klasse, 1938b.
——.“Diocles. A New Pupil of Aristotle.” Philosophical Review 49 (1940): 393–414.
——. “Diokles von Karystos. Ein neuer Schüler des Aristoteles.” Zeitschrift für philosophische Forschung 5 (1951): 25–46.
——. “Diokles von Karystos und Aristoxenos über die Prinzipien.” In Hermeneia. Festschrift für Otto Regenbogen. Heidelberg, 1952.
Kudlien, Fridolf. “Probleme um Diokles von Karystos.” Sudhoffs Archiv 47 (1963): 456–464.
——. “Diokles von Karystos.” Der kleine Pauly 2 (1967): 56–57.
Kullmann, Wolfgang. Wissenschaft und Methode. Interpretationen zur aristotelischen Theorie der Naturwissenschaft. Berlin: De Gruyter, 1974.
Longrigg, James. Greek Rational Medicine, London: Routledge, 1993.
——. Greek Medicine from the Heroic to the Hellenistic Age. New York: Routledge, 1998.
Mansfeld, Jaap, and Runia, David T., Aetiana. The Method and Intellectual Context of an Ancient Doxographer, Vol. 1: The Sources. Leiden, Germany: Brill, 1996.
Nutton, Vivian. “Diokles von Karystos.” Der neue Pauly 3 (1997): 610–613.
——. Ancient Medicine. London: Routledge, 2004.
Sconocchia, Sergio. “La lettere di Diocle ad Antigono e le sue traduzioni latine.” In Prefazioni, prologhi, proemi di opere technico-scientifiche latine, edited by Carlo Santini and Nino Scivoletto, Vol. 3. Rome: Herder, 1998.
Staden, Heinrich von. “Jaeger’s ‘Skandalon der historischen Vernunft’: Diocles, Aristotle, and Theophrastus.” In Werner Jaeger Reconsidered: Proceedings of the Second Oldfather Conference, University of Illinois, 26–28 April 1990, edited by William M. Calder III. Atlanta: Scholar’s Press, 1992.
Torraca, Luigi. “Diocle di Caristo, il Corpus Hippocraticum ed Aristotele.” Sophia 33 (1965): 105–115.
Vallance, J. “Diocles (3).” In Oxford Classical Dictionary, edited by Simon Hornblower and Anthony Spawforth. Oxford: Oxford Unviersity Press, 1996, 470.
Wöhrle, Georg. Studien zur Theorie der antiken Gesundheitslehre(Hermes Einzelschriften 56). Stuttgart, Germany: Steiner Verlag, 1990.
Philip J. van der Eijk
(fl. ca. 190 b.c.)
The few facts known about Diocles’ life are derived entirely from his one surviving work. On Burning Mirrors (Π∊ρί πνρίων). His date can be determined approximately from his acquaintance with the mathematician Zenodorus, who is known to have lived in the early second century b.c. This date accords well with the terminology and treatment of conic sections in Diocles’ work, which shows little or no trace of influence by Apollonius’ Conics, since it makes him an exact contemporary of Apollonius. Diocles mentions only mathematicians contemporary with or earlier than Archimedes (except for Zenodorus). He was living in Arcadia when Zenodorus visited him.
Until the recent discovery of the Arabic translation of On Burning Mirrors, it was lost except for three excerpts in the commentary by Eutocius on Archimedes’ Sphere and Cylinder. Since Eutocius did not quote Diocles verbatim, but reformulated his proofs (for instance, introducing references to Apollonius’ Conies), modern inferences about Diocles’ date and place in the history of the theory of conics are misleading and usually wrong. The following account is based on the Arabic text.
The work consists of an introduction and sixteen propositions, of which numbers 6, 9, and 14 are spurious (probably interpolated in the Arabic transmission). The title On Burning Mirrors is somewhat misleading, as it applies only to the first five propositions. Numbers 7 and 8 deal with a problem in Archimedes’ Sphere and Cylinder, and propositions 10- 16 with the problem of “doubling the cube.” The book as a whole has no unity except that it deals with” higher geometry“; as is natural for a Hellenistic Greek work, much of it is concerned with conic sections.
Diocles starts from two problems, the first posed by one Pythion (otherwise unknown) to Conon of Samos: What mirror surface will reflect the sun’s rays to the circumference of a circle? The second was posed by Zenodorus to Diodes: What mirror surface will reflect the sun’s rays to a point? Diocles says that the second problem was solved by Dositheus (well-known as a correspondent of Archimedes). This implies that the focal property of the parabola was recognized by about the middle of the third century b.c. Diocles indicates, however, that he himself is the first to give a formal proof of the property. After an obscure but historically interesting discussion of the application of burning mirrors to sundial construction, he proves (prop. 1) the focal property of the parabola, and shows how Pythion’s and Zenodorus’ problems can be solved by suitable rotation of the parabola.
In propositions 2 and 3 Diocles shows that it is useless to construct a spherical burning mirror from an arc greater than 60°. and proves that all rays reflected from such an arc will pass through a section less than 1/24 the diameter of the mirror.
Propositions 4 and 5 are of great historical interest. They address the problem of constructing a parabolic mirror of given focal length. Diocles’ solution is as follows (see Figure 1). If the given focal length is AB, complete the square ABEF, extend AF to K so that FK =AF, join KE. and produce it to meet AB produced in R. Take arbitrary points D,G on AB. draw DH, GZ parallel to AF, and produce them to meet KE in L,M. Then, with center A and radius DL, draw a circle to cut DL, in N and (on the other side in)φ. Similarly, with center A and radius GM, draw a circle to cut GM in Θ and Ψ Make AX = AK. Then points K, N,Θ,B,Ψ,X lie on a parabola with A as focus. This construction is equivalent to the construction of the parabola from focus and directrix– as is obvious if, like Diocles, we complete the square ARSK and drop onto SR the perpendiculars LQ, NO, MC,ΘP. For NA=LD by construction, and NA=LQ=LD,so NA=NO, or the distance of the point N from A (the focus) is equal to its vertical distance from the line SR (the directrix). Similarly for the other points K, Θ, and so on. It is noteworthy that Diocles proves not only this, but also that a curve so generated is indeed a parabola (according to the classical Greek definition, by the equivalent of the relationship y2=px The obvious inference is that Diocles himself discovered the focus-directrix property of the parabola.1
In propositions 7 and 8 Diocles discusses a problem arising out of Archimedes’ Sphere and Cylinder II,4: to divide a sphere in a given ratio. The problem involves, in modern terms, a cubic equation, which Diocles solves by the intersection of a hyperbola and an ellipse. His solution was already known from Eutocius, who also gives solutions by Dionysodorus and (possibly) Archimedes that likewise employ the intersection of two conies.
The rest of the book is devoted to the problem of doubling the cube, to which much attention was paid by Greek mathematicians from the fifth century b.c. on. Like everyone else in antiquity. Diocles in fact solves the equivalent problem of finding two mean proportionals between two given magnitudes.2 His first solution, employing the intersection of two parabolas, was already known (in an altered form) from Eutocius; but since Eutocius did not mention the author, in modern times it has been almost universally misattributed to Me-naechmus.
The second solution is both more interesting and more influential (see Figure 2). In a circle, with diameters AB, GD intersecting at right angles, there are marked off from D equal small arcs DZ, ZH,HΘ … and DN, NS,SO… on the other side of D. Drop onto AB the perpendiculars ZK, HL,ΘM.… and joinBN, US, BO.… Mark the points P, Q, R… where BNcuts ZK.… Then it can be shown that.
That is, KZ and KB are two mean proportionals between AKand KP. Similarly for point Q, LH and LB are two mean proportionals between AL and LQ, and so on. The points P,Q,R … are joined in a smooth curve DPQRB, which can be used to find two mean proportionals between any two magnitudes, and to solve related problems, as Diocles demonstrates at length in propositions 11-16.
Despite his contributions to the theory of conics,
there is no mention of Diocles in surviving Greek mathematical works until late antiquity. In the sixth century his work was used by Eutocius and. about the same time, by the unknown author of the “Bobbio Mathematical Fragment.”3 It seems likely that Diocles had a considerable indirect influence on medieval discussions of the parabolic burning mirror. There is only one known explicit reference to his work in Islamic literature.4 It is, nevertheless, very probable that it is one of the sources of Ibn al-Haytham’s On Paraboloidal Burning Mirrors,5 which was well-known not only in the Islamic world but also in the West after its translation into Latin. Diocles was known by name in the West, however, only through the extracts in Eutocius, whose commentary on Archimedes attracted the attention of mathematicians of the late sixteenth and seventeenth centuries particularly for its discussion of curves used by the ancient geometers to solve the problem of doubling the cube. Among these was Diodes’ curve (see Figure 2 above), part of the discussion on which had been excerpted by Eutocius. This curve was dubbed “cissoid” in the seventeenth century,6 It was discussed by some of the most notable mathematicians of that time, including Fermat, Descartes, Roberval, Huygens, and Newton. To them we owe the generalization of the curve, the discovery of its infinite branch, and the revelation of many of its beautiful properties.
1. The extension to all three conic sections is found in Pappus. VII, 312-3 IS. Hulisched., II, 1004-1014. It was probably made in the later Hellenistic period. The argument that it was known as early as Euclid cannot be sustained: see Toomer ed. of Diodes, 17.
2. The problem of doubling the cube had been reduced to finding two mean proportionals between two lines, one of which was double the other, by Hippocrates of Chios (late fifth century b.c.)
3. The author mentions a work, “On the Burning Mirror.” which he attributes to Apollonius. For arguments that this is in fact Diodes’ work, see Toomer ed. of Diocles, 20- 21.
4. In the encyclopedic work by al-Akfānī (fourteenth century). Sprenger ed.. 82. All other references known to me are derived from Entocius, whose commentary on Archimedes’ Sphere and Cylinder was also translated into Arabic.
5. For arguments in favor of this see Toomer ed. of Diodes. 22.
6. Because it was identified with a class of curves known as κισσoειδής (“ivy-shaped”) from ancient sources. The identification is almost certainly wrong, as I have argued in my ed. of Diocles, 24.
The Arabic trans, of Diocles’ On Bunting Mirrors is edited by G. J. Toomer, with English trans, and commentary (New York, 1976), as Sources in the History of Mathematics and the Physical Sciences, no. I. This includes the extracts by Eutocius. which are found in his commentary on Archimedes’ Sphere and Cylinder, bk. II:Archimedis Opera Omnia, J. L. Heiberg, ed., 2nd ed.. Ill (Leipzig. 1915), 66-70, 82-84, 160-174. My intro, to the ed, (1 - 17) gives a full discussion of the evidence for the date of Diocles and his place in the history of the theory of conies. On the date of Zenodorus, see G. J. Toomer, “The Mathematician Zenodorus,” in Greek, Roman and Byzantine Studies, 13 (1972), 177- 192. Pappus’ discussion of the focus-directrix properties of the three conic sections is in Pappi Alexandri-ni Collectionis quae supe sunt, Hultsch, ed., 3 vols. (Berlin 1875- 1878). II, 1004- 1014. On the solutions to Archimedes’ problem, see T. L. Heath. A History of Greek Mathematics, II (Oxford, 1921). 45-49. On the history of the problem of doubling the cube, see ibid.. I. 244-270. The “Bobbio Mathematical Fragment” is edited by J. L. Heiberg in his Mathematics graeci minores, which is Kongelige Danske Videnskabernes Selskab, Historisk-filologiske Meddelelser, 13 , no. 3 (Copenhagen, 1927). 87-92.
The work of al-Akfānī is edited by A. Sprenger, Two Works on Arabic Bibliography, which is Bibliotheca Indica, VI (Calcutta, 1849), 14-99. The passage referring to Diocles is translated by Eilhard Wiedemann in Aufsätze zur Arabischen Wissenschafisgeschichte, I (Hildesheim, 1970), 119-120. Ibn al-Haytham’s work on the parabolic burning mirror is printed as the third treatise in his Majmūʿ al-Rasāʾil (Hyderabad, 1938). A German trans, from the Arabic, together with the medieval Latin trans., was published by J. L Heiberg and E. Wiedemann, “lbn al-Haitarns Schrift über parabolische Hohlspiegel,” in Bibliotheca mathematica, 3rd ser., 10 (1910), 201-237. For a modern mathematical treatment of the cissoid and an account of the discoveries of its properties by seventeenth-century mathematicians, see F. Gomes Teixeira, Traité des courbes spéciates remar-quables planes et gauches, I, which is his Obras sohre mathematics IV (Coimbra. 1908), 1-26; and Gino Loria, Spezielle algebraische und transzendente ebene Kurven, 2nd ed., I (Leipzig-Berlin, 1910), 36-51.
G. J. Toomer
Diocles of Carystus
Diocles of Carystus
(b. Carystus, Euboea; fl. Athens, late fourth century b.c.)
Diocles, the son of Archidamus, also a physician, was still alive shortly after 300 b.c. The Athenians called him a “second Hippocrates” and Pliny the Elder (Natural History, XXVI, 10) wrote that Diocles came “next after Hippocrates in time and reputation.” Galen and Celsus place him as an equal with Hippocrates, Praxagoras, Herophilus, and Erasistratus. The last three of these physicians were contemporaries of Diocles, and these four raised Greek medicine to a high point in its history. Diocles was a pupil of Aristotle, and he was also a contemporary of such Peripatetics as Theophrastus and Strato. By some, Diocles is considered the leading representative of the dogmatic school, which introduced philosophical speculations into the Hippocratic materials and formalized the medical systems. Diocles saw, however, that philosophical theory could not explain everything, and he is best considered as independent of any school.
Diocles’ writings were considerable. The titles of seventeen works are known and more than 190 fragments have been preserved. Unlike the physicians of his time, he wrote in Attic Greek. His writings show a well-polished if simple style, and his language and terminology show the influence of the literary style of Aristotle in scientific writing. The subjects covered in his books range widely.
Diocles’ medical writings show the influence of the Aristotelian teleological view of nature. They also indicate that he was the first physician to use a collection of Hippocratic writings, which he may have assembled himself. According to Galen, he was the first to write a book on anatomy and to use that term in the title. While he did not distinguish the nerves from the veins, he did recognize more of the latter than his predecessors. The heart was the source of the blood, which was carried through the aorta and the vena cava. He also described the lungs, ureters, ovaries, fallopian tubes, ileocecal valve, cecum, and the gall bladder with the tube leading to it from the liver. He distinguished between pleurisy and pneumonia and described hepatic and splenic ascites.
In his views on embryology Diocles followed Empedocles. In generation both the man and the woman furnished seed, which contributed to the development of the embryo. The seed, originating in the brain and spinal marrow, was a product of nourishment. Excessive coition was detrimental to the eyes and spinal marrow. In agreement with Empedocles, he felt that the full development of the embryo occurred in forty days, and as the male child grew in the right (i.e., warmer) side of the uterus, it developed quicker than the female. He described human embryos of twentyseven and forty days. In his studies of sterility, he was especially interested in the mule, and according to Galen, he dissected such animals. Again following Empedocles, he asserted that menstruation occurred during the same period of life for all women, beginning at age fourteen and lasting until sixty. He felt that broad hips, freckles, auburn hair, and manly appearance were certain indicators of fertility. Sterility in the female was attributed to displacement of the uterus.
Diocles’ physiology was similar to that of Philistion and was based on the four basic elements of Empedocles—fire, water, air, and earth. The human body also had the four qualities of heat, moisture, cold, and dryness. Health was dependent upon the proper equilibrium of the four elements in the body. Warmth was especially important in the formation of the four humors of blood, phlegm, yellow bile, and black bile. The proper movement of the pneuma, seated in the heart and spreading through the body by means of the veins, had a most important place in health and illness, the latter being independent of outside causes. Fever, disease, or death occurred if the pneuma was hindered by phlegm or bile. Respiration took place through the pores of the skin as well as through the nose and mouth. The Pythagorean number seven was evident in Diocles’ view that the seventh, fourteenth, twenty-first, and twenty-eighth days were most critical during illness. Fever was not a disease itself but symptomatic of some morbid condition. He distinguished between continuous and intermittent fevers and also quotidian, tertian, and quartan forms. Like Hippocrates, he stressed practical experience, observation, and the importance of diagnosis and prognosis.
Some indication of Diocles’ prominence is seen in the fact that he was known to the rulers of his time. A work on hygiene, written after 300 b.c., was dedicated to the Macedonian prince Pleistarchus, the son of the famous general Antipater. Diocles’ letter on hygiene, written between 305 and 301 b.c. and addressed to King Antigone, one of the generals of Alexander the Great, was fortunately preserved by Paul of Aegina, a Greek physician of the late seventh century a.d. Many editions of this work were printed in the sixteenth century in Latin, French, and English.
One of Diocles’ works is entitled Archidamos in dedication to his dead father. His father had condemned the then current practice of massaging the body with oil, because to do so heated the body too much and made it too dry by rubbing. While refuting his father’s arguments, Diocles proposed a compromise: he suggested that in summer a mixture of oil and water be used and in winter pure oil. In the use of oil and water, he is apparently following the idea of a slightly earlier and anonymous work on diet.
Lengthy fragments of Diocles’ own work on diet were preserved by Oribasius, physician to Emperor Julian. In this work the Greek physician looked at human life as a whole and by describing the routine of one summer’s day prescribed what is suitable and beneficial for men. He made allowances for various ages and changes of seasons. His descriptions are given as ideal standards, dictated by suitable and tasteful behavior—the Aristotelian ethic. He does not describe the various physical exercises, but his whole plan for the day is based on exercise in the morning and the afternoon, revolving around the gymnastics of Greek civilization. His exposition of diet described well the Greek ideals of health, harmony, and balance.
In the history of medical botany or pharmacy, Diocles also deserves recognition. Here, like his colleague Theophrastus, he was probably stimulated to the study of botany by his teacher Aristotle. Diocles was the first scientist to write a herbal on the origin, recognition, nutritional value, and medical use of plants; thus he can be considered the founder of pharmacy. His work was used as a source for all later works until Dioscorides. Two other botanical works, dealing with vegetables and with healing, are practical in nature, but apparently they also advanced the study of plants. Theophrastus, the founder of scientific botany, seems to have made extensive use of the botanical works of Diocles; although he does not name his colleague in his botanical works, in his work On Stones he does refer to Diocles as an authority on a certain mineral.
Diocles is credited with two inventions—a bandage for the head and a spoonlike device for the extraction of arrows.
1. Original Works. Fragments of Diocles’ works are in C. G. Kühn, Diocles Carystius fragmenta collegit (Leipzig, 1827); Mauritz Fraenkel, Dioclis Carystius fragmenta quae supersunt (Berlin, 1840); Werner Jaeger, “Vergessene Fragmente des Peripatetikers Diokles von Karystos,” in Abhandlungen der Deutschen Akademie der Wissenschaften zu Berlin, Phil-hist. Kl., no. 2 (1938); and Max Wellman, “Die Fragmente des sikelischen Aerzte, Akron, Philistion, und des Diokles von Karystos,” in Fragmentsammlung der griechischen Aerzte, I (Berlin, 1901), 117–207.
II. Secondary Literature. On Diocles and his work, see Gustav A. Gerhard, “Ein dogmatischer Arzt des vierten Jahrhunderts vor Christ,” in Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Phil.-hist. Kl. (1913); W. Haberling, “Die Entdeckung einer kriegschirurgischen Instrumentes des Altertums,” in Deutsche militärärztliche Zeitschrift, 40 (1912), 658–660; Werner Jaeger, Paideia, Die Formung des griechischen Menschen, 3 vols. (Berlin-Leipzig, 1934–1947), trans, into English by G. Highet as Paideia: The Ideals of Greek Culture, 3 vols. (New York, 1960), III, 41–44, passim; Werner Jaeger, Diokles von Karystos. Die griechische Medizin und die Schule des Aristoteles (Berlin, 1938, 1963); George Sarton. Introduction to the History of Science, I (Baltimore, 1927), 121; and Max Wellman, Die pneumatische Schule bis auf Archigenes, in ihrer Entwicklung (Berlin, 1895); “Das ältesie Kräuterbuch der Griechen,” in Festgabe für Franz Susemihl. Zur Geschichte griechischer Wissenschaft und Dichtung (Leipzig, 1898); and “Diokles von Karystos,” in Pauly-Wissowa, Real Encyclopädie.
Karl H. Danwinfeldt
Diocles of Carystus
Diocles of Carystus
fl. fourth century b.c.
Diocles of Carystus was a philosopher and pioneer in Greek medicine, acclaimed by the historian Pliny to be second only to Hippocrates (c. 460-c. 377 b.c.) in reputation and ability.
Born in the late fourth century b.c. at Carystus, Euboea, he was the son of Archidamus, a physician. He moved to Athens and became a pupil of Aristotle (384-322 b.c.). Although Aristotle was known as a philosopher, he influenced many physicians of his time because of his inquiries into body physiology. His anatomy was adapted by Diocles and three prominent doctors of the Alexandrian school: Herophilus (c. 335-280 b.c.), Erasistratus (c. 304-250 b.c.), and Praxagoras (fl. fourth cent. b.c.). The four raised Greek medicine to its highest point. Another of Aristotle's pupils, Alexander the Great, died in 323 b.c. exclaiming, "I die by the help of too many physicians"—showing there were a number of physicians in his court.
Physicians including Diocles became prominent in the late fourth century b.c., and when Alexander conquered Egypt and founded the city of Alexandria, he set the stage for the advent of the famous Alexandrian School of Medicine. His successor, Ptolemy I, collected a library of 70,000 manuscripts with information on medicine and drugs.
Diocles was a leading proponent of the dogmatic or logical school and sought to combine philosophy with the medical ideas of Hippocrates. While other physicians were engaged in speculation and superstition, Diocles formalized and organized medicine.
A prolific writer, Diocles was the first to use Attic Greek, the polished Greek of Athens, and showed the influence of Aristotle's literary style in writing. (Most physicians of the time wrote in Ionic Greek, a coarse vernacular style.) The subjects addressed were wide and varied. Only a few fragments of his writing are extant. Diocles carefully assembled the writings of Hippocrates. The Roman physician Galen (c. 129-216 a.d.) stated that Diocles was the first to use the term "anatomy." Like Aristotle, Diocles did not distinguish the nerves from veins and believed that the heart, not the brain, was the seat of intelligence.
Empedocles (c. 492-432 b.c.) greatly influenced the Greek physicians, including Diocles. For example, he was interested in reproduction and asserted that both man and woman furnished the seed that became the embryo, which was fully developed in 40 days. The male developed on the warmer, right side of the uterus and grew faster than the female. Menstruation was the same for all females—beginning at age 14 and ending at 60. Empedocles also influenced Diocles's physiology, in that the latter believed in the four basic elements: air, water, fire, and earth. Health was the proper balance of the system. The four humors—blood, phlegm, yellow bile, and black bile—that corresponded to the elements must also be in balance.
Diocles was close to political rulers. He dedicated a work on hygiene to the Macedonian prince Pleistarchus, son of the famous Greek general Antipater. He wrote a letter also on hygiene to King Antigone, a general of Alexander. The letter was preserved by the seventh-century Greek physician Paul of Aegina (c. 625-690 a.d.). A work called Archidamus was dedicated to his father. When the Greek classics were rediscovered in the sixteenth century, translations of the works of Diocles were made into Latin, French, and English. Also, large fragments of his work on diet were preserved by Oribasius (325-403 a.d.), physician to Emperor Julian.
Inspired by Aristotle and his studies of plants, Diocles was the first scientist to write on nutrition and the medical use of plants. He is considered to be the father of pharmacy. Two inventions were credited to Diocles: a bandage for the head and a spoon-like device for scooping arrows out of the flesh.
Diocles was second only to Hippocrates in the annals of Greek medicine. His work influenced many Greek physicians and scientists, such as Theophrastus (c. 372-287 b.c.) and Dioscorides (c. 40-90 a.d.).
EVELYN B. KELLY
(second century b.c. [?])
Nothing is known of the life of this Greek mathematician, but he must have lived after Archimedes (d. 212 b.c.) and before Geminus of Rhodes [fl. 70 b.c.). Eutocius, a Byzantine mathematician of the fifth and sixth centuries, preserved two fragments of Diocles’ work On Burning Mirrors in his commentary on Archimedes’ On the Sphere and Cylinder.
One of these fragments deals with the solution of the problem of the two mean proportionals by means of the cissoid, which Diodes invented. The problem of doubling the cube, the celebrated Delian problem of ancient geometry, had been the subject of mathematical investigation at least as early as the fifth century B.C. Hippocrates of Chios is reported to have discovered that a solution could be found if a way could be devised for finding two mean proportionals in continued proportion between two straight lines, the greater of which line is double the lesser. The question was studied by Plato’s Academy and a mechanical solution is even attributed, erroneously, to Plato. Before Diocles, solutions were offered by Archytas, Eudoxus, Menaechmus, Eratosthenes, Nicomedes, Apollonius, Hero, and Philo of Byzantium. All of these, and later solutions, are preserved by Eutocius.
Proposition 4 of Book II of Archimedes’ On the Sphere and Cylinder presents the problem of how to cut a given sphere by a plane in such a way that the volumes of the segments are in a given ratio to one another. Diocles’ solution to the problem, as given in the fragment preserved by Eutocius, was an ingenious geometrical construction that satisfied, by means of the intersection of an ellipse and a hyperbola, the three simultaneous relations which hold in Archimedes’ proposition.
Diocles’ work On Burning Mirrors, judging from the time at which he lived and the work of his predecessors, must have been of considerable scope. It can be assumed that it discussed concave mirrors in the forms of a sphere, a paraboloid, and a surface described by the revolution of an ellipse about its major axis. Apollonius of Perga, a mathematician who was born about 262 b.c., had earlier written a book on burning mirrors, but Arabic tradition associated Diocles with the discovery of the parabolic burning mirror. The Greek Fragmentum mathematicum Bobiense contains a fragment of a treatise on the parabolic burning mirror, and some authorities have attributed this work to Diocles. Others consider this attribution very doubtful. William of Moerbeke translated into Latin the fragments of Diocles on mean proportionals and the division of the sphere as a part of his general translation from the Greek of the works of Archimedes and Eutocius’ commentaries on them.
The fragments of Diocles’ work can be found in Archimedis Opera omnia cum commentariis Eutocii iterum, J. L. Heiberg, ed., III (Leipzig, 1915), 66–70, 160–176.
On Diocles or his work, see Moritz Cantor, Vorlesungen über Geschichte der Mathematik, I (Stuttgart, 1907, repr. New York, 1965), 350, 354–355; Thomas Heath, A History of Greek Mathematics, 2 vols. (Oxford, 1960), I, 264–266;II, 47–48, 200–203; George Sarton, Introduction to the History of Science, I (Baltimore, 1927), 183; Moritz Steinschneider, Die europäischen Uebersetzungen aus dem arabischen bis Mitte des 17. Jahrhunderts (Graz, 1965), p.17; and E. Wiedemann, “Ibn al Haitams Schrift über parabolische Hohlspiegel,” in Bibliotheca mathematica, 3rd ser., 10 (1909–1910), 202.
Karl H. Dannenfeildt
c. 240-c. 180 b.c.
The details of his life are virtually unknown, and Diocles is remembered almost entirely for a fragmentary manuscript entitled On Burning Mirrors. In it he discussed not only the physical problem referred to in the title, but such subjects as cutting sphere with a plane, as well as the famous Delian problem of doubling the cube.
On Burning Mirrors may actually have been a collection of three separate short works, combined under a single title that does not reflect the whole. In any case, the book consisted of 16 geometric propositions, most of which involved conics.
In the first proposition, Diocles put forth the focal property of the parabola, and in propositions 2 and 3 presented the properties of spherical mirrors. The next two propositions show the focus directrix parabola construction, which as with the statement on the focal property of that shape are innovations credited to Diocles. In propositions 7 and 8, Diocles examined a problem first put forth by Archimedes (c. 287-212 b.c.), on how to cut a sphere with a plane.
Propositions 10 through 12 offered Diocles's solution to the Delian problem. The latter had long perplexed Greek mathematicians who, using only a straightedge and compass, attempted to double the volume of a cube. Today this problem could be expressed in simple terms: find length x for a cube such that x3 = 2a3, where a is the length of the known cube. The Greeks, however, possessed no such sophisticated algebraic notation, and attempted to work the problem geometrically. Diocles's solution to the problem involved a special curve called a cissoid, which he used to find two mean proportionals. The use of the cissoid, however, involved more than a straightedge and compass, and in fact by the nineteenth century a.d. mathematicians recognized that it was impossible to solve the problem using only those tools.
In a 1976 article for a journal on the history of mathematics and the sciences, G. J. Toomer made a case for Diocles, and not his contemporary Apollonius of Perga (c. 262-c. 190 b.c.), as the originator of such familiar terms as hyperbola, parabola, and ellipse. In any case, On Burning Mirrors discussed a number of interesting subjects, not least of which was the one contained in the title. In this regard, Diocles put forth the problem of finding a mirror such that the reflected light forms certain curves, questions that would have implications for the creation of sundials.
fl. second century b.c.
Greek mathematician who, according to Arabic tradition, discovered the parabolic burning mirror. Diocles discusses the theory of spherical and parabolic burning mirrors in his only extant work, On Burning Mirrors. However, this treatise is primarily a collection of theorems in higher geometry. Among those mentioned are two solutions to the problem of doubling of the cube: (1) using the intersection of two parabolas, and (2) using a spherical curve known as the cissoid.