Acknowledgements. I am most grateful to Dr. A. M. Snodgrass for reading a preliminary draft of this paper. He has at many points improved its clarity, but is not, of course, responsible for any errors or inanities it may contain.

¹ Even Bundgaard's ploneering study of Greek design methods is based on a study of the Propylaia (J. Bundgaard, Mnesicles, A Greek Architect at Work (1957)).

² Vitruvius (De Arch. vii. praef. 12) refers to accounts of their work by sixth-century architects, but such accounts could have dealt only with construction, not with design. On the second hypothesis it is to be noted how similar in nature is the design of Arsenal, Philo's (IG ii ²1668)Google Scholar to that of Vitruvius' basilica (De Arch. v. 1. 6–10)—main measurements in round numbers of feet, with simple dimensions overriding simple proportions where necessary (virtually all dimensions expressible in quarter feet).

³ Note Rhys Carpenter's criticism of Hambidge's principles of dynamic symmetry, which were intended to explain the basis of all Greek design (AJA xxv (1921) 16–36).

⁴ The great recurring difficulty in any statistical assessment of probability is in finding sufficient buildings which one can reasonably assume to have been built according to the same rules of design.

⁵ On this, see J. Bundgaard, Mnesicles (1957). The conclusions of this paper tend to support Bundgaard's position, although I believe that many Greek architects had more interest in, and responsibility for, the aesthetic side of architecture than Bundgaard allows (ibid. 184).

⁶ De Arch. iii. 3. 7 (Ionic); iv. 3. 3, iv. 3. 7 (Doric); iv. 7. 2 (Tuscan); iv. 8. 1, iv. 8. 2 (Circular).

⁷ It is hoped to discuss these matters more fully in a later paper.

⁸ On proportion, Vitruvius, De Arch. iii–vi, passim; on dimension, De Arch. v. 1. 6–10; IG ii.² 1668, and elsewhere.

⁹ On this system: Heath, T., A History of Greek Mathematics i (1921) 30–1Google Scholar; BSA xviii (1911–12) 98–132; xxviii (1926–7) 141–57; xxxvii (1936–7) 236–57. Signs for fractions of a drachma or of an obol are not, of course, signs for fractions.

¹⁰ Heath, op. cit. i (1921) 41–2; examples of rounding off in fractional calculations can be found in [Heron], Geometrica, Stereometrica, and De Mensuris.

¹¹ De Arch. iii. 5. 8.

¹² On the use of the abacus see Herodotos ii. 36; Aristophanes, Vesp. 656; RE Supp. iii. 4–13. For the Salamis Table as a gaming board not an abacus see Hesperia xxxiv (1965) 131–40.

¹³ Dinsmoor, W. B., The Architecture of Ancient Greece, 2nd edn. (1960) 54Google Scholar n. 4. Id. in Atti del vii Congresso Internazionale di Archeologia Classica (1961) i. 360.

¹⁴ On changes in standard and variations from standard in Attic weights see M. Lang, M. Crosby, The Athenian Agora x, Weights, Measures, and Tokens (1964).

¹⁵ None has survived. On the accuracy of calibration of measuring-rods from Egypt see Petrie, F., Ancient Weights and Measures (1926) 38–40.Google ScholarMost Roman foot rules are divided into 16 or 12 parts, some only into two (cf. op. cit. 48–9). The accuracy of official length standards cannot be taken as typlcal of that of the measuring rods in actual use.

¹⁶ For ancient references to the use of models see Orlandos, A., Ta Ilika Domes tōn archaiōn Ellēnōn ii (1958) 268 n. 3Google Scholar; Bundgaard, J., Mnesicles (1957) 216–19Google Scholar, n. 217.

¹⁷ Measurements from the model would presumably be taken with dividers, not with a rule.

¹⁸ e.g. W. B. Dinsmoor, op. cit. 73 n. 3, 76, 77, 80, 89, 93, 98, 99, 101; Robertson, D. S., Greek and Roman Architecture (2nd edn., 1943) 75Google Scholar; Lawrence, A. W., Greek Architecture (2nd edn., 1967) 120, 122, 123, 127.Google Scholar

¹⁹ For this general rule cf. Hesperia ix (1940) 2 n. 7, 45.

²⁰ The following abbreviations for parts of buildings are used throughout this paper and the accompanying Tables:

AW Architrave thickness from back to front.

C Number of columns (= N+I).

D Lower diameter of column on arrises.

H Column height.

IAxial intercolumniation.

LLength over stylobate.

N Number of intercolumniations (= C−1).

OL Over-all length.

OW Over-all width.

S Stylobate breadth from back to front.

T Triglyph width.

W Width over stylobate.

Note that where the size of an element on the flank of a building differs from the size of the corresponding part on the front, the abbreviations referring to the front and flank are distinguished by W and L respectively, written as a subscript. Similarly where the size of an element near the corner of a building differs from that of the corresponding parts elsewhere, the part next to the corner is distinguished by A written as a subscript.

I_L Axial intercolumniation on flanks.

N_W Number of intercolumniations on fronts.

D_A Lower diameter of columns at corners.

l_WA Axial intercolumniation next to corners on fronts.

²¹ Strictly this should be twice the distance from the front edge of the stylobate to the axis of the colonnade, since the axis of the colonnade does not always coincide with the centre of the stylobate.

²² See Bundgaard, J., Mnesicles (1957) 134–6.Google Scholar

²³ A. W. Lawrence, op. cit. 116.

²⁴ W. B. Dinsmoor, op. cit. 93 comments ‘somewhat perversely’.

²⁵ A. W. Lawrence, op. cit., 127.

²⁶ W. B. Dinsmoor, op. cit. 161–2.

²⁷ Hesperia Suppl. iii (1940) 4; the entrance to most Greek sanctuaries was placed so that visitors got first an angle view of the temple, although that may have been for functional rather than aesthetic reasons; (Bergqvist, B., The Archaic Greek Temenos (Opuse. Ath. iv (1967) 13).Google Scholar

²⁸ The increase necessary, c. 0·10 m., would give the Parthenon columns a height of about 5·54 lower diameters, midway between the proportions of the east and the west porticoes of the Propylaia.

²⁹ See above, p. 63.

³⁰ De Arch. i. 1. 4; i. 2. 2.

³¹ See note 6 above.

³² The interior of a basilica presents a different problem from the pteron of a temple, of course, but it is interesting that the central space of Vitruvius' basilica at Fano is surrounded by 4 × 8 columns, while the ratio of its width to its length is 60:120 = 4:8; i.e. the length is related to the width by Rule 1 proposed below (p. 69).

³³ De Arch. iii. 5. 5; iii. 5. 8. (trans. M. H. Morgan).

³⁴ On the probability of a general lack of detailed preliminary planning in Greek architecture see J. Bundgaard, Mnesicles (1957), but the procedure adopted in this investigation does not demand such an assumption.

³⁵ See above, p. 61.

³⁶ For the exceptions: Dinsmoor, W. B., The Architecture of Ancient Greece (2nd edn., 1950) 337 n. 1.Google Scholar For the Temple of Askleplos at Epldauros, which was one of the exceptions, the figures are all taken from the recent study by G. Roux (see n. 37). It is unfortunate that in his most recent study of Greek temple design (Atti del vii Congresso Internazionale di Archeologia Classica (1961) i 355–68), Dinsmoor gives without explanation figures differing significantly from those given in The Architecture of Ancient Greece (2nd edn., 1950), which are used here.

³⁷ The data given by Dinsmoor is supplemented from the following sources: for the temples of Sicily and south Italy, R. Koldewey, O. Puchstein, Die griechischen Tempel in Unteritalien und Sicilien (1899); P. Marconi, Himera (1931); for Aigina, A. Furtwaengler, Aegina: Das Heiligtum der Aphaia (1906); for the Argive Heraion, C. Waldstein, The Argive Heraeum (1902–5); for Assos, F. H. Bacon, J. T. Clark, R. Koldewey, Investigations at Assos 1881–3 (1902–21); for Athens, T. Wiegand, Die archaische Poros-Architektur der Akropolis zu Athen (1904); Hill, B. H., AJA xvi (1912) 535–58CrossRefGoogle Scholar; Penrose, F. C., An Investigation of the Principles of Athenian Architecture (2nd edn., 1888)Google Scholar; R. Bohn, Die Propyläen der Akropolis zu Athen (1882); H. Koch, Studien zum. Theseustempel (1955); Dinsmoor, W. B., Hesperia ix (1940) 1–52CrossRefGoogle Scholar; for Bassai, C. R. Cockerel, The Temples of Jupiter Panhellenius at Aegina and of Apollo Eplcurius at Bassae … (1860); for Delos, , Délos xii (1931)Google Scholar; for Delphi, FdD (1915, 1923, 1933); for Eleusis, Society of Dilettanti, Unedited Antiquities of Attica (1817); for Epldauros, G. Roux, L'Architecture de l'Argolide aux iv^e et iii^e siècles av. J.-C. (1961); for Nemea, B. H. Hill, C. K. Williams, The Temple of Zeus at Nemea (1966); for Olympla, Curtius, E., Adler, F., Olympla: die Ergebnisse … 2 (1892)Google Scholar; for Pergamon, Altertümer von Pergamon ii and iii. I (1885, 1906); for Rhamnous and Sounion, Doerpfeld, W., AM ix (1884) 329–37Google Scholar, Plommer, W. H., BSA xlv (1950) 78–109Google Scholar; for Stratos, C. Plcard, F. Courby, Recherches archéologiques à Stratos d'Acarnanie (1924); and for Tegea, C. Dugas, J. Berchmanns, M. Clemmensen, Le Sanctuaire d'Aléa Athéna à Tégée (1924).

³⁸ For the accuracy of fit of this rule see Table 2, Cols. 2–4. The rule W:L= C_W:C_L seems to be implied by Krauss, F., Paestum, Die Griechischen Tempel (2nd edn., 1943) 34Google Scholar, but the idea is not followed up.

³⁹ MA xli (1951) 813: 22·50 × = 56·25; error = 0·05 m. The significance of this step is also suggested by the fact that the flank intercolumniation equals of the length of the temple at this level (56·30/17= 3·312 m.; I _L = 3·331 m.), while the normal front intercolumniation equals of the width at the same level (22·50/6= 3·75; I_W = 3·772 m.). These are much simpler proportions than those relating intercolumniation and stylobate size (cf. Table 1, Col. 21, 24).

⁴⁰ See also MA xiii (1902) pl. 18.

⁴¹ For the sources of these figures see p. 67 n. 37.

⁴² It has been often suggested that the colonnade of the Peisistratid Temple was constructed around a pre-existing cella (JHS lxxx (1960) 129–34). If so, the architect will not have been entirely free in his choice of stylobate proportions.

⁴³ For other dimensions of this temple expressed in terms of this foot see Table 3, Cols. 12–23.

⁴⁴ ft. would be a closer approximation, but both Philo and Vitruvius liked to have dimensions expressible in quarter feet, not smaller fractions. See p. 61 n. 2 above.

⁴⁵ Both these calculations can also be worked in reverse. If the length of the temple rather than the width was decided first: L = 170 ft.; 170/14 = —say 12; 12×6 = 72; W = 72 ft.; L = 190 ft.; 190/15 = —say ; × 6 = 75; W = 75 ft.

⁴⁶ Possible cases are the Older and later Parthenon (see below pp. 75–6).

⁴⁷ It is also unusual for Sicily in having enlarged front columns, although that was the normal mainland practice (Table 1, Cols. 10–11).

⁴⁸ See below, p. 80.

⁴⁹ See, for instance, D. S. Robertson, op. cit. 106–9. AW is used for the thickness of the architrave from back to front here, to avoid confusion with the architrave height.

⁵⁰ De Arch. iv. 3. 2.

⁵¹ See above, p. 72.

⁵² AJA xvi (1912) pl. 9. The over-all size of the whole platform was 31·39×76·816 m.—approximately 7:17 or (C _W+1): (C _L+1).

⁵³ See above, p. 66. The Temple of Zeus at Kyrene, also with 8×17 columns, has a stylobate with the same proportions (30–40×68·35 m.; 30·40 × 18/8 = 68·20 m.), but the angle contraction is here not so strong, and as a result the front and flank intercolumniations are different: W. B. Dinsmoor, The Architecture of Ancient Greece (2nd edn., 1960), 86; but according to Pesce (BCH lxxi–lxxii (1947–8) 319 pl. 56) although the columns are rather irregularly spaced there is no systematic difference between front and flank intercolumniations.

⁵⁴ W. B. Dinsmoor, op. cit., 2nd ed., 217.

⁵⁵ The effectiveness of the rule W= I_W(N_W+) in producing approximately the right amount of angle con traction depends very much on the relation of the stylobate breadth (S) to the axial intercolumniation (see p. 83 n. 64 below). If the stylobate is too narrow the angle contraction will be too little, or even non-existent. Since the lower column diameter must be closely tied to S, columns de signed for a temple where I_W = W/(N_W+) could not be used in a temple of virtually the same size where I_W = W/(N_W+). We must assume that the decision to reuse the old columns was taken after the lengthening of the foundations, but before the detailed setting out of the stylobate and column positions, i.e. before the construction of the visible krepls.

⁵⁶ The stylobate width of the Alkmaionid Temple was was probably much the same as that of the fourth-century temple, c 21·68 m., since the over-all width is virtually the same in both buildings; and if the stylobate was set back from the ends of the foundation by the same amount as at the sides, the stylobate length would be c. 57·38 m. Using the formulae I_W = W/(N_W+), I_L = L/(N_L+), we get I_W = 4·130 m., I_L = 4·027 m. Using the formulae I_W = W/(N_W+), IL = L/(N_L+), we get I_W = 4·065 m., I_L = 4–003 m. The estimate of the French publication, based on other evidence, is I_W = c. 4·104 m., I_L = 3·95—4–00 m. (FdD ii, Courby, F., La Terrace du Temple i (1915) 96–7Google Scholar).

⁵⁷ Dinsmoor suggests (The Architecture of Ancient Greece (2nd edn., 1950) 108–9) that the discrepancy arose from a last-minute decision to give the flanks double rather than single contraction. However, that does not explain why the total angle contraction on the flanks was increased, instead of the same amount being spread over two intercolumniations. For if the two intercolumniations nearest each corner on the flanks had been made equal to the two nearest each corner on the fronts, the discrepancy between the normal front and flank intercolumniations would have been only 0·011 m., rather than the actual 0·023 m.

⁵⁸ The advantage of the second formulation is that it explains the odd foot in the stylobate width, which must otherwise be put down to error in measurement. For by the first formulation, with I = 8 ft., the stylobate size ought to be ×8 by ×8 ft., that is 44×100 ft.

⁵⁹ The discrepancy could be reduced by calculating in feet of 0·326846 m. as follows: W = 75 ft.; 75/9 = —say ; ×20 = —say 166; L= 166 ft.

⁶⁰ see above, p. 71.

⁶¹ Krauss, F., Paestum, Die Griechischen Tempel (2nd edn., 1943) 62–3Google Scholar, suggests that the aim was to give the temple the proportion 1: at the level of the architrave taenia. This seems rather a strange place to embody such a proportion if it was meant to be appreciated by the onlooker. If the plan of a Doric temple is taken at every possible level, it will produce so many concentric rectangles of slightly varying proportion, that this result could easily be due to chance.

⁶² Cf. the Temples of Athena at Syracuse and ‘Victory’ at Himera and the Olympleion at Akragas, pp. 78–9 above.

⁶³ For the abbreviations used here and throughout this paper see n. 20.

⁶⁴ With no allowance for tilting the columns, this rule will give the correct angle contraction (I—I _A = AW— T/2) when S = I ₃+AW—T; that is, if AW = 2T and T = I/₅, then I must equal about 2D. The use of the same rule with three-metope spans, and therefore much smaller columns, in the Stoa at Brauron (I = c. 4·0 D) means that the angle intercolumniations are enlarged not contracted—conveniently, since the angles are re-entrant not external ones.

⁶⁵ This rule will give the correct angle contraction (I—I _A = AW—T)/2) when S = +AW—T; that is, if AW = 2T and T = , then I must equal about D.

⁶⁶ See p. 81 and n. 58 above.

⁶⁷ See, for example, the differences of opinion between he reviewer and authors in O. Broneer's review of B. H. Hill, C. K. Williams, The Temple of Zeus at Nemea (1966) in AJA lxxii (1968) 188–9.