¹ Friederichs, K., ‘Der Doriphoros des Polykle’ in 23 Berl. Winckelmannsprogramm (1863)Google Scholar; Benndorf, O., ‘Der Kanon des Polykle’ in Zeits. Oest. Gymn. xx (1869) 260–8Google Scholar.

² Gardner, E., A Handbook of Greek Sculpture (1915) 360Google Scholar; Furtwängler, A., Masterpieces of Greek Sculpture (1894) 226Google Scholar.

³ Anti, C., ‘Monumenti policletei’ in MAL xxvi (1920–1921) 501–792Google Scholar.

⁴ E.g. Ferri, S., ‘Nuovi contributi esegeteci al ‘Canone’ della scultura greca’ RIA vii (1940) 117–52.Google Scholar

⁵ Gordon, D. E. and Cunningham, D. E. L., ‘Poly kleitos' “Diadoumenos”—Measurement and Animation’ in Art Quarterly (summer 1962) 128–42Google Scholar; Arias, P. E., Policleto (1964)Google Scholar; Hiller, F., ‘Zum Kanon Polyklets’ in Marburger Winckelmannsprogramm (1965) 1–15Google Scholar; EAA (1966) s.v. ‘Canone’, ‘Embater’, ‘Policleto’, ‘Quadratus’ (L. Beschi, S. Ferri); Encyclopedia of World Art (1966) s.v. ‘Polykleitos’ (Berger, E.)Google Scholar; Lorenzen, E., Technological Studies in Ancient Metrology (1966), passim, but esp. 48–9 and 97–100Google Scholar; Linfert, A., Von Polyklet zu Lysipp (1966)Google Scholar; Arnold, D. ‘Die Polykletnachfolge’, JdI Ergänzungsheft xxv (1969)Google Scholar; Vermeule, C., Polykleitos (Boston 1969)Google Scholar; Lorenz, T., Polyklet (1972)Google Scholar; Steuben, H. von, Der Kanon des Polyklet (1973)Google Scholar; Pollitt, J. J., The Ancient View of Greek Art (1974) 14–22Google Scholar and s.v. ‘ἀριθμός’, ‘συμμϵτρία’ and ‘τϵτράγωνος-quadratus’; Tobin, R., ‘The Canon of Polykleitos’, AJA lxxix (1975) 307–22CrossRefGoogle Scholar; Philipp, H., ‘Zum Kanon des Polykleitos’ in Wandlungen: E. HomannWedeking gewidmet (Waldsassen 1975) 132–40Google Scholar; Schindler, W., ‘Der Doryphoros des Polyklet. Gesellschaftliche Funktion und Bedeutung’ in Der Mensch als Mass der Dinge (ed. Müller, R., Berlin 1976) 219–37Google Scholar.

⁶ E.g. those of Richter, G. M. A., The Sculpture and Sculptors of the Greeks ⁴ (1970) 190Google Scholar and Lawrence, A. W., Greek and Roman Sculpture (1972) 153Google Scholar.

⁷ As a basis for study, I would suggest the following treatises from the Hippocratic corpus: De articulis, De fracturis, De locis in homine, plus the pseudo-Aristotelian Physiognomonica and two later books, Philostratus' De Gymnastica and Rufus' Onomasticon. Recent literature: Benveniste, E., ‘Termes greco-latins ďanatomie’ in RPh xxxix (1965) 7–13Google Scholar; Herrlinger, R., ‘Die Rolle von Idee und Technik in der Geschichte der Anatomie’ in AGM xlvi (1962) 1–16Google Scholar; Jüthner, J., Körperkultur im Altertum (1928)Google Scholar; Kudlien, F., ‘Antike Anatomie und menschlicher Leichnam’, Hermes xcvii (1969) 74–94Google Scholar; and Premuda, L., Storia delľ iconografia anatomica (1957)Google Scholar. I thank Professor I. M. Lonie for his assistance in an unfamiliar field.

⁸ The difficulty of deciding exactly where a measuring point is to be located may be illustrated by comparing Kalkmann's, Lorenzen's and Tobin's estimates of the distance from the centre of the mouth to the chin of the Naples Doryphoros—4·975, 6·0 and 5·02 cms respectively (Kalkmann, A., ‘Die Proportionen des Gesichts in der Griechischen Kunst’, 50 Berl. Winckelmannsfeste (1893) vol. liii, 36–7Google Scholar; Lorenzen, op. cit., 48; Tobin, op. cit., 315–16); a non-initiate might justifiably conclude that results obtained from data so erratically and subjectively assessed can hardly be called ‘scientific’ in any generally accepted sense of the term.

⁹ See esp. Richter, , ‘How were the Roman copies of Greek portraits made?’ in MDAI (R) lxix (1962) 52–8Google Scholar.

¹⁰ E.g. by von Steuben, op. cit. 12–26.

¹¹ Schmidt, E., ‘Der Kasseler Apollon und seine Repliken’ in Ant. Plastik v (1966) 38–9Google Scholar; my own measurements of the two runners in the Galleria of the Palazzo dei Conservatori (Stuart-Jones, H., The Sculptures of the Palazzo dei Conservatori [1926] nos. 49 and 52Google Scholar; Helbig, W., Führer durch die … Sammlungen … in Rom [4th edn. by Spier, H. 1966] ii no. 1518Google Scholar), usually considered to be copies of a work of the Polykleitan school from c. 400–c. 380, confirm this estimate. A computer-controlled multivariate analysis (standard practice in the analysis of prehistoric artifacts) of every possible measurement from every known copy would be the only truly scientific approach, but even here there is, again, no guarantee that the points selected would coincide with those chosen by the sculptor of the original.

¹² Cf. Lippold, G., in EAA s.v. ‘Copie’ 806Google Scholar; the sole properly attested case is the torso in Florence, Richter, , Kouroi ³ (1970) no. 195 and figs. 585–8Google Scholar.

¹³ On the process see Lippold, loc. cit.; Kluge, K. von and Lehmann-Hartleben, K., Die antiken Grossbronzen (1927) i 88–9Google Scholar (with direct reference to Apollonios's herm).

¹⁴ Comparing the measurements of the head of the Naples statue published by Kalkmann, loc. cit. (n. 8) with those of Apollonios's herm given by von Steuben (op. cit., 12–21) one finds about a 1–2% discrepancy in most cases; for further comparison with n. 8, von Steuben's estimate from the centre of the mouth to the chin of the bronze would be about 4·72 cm (ibid., 19 and fig. 1:dB + ).

¹⁵ Schuchhardt, W.-H., ‘Antike Abgüsse antiker Statuen’ in AA (1974) 631–5Google Scholar; I have examined casts of the fragments in Munich, and thank Dr R. Wünsche and Dr H. Sichtermann for pointing out the relevant pieces.

¹⁶ On the Amazon, see von Bothmer, D., Amazons in Greek Art (1957) 216–22Google Scholar and pl. 89; von Steuben, op. cit. 56–68 and pls. 40–3, 45, 48–51 (with bibliography). Cf. however Weber, M., ‘Die Amazonen von Ephesos’ in Jdl xci (1976), 28–96Google Scholar, esp. 86 ff. (Sciarra type by Polykleitos).

¹⁷ On the Aristogeiton, Baiae see AJA lxxiv (1970) 296–7Google Scholar and pl. 74; the best studies concerning ‘modernised’ Roman copies are Wünsche's, R. short article ‘Der Jüngling vom Magdalensberg: Studie zur römischen Idealplastik’ in Festschr. L. Düssler (Vienna 1972) 45–80, esp. 62 ff.Google Scholar, which discusses the principles, and Zanker, P., Klassizistische Statuen (Mainz 1974)Google Scholar, which explores the practice.

¹⁸ Quoted in Broneer, O., Isthmia i: The Temple of Poseidon (1971) 181Google Scholar; this whole Appendix (i.e. pp. 174–81) should be prescribed reading for those who would venture into the perils of metrology. Thus, not only Carpenter's but both Lorenzen's ‘Archaic’ and ‘Classical’ canons fit the ‘Blond Boy’, Akr. 689, quite well—given, of course, the latitude usually and conveniently allowed in the selection of measuring-points and rounding-off of measurements (cf. Greek Sculpture [1960] 93; Technological Studies 46–7).

¹⁹ ‘Many, though, have begun the construction of weapons of the same size, have made use of the same system of rules, the same types of wood and the same amounts of iron, and have kept to the same weight; yet of these some have made machines that throw their missiles far and with great force, while those made by others have lagged behind their specifications. When asked why this happened, the latter have not been able to give an answer. So, it is appropriate to warn the prospective engineer of the saying of Polykleitos the sculptor: beauty, he said, comes about para mikron through many numbers. And in the same way, as far as concerns our science, it happens that in many of the items that go to make up the machine a tiny deviation is made each time, resulting in a large cumulative error.’

²⁰ ‘But those who are making progress, of whose life already, as of some temple or regal palace “the golden foundation has been wrought”, do not indiscriminately accept for it a single action, but using reason to guide them they bring each one into place and fit it where it belongs. And we may well conceive that Polykleitos had this in mind when he said that the task is hardest for those whose clay has come to the fingernail.’ (Trans. F. C. Babbitt, Loeb, slightly adapted.)

²¹ ‘And in the arts, formless and shapeless parts are fashioned first, then afterwards all details in the figures are correctly articulated; it is for this reason that the sculptor Polykleitos said that the work is hardest, when the clay is at [or on] the finger-nail.’ (Trans. P. A. Clement, Loeb, slightly adapted.)

²² ‘This, then, is the mode of inquiry: to train to be able to recognise the mean readily in each class of living thing, and indeed in all things, is not the task of any common man, but for the most industrious, who through long experience and comprehensive and detailed knowledge of everything are alone able to discover the mean. Thus do modellers, sculptors, painters, and, indeed, image-makers in general, paint or model the most beautiful likenesses in each case (that is, the most beautiful man, horse, cow or lion), by observing the mean in that case. And one might comment upon a certain statue, the one called the ‘Canon’ of Polykleitos, since it received this name from its having a precise commensurability of all the parts to one another.’

²³ ‘For Chrysippos showed this clearly in the statement from him quoted just above, in which he says that the health of the body is identical with due proportion in the hot, the cold, the dry and the moist (for these are clearly the elements of bodies), but beauty, he thinks, does not reside in the proper proportion of the elements but in the proper proportion of the parts, such as for example that of finger to finger and of all these to the hand and wrist, of these to the forearm, of the forearm to the whole arm and of everything to everything else, just as described in the Canon of Polykleitos. For having taught us in that work all the proportions of the body, P. supported his treatise with a work of art, making a statue according to the tenets of the treatise and calling it, like the treatise itself, the Canon. So then, all philosophers and doctors accept that beauty resides in the due proportion of the parts of the body’. See further n. 25 for the translations of μϵτακάρπιον and βραχίων adopted here.

²⁴ ‘Now in every piece of work, beauty is the product of many numbers, so to speak, that come to a kairos through some system of proportion and harmony, whereas ugliness is ready to spring into being immediately if only one chance element is omitted or added out of place. And so, in the particular case of a lecture, not only frowning, a sour face, a roving glance, twisting the body about, and crossing the legs, are unbecoming, but even nodding, whispering to one another, yawns, bowing the head, and all like actions are culpable and need to be carefully avoided.’ (Trans. F. C. Babbitt, Loeb, slightly adapted.) This passage was added by Schulz, D., ‘Zum Kanon Polyklets’ in Hermes lxxxiii (1955) 200–20Google Scholar.

²⁵ The exact meaning of passage 4 is unclear. Μϵτακάρπιον is usually translated ‘palm’ (e.g. by Tobin, op. cit. [n. 5] 308–9 n. 9: ‘from the knuckle or origin of the little finger to the head of the ulna’), but as Iversen, E. points out in The Legacy of Egypt ² (1971) 76Google Scholar n. 3, the parallel passage in Vitr. iii 1.2 defines ‘manus palma’ as the area from the wrist to the tip of the middle finger: this is the translation adopted here. Also, βραχίων could refer either to the whole arm or to the upper arm only: in the former case the ratios would be a:a + b, b:b + c, etc., and in the latter a:b, b:c, etc. Cf. Panofsky, E., Meaning in the Visual Arts (1955) 64Google Scholar and Gordon and Cunningham, op. cit. 129, 134 and n. 13 for the arguments on each side. Most would-be reconstructors of the canon either ignore these problems with the text or translate it to suit their own convenience. But see my Postscript p. 131.

²⁶ Proposed by Diels, in DK⁶ i 392, and accepted, e.g., by Hiller, op. cit. 13 n. 8.

²⁷ Kranz, in DK loc. cit.; Pollitt, op. cit. (n. 5) 89.

²⁸ Jones, H. Stuatt, Ancient Writers on Greek Sculpture (1895) 129Google Scholar; Anti, loc. cit. (n. 3); Beschi, , in EAA s.v. ‘Policleto’, 273Google Scholar; Tobin op. cit. (n. 5) 319 n. 16.

²⁹ Carpenter, Rhys, The Esthetic Basis of Greek Art (1921) 124Google Scholar; id., Greek Sculpture 101; Schulz, op. cit. 215. Von Steuben, op. cit. 50–1 proposes a variant of this, whereby the ἀριθμοί do not cover every part of the body; yet does not this flatly contradict Galen's repeated assertion in passages 3 and 4 that everything must be in proportion to everything else?

³⁰ Op. cit. 200–8, 214–19.

³¹ Cf. von Steuben, 's criticism of Schulz, in Der Kanon des Polyklet 50–3Google Scholar; Galen's remark noted above (n. 29) points the same way.

³² That sculptural canons did not always ‘come to a καιρός’ is implied by the criticism of Euphranor, 's preserved in Plin. N.H. xxxv 128Google Scholar.

³³ Cf. here Iversen, op. cit. (n. 25) 69: ‘it is curious to observe that the self-imposed restrictions of the canon had never hampered the creativeness of Egyptian artists or lowered the standard of their work. Rather the opposite would seem to have been the case, for the most rigorously canonical representations in Egyptian art are also as a rule those of the highest artistic perfection.’

³⁴ ‘Polyclitus and Pythagoreanism’ in CQ xlv (1951) 147–52; summary and discussion in Pollitt, op. cit. (no. 5) 14–22; the possibility was first raised by Diels, , AA (1889) 10Google Scholar, and Antike Technik (1914) 15.

³⁵ Aët. i 3.8; DK⁶ i 454 line 35. ‘Pythagoras was the first to call philosophy by this name, [laying down] as its principles numbers and proportions in these things, which he also calls harmonies…’

³⁶ Arist. Metaph. 985b30, 990a23, 1078b21; Philolaos, fr. 20Google Scholar (DK⁶ i 416 line 8; 452 lines 5, 25; 456 line 36); on the significance of καιρός to the Pythagoreans see esp. Burkert, W., Lore and Science in Ancient Pythagoreanism (1972) 467Google Scholar.

³⁷ Vitr. iii 1.2–7. This passage forms the basis of Lorenzen's work (n. 5) further discussed in the following notes.

³⁸ Observed by Schlikker, F. W., Hellenistische Vorstellungen von der Schönheit des Bauwerks nach Vitruv (1940) 55Google Scholar and 66. That Vitruvius's fractions are self-contradictory was recognised as early as Leonardo: only two of them in any sense fit the Doryphoros. Panofsky, op. cit. 67 n. 16, von Steuben, op. cit. 68–71, and Iversen, op. cit. (n. 25) 78–9 investigate the problem of possible textual corruption, all suggesting various emendations, and the last a general conformity with the later Egyptian canon. The enormous complexity of Lorenzen's system, involving no fewer than two sets each of two basic modules, each applicable to two further sets each of twenty or so ‘flexible’ scales, all apparently available to the sculptor of the classical period in any combination or permutation, enables him to circumvent such niceties of interpretation as these. It should perhaps also be remarked that, in the opinion of this writer at least, Iversen's work on the Egyptian canon (op. cit. 55–82 passim) has more-or-less invalidated many of Lorenzen's basic assumptions.

³⁹ This would appear to militate against the improved modular system proposed by Ferri and Beschi (nn. 4 and 5) also Lorenzen's conclusion that Polykleitos returned to the older Egyptian canon (op. cit. 89–9).

⁴⁰ Lang, M., Hesp. xxvi (1957) 271–87CrossRefGoogle Scholar shows how the Greeks could manage abacus calculations of sums in the millions by Herodotus' day—but not always without error.

⁴¹ This excludes Tobin's solution from consideration, for here the entire head (!) does not fit the reconstructed canon (op. cit. 314–15, 32.1), and also does some damage to Lorenzen's (op. cit. 48–9: the top of the head is 3 cms lower than predicted) and to von Steuben's (op. cit. 51–2), where several measurements again do not come up to expectations. Tobin does not seem to have noticed that Pliny, 's remarks on Lysippos in N.H. xxxiv 55Google Scholar, which he quotes as supporting his case, specifically exclude the opticallybased adjustments to the canon which he proposes. On this passage see further Moreno, P., Testimonianze per la teoria artistica di Lisippo (1973) 123–4Google Scholar, 133, 139–43.

⁴² Cf. Plin., N.H. xxxiv 55Google Scholar: Polyditus … diadumenum fecit molliter iuvenem … et doryphorum uiriliter puerum [et] quem canona artifices uocant liniamenta artis ex eo petentes ueluti a lege quadam …, also ibid. 53 on the Amazon and e.g. Paus, ii 17.4 on the Hera. The Baiae casts show how different his styles could be for male and female subjects.

⁴³ See Pliny's comment in the previous note, also, in gen., Arnold, op. cit. (n. 5) passim.

⁴⁴ See Panofsky, op. cit. 56–62 and fig. 1, with the references there cited, and also Diod. i 98; Iversen, E., ‘The Egyptian origin of the archaic Greek canon’ in MDAI (Kairo) xv (1957) 134–47Google Scholar; Ridgway, B. S., ‘Greek Kouroi and Egyptian methods’ in AJA lxx (1966) 68–70CrossRefGoogle Scholar; cf. Richter's, I. A. study in Brunn-Bruckmann, , Denkmäler Griechischer und Römischer Sculptur cli (1934) 27Google Scholar, also ead., ‘The Archaic Apollo in the Metropolitan Museum’, Metr. Mus. Stud. v (1934) 51–6; Lorenzen, op. cit. (n. 5) passim; Ahrens, D., ‘Metrologische Beobachtungen am “Apoll von Tenea”’, JÖAI xlix (1968–1971) Beibl. 117–32Google Scholar; Carpenter, , Greek Sculpture 94–5Google Scholar; Iversen, , ‘The Canonic tradition’ in The Legacy of Egypt (1971) 55–82Google Scholar, with further bibliography; Guralnick, E., ‘The proportions of some archaic Greek sculptured figures: A computer analysis’ in Computer and the Humanities x 3 (1976) 153–69CrossRefGoogle Scholar. One fairly firm piece of evidence for the use of the second Egyptian canon by the Greeks as late as the mid fifth century is the metrological relief in Oxford (Michaelis, A., JHS iv (1883) 335–50CrossRefGoogle Scholar; Lorenzen, op. cit. 28–30, 39, 47–8, 60 ff. [but cf. n. 38]; Iversen, op. cit. 75)—though I am well aware that with the partial exception of those concerning the Oxford relief all these studies are still open to the objection stated on p. 122 above, that we still do not know the points considered significant by the Greeks. See, in gen., Ferri in EAA s.v. ‘Canone’ and ‘Embater’.

⁴⁵ Corbett, P. E., JHS lxxxvi (1966) 275–6Google Scholar; cf. Coulton, J. J., BSA lxx (1975) 59–99Google Scholar, esp. 89–98.

⁴⁶ Proposed by Bieber, M. in Thieme-Becker, , Allgemeines Lexikon der bildenden Künstler (1933) s.v. ‘Polykleitos’ 225Google Scholar; AJA lxvi (1962) 242; ibid. lxxiv (1970) 90; Gordon and Cunningham, op. cit. (n. 5) passim.

⁴⁷ Proclus on Eucl. i p. 67, 6; cf. ibid. p. 60, 16–19: ‘common to both sciences [geometry and arithmetic] are the theorems regarding sections… with the exception of the division of a line in extreme and mean ratio’. See Heath, T., A History of Greek Mathematics (1921) 87Google Scholar, 160, 304, 324–5; id., Euclid ² (1926) i 137; ii 97–101—though the present tendency, following Heidel, 's fundamental ‘The Pythagoreans and Greek Mathematics’ (AJP lxi [1940] 1–33)Google Scholar is to downgrade the Pythagorean contribution to the science, and to down-date it as well: cf. Philip, J. A., Pythagoras and Early Pythagoreanism (1966) 204–5Google Scholar, and esp. Burkert, op. cit. 401 ff., and in particular pp. 452–3; this is the stance adopted in the present study.

⁴⁸ Nicom. Ar. 2. 28; Papp. p. 102; lamb, in Nic. p. 116, 4–6 (DK⁶ i 445 line 3): cf. Heath, , History of Greek Mathematics 87Google Scholar. Tobin's canon, based on the ratio of (1: 1·4142136 …: cf. Heath, op. cit. 90–1, 154–7), is open to several of the objections already levelled at the Golden Section, plus the additional one that if there is anything at all in the postulated connection between Polykleitos and the Pythagoreans, the latter could not for a moment have entertained a canon grounded in the ultimate in irrationals, the one, in fact, that was eventually to contribute much to bringing down their entire world system.

⁴⁹ Archyt. ap. Porph. in Harm. p. 92 (DK⁶ i 435–6); Heath, op. cit. 85–6.

⁵⁰ Von Steuben, op. cit. 16–20.

⁵¹ Op. cit. (n. 34) 150–1.

⁵² ‘Moreover they collected from the members of the human body the proportionate dimensions which appear necessary in all building operations: the finger [inch] the palm, the foot, the cubit. These they distributed into the perfect number, which the Greeks call teleon, for the ancients determined as perfect the number which is called ten.’

⁵³ Heath, op. cit. 78–9.

⁵⁴ Hdt. ii 109; Souda, s.v. ‘γνώμων’. Gnomons (builders' set squares in the form of a cross) are illustrated on red-figured vases from c. 490 onwards: Beazley, J. D., Attic Red-figure Vase-painters ² (1963) 348/3Google Scholar, 431–2/48, 892/7; (Pottier, E., Vases Antiques du Louvre [1897–1922] pl. 135Google Scholar; Beck, F. A. G., Greek Education, 450–350 B.C. [1964] pls. 4, 8Google Scholar): cf. Heath, loc. cit.; Burkert, op. cit. 33 n. 27, 274 n. 172. 419 n. 104 (with references). For a sample of technical terms in geometry and mensuration in use at the beginning of the fifth century cf. Simon. fr. 542. 3 (Page); Thgn. 805; Plin. N.H. vii 198; Coulton op. cit. (n. 45) passim.

⁵⁵ Collections of sources and discussion in Heidel, op. cit. 16–17, 30–3; Burkert, op. cit. 407–20, 429, 433, 442; on Babylonian science see esp. Neugebauer, O., The Exact Sciences in Antiquity (1957) 29–52Google Scholar, and 157 ff. for its influence on Greece.

⁵⁶ Richter, , ‘Greeks in Persia’, AJA 1 (1946) 15–30CrossRefGoogle Scholar.

⁵⁷ Bronowski, J., The Ascent of Man (1973) 161Google Scholar; N.B. that at least one fifth-century non-Pythagorean used the term κανών to describe the musical scale: Porph. V.P. 3 (DK⁶ i 444 lines 33–445): Burkert, op. cit. 455 n. 40.

⁵⁸ Op. cit. (n. 34) 151–2.