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Entasis in Fourth-Century BC Doric Buildings in the Peloponnese and at Delphi

  • Jari Pakkanen (a1)


Entasis, the slight outward curve of the column shaft, is a constant feature of fourth-century BC Doric architecture in the Peloponnese and at Delphi. In the buildings studied, the position of the maximum entasis is usually in the middle of the column shaft and, even though the degree of entasis varies, it is usually less pronounced than in fifth-century Doric columns. The relation between different mathematical formulae and entasis curves is also discussed; if Greek architects used a particular curve in the design of the column shaft, is it currently possible to identify it? The curves considered are the conic sections (circle, ellipse, parabola, and hyperbola) and the vertical projection of a helix. All the conic sections fit almost equally well in each individual building; therefore, if any particular curve was employed in the design, the simplest method of stretching a circle into an ellipse is a likely candidate.



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1 I wish to thank Professor Richard A. Tomlinson for reading the manuscript and for very valuable discussions. The current outline of the paper is largely due to the comments I received from the anonymous reader of the paper; especially the section on the Tholos at Epidauros has benefited from the criticism. Needless to say, I am very grateful to this person. I also owe my gratitude to Dr Petra Pakkanen and Mr Kimmo Vehkalahti for their comments on my work. The following short titles and special abbreviations are used in this article:

Amandry and Bousquet 1940–1 = Amandry, P. and Bousquet, J., ‘La colonne dorique de la Tholos de Marmaria’, BCH 64–5 (19401941), 121–7.

Bousquet 1952 = Bousquet, J., J. Trésor de Cyréene. (FD; Paris, 1952).

Hill 1966 = Hill, B. H., The Temple of Zeus at Nemea. rev. and suppl. by Williams, C. K., II (Princeton, 1966).

Michaud 1977 = Michaud, J.-P., Le Temple en calcaire. (FD; Paris, 1977).

Roux 1961 = Roux, G., L'Architecture de l'Argolide aux IVe et IIIe siècles avant J.-C. (BEFAR 199; Paris, 1961).

Stevens 1924 = Stevens, G. P., ‘Entasis of Roman Columns’, MAAR 4 (1924), 121–52.

ƒ = half of the flute width (see Fig. 4).

FIW = flute width.

r1 = column drum radius from centre to arris (see fig. 4).

r2 = column drum radius from centre to bottom of flute (see Fig. 4).

2 Although the treasury of Kyrene is located at Delphi, stylistically the building belongs to the sphere of Kyrenian architecture; see below section 7.

3 The temples of Athena Alea at Tegea and of Zeus at Nemea.

4 At Delphi and Epidauros.

5 The fourth century temple of Athena at Delphi.

6 The treasury of Kyrene at Delphi.

7 The earliest of the buildings is the Tholos at Delphi and the latest the temple of Zeus at Nemea.

8 See e.g. Penrose, F. C., An Investigation of the Principles of Athenian Architecture (London, 1851), 40 and Stevens 1924, 123.

9 As the curve fitting method I have used least squares approximation: on the method see Pakkanen, J., ‘The Entasis of Greek Doric Columns and Curve Fitting. A Case Study Based on the Peristyle Column of the Temple of Athena Alea at Tegea’, Archeologia e calcolatori 7 (1996), 694–5.

10 Amandry, P., ‘Observations sur les monuments de l'Héraion d'Argos’, Hesperia, 21 (1952), 272 n. 94: c. 380 BC; Roux 1961, 413, 415, and 418: c. BC: Bernard, P. and Marcadé, J., ‘Sur une métope de la Tholos de Marmaria à Delphes’, BCH 85 (1952), 469–73: c. 375 BC. See also Marcadé, J., ‘Les métopes de la Tholos de Marmaria à Delphes’, CRAI 1979, 168–70. F. Seiler has recently suggested that there could have been two building phases in the Tholos: the main part would have been finished by the beginning of the 4th cent. BC, and only the roof elements would belong to the second later phase (Seiler, F., Die griechische Tholos (Mainz, 1986), 65–7). The suggestion explains part of the contradictory dates given for the building, but redating the metopes to the 5th cent. BC would perhaps require a closer study of the objects themselves.

11 Amandry and Bousquet 1940–1, 121–7.

12 For the data, see Amandry and Bousquet 1940–1, 124 and 125 n. 2. The plotted points are (0, 0), (0.015, 1.115), (0.028, 2.230), (0.0455, 3.345). (0.065, 4.460). (0.085, 5.575); the formula of the fitted curve is y = −0.013 + 85.3x− 291.5x 2 + 718.6x 3. H. Ducoux discovered that the columns were inclined toward the interior (Amandry and Bousquet 1940–1, 123). Therefore, the shaft profiles on the sides and the exterior and interior faces of the column are slightly different: in Fig. 1 is given the side profile. The x co-ordinate points for this profile are calculated from the drum diameter measurements.

13 Athenian influences and features: proportions (see Charbonneaux, J., La Tholos du sanctuaire d'Athéna Pronaia à Delphes (FD; Paris, 1925), 32); building materials (Pentelic marble and limestone from Eleusis; J. Bousquet, ‘La destination de la Tholos de Delphes’, Revue historique, 1960, 287 n. 2). Peloponnesian influences: Corinthian capitals and ceiling coffers (for parallels with the temple of Bassai, see e.g. Dinsmoor, W. B., The Architecture of Ancient Greece (London, 19503 (1975)), 234). According to J. Charbonneaux the carved sima of the Tholos displays Ionic influences (see J. Charbonneaux, op. cit, 7 and 31), but it actually has a fairly close—even though less elaborate—predecessor in the 5th cent, sima of the temple of Poseidon at Isthmia (date and description: Broneer, O., Temple of Poseidon, Isthmia I (Princeton, NJ, 1971), 150–2, and compare pls. 24–6 with the Tholos sima in Charbonneaux op. cit., figs. 9–11).

14 Bernard and Marcadé (n. 10), 461–9; J. Marcadé (n. 10), 170.

15 The proportional emphasis of the maximum entasis (the maximum projection between the shaft profile and the straight line connecting the bottom and the top divided by the shaft height) is in the Parthenon 0.18% and the Tholos 0.09%; the proportional position of the maximum entasis in the shaft (the height of the maximum entasis from the stylobate divided by the shaft height) is in the first one 0.439 and the latter 0.534 (for the Tholos see Table 2; figures on the Parthenon are calculated from measurements given in F. C. Penrose (n. 8), 40).

16 For a synopsis of the argument, see Cooper, F. A., The Temple of Apollo at Bassai. A Preliminary Study (New York and London, 1978), 103–4 (this is an unaltered publication of his 1970 dissertation). Cooper himself detected no entasis with an optical instrument (p. 104), but one should be very cautious using e.g. a theodolite to measure column entasis (see below section 8 and Figs. 10–11). W. B. Dinsmoor was convinced that the temple columns do have entasis: ‘the swelling outline of the shaft known as the entasis is certainly present, and is quite apparent as sighted from pavement or capital’ (‘The Temple of Apollo at Bassai’, MMS 4 (1932–3), 207).

17 The proportional emphasis of the maximum entasis is in the Tholos 0.09% and in the Ionic columns of the Erechtheion North Porch 0.092%; the corresponding figures for the proportional position of the maximum entasis in the shaft are 0.534 and 0.493 (for the Tholos see Table 3; figures on the Erechtheion are calculated from measurements given in F. C. Penrose (n. 8), 40).

18 J.-P Michaud dates the temple approximately 10 years later than the treasury of Thebes at Delphi (c. 370 BC) on building-technical and stylistic reasons; see Michaud 1977, 117–18.

19 Michaud 1977, 30–6, 50–1.

20 The data are based on the actual measurements as reported in Michaud 1977; this and rounding of figures explains e.g. the height discrepancy of the fourth level drum in Figs. 2 (1.02 m) and 3 (1.01 m); Michaud himself uses a system where the measurements are first converted into theoretical dimensions of a foot unit of 0.2976 m and then these figures are used in the further study. The plotted points in Fig. 2 are (0, 0), (0.016, 1.163), (0.028, 2.034), (0.0425, 3.084), (0.0615, 4.102), (0.0755, 4.920); the formula of the fitted curve is y = −0.006 + 76.856x − 110.522x 2 − 598.710x 3. Fig. 3 is drawn to smaller scale in order to emphasize that the reconstruction in Fig. 2 is to be preferred.

21 Michaud 1977, 35.

22 IG 4.1 103. On the inscription see Roux 1961, 171–6; Burford, A., ‘Notes on the Epidaurian Building Inscriptions’, BSA 61 (1966), 275–81; and eadem, The Greek Temple Builders at Epidauros (Toronto, 1969), 64–8.

23 Burford, A.Greek Temple Builders (n. 22), 63–4.; Tomlinson, R. A., Epidauros (Austin, 1983), 29. F. Seiler (n. 10), 80–4. suggests a longer building period and the date as c. 370–320 BC.

24 The capital profile presented in Roux 1961, fig. 16 is a compilation derived from the preserved capital fragments.

25 Roux 1961, 138–40, figs. 30–1.

26 Roux 1961, 147 and 153.

27 Tomlinson (n. 23), 62–4; F. Seiler (n. 10), 76; Büsing, H., ‘Zur Bauplanung der Tholos von Epidauros’, AM 102 (1987), 251–2. Büsing also argues that the inclination of the columns towards the interior of the building can neither be ruled out nor verified on the basis of preserved material (249–50 and fig. 6), but as I have elsewhere demonstrated, the columns must be reconstructed vertical and not inclined; see Pakkanen, J., ‘The Height and Reconstructions of the Interior Corinthian Columns in Greek Classical Buildings’, Arctos 30 (1997). 152–3.

28 Roux 1961, 140.

29 On the study of the building blocks at Tegea, see n. 46. below.

30 Because of the structural similarities G. Roux suggests that the same workers could have been employed both at Epidauros and Tegea; Roux 1961, 138 and 184. Tegean workmen are also listed in the building accounts of the Tholos at Epidauros; see IG 4.1 103B lines 51–4.

31 The measurements in the table are taken from my fieldwork (n. 46), except for those in column (4) for blocks 501 and 562, which have been adopted from Dugas, Ch., Berchmans, J., and Clemmensen, M., Le Sanctuaire d'Aléa Athéna à Tégée au IVe siècle (Paris, 1924), pl. 35 (block 501) and pl. 36 (block 562). For the latter variation of 1.209–1.213 m is given, and in Table 1 I have given the mean 1.211 m.

32 The formula used is r 1 = (F1W / 2) / sin(α), where α = 9°.

33 Roux 1961, 140.

34 Roux 1961, 138 lists the ‘theoretical’ diameters as follows: 0.947–0.936 m; 0.936–0.922 m; 0.922–0.905 m; 0.905–0.888 m; 0.888–0.872 m; 0.871–0.854 m; (0.854–0.835 m); 0.834–0.812 m. The measurements are taken between the bottoms of two opposite flutes. The capital radius from centre to arris is calculated from the preserved flute width as 0.386 m; the radius from centre to bottom of flute he gives as 0.37 m and the diameter as 0.74 m; see Roux 1961, 140. For the joint between the fifth and sixth drums the average value of 0.8715 m has been used in Figs. 5 and 6 and for the joint between the missing seventh drum and the eighth drum I have preferred to use the actual measurement of 0.834 m. The fact that I use Roux's measurements exact to the millimetre does not actually imply that I think that weathered and broken poros stone can necessarily be measured so precisely: as Roux himself states, the figures are theoretical.

35 The plotted points in Fig. 5 are (0, 0), (0.0055, 0.591), (0.0125, 1.182), (0.021, 1.773), (0.0295, 2.364), (0.03775, 2.955). (0.0465, 3.546), (0.0565, 4.137), (0.0675, 4.728), (0.1035, 6.5); the formula of the fitted curve is y = 0.05 + 90.5x −393.8x 2 + 1173.1x 3.

36 The plotted points in Fig. 6 are the same as in Fig. 5, except for the last, which is (0.1145, 7.1); the formula of the fitted curve is y = 0.05 + 90.8x − 408.90x 2 + 1338.9x 3.

37 The residual sum of squares is in the case of the twelve drum reconstruction 0.0138 and in Roux's reconstruction 0.0141. For other curves fitted to shaft data, see Table 2.

38 All the seventh drums are lost, but because the eighth drum fits to the shaft profile 0.59 m above the sixth drum, it is very likely that the seventh drum was also c. 0.59 m high (see Figs. 5 and 6). As cited at the beginning of this section, Roux gives the average height of the 12 consistent drums as 0.591m and the two exceptions as 0.585 and 0.574 m. He does not give the range of his measurements, but because 0.585 m is listed as an exception, I reconstruct a hypothetical range of 0.588–0.594 m. If the heights of the 12 drums were distributed roughly normally within this range (four cases of 0.591 m, two each of 0.590 m and 0.592 m, and one each of 0.588 m, 0.589 m, 0.593 m, and 0.594 m), we get a sample we can use in statistical computation. The sample mean (X) is 0.589 m, the t-value corresponding to 13 degrees of freedom (t n−1) 3.012, the sample standard deviation (S) 0.0049, and the sample size in) 14. Substituting these into the formula X ± (t n−1) S/√n, we get the 99% confidence interval: 0.585–0.593 m. In other words we can be 99% sure that the mean drum height for the eight lowest drums is between 0.585 and 0.593 m, and that the column shaft height at the level of the eighth drum is between 4.68 and 4.75 m (on confidence intervals, see e.g. Siegel, A. F. and Morgan, C. J., Statistics and Data Analysis. An Introduction, 2nd edn (New York, 1996), 321–30).

39 The Tholos at Delphi has drums of almost regular height (see above section 3), but this is an exception. The other buildings discussed in this paper have varying drum heights, as do the 4th-cent. temples of Asklepios and Artemis at Epidauros: in the temple of Asklepios the range of preserved drums is 0.51–0.57 m and in the temple of Artemis 0.73–0.84 m (Roux 1961, 93 and 206–8). The temple of Athena Alea is an interesting example: the two lowest drums are of uniform height, but in the next four levels there is a difference of 0.39 m between the shortest and tallest drum (see below section 6). The 4th-cent. temple of Apollo at Delphi has 107 preserved drums, 98 of which are c. 0.75 m, three 0.660–0.695 m, and six 0.807–0.835 m: the nine odd drums are from different levels (Courby, F., La Terrasse du temple (FD; Paris, 1927), 1517).

40 Epidauros: column height 6.5 + 0.38 ≈ 6.9 m, lower diameter 0.998m (Roux 1961, 138 and 140); for Delphi see Table 3.

41 Column height 7.1 + 0.38 ≈ 7.5 m.

42 Roux 1961, 321; R. A. Tomlinson (n. 23), 64. It should also be remembered that W. B. Dinsmoor rejected the reconstruction of five drums per column for the Tholos at Delphi and suggested, against factual information, a reconstruction of four drums, only because the latter better fitted the usual proportions of the 4th-cent. columns (W. B. Dinsmoor (n. 16), 234 n. 3). Rejection solely on the ground of proportional reasons is quite hazardous.

43 On the connection of Corinthian order to the exterior order, see Roux 1961, 153–7 and fig. 31. The height of the Corinthian column in Roux's reconstruction is c. 6.74 m, and with the additional drum of 0.59 m we get 7.33 m.

44 The Tholos at Delphi: at least 10.9, but very likely 11.7. lower diameters; see J. Pakkanen (n. 27), 146–49.

45 This makes it perhaps necessary to consider the recently started restoration project for the Tholos. Two alternatives were presented before starting the project. The first was to restore a section of the superstructure over the area of four columns, also incorporating the marble ceiling from the museum. The second was to partially build the peristyle and cella walls to a lower level (see Lambrinoudakis, V., ‘Excavation and Restoration of the Sanctuary of Apollo Maleatas and Asklepios at Epidauros’, in Πραϰτιϰὰ του Γ' Διεθνοῡς συνεδρίου πελοποννησιαϰῶν σπουγῶν. Πελοποννησιαϰὰ 13:2 (Athens, 19871988), 303). If the conclusion of this chapter is accepted, the second restoration alternative becomes preferable.

46 Since 1993 I have studied the building blocks at Tegea as part of the new work in the sanctuary; the excavation project has been conducted by the Norwegian Institute at Athens as an international co-operation under the direction of Professor Erik Østby; on the excavations, see Østby, E., Luce, J.-M., Nordquist, G. C., Tarditi, C., and Voyatzis, M. E., ‘The Sanctuary of Athena Alea at Tegea: First Preliminary Report (1990–1992)’, OpAth 20 (1994).

47 Paus. 8.45.4–5.

48 For the date, see Norman, N.J., ‘The Temple of Athena Alea at Tegea’, AJA 88 (1984), 191–3; dating the building to the second half of the 4th cent, is also supported by the pottery discovered to die north of the temple in die Norwegian excavations.

49 The peristyle consisted of 36 columns of 6 drums each, so the building had originally 216 drums. With 50 drums 23% of the original material is well or fairly well preserved; in addition to these drums there are 29 blocks at the site which have at least one missing critical dimension.

50 M. Clemmensen gives a list of forty-seven drums, but some of them are only fragmentary (Dugas et al. (n. 31), 131–3). All these, with the exception of two, have been able to be identified with certainty in the sanctuary on the basis of the published measurements and Dugas' and Clemmensen's systematic numbering of the drums.

51 It is possible to determine 99% confidence intervals of the mean height for each level of drums (these intervals do not exactly match the measured height variation) as following: level A, 1.465–1.475 m; level B, 1.471–1.481 m; level C, 1.392–1.589 m; level D, 1.390–1.651 m; level E, 1.268–1.619 m; level F, 1.282–1.602 m. Adding together the lower and upper limits we get c. 8.27—9.42 m as the column height interval. On determining confidence intervals, see n. 38 above.

52 Within this range the height position of maximum entasis in the shaft varies slighdy, from 4.25 m to 4.47 m; in Table 3 I have used c. 4.4 m in the calculations. For a more detailed account of the research process, see J. Pakkanen (n. 9) 695–702.

53 See J. Pakkanen (n. 27), 163–4.

54 The plotted points in Fig. 7 are (0, 0), (0.017, 1.469), (0.039, 2.933), (0.064, 4.576), (0.091, 6.069), (0.120, 7.649), (0.148, 8.980) and the formula of the fitted curve is y = 0.019 + 84.611x − 247.634x 2 + 577.145x 3.

55 The temenos wall behind the treasury was restored c. 334 BC (Bousquet, J., Les comptes du quatrième et du troisième siècle (Corpus des inscriptions de Delphes II; Paris, 1989), 81A; for a discussion of the inscription see pp. 181–2.; see also Laroche, D., ‘L'emplacement du trésor de Cyrène à Delphes’, BCH 112 (1988), 296–7.). The fragmentary inscription from the north-west anta (Bousquet 1952, pl. 51.6 and p. 70, fig. 11) can be dated to 323/322 Be (Bousquet, J., ‘Inscriptions de Delphes’, BCH 109 (1985), 249).

56 Bousquet 1952, 47–8, fig. 3 and pls 17–19. The height of the column is partially derived from the wall height. The recent relocation of the building from foundation XIII to XIV (D. Laroche (n. 55), 291–305) does not affect Fomine's. column and anta reconstruction.

57 ‘En relevant les cotes de ces fragments de Pentélique et celles des demi-colonnes en Paros, M. Fomine a cru pouvoir déceler l'existence d'une entasis légère. Mais je dois dire que je n'en suis pas absolument sûr, d'abord à cause de la difficulté singulière de la mesure, ensuite parce qu'une règle posée dans les cannelures, à l'endroit où elles sont le mieux conservées, ne donne qu'une indication incertaine, et les blocs de demicolonne, parfois moins dégradés, ne sont pas assez hauts pour que Ventasis y soit sensible à la règle. Il arrive sur un même bloc que la règle bascule légèrement dans une cannelure et ne bouge pas dans la cannelure voisine’ (Bousquet 1952, 47–8).

58 Radius measurements (from centre to arris) of the engaged half-columns by Y. Fomine (in pls. 17–19 of Bousquet 1952): 278 mm (block no. 5 in pl. 17); 272 mm and 269 mm (no. 3 in pl. 17); 269 mm and 266 mm (no. 2 in pl. 18); 259 mm and 255 mm (no. 5 in pl. 18); 238 mm and 231 mm (no. 1 in pl. 18). The radius of the top of the shaft can be measured from a preserved top drum (no. 7 in pl. 19) as 229 mm. The three radii given by Fomine and not directly measurable are 263 mm (at the height of 1.76 m; see Fig. 8 in this paper), 249.5 mm (at 2.65 m), and 244 mm (at 2.95 m).

59 The plotted co-ordinate points (x, y) are (0, 0), (0.006, 0.846), (0.009, C164), (0.012, 1.461), (0.015, 1.758) (0.019, 2.055), (0.0235, 2.352), (0.0285, 2.649), (0.034, 2.946), (0.040, 3.243), (0.047, 3.540), (0.049, 3.643) and the fitted curve is y 143.9x 2−2312.1x 2 + 17873.4x 3.

60 Bousquet 1952, 41–6, 51, pls 17–18, pl. 20.

61 Slanting vertical face of abacus: the extra-mural Doric temple at Apollonia (4th cent. Wright, G. R. H., ‘The Extramural Doric Temple’, in Goodchild, R. al., Apollonia, the Port of Cyrene. Excavations by the University of Michigan 1565–1067. (Supplements to Libya Antiqua, 4; Tripoli, [1976]), 4951. and figs. 5–6); annulets not separated: the second temple of Apollo at Kyrene (end of 4th cent.; Stucchi, S., Architettura Cirenaica. Monografie di archeologia Libica 9 (Rome, 1975), 9293, fig. 83); moulding at the top of the metope: Strategheion at Kyrene (last quarter of 4th cent.; (Stucchi (op. cit.), 95); Gismondo, I., ‘Il restauro dello Strategheion di Cirene’, QAL 2 (1951) fig. 13: the moulding is actually cyma recta, not just cavetto); half-columns attached to antae: the temple at Eluet Gassam (first half of 4th cent.; Stucchi (op. cit.), 51–2, Fig. 39) and the temple of Isis in the sanctuary of Apollo at Kyrene (end of 4th cent.; Stucchi (op. cit.), 100–1, Fig. 89).

62 Crown moulding of the abacus and apophyge in Doric flutes: Portico O2 in the Agora of Kyrene (middle of 3rd cent.; L. Bacchielli, L' Agorà di Cirene II.1 (Monografie di archeologia libica 15; (Rome, 1981), fig. 85).

63 L. T. Shoe dates the moulding profiles to c. 330 BC (The Profiles of Greek Mouldings (Cambridge, Mass., 1936), 71, 159, 164, 166). In 1964 a kiln used to produce the roof tiles of the temple was discovered at Nemea; on the basis of coin and pottery finds it can be dated between 340 and 320 BC, and likely towards the end of the period (Hill 1966, 46).

64 Hill 1966, 22. The plotted co-ordinate points (x, y) are (0, 0), (0.007, O.840), (0.013, 1.543), (0.022, 2.478), (0.030, 3.168), (0.041, 3.955), (0.051, 4.745), (0.060, 5.430), (0.072, 6.215), (0.084, 7.005), (0.096, 7.775), (0.108, 8.440), (0.119, 8.985); the fitted curve is y = 0.03 + 119.8x − 601.6x 2 + 1949.9x 3.

65 Cooper, F. A., ‘The Temple of Zeus at Nemea and Architectural Refinements of the Fourth Century BC’, in Πραϰτιϰά του XII γιεθνούς Συνεγρίου Κλασιϰής Αρχαιολογίας 4–10 Σεπτεμβρίου 1983 (Athens, 1988), 38.

66 Hill 1966, 22 n. 53.

67 ‘The telescope of a theodolite is adjusted so that the vertical hair is made to tilt along a straight line that scans parallel to the cord which intercepts the arc of entasis. With the use of rulers offset ordinates are read directly and at regular intervals between the scanning telescope hair and the curve of entasis’ (Cooper (n. 65), 38 n. 4).

68 Hill 1966, 22.

69 See e.g. Bundgaard, J. A., Mnesicles. A Greek Architect at Work (Copenhagen, 1957), 137–8, and Coulton, J. J., Ancient Greek Architects at Work. Problems of Structure and Design (Ithaca, 1977), 107–8.

70 The first edition of F. C. Penrose's The Principles of Athenian Architecture was published in 1851. For other studies on curves and entasis, see e.g. Pennethorne, J., The Geometry and Optics of Ancient Architecture (London and Edinburgh, 1878), 153–6; Stevens 1924, 121–52; and recently, Mertens, D. and Welsch, H., ‘Zur Entstehung der Entasis griechischer Säulen’, in Büsing, H. and Hiller, F. (eds), Bathron (Saarbrücken, 1988), 307–18, esp. 317.

71 For this study Chris Rorres and David Romano's paper ‘Finding the Center of a Circular Starting Line in an Ancient Greek Stadium’ (forthcoming in SLAM Review) has been of great inspiration. I wish to thank both of the authors for giving me a copy of the paper.

72 Heath, T., A History of Greek Mathematics, (1921; New York, 1981), i. 179–80, 251–5.

73 On the vertical projection of a helix (Tschirnhausen's quadratrix), see Stevens 1924, 126, 129–33. Deinostratos ƒl. mid 4th cent. BC) was the brother of Menaichmos. Recently, W. R. Knorr has quite conclusively argued against the traditional attribution of the ‘quadratrix’ to Hippias (c. 420 BC); see Knorr, W. R., The Ancient Tradition of Geometric Problems (1986; New York, 1993), 80–6. On the curve, see also T. Heath (n. 72), i 219, 225–30.

74 See e.g. Stevens 1924, 126–7, 130.

75 Haselberger, L., ‘Werkzeichnungen am Jüngeren Didymeion’, IstMitt (1980), 191215.

76 For the drawing compared to the unfinished column with inscribed measurements, see Haselberger, L., ‘Berichtüber die Arbeit am Jüngeren Apollontempel von Didyma’, IstMitt 33 (1983), 115–21.

77 The reason for selecting the treasury of Kyrene is that it is the only building which is sensible to illustrate with the vertical projection of a helix.

78 Estimate-operation can be used for both linear and non-linear regression analysis: I have selected to use in the estimation process ordinary least squares approximation and in the minimization of the residual sum of squares the Davidon-Fletcher-Powell variable metric method; see Mustonen, S., SURVO. An Integrated Environment for Statistical Computing and Related Areas (Helsinki, 1992), 178196. I wish here to take the opportunity to thank Professor Seppo Mustonen from the Department of Statistics at the University of Helsinki for his support and especially for providing the possibility to use Survo, the main tool I have used throughout this study.

79 This simple process can be applied to the circle, the parabola, and the vertical projection of a helix. Estimating the ellipse and the hyperbola is not quite so easy because there are four parameters to be evaluated. I decided to cope with this problem by writing a small computer program which actually reduces the number of estimated parameters to three: a range of x 0 values is given as input for the program, and then the program searches, using the estimate-operation, for a local minimum for the residual sum of squares.

80 On calculation and the ancient architect, see Coulton, J. J., ‘Towards Understanding Greek Temple Design: General Considerations’, BSA 70 (1975), 7485.

81 See L. Haselberger (n. 76), 201.

82 Stevens 1924, 148–51.

83 Stevens 1924, 126.

84 In Fig. 17 I have placed also the shaft of the Parthenon for comparison.

85 See columns B, G and I in Table 3.

86 See columns H and I in Table 3.

87 F. Courby (n. 39), 17, reports that the columns had no entasis but his work on the columns has been corrected by H. Ducoux in this and many other aspects; see Ducoux, H., ‘Restauration de la façade du temple d'Apollon’, BCH 64–5 (19401941), 266–7.

88 Broneer, O., The South Stoa and its Roman Successors, Corinth I.4 (Princeton, 1954), 30. The recently conducted excavations in the Stoa area have changed the date of the building to c. 300 BC or perhaps later; see Williams, C. K., Fischer, H and Fischer, J. E., ‘Corinth, 1971: Forum Area’, Hesperia, 41 (1972), 171.

Entasis in Fourth-Century BC Doric Buildings in the Peloponnese and at Delphi

  • Jari Pakkanen (a1)


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