Hippias of Elis

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Hippias of Elis

(b. Elis, Greece; fl. 400 b.c.)

philosophy, mathematics.

Elis was a small state in the northwest of the Peloponnesus whose inhabitants had charge of the Olympic festival. Hippias’ father was named Diopeithes,1 but his ancestry is otherwise unknowns.2 In the Platonic dialogue Hippias Major3 he is made to say that he was yough when Protagoras was old, and in the Protagoras Plato represents him as present at a philosophic discussion with that eminent Sophist about 432 b.c.4 The date o the birth of Protagoras is uncertain but is usually placed from 488 to 485. In Plato’s Apology5 set in 399, Hippias is mentioned as a teacher o youth along with Gorgias and other famous Sophists, and may then be presumed to have been at the hight of his fame. He was therefore a contemporary of Plato. His wife Platane bore him three sons; and when she was left a widow, the orator Isocrates in extreme old age took her in marriage and adopted her youngest son, Aphareus,6 who achieved some fame as a tragic poet. Isocrates died in 338 These facts would suggest that Hippias had a long life; and the belief is made certain if, with Mario Untersteiner, the preface to the Characters of Theophrastus is at tributed to Hippians, for he is there made to say that he has reached ninety-nine years of age.7 The old notion that he was killed while weaving plots against his native land must be abandoned now that the correct name in the text of Tertullian has been established as Icthyas.8

Hippias was taught by an otherwise unknown Aegesidamus, and he emerged as a polymath who wrote and lectured over a wide range of disciplines: rhetoric, politics, poetry, music, painting, sculpture, and astronomy, as well as the philosophy and mathematics on which his fame chiefly rests.9 The secret of his wide knowledge appears to have neen an exceptional memory. According to Philostratus, he hand a system of mnemonics ushc that if he once heard a sting of fifty names, he could repeat them in correct order.10 Most of what is kown about Hippias’ life and character comes from a dialogue between Socrates ad Hippias recorded by Xenophon11 and from the two Platonic dialogues that bear hsi name, the Hippias Major and Hippias Minor Their authenticity has been disputed, but even if not genuine they still correctly reflect Plato’s attitude; in these dialogues Hippias is represented as a naive and humorless boaster who cannot stand up to the remorseless logic of Socrates. xenophon’s portrait is not so ruthless, but there also Hippias is reduced to silence by Socrates’ arguments. Hippias was a second-generation Sophist, and Plato had no love or the Sophists as a class. Apart from more fundamental diferences, Plato’s aristocratic soul was offended by their professional teaching; and Hippias was especially successful in negotiating lecture fees, particularly in Sicily, Although he recevived none in Sparta, where the law forbade a oreign education.12

The picture in the Platonic dialogues is no doubt a caricature; but in the light of Plato’s more sympathetic treatment of other individual Sophists, there must have been enough truth in the caricature for it to be recognizable as a portrait.13 Hippias is made to accept lattery even when laid on with a trowel, acknowledging that he had never found any man to be his superior in anything.14 At the Olympic festivel it was his custom to ofer to discourse on any subject proposed to him out of those which he had prepared and to answer any questions.15 He once appeared at the estival with everything that he wore made by himself, not merely his clothes but also a ring, an oil flask, and an oil scaper—which bears out the statement in the Suda Lexicon that he made self suficiency the end of life—and he brought with him poems, epics, tragedies, dithyrambs, and all kinds of prose works.16

Hippias could not have been such a figure of fun as the Platonic dialogues make him out to be, for he was frequently asked to represent his native state on missions to other states, notably Sparta.17 He was widely traveled—two visits to Athens are recoded— and in Sicily his influence was lasting if, as Untersteiner believes, he was the mentor of Dionysius the Younger and inspired the work known as the Dissoi logoi.18

The Suda Lexicon tersely records that Hippias “wrote many things.” None of his voluminous works has survived, but some of the titles and hints of the contents are known. His Synagoge, known through Athenaeus, has usually been thought, on the strength of a passage in Clement of Alexandria which seems to refer to it, to have been merely a miscellany in which he put together sayings of poets and prose writers, both Greek and foreign.19 But Bruno Snell has advanced the theory that through this work Aristotle derived his knowledge of Thales; that the views of Thales about the All being water and about the souls of inanimate objects are thereby shown to be derived from earlier mythological speculations; and that the Synagoge is to be looked upon as the earliest work in both the history of Greek philosophy and the history of Greek literature.20 If this is so, it encourages the thought that Hippias’ Nomenclature of Tribes21 may not have been a mere catalog but an expression of his belief in the fundamental unity of all mankind. His Register of Olympic Victors was no doubt a piece of Elian patriotism. It was the first such list to be drawn up; and Plutarch notes that, since it came so late after the enents recorded, too much authority should not be attached to it.22 Among his epideictic or set speeches, the one known as The Trojan may have been in dialogue form; in it Nestor suggests to Neoptolemus many lawful and beautiful pursuits by which he might win fame.23 Hippias wrote an elegiac inscription for the statues made by Calon at Olympia in memory of a boys’ choir from Messina drowned in crossing to Rhegium.24 More important in its ultimate significance than any of these compositions is a work on the properties of the geometrical curve he discovered, since known as the quadratrix.

Hippias’ teaching has to be reconstructed from the scattered references to him in Greek and Latin authors. Untersteiner has argued that Hippias was the author not only of the preface to Theophrastus’ Characters but also of a spurious chapter in Thucydides (III, 84) dealing with events in Corcyra and of the epideictic speech known as the Anonymus Iamblichi; that the Dissoi logoi, a work drawing on Pythagorean and Sophistic sources, reflects the teaching of Hippias; and that the philosophical digression in Plato’s seventh letter is an attack upon Hippias’ doctrines.25 If this were established, it would enable a clearer picture of Hippias’ philosophy to be drawn; but Untersteiner’s theories are too conjectural for any conclusions to be based on them. It is therefore to the dialogues between Socrates and Hippias as recorded by Xenophon and Plato, and to a passage in Plato’s Protagoras which may well be an imitation of the Sophist’s style, that we must look in the main for Hippias’ teaching.26

The core of it would appear to be a distinction between νόμος and ϕύσις,27 that is, betwen positive law and nature, with a corresponding belief in the existence of unwritten natural laws which are the same for all men in all places and at all times. Reverence for the gods and honor for parents are among such natural laws.28 It was one of Hippias’ fundamental beliefs that like is kin to like by nature, and he extended it to mean that men are neighbors and kinsmen. Positive law is a matter of human agreement and can be altered; it can be a great tyrant doing violence to human nature. It is a pity that Hippias’ teaching has to be seen through the distorting mirrors of Plato and Xenophon, for he would appear to have been a progenitor of the doctrine of natural law, of the social-contract theory of the state, and of the essential unity of all mankind—in fact, no mean thinker.

It is clear from Plato’s raillery that Hippias claimed proficiency in arithmetic, geometry and astronomy,29 and one important discovery is attributed to him: the transcendental curve known as the quadratrix.

The evidence comes from two passages in Proclus which are probably derived from Geminus. The first is “Nicomedes trisected every rectilineal angle by means of the conchoidal curves... . Others have done the same thing by means of the quadratrices of Hippias and Nicomedes, making use of the mixed curves which are called quadratices.”30 The second is “In the same manner other mathematicians are accustomed to treat of curves, setting forth the characteristic property of each type. Thus Apollonius shows what is the characteristic for each of the conic sections, Nicomedes for the conchoids, Hippias for the quadratrices, and Perseus for the spiric curves.”31

Who is this Hippias? The natural assumption is that he is Hippias of Elis, who is mentioned in an earlier passage by Proclus,32 this time in the summary of geometry derived from Eudemus, as having recorded that Mamercus (or perhaps Ameristus), brother of the poet Stesichorus, acquired a reputation for geometry. No other Hippias is mentioned by Proclus; and it is in accordance with his practice, have once referred to a person in full, to omit the patronymic on subsequent mention.33 Hippias of Elis, as shown by the references of Plato and Xenophon, had mathematical qualifications; and among the many bearers of the name Hippias in antiquity there is no other of whom this can be said.34 It is therefore natural to identify the Hippias who is mentioned in connection with quadratrices as Hippias of Elis; and most historians of Greak mathematics, from J. E. Montucla to B. L. van der Waerden, have done so.35

The objections made can easily be discounted.

1. If he made so important a discovery as the quadratrix, it has been argued, Hippias would be recorded in Proclus’ “Eudemian Summary”; but the omission is accounted for by the Platonic prejudice against the Sophists, and the omission of Democritus is even more remarkable.

2. Diogenes Laertius says that Archytas was the first to use an instrument for the description of a curve,36 and the quadratrix requires an instrument for its decription. Yet, on the one hand, an indefinite number of points on the quadratrix can be obtaineed by the ruler and compass and, on the other hand, Diogenes is not a trustworthy guide in this matter, since (a) there is no suggestion of an instrument in Eutocius’ description of the curve found by Archytas to solve the probelm of doubling the cube;37 and (b) Eratosthenes specifically states that Archytas was not able to realize his solution mechanically.38

3. Hippias is not mentioned by Pappus and Iamblichus in their accounts of curves used for squaring the circle;39 but this is explained if, as seems probable, Hippias did not use the curve for that purpose but only for trisecting an angle.

It may therefore be taken that the Hippias who is mentioned by Proclus in connection with the quadratrix is Hippias of Elis; and, if so, he was its discover, since he preceded Nicomedes. But did he use it for squaring the circle? And did he give it the name quadratrix? This is more doubtful. Proclus implies that the curve was used by Hippias for trisecting an angle, saying nothing about squaring the circle; and those Greek authors who write about the squaring of the circle do not mention Hippias. A fundamental and obvious property of the curve is that it can be used to divide an angle in any given ratio, and therefore to trisect it; but to use it for squaring the circle is a more sophisticated matter and might not be obvious to the original discoverer. This can be seen from the way the curve is generated, as described by Pappus.40

Let ABCD be a square and BED a quadrant of a circle with center A. If the radius of the circle moves uniformly from AB to AD and in the same time the line BC moves parallel to its original position from BC to Ac, then at any given time the intersection

of the moving radius and the moving straight line will determine a point F. The path traced by F is the curve. If it is desired to trisect the angle EAD, let H be taken on the perpendicular FK to Ad such that FK =3 HK. Let a straight line be drawn through H parallel to AD, and let it meet the curve at P. Let AP be produced to meet the circle at Q. Then, by the definition of the curve,

and therefore ∠QAD is one-third of ∠EAD. It is obvious that the curve can be used not merely to trisect an angle but also to divide an angle in any given ratio; trisection is specified because this was one of the great problems of Greek mathematics when Hippias flourished.

If a is the length of a side of the square, ρ is any radius vector AF, and ϕ is the angle EAD, the equation of the curve is


The use of the quadratrix to square the circle is a more complicated matter, requiring the position of G to be known and an indirect proof per impossibile. (For this the article on Dinostratus may be consulted.)

The ancient witnesses can therefore be reconciled if Hippias discovered the curve and used it to trisect an angle, but its utility for squaring the circle was perceived only by such later geometers as Dinostratus and Nicomedes. In that case Hippias could not have called his curve the quadratrix, and we do not know what name he gave it. It is no objection that Proclus refers to “the quadratrices of Hippias and Nicomedes,” for we have no hesitation in saying that Menaechms discovered the parabola and hyperbola, although these terms did not come into use until Apollonius; Menaechmus would have called them “section of a right-angled cone” and “section of an obtuse-angled cone.” There is, however, a more serious objection. From the second of the Proclus passages quoted above if could, without straining the sense, be inferred that Hippias wrote a whole treatise on the curve, setting forth its special properties; and in that case the probability increases that he was aware of its use for squaring the circle. Paul Tannery was of this opinion, and T. L. Heath thinks it “not impossible”; but no balance it seems preferable to hold, with C. A. Bretschneider and Moritz Cantor, that the circle-squaring property was discovered, and the name quadratrix given, later than Hippias.41

The citation of Hippias as the authority for Mamercus’ mathematical proficiency has led some to suppose that Hippias wrote a history of geometry.42 If so, it would be the first, antedating Eudemus by perhaps three-quarters of a century. But this is to read too much into the Greek word ίστόρησεν, translated above as “related.” It does not necessarily imply a full-scale treatise, but only that Hippias mentioned the fact in one of his many works.


1.Suda Lexicon, “Iππίας,” Adler ed., pt., pt. 2 (Leipzig, 1937), Iota 543, p. 659.

2. Apuleius, Florida 9, Helm ed., p. 12.1.

3. Plato, Hippias Major, 282 d-e.

4. Plato, Protagoras, 337c6–338b1. The scene is usually assigned to 432 b.c. but—as Ahteaneus, V.218c-d. gulick ed. (Loeb), II (London-New York, 1928), 428, points out—in antiquity Hippias could not have safely stayed in Athens until an annual truce was concluded in the archonship of Isarchus (423), and the chronology of what is presumably a fictitious gathering cannot be pressed.

5. Plato, Apology, 19e1–4.

6. [Plutarch], Lives of the Ten Orators, 838a-839c, Fowler ed. (Loeb); and Moralia 10, pp. 376–385 (the author makes Platane the daughter and not the widow of Hippias); Harpocration, Lexicon, “Αϕαρεύς,” Dindorf ed., I (Oxford, 1853), 68.18; Zosimus, Historia nova V, Mendelssohn ed. (Leipzig, 1887). Isocrates’ marriage followed his liaison—when already an old man—with the courtesan Lagisca; hence “in extreme old age.”

7. Theophrastus, Characters, pref. 2, Diels ed. (Oxford, 1909). See Mario Untersteiner, “II proemio dei ‘Caratteri’ di Teofrasto e un probabile frammento di Ippia,” in Rivista di filologia classica, n.s. 26 (1948), 1–25. In I sofisti, 2nd ed., fasc. 2, p. 115, translated by Kathleen Freeman in The Sophists, p. 274, he says the preface is “definitely a work of Hippias.” But it is incredible that the author should have been still writing—even banalities—at the age of ninety-nine; and the figure must be treated with reserve. Perhaps there is a textual error. The preface is certainly not the work of Theophrastus; but the only reason for attributing it to Hippias is that it is such a work as the boastful Hippias of Plato’s dialogues might have written, which is not a sufficiently strong ground.

8. The printed texts of Tertullian, Apologeticum, 46.16, until 1937 read: “et Hippias, dum civitati insidias dispoint, occiditur.” There was some dispute whether this referred to Hippias, son of Pisistratus; but since Tertullian is cataloging the misdeeds of pagan philosophers, there can be little doubt that the reading, if correct, would refer to Hippias of Elis. But H. Emonds, “Die Oligarchenrevolte zu Megara im Jahre 375 und der Philosoph Icthyas bei Tertullian Apol. 46.16,” in Rheinisches Museum für Philologie, n.s. 86 (1937), 180–191, shows that the reading “et Hippias” has no MS authority and that “Icthyas” (Icthyas of Megara) should be substituted. Emonds has been followed by H. Hoppe (Vienna, 1939) and E. Dekkers (Tournai, 1954) in their subsequent eds.

If the reading “Hippias” had been correct, the event could be referred, as in Untersteiner, to the war waged in 343 by the democrats of Elis, among whom Hippias might be numbered, in alliance with the surviving soldiers of the Phocian adventurer Phalaecus. With this peg gone, the case for giving Hippias an exceptionally long life is weakened, particularly if Platane is regarded as daughter and not wife of Hippias (see note 6) and the evidence for ascribing the Theophrastian preface to Hippias is regarded as unconvincing.

9.Suda Lexicon, “Iππiας,” Otto Apelt, Beiträge zur Geschichte der griechischen Philosophie, pp. 382–384, 391–392, gives no convincing reasons for thinking that Aegesidamus is a mistake for Hippodamus of Miletus.

Xenophon, Memorabilia IV.6, has Socrates apply the word “polymath” to Hippias; and Plato, Hippias Minor, 368b, makes Socrates call him, no doubt sarcastically, “the wisest of men in the greatest number of arts.”

10. Philostratus, Lives of the Sophists 1.11, Kayser ed., II (Leipzig, 1871), 13.27–30. See also Xenophon, Symposium 4.62; Plato, Hippias Major, 285E. According to Cicero, De oratore 2.86–351-354, the first to work out a mnemonic was Simonides, who is mentioned along with Hippias by Aelian, On the Characteristics of Animals VI.10, Scholfield ed. (Loeb), II (London-Cambridge, Mass., 1959), 22.9–13. Ammianus Marcellinus XVI.5.8, Clark ed., I (Berlin, 1910), 76.17–20, notes the belief of some writers that his feats of memory, like those of King Cyrus and Simonides, were due to the use of drugs.

11. Xenophon, Memorabilia IV.4. 19–20.

12.Hippias Major, 282d-e, 283b-284c. In the former passage Hippias boasts that although Protagoras was in Sicily at the time, he made more than 150 minas—at one small place, Inycus, taking in more than 20 minas.

13. See W. K. C. Guthrie, A History of Greek Philosophy, III (Cambridge, 1969), 280.

14. Plato, Hippias Minor 364a; compare Hippias Major 281d.

15. Plato, Hippias Minor 363c.

16.Ibid., 368b-c; Apuleius, Florida 9, Helm ed., pp. 12.3–13.6.

17. Plato, Hippias Major 281a-b; Xenophon, Memorabilia IV.4.5.

18. The visits are recorded in Plato, Hippias Major 281a; and Xenophon, Memorabilia IV.4.5. See Mario Untersteiner, “Polemica contra Ippia nella settima epistola di Platone,” in Rivista di storia della filosofia, 3 (1948), 101–119. The text of the Dissoi logis is given is Diels-Kranz, Vorsokratiker, II, 90, pp. 405–416, and by Untersteiner, Sofisti, fasc. 3, pp. 148–191.

19. Athenaeus, XIII,608f-609a, Gulick ed. (Loeb), VI (London-Cambridge, Mass., 1937), 280; Clement of Alexandria, Stromata VI.c.2, 15.2, Stählin ed., Clemens Alexandrinus (in the series Die Griechischen Christlichen Schriftsteller), 3rd ed., II (Berlin, 1960), 434.23–435.5. Clement is making the point that the Greeks were incorrigible plagiarists, as shown by Hippias.

20. Bruno Snell, in Philologus, 96 (1944), 170–182. G. B. Kerferd, in Proceedings of the Classical Association, 60 (1963), 35–36, has adopted and extended Snell’s views, and in particular has attributed to Hippias the doctrine of “continuous bodies” mentioned in Hippias Major 301b-e. (This passage would seem to have anticipations of Smuts’s “holism”—τά óλα τσν πραγμάτων.)

21. Scholium to Apollonius of Rhodes, III.1179, Scholia in Apollonium Rhodium vetera, Wendel ed. (Berlin, 1935), p. 251.13–14.

22. Plutarch, Numa 1.6, Ziegler ed., Vitae parallelae, III, pt. 2 (Leipzig, 1926), 55.7–9.

23. Plato, Hippias Major 286a.

24. Pausanias, V.25.4, Spiro ed. (Teubner), II (Leipzig, 1903), 78.4–13. Another statue made by Calon is dated 420–410 b.c.; but this does not have much bearing on Hippias’ date, since his verses were added some time after the statues were made, in place of the original inscription.

25. See final paragraph of Bibliography. The Anonymus Iamblichi is reproduced in Diels-Kranz, Vorsokratiker, II, 89, 400–404.

26. Xenophon, Memorabilia IV. 4.5–23. This passage purports to record a discussion between Socrates and Hippias in which Socrates identifies the just with the lawful—a view difficult to reconcile with Plato’s Socrates—and discomfits Hippias.

In Protagoras 337c-338b, Hippias mediates between Socrates and Protagoras, urging Socrates not to insist on brief questions and answers, and Protagoras not to sail off into an ocean of words. This pleases the company. In the opening sentence Plato would appear to have packed the main tenets of Hippias’ thought: “Gentlemen, I look upon you all as kinsmen and neighbors and fellow citizens by nature, not by law; for by nature like is akin to like, but law, tyrant of men, often constrains us against nature.”

27. Regarding these as key words, and in the fourth and fifth centuries as catch words, W. K. C. Guthrie devotes a chapter to the antithesis in A History of Greek philosophy, III, 55–134.

28. Xenophon, Memorabilia IV.4.19–20.

29. Plato, Protagoras 318e; Hippias Major 366c-368a. The former passage deserves citation because it implies that Hippias believed in compulsory education in the quadrivium at the secondary level. Protagoras is the speaker: “The other [Sophists] mistreat the young, for when they have escaped from the arts they bring them back against their will and plunge them once more into the arts, teaching them arithmetic, astronomy, geometry and music—and here he looked at Hippias—whereas if he comes to me he will not be obliged to learn anything except what he has come for.”

30. Proclus, In primum Euclidis, Friedlein ed. (Leipzig, 1873; repr., 1967), 272.3–10.

31.Ibid., p. 356.6–12

32.Ibid., p.65.11–15. The objection by W. K. C. Guthrie, op.cit., III. 284, that it is “nearly 200 Teubner pages” earlier is not convincing.

33. He so treats Leodamas of Thasos, Oenopides of Chios, and Zeno of Sidon; and if he departs from this practice in the case of Hippocrates of Chios, it is only to avoid confusion with Hippocrates of Cos.

34. The Hippias described by the pseudo-Lucian in Hippias seu Balneum as a skillful mechanician and geometer is a fictional character.

35. J. E. Montucla, Histoire des mathematiques, I, 181; B. L. van der Waerden, Science Awakening, 2nd ed. (Groningen, n.d.), p. 146. Also C. A. Bretschneider, Die Geometrei und die Geometer vor Euklides, pp, 194–196; but H. Hankel, Zur Geschichte der Mathematik, p. 151, note, thought him “sicherlich nicht der Sophist Hippias aus Elis.” After initial disbelief in the identification, G. J. Allman, Greek Geometry From Thales to Euclid, pp. 92–94, 189–193, was converted by Paul Tannery, in Bulletin des sciences mathématiques et astronomiques, 2nd ser., 7 (1883), 278–284; and by Moritz Cantor, Vorlesungen über Geschichte der Mathematik, 3rd ed., 1, 193–197. After a thorough examination. A. A. Björnbo, in Pauly-Wissowa, VIII, cols. 1706–1711, accepted the identification; but Gino Loria, Le scienze esatte nell’ antica Grecia, 2nd ed., p. 69, would say only: “Pesando dunque gli argomenti pro e contro l’identificazione, sembra a noi che i primi vincono per valore i secondi.” T. L. Heath A History of Greek Mathematics, 1, 2, 23, 225, takes the identification for granted; but U. von Wilamowitz, Platon, I, 136, note, thinks that the name is so common that it is a matter of discretion; and W. K. C. Guthrie, loc.cit., is undecided.

36. Diogenes Laertius VIII.iv, Cobet ed., p. 224.

37. Archimedes, Heiberg ed., 2nd ed., III, 84.12–88.2.

38.Ibid., p. 90.4–11.

39. Pappus, Collection, Hultsch ed., pp. 250.33–252.3: “For the quadrature of the circle a certain curve was assumed by Dinostratus and Nicomedes and certain others more recent, and it takes its name from its property, for it is called by them quadratrix.”

Iamblichus as recorded by Simplicius, In Aristotelis Categorias, Kalbfleisch ed., p. 192.19–24: “Archimedes succeeded by means of the spiral-shaped curve, Nicomedes by means of the curve known by the special name quadratrix, Apollonius by means of a certain curve which he himself terms ’sister of the cochloid’ but which is the same as the curve of Nicomedes, and lastly Carpus by means of a certain curve which he simply calls ’the curve arising from a double motion.’ “When W. K. C. Guthrie, op. cit., III, 284, note 2, finds significance in “the silence of Simplicius, who at Physics 54 ff (Didls ed.) seems to be giving as complete an account as he can of attempts to square the circle, “it must be objected that Simplicius’ aim in that passage was much more limited: the efforts of Alexander and Hippocrates.

40. Pappus, op. cit., p. 252.5–25.

41. For references see Bibliography.

42. Kerfered, op. cit., appears to hold this view.


I. Original Works. None of Hippias’ many works has survived. The titles of the following are known: ’Eθνω̂νòνoμαìαι, Nomenclature of Tribes; ’Ολνμπìανγραω̂v, Register of Olympic Victors; Σνναγωγή, Collection; and Tρωικóς (sc, λóγoς or διάλoγoς), The Trojan. Hippias is also known to have composed an elegiac inscription for the statues at Olympia in memory of a boys’ choir from Messina drowned in crossing to Rhegium. He probably wrote a treatise on the quadratrix, of which he was the discoverer.

References to these works, and other witnesses to Hippias, are collected in H. Diels and W. Kranz, Die Fragmente der Vorsokratiker, 6th ed., II (Dublin-Zurich, 1970), 86, 326–334; and Mario Untersteiner, Sofisti: Testimonianze e frammenti, vol. VI in Biblioteca di Studi Superiori, fasc. 3 (Florence, 1954), 38–109.

It is conjectured by Untersteiner that Hippias was also the author of the preface to the Characters of Theophrastus; the Anonymus Iamblichi; and a spurious chapter in the third book of Thucydides’ history, III, 84.

II. Secondary Literature. In Greek literature the main secondary sources for Hippias are Plato, Protagoras 315C, 337C-338B; Plato (?), Hippias Major and Hippias Minor, Burnet ed., III (Oxford, 1903; repr., 1968); and Xenophon, Memorabilia IV.4.5–25, Marchant ed. (as Commentarii), in vol. II of Xenophon’s Works (Oxford, 1901; 2nd ed., 1921). Other scattered references will be found in the notes.

The best recent accounts of Hippias as a philosopher are W. K. C. Guthrie, A History of Greek Philosophy, III (Cambridge, 1969), 280–285; and Mario Untersteiner, I sofisti (Milan, 1948; 2nd ed., 1967), II, 109–158, translated by Kathleen Freeman, The Sophists (Oxford, 1954), pp. 272–303.

Hippias’ mathematical work may be studies in G. J. Allman, Greek Geometry From Thales to Euclid (Dublin, 1889), pp.92–94, 189–193; A. A. Björnbo, “Hippias 13,” in Paulty-Wissowa, Real-Encyclopädi, VIII (Stuttgart, 1913), cols. 1706–1711; C. A. Bretschneider, Die Geometrie und die Geometer vor Euklides (Leipzig, 1870), pp. 94–97; Moritz Cantor, Vorlesungen über Geschichte der Mathematik, 3rd ed., I (Leipzig, 1907), 193–197; James Gow, A Short History of Greek Mathematics (Cambridge, 1884), pp. 162–164; T. L. Heath, A History of Greek Mathematics, I (Oxford, 1921), 225–230; Gino Loria, Le scienze esatte nell’ antica Grecia, 2nd ed. (Milan, 1914), pp. 67–72; and Paul Tannery, “Pour l’histoire des lignes et surfaces courbes dans l’antiquité,” in Bulletin des sciences mathématiques et astronomiques, 2nd ser., 7 (1883), 278–291, repr. in his Mémoires scientifiques, II (Toulouse-Paris, 1912), 1–18.

Among other noteworthy assessments of Hippias are the following, listed chronologically: J. Mahly, “Der Sophist Hippias von Elis,” in Rheinisches Museum für Philologie, 15 (1860), 514–535, and 16 (1861), 38–49; O. Apelt, Beiträge zur Geschichte der griechischen Philosophie, VIII, “Der Sophist Hippias von Elis” (Leipzig, 1891), 367–393; W. Zilles, “Hippias aus Elis,” in Hermes, 53 (1918), 45–56; D. Viale (Adolfo Levi), “Ippia di Elide e la corrente naturalistica della sofistica,” in Sophia (1942), pp. 441–450; Bruno Snell, “Die Nachrichten über die Lehren des Thales und die Anfäange der griechischen Philosophie-und Literaturgeschichte,” in Philologus, 96 (1944), 170–182; and G. B. Kerferd, in Proceedings of the Classical Association, 60 (1963), 35–36.

Mario Untersteiner has put forward his conjectures about Hippias in “Un unovo frammento dell’ Anonymus Iamblichi. Identificazione dell’ Anonimo con Ippia,” in Rendiconti dell’ Istituto lombardo di scienze e lettere, classe di lettere, 77 , fasc. II (1943–1944), 17; “Polemica contro Ippia nella settima epistola di Platone,” in Rivista di storia della filosofia, 3 (1948), 101–119; and “il proemio dei ‘Caratteri’ di Teofrasto e un probabile frammento di Ippia,” in Rivista di filologia classica, n.s. 26 (1948), 1–25.

Ivor Bulmer-Thomas

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Hippias of Elis, the Greek Sophist and polymath, was probably born before 460 BCE. The date of his death is not known, but Plato speaks of him as one of the leading Sophists at the time of the death of Socrates in 399 BCE. On a number of occasions he acted as ambassador for his native city and also traveled widely, earning very large sums of money. He claimed to be a master of all the learning of his day, and his teaching and writings included elegies, tragedies, dithyrambs, historical works, literary discourses, epideictic speeches, discussions of astronomy, geometry, arithmetic, music, painting, sculpture, and ethics, and a technical system of mnemonics. None of his writings survives, but a reference in a papyrus book list of the third century suggests that at least one of his works survived until that date. Our knowledge of his teaching rests above all upon the picture of him given in Plato's dialogues, the Hippias Major (now generally accepted as written by Plato), the Hippias Minor, and the brief sketch in the Protagoras.

His polymathy invites comparison with Plato's more philosophic approach to reality, and Hippias has often been presented as standing for a superficial encyclopedic approach to knowledge, in contrast with the more profound penetration of the genuinely philosophic search for truth. This is the way Plato came to view all the Sophists, but it is probably unfair to Hippias, who in some ways anticipated Aristotle's approach to the whole range of human knowledge. Mathematics and astronomy in the sophistic period were certainly not studied for their practical application in everyday life but, rather, in the pursuit of knowledge for its own sake. Hippias made a really important contribution to mathematical development through his discovery of the curve known as the quadratrix, used for the trisection of an angle and later in attempts to square the circle. He was also used by Eudemus as a source for the early history of geometry, which would suggest that he himself may have written a history of mathematics. He was fairly certainly the source of Aristotle's information about the doctrines of Thales, and he may also have been responsible for the main lines of the schematized picture of the history of the pre-Socratics found in Plato's Sophist (242d).

Whether he had any general theory of the nature of reality is not certain, but it is probable that he did. In the Hippias Major, Plato attributes to him a "continuous doctrine of being," which implies that some particular doctrine was regularly attributed to him. This doctrine dealt with "continuous physical objects that spring from being" (301b), and was opposed to Socrates's attempt to distinguish "the beautiful" from "beautiful objects." While the details of the doctrine are not given, it seems clear that Hippias objected to attempts to explain phenomena in terms of qualities or entities whose existence does not lie wholly within the phenomena that exemplify them. If this is so, then he held to the standard sophistic rejection of the position of Parmenidesfor Hippias, phenomenal reality was the whole of reality. If Plato presents the matter correctly, Hippias regarded reality as composed of concrete physical objects such that all qualities applicable to any group will also apply individually to each member of the group, and all qualities found in each of the individual objects will also apply to the group as a whole.

In ethics Hippias propounded an ideal of individual self-sufficiency. Plato's evidence in the Protagoras, together with that of Xenophon in the Memorabilia (Book IV, Ch. 4, Sec. 5), shows that Hippias made free use of the opposition between nature and convention and that he accepted the overriding claim of Nature in cases of conflict. That he originated this antithesis has often been asserted, but no ancient source suggests this; and there is good evidence that the origins of the doctrine are earlier than Hippias. In the Protagoras, Hippias declares that his listeners are kinsmen, friends, and fellow citizens by Nature because the friendship of like to like comes by Nature, not by convention. While this clearly contains the seeds of a doctrine of cosmopolitanism, it should be remembered that Hippias's listeners in the dialogue are all Greeks and are all alike in their interest in sophistic discussion.

See also Sophists.


Fragments and testimonia are found in H. Diels and W. Kranz, Fragmente der Vorokratiker, 10th ed. (Berlin, 1961).

Guthrie, W. K. C. The Sophists. Cambridge, U.K.: Cambridge University Press, 1971.

Kerferd, G. B. The Sophistic Movement. Cambridge, U.K.: Cambridge University Press, 1981.

Sprague, Rosamond Kent, ed. The Older Sophists; A Complete Translation by Several Hands of the Fragments in die Fragmente der Vorsokratiker, ed. by Diels-Kranz. Columbia: University of South Carolina Press 1972.

Woodruff, Paul, ed. and trans. Hippias Major (Plato). Indianapolis: Hackett, 1982.

G. B. Kerferd (1967)

Bibliography updated by Paul Woodruff (2005)

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Hippias of Elis

c. 460-c. 400 b.c.

Greek Mathematician

Rarely does the personality of a mathematician or scientist play a significant role in his biography, but Hippias was so widely known as a braggart that he can hardly be mentioned without making note of this fact. Yet he also seems to have had cause for boasting, being a man knowledgeable in a wide variety of areas, including several with a mathematical application. In this regard it is notable that he may have developed the quadratix as a means of doubling the cube, trisecting angles, and squaring the circle nearly a century before Dinostratus (c. 390-c. 320 b.c.) used it for the latter purpose.

From Elis on the Greek mainland, Hippias made his living as a traveling philosopher, perhaps a member of the Sophists. Among the areas on which he lectured were poetry, grammar, history, politics, and archaeology, as well as mathematics and related fields. The latter included calculation, geometry, astronomy, and the application of math to music as pioneered by the Pythagorean school.

He was also, according to Plato (427-347 b.c.), a boastful man. T. L. Heath, in A History of Greek Mathematics (1921), related that Hippias "claimed...to have gone to the Olympian festival [i.e., the Olympic Games] with everything that he wore made by himself, ring and sandal (engraved), oil-bottle, scraper, shoes, clothes, and a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works." Clearly Hippias was something of a showoff, but he was also quite talented, having developed a mnemonic system that made him able to repeat a catalogue of 50 names after hearing it only once.

His single contribution to mathematics, which certainly would have been a great one, seems to have been the quadratix. This curve, which Dinostratus later used for the squaring of the circle, could also be used to trisect angles, which was Hippias's purpose. Also, as Heath wrote, Archytas (c. 428-350? b.c.) used it for the duplication of the cube.

The quadratix may be described in terms of a square ABCD. Point A marks the center of a circle, the radius of which is labeled as AE, and the curve BED forms a one-quarter arc of the circle with A at its center. If AE moves uniformly from AB toward AD (rather like a windshield wiper moving clockwise from top to bottom), and BC descends straight downward toward AD, the radius AE and the line BC will intersect at point F. A segment may then be drawn from F downward to intersect AD at point H.

Using these materials, Hippias developed a set of ratios that established an equivalence between angle EAD, arc ED, and (FX × π/2). He was then able to graphically illustrate means for dividing angle EAD by a given ratio. However, Sporus (c. 240-300) criticized Hippias's method as a circular one, and in any case, to use it involved more than the two basic tools of compass and straightedge.