LEUCIPPUS AND DEMOCRITUS

Leucippus and Democritus were the earliest Greek atomists. The originator of the atomic theory, Leucippus (fifth century BCE), must be considered a speculative thinker of the first order, but to Democritus (c. 460–c. 370 BCE) must go the credit for working out the detailed application of the theory and supporting it with a subtle epistemology. Moreover, the range of Democritus's researches surpassed that of any earlier philosopher, and he appears to have been an original and, for his day, advanced ethical thinker.

We have very little biographical data for Leucippus. Epicurus is even reported to have said that there was no philosopher Leucippus, but the evidence of Aristotle decisively refutes this opinion (if, indeed, Epicurus did not merely intend to deny Leucippus's philosophical importance). Leucippus was probably born at Miletus; reports associating him with Elea or Abdera should be taken as reflecting views concerning his philosophical affiliations rather than as reliable evidence for his birthplace. He was presumably older than Democritus. His book On Mind may have been directed partly against Anaxagoras, and according to Theophrastus, Diogenes of Apollonia derived some of his theories from Leucippus. All this suggests that Leucippus was a slightly younger contemporary of Anaxagoras and that his main philosophical activity fell some time within the broad limits of 450–420 BCE.

Democritus was born at Abdera. He described himself in the Little World-System as a young man in the old age of Anaxagoras; Diogenes Laërtius says that he was forty years younger than Anaxagoras. On this evidence the date given for his birth by Apollodorus (in the 80th Olympiad, 460–456 BCE) is generally preferred to that suggested by Thrasylus (the third year of the 77th Olympiad, 470–469 BCE). He is variously reported to have lived between 90 and 109 years. To judge from the number of his writings, his literary activity extended over a considerable period, but we have no means of assigning different works to different times in his life. His statement that he wrote the Little World-System 730 years after the fall of Troy (Diogenes Laërtius, Lives IX, 41) is of little value since we cannot tell which of several possible chronologies for the Trojan War Democritus accepted.

Many stories, most of them apocryphal, relating to Democritus's life and character circulated in antiquity. There are the accounts of his saving the Abderites from a plague, of his dying by voluntarily abstaining from food, and of his reputation as the "Laughing Philosopher." The tradition that he traveled extensively is, however, more plausible and better grounded. The authenticity of the fragment (299) in which he claimed to be the most widely traveled of his contemporaries is disputed, and the genuineness of the five books dealing with foreign travel mentioned by Diogenes Laërtius (for example, A Voyage round the Ocean ) has also been doubted. But evidence concerning his travels goes back to Theophrastus (see Aelian, Varia Historia IV, 20), and the reports that he visited such places as Egypt, Chaldea, and the Red Sea (see Diogenes Laërtius, Lives IX, 35) may well have a sound basis in fact.

All that has been preserved of the original writings of Leucippus and Democritus is a poor selection of isolated quotations, most of which derive from the ethical works of Democritus. For the atomic theory itself we rely on reports in Aristotle, Theophrastus, and later doxographers, who were often unsympathetic to the views of the atomists. In most of the principal texts referring to Leucippus, his doctrines are not clearly distinguished from those of Democritus, and the precise contribution of each philosopher is in question. Aristotle, however, undoubtedly treated Leucippus as the founder of atomism (De Generatione et Corruptione 325a23ff.), and we may reasonably attribute both the principles of the physical theory and a fairly complex cosmogony to him. Democritus evidently elaborated the atomic theory and was responsible for the detailed account of sensible qualities, besides going far beyond Leucippus both in the range of his scientific inquiries and in his interest in moral philosophy.

Writings

Only two works are ascribed to Leucippus, On Mind, from which our sole surviving quotation comes, and the Great World-System, which may be attributed to Leucippus on the authority of Theophrastus (Diogenes Laërtius, Lives IX, 46), although Thrasylus later assigned it to Democritus.

Democritus, on the other hand, wrote some sixty-odd works, the titles of which provide valuable evidence of the scope of his interests. The main works were cataloged by Thrasylus into thirteen tetralogies. Two tetralogies are devoted to ethics and four to physics (including Little World-System, On the Planets, On Nature, On the Nature of Man, On the Senses, and On Colors ). These were followed by nine works not arranged in tetralogies—for example, Causes of Celestial Phenomena, Causes concerning Seeds, Plants and Fruits, and three books of Causes concerning Animals. Three tetralogies are classified as mathematics, two deal with music and literature, and two consist of technical works, including treatises on medicine, agriculture, painting, and warfare. Nine other miscellaneous works, mostly concerning travel, are also mentioned by Diogenes Laërtius but are less certainly authentic as they were not included in Thrasylus's catalog.

Democritus's style is described by Cicero as elegant (De Oratore I, 11, 49) and lucid (De Divinatione II, 64, 133), and an anecdote recorded by Diogenes Laërtius (Lives IX, 40) implies that his works already had wide circulation by the time of Plato.

Tthe Atomic Theory

The basic postulate of Greek atomism in its original form was that atoms and the void alone are real. The differences between physical objects, including both qualitative differences and what we think of as differences in substance, were all explained in terms of modifications in the shape, arrangement, and position of the atoms. Aristotle illustrates these three modes of difference with the examples A and N, AN and NA, and ⌶ and H.

This theory was already interpreted by Aristotle as an answer to the Eleatic denial of change and movement. Other post-Parmenidean philosophers had countered this denial in different ways, but both Empedocles and Anaxagoras had assumed a variety of elemental substances, on the one hand, the four "roots," on the other, an original mixture containing every kind of natural substance. In postulating a single elemental substance, Leucippus remained closer to Parmenides' own conception. In common with Parmenides' One Being the individual atoms are ungenerated, indestructible, unalterable, homogeneous, solid, and indivisible. Leucippus may be said to have postulated an infinite plurality of Eleatic ones, and he may even have been directly influenced by Melissus's argument (Fr. 8) that "if there were a plurality, they would have to be as the One is." Leucippus also agreed with the Eleatics that without void movement is impossible. Yet whereas the Eleatics denied the existence of the void, or "what is not," Leucippus maintained that not only "what is" (the atoms), but also "what is not" (the void), must be considered real. Leucippus thereby reinstated both plurality and change; the void is that which separates the atoms and that through which they move.

The atoms are infinite in number, dispersed through an infinite void. Their shapes are infinitely various, there being no reason that any atom should be of one shape rather than another. Democritus, at least, also allowed differences in the sizes of the atoms, but whether he thought any atom large enough to be visible seems doubtful. Late sources that report that atoms are unlimited in size as well as number (Diogenes Laërtius, Lives IX, 44) or which suggest the possibility of an atom the size of the world (Aëtius, Placita I, 12, 6) are difficult to credit in view of the testimony of Aristotle, who apparently believed that for both Leucippus and Democritus the atoms are all so small that they are invisible.

The atoms are in continuous motion. Aristotle, among others, objected that the atomists did not explain the origin of movement or say what kind of movement is natural to the atoms. However, they evidently assumed that the motion of the atoms is eternal, just as the atoms themselves are, and they perhaps drew no clear distinction between original and derived motion. Although Epicurus was later to suggest that atoms naturally fall vertically, the earlier atomists probably did not consider movement in any particular direction prior to movement in any other. Weight for them, it seems, was not a primary property of the atoms nor a cause of their interactions, although in a developed cosmos the atoms have "weight" corresponding to their size (and the weight of compound bodies varies according to the proportion of atoms and void they contain).

The movements of the atoms give rise to constant collisions whose effects are twofold. Sometimes, the atoms rebound from one another; alternatively, when the colliding atoms are hooked or barbed or their shapes otherwise correspond, they cohere and thus form compound bodies. Change of all sorts is accordingly interpreted in terms of the combining and separating of atoms, which themselves remain unaltered in substance. The compound bodies thus formed possess various sensible qualities—color, taste, temperature, and so on—and Democritus undertook a detailed exposition relating these qualities to specific atomic configurations.

cosmogony

Evidence concerning Leucippus's cosmogony comes mainly from Diogenes Laërtius (Lives IX, 31ff.). The process begins when a large group of atoms becomes isolated in a great void. There they conglomerate and form a whirl or vortex in which atoms of similar shape and size come together. In this vortex the finer atoms are squeezed out into the outer void, but the remainder tend toward the center, where they form a spherical mass. More atoms are drawn into this mass on contact with the whirl, and some of these are ignited by the speed of the revolution, thus forming the heavenly bodies. Earth is formed by atoms that cohere in the center of the mass. The cosmogonical process is not unique. The atomists argued that since atoms and the void are infinite, there are innumerable worlds. These worlds are not all alike, however; Democritus held that some worlds have no sun or moon and that some lack moisture and all forms of life (Hippolytus, Refutatio I, 13, 2f.).

Several features of this account are obscure, and two apparently conflicting criticisms were leveled against it in antiquity—first, that although the atomists asserted that the cosmogonical process came about by necessity, they did not explain what this necessity was (Diogenes Laërtius, Lives IX, 33); second, that they maintained that it occurred spontaneously (Aristotle clearly has the atomists in mind when he considered this view, Physics 196a24ff.).

But Aristotle's judgment should be taken as referring primarily to the atomists' exclusion of final causes; in Aristotelian terms the atomists held that the world arose spontaneously because they denied that it was intelligently planned. Leucippus explicitly stated that "nothing happens at random, but everything for a reason and by necessity" (Fr. 2), and throughout their cosmology the atomists not only excluded purpose or design but also assumed that every event is the product of a definite, theoretically determinable cause. Thus, they doubtless conceived the vortex to arise from certain mechanical interactions between the colliding atoms, although it is unlikely that they attempted to say precisely how this came about. Democritus illustrated his doctrine that like things tend to come together with examples drawn from both the inanimate and the animate sphere (Fr. 164). And like many of the pre-Socratics, the atomists constructed their cosmogony in part on an embryological model. The outer envelope of the world was likened to a membrane, and in both Leucippus's cosmogony and Democritus's embryology the process of differentiation apparently takes place from the outside (see Aristotle, De Generatione Animalium 740a13ff.).

astronomy and biology

Leucippus's astronomical theories are surprisingly retrograde. He accepted the old Ionian picture of a flat earth, tilted toward the south, and he believed that the sun is the most distant of the heavenly bodies. Democritus's theories were generally less crude, and he attempted rational explanations of a wide variety of obscure phenomena. He accepted Leucippus's account of Earth with only minor modifications (Aëtius, Placita III, 10, 5) but corrected his notion of the relative positions of the heavenly bodies, observing, for example, that the planets are not equidistant from Earth and placing Venus between the sun and moon. Among other topics on which some of Democritus's theories are recorded are the behavior of the magnet, the nourishment of the embryo, and the relative longevity of different types of plants. Of his biological doctrines the notion that the seed is drawn from the whole of the body (the pangenesis theory) was particularly influential (Aëtius, Placita V, 3, 6).

soul, knowledge, and sensation

Our evidence concerning the atomists' psychological and epistemological doctrines derives very largely from Democritus, although his theory of the soul was probably developed from ideas outlined by Leucippus. This theory was a materialist one in line with the principles of atomism. Democritus conceived of the soul as consisting of spherical atoms, this being the shape best adapted to penetrate and move things. Fire, too, is composed of spherical atoms, and he evidently subscribed to the common Greek belief in the connection between life and heat, now interpreted in terms of the similarity in the shapes of soul atoms and fire atoms. The soul atoms tend to be extruded from the body by the pressure of the surrounding air, but this process is counteracted by other soul atoms that enter the body with the air we breathe; life depends on this continuous replenishment.

Our main source for Democritus's theory of knowledge is Sextus Empiricus. Several of the fragments that he quotes appear to express an extreme skepticism—for instance, "We know nothing truly about anything" (Fr. 7). However, Fragment 11 shows that Democritus was no outright skeptic. There he distinguished between two modes of cognition; the senses provide what is called a "bastard" knowledge but contrasted with this is a "legitimate" knowledge, which operates on objects too fine for the senses to perceive. Clearly, "legitimate" knowledge relates to atoms and the void, which alone are real; the objects of sensation, on the other hand, exist "by convention" (Fr. 9). The doctrine enunciated in the fragments is that sense perception is not trustworthy, and Aristotle's repeated statement that the atomists found truth in appearance (De Generatione et Corruptione 315b9ff.) should be understood as an interpretative comment based on Aristotle's own conception of the distinction between sensibles and intelligibles. Yet although we must rely on reasoning to attain knowledge, Democritus acknowledged that the mind derives its data from the senses (Fr. 125). Not a pure intellectualist like Parmenides, a crude sensationalist like Protagoras, nor a complete skeptic as Gorgias made himself out to be, Democritus advocated critical reflection on the evidence of the senses as our best means of approaching the truth; yet since thought itself, like sensation, involves physical interactions between atoms, it, too, is subject to distortion, and even "legitimate" knowledge is at best, it seems, only opinion (Fr. 7).

Democritus's detailed accounts of the five senses were reported and criticized at length by Theophrastus (De Sensibus 49–82). According to Alexander (In Librum de Sensu 24, 14ff.), Leucippus already held that physical objects constantly emit images that effect vision on entering the eye. Democritus modified and complicated this doctrine by suggesting that images from both the object and the eye itself meet and imprint the air in front of the eye. Each of the other senses, too, is produced by contact between the organ and images deriving from the object, and thought was analogously explained as the contact between soul atoms and images coming from outside the body. But not content merely to assert in general terms that secondary qualities are due to differences in the shapes and sizes of the atoms, Democritus also proposed a detailed account relating specific tastes, colors, smells, and so on to specific shapes. Thus, an acid taste is composed of angular, small, thin atoms and a sweet taste of round, moderate-sized ones. Democritus's primary colors—black, white, red, and greenish yellow—were similarly associated with certain shapes and arrangements of atoms, and other colors were derived from combinations of these four. For all its crudities Democritus's theory may claim to be the first fully elaborated account of the physical basis of sensation.

mathematics

Democritus's interest in mathematics is apparent from the titles of fives works dealing with mathematical subjects, and we are told, for example, that he discovered the relation between the volumes of a pyramid and a prism with the same base and equal height. We have, however, little evidence on the part of his mathematical work that related directly to the atomic theory. The atoms are definitely conceived of as physically indivisible (on the grounds that they are solid and contain no void), but it is not clear whether they are absolute minima in the sense of being mathematically indivisible. Epicurus later distinguished between atomic bodies (which are physically indivisible but logically divisible) and the "minima in the atom." But Aristotle appears to have assumed that Leucippus and Democritus themselves drew no distinction between the limits of physical and mathematical divisibility (De Generatione et Corruptione 315b28ff.), and he considered that their atomic theory necessarily conflicted with the mathematical sciences (De Caelo 303a20ff.). Unless Aristotle has completely misrepresented the atomists, it would appear that Democritus was unaware of any inconsistency in holding both (1) that the atoms have different shapes and sizes and (2) that they are mathematically as well as physically indivisible. But it must be repeated that the evidence on which to convict or absolve Democritus of this gross confusion is scanty.

ethics

Although serious doubts have been raised concerning the transmission of the ethical fragments of Democritus, most scholars now consider that the majority of those accepted by Hermann Diels and Walther Kranz may be used as a basis from which to reconstruct his ethics. There remain, however, wide disagreements on the nature and value of his moral teaching. Alongside the fragments that convey traditional sentiments (for example, on the dangers of fame and wealth if not accompanied by intelligence) we find others that expound notions far in advance of the popular morality of the day, as, for instance, the doctrine that it is one's own consciousness of right and wrong, not fear of the law or public opinion, that should prevent one from doing anything shameful (Frs. 181, 264). And sayings such as Fragment 45 ("The wrongdoer is more unfortunate than he who is wronged") express views more commonly associated with Socrates than with Democritus.

The ethical ideal is termed "well-being" or "cheerfulness," which is to be gained through uprightness and a harmonious life. Although Democritus clearly implied that life without pleasure is not worth living and even said that pleasure is the mark of what is expedient (Fr. 188), it is the higher pleasures of the soul that we should cultivate, not those of the body. Sensual pleasures are condemned as short-lived. He repeatedly stressed that we should moderate our desires and ambitions, become self-sufficient, and be content, in the main, with simple pleasures. Yet Democritus was no quietist. Rather, he recognized that worthwhile objects are to be achieved only through effort (Frs. 157, 182).

One of the salient features of Democritus's ethics is his rejection of supernatural sanctions of behavior. In part, he seems to have rationalized belief in the gods as a mistaken inference from terrifying natural phenomena (Sextus, Adversus Mathematicos IX, 24), and yet he did not dismiss notions of the gods entirely, for he appears to have related certain such ideas to images, some beneficent, some harmful, that visit humans (Fr. 166). Religious sanctions are, however, rigorously excluded from his ethics. He refuted those who concocted fictions concerning the afterlife (Fr. 297), and he spoke with apparent irony of those who prayed to Zeus as "king of all" (Fr. 30). Equally, he castigated those who invented chance as an excuse for their own thoughtlessness or who failed to recognize that their misfortunes stemmed from their own incontinence (Frs. 119, 234). Throughout his ethics he may be said to have set high standards of personal integrity and social responsibility.

The question of the relation between Democritus's ethics and his physics has been much debated. In some respects, such as in the idea that excesses "cause great movements in the soul"—that is, presumably, in the soul atoms (Fr. 191)—his ethics reflect a psychology that is based on his physical theories. Whether we should expect other aspects of the atomic doctrine to be in evidence in the ethical fragments seems very doubtful. Democritus clearly did not feel (nor need he have felt) that the notion of necessity in his physics (the belief that every event has a definite cause to be sought in the interactions of the atoms) conflicted with his doctrine of moral responsibility in the sphere of human behavior. His denial of supernatural sanctions in his ethics parallels his rejection of teleology in his cosmology. And his ethics have in common with his epistemological theory that he argued against an unreflecting acceptance of the evidence of the senses concerning what is pleasant just as much as concerning the nature of reality as a whole.

sociology and politics

The only indication we have of Democritus's political leanings is the idealistic but otherwise rather inconclusive Fragment 251: "Poverty under democracy is as much to be preferred to so-called happiness under tyrants as freedom to slavery." It has, however, been conjectured that the account of the origin of civilization preserved in Diodorus (Bibliotheca Historica I, 8) owes much to Democritus. According to this, primitive peoples originally gathered in groups for the sake of mutual protection from wild animals, and subsequently language and the arts were also invented under the spur of human needs. It is very uncertain how far this reproduces Democritus's ideas, but there is some evidence in the fragments that he maintained a naturalistic theory of civilization and progress and excluded teleological explanations here, as he did elsewhere in his philosophy. Fragment 144 may be taken to suggest that he believed that the earliest arts (although not some of the later ones) were products of necessity, and in Fragment 154 he argued that humans learned many of their skills by copying the behavior of animals.

The theory founded by Leucippus and developed by Democritus was the most coherent and economical physical system of its day, and the history of its influence can be traced from the fourth century BCE to modern times. Although Plato mentioned neither Leucippus nor Democritus, the Timaeus is markedly indebted to their thought. Even Aristotle, who rejected atomism outright, conceded that of all his predecessors Democritus was the most notable physicist. Later, the Epicureans championed atomism against the continuum theory of the Stoics. Leucippus's theory, in origin primarily an answer to the Eleatic arguments against change, was the first clear formulation of the doctrine that matter exists in the form of discrete particles, and as such it may legitimately be considered the prototype of modern theories of the discontinuous structure of matter, even though the nature of such theories, the problems they are intended to resolve, and the methods used to establish them all differ fundamentally from those of ancient atomism.

See also Alexander of Aphrodisias; Anaxagoras of Clazomenae; Aristotle; Atomism; Cosmology; Diodorus Cronus; Diogenes Laertius; Diogenes of Apollonia; Empedocles; Epicureanism and the Epicurean School; Epicurus; Gorgias of Leontini; Parmenides of Elea; Plato; Pre-Socratic Philosophy; Protagoras of Abdera; Quantum Mechanics; Sextus Empiricus; Stoicism; Theophrastus.

Bibliography

The extant fragments of Leucippus and Democritus and the principal reports and commentaries in ancient authors are collected in Hermann Diels's Die Fragmente der Vorsokratiker, Vol. II, 6th ed., with additions by Walther Kranz, ed. (Berlin, 1952). There is an English translation of the fragments in Kathleen Freeman, Ancilla to the Pre-Socratic Philosophers (Cambridge, MA: Harvard University Press, 1948).

More recent collections of texts are S. Luria, Democritea (Leningrad, 1970; original texts of fragments and testimonia with Russian translation and commentary) and C. C. W. Taylor, The Atomists: Leucippus and Democritus. Fragments: A Text and Translation with a Commentary (Toronto: University of Toronto Press, 1999; fragments in Greek with facing English translation, testimonia in translation, commentary).

A classic monograph on Leucippus and Democritus is Cyril Bailey, The Greek Atomists and Epicurus (Oxford, 1928). Lucid, brief expositions of their thought are found in G. S. Kirk, J. E. Raven and M. Schofield, The Presocratic Philosophers, 2nd ed. (Cambridge, U.K., 1983) and in R. D.McKirahan Jr., Philosophy before Socrates (Indianapolis and Cambridge, 1994). W. K. C. Guthrie, A History of Greek Philosophy, vol. 2 (Cambridge, U.K.: Cambridge University Press, 1965), contains a full discussion and extensive bibliography. Other significant general studies are E. Zeller, Die Philosophie der Griechen, 6th ed., revised and enlarged by W. Nestle (Leipzig, 1920), Part I, Sec. 2, pp. 1038–1194; Wilhelm Schmidt in Wilhelm Schmidt and Otto Stählin, eds., Geschichte der griechischen Literatur (Munich, 1948), Part I, Sec. 5, pp. 224–349; V. E. Alfieri, Gli atomisti (Bari, 1936) and Atomos Idea, 2nd ed. (Florence, 1979); J. Barnes, The Presocratic Philosophers, 2nd ed. (London and Boston: Routledge and K. Paul, 1982), Chapters 17, 19 (b ), 20, 21 (c ), 23 (d ), 24 (e ); P.-M. Morel, Démocrite et la recherche des causes (Paris, 1996); and J. Salem, Démocrite: Grains de poussière dans un rayon de soleil (Paris: CNRS-Éditions, 1996).

A full discussion of the chronological and biographical data on Democritus is contained in D. O'Brien, "Démocrite d'Abdère" in R. Goulet, ed., Dictionnaire des Philosophes Antiques, vol. 2, pp. 649–715 (Paris: CNRS-Éditions, 1994).

physics and cosmology

Barnes, J. "Reason and Necessity in Leucippus." In Proceedings of the First International Conference on Democritus, edited by L. Benakis. Vol. 1, pp. 141–158. Xanthi, 1984.

Bury, R. G. "The Origin of Atomism." Classical Review 30 (1916): 1–4.

Dyroff, Adolf. Demokritstudien. Leipzig: Dieterich, 1899.

Fritz, Kurt von. Philosophie und sprachlicher Ausdruck bei Demokrit, Plato und Aristoteles. New York, n.d. [1939].

Furley, D. J. Cosmic Problems. Cambridge, U.K., and New York: Cambridge University Press, 1989. Chapters 7–9.

Furley, D. J. The Greek Cosmologists. Cambridge, U.K., and New York: Cambridge University Press, 1987. Chapters 9–13.

Furley, D. J. "Weight and Motion in Democritus' Theory." Oxford Studies in Ancient Philosophy 1 (1983): 193–209.

Hammer-Jensen, Ingeborg. "Demokrit und Platon." Archiv für Geschichte der Philosophie, n.s. 16 (1910): 92–105, 211–229.

Kersehensteiner, Jula. "Zu Leukippos A 1." Hermes 87 (1959): 441–448.

Löbl, R. Demokrits Atomphysik. Darmstadt: Wissenschaftliche Buchgesellschaft, 1987.

Löwenheim, Louis. Die Wissenschaft Demokrits. Berlin: Simion, 1914.

Makin, S. Indifference Argument. Oxford and Cambridge, MA: Blackwell, 1993.

McDiarmid, J. B. "Theophrastus De Sensibus 61–62: Democritus' Theory of Weight." Classical Philology 55 (1960): 28–30.

Mugler, Charles. "Sur Quelques Particularités de l'atomisme ancien." Revue de philologie, 3rd series 27 (1953): 141–174.

O'Brien, D. Theories of Weight in the Ancient World. Vol. 1: Democritus: Weight and Size. Paris: les Belles Letters; and Leiden: Brill, 1981.

Sambursky, Samuel. "Atomism versus Continuum Theory in Ancient Greece." Scientia 96 (1961): 376–381.

epistemology

McDiarmid, J. B. "Theophrastus De Sensibus 66: Democritus' Explanation of Salinity." American Journal of Philology 80 (1959): 56–66.

McKim, R. "Democritus against Scepticism: All Sense-Impressions Are True." In Benakis op. cit., vol. 1, pp. 281–290.

Weiss, Helene. "Democritus' Theory of Cognition." Classical Quarterly 32 (1938): 47–56.

mathematics

Bodnár, I. M. "Atomic Independence and Indivisibility." Oxford Studies in Ancient Philosophy 16 (1998): 35–61.

Furley, D. J. Two Studies in the Greek Atomists. Princeton, NJ, 1977. Study 1, "Indivisible Magnitudes."

Makin, S. "The Indivisibility of the Atom." Archiv für Geschichte der Philosophie 71 (1989): 125–149.

Luria, Salomo. "Die Infinitesimaltheorie der antiken Atomisten." Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Part 2 (Studies) 2, Sec. 2 (Berlin, 1932–1933): 106–185.

Mau, Jürgen. Zum Problem des Infinitesimalen bei den antiken Atomisten. Berlin: Akademie-Verlag, 1954.

Philippson, Robert. "Democritea." Hermes 64 (1929): 175–183.

Vlastos, Gregory. "Minimal Parts in Epicurean Atomism." Isis 56 (1965): 121–147.

Zubov, V. P. "K Voprosu o Matematicheskom Atomisme Democrita." Vestnik Drevnei Istorii 4 (1951): 204–208.

ethics

Kahn, C. H. "Democritus and the Origins of Moral Psychology." American Journal of Philology 106 (1985): 1–31.

Langerbeck, Hermann. ΔΟΞΙΣ ΕΠΙΡΥΣΜΙΗ. Neue Philologische Untersuchungen, Vol. X. Berlin, 1935.

Laue, Heinrich. De Democriti Fragmentis Ethicis. Unpublished dissertation, University of Göttingen, 1921.

Laue, Heinrich. "Die Ethik des Demokritos." Sokrates, n.s. 11 (1923–1924): 23–28, 49–62.

Luria, Salomo. Zur Frage der materialistischen Begründung der Ethik bei Demokrit. Berlin: Akademie-Verlag, 1964.

McGibbon, Donal. "Pleasure as the 'Criterion' in Democritus." Phronesis 5 (1960): 75–77.

McGibbon, Donal. "The Religious Thought of Democritus." Hermes 93 (1965): 385–397.

Natorp, Paul. Die Ethika des Demokritos. Marburg: Elwert, 1893.

Philippson, Robert. "Demokrits Sittensprüche." Hermes 59 (1924): 369–419.

Stewart, Zeph. "Democritus and the Cynics." Harvard Studies in Classical Philology 63 (1958): 179–191.

Taylor, C. C. W. "Pleasure, Knowledge and Sensation in Democritus." Phronesis 12 (1967): 6–27.

Vlastos, Gregory. "Ethics and Physics in Democritus." Philosophical Review 54 (1945): 578–592; 55 (1946): 53–64.

Warren, J. Epicurus and Democritean Ethics. Cambridge, U.K., and New York: Cambridge University Press, 2002.

sociology and politics

Aalders, G. J. D. "The Political Faith of Democritus." Mnemosyne, 4th series, 3 (1950): 302–313.

Procopé, J. F. "Democritus on Politics and the Care of the Soul." Classical Quarterly, n.s. 39 (1989): 307–331 and 40 (1990): 21–45.

Vlastos, Gregory. "On the Pre-history in Diodorus." American Journal of Philology 67 (1946): 51–59.

G. E. R. Lloyd (1967)

Bibliography updated by C. C. W. Taylor (2005)