CORNELL UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 189I BY HENRY WILLIAMS SAGE DATE DU| % 7194^ ^Mm't'mB§"& I 7 2005 NK4645 .cm" ""'™™"^ ^'*'""^ j o„„ 3 1924 030 679 280 Overs M XI Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924030679280 GEOMETRY OF GREEK VASES MUSEUM OF FINE ARTS, BOSTON COMMUNICATIONS TO THE TRUSTEES, V GEOMETRY OF GREEK VASES ATTIC VASES IN THE MUSEUM OF FINE ARTS ANALYSED ACCORDING TO THE PRINCIPLES OF PROPORTION DISCOVERED BY JAY HAMBIDGE BY L. D. CASKEY CURATOR OF CLASSICAL ANTIQUITIES BOSTON MCMXXII mRii^ (; / / Qi-MiJi COPYRIGHT, 1922, BY MUSEUM OF FINE ARTS, BOSTON ALL RIGHTS RESERVED NOTE Beginning in 1904 the Museum printed four volumes of Communica- tions, consisting of studies on problems which confronted the Trustees on the occasion of the erection of the present building, viz. : I. Papers on the building and installation of a Museum of Art, 1904. II. The Exhibition of Casts; Museum Methods, 1904. III. The Museum Commission in Europe, 1905. IV. The Experimental Gallery, 1906. These volumes have long been out of print. The Museum has met fre- quent demands for them by the use of lending copies, which are available for periods of three months or longer if not otherwise in demand. In resuming the series of Communications the Museum undertakes again the publication of studies by officers of the Museum on subjects connected with the branches of art or of Museum activity which the writers represent. A quarto form has been adopted as most convenient and most suitable for illustration. The series will not be a periodical pubUcation, but studies will be printed occasionally and without regard to their compass. CvD /^ PREFACE In Mr. Hambidge's book, Dynamic Symmetry: The Greek Vase, a remarkable connection is traced between the proportions of Attic vases and those of certain rectangles derived from the square by a simple process which was familiar to Greek geometers. Mr. Hambidge finds that if the over-all proportion of a vase, that is, the ratio existing between its height and greatest width, is expressible in terms of one of these rectangles, then the heights and widths of all its parts can be expressed in terms of that rectangle, and of no other. The vase possesses sym- metry in the sense that all its elements are commensurable, the common factor or coordinating principle being a rectangle whose sides illustrate one of the simple ratios 1 :\/l, 1 :\/2, 1 :\/3, 1 :\/5- Except for the first, these proportions are com- mensurable in square, but not in line ; they can be studied in terms of geometry, but not in terms of a,rithmetic. Having observed this phenomenon not only in Attic pottery and other prod- ucts of the minor arts having an architectonic character, but also in Greek temples, notably the Parthenon, Mr. Hambidge has advanced the theory that Greek artistic design was based on geometrical principles. In the absence of clear and reliable literary evidence this conclusion cannot be rigorously proved, or dis- proved. But the coincidences which Mr. Hambidge has brought to light are too striking to be ignored; and progress towards a solution of the problem they pre- sent can be made only by a thorough study of as many extant monuments as possible. In this book I have tried to present in an intelligible form all the evidence af- forded by the collection of Attic black-figured and red-figured pottery in the Museum of Fine Arts. With some exceptions, which are noted in the introduc- tion, I have investigated every example which lent itself to this sort of analysis, and have arranged the material in the form of a catalogue, in which every type of vase in the collection, except the rhyton, the phiale and the plate, is represented. The proportions are expressed in two ways — by drawings and by tables of ratios. The former are reproductions, usually much reduced, of full-size eleva- tions. More than three-fifths of the vases have been measured and drawn by my- self. The rest of the drawings have been executed by Mr. E. G. Go wen under my supervision; I have checked his measurements in every case, and am responsible for their accuracy. The outUne of each vase is enclosed in a rectangle belonging to one of the systems of dynamic symmetry, the margin of error allowed averaging less than one millimetre. The interrelation of details is shown by subdivisions of the containing rectangles and by intersections of diagonals. It has often proved impossible to analyse vases of complicated structure without using a confusingly [vii] PREFACE great number of lines. This might have been avoided by publishing several draw- ings of each vase, or by including only those proportions which could be most simply expressed. The former alternative has been rejected from motives of economy; the latter from a desire to achieve thoroughness, even at the expense of clearness. In a few cases I have shown some of the more striking coincidences by means of subsidiary diagrams. The text accompanying each drawing gives the chief dimensions of the vase, with a description and bibliography sufficient to identify it, a brief explaiiation of the geometrical analysis, and a table of ratios in which the proportions are expressed arithmetically — usually in the form of inde- terminate fractions carried to three or four decimal places. These ratios are given chiefly because of their practical usefulness as labels. In the introduction I have explained the rectangles used in the analyses, the methods of subdivision, and the resulting ratios. I have also submitted some statistics regarding the margin of error allowed, and the number of occurrences of the more important proportions. The title of the book was suggested by a remark of M. Edmond Pottier, which is quoted on page 32 of the introduction. Whether arithmetic or geometry can be of aid to the artist in achieving symmetry, i. e., harmony of proportions, is open to question. But, if either is to be used, there can be no doubt that geometry is the simpler instrument. Artists cannot dispense with the use of linear units for certain practical purposes. A few proportions, such as 1:1, 1:2, can be determined as easily by arithmetic as by geometry. But the use of numbers if carried farther, leads either to monotonous, stereotyped designs, or to hopeless confusion. For example, the skjrphos, no. 104, has dimensions which can be expressed by the numbers 8, 15, 11, 5, 7. It would take a Pythagoras to explain the proportional value of these numbers, whereas the geometrical analysis of the vase is readily understood and appreciated by a child. Many of the analyses in this book are equally simple; many others are much more compUcated. The cups, nos. 132 and 136, may be cited as extreme cases, the former telHng in favor of the theory, the latter against it. A judgment as to its probability or improbability must take account of all the examples. My own position may be stated baldly as follows: (1) The coincidences are in so many cases, so accurate, simple and logical that I find it less difficult to befieve them due, in part at least, to conscious design, than to instinctive obedience to a mysterious aesthetic law, or to mere accident. (2) The proportion obtained by dividing a line in extreme and mean ratio, which plays an important part in Euclidean geometry, has for ages been recognised as an ever-recurring phenomenon in artistic design. It has been called by various names — divine proportion, golden section, ratio of Phidias, and the like; and it has been studied in many ways. By considering it as an area, rather than as a divi- sion of a line, and by emphasising its relation to the VS rectangle, Mr. Ham- bidge has immensely simplified the problem of investigating its significance. He has not only made it easier to detect the occurrences of this proportion; he has revealed its more important function as a coordinating principle in designs in which it does not itself necessarily occur. For this reason alone Mr. Hambidge's work is of permanent value. C viii ] PREFACE A criticism of Dynamic Symmetry, published by Professor Rhys Carpenter in the American Journal of Archaeology, 'XXV, 1921, pp. 18-36, concludes that "a priori, the probabilities are all against its being true." A different explanation of the coincidences must therefore be sought. Mr. Carpenter falls back on the theory that Athenian potters made the dimensions of their vases conform to units of a foot-rule. This would be a simple and legitimate solution, if it could be sihown to fit the evidence satisfactorily. It was rejected by me (as it had been by Mr. Hambidge) after a more thorough study than appears to have been devoted to it by Mr. Carpenter. A reply to his article, which I prepared at Mr. Ham- bidge' s suggestion, has been declined, unread, by the editor of the Journal. Since access to this periodical has been denied me, I have revised the concluding para- graphs of the following introduction in the hope of throwing some light on the question at issue. I desire to express my thanks to the Trustees of the Museum for publishing this book in the series of Communications, and among them especially to Dr. Denman W. Ross for his interest in the work, and for furnishing the services of a draughtsman. While the investigation was in progress I had the privilege of consulting Mr. Hambidge almost daily. The determination of the over-all proportions of the vases is due to him in most cases, the geometrical analysis of details in many cases. I am indebted to him above all for explaining his theory to me in all its phases, and for giving me access to his vast collection of unpubMshed material — the result of minute and laborious researches carried on for many years. That Mr. Hambidge possesses the imaginative insight to construct daring and sugges- tive hypotheses is apparent to all who read his book; only those who have worked with him can fully appreciate his industry in collecting facts, his accuracy in recording them, his cautious and critical attitude towards them as evidence. Cix] CONTENTS Preface . . vii Introduction 1 Analyses of Vases: Amphora (1-44) 35 Pelike (45-50) 83 Stamnos (51-56) 91 Hydria-Kalpis (57-69) 101 Deinos-Krater (70-83) 116 PSYKTER (84) 131 Oinochoe-Olpe (85-100) 132 Skyphos (101-117) 148 Cup with Impressed Decoration (118) 159 Kantharos (119-123) 160 Kylix (124-162) 167 Lekythos (163-180) 209 Pyxis (181-182) 226 Perfume Vase (183-185) 230 [xi] GEOMETRY OF GREEK VASES INTRODUCTION No attempt is made in this book to give a new statement, a r4sum6, or a criti- cism of Mr. Hambidge's revolutionising theory in regard to Greek design. Study of his book, Dynamic Symmetry: The Greek Vase, and of his periodical, The Diagonal, is indispensable to the understanding of the principles of proportion which he has discovered. In the present work these principles are accepted as a starting point and apphed to the examples of Attic pottery in the Museum. All that is required by way of introduction is an explanation, in as simple language as possible, of the rectangles which are used in analysing the vases. The first point to be grasped — and its importance cannot be overemphasised — is that Mr. Hambidge has opened up an entirely new method of approach to the problem of determining the proportions of Greek works of art. And this method, though at first sight strange to us of the modern world, is just the one which the Greeks might natxirally be expected to have employed. Since the time of Vitruvius attempts to analyse Greek proportions have generally been based on units of linear measurement. And every one will admit that the results ob- tained — ■ whether the unit experimented with was a Greek foot, or a modulus taken from the object analysed — have been meagre, and for the most part un- convincing. What the investigators have failed to take into account is that the science of numbers was still in its infancy during the culminating period of Greek art, whereas geometry was a highly developed science long before the days of Euchd. If the Greeks consciously employed any system of proportions, it is a priori more probable that they based it on relations of areas rather than on relations of fines. In other words they would have used geometry rather than arithmetic. Having realised this Mr. Hambidge made the startUng discovery that the proportions found in Greek works of art, which are in perhaps nine cases out of ten incommensurable in terms of a Unear unit, or modulus, can in the large majority of cases be accurately and intelligibly expressed in terms of areas possess- ing certain clearly definable properties. Nine (and possibly a few more) vases studied in this book have proportions which can be expressed exactly in whole numbers. The two deinoi (nos. 70, 71), for example, have a relation of height to diameter of 4 to 5. It is obvious that this proportion can also be expressed geometrically by enclosing the vases in a rec- tangle with sides in the ratio 4 : 5. Such a rectangle is made up of twenty squares. And the proportions of details can be expressed in simple subdivisions of these squares. The two vases illustrate what Mr. Hambidge has called static symmetry. [1] GEOMETRY OF GREEK VASES A more complicated example of static sylnmetry is furnished by the lekythos, no. 167. Its height equals 2 J times its diameter. If the diameter is regarded as unity the proportions are as follows (column A) : A B Height 2i 150 units Height to shoulder 1| 105 " Height of lip, neck and shoulder | 45 " Height of lip w 18 Height of neck lo 18 Height of shoulder ii 9 Diameter 1 60 Diameter of lip s 36 Diameter of bottom of body i 20 Diameter of foot i 45 " In order to express all these proportions in whole numbers the diameter of the vase must be divided into no less than 60 units. The proportions in terms of these units are given in column B in the table. It is of course inconceivable that a potter should have used a unit 2 mm. long in working out the proportions of a vase, or that he should have used the diameter as a unit, and have divided it into halves, thirds, quarters, fifths, tenths, and twentieths. The alternatives are to suppose that these relations of part to whole and of part to part were arrived at unconsciously, or that they were worked out by a geometrical construction similar to that shown on page 213. Whichever hypothesis is preferred, the fact remains that the proportions of the lekythos can be more clearly and simply expressed in terms of squares than in terms of units of length. The vast majority of vases investigated have proportions which cannot be accurately expressed either in linear units or in squares, but which can be clearly analysed in terms of rectangles derived geometrically from the square. These ' \ ' ' \ - ' ' \- ' ' ' ■ 1.000 ■ V2- A/3- V4 V5 Diagram I. The root rectangles [2] INTRODUCTION vases illustrate what Mr. Hambidge has called dynamic symmetry. The rec- tangles of dynamic symmetry fall into two classes: (1) Those derived from the diagonal of a square; (2) those derived from the diagonal of two squares. The second class is of course related to the first, but possesses some remarkable quali- ties which justify its being treated separately. The first class is composed of the "root-rectangles" and their derivatives. The root-rectangles are those whose short side is unity and whose long side is equal to \/2, VS, -\/4, VS, etc., respec- tively. A simple method of constructing these rectangles is shown in diagram I. The diagonal of a square whose side is unity equals y/2, which can be expressed approximately by the irrational fraction 1.4142 A ■\/2 rectangle is con- structed by using the diagonal of the generating square as the radius of a circle. This circle cuts the base of the square produced so as to fix the length \/2. A per- pendicular from this point to the top of the square produced completes a ■\/2 rec- tangle. The diagonal of a \/2 rectangle equals -\/3 (1.732 ....). The diagonal of a -\/3 rectangle equals -v/4 (2.000). The diagonal of a ^4 rectangle equals \/5 (2.236 ....), and so on. Each root-rectangle is constructed in the same way from the previous one. For the purposes of this investigation the \/4 rectangle may be disregarded since it is composed of two squares. The root-rectangles beyond -s/b are also omitted from consideration. Of the vases here published nine or more are based on the square, eighteen on the \/2 rectangle, six on the \/3 rectangle. The remainder, some 136 in all, have proportions based on the \/5 rectangle, or on a rectangle intimately related to it, which remains to be described. It will readily be seen that the \/5 rectangle is derived from the diagonal of two squares, since \/4 = 2. Diagram II shows a second method of describing the \/5 rectangle. If a square with sides equal to unity is bisected vertically, each half is composed of two squares with sides equal to | or .500. The diagonal of half the larger square may thus be regarded as the diagonal of two squares, and its length is -— , or 1.118 A semicircle with the centre of the base of the -v/S • • large square as centre and the diameter of half that square, -^, as radius will Diagram II. The V 5 rectangle [3] GEOMETRY OF GREEK VASES have a diameter equal to -\/5. And a rectangle with that diameter as base and unity as height will be a \/5 rectangle. This rectangle is composed of a square •\/5 — 1 flanked by rectangles with unity as height and — - — as width. This may also , 2.236-1.000 „.^ be expressed as , or .olo At this point it is convenient to consider the methods of obtaining the recip- rocal of a given rectangle. The reciprocal is a rectangle of the same shape as the original rectangle, and with its long side equal to the short side of the original rectangle. A reciprocal may be obtained by drawing a diagonal of the rectangle and erecting a perpendicular to the diagonal at one of its ends. The point of inter- section of this perpendicular with the opposite side of the rectangle produced determines the width of the reciprocal. In diagram III the rectangle 5 C is the DiAGEAMS III AND IV. Reotangles and their reciprocals reciprocal of the rectangle A B. In this case the reciprocal is added to the end of the rectangle. A reciprocal may be cut off from, or, as the Greeks expressed it, applied to a rectangle, by drawing the diagonal of the rectangle, and dropping a perpendicular upon it from one of the opposite angles. This perpendicular pro- duced intersects the opposite side so as to determine the width of the reciprocal. In diagram IV the rectangle C D is the reciprocal of the rectangle A B. The root-rectangles and their reciprocals have a relation to one another which is of importance in the geometric analysis of proportions based on these rec- tangles. If a reciprocal is applied to a \/2 rectangle, the remaining or excess lily \ V3 X \ V4 M^^|^^ \ V5 \ V5 J^VS'^S K \ -V2- -V3- ■A/5- DiAQBAM v. The root rectangles and their reciprocals area is also a reciprocal of the rectangle. In other words the \/2 rectangle is composed of two •\/2 rectangles. Similarly the VS rectangle is composed of three V3 rectangles, the V4 rectangle of four \/4 rectangles, the \/5 rectangle of five \/5 rectangles, and so on. Cf. diagram V. Returning to the \/5 rectangle subdivided, as in diagram II, into a square a/5 — 1 flanked by two rectangles of the proportions 1:^^--^^ — , or 1: .618, it is obvious [4] INTRODUCTION that in diagram VI the diagonal of the area A+B meets the diagonal of the area C at a right angle, since all angles in a semicircle are right. The rectangle C is thus seen to be a reciprocal of the rectangle A -\-B. The rectangle A, being equal to C, is also a reciprocal of the rectangle A-\-B. The rectangle A-\-B is thus seen to have the interesting property that, if a reciprocal {A) be applied to it, the excess area {B) is a square. A reciprocal applied to the reciprocal again leaves a square, and so on. If this process is continued indefinitely a series of squares is produced, DiAGBAM VI. The V 5 rectangle and the rectangle of the whirling squares which continually decrease in size and revolve about the point of intersection of the diagonal with its perpendicular in a logarithmic spiral to infinity. Cf. dia- gram VII. Because of this property Mr. Hambidge has named this rectangle the rectangle of the whirling squares. The appUcatibn of the reciprocal to a whirling square rectangle divides the base of the rectangle in extreme and mean ratio. That is, the smaller part is to the larger part as the larger part is to the whole line. .618 :1.000 = 1.000 :1.618, or expressing it in terms of -y/S : — ^ — : 1 = 1 V5 + 1 This division of a line is of 2 2 course the well-known divine section or divine proportion of Paccioli, Kepler, and Leonardo da Vinci, more generally called in recent years the golden section. Eleven vases published in this book are contained in the whirling square rec- tangle, i. e., their diameter is to their height in the ratio of 1 :1.618. Cf. diagram s ^^ V y^^ 5\ 1.000- .618- 1.618- DiAGEAM VII. The rectangle of the whirling squares [5] GEOMETRY OF GREEK VASES VIII in which ten of these vases are shown. In eight cases the rectangle encloses the complete vase ; in three it contains the vase without its handles. The eleventh example, no. 93, is omitted because it is practically a duplicate of the oinochoe no. 92. The whirling square rectangle is one of the most common but by no means a predominating shape. Its aesthetic significance as Mr. Hambidge has observed, lies rather in its value as a coordinating factor. A very large proportion of the vases studied are contained in rectangles derived in simple ways from the whirl- ing square rectangle. Some of these will be described below. In many examples ^ J ^ h 1 1 17 86 87 92 97 Diagram VIII. Examples of the rectangle 1.618 = .618 185 the ratio .618 occurs as the proportion of some detail. About thirty instances of this have been noted, and it is probable that a more exhaustive study would reveal stiU others. But the occurrences of this rectangle as the containing area of the whole or a portion of a vase, or as the proportion of a detail by no means exhaust its possibilities. It can itself be subdivided in various ways, producing new proportions related to the generating form, which appear repeatedly in the analyses of the vases based on this system. Some of the more common subdivisions of the whirhng square rectangle are illustrated in the following dia- grams by simple geometrical constructions. They should be studied in connec- tion with the numerical ratios, which will be found extremely useful, if not indispensable, in memorising them. Diagram IX represents a whirling square rectangle placed horizontally, the width being regarded as unity, the height as .618. By applying squares at either end, and by applying squares at the bottom of each reciprocal the area may be divided horizontally into two rectangles, the upper one of which is composed of a square flanked by horizontal whirUng square rectangles, while the lower one is composed of a vertical whirUng square rectangle flanked by squares. The height of the upper rectangle is .236, that of the lower .382. The former is the reciprocal [6] INTRODUCTION of 4.236, the latter of 2.618. All these ratios — .236, .382, .618, 1.000, 1.618, 2.618, 4.236 — are terms in a continuous proportion. As many as forty-five occurrences of the proportion .382 have been noted in this book. The rectangle .382 = 2.618 will be further considered below. The ratio .236 occurs at least thirty-one times. In diagram X a, perpendiculars dropped from the centres of the applied squares divide the base into the ratios .309 -|-. 382 +.309. Perpendiculars from the centres T -.382- -.236- -.382- DiAGBAM IX. Subdivisions of the whirling square rectangle of the reciprocals divide the base into .191+.618-1-.191. Simpler geometrical expressions of these ratios are given in diagram X, b and c. It is worth noting that .309 is half of .618 and that .191 is half of .382. In diagram XI the process of applying squares is carried a step farther, and produces the ratios .236-1-. 528+. 236, and .146+.708+.146. Many of the vases analysed show the proportions .528 and .708. The latter equals .236X3. The ratio .5669 which occurs several times is obtained from the intersection of the diagonal of half a whirling square rectangle with the diagonal of the reciprocal, as shown in diagram XII. Diagram XIII shows the relation of the important ratio .4472 (the reciprocal of -v/5, or 2.236) to the whirling square rectangle. If a horizontal line is drawn A i \ x X, f\ / s w 5 S S w s w w w ' ^.309- -.382- -.' '\ ^ a I _/. ,-"'■•. *^ -.l4«-«- -.706- -X-.I44- .5669 DiAGEAMS XI AND XII. Subdivisions of the whirling square rectangle ratios .2764 + .4472 + .2764. The diagram also accounts for the ratios .1382, .7236, and .1708. The ratio .4472 occurs at least thirty times in the vases analysed. In diagram XIV the \/5 rectangle is shown with two whirling square rectangles in the centre and half a square on each side. This suppHes the ratios .2236 and .5528. These examples will suffice to show the importance of the whirling square rectangle as a containing area, and as a coordinating factor for determining other toi - *— .2744 — » .4472 «— .2764 — » *.f3az<« .7236 fXiVir* DiAGBAM XIII 5 5 W W 5 5 HlZit—— .5528- -*-:2236- DlAGKAM XIV w w DiAQEAM XV. The rectangle 1.236 = .809 rectangles and proportions occurring repeatedly in Attic vases. It remains to consider some of the more common areas derived from the whirUng square rectangle and the V5 rectangle. [8] INTRODUCTION Perhaps the commonest shape of all is that composed of two whirling square rectangles placed side by side, which can also be regarded as half a whirling 109 110 111, 112, 113, 115 120 Diagram XVI. Examples of the rectangle 1.236 = .809 185 square rectangle. Cf . diagram XV. The ratio of this rectangle is 1.236, or .618 X 2. Its reciprocal is .809, or 1.618-^2. Diagram XVI shows some of the vases which are contained in this area. Cf. also the pelike no. 46, the red-figured skyphoi nos. Ill, 112, 115, the black-figured amphorae nos. 18, 19 (up to the shoulder), the stamnos no. 50 (up to the shoulder), the stamnos no. 56 (omitting the handles), the black-figured krater no. 73 (omitting the handles), the bowl of the same krater, the red-figured krater no. 76 (up to the hp), twenty-one examples in all. The ratio .809 also occiu-s a number of times. 104 108 fl ^ ~^ J V 109 183 131 Diagram XVII. Examples of the rectangle 1.854 = .5394 The rectangle composed of three whirling square rectangles placed side by side (ratio: 1.854; reciprocal: .5394) occurs seven times. Cf. diagram XVII, and nos. 37 and 88. [9] GEOMETRY OF GREEK VASES The rectangle 2.854 contains the black-figured kylix, no. 132, with its handles, and the red-figured kylix, no. 161, without its handles. The latter, with its handles included, fits the rectangle 3.854. Four whirling square rectangles placed side by side make an area with the ratio: 2.472 (reciprocal: .4045). The two red-figured kyUkes, nos. 140, 141, are contained in this shape, if the handles are omitted. Cf. diagram XVIII. ^ 1- K ^ .. ^ 140, 141 DiAGBAM XVIII The rectangle 2.472 = .4045 138, 139, 140 DiAGBAM XIX The rectangle 3.090 = .3236 141, 142, 143, 144, 145, 146, 147 Diagram XX. The rectangle 3.236 = .309 The area composed of five whirling square rectangles placed side by side (ratio: 3.090; reciprocal: .3236) occurs as the over-all shape of the three red- figured kylikes, nos. 138-140. Cf. diagram XIX. The areas just described are obtained by placing vertical whirling square rec- tangles side by side. If two horizontal whirling square rectangles are placed side 47 68 92, 93 117 DiAGBAM XXI. Examples of the rectangle 1.309 = .764 by side the resulting area has the ratio 3.236 (reciprocal : .309). This is one of the two most common containing areas of red-figured kyhkes. Cf . diagram XX, nos. 141-147. This rectangle added to a square produces an area with the ratio 1.309 (recip- rocal: .764 = .382X2), which is found several times. Cf. diagram XXI. The same rectangle added to three squares (3.309) produces the containing area of the lekythos, no. 175. [10] INTRODUCTION Three pairs of whirling square rectangles placed one above the other produce an area with the ratio 1.0787, more simply expressed in terms of its reciprocal, .927 = .309X3. Three examples are shown in diagram XXII. The hydria no. 55 and the kalpis no. 65 are also perhaps contained in this shape. The ratio .382, which is the reciprocal of 2.618, was noted above as being one of the most common. Its use as a containing rectangle seems to have been com- 51 55 66 Diagram XXII. Examples of the rectangle 1.0787 = .927 paratively rare. Four red-figured kylikes without their handles (nos. 145-147, 156, and the lid of the pyxis, no. 181) illustrate this shape. Cf. diagram XXIII. This ratio (.382) combined with .809 produces the rectangle 1.191 which occurs at least three times. Cf. diagram XXIV, and no. 51. The ratio .382 added to a square produces one of the most important areas of dynamic symmetry — the rectangle 1.382 (reciprocal: .7236). It has an intimate r—^ — s rx \ A 145, 146, 147, 155 181 Diagram XXIII. Examples of the rectangle 2.618 = .382 relation to the equally important area 1.4472, composed of a square and a s/b rectangle, which will be discussed below. Two black-figured amphorae, nos. 4, 5; a pelike, no. 49, a kalpis, no. 69, a black-figured skyphos, no. 104, a kantharos, no. 123, two black-figured kyUkes, nos. 130, 131, and a black-figured perfume vase, no. 183, illustrate this shape. Cf. diagram XXV. 181 51 Diagram XXIV. Examples of the rectangle 1.191 = .8396 [11] GEOMETRY OF GREEK VASES 131 183 Diagram XXV. Examples of the rectangle 1.382 = .7236 The rectangle 2.382 is found as the containing area of two kyhkes without their handles (nos. 137, 139). Two others, including their handles (nos. 155, 156), are enclosed in the rectangle 3.382. The ratios .382 and .500 (half a square), added together, make with unity a rectangle with the ratio 1.1338, which is more easily intelligible in terms of the reciprocal .882. Two kraters, nos. 79, 80, are contained in this rectangle. Cf. diagram XXVI. DiAGBAM XXVI. Examples of the rectangle 1.1338 = .882 The VS rectangle occurs only three times as a containing area among the vases analysed in this book. These are the kyhkes nos. 132, 138, without their handles, and the black-figured lekythos, no. 163. In both the kyhkes the diam- [12] INTRODUCTION eter of the foot equals the height of the vase. In other words, it is exactly con- tained in the central square of the \/5 rectangle. Cf. diagram XXVII. Two \/5 rectangles placed vertically side by side constitute an important Its ratio is 1.118 (2.236 -f- 2). Its reciprocal is .8944 (.4472X2). Seven area. 13S 163 DiAGBAM XXVII. Examples of the V 5 rectangle (2.236 = .4472) examples are illustrated in diagram XXVIII. They are the stamnos, no. 51, the kalpis, no. 67, the black-figured krater, no. 73, the red-figrxred olpe, no. 100, the black-figured skyphos, no. 108, the red-figured skyphos, no. 116, the red-figured kantharos, no. 121. The bowl of the black-figured skyphos, no. 104, is enclosed in three \/5 rec- tangles placed side by side (ratio: 1.3416); and the same vase, with its handles. 108 116 121 DiAGBAM XXVIII. Examples of the rectangle 1.118 = .8944 is contained in an area made up of four \/5 rectangles (ratio: 1.7888). The skyphos, no. 107, also has the ratio 1.7888. The area made up of a square plus a \/5 rectangle, which has been foimd by Mr. Hambidge'to be the basis of the proportions of the Parthenon, occurs at [13] GEOMETRY OF GREEK VASES least seven times among the vases studied in this book. Its ratio is 1.4472 (recip- rocal: .691). The examples illustrated in diagram XXIX are two black-figured amphorae, nos. 7, 8, a red-figured hydria, no. 63, a black-figured deinos and 118 123 129 Diagram XXIX. Examples of the rectangle 1.4472 = .691 stand, no. 72, a cup with impressed decoration, no. 118, a kantharos, no. 123, and a black-figured kylix with jts handles omitted, no. 129. One half of 1.4472 equals .7236. This is the reciprocal of 1.382, showing that the 1.382 rectangle is composed of two 1.4472 rectangles. Cf. diagram XXX. The relation appears also in thte following equation: 1 /r.. 1-382 1:4472 = -^^^ = -^ It is noteworthy that one of the two amphorae signed by Amasis, which are technically among the finest pieces in the collection, is enclosed in the rectangle 1.382 (no. 4), while the other (no. 7) is contained in the rectangle 1.4472. W 5 s V5 s V5 Diagram XXX. Relation between the rectangles 1.4472 and 1.382 Two 1.4472 shapes, placed one above the other, have the ratio 2.8944 (recip- rocal: .3455 = .691-f-2). Two lekythoi, nos. 166, 170, exhibit this proportion. The lekythoi are illustrated in diagram XXXI. The lid of the stamnos, no. 51, is also contained in this rectangle. [14] INTRODUCTION If a square is subtracted from this rectangle, the ratio becomes 1.8944. The reciprocal of this is .52787, generally given as .528 in this book. A skyphos, no. Ill, and two lekythoi, nos. 165, 179, fit this rectangle. The lekythoi are illus- trated in diagram XXXI. Several areas which are close to a square are sometimes difl&cult to distinguish from one another, though in most examples a thorough study of the details V5 a/5 DiAGBAM XXXI. Examples of the rectangles 2.8944 and 1.8944 establishes clearly the ratio of the over-all rectangle. The shapes 1.0787 (.927), 1.118 (.8944) and 1.1338 (.882) have already been discussed. A fine red-figured kalyx krater, no. 76, is contained in the rectangle 1.090, or, more accurately, 1.0902, which is made up of .618 and .472 (better: .4722). A stamnos, no. 56, and a colimm krater, no. 74, are enclosed in the rectangle 1.0225 (.618-I-.4045), the reciprocal of which is .978. The analysis of the stamnos is unusually clear. The shape 1.0557 is more easily comprehended in terms of its reciprocal, .9472, which is made up of .4472 and .500 — a •\/5 rectangle added to half a square. n f^ i\ /;/ ^ ^ r 122 Diagram XXXII. Examples of the rectangle 1.0557 = .9472 A stamnos, no. 52, and a kantharos signed by Hieron, no. 123, are contained in this area. Cf. diagram XXXII. A ■\/5 rectangle (.4472) combined with a whirling square rectangle (.618) make the area 1.0652 (reciprocal: .9388) which is illustrated by the kalpis, no. 64, and the olpe, no. 98. The red-figured oinochoe, no. 89, is enclosed in the rectangle [15] GEOMETRY OF GREEK VASES 2.0652. The ratio appears also in the Corinthian skyphos, no. 103, and the black- figured kylbc, no. 125. Nos. 64, 89, and 98 are shown in diagram XXXIII. The area 1.1708, more easily recognisable in terms of the reciprocal, .854, occurs several times, but the analysis of this rectangle is unusually difficult. The ratio .854 equals (.618X3) -1.000, or .618+.236. This is the containing area of the kantharos, no. 120, and probably of two or three other vases in the collection. 64 89 98 Diagram XXXIII. Examples of the rectangles 1.0652 and 2.0652 Two other shapes which are close to a square — 1.0355 and 1.0606 — are based on the \/2 rectangle, and will be considered below. The area 2.472 (.618X4) was mentioned above as the containing shape of two red-figured kylikes. If a square is subtracted, the rectangle 1.472 (reciprocal: .6793), is obtained. It is composed of a square plus two .236 ( = 4.236) rectangles, and can be subdivided in various ways. Five black-figured amphorae, nos. 9-13, a black-figured oinochoe, no. 85, a Corinthian kylix, omitting its handles, no. 103, 13 184 85 Diagram XXXIV. Examples of the rectangle 1.472 = .6793 and a perfume vase of the black-figured period, no. 184, illustrate this shape. Cf. diagram XXXIV. Another favorite area is the rectangle 1.528 (reciprocal: .6545). This may be subdivided in various ways. In diagram XXXV, figure a shows the area divided [ 16] INTRODUCTION into four .382 ( = 2.618) shapes. 1.528 -4- 2 = .764. This is the reciprocal of 1.309: of. diagram b. The two 1.309 rectangles can be further subdivided as shown in diagram c. The division 1.000+.528 is shown in diagram d. The ratio .528 is the reciprocal of 1.8944, and is composed of a square plus two -\/5 rectangles. A further subdivision is shown in diagram e. A black-figured amphora, no. 15, a red-figured amphora, no. 40, a hydria with its handles omitted, no. 63, a red- 5 W 5 5 W 5 5 W 5 5 W 5 5 W W 5 w w w w V5 W W 5 w w 5 W W V5 V5 V5 V5 5 ^5 5 V5 vS 5 V5 S V5 5 V5 Diagram XXXV. Subdivisions of the rectangle 1.528 = .6545 figured oinochoe, no. 95, and a Corinthian skyphos, no. 102, are enclosed in this area. The rectangle 2.528 is illustrated by the kylikes, nos. 142, 143; the rec- tangle 3.528 by the kyUx, no. 157. The rectangle 1.809 (reciprocal: .5528), made up of a square plus half a whirl- ing square rectangle, is the containing area of four Nolan amphorae, nos. 33-36, and two red-figured skyphoi, nos. 115, 117. Cf. diagram XXXVI. This area with a square added, i. e., 2.809, encloses two red-figured lekythoi, nos. 168, 169. The generating form of all the rectangles — the square — occurs less often than might be expected. Nine examples are illustrated in diagram XXXVII as follows: two stamnoi, nos. 52, 53; a red-figured hydria, no. 63; a red-figured 35 117 DiAGBAM XXXVI. Examples of the rectangle 1.809 = .5528 kalpis, no. 64; a black-figured krater, no. 73; a red-figured krater, no. 77; a red- figured oinochoe, no. 92; a cup with impressed decoration, no. 118; a polychrome pyxis, no. 182. The oinochoe, no. 93, is omitted because it is a duplicate of no. 92. The small olpe, no. 99, brings the total number of examples up to eleven. It is [17] GEOMETRY OF GREEK VASES probable that many more "hidden squares" would be revealed by further study. Seven of the eleven examples have proportions based on the whirling square rectangle (nos. 52, 63, 64, 73, 92, 93, 118). The proportions of the krater, no. 77, and of the pyxis, no. 182, are in terms of y/2. The stamnos, no. 53, is an example of static symlnetry. It is noteworthy that in this example not only the over-all rectangle, but also the rectangle containing the vase up to the shoulder and omitting the handles is a square. The \/4 rectangle, composed of two squares (ratio: 2.000; reciprocal: .500) occurs three times as the containing area of a complete vase, and six times as the I . , => m (^ \V\ rii M Ml V J 77 92 Diagram XXXVII. Examples of the square 118 containing area of an important part of a vase. The examples are : the kantharos, no. 122 (to lip); the black-figured kyUkes, nos. 128, 129; the black-figured lekythos, no. 164; the red-figured lekythoi, nos. 168, 170, 171, 174 (to shoulder, in each case); the black-figured perfume vase, no. 183 (without its foot). Cf. diagram XXXVIII. Nos. 128 and 164 are examples of static symmetry. The others have proportions bafeed on the whirling square rectangle. The ratio .500 occurs very often among the proportions of details. The \/9 rectangle, made up of three squares, ratio : 3.000; reciprocal: .333, is found seven times among the vases studied. The examples are: the loutrophoros, no. 43; the red-figured kylikes, nos. 135-137; the red-figured lekythoi, nos. 172- 174. Cf. diagram XXXIX. All these examples have the proportions of their details in terms of the whirhng square rectangle. The rectangle composed of a square and a half (ratio: 1.500; reciprocal: .666) encloses several black-figured amphorae with apparently "static" proportions. [18] INTRODUCTION The black-figured kylix, no. 128, without its handles, is contained in the same area; its proportions are also static. The midget pointed amphora, no. 44, up to its shoulder, is another example. The proportions of this are in terms of \/2. Two hydriae, nos. 61 and 62, without their handles, are also enclosed in this shape. 129 164 Diagram XXXVIII. Examples of the V 4 rectangle (2.000 = .500) The red-figured lekythos signed by the potter Gales, no. 167, is contained in a rectangle composed of two and a half squares (ratio: 2.500; reciprocal: .400). Its proportions are static throughout (cf. above, page 2). Consideration of the occurrences of squares and multiples of squares in the vases here published has brought us to the subject of static symmetry, about ' 43 135 174 Diagram XXXIX. Examples of the V 9 rectangle (3.000 = .333) which Uttle need be said, because of the scanty and unsatisfactory material offered in this book. Nine vases have proportions which can be expressed in simple fractions as well as in geometrical constructions (cf. the table below, page 27). The black-figured amphorae nos. 23-25, on the other hand, though their [19] GEOMETRY OF GREEK VASES diameter is to their height as 1 is to 1.5, have proportions which I have been un- able to express intelUgibly either in areas or in linear units. Certain "static" ratios, especially .500, and, less frequently, .333, 1.000, .750, .666, .250, occur in vases belonging to the "dynamic" class. It remains to examine the vases with proportions based on the \/2 and \/3 rectangles. The \/2 rectangle (ratio: 1.4142; reciprocal: .7071) occurs six times as a containing area, the examples being a black-figured amphora, no. 22; a red- figured pelike, no. 50; two red-figured oinochoai, nos. 90, 96; a black-figured 90 96 105 Diagram XL. Examples of the V2 rectangle (1.4142 = .7071) skyphos, no. 105, omitting its handles. The black-figured hydria, no. 57, up to its shoulder, is also contained in a ■\/2 rectangle. Cf. diagram XL. Eight areas derived from the y/2 rectangle are of sufficient interest to be mentioned here. Three \/2 rectangles placed vertically side by side compose an area with the ratio 2.1213 (reciprocal: .4714). A fine black-figured kylix signed by Tleson, no. 124, and the lid of the polychrome pyxis, no. 182, illustrate this shape. Cf. diagram XLI. If this rectangle is bisected vertically each half has the ratio 1.0606 (recip- rocal: .9428), and is composed of six V2 rectangles placed in two rows of three each. This is the area in which the black-figured hydria, no. 57, is enclosed, if the projection of the vertical handle above the lip is omitted. Cf . the small diagram C on page 103. If two squares are added, the ratio just mentioned is changed to 3.0606. This is the rectangle enclosing the kylix decorated by Oltos, no. 134. [20 ] INTRODUCTION A square with a \/2 rectangle added has the ratio 1.7071 (reciprocal: .5858). The red-figured lekythos, no. 178, is enclosed in this area. Two squares plus a \/2 rectangle make an area with the ratio 2.7071 (recip- rocal: .3693). Three red-figured kylikes without their handles, nos. 151, 152, 154, are placed in this rectangle. One of the most interesting areas based on the -\/2 rectangle is the rectangle 1.2071 (reciprocal: .8284), which is composed of a \/2 rectangle plus half a square (.7071 + .500). The black-figured amphora, no. 2, the black-figured hydria, no. 57, with its lateral handles omitted, and the polychrome pyxis, no. 182, are con- 124 182 Diagram XLI. Examples of the rectangle 2.1213 = .4714 V2 2 57 Diagram XLII. Examples of the rectangle 1.2071 = .8284 182 tained in this area. Cf . diagram XLII. (Cf. also the pelike no. 45 and the hydria no 61.) This shape with a square added (ratio: 2.2071; reciprocal: .4531) encloses the small, pointed amphora, no. 44. The ratio .2071, which is the reciprocal of 4.8284, represents an area made up of two squares and two -\/2 rectangles. If five of these areas are placed vertically side by side, the resulting rectangle has the ratio 1.0355 (.2071 X 5), the reciprocal of which is .9657. Cf. diagram XLIII. Th'e black-figured hydria, no. 57, is contained in this shape. A rectangle made up of a y/2 rectangle and two squares (ratio: 3.4142; recip- rocal: .2929) contains the kylikes, nos. 148-152, with their handles included. Without their handles most of these kylikes have the ratio 2.7071, mentioned above. One of them, no. 148, has the ratio 2.4714, which can be obtained by subtracting .9428, the reciprocal of 1,0606 from 3.4142. The ratio .4714 is the reciprocal of 2.1213, or three \/2 rectangles placed vertically side by side. Cf. diagram XLI. [21] GEOMETRY OF GREEK VASES s s s S V2 V2 V2 >/2 Diagram XLIII. The rectangle 1.0355 = .9657 The \/3 rectangle occurs as the containing shape of three Nolan amphorae, nos. 29-31. Another red-figured amphora, no. 43, is enclosed in an area composed of a square with a y/S rectangle added to one side. Ratio 1.5773. A black-figured kylix signed by Xenokles, no. 126, is apparently contained in an area based on the -\/3 rectangle. The ratio of width to height is 2.0206, or \/3+^, i.e., 1.732+. 2886. 6 In the foregoing pages fifty-four areas have been defined which occur more than two hundred times among the vases analysed in this book. Speaking in general terms, each area is found, on an average, four times. Many of the vases, however, illustrate two, three, four, or even five different rectangles, according as the whole or some important part is considered. A striking instance of this is fur- nished by the stamnos, no. 51, which exhibits five famiUar areas: The complete stamnos with its lid . . The complete stamnos without its lid . The stamnos up to the shoulder . The stamnos without lid and handles The lid considered separately 1.118 = .8944 1.0787 = .927 1.236 = .809 1.191 = .8396 2.8944 = .3455 The number of different vases which illustrate these fifty-four areas is con- sequently only one hundred and thirty-four. Since one hundred and eighty-five vases are published, it is obvious that the total number of areas must be far greater than fifty-four. As a matter of fact the number is at least ninety. Judging the problem merely on the basis of these statistics, one might be justified in inferring that the shapes described are devoid of significance. Every object under the sun can be enclosed in a rectangle ; and, if a sufficient number of objects are measured, recurrences of some of the rectangles are bound to appear. In the same way it might be argued that the ratios which describe the proportions of details of the vases with reference to the complete height or width are so numerous that every detail must fit approximately some one of the ratios. But that the C 22 ] INTRODUCTION matter cannot be dismissed so lightly will be apparent to any one who seriously examines the geometrical analyses of the vases and the tables of ratios which accompany them. Consideration of the following points may perhaps facilitate the study of the evidence. 1. The accuracy with which the vases conform to the rectangles used in ana- lysing them is a question of some importance. Since the main dimensions are given in every case the margin of error can easily be calculated. But to spare the reader this labor the results of such a calculation are presented here in tabular form. In each case I have multiplied the smaller dimension by the ratio, thus getting the maxim error. Multiplying the larger dimension by the reciprocal would have decreased the error; and by distributing the error between height and width, it could have been still farther reduced. The total number of proportions thus investigated is 277. Many of the vases furnished more than one rectangle. It seemed worth while, in vases like the stamnos, hydria, oinochoe, skyphos, kantharos, and kylix to examine the proportions with and without the handles, and where a vase had a lid to consider it with and without the lid. Fifteen ex- amples were eliminated either because restorations made them unreliable, or be- cause they did not fit any recognisable proportions. The remaining 263 examples were grouped as follows : A Error indistinguishable. B Error less than one millimetre. C Error between one and two millimetres. D Error between two and three millimetres. E Error between three and four millimetres. F Error greater than four millimetres. The results of the calculation may be summarized as follows: A-C Error less than two millimetres. 195 examples. D-F Error greater than two millimetres. 68 examples. The examples in the second group require further scrutiny. Of the fifteen amphorae included foiir could not be analysed (nos. 6, 21, 24, 25), and in four other cases (nos. 16, 19, 22, 31) the analysis is to be regarded as merely tentative. The stamnos no. 54 is to be eliminated as a failure. The krater no. 77, is a doubt- ful example. Nos. 73 and 79, on the other hand, are admitted because the correctness of the schemes adopted was strikingly confirmed by the details. The red-figured kylikes are a special case. The widely spreading bowls are often not set quite horizontally upon the slender stems, so that the intended height is difficult to determine. A variation of one millimetre in the height changes the width and the diameter of the bowl about three millimetres. In the analyses a theoretical height is usually taken; and this is carefully noted in every case. Forty-two ex- amples may, therefore, fairly be removed from the second group, which is thus reduced to twenty-six examples. In general the maximvun error admitted is under two millimetres. I have no figures to show the accuracy of details, but I am [23 ] GEOMETRY OF GREEK VASES confident that in the very large majority of cases the error is well under two milli- metres. At first glance this margin of inaccuracy perhaps appears too small. When one takes into account the possible carelessness of so humble an artisan as a potter in conforming to a given scheme, the difiiculty of accurately assembhng the various parts of a vase before the clay has hardened, the shrinkage in firing, Number of examples Amount of Error Amphora Pelike Stamnos Stamnos details Hydria Hydria details Deinos Deinos and stand Deinos, stand alone Krater Krater details Psykter Oinbchoe Oinochoe without handle . . . Skyphos Skyphos without handles . . . Kantharos Kantharos without handles . Kyhx Kylix without handles Lekythos Pyxis Pyxis without lid Perfume vase Perfume vase without lid Total. 40 6 6 7 12 11 3 1 1 10 2 1 16 8 15 15 4 3 37 38 17 2 2 3 3 263 4 2 1 2 1 1 4 3 3 1 1 3 3 2 33 8 1 1 5 7 6 1 3 1 1 9 5 8 10 3 1 9 14 9 2 2 3 3 13 3 3 1 2 2 1 7 7 4 6 1 2 3 2 3 112 50 31 6 1 6 8 1 23 14 one is inclined to admit a more considerable error. And by doing so the confus- ingly large number of ratios could very likely be reduced. On the other hand, any one who has carefully examined masterpieces of pottery like the Brygan kan- tharos, no. 121, will agree that such a vase must have been fashioned with almost as much care as the vases of precious metal whose forms it imitates. It was not merely "thrown" on the wheel, but carefully "turned" after the clay had be- come "leather hard." The clay is of extremely fine texture, the walls astonish- ingly thin; and it is a question whether the firing would appreciably affect the C 24 ] INTRODUCTION proportions, though it would decrease the dimensions. All things considered, it is safer to follow the evidence we possess, rather than to juggle with it in the hope of achieving more striking results. 2. As was remarked above there is no significance in the fact that Greek vases can be contained in the rectangles of dynamic symmetry. It may, however, be significant that (1) a large proportion of the vases conform accurately to a limited number of comparatively simple rectangles, that (2) in a large proportion of ex- amples all the details can be accurately expressed in terms of the containing rec- No. of Ratio occur- rences A B 1.236 = .809 21 46, 51, 55, 56, 73, 91, 95, 107, 109, 110,111,112,113,115,120,185 18,19,73,76,80 1.000 15 53, 55, 73, 77, 99, 118 16,52,53,63,64,92,93,182 1.618 = .618 12 9, 17, 18, 41, 69, 85, 86, 87, 92, 93, 97, 185 1.118 = .8944 11 51, 67, 68, 73, 100, 108, 116, 121, 121 9,17 1.382 = .7236 10 4, 5, 49, 67, 69, 104, 123, 130, 131, 183 2.000 = .500 9 122, 128, 129, 164 168, 170, 171, 174, 183 1.309 = .764 8 47,65,68,92,93,117 35,41 1.4472 = .691 8 6, 7, 8, 63, 72, 118, 123, 129 3,236 = .309 8 133, 141, 142, 143, 144, 145, 146, 147 1.472 = .6793 7 9,10,11,12,13,85,103 1.500 = .666 7 14,21,23,24,44,61,62 1,854 = .5394 7 37, 88, 104, 108, 109, 171, 183 3.000 = .333 7 43, 135, 136, 137, 172, 173, 174 1.809 = .5528 6 33,34,35,36,115,117 2.618 = .382 6 133, 145, 146, 147, 155, 181 1.0787 = .927 5 51, 55, 58, 65, 66 1.2071 = .8284 5 2,45,57,61,182 1.4142 = .7071 5 20, 22, 90, 96, 105 1.528 = .6545 5 18, 39, 63, 95, 102 3.4142 = .2929 5 148, 149, 150, 151, 153 20 Ratios 167 tangles, and that (3) a large majority of the details coincide accurately with a small number of simple ratios. In connection with the first point it seems worth while to give a full list of the occurrences of the twenty most common containing areas, classifying them according as they contain (A) whole vases with or without their lids and handles, and (B) important parts of vases. The significance of the second point can only be judged by studying the draw- ings with the accompanying explanations. As regards the third point the follow- ing statistics are submitted. In compiling them I have limited myself to the vases analysed in terms of \/5. These furnished 99 ratios less than unity, which [25] GEOMETRY OF GREEK VASES occurred in all 591 times. The thirty most popular ratios are given here with the number of occurrences of each: Ratios Occurrences Ratios Occuirences Ratios Occurrences .382 52 .5669 14 .7888 7 .500 46 .5528 13 .191 7 .309 40 .708 13 .354 7 .4472 32 .4045 13 .118 6 .236 32 .6584 11 .2236 6 .618 30 .250 11 .333 6 .764 21 .472 10 .1459 5 .528 19 .854 9 .264 5 .691 18 .7236 8 .5394 5 .809 15 .2764 8 .1708 4 10 305 10 110 10 58 The ten ratios in the first column occur more frequently than the remaining eighty-nine ratios. And the thirty ratios here Usted occur 473 times, whereas the remaining- sixty-nine ratios occur only 118 times. Of the twenty-nine intervals between these ratios only nine are small enough to cause confusion in fixing the ratio in an example whose smaller dimension is 0.20 m. In actual practice I have found that doubt as to the choice between two ratios very rarely arises. In twenty-eight amphorae, for example, whose average diameter was 24 cm., the difference between thirty-three pairs of adjacent ratios was over 3 mm. in twenty-seven cases. 3. The following table gives a classification of the vases according to the sys- tem of analysis used. The examples which worked out successfully are marked A; those which worked out fairly well are placed in colimin B; and those which were regarded as unsatisfactory in column C. 4. In a criticism of Dynamic Symmetry, pubUshed in the American Journal of Archaeology, XXV, 1921, pp. 18 ff.. Professor Carpenter has raised the question whether the analyses in Mr. Hambidge's book are not for the most part "mere adroit manipulation, combined with a mystifying conversion of very simple linear ratios into a guise of root-rectangles. " To illustrate his pomt he places side by side Mr. Hambidge's "dynamic" analysis of a lekythos at Yale and his own static analysis of the same vase. He justly remarks that the issue here involved is crucial. He also notes that many of the most frequent and important "effective numbers" happen to fall very close to certain simple "static" ratios, as follows: 2.236 (V 5) is scarcely distinguishable from 2.25 = 9 : 2.000 is identical with 2 =8: 1.732 is scarcely distinguishable from 1.75 = 7 ; 1.618 (the redoubtable "whirling square" ratio) and. . 1.6 =8: .618 (its reciprocal) closely approximate 625 = 5: 1.309 agrees very nearly with 1.333 = 4: 4 4 = 4 5 2:1 [26] INTRODUCTION CO 00 o 1 : 00 : > 00 (N . . 00~ . . IM . 1-H ■ co" CO CO • • »< 00 "2 o ^ CO l:~ 'i* M ■* ffl s" rH T-l 1-1 tH rH (n" i-Tcq" i>r i>r lO" «D CO -^ •^ t^ T— ( 1-H 1-H r^ 1-H «,-fc . O m . CO CO o cT os" CO CO 4 ~ 03 '. oT tH rH 1-H rH 1-H U3 > T— < T— 1 i-H lO co" o in !>" co" «> CO OS Tt* - <© oo" to o (M C* CO 1* in CO t- 00 O tH 1-H lO to 1—1 T-H 1-H 1-H rH 1-H rH tH I-H CO ^ -X ^ ^ CO ~ - •s rN OS CO 00 CO Os" OS in T-l Tjl 05 05 o 05 in o ^ CO TjH in CO t^ •^ CO CO CO 00 00 05 1— 1 1-1 1— 1 1-H 1-H T-H T-H '^ -" _- T-l CO 00 <© 00 05 00 ■<*< 00 (N t> CO in m 00 00 CO ^ CO CO lO « t> 00 05 O "-I 1-1 (M CO TtH in CO t> 05 - - O t-l 1-1 1—1 1-H T— 1 1-H 1— 1 i-H rH CO 05 kC t^ CO l> CO 00 (N CO •>* ■* lO CO O 00 OS 1— I i>r T— t CO of Co" i-h" CO th in" r-T m ^ O 1-1 1—1 (M CO Th in CO CO IV 00 l> 00 «o ^ t^ (N ■* Tfl CO (M 00 i-l i-( 1— 1 1-H 1-H 1-H r~i rH I-H I-H I-H ~ ,-1 CO Tt< -* in i» I> 00 05 OS - ~ ^ »c ., - ^ K •^ -s -. ^ ^ ~ Tt* o in 00 i-T (n" o in o co" O 1-H CO" „ t^ lO o ■^ 1-1 CO ■rt* o •* ss (M CO ■* in rH I> t^ 00 00 05 t^ O 1-1 1-H rt !N CO Tt< in CO CO t^ 00 00 g O^ T-H rH i-H 1-H 1-^ i-H 1-H ""^ 1-H 1-H 1-H oT r-T CO (N CO . . ^ T to O (N (N CO Tt< 1-H • CO (N o (N lO O o in '^ (N IM ■* -. -N ^ ■?• < ^ . CO ! CO . OS . 1-H 1-H in T-H 1-H in : : Ttf oo" o cq'oo 00 00 (N lO t- OS 1-1 1-H 1-H 1-H rH rH T— i I— 1 Q o 4,21 3,24 Tj< • • • OS OS T— 1 o 1— 1 ?2 • —1 iM lO T— 1 I-H OS -> IM t^ . . CO CO 1— 1 ' ' " . T-l r~i < «> . . . . . . 1-H in rH 1-H CO & P m (D o 'S, a t? m [2 1 •fH Stamnos Hydria-Kal Deinos Krater Psykter Oinochoe-0 02 o Two-handle Kantharos Kylix Oi O CD > a II > o 1> CO (M Q (M in s •^ O CO 03 (N 00 O 1-1 (M CO 00 00 00 1 T< "P f ? 1—1 I> o CO ■* in 1—1 1 1-H T-H 1 1 1-H r I-H 1-H j 1 3 1— ( 1 1 1-H 00 OS 1^ cij J^ c4 in iz; •* in in l> ir- 00 00 o 1-H 1-H IM CO 00 00 00 1— 1 rH 1-H 1-H T— 1 rH rH tH [27] GEOMETRY OF GREEK VASES He finds that "here more than anywhere else hes the key to Mr. Hambidge's ingenious magic," and reminds us that "a ratio approximating 5: 8 has in all ages been a recurring favorite in artistic composition and artistic design. . . . Some- where in the neighborhood of that ratio, man has an inveterate tendency to locahse his sense for beauty of proportions. For the old potter working with a simple rule, that ratio was a natural one to employ. Continued bisection of his rule would give him 8 parts or 16 parts with which to lay out and measure. It was only to be expected that he should often avail himself of that harmonious division into a little more and a Httle less than half which | or H would give him. Wherever he used this ratio, the dynamic analyst will be able to discover 'whirhng squares,' since f is a remarkably close approximation to the division into extreme and mean proportion from which the 'whirling square' rectangle derives its peculiar prop- erties of subdivision." In short Mr. Carpenter prefers to return to the Vitruvian method of studying Greek proportions, which, as regards architecture, had long since been abandoned as a hopeless failure, and, for that reason perhaps, has never been seriously ap- plied to Attic pottery. His remarks call for consideration here, if only to make clear the purpose of this book. It is not published as an argument for or against the theory that the Attic potters consciously used the systems of proportion dis- covered by Mr. Hambidge, nor as an argument for or against the theory that a work of art designed according to these systems is "better" than one designed according to another system, or according to no system at all. Its aim. is to present in as complete and accurate and intelUgible a form as possible the evidence fur- nished by the whole collection of Attic pottery in the Museum of Fine Arts. Many pieces had to be rejected because they were incomplete, some others because they were badly made, and consequently could not be accurately measured. A considerable number of lekythoi were omitted because this type of vase is not a satisfactory subject for analysis. Most red-figured lekythoi are nearly three times as high as they are wide; a small change in the diameter produces a three-fold change in the ratio of the height. And the details, except for the height of the body, are all at a very small scale. A few black-figured amphorae and a dozen red- figured kylikes have been excluded because these classes of vases are represented in the collection in much greater numbers than any other shapes. The pieces omitted are those which could not be satisfactorily analysed. With these excep- tions, however, the vases are not select pieces, but a representative collection of admittedly high quahty. The reader is left to draw his own conclusions as regards the probability or improbability of Mr. Hambidge's theory. Sufficient measurements are given to enable him to test also the vahdity of the theory of Vitruvius and of Mr. Carpenter. My own experiments along this line have not been successful. It is true that the proportions of a certain number of vases can be expressed in sunpl§ linear units, but in every case one or more or all of the follow- ing obstacles are encountered: (1) The unit chosen must be arbitrary, not some simple division of the Greek foot. (2) The unit must be made very small, so that the proportions have little more significance than a mere record of the dimensions [28 ] INTRODUCTION would have. (3) A large margin of error must be admitted. (4) Even if the pro- portions can be expressed in fairly large divisions of the Greek foot no reason appears why those particular lengths were chosen rather than others. To illustrate these points let us examine some of the examples cited by Mr. Carpenter. The proportions of the lekythos (I.e. p. 33, fig. 7) seemingly work out as well in terms of units 0.013 m. long as they do in terms of the \/2 rectangle. And I agree that "on the reader's judgment of the issue here involved will hang his whole faith in, or distrust for, dynamic symmetry." But an issue of such importance cannot be decided on the evidence of one example. The skyphos in New York, to which he refers (I.e. p. 35, note 1) furnishes a better example, because there is another skyphos with identically the same proportions in Boston (no. 104) and because the height of each is very close to half of the Aeginetan-Attic foot. The dimen- sions of the skyphoi are as follows: New York Boston Height 0.164 m. 0.160 m. Width ; 0.304 m. 0.298 m. Largest diameter of bowl 0.2255 m. 0.221 m. Smallest diameter of bowl 0.100 m. 0.099 m. Diameter of foot 0.137 m. 0.134 m. The static analysis is as follows: Height Width Greatest diameter of bowl . Smallest diameter of bowl . Diameter of foot Units (1 dactyl) 8 15 11 5 7 The dynamic analysis is as follows: Ratios Height 1.000 Width 1.854 Greatest diameter of bowl 1.382 Smallest diameter of bowl 618 Diameter of foot 854 Error New York Boston -0.0035 m. -0.002 m. -0.001 m. -0.0025 m. +0.001 m. -0.0065 m. -0.006 m, Error New York Boston -0.00006 m. -0.00136 m, -0.00115 m. -0.00012 m, -0.00135 m. -1-0.00012 m, -0.003 m. -0.00264 m, The static ratios are seen to be less accurate than the dynamic, especially that of the diameter of the foot. The static analysis shows plainly that the smallest diameter of the bowl is one-third of the total width, and that it is to the height in the proportion of 5:8 (= 1.600). Mr. Carpenter notes also that the lower diam- eter plus the projection of the bowl equals the total height, and suggests that this would seem to have been a common potter's formula, just as in speaking of an- other skyphos (I.e. p. 30; fig. 5; no. 117 in this book) he observes that "the maxi- mujn width is equivalent to width of base plus height of vase, and the width of the bowl is equivalent to width of base plus half the height of vase, — which looks [29] GEOMETRY OF GREEK VASES like a convenient potter's formula." One wonders how many "convenient" formulae like these the Attic potter kept in his memory. The dynamic analysis shows clearly the same pair of proportions 1:3 and 5:8; but in the latter case the exact ratio is substituted for the approximation. It re- veals also other instances of extreme and mean proportion, which are not at once apparent from the Ust of static units. For example, the diameter of the foot is to the greatest diameter of the bowl as 1 is to 1.618. In the static analysis this ap- pears as a new, but not very close approximation, 7:11 (= 1.5714). The height is to the greatest diameter of the bowl plus the projection of one handle as 1 is to 1.618. To express it geometrically, the whole vase is contained in two overlapping whirling square rectangles, and the bowl is contained in the overlapping portion (cf. the diagram on page 150). In the static analysis this appears as still another approximation, 8 :13 ( = 1.625). The smallest diameter of the bowl is to its largest diameter as 1 is to a/S. This appears in the static analysis in the form 5:11 (1.200) instead of 4:9 (1.250). The relation of height to largest diameter of bowl is the same as that of smallest diameter of bowl to diameter of foot (1.382). In the static analysis this appears as 8:11 (1.375), and as 5:7 (1.400). To sum up, all the ele- ments of the skyphos are related to one another in extreme and mean proportion and other proportions intimately connected with it, which could be arrived at with the greatest ease by enclosing the preliminary sketch of the vase in a 1.854 rectangle, and drawing the simplest subdivisions of one of the three whirling square rectangles of which it is composed. It is difficult to see how a potter could have created so perfectly coordinated a design with the help of the nimibers 8, 15, 11, 5, 7. The proportions of this skyphos could be expressed with a fair amount of ac- curacy in terms of the Greek foot. This is, however, by no means always the case. Let us take as an example the Brygan kantharos, no. 121, which is perhaps the finest piece of Attic pottery in the collection. Mr. Carpenter remarks (I.e. p. 34, note 1) that "the whole kantharos can be constructed "statically" on a meas- ure divided into eight parts," and in response to an inquiry he has explained his proposed analysis in detail. The vase is in excellent preservation, except for one handle which has been broken and has a small piece missing at the top restored in plaster. The other handle is intact. The dimensions are : Height of unbroken handle 0.241 m. (Height of restored handle 0.246 m.) Height of lip 0.1675 m. Height of stem 0.075 m. Width 0.270 m. Largest diameter of bowl 0.1885 m. Diamieter of lower member of bowl 0.121 m. Diameter of top of stem 0.031 m. Diameter of foot 0.0985 m. Width of handles ±0.0275 m. [30] INTRODUCTION The height of the handles is assumed to be 0.242 m., which is twice the diam- eter of the lower member of the bowl. The ratios, with the amount of error are as follows: Katios Error Height to top of handles 1.000 -0.001 m. Height tolip 691 -0.00028m. Projection of handles above lip 309 —0.0003 m. Height of stem 309 +0.0002 m. Height of bowl alone 382 +0.00005 m. Width 1.118 -0.0005 m. Diameter of lip 7725 +0.0016 m. Diameter of lower member of bowl 500 Diameter of top of stem 118 +0.0024 m. Diameter of foot 4045 +0.0006 m. Width of handles 118 —0.001 m. The unit in Mr. Carpenter's analysis is one-eighth of the diameter of the Up, 0.02356 m., which has no recognisable relation to the Greek foot. His analysis is as follows : Units Ratios Error Height to top of handles 10 1.000 0.0054 m. Height to lip 7 .700 0.00358 m. Projection of handles above Up 3 .300 0.00282 m. Height of stem 3 .300 0.00432 m. Height of bowl alone 4 .400 0.00024 m. Width 12 1.200 0.01272 m. Diameter of Up 8 .800 Diameter of lower member of bowl 5 .500 0.0037 m. Diameter of foot 4 .400 0.00426 m. The first thing that strikes one in studying this scheme of proportions is that the vase does not fit the scheme. The height is half a centimetre too small, the width more than a centimetre too great. Mr. Carpenter accounts for this by assuming that the handles have been wrongly adjusted, and that the moulding at the junction of bowl and stem was not included in the scheme, but was added when the two elements were joined. We have noted in the case of the kantharos no. 119 that the weight of the handles pulled the bowl out of shape, so that it is oval rather than circular. In the present example there is nothing to suggest any such disturbance of the scheme ; the bowl is perfectly circular. Moreover the list of units does not reveal a logical and consistent "theme." Five is half of 10; 4 is half of 8, and one-third of 12; 3 is one-quarter of 12. Extreme and mean proportion apparently appears in the ratio of the diameter of the lower member of the bowl to the diameter of its lip. But actuaUy this ratio is 1.545, not 1.618. The unit, as already remarked, bears no relation to the Greek foot. The height of the vase is actually about half a centimetre less than twelve dactyls, and the discrepancy can be explained as due to shrinkage in firing. But I have been imable to find a plausible static scheme using the dactyl as a unit. [31] GEOMETRY OF GREEK VASES Mr. Hambidge's analysis, on the other hand, shows that every element of the vase conforms accurately to a logical and consistent "theme." This has been made sufficiently clear in his book and in the drawing and table of ratios pubHshed below, page 162. It appears still more clearly in the four diagrams here repro- duced. The first shows that the containing area is composed of two y/d rectangles. Each of these may be regarded as two whirhng square rectangles overlapping to the extent of a square. The diameter of the lower member of the bowl equals the side of this square, which is half of the total height. The diagonals in the second diagram show that the main proportions of the kantharos are the same whether the handles are included or omitted. This is a relationship worth observ- ing, and it would hardly appear from a study of Mr. Carpenter's units. Attic pot- ters do not seem to have used it frequently; at least only two other instances have been noted in this book (nos. 53 and 79). The third diagram shows that if whirUng square rectangles are applied at the top and bottom of the containing area the bowl is contained in the overlapping portion. In the fourth diagram the kantharos has been revolved 90°. If whirhng square rectangles are apphed at each side, the handles and the top of the stem are seen to be contained in the overlapping por- tion. It is also noteworthy that if this overlapping portion were removed the con- taining rectangle would become a square. The proportions of the kantharos can all be simply and accurately expressed in terms of extreme and mean proportion, i. e., the ratio 1.618, or the whirhng square rectangle. This example alone does not prove that Greek potters consciously employed such a scheme, just as the lekythos studied by Mr. Carpenter does not prove the contrary. But it does seem to the writer to furnish stronger evidence in favor of the theory than his ex- ample furnishes against it, and therefore to justify a geometrical investigation of the proportions of Attic pottery. It is unjust to characterise such a procedure as "ingenious magic," "adroit manipulation" and "a mystifying conversion of very simple Hnear ratios into a guise of root-rectangles." There is no such thing as magic. The word is a euphemism for such terms as quackery, charlatanism, which imply dehberate or unconscious deception on the part of the manipulator. The only opportunity for deception in the present case is in the recording of the dimensions of the vases. And, so far as I know, no one has yet charged Mr. Hambidge and his collaborators with dehberate falsification of the evidence. 5. Before the pubhcation of Mr. Hambidge's book the question whether Attic potters constructed their vases according to a predetermined design had hardly been raised. In his catalogue of the Attic pottery in the Louvre, pubhshed in 1906, M. Pottier had indeed remarked: "Les proportions des vases, les rapports de mesures entre les diff^rentes parties de la poterie paraissent avoir 6t6 chez les Grecs I'objet de recherches minutieuses et d^hcates . . . Je crois qu'un examen attentif du sujet m^nerait k des observations int6ressantes sur ce qu'on pourrait appeler la 'g^ometrie' de la c^ramique grecque" (Vases antiques, III, p. 658). Apparently Mr. Hambidge was the first to undertake such a careful examination of the subject. It could only be done on the basis of accurate measurements of all the elements of a large number of vases; and the labor of procurmg these measure- [32 ] INTRODUCTION ments did not seem worth while so long as no method of coordinating the propor- tions had been discovered. That the geometric analysis of the vases in terms of the root-rectangles furnishes such a method can hardly be disputed. If the pro- portions thus revealed cannot be satisfactorily explained either as coincidences, or as "a mystifying conversion of very simple linear ratios into a guise of root- rectangles," the only alternative is to suppose that the potters worked from a ^ M'i.,i.;mSii^M.'^!^hw.x ? u AS > ^^>^ y^ ^^HSgppp^ \^ /600 .236 .472 35 1.000 1.809 1.309 .500 .809 .4472 .500 .236 .500 36 1.000 1.809 1.2696 .5394 .7236 .4472 .5394 .250 .500 37 1.000 1.854 1.354 .500 .764 .4472 .528 .264 .472 38 1.000 1.9045 1.2865 .618 .854 .4472 .528 .2236 .4472 The twelve amphorae are of seven different shapes, six of which belong to the dynamic class. The shape 1.809 occurs four times, the -y/S rectangle three times, the others only once. The simple geometrical divisions of these seven rectangles are made clear by the small diagrams at the foot of this page. The analysis of details in the larger drawings will in most cases be intelligible, if examined in connection with the small diagrams and the table of ratios. Considered from the point of view of chronology, it is noteworthy that the Nolan amphorae become slenderer as time goes on. The first five examples belong to the ripe archaic period, the last seven, with one exception (no. 36) to the late archaic or early free style. It is also noticeable that the elongation is all in the upper part of the vase — the shoulder and neck. In every example except no. 1, the body attains its greatest diameter at approximately the level of the top of a square inscribed on the base of the rectangle. Another point brought out by the table is the very frequent occurrence of the simple ratio .500 (one-half of the width). It is found four times as the height of neck and lip, seven times as the diameter of the shoulder, four times as the diam- eter of the foot. The diameter of the bottom of the body is .250, or one-quarter of the width, in three cases. In seven examples the diameter of the neck is to the total width as the height of a \/5 rectangle is to its length (.4472: 1.000) and only in one case (no. 27) is there a marked departure from that proportion. ^ y D CI «lw w| w> w ( 1 s J [75] GEOMETRY OF GREEK VASES B. Amphora of Panathbnaic Shape 39 Inv. 95.19. Beazley, 7. A., p. 25. (A, B) Figure of Athena. Painter." Height, 0.435 m. Diameter, 0.284 m. Ratio, 1.528. The proportions are : 'Nikoxenos Height 1.528 Height to shoulder 1.219 Height of neck and hp 309 Diameter 1.000 Diameter of hp 5669 Diameter of shoulder 382 Diameter of bottom of body 292 Diameter of foot 472 The proportions of the foot are obtained by simple subdivisions of the rectangle .528, as shown in diagram a. The height of the neck and lip and the diameter at the shoulder and the top of the neck are obtained from the rectangle .309, or two whirling square rectangles. One of these is shown in the upper left-hand corner of the drawing. The diameter of the lip is most simply expressed in terms of the rectangle .3504 ( = 2.854), as shown in diagram b. Only the right-hand half of this rec- tangle is shown in the drawing. s vs vs s s 5 w w w [76] AMPHORA t 77] GEOMETRY OF GREEK VASES 40 Inv. 96.719. Beazley,7.A., p. 122, no. II, 4. (A) Citharode and Athena. (B) Hermes and Poseidon. "Nausikaa Painter." Height, 0.512m. Diameter, 0.323 m. ^ _ The ratio is 1.5854. In the ' '• ^^^ ^ -^ ' ' drawing a square is appUed at the top of the containing rectangle. The excess area at the bottom (.5854 = 1.708) accounts for the proportions of the foot. The proportions of the lip, neck, and shoulder are expressed simply and accurately in terms of a whirling square rectangle applied at the top. The ratios are: Height 1.5854 Height to shoulder 1.2034 Height of neck and lip 382 Diameter 1.000 Diameter of Hp 5528 Diameter of shoulder 382 Diameter of bottom of body .2764 Diameter of foot 4146 The ratio 1.5854 is very close to another ratio, 1.5858, which belongs to the ■\/2 system. The rectangle .5858 (=1.7071) ac- counts satisfactorily for the diam- eter of the foot, the ratio being .4142 (1.000-.5858), or V2-1. But the ratio .4146 (1.000-.5854) is hardly distinguishable in a drawing from .4142. The rectangle .5854 also accounts simply for the diameter of the bottom of the body. All the proportions can thus be expressed in terms of the whirling square rectangle, whereas only one of the details fits a simple ratio derived from ■\/2. 41 Inv. 10.178. Beazley, V. A., p. 42. Hambidge, p. 130. (A) Athlete with presents. (B) Youth offering wreath. " Kleophrades Painter." Height, 0.454 m. Diameter, 0.2815 m. Ratio, 1.618, or a whirling square rectangle. The proportions are : ' Height 1.618 Height to shoulder 1.309- Height of neck and lip 309 -[- Height of hp 1382 Height of neck 1708 Diameter 1.000 Diameter of hp 5528 Diameter of shoulder 3416 Diameter of bottom of body 236 -|- Diameter of foot 427 [78] AMPHORA rN y' ^T~ ''■■■■>rU . 1 .'' ,'\ I- ) ■••'.. \? V 7 ,. - . \7 41 There is an inaccuracy of 0.00175 m. in the height of the shoulder. It is note- worthy that the Up is inclosed in four squares, and the neck in two squares. [79] J GEOMETRY OF GREEK VASES C. Amphora wiTHOtrT Division between Neck and Shoulder 42 Inv. 98.882. Beazley, 7. A., p. 59, no. I, 4. (A) Seilen holding his son on his shoulders. (B) Seilen with phallos. " Flying-angel Painter." Height, 0.4085 m. Diam- eter, 0.2604 m. This amphora is not a masterpiece of pottery; and the dimensions given are averages obtained from sev- eral measurements which vary more than 0.005 m. in the case of the height, and 0.003 m. in the case of the diameter. If the height is taken as 0.41 m. and the di- ameter as 0.26 m., the result- ing ratio is 1.5773. .5773 is the reciprocal of 1.732, or -\/3. The containing rectangle is therefore close to the area composed of a square and a VS rectangle. The propor- tions of details are expressible in simple ratios, three of which occur in "static sym- metry." \\ 1 \ ',.'/' \,, \ J " / \ I ! / ■> 4 ■ ^ /I ( / 1 * ■ V \ y r' \y s 56 56 Stamnos in the collection of Professor J. C. Hoppin at Pomfret, Conn. Height, 0.450 m. Width (handle to handle), 0.443 m. Diameter of bowl, 0.365 m. Diameter of lip, 0.271 m. Diameter of foot, 0.1595 m. The two accompanying drawings were made from measurements and a full- size elevation furnished by Professor Hoppin. His drawing did not include the handles; these have been added in the diagrams in dotted lines to suggest ap- proximately their appearance. The enclosing area is a rectangle with the ratio 1.0225, i. e., .618-f-.1545, or a horizontal whirling square rectangle with four vertical whirling square rectangles placed above it. The height of the shoulder, and the diameters of lip, shoulder, and bottom of body are obtained from simple subdivisions of the containing shape. The stamnos without its handles is contained in the familiar rectangle .809 or 1.236, i. e., two horizontal whirling square rectangles. The diagonals in the lower half of the second drawing show the relation of the lower of these two whirling square rectangles to the whirling square rectangle of the over-all shape. The diameter of the neck is seen to be one-half the diameter of the body. The ratios are: Height 1.0225 Diameter of lip 618 Height to shoulder.. .868 = .618 + .250 Smallest diameter of neck 4136 Height of neck and lip .1545 = .618 -f- 4 Diameter of shoulder 427 Width 1.000 Diameter of bottom of body . . . .236 Diameter of bowl . . . .8272 Diameter of foot 354 [ 100] HYDRIA-KALPIS HYDRIA-KALPIS Two types of water jar are distinguished by these names. The hydria is a jar with two horizontal handles at the sides for lifting and a vertical handle at the back for pouring. The junction between neck and shoulder is definitely marked, and, except in no. 57, the sloping shoulder is also definitely distinguished from the body. This type lasted down into the ripe archaic red-figured period. In the later red-figured period its place was taken by the kalpis, a jar broader in proportion to its height, and differing also in the shapes of its handles and foot, and in the absence of lines of demarcation between neck, shoulder and body. The geometrical schemes of the hydriae nos. 58-60, 62, and of the kalpis no. 65 have not been worked out satisfactorily. The variation in the main proportions is shown in the following table: Ratio with liandles Ratio without handles No. 57 B. F. Hvdria 1.0355 1.0787 1.1056 1.1708 1.2071 1.3455 1.4472 1.0652 1.0787 1.0787 1.118 1.118 1.382 1.178 58 u 1.3455 59 u 1.3455 60 u 1.3618 61 u 1.500 62 u 1.500 63 R. F. Hydria 1.528 64 R. F. Kalpis 1.266 65 a 1.309 66 u 1.333 67 u 1.382 68 u 1.309 69 u 1.618 [101] GEOMETRY OF GREEK VASES BLACK-FIGURED PERIOD 57 Hydria. Inv. 95.62. Collection Van Branteghem, No. 6. Hambidge, p. 55, fig. 16; photograph opp. p. 56. On shoulder, bacchic dance. On body, Dionysos, with seilens, maenads, and a horseman galloping. The vase is tech- . nically a masterpiece. The only blemish is the unevenness of the lip. Height to top of handle, 0.325 m. Height of lip, 0.316 m. Height to shoulder, 0.238 m. Width, mcluding handles, 0.3365 m. Diameter of body, 0.267 m. Diameter of Up, 0.191 m. Diameter of foot, 0.1475 m. The proportions of this remarkable vase were worked out by Mr. Hambidge on the basis of a drawing made by the writer in 1918. In view of its importance a new drawing based on revised measurements is here published. The only varia- tions are a shght reduction of the diameter of the bottom of the neck and the enlargement of the diameter of the bottom of the body from 0.069 m. to 0.073 m. The scheme of proportions remains essentially unchanged. Though complicated, it is obviously in terms of the •\/2 rectangle throughout. If the height to the top of the handle is taken as unity, the width has the ratio 1.0355 (= .2071X5). The amount of projection of each handle from the body is .10355 (=.2071 -=-2), or one-tenth of the total width. The diameter of the body is .8284 (= .2071 X 4). This is the reciprocal of 1.2071, a familiar ratio represented geometrically by half a square plus a ■\/2 rectangle. If the lateral handles are removed, the vase fits this area fairly accu- rately. Cf . diagram A. The diameter of the lip has the ratio .5858, which is the reciprocal of 1.7071, again a familiar ratio represented by a square and a \/2 rectangle. Cf . diagram B. The diameter of the foot has the ratio .4531, the reciprocal of which is 2.2071 expressed geometrically as a square plus half a square plus a \/2 rectangle. The diameter of the ring at the bottom of the body equals .2265, or half the diameter of the foot. The smallest diameter (just above the ring) is .2071, or one-fifth of the total width. It is noteworthy that the main horizontal division of the enclosing rectangles in diagrams A and B coincides exactly with the painted band immediately below the lower zone of figures. The height of this zone exactly equals the diameter of the ring at the bottom of the body (.2265), or half the diameter of the foot. The interrelation of a number of these proportions is shown by the dotted fines in the large drawing. It is difficult to determine the height of the lip in simple terms of the containing rectangle. This height varies between 0.314 m. and 0.318 m. If the height is taken as 0.31728 m. the vase, omitting the projection of the vertical handle, fits the rectangle 1.0606 (= .3535 X 3), or — — . Cf. diagram C. This area is com- posed of six \/2 rectangles. The shoulder is seen to be at three-quarters of the total height; and the vase up to the shoulder is contained in a y/2 rectangle. C 102 ] HYDRIA-KALPIS \ ^L^ —1 .- M /••■' /^ ^ ^ f — ' f* \' — ^ ^ '■■■ ^ = \ I \ 1 y .'ilO-^ J/"'' ,■' ■• >\,-' ^ " ^'. /■'''P \ .'' J \- '\fJ2'y • , / yzV . ■'' '^.i'~ . /4 ■■■'' c ^fe^kr s GEOMETRY OF GREEK VASES r ~c 58 Hydria. Inv. 01.8125. On body, four women carrjdng water jars on their heads. On shoulder, a warrior attacked by two horsemen. Greatest height, 0.36 m. Width, 0.334 m. Height of Up, 0.349 m. Diameter of body, 0.259 m. The enclosing rectangle apparently has the ratio 1.0787 ( = .927). The diameter of the lip is .618, the greatest diameter of the body, .764, the diameter of the bot- tom of the body, .382, the diameter of the foot, .4045. The geometrical analysis is complicated, and is therefore omitted. It is possible that the ratio is 1.073, i. e., .309-I-.764, or .691 + .382. If the three handles are omitted, the ratio is 1.3455. 59 Hydria. Inv. 89.562. On body, Herakles mounting a quadriga in the presence of Athena, Apollo, and five other figures. On shoulder, Theseus killing the Minotaur. Greatest height, 0.4505 m. Width, 0.407 m. Height of lip, 0.43 m. Greatest diameter of body, 0.32 m. The ratio is apparently 1.1056. This is .5528X2. The reciprocal of .5528 is 1.809. The enclosing rectangle is thus composed of two 1.809 rectangles placed horizontally one above the other. The shoulder is at three-quarters of the total height (.8292). The height of neck, lip and handle is thus .2764, or .5528 -^ 2. The diameter of the Up is .5669, that of the shoulder, .3455, that of the body, .7888, that of the foot, .3504. The hydria without the handles fits the rectangle 1.3455. The geometrical analysis is omitted. 60 Hydria. Inv. 99.522. On body, Herakles and Triton. On shoulder, three youths, each leading a horse. Greatest height, 0.409 m. Width, 0.349 m. Height of lip, 0.3955 m. Greatest diameter of body, 0.29 m. The ratio is apparently 1 . 1708 ( = .854) . Without its handles the hydria fits the rectangle 1.3618. The shoulder is at three-quarters of the height. Most of the [ 104 ] HYDRIA-KA.LPIS details are expressible in simple ratios, though the geometrical analysis has again proved complicated, and has been omitted. The ratios are: Height 1.1708 Height of neck, lip and handle 2927 = 1.1708 -^ 4 Height to shoulder 8781 = .2927 X 3 Width 1.000 Diameter of lip 5854 = 1.1708 -^ 2 Diameter of shoulder 4146 Diameter of body 8292 Diameter of bottom of body 250 Diameter of foot 4472 C 105 ] GEOMETRY OF GREEK VASES 61 Hydria. Inv. 01.8060. On body, eight horses being watered at a trough. On shoulder, three horsemen. Greatest height, 0.456 m. Width, 0.3775 m. Height of Up, 0.436 m. Greatest diameter of body, 0.291 m. The hydria fits the familiar rectangle 1.2071. Without its handles it is enclosed in the rectangle 1.500. The height of neck, lip and handle is one-third of the total width (.333), shown in the drawing by three squares. The diameter of the shoulder is somewhat less than this, i. e., it nearly coincides with the central square. The diameter of the lip is simply obtained by describing arcs of circles with the comers of the central square as centres and its diagonal as radius. The ratio is .6094. The diameter of the foot is .4142. [ 106] HYDRIA-KALPIS RED-FIGURED PERIOD 62 Inv. 13.200. Hydria. Beazley, V. A.,-p. 52. On body, carpenter working on chest in presence of Akrisios, Eurydike, and Danaewith Perseus in her arms. On shoulder, three youths baiting a bull. Greatest height, 0.458 m. Width, 0.34 m. Efeight of Up, 0.4185 m. Greatest diameter of body, 0.279 na. Diameter of lip, 0.1915 m. Diameter of foot, 0.132 m. The over-all ratio is 1.3455. If the handles are omitted the ratio is 1.500. Most of the details can be accurately expressed in simple ratios, but the geometrical analysis is too complicated to be easily intelligible. The diagonal lines in the drawing are diagonals of squares, half squares, whirling square rectangles and half whirling square rectangles. The height of the shoulder has not been deter- mined. The diameter of the lip is .5669, that of the shoulder, .309, that of the bottom of the body, .236, that of the foot, .382. The projection of the vertical handle is .118. [107] GEOMETRY OF GREEK VASES 63 Hydria. Inv. 98.878. On body, a young warrior and a woman making a libation. On shoulder, two lions attacking a bull. Ripe archaic style. The foot, which is missing, has been incorrectly restored in plaster, the type of foot found in kalpides having been used in place of the simple disc proper to hydriae of the black-figured type. The dimensions are: Height, omitting foot, 0.565 m. Height from bottom of neck to top of vertical handle, 0.181 m. Width, 0.4045 m. Diameter of bowl, 0.339 m. Diameter of lip, 0.223 m. The width, 0.4045 m., divided by the height from the bottom of the neck to the top of the vertical handle, 0.181 m., gives the ratio 2.236, or ■\/5. All the details of the vase above the shoulder can be easily obtained from subdivisions of the \/5 rectangle, as shown in the drawing. If a square be added to this rectangle, giving the familiar shape 1.4472, exactly the right amount of space is obtained to restore a foot of the normal type. The body, with the horizontal handles, is enclosed in a square. The body without the handles is contained in the rectangle 1.191 ( = .8396), another shape which occurs frequently. The vase omitting the three handles is placed in the rectangle 1.528. The diameter of the missing foot cannot of course be determined with certainty. In the drawings it has been assumed to be .4472. This is, however, probably too great. The relation between the 1.4472 rectangle and the 1.528 rectangle has not been shown in the geometrical analysis. The ratios are : Height, including vertical handle 1.4472 Diameter of lip 5528 Height, omitting vertical handle . 1.2764 Diameter of neck at bottom 3292 Height to shoulder 1.000 Greatest diameter of body 8396 Height of neck 2236 Diameter of bottom of body 2236 Width 1.000 Diameter of foot, restored (.4472?) C 108 ] HYDRIA-KALPIS C 109 ] GEOMETRY OF GREEK VASES 7^^ Xn^r: sz=3_-,j. i:j 64 64 Kalpis. Inv. 90.156. Beazley, V. A., p. 148, no. VIII, 32. Hambidge, p. 87. The death of Orpheus. "Niobid Painter." Height, 0.403 m. Width, 0.3785 m. Diameter of body, 0.316 m. Height to neck, 0.3175 m. The ratio is 1.0652, or .618+.4472, showing that the enclosing area is com- posed of a whirhng square rectangle plus a \/5 rectangle. In the drawing the \/5 rectangle has been placed above the whirling square rectangle. The shoulder, marked by a painted band, is at half the height of the -\/5 rectangle, and the smallest diameter of the neck equals the height of the neck and lip. The vase up to the shoulder, and omitting the handles, is contained in a square (error: 0.0015 m.). The top of the maeander band on which the figures rest is at half the height of this square. The ratios are: Height 1.0652 Height of neck and lip 2236 Height to shoulder 8416 Height of top of maeander band . .4208 Width 1.000 Diameter of lip 382+ Smallest diameter of neck 2236 Diameter of body 8416 — Diameter of bottom of body . . . .236 Diameter of foot 382 [110] HYDRIA-KALPIS 65 Inv. 03.792. Kalpis, Beazley, V. A., p. 162. Danae with Perseus seated in chest; a man and two women looking at them. Height, 0.404 m. Width, 0.374 m. Greatest diameter of body, 0.308 m. Diam- eter of lip, 0.1585 m. Diameter of foot, 0.1505 m. The enclosing rectangle apparently has the ratio 1.0787 ( = .927), but no con- vincing analysis has been found. [Ill] GEOMETRY OF GREEK VASES 66 Kalpis. In v. 10.183. Eros flying towards two seated women, one of whom is plajdng with a panther. Free style. Height, 0.1775 m. Width, 0.1645 m. Diameter of bowl, 0.133 m. Diameter of lip, 0.077 m. Diameter of foot, 0.0745 m. The enclosing rectangle has the ratio .927, or 1.0787. The proportions are as follows: Height 1.000 Height, omitting foot. . . .927 Width 927 Diameter of bowl 750 = f Diameter of hp 427 = .927 — .500 Smallest diameter of neck. . .250 = J Diameter of bottom of body .309 = 927 4- 3 Diameter of foot 427 — = .927 — .500 67 Kalpis. Inv. 91.224. Beazley, V. A., p. 175. Hambidge, p. 72, fig. 14. Youth between woman and man. Free style. Height, 0.2775 m. Width, 0.2485 m. Diameter of bowl, 0.205 m. C 112] HYDRIA-KALPIS If the width is regarded as unity, the height has the ratio 1.118. The enclosing area is thus composed of two \/5 rectangles. Without its handles the vase fits inaccurately the familiar area 1.382. The ratios of the details are as follows: Height 1.118 Height to top of upper painted band . . . .9472 Height to bottom of upper painted band .736 Height to top of lower painted band 4472 Height to bottom of lower painted band .382 Width 1.000 Diameter of bowl 809 Diameter of lip 4472 Diameter of bottom of body . .236 Diameter of foot 382 Another kalpis, Inv. 91.225, Beazley, V. A.,p. 175, painted by the same hand, is of approximately the same proportions, but none of the details fits the same scheme, nor has it been possible to analyze this vase in terms of any other rec- tangle. Its height is 0.255 m., its width 0.23 m. C 113] GEOMETRY OF GREEK VASES 3 68 Kalpis. Inv. 08.417. Beazley, V. A., p. 121. Hambidge, p. 71. The death of Argos. Ripe archaic period. Height, 0.3685 m. Width, 0.332 m. Diameter of bowl, 0.28 m. The enclosing rectangle has the ratio 1.118, or ^; but some of the propor- tions of details are more simply derived from the rectangle containing the vase without its handles. This has the ratio 1.309. The vase up to the neck is contained approximately in a square; the neck and lip have the height .309. This is, how- ever, uncertain, as the shoulder is not marked by a painted band, as often happens. The diameter of the lip equals half the diameter of the bowl. The relation of the over-all rectangle to the rectangle of the bowl is indicated, as Mr. Hambidge has pointed out, by the diagonal of the whole. The chief ratios, regarding the total width as unity, are: Height 1.118 Diameter of lip 427 Width 1.000 Diameter of bottom of body 2764 Diameter of bowl 854 Diameter of foot 4045 C 114] HYDRIA-KALPIS \^n \ \ i ■-^1 j_.\j 69 69 Kalpis. Inv. 95.22. Dionysos, Ariadne, Eros, Hermes, a nymph and a seilen. Free style. Average height, 0.315 m. Width, 0.226 m. to 0.233 m. Diameter of bowl, 0.192 m. to 0.1935 m. In the drawing the vase has been placed in a 1.382 rectangle. The average height multiplied by .7236 (the reciprocal of 1.382) gives .227934. The height multiplied by .618 gives .19467, showing that the vase without the handles is contained fairly accurately in a whirling square rectangle. The ratios are: Height 1.382 Smallest diameter of neck . . .250 (or .236) Width 1.000 Diameterof bottom of body .236 Diameter of bowl 854 Diameter of foot 382+ Diameter of lip 500 [115] GEOMETRY OF GREEK VASES DEINOS-KRATER Under this heading are included vases which were used for mixing wine and water. The collection contains three examples of the deinos, or lebes, a bowl with- out foot or handles which was set on a tall stand. No. 70 is an example of Ionic pottery of the sixth century, B.C.; no. 71, which has the same main proportions, belongs to the later Attic red-figured period. No. 72, of the black-figured period, has its stand preserved. The large krater with volute handles, from Orvieto, no. 73, is the only black- figured krater in the collection which is worthy of study. The red-figured kraters are of four types. No. 74 is an attic example of the column-handled krater which was most popular in Corinthian times. No. 75 is included as an example of the volute-handled krater as it appears in the red-figured period. Its scheme of pro- portions remain obscure. Three kalyx kraters (nos. 76-78) and five bell kraters (nos. 79-83) complete the fist. Nos. 81-83 have not been analysed; and the analysis of no. 78 is not entirely satisfactory. Nos. 72, 73, 76, 79 are especially worthy of study. 70 Deinos. Inv. 13.205. Fairbanks, American Journal of Archaeology, XXIII, 1919, p. 279. Hambidge, p. 127. Height, 0.2215 m. Diameter, 0.277 m. The ratio is 1,250, or 4:5. The enclosing area divides into twenty squares, the height being four squares, the diameter five squares. The diameter of the lip is determined by the intersection of the diagonal of one of these squares with the diagonal of two and a half of the squares. The contour of the bowl has an interest- ing relation to the small squares, as is shown by the diagonals of the two pairs of squares at the lower angle, and the diagonal of half the square at the upper angle of the over-all rectangle. It is also noteworthy that the diameter of the smallest of the painted rings equals the side of one of the small squares. 71 Deinos. Inv. 96.720. Beazley, 7. A., p. 175. Athletic scenes. Freestyle. Height, 0.221 m. Diameter, 0.276 m. Diameter of lip, 0.221 m. This deinos of the late red-figured period has not only the same proportions as the preceding example, but is of the same size, the difference being less than a millimetre. The diameter of the lip is in this case exactly equal to the height. The variation in the contour is greatest at the shoulder and the lower part of the bowl. [116] DEINOS-KRATER -■:. ■; K ■■■ =Pn '"•■-.- i : \ , '■ h ^\ : p \ Ml I / ,,.......^^ ' ' :/ \ X. ill ^\ L. "--^1 ^^^ \ 70 ^■.. /^ —3 H--^ / ^ \ / .... ^ / s^ / _J 71 [117] GEOMETRY OF GREEK VASES 72 Deinos and Stand. Inv. 90.154. Ivy pattern on the neck of the deinos and on the upper member of the stand. Inside the rim of the deinos, four ships, painted in the black-figured technique. Hambidge, p. 81, fig. 9; photograph opp. p. 82. Height of deinos and stand together, approximately 0.5685 m. Height of deinos, 0.3165 m. Height of stand, 0.299 m. Diameter of lip of deinos, 0.281 m. Largest diameter of deinos, 0.388 m. Diameter of top of stand, 0.276 m. Diam- eter of foot, 0.311 m. In view of the fact that two separate pieces of pottery are combined to form one whole, and that the stand is not exactly perpendicular, some of these measure- ments are uncertain. The two pieces together, however, fit a simple scheme fairly accurately. The containing shape is made up of a square and a \/5 rectangle (1.4472). The deinos alone is placed in a rectangle which has nearly the ratio .809, i. e., it is composed of two whirling square rectangles placed vertically side by side. The diameter of the hp (.7236) and the diameter of the bowl where it joins the lip (.618) are simply derived from the .809 rectangle. The former also equals half of the total height (.7236X2 = 1.4472). The portion of the over-all rec- tangle in which the stand is contained has the ratio .7764, or .500+. 2764. The ratio .500 is represented by two squares. The reciprocal of .2764 is 3.618, or two squares plus a horizontal whirling square rectangle. The height of the upper member of the stand (two toruses with a scotia between) is half of .2764, or .1382. Many of the proportions of the details of the stand are simply expressed in terms of a whirling square rectangle applied at the bottom of the over-all shape. The smallest diameter of the stand is close to .236. The diameter of the top of the stand is .236X3, or .708. This is also the diameter of the foot above the torus at the bottom. The height of the foot is half of .236, or .118. The height of the foot and stem (up to the first moulding) is .4472. The greatest diameter of the foot is equal to the height of the deinos (.809). As Mr. Hambidge has noted, a rectangle with the diameter of the foot as the short side and the total height as the long side has the ratio 1.7888, or .4472X4. [118] DEINOS-KRATER [119] GEOMETRY OF GREEK VASES KRATER BLACK-FIGURED PERIOD 73 Volute-handled Krater. Inv. 90.153. Ann. Rep. 1890, p. 16, no. 1. Ham- bidge, p. 73. Average height of handles, 0.773 m. Width (distance apart of handles) 0.684 m. Height without handles, 0.6843 m. Height to shoulder, 0.494 m. Diameter of hp, 0.559 m. Diameter of shoulder, 0.428 m. Diameter of bowl, 0.540 m. Diameter of bottom of bowl 0.181 m. Diameter of foot, 0.307 m. The greatest height can be determined only approximately, since the four volutes of the handles are all at different levels. The ratio 1.118 gives a height of 0.7644 m., or about 8 mm. less than the average height of the handles. The pro- portions of several details (diameter of lip, division in neck, shoulder, step on foot, height of foot) are obtained simply from divisions of this 1.118 rectangle. Other details can be more directly derived from the square in which the krater is en- closed if the projection of the handles above the lip is omitted. This square is subdivided in the drawing by placing ■\/5 rectangles vertically and horizontally across its centre. The resulting figures are clearly shown in the second of the small diagrams. It is noteworthy that the krater without its handles is contained in the familiar rectangle 1.236 (= .809), i. e., two whjirhng square rectangles placed horizontally, one above the other. The body, considered separately, i. e., omitting the neck, lip, and foot, is contained in a similar area, the two whirling square rectangles in this case being placed vertically, side by side. Cf. the third and foiKth of the small diagrams. The study of this krater is interesting because of its monumental size and because of the accuracy with which most of the details are expressible in simple ratios, as follows : Height of handles (inaccurate) 1.118 Height of lip 1.000 Height to lower line of lip 9655 Height to division in neck (A) upper line 868 (B) lower line 8618 Height to shoulder 7236 Height to painted band near bottom of body 191 Height of foot (A) upper line 090 (B) lower line 0854 Width (handles) l.ftOO Diameter of Up (inaccurate) 809 Diameter of division in neck 691 Diameter of shoulder (inaccurate) 618 Greatest diameter of bowl 7888 Diameter of bottom of bowl 264 Diameter of step on foot 382 Diameter of foot 4472 C 120] KRATER '&) «si' \V\ 1 li ^\ i/j Ta/5 1 [121 ] GEOMETRY OP GREEK VASES RED-FIGURED PERIOD 74 Column Krater. Inv. 89.274. Beazley, F. A., p. 134, no. I, 6. (^) Woman with torch, and man. (B) Youth between women. " Orchard Painter." Height, 0.355 m. Width, 0.365 m. Diameter of bowl 0.282 m. The enclosing area is very close to the rectangle 1.0225, or .978, made up of a vertical whu-ling square rectangle with four horizontal whirling square rectangles beside it. In the drawing most of the proportions are obtained from subdivisions of the large whirling square rectangle. The diameter of the foot is contamed in the central square of a \/5 rectangle described on the base of the over-all shape. The ratios appear in a more familiar form when the width is regarded as unity: Height 978 Diameter at shoulder 528 Height to shoulder 691 Largest diameter of bowl '. 764 Width 1.000 Diameter of bottom of bowl 309 Diameter of lip 820 Diameter of foot 4472 [ 122 ] KRATER 75 Volute-handled Krater. Inv. 00.347. Beazley, F. A., p. 151. (JL) Apollo, Artemis, and Leto. (B) Three women sacrificing at an altar. " Painter of the Berlin Nike Hydria." Greatest height, 0.525 m. Width, 0.3915 m. Height of lip, 0.4482 m. Diame- ter of lip and of bowl, 0.322 m. Height of shoulder, 0.343 m. Diameter of foot, 0.175 m. No plausible analysis of this krater has been found. [ 123 ] GEOMETRY OF GREEK VASES 76 76 Kalyx Krater. Inv. 95.23. (A) Zeus pursuing Aegina; second woman flee- ing. (B) An old man standing, with sceptre, is approached by a woman with hands extended. Between them an altar. Ripe archaic period. Height, 0.438 m. Diameter, 0.4785 m. Diameter of foot, 0.2195 m. If the height is taken as 0.439 m., which equals twice the diameter of the foot, the containing rectangle has exactly the ratio 1.0902. This is made up of the ratios 618 and .4722. The latter is usually written .472 in this book. The ratios are as follows : Height 1.000 Height to lip 882 Height of lip 118 Height of lower member of krater .4045 Diameter 1.0902 Diameter of bottom of lip 8992 Diameter of lower member 708 Diameter of bottom of body 2764 Diameter of foot 500 IS The krater without the lip is contained in two whirling square rectangles as L shown by the diagonals. The intersections of these diagonals with the top of the lower member fix the diameter of the foot. [ 124 ] KRATEE 77 Kalyx Krater. Inv. 03.817. Beazley, V. A., p. 163. Aegina. (B) Draped youth. "Achilles Painter." Height, 0.2405 m. Diameter, 0.244 m. In the drawing the krater has been placed in a square, the height and diameter being taken as 0.242 m. If ■\/2 rectangles are applied at each side, the foot is contained in the overlapping portion (.4142). The diameter of the bottom of the body is half that of the foot (.2071). The height of the lower member of the krater is slightly less than .4142. Other details are ac- counted for by geometrical divi- sions of the VS rectangle. But the analysis is not altogether convincing. (A) Zeus pursuing 78 Kalyx Krater (B) Draped youth. Height, 0.189 m. Diameter, 0.194 m. The ratio is assumed to be 1.0225 (.978), the height being regarded as unity. The diameter of the foot is obtained by apply- ing whirling square rectangles at each side, the foot being con- tained in the overlapping portion (.4045). The diameter of the bottom of the body is .20225, or half of .4045. The height of the lower member is also .4045. No other simple ratios have been found, and the analysis is not to be regarded as satisfactory. Inv. 03.796. (A) Youth with a goat, approaching Hermes. C 125 ] GEOMETRY OF GREEK VASES 79 BellKrater. Inv. 10.185. Beazley, V. A.,p. 113. Hambidge, p. 88. (A) Artemis killing Aktaeon. (B) Pan pursuing a shepherd. " Pan Painter." The height varies between 0.370 m. and 0.372 m., the diameter between 0.420 m. and 0.428 m. The diameter of the foot is 0.211 m. If the diameter of the vase be taken as twice the diameter of the foot, 0.422 m., and the height be taken as 0.372 m., the over-all rectangle has the ratio 1.1338, or .882. The latter is .382+. 500, a simple shape which accounts for all the details. The main proportions are made clear in the four small diagrams. In diagram A the two squares (.500) are placed above the .382 shape which may be expressed as a square flanked by two whirling square rectangles on each side. The diameter of 5 5 the bottom of the bowl equals the side of this square, or .382. In diagram B the two squares are placed below; perpendiculars dropped from their centers deter- mine the diameter of the foot. The square .382 is used above as the central square of a -\/5 rectangle the ends of which determine the largest diameter of the bowl. Diagram C shows that diagonals of half the shape intersect the sides of the rec- tangle enclosing the bowl at the level of its top. This proves that the vase without the lip and handles is of the same shape as the complete vase, a phenomenon which appears also in the kantharos, no. 121. Diagram D shows that the height of the krater to the level of the greatest projection of the lip is equal to the diameter of C 126 ] KRATER the bowl. The height of the foot, and the levels of the top and bottom of the maeander band on which the figures on side B rest are obtained by simple sub- divisions of the square .382, as is shown in the large drawing. The ratios, regarding the diameter as unity, are as follows: Height 882 Height of foot 090 Height to greatest projection of lip .854 Diameter 1.000 Height without lip 7532 Diameter of top of bowl 854 Height to top of maeander band. . .236 Diameter of bottom of bowl 382 Height to bottom of maeander band .191 Diameter of foot 500 C 127] GEOMETRY OF GREEK VASES 80 Bell Krater. Inv. 00.348. (A) Athena playing the flutes in the centre of a group. (B) Three seilens and a maenad. Free style. Height, 0.34 m. Diameter, 0.387 m. The rectangle has the ratio .882 (1.1338), i. e., 5.000+.382. The proportions are: Height 882 Height to top of bowl 809 Height of foot 090- Diameter 1.000 Diameter at top of bowl 764 = .382 X 2 Diameter of bowl at level of handles 708 Diameter of bottom of bowl 382 Diameter of foot 416 C 128 ] KRATER 81 81 Bell Krater. Inv. 95.24. Beazley, V.A.,p. 184. Sacrifice of a sheep. Free style. Height, 0.4136 m. Width (handle to handle), 0.423 m. Diameter of lip, 0.411 m. The containing shape appears to be a 1.0225 rectangle. If the diameter of the lip is regarded as the width, the shape is very close to a square. No satisfactory analysis of the details has been found. [ 129 ] GEOMETRY OF GREEK VASES 82 BellKrater. Inv. 00.346. Beazley, V. A., p. 173. (A) Aktaeon attacked by dogs in the presence of Artemis, Lyssa, and Zeus. (B) Youth standing between two women. "Lykaon Painter." Height, 0.377 m. Width (handles), 0.441 m. Diameter of Up, 0.413 m. Diameter of foot, 0.187 m. The over-all ratio is 1.1708 (= .854). The diameter of the foot is close to half the height. No satisfactory anal- ysis has been found. 82 83 BellKrater. Inv. 76.50. (A) Bearded man, woman with flutes and youth. (B) Three youths. Free style. Height, 0.2935 m. Width (handles), 0.341 m. Diameter of lip, 0.322 m. The enclosing rec- tangle has the ratio 1.1618, which is that of the ground plan of the Erechtheum. No significant proportions have been noted. 83 C 130 ] PSYKTER RED-FIGURED PSYKTER (WINE-COOLER) 84 Inv. 01.8019. Beazley, V. A., p. 29, no. V, 8. Hambidge, p. 99. Athletes and trainers. Height, 0.3435 m. Diameter, 0.269 m. The ratio is 1.2764, expressed geometrically as a square with a 3.618 (.2764) rectangle added. The proportions of the lower part of the vase can be expressed in simple terms of a whirling square rectangle applied at the bottom of the containing area. The top of this rectangle coincides with the painted band on which the figures stand. The diameter of the Up equals one-half of the diameter of the vase. The height and diameter of the shoulder are clearly expressed in terms of a .309 rectangle applied at the top of the containing area. [131] GEOMETRY OF GREEK VASES The ratios are: Height 1.2764 Diameter of lip 500 Height to shoulder 1.0854 Diameter of shoulder 382 Height of neck and Up 191 Smallest diameter of stem 382 Height to painted band below figures .618 Diameter of ring at top of foot . . .4472 Diameter 1.000 Diameter of foot 5669 OINOCHOE (WINE JUG) Under this name are included sixteen vases, four of the black-figured and twelve of the red-figured period. Of the black-figured examples three have a tre- foil lip; in two of them the line of the shoulder is accentuated plastically; in the third it is marked only by a painted band. The fom-th is of the shape to which the name " olpe " is often given. It has a large cylindrical mouth, and the neck merges gradually into the body. The red-figured examples include three oinochoai with trefoil lip, but differing widely from one another in proportions, and nine vases, which might be designated by the name "olpe," in that all of them have large cylindrical mouths, but which are otherwise very dissimilar. The last three examples, because of their shape and small size, appear to have been drinking mugs rather than wine pitchers. BLACK-FIGURED PERIOD 85 Oinochoe with Trefoil Lip. Inv. 98.924. Herakles leading a sphinx. Height of handle, 0.173 m. Height of lip, 0.157 m. Diameter, 0.107 m. The vase is enclosed in a whirling square rectangle, but the proportions of de- tails are obtained more clearly from the rectangle 1.472, which is obtained by dividing the diameter into the height of the lip. If a horizontal whirling square rectangle is applied at the bottom of the over-all shape, the intersection of its diagonal with the diagonal of the 1.472 rectangle fixes the diameter of the bottom of the body. The diameter of the foot is obtained from the intersection of the 1.472 diagonal with the horizontal line at half the height of the whirling square rectangle. The diameter of the shoulder is given by the intersection of the 1.472 diagonal with the horizontal line at the level of the shoulder. This level is at the height 1.090, or 1.472 -.382. The height of the neck and lip is thus .382, and the width of the lip is seen to be .382 X 2, or .764. The smallest diameter of the neck is also .382; and the width of the spout is .236, i. e., it is enclosed in the whirling square rectangle in the centre of the .382 (2.618) shape. The ratios are: Height to top of handle 1.618 Width of spout 236 Height to top of Hp 1.472 Smallest diameter of neck 382 Height of neck and lip alone 382 Diameter of shoulder 481 Height to shoulder 1.090 Diameter of bottom of body 4086 Diameter 1.000 Diameter of foot 5801 Width of lip 764 C 132 ] OINOCHOE [ 133 ] GEOMETRY OF GREEK VASES 86 Oinochoe with Trefoil Lip. Inv. 99.527. Butcher cutting up meat. Height, with handle, 0.25 m.; without handle, 0.211 m. Diameter, 0.154 m. This vase is also enclosed in a whirling square rectangle, and its analysis is remark- ably simple. The proportions can be expressed arithmetically as follows : Height to top of handle 1.618 Projection of handle above lip . . .250 Height to top of lip 1.368 Height to shoulder 9635 Diameter 1.000 Width of mouth 750 Width of spout 236 Width of handle 191 Diameter of shoulder 500 Diameter of bottom of body .... .4472 Diameter of foot 5669 [ 134 ] OINOCHOE 87 Oinochoe with Trefoil Lip. Inv. 13.74. Dionysos reclining under a grape- vine. Height to top of handle, 0.228 m.; to top of lip, 0.216 m. Diameter, 0.1335 m. The over-all ratio is 1.708. If the projection of the handle above the lip is dis- regarded the ratio is 1.618. This whirling square rectangle has been used in the analysis because of its greater simplicity. The ratios are : Height to top of handle 1.708 Height to top of Up 1.618 Height to shoulder 1.146 Height of neck and lip 472 Diameter 1.000 Width of lip 764 Smallest diameter of neck 382 Diameter of shoulder 528 Diameter of bottom of body 333 Diameter of foot 5669 C 135] GEOMETRY OF GREEK VASES 88 88 Olpe. Inv. 03.783. Herakles in the beaker of Helios, Height to top of handle, 0.219 m. ; to top of lip, 0.2096 m. Diameter, 0.113 m. If the handle is included the enclosing rectangle is of the shape 1.927, = 1.000+ .618+.309. Without the handle the ratio is 1.854, = .618X3. The proportions are as follows: Height to top of handle 1.927 Height to top of lip 1.854 Diameter 1.000 Diameter of Kp 7888 Smallest diameter of neck 618 Diameter of bottom of body 4472 Diameter of foot 6584 [136] OINOCHOE RED-FIGURED PERIOD 89 Oinochoe with TrefoU Lip. Inv. 97.370. Beazley, V. A., p. 145, no. 24. Apollo and Artemis. "Altamura Painter." Height to top of handle, 0.3465 m. Diameter, 0.168 m. The rectangle has the ratio 2.0652, or 1.000 + .618 + .4472. The propor- tions are: Height 2.0652 Height to lip at ends . 1 .7562 =2.0652 - .309 Height to shoulder . . 1.2764 Diameter 1.000 Diameter of hp 666 Distance apart of ends of handle 618 Width of handle 1545 = .309 h-2 Width of spout 309 Smallest diameter of neck 382- Diameter of shoulder . .5528 Diameter of bottom of body 427 Diameter of foot 618 C 137 ] GEOMETRY OF GREEK VASES 90 Oinochoe with Trefoil Lip. Inv. 03.794. Hambidge, p. 56. Young warrior standing before seated old man. Height to top of handle, 0.186 m. Diameter, 0.1325 m. The vase is contained in a -\/2 rectangle. The ratios are: Height to top of handle 1.4142 Height to top of spout 1.3685 Height to shoulder 1.0355 Height to bottom of painted band on shoulder 1.000 Diameter 1 000 Width of lip 6465 — Smallest diameter of neck 333 Diameter of shoulder 4142 Diameter of bottom of body 60952 Diameter of foot 6465— C 138] OINOCHOE 91 91 Oinochoe with Trefoil Lip and Squat Body. Inv. 00.352. Beazley, F. A., p. 177. Seilen and two women. Free style. Height, 0.212 m. Diameter, 0.1725 m. These dimensions give the ratio 1.236, i. e., 618X2, or two whirhng square rectangles. The proportions are: Height 1.236 Smallest diameter of neck 5000 Diameter 1.000 Diameter at bottom of body 708 Width of lip 7236 Diameter of foot .7888- [ 139 ] GEOMETRY OF GREEK VASES 92-95 Four Oinochoai with Cylindrical Mouth. Inv. 13.192, 13.191, 13.196, 13.197. Beazley, 7. A., p. 157. 92. Twokomasts. 93. Two athletes. 94. Greek and Persian. 95. Seilen and maenad. Early free style. The paintings, according to Beazley, are perhaps by the painter of the Chicago stamnos. These four jugs should be studied as a group. The paintings are all by the same hand, and the vases are evidently from the same factory. They resemble one another very closely in dimensions, contour, and perfection of workmanship. Their walls are so thin as to make their weight astonishingly slight. All are covered by a deep, lustrous black varnish. The actual dimensions are given in table 1. The principal differences are seen to be in the height of the body which is equal to the diameter in nos. 4 and 5, less by 4 mm. in no. 6, and by 9 mm. in no. 7. The diameter of the lip remains almost the same, but the diameter of the foot varies considerably. The mouldings of the lip and foot are nowhere exactly duplicated. Nos. 92-94 have one indented ring around the neck below the lip; no. 95 has two such rings. All four vases have a band with a painted tongue pattern marking the shoulder, and a maeander band encircling the body, as a ground for the pictures. [ 140] OINOCHOE TABLE 1. DIMENSIONS 92 94 95 Height to top of handle Height to top of Hp Height to top of painted band on shoulder . Diameter Diameter of lip Smallest diameter of neck Diameter of bottom of body Diameter of foot 0.247 m. 0.200 0.1525 0.1525 0.098 0.078 0.057 0.082 0.245 m. 0.199 0.152 0.152 0.097 0.078 0.061 0.086 0.240 m. 0.193 0.1475 0.1515 0.0965 0.0765 0.0575 0.0825 0.229 m. 0.1855 0.141 0.150 0.094 0.065 0.0685 0.086 In view of the fact that the diameter decreases from 0.1525 m. in no. 92 to 0.150 m. in no. 95, the correspondences and variations in proportions can be more easily noted in the following table of ratios, in which the diameter is taken as unity: C 141 ] GEOMETRY OF GREEK VASES TABLE 2. RA.TIOS 92 93 94 1.618 1.618- 1.5854 1.309 1.309 1.2764 1.000 1.000 .9674 1.2135 1.2135 1.1809 .309 .309- .309 .309 .309 .309- .4045 .4045 .382 1.000 1.000 1.000 .618+ .618+ .618+ .500+ .500+ .50,0+ .382 .4045 .382 .528+ .5669 .5528 95 Height to top of handle Height to top of lip Height to top of painted band on shoulder . Height to indented ring on neck Projection of handle above Up Height of neck and lip Height to top of maeander band on body . . Diameter Diameter of Up Smallest diameter of neck Diameter of bottom of body Diameter of foot 1.528 1.236 .944 1.146 .292 .292 1.000 .618+ .500 .4472 .5669 C 142 ] OINOCHOE " F ■> — J) 'i ^ , ■ "r-- — ^ -r \ \ ■ ■ I 'M= ^v I '7^" i \ ; \ ,' \ ""•■ 1 'K 1 \ '■. y '■■-..; !"'■, // ■ \ /t-. : '■'• V : ■••■ y / )h \ \ '/lix^^ ■ ^•■■ i\ ••--■ ;/ I :) ••••, 95 The table shows that nos. 92 and 93 are enclosed in a whirling square rec- tangle. And, with the exception of the foot, the proportions of details fit exactly the same simple scheme obtained from the containing shape. The body in each case is a perfect square. The portion above the shoulder is enclosed in the reciprocal of the over-all rectangle, and the Up is at half the height of the recip- rocal. The diameter of the neck is half the diameter of the body. The indented ring on the neck is at three-quarters of the total height, the top of the maeander band is at one-quarter of the height. No. 94 is identical with nos. 92 and 93 as regards its handle, neck and Hp, even to the placing of the indented ring, but the height of the body has been reduced. No. 95 is of a still more squat shape, and the height of neck and handle has been reduced proportionally to that of the body. C 143] GEOMETRY OF GREEK VASES 96 Olpe. Inv. 13.93. Comic actor. Height to top of lip, 0.182 m. Diameter, 0.129 m. The ratio is that of the y/^ rectangle, 1.4142. The proportions of details are: Height to top of lip 1.4142 Height to top of ring on shoulder. .9428 Height to bottom of ring on shoulder .9142 Diameter 1.000 Diameter of hp 7777 Smallest diameter of neck 5858 Diameter of shoulder 6394 Diameter of bottom of body 6394 Diameter of foot 6666 C 144 ] OINOCHOE i^H U-" , ,. . , ^ \; >-\i / '''' ,-*' i \ 1 A -- . N \ N 1 7 »> \ 1 /\ ''. \ 1 i / ^ '^^. \ I / \ '• \ / \ * \ / \ \/ / \ / \ / \ \ , \ \ •• ■^-- - '1 *^» ! ^ /I |\ '••, : \ /I i \ I \ / 1 \ I \ / \ ^* ' '^ / '■•, 1 \ / \ >i, \../.... \ •y . \ : "'■-,, y \/ : /^^if-^-'i 1 c ; 1-' i~'-\' 97 97 Olpe. Inv. 76.51. Hambidge, p. 128. No painted decoration. Height, 0.1995 m. Diameter, 0.1235 m. These figures are very close to 2.000 and 1.236, which give the ratio of the whirUng square rectangle, 1.618. The ratios are: Height to top of lip 1.618 Diameter 1.000 Diameter of lip 7888 Smallest diameter of neck 528 Diameter of foot 708 C 145 ] GEOMETRY OF GREEK VASES 98 98 Olpe. Inv. 95.56. Beazley, V. A., p. 158. Dancing seilen. "Euaion Painter." Height, 0.1065 m. Diameter, 0.10 m. The ratio is obviously 1.0652, i. e., .618 +.4472, a whirling square rectangle plus a V5 rectangle. Cf . the oinochoe no. 89, whose ratio is 2.0652. On account of the diminutive size the geometrical analysis of some of the de- tails has been omitted. The ratios are : Height 1.0652 Height omitting base 1.000 Height of lower termination of handle 618 Diameter 1.000 Diameter of lip 9472? Smallest diameter of neck 764 Diameter of bottom of body 8292? Diameter of foot 8944 Width of handle 236 C 146 ] OINOCHOE 99 Olpe. Inv. 97.606. Eros flying. Height and diameter, 0.075 m. The vase is enclosed in a square. 100 Olpe. Inv. 00.339. Beaz- ley, V. A., p. 92. Youth dancing with castanets; woman playing flutes. ' ' Brygos Pain ter . " Height to top of lip, 0.0805 m. Diameter, 0.0905 m. Theratiois 1.118, or — . The proportions are: Height 1.000 Height to shoulder 750 Diameter ., 1.118 Diameter of lip 1. US- Diameter of neck 882 Diameter of base 944 r -A Mi 100 C 147 ] GEOMETRY OF GREEK VASES SKYPHOS This form of drinking cup has a long history. In addition to the fourteen Attic examples here published, three earlier skyphoi are illustrated for the sake of com- parison. The first is of the so-called Proto-Corinthian style, — a small cup of very fine workmanship with walls of egg-shell thinness, and decoration consisting of numerous horizontal bands. The second is a Corinthian skyphos. The third example, of the same general shape as the preceding one, is presumably to be classed as Corinthian. 101 Inv. 03.809. Height, 0.0875 m. Width, 0.138 m. Diameter of bowl, 0.1005 m. Diameter of foot, 0.035 m. The diameter of the bowl is two and one-half times that of the foot (0.035X2.5 = 0.875). The total width is close to four times the diameter of the foot (0.035X4 = 0.140). If any scheme of proportion was used it appears to have been of the "static" type. 102 Inv. 95.14. Fowler and Wheeler, Greek Archaeology, p. 449, fig. 365. {A) Two warriors. {B) Two leopards with a palmette-lotus design between them. Height, 0.091 m. Width, 0.198 m. Diameter of bowl, 0.139 m. Diameter of foot, 0.07 m. The over-all ratio is close to 2.1708. The skyphos without the handles fits a 1.528 rectangle, and the diameter of the foot is sHghtly greater than one-half the diameter of the bowl. 101 102 C 148] SKYPHOS "^ ! ■• • ..■■ I /' UT^'>? fv-i/: : y--' -: ^^ • 1 103 103 Inv. 97.366. (A) Two lions. (5) Palmette-lotus design, fine workman- ship. Height, 0.112 m. Width, 0.2315 m. Greatest diameter of bowl, 0.165 m. Diameter of lip, 0.162 m. Smallest diameter of bowl, 0.793 m. Diameter of foot, 0.095 m. The enclosing area apparently has the ratio 2.0652 (cf. the oinochoe no. 89). If the handles are omitted the rectangle has the familiar ratio 1.472. The diam- eter of the foot has the ratio .854. If an .854 rectangle is constructed in the centre of the containing area, and a whirHng square rectangle applied at the bottom, the top of the upper painted band is seen to be at half the height of the whirling square rectangle. The ratio of the height of this whirling square rectangle is .528. This equals 1.000-.472. The ratio of the height of the bowl without the foot is .944, or .472X2. The area on each side between the edge of the foot and the edge of the bowl is made up of two whirling square rectangles (.309). The ratios are: Height 1.000 Diameter of bowl 1.472 Height of bowl without foot 944 Diameter of bottom of bowl . . . .708 (?) Width 2.0652 Diameter of foot 854 In view of the small size of these two Corinthian skjrphoi, the complicated sys- tems of proportion which they reveal are probably not to be regarded as due to conscious design. The Attic black-figured and red-figured skyphoi, on the other hand, have proportions most of which can be expressed in simple ratios and by means of simple geometrical constructions. BLACK-FIGURED PERIOD 104 Large Skyphos. Inv. 20.18. Robinson, Catalogue, no. 372. Hambidge, p. 106. (A) Frieze of warriors riding on dolphins. (5) Frieze of youths riding on ostriches. Height, 0.16 m. Width, including handles, 0.298 m. Greatest diameter of bowl, 0.221 m. Smallest diameter of bowl, 0.099 m. Diameter of foot, 0.134 m. C 149 ] GEOMETRY OF GREEK VASES 104 ^1 / / \ r w / -1 .618- .618- -.618- -UI8. ^ I ( ■^ V W ' 5 V J S W sV A W !- -I if236 K382HI s— .618 —J f'.382-*236« The ratios are : Height 1.000 Width including handles 1.854 (error, 0.00128 m.) Greatest diameter of bowl 1.382 Smallest diameter of bowl 618 Diameter of foot 854 (error, 0.00264 m.) The proportions are clearly shown in the large drawing, and the simplicity of the scheme appears in the three small diagrams. This skyphos and one in the MetropoUtan Museum furnish one of the rare instances in which two vessels of the same type are constructed on identically the same scheme throughout, except for the disposition of the painted ornament. In the present example there is a very shght error in the total width, and a slightly greater error in the diameter of the foot. Cf. above, p. 29 and Hambidge, op. cit., where drawings of both skyphoi are published. 105 Large Skyphos. Inv. 99.523. Ann. Rep. 1899, p. 64, no. 21. (A) Mounted Amazon in combat with an egg-shaped monster. (B) Lion attacking four bulls. Height, 0.157 m. Width, 0.295 m. Diameter of bowl, 0.2205 m. Diameter of foot, 0.130 m. [ 150] SKYPHOS Without the handles the skyphos is contained in a -s/2 rectangle. The ratios are : Height 1.000 Width 1.8787 = 1.00 + (.2929 X 3) Diameter of bowl -...'. 1.4142 = \/2 Diameter of bottom of bowl 6143 = .2071 X 3 Diameter of foot 8284 = .2071 X 4 ^ ^~^ '"^ ^ — h. r V : :/ n \ n 106 Large Skyphos. Inv. 99.524. {A) Amazon holding horse. (5) Man holding horse. Height, 0.168 m. Width, 0.3005 m. Diameter of bowl, 0.2255 m. Diameter of foot, 0.141 m. The whole vase is apparently contained in four \/5 rectangles, the ratio being 1.7888 = .4472X4; and the bowl is contained in three s/h rectangles, having the [151] GEOMETRY OF GREEK VASES ratio 1.3416 = .4472X3. The diameter of the bottom of the bowl has the ratio .618. The foot apparently has the ratio .8416 = 1.3416 -.500. TABLE OF RATIOS OF THE THREE LARGE SKYPHOI No. Height Width Diameter of bowl Diameter of bottom of bowl Diameter of foot 104 105 106 1.000 1.000 1.000 1.854 1.8787 1.7888 1.382 1.4142 1.3416 .618 .6143 .618 .854 .8284 .8416 1 \ ' ' '• * 1 : -'Y N ■' ■■■...^■•■'■•■- /i I ' ■J 107 107 Small Skyphos. Inv. 88.319. No figure decoration. Height, 0.105 m. Width, including handles, 0.188 m. Diameter of bowl, 0.1315 m. Diameter of foot, 0.076 m. The ratio is 1.7888 ( = .4472X4), four V5 rectangles. The ratio, excluding the handles is 1.236, i. e., two whirhng square rectangles. The diameter of the bottom of the bowl is one-third of the total width, expressed geometrically in the drawing by the intersection of the diagonal of the whole shape with the diagonal of half of the shape. The ratio of the height to the diameter of the foot is 1.382. If a square be subtracted from this shape, the remainder, .382 ( =2.618), with its simple sub- divisions, expresses the proportions of the details of the foot. The ratios are : Height 1.000 Width 1.7888 Diameter of bowl 1.236 Diameter of bottom of bowl 5963 Diameter of foot 7236 108 Small Skyphos. Inv. 97.373. No figure decoration. An incised line near the middle of the bowl; painted bands near the bottom. Height, 0.083 m. Width, 0.1535 m. Diameter of bowl, 0.093 m. Diameter of foot, 0.049 m. [ 152 ] SKYPHOS 108 The enclosing area has the ratio, 1.854, i. e., it is composed of three whirling square rectangles. The ratio of the bowl is 1.118, or — . The proportions of de- tails can all be accurately expressed in terms of the V5 rectangles composing the area of the bowl. The ratios are: Height 1.000 Width 1.854 Diameter of bowl 1.118 Diameter of bottom of bowl 500 Diameter of foot 591 109 109 Small Skjrphos, with off-set lip, and without foot. Inv. 76.48. {A) and {B) Seilen and two maenads. C 153 ] GEOMETRY OF GREEK VASES Height, 0.0845 m. Width, including handles, 0.1555 m. Greatest diameter, 0.104 m. Diameter of base, 0.037 m. The ratios are : Height 1-000 Width 1-854 =.618X3 Greatest diameter of bowl 1-236 = .618 X 2 Diameter of base 4472 = 110 110 Skyphos of unusual shape, without foot, and with vertical handles. Inv. 08.292. Ann. Rep. 1908, p. 61. Erotic scenes. Height, 0.108-0.111 m. Width, 0.187 m. Diameter of bowl at top, 0.135 m., at bottom, 0.102 m. The enclosing rectangle apparently has the ratio 1.691. If the handles are omitted the area has the familiar ratio, 1.236, or two whirling square rectangles. The ratios are: Height 1.000 Width 1.691 Diameter of bowl at top 1.236 Diameter of bowl at bottom 927 RED-FIGURED PERIOD 111 Skyphos, Inv. 76.49. Hambidge, p. 108. Laurel wreath below lip; near bottom a reserved band with cross-hatching. 112 Skyphos. Inv. 01.8032. Beazley, V. A., p. 130. (A) The rising of Kore. (B) Maenad and seilen. "Penthesilea Painter." 113 Skyphos. Inv. 01.8076. Beazley, 7. A., p. 65. Hambidge, p. 109. (A) Herakles and man. (B) Man addressing youth. [ 154 ] SKYPHOS ._J 112 114 Skyphos. Inv. 01.8097. Beazley,F. A.,p. 65. (A) Nestor and Euaichme. (B) Aktor and Astyoche. 115 Skyphos. Inv. 10.176. Beazley, V. A., p. 90. Hambidge, p. 111. Ath- letes practising the jump. " Brygos Painter." 116 Skyphos. Unregistered. Indecent. 117 Skyphos. Inv. 13.186. Signed by Hieron and Makron. Beazley, V. A., p. 101. Hambidge, p. 109. The seduction and return of Helen. The measurements of the skyphoi are noted in Table 1. Table 2 gives the ratios of the various parts, the height being taken as unity in each case. [155] GEOMETRY OF GREEK VASES 114 TABLE 1. DIMENSIONS No. Height Width including handles Greatest diameter of bowl Smallest diameter of bowl Diameter of foot Ill 0.096 m. 0.182 m. 0.118 m. 0.0715 m. 0.0785 m. 112 0.228 0.398 0.279 0.177 0.194 113 0.183 0.3245 0.2255 0.137 0.1455 114 0.177 0.308 0.213 0.132 0.1415 115 0.145 0.2605 0.179 0.109 0.1225 116 0.0905 0.161 0.101 0.065 0.072 117 0.215 0.389 0.279 0.159 0.180 C 156 ] SKYPHOS lis 116 TABLE 2. RATIOS No. Height Width including handles Greatest diameter of bowl Smallest diameter of bowl Diameter of foot Ill 1.000 1.8944 1.236 .764 .809 112 1.000 1.750(?) 1.236 .764(?) .854 113 1.000 1.764 1.236 .764(?) .7888 (?) 114 1.000 1.750 1.200 .750 .800 115 1.000 1.809 1.236 .764 .8396 116 1.000 1.764 1.118 .736 .809 117 1.000 1.809 1.309 .736 .8396 C 157] GEOMETRY OF GREEK VASES The most striking result of the study of these skyphoi is the reappearance in four of them of the famiUar ratio 1.236, i. e., two whirhng square rectangles, as the rectangle enclosing the bowl. The same shape was also used for skyphoi of the black-figured period, as we have seen (nos. 107, 109, 110) and Mr. Hambidge has noted several other examples in New York and New Haven {op. cit, pp. 110 ff., figs. 13, 14, 16, 17, 18 in addition to the skyphos by Hieron in the British Museum, The Diagonal I, p. 114). In the four examples here published the projection of the handles varies, as does the diameter of the foot. But the diminution of the diameter of the bowl is practically the same in all four, the lower diameter being expressed by the ratio .764 exactly in two cases, less accurately in the other two. The skyphos, no. 4, though its proportions fall in the " static" class, being expressed in squares, does not differ noticeably in appearance from the four ex- amples just mentioned. The last two skyphoi, on the other hand, vary strikingly from what may perhaps be called the normal shape. No. 6, a small, but beauti- fully made skyphos, is slenderer with a slighter diminution from top to bottom. No. 7, the famous skyphos signed by Hieron and Makron, is abnormal in its breadth, in the excessive diminution, and in its contours. From the point of view of form it is perhaps the least pleasing of the seven. The shape seems to have been chosen with a view to affording a more advantageous field for the pictorial com- position — a frieze of numerous large, closely spaced figures. [ 158] TWO-HANDLED CUP 118 TWO-HANDLED CUP 118 Inv. 01.8023. Ann. Rep. 1901, p. 34, no. 23. Impressed decoration. A and B, Perseus and Medusa. The whole surface of the vase is covered with black varnish. Height, 0.117 m. Width, 0.170 m. Diameter of lip, 0.119 m. Diameter of bowl, 0.1175 m. The over-all ratio is 1.4472, i. e., a square plus a \/5 rectangle. If the handles are omitted the enclosing shape is very close to a square. The proportional rela- tion of details is shown in the drawing. The ratios are as follows: Height 1.000 Width 1.4472 Diameter of lip 1.000 Diameter of bowl 1.000 Diameter of shoulder 8944 = .4472 X 2 Diameter of bottom of bowl 500 Diameter of foot 5528 (inexact) C 159] GEOMETRY OF GREEK VASES KANTHAROS As Professor Tarbell has pointed out in his publication of no. 121, this type of wine-cup is very frequently represented in vase paintings and on marble reliefs, while comparatively few actual examples of the best period have survived. This fact suggests that kantharoi were generally made of another material than terra- cotta; and the delicacy of handles and stems leaves little doubt that this material was metal — gold, silver or bronze. Of the five kantharoi here published, all, except possibly the first, appear to follow closely metal prototypes. It happens that three of them are signed, nos. 119, 120 being from the famous pottery of Nikosthenes, no. 121 from the equally famous pottery of Hieron. No. 122, though unsigned, is in all probability to be assigned to Brygos, since the decora- tion is by the "Brygos Painter." The name SENO#ANTO, incised on the bottom of no. 123, is probably that of the owner rather than the maker. Numbers 121, 122, 123 are among the most interesting examples of "dynamic symmetry " in the collection. 119 119 Kantharos. Inv. 00.334. Beazley, V. A., p. 23. (A) Above, Dionysos resting. Below, Herakles and the lion. (B) Above, Sacrifice. Below, Herakles and the bull. Signed by Nikosthenes as potter. " Painter of the London Sleep and Death." The proportions of this vase cannot be determined, because the weight of the handles pulled the bowl out of shape before the firing. The lip is oval instead of circular, and the enclosing rectangle is lower and wider than was intended. In the drawing an attempt has been made to restore the original appearance by increas- ing the height of the handles by 4 mm., and decreasing the width proportionally. The dimensions thus obtained are: height of handles, 0.237 m. width, 0.294m. This rectangle is close to the familiar shape 1.236. The rectangle in which the kantharos without its handles is contained is also nearly of this shape. C 160] KANTHAROS 120 120 Kantharos. Inv. 95.61. Beazley, 7. A., p. 23. Erotic scenes. Signed by Nikosthenes as potter. " Painter of the London Sleep and Death." Height of one handle, 0.238 m., of the other 0.235 m. Average: 0.2365 m. Width (handle to handle) 0.278 m. Height of lip, 0.1635 m. (average). Diameter of lip, 0.2025 m. Diameter of foot, 0.1015 m. The over-all ratio is .854 (1.1708) ; and the ratio of the rectangle withthe area above the lip omitted is .5854 (1.708). That these two ratios are related is ap- parent, but the geometrical analysis is complicated. The vase without the handles is enclosed in the simple rectangle 1.236 (.809). This shape has bee n used as the basis of the analysis. It can be seen at once that the height of the bowl (including the ring at the top of the stem) equals half its diameter, and that the diameter of the foot also equals half the diameter of the bowl. The upper of the two horizontal lines near the bottom of the bowl is at half the height of the lip. The ratios, regarding the height of the bowl as unity, are as follows : Height to Up 1.000 Height of upper horizontal line near bottom of bowl 500 Height of bowl without stem 618 Height of stem to bottom of ring 382 Diameter of bowl , 1.236 Diameter at horizontal Hne near bottom 708 ' Diameter of stem at top 146 Diameter of foot 618 [161] GEOMETRY OF GREEK VASES 121 Kantharos. Inv. 95.36. Beazley, V. A., p. 90. Tarbell, University of Chicago Decennial Publications, 6 (1902), p. 3, pi. 2-3. Hambidge, p. 68. (A) Zeus pursuing Ganymede. (B) Zeus pursuing a nymph. By the Brygos painter, and, in all probability, from the factory of Brygos. Height to top of unbroken handle, 0.241 m. Height to lip, 0.1675 m. Greatest width (handle to handle), 0.270 m. Diameter of bowl, 0.1885 m. Cf . introduction, pp. 30 ff. The total width divided by 1.118 (-^) equals 0.24148, which is within half a millimeter of the height of the unbroken handle. Every detail of the vase is clearly expressible in terms of the rectangle 1.118. If the area above the lip is omitted, the remainder is a whirling square rectangle. If the area below the bottom of the bowl is omitted, the remainder is again a whirling rectangle. In other words the total area may be regarded as two overlapping whirling square rectangles, and the bowl is contained in the overlapping portion. If a vertical whirling square rec- tangle be applied at either end it defines the diameter of the stem at its junction with the body, the stem being contained in the overlapping portion. The diam- eter of the hp is fixed by the intersection of the diagonals of half the shape with the horizontal line at the level of the lip. This shows that the main proportions of the kantharos without its handles are the same as those of the kantharos with its handles. The diameter of the ridge near the bottom of the bowl equals half the total height. It coincides with the central square of each of the two \/5 rectangles [162 ] KANTHAROS of which the area is composed. The diameter of the foot is obtained from the intersection of two whirUng square rectangle diagonals. It is noteworthy that, as in the kantharos no. 122, the diameter of the stem at its junction with the body is also determined by the intersection of the diagonals of the large whirling square rectangle applied at the bottom of the total area. These points can also be determined in a third manner. If a semicircle be described with the centre of the lip as centre and the radius of the lip as radius its intersections with the cen- tral square of the lower VS rectangle determine both the height and the diameter of the ridge near the bottom of the bowl, and its intersections with the sides of the two overlapping vertical whirling square rectangles determine the height and diameter of the top of the stem. The bowl is practically contained in this semi- circle (cf. the semicircle in no. 120). The following table of proportions gives the ratios, (a) with the height re- garded as unity, (6) with the width regarded as unity. A B Height to top of handles 1.000 .8944 Projection of handles above lip 309 .2764 Height of lip 691 .618 Height of upper line near bottom of bowl 4045 .3618 Height of lower line near bottom of bowl 3944 .3527 Height of stem 309 .2764 Width 1.118 1.000 Diameter of bowl 7725 .691 Diameter of ridge near bottom of bowl 500 .4472 Diameter of stem at junction with bowl 118 .1055 Diameter of foot 4045 .3618 122 Kantharos. Inv. 98.932. Signed by Hieron as potter. Gigantomachy, Pollack, Zwei Vasen aus der Werkstatt Hierons, p. 28, pi. 4, 5. Beazley, V. A., p. 109. The vase has been put together from many fragments, and some pieces are missing, especially the greater portion of the bottom of the bowl and the upper part of one handle. But the stem is complete, and enough of the bottom of the bowl is preserved to give almost its entire contour; and the incomplete handle has been accurately restored to match the other. Under these chcumstances an in- vestigation of the proportions seemed feasible. The height of the complete handle is 0.262 m. The greatest width (handle to handle) is 0.277 m. These dimensions make a rectangle of the shape 1.0557, more easily intelligible in terms of its reciprocal, .9472, i. e., 500 +.4472 or half a square plus a VS rectangle; and the geometrical scheme, though necessarily complicated because of the numerous elements to be accoimted for, is clear and consistent throughout. The three small diagrams will help to make this scheme intelligible. In the first of these diagrams the \/5 rectangle has been placed above the two squares. The lip is seen to be at half the height of the V5 rectangle. Perpendiculars from [ 163 ] GEOMETRY OF GREEK VASES 122 the centres of the two squares give the diameter of the upper of the two hori- zontal lines near the bottom of the bowl. This diameter is, therefore, one-half of the total width. The diameter of the foot is one-third of the total width, and can be determined geometrically by the intersections of diagonals of each of the two squares with the diagonal of the rectangle composed of both the squares. If two diagonals of half the over-all shape are drawn from the top of the vertical axis to the lower corners their intersection with diagonals of the squares determine the diameter of the lip. In the second small diagram a whirling square rectangle has been applied at the bottom of the over-all shape. By applying the reciprocals, and the reciprocals of the reciprocals, the simple figure is obtained which is illustrated by diagram IX, page 7. The upper of the two horizontal lines near the bottom of the bowl coin- cides with the horizontal division in this figure. The greatest diameter of the C 164 ] KANTHAROS lower part of the bowl is determined by the intersections of the diagonals of the applied squares with this line. The diameter of the lip can also be determined by the intersections of the diagonal of half the over-all shape with the top of the whirling square rectangle. The diagonals of the whirling square rectangle fix the diameter of the top of the stem. In the third diagram the area above the lip has been omitted. The remaining rectangle is easily seen to be made up of two squares plus two \/5 rectangles, i. e., .7236, or 1.382. The intersections of the diagonals of half this shape with the upper of the two lines near the bottom of the bowl, determine a smaller 1.382 rec- tangle, the width of which is equal to the diameter of the lower of the two lines near the bottom of the bowl (the ridge). If squares be applied at either end of this rectangle their centres determine both the height and the diameter of the ridge in the stem. The intersection of their diagonals at the axis of the vase fixes the height of the stem. The kantharos without its handles is enclosed in the rectangle .9045, i. e., .500-1-. 4045. The reciprocal of .9045 is 1.1056, which also can be divided into famihar ratios such as .5528X2, or .382-f.7236. In the following table are given: (A) the ratios, (B) the dimensions computed on the assumption that the total width is 0.2764 m., (C) the actual dimensions as measured by the writer, (D) the dimensions published by Pollack, op. dt. The total width is taken as .2764 m., instead of 0.277 m., because it happens to coin- cide with a familiar ratio. The greatest variation between the figures in columns B and C is less than a millimetre. Height to top of handles Height of lip Height of upper line near bottom of bowl. . . Height of lower line near bottom of bowl . . . Height of stem Height of ridge in stem Total width Diameter of lip Diameter of upper line near bottom of bowl Diameter of lower line near bottom of bowl Diameter of stem at top and at' ridge Diameter of foot A B c .9472 0.2618 m 0.262 m .7236 0.200 0.2005 .382 0.1056 0.106 .354 0.0978 0.0975 .264 0.07295 0.073 .191 0.528 0.053 1.000 0.2764 0.277 .6584 0.1819 0.181 .500 0.1382 0.1375 .528 0.1459 0.145 .1459 0.0403 0.0405 .333 0.092 0.0925 0.26 m 0.20 0.18 0.091 123 Kantharos. Inv. 01.8081. Hambidge, p. 123. No figure decoration. The whole surface, except for one member of the base, covered with a fine black varnish. The vase is intact, and of extremely careful workmanship. Height of handles, 0.1535 m. Height of bowl, 0.111m., width (handles), 0,222 m. Diameter of bowl, 0.146 m. Diameter of base, 0.0995 m. C 165 ] GEOMETRY OF GREEK VASES This kantharos fits very accurately the familiar shape composed of a square and a y/h rectangle — 1.4472, or .691. If the projection of the handles above the lip is omitted the rectangle is composed of two squares. The ratio of the diameter of the base to the total width is .4472:1.000, that is if a VS rectangle is applied at the bottom of the over-all rectangle, the diameter of the base of the kantharos coincides with the side of the central square of the VS rectangle. All the details are accounted for by simple subdivisions of the over-all rectangle or the applied a/5 rectangle. The proportioning of the mould- ings of the base is noteworthy, each member being twice as wide as the member immediately below it. The proportions of details are given below in two forms, the width being taken as unity in the first colunrn, the height in the second. Height of handles 691 1.000 Height of lip 500 .7236 Height of ridge near bottom of bowl 1236 .1788 Width 1.000 1.4472 Diameter of bowl 6584 .9528 Diameter of ridge near bottom of bowl 5236 .7577 Smallest diameter 382 .5367 Diameter of foot .4472 .6472 Cl66] KYLIX KYLIX BLACK-FIGURED PERIOD (A) Kylix with deep bowl, off-set Up, and high stem. Of the eight examples here pubHshed two bear the signature of the potter Tleson, son of Nearchos, three that of Xenokles, one that of Hermogenes. Their shapes, the thinness of their walls, the dehcacy of handles and feet suggest that they follow metal prototypes. The painted decoration on the exterior is hmited to palmettes adjoining the handles, and in some examples a small design on each side of the lip. The drawing of the kylix by Tleson, no. 124, shows which portions are normally left in the color of the clay and which are covered with black varnish. It is noteworthy that the bottom of the foot curves up slightly at the edge to prevent breaking when the cup is set down. The kylix, no. 124, is proportioned on the basis of the \/2 rectangle. No. 128 works out in squares. The proportions of no. 126 seem to be related to the V3 rectangle. The remaining five exhibit famihar shapes derived from the rectangle of the whirUng squares. For conveni- ence of comparison the four most important ratios are given here in tabular form: Height Width Diameter of bowl Diameter of foot 124 Tleson 125 Hermogenes 126 Xenokles. . . 127 Xenokles... 128 Unsigned... 129 Tleson 130 Unsigned . . . 131 Xenokles. . . 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2.1213 2.0652 2.0206 2.045 2.000 2.000 1.882 1.854 1.5988 1.5652 1.4433 1.427 1.5000 1.4472 1.382 1.382 .7071 .6708 .7217 .750 .6666? .5647 .6753 [167] GEOMETRY OF GREEK VASES 124 124 Kylix. Inv. 98.920. Hambidge, p. 53. Signed by Tleson, son of Nearchos, as potter. Interior picture, a wounded stag. Average height, 0.14235 m. Width (handle to handle), 0.30 m. Diameter of bowl, 0.227 m. Diameter of foot, 0.10 m. The enclosing area is made up of three \/2 rectangles placed vertically side by side, the ratio being 2.1213, or .7071 X 3. It can readily be seen that if the width of the over-all area is taken as 30 cm., the width of each of the y/2 rectangles is 10 cm., and their height is 14.142 cm. This comes within a milUmetre of the average height obtained from eight measurements (.1425, 141, 142, 1435, 142, 1435, 143, 1415). If a square be applied at the bottom of the three \/2 rectangles the intersection of its diagonal with the diagonal of the -s/2 rectangle determines the height of the stem. The diameter of the bowl is determined by the intersec- tion of the diagonal of the area which is in excess of the applied square with the diagonal of half the \/2 rectangle. The ratios are : Height 1.000 Height of bowl 5858 Height of stem 4142 Width 2.1213 Diameter of bowl 1.5988 Diameter of foot 7071 C 168] KYLIX ^K ■// 1 * 1 v' ". \l\ Jl'i. — — -■> ! ^\^ y. 1 ^ ^....^^^ _^ ■^f 1 .• "•■'f '■> V .,. ..^._ly ^ ;■ .. /■ « I i * / '\ 1, 1 .' \ ■._ 1 "■ .,.< , -,,.> ,/ V L. 1 '••I- _ %_ L- \y 125 125 Kylix. Inv. 95.17. Signed by Hermogenes as potter. On each side of lip, a hen. Height, 0.1425 m. Width, 0.2895 m. Diameter of bowl, 0.220 m. Diameter of foot, 0.094 m. The enclosing area is the rectangle 2.0652 (= 1.000+.618+.4472). If the projection of the handles is omitted the remaining rectangle has the ratio 1.5652, or 2,0652 -.500. This rectangle can also be regarded as 1.118+ -4472. If .4472 is cut off from each end the remainder is .6708. This is the ratio of the diameter of the foot. The height of the stem is .4472. The ratios are: Height 1.000 Width 2.0652 Diameter of bowl 1.5652 Height of bowl with lip 5528 Height of lip alone 1708 Height of bowl alone 382 Height of stem 4472 Diameter of foot ^6708 C 169 ] GEOMETRY OF GREEK VASES 126 Kylix. Inv. 99.529. Signed by Xenokles as potter. Lip black. On re- served band below lip, the signatures between palmettes. Height, 0.141 m. Width, 0.284. Diameter of lip, 0.207 m. Diameter of foot, 0.101 m. This vase is apparently composed on a s/S theme. The over-all area is a VS rectangle plus one-sixth of a \/S rectangle. The ratio is 2.02065. No simple ratio can be obtained for the diameter of the bowl, but if the diameter at the top and bottom of the moulding which forms the upper member of the lip be taken instead (as is done in the drawing) the ratio is 1.4433, or a \/S rectangle minus one-sixth of a V3 rectangle. The diameter of the foot is half this diameter. The kylix may be regarded as composed within two overlapping \/3 rectangles, the bowl being placed in the overlapping portion. The intersection of diagonals of these V'3 rectangles fixes the height of the junction of the bowl and stem. The height ratio of the bowl is .58337. If twice this portion be subtracted from the ratio of the diameter of the bowl, the remainder, .2766 is the ratio of the diameter of the ring at the top of the stem. This is expressed geometrically in the drawing by applying squares at either end of the rectangle enclosing the bowl. The ratios are : Height 1.000 Width 2.0206 = V3 + Diameter of bowl (excluding moulding of lip) 1.4433 = ^3 — Projection of each handle 2886 = ^^ Height of bowl 58337 ^ Height of stem 41663 Diameter of ring at top of stem 2766 Diameter of foot 7217 C 170] V3 6 V3 6 KYLIX 127 Kylix. In v. 98.921. Signed by Xenokles as potter. Interior, forepart of horse and rider. One handle missing. Height, 0.092 m. Diameter of bowl, 0.132 m. Width, including handles (estimated) 0.188 m. The over-all rectangle appears to be 2.045, the rectangle with the handles omitted being 1.427. The projection of each handle is .309 or .618-f-2. The pro- portions of details cannot be expressed in simple subdivisions of these shapes. The diameter of the foot is perhaps to be obtained from the intersection of the diagonal of the rectangle 1.736 (2.045 — .309) with the diagonal of the rectangle 1.0225 (2.045-=- 2). [171] GEOMETRY OF GREEK VASES 128 128 Kylix. Inv. 89.268. Lip black. On the reserved band below the hp, seilens pursuing maenads. Height, 0.0946 m. Width, 0.1895 m. Diameter of bowl, 0.141 m. Diameter of foot, 0.0705 m. The over-all area is composed of two squares, and most of the details work out simply in squares. Height Width Diameter of lip . . Diameter of foot . 1 Projection of each handle. 2 Diameter of stem 1| Height of Up \ / ^^ ^ ■■•;. ___,.^--' V ■ ■ ^ : ^ 129 [ 172] KYLIX 129 Kylix. Inv. 92.2655, Signed by Tleson, son of Nearchos, as potter. Height, 0.1335 m, Width, 0,267 m. Diameter of bowl, 0,1925 m. Diameter of foot, 0,09 m. With the handles the kylix is enclosed in a \/4 rectangle, i. e,, two squares. Without the handles it is enclosed in a 1,4472 rectangle, i. e., a square plus a \/5 rectangle. The diameter of the foot is nearly one-third the width. The height of the lip, and the diameter of the stem are each equal to one-fifth the height of the vase. The bowl without handles or stem fits a 2.764 rectangle. The geometrical analysis is omitted, since no simple method of expressing these ratios has been found. 130 Kylix. Inv. 92,2654. Hambidge, p. 117. Meaningless lettering. On each side of the lip, a swan. Height, 0,148 m. Width, 0.281 m. Diameter of bowl, 0.204 m. Diameter of foot, 0.0865 m. The area in which the kylix without its handles is enclosed has the familiar ratio 1.382. With the handles added the rectangle has nearly the shape, 1.882, or 1. 382 -|-. 500. The diameter of the foot is apparently determined by the inter- section of the diagonal of half the 1.382 rectangle with the diagonal of the ap- plied square. The ratio of this diameter is .5647. The height of the stem is .427, that of the bowl .528. If the bowl without stem or handles is considered sepa- rately, it is seen to conform to the shape, 2.618 (= .382); and the height of the bowl to the junction with the Up is to the height of the lip as 1.000 is to .618. Cf. Mr. Hambidge's analysis, which brings out some further points. C 173 ] GEOMETRY OF GREEK VASES 131 Kylix. Inv. 95.18. Signed by Xenokles as potter. Interior, a sphinx. Exterior, {A) Two centaurs. (B) Lion and faun. Height, 0.109 m. Width, 0.2025 m. Diameter of bowl, 0.153 m. Diameter of foot, 0.0735 m. The enclosing area is 1.854, or three whirling square rectangles. Without the handles the vase is contained in a 1.382 rectangle. The main proportions are thus exactly those of the black-figured skyphos, no. 104. The shape can also be considered as two horizontal whirling square rectangles overlapping, the over- lapping portion being the rectangle of the bowl. The intersection of the diagonal one of these whhling square rectangles with the diagonal of half the over-all area determines the diameter of the foot. The ratios are : Height 1.000 Width 1.854 Diameter of lip 1.382 Diameter of bowl at junction with lip . 1 .236 Diameter of foot 6753 Height of stem 472 Height of bowl and lip 528 C 174 ] KYLIX (B) Eye Kylix, with shallower bowl, and shorter and heavier stem. This type of kylix, apparently adapted by the Athenian potters of the late black-figured period from an Ionian type, became the favorite shape in the red- figured period, in which it underwent further modifications. All three of the examples here published have large pairs of eyes painted on them in the black- figured technique. The remaining painted decoration is black-figured on no. 132, red-figured on nos. 133 and 134. ■■■. I /■ 1-- ...■■r--. •■■■) 132 132 Kylix. Inv. 03.784. Hambidge, p. 116. The painted decoration, which is confined to the exterior, is in the black-figured technique. On each side, between two large eyes, a seilen grasping vines. Beneath each handle, a siren. Below, a frieze of lions, pxursuing pegasi and a deer. Height, 0.1205 m. Width, 0.346 m. Diameter of bowl, 0.274 m. Diameter of foot, 0.121 m. If the height is taken as 0.1212 m., the over-all ratio is, 2.854. The ratio of the bowl is fairly close to 2.236, i. e., a \/5 rectangle. The error is 3 mm. The diam- w <\ w w /> w 5 5 vv ^^ s s eter of the foot is equal to the height. The junction of bowl and stem is deter- mined by the intersection of the diagonals of the two overlapping whirhng square rectangles in the \/5 rectangle. The ratios are: Height 1-000 Height of stem 309 Height of bowl alone 691 Width 2.854 Diameter of bowl 2.236-f Projection of each handle 309 — Diameter of ring at top of stem . .500 Diameter of foot 1.000 [ 175 ] GEOMETRY OF GREEK VASES o •a •a o CD +2 73 : -^ 00 > 03 CI f^ CO TJ O H-H ^ ° OJ 03 . S o Si CO ^3 bC 1^ •!— I ;^ 03 <-! M «= M o3 t-H 03 a o O o3 ^ 03 ^ 03 bD «, • fh 03 03 JH 03 bC rSH q -t^ o3 li- -*^ o 03 ^ 05 03 O g CO • 1—1 _a t.l CO 73 ll' « i^ CO '^ bC'g I S3 03 !» "^ a £ «= T3 .2 ^ 03 g rd "^ 4J O 03 o ■* O ■=3 00 >o ■* o 03 O 1 aw ^t-3 O 02 4^ O 03 03 += 1 8 o S O 03 03 Sh • I— s H O ^ a o3 t! += 03 CO += 03 43 ^ I as r-t 02 O CO CO CO 00 O CO CD CO rH O '^ lO (N CD >-! ' ' CO N 03 O ^ o O O -t^ +^ -t^ o J3 Jd ,i3 rC 'S "S 'S •^r' •« W W W ^Q 03 C 176] KYLIX Qj (D m W W M [ 177 ] GEOMETRY OF GREEK VASES RED-FIGURED PERIOD The collection is especially rich in examples of kylikes of the archaic period. A number of them are, however, so incomplete as to make the analysis of their proportions impossible, or at least unreliable. A few others have been omitted because they seemed to fit no geometric scheme convincingly. The examples chosen for publication are fairly representative. They fall into two main classes, the kylix with a fairly tall, slender stem, nos. 135-160, and the stemless kylix found in the period of the free style, nos. 161, 162. Examination of the table of ratios shows that certain simple ratios occur repeatedly. The total width of the kylikes with stems varies between 3.000 and 3.618. The two most common widths are 3.236, or two whirling square rectangles, and 3.4142, or two squares plus a ^2 rectangle. The commonest ratios for the diameter of the bowl are 2.618 ( = .382) and 2.7071, or 2 + ^^^- The former occurs four times, three times in connection with the over-all ratio 3.236. The latter is found three times in connection with the shape 3.4142. The diameter of the foot varies between .927 and 1.236. In eleven examples out of twenty-six it is exactly, or almost exactly, equal to the height of the kylix. The projection of the handles is .309, or '—--, in six cases. The diameter of the raised ring on the foot has the ratio .691 in six cases. If one were to choose an ideal scheme it might well be the following : Width 3.236 Diameter of ring on foot 691 Diameter of bowl 2.618 Diameter of foot 1.000 Diameter of stem 236 Projection of handles 309 The level of the junction between bowl and stem cannot be measured exactly. It will be found to be at about half the height in the majority of large kylikes. The two kylikes, nos. 146, 147 conform to this ideal scheme in their main propor- tions. The former happens to be signed by the potter Pamphaios. Nos. 140, 151, 155 are also perhaps from his factory, though this cannot be proved. They are among the best examples of dynamic symmetry in the series. It is noteworthy that two of the kyhkes decorated by the " Panaitios Painter " who worked in the factory of Euphronios have identically the same proportions throughout (nos. 159, 160). Of the two stemless kylikes one conforms to a simple scheme based on the whirling square rectangle, while the other belongs to the static class. The ratios, so far as they have been determined, are given in the following table: C 178 ] KYLIX No. Height Width Diameter of Diameter of Diameter of Diameter of Projection of bowl stem ring on foot foot each handle 135 1.000 3.000 2.191 .4472 1.000 .4045 136 1.000 3.000 2.3504 .... .5669 1.0512 .3248 137 1.000 3.000 2.382 .1708? .691 .8944 .309 138 1.000 3.090 2.236 .309 1.000 .427 139 1.000 3.090 2.382 .5609? .927? .354 140 1.000 3.090 2.472 .236 .5652 .8944- .309 141 1.000 3.236 2.472 .691 1.000- .382 142 1.000 3.236 2.528 .236 .764 1.000 .354 143 1.000 3.236 2.528 .236 .691 1.146 .354 144 1.000 3.236 2.545 .236- .691 1.000 .3455 145 1.000 3.236 2.618 .764 .927 .309 146 1.000 3.236 2.618 .236 .691 1.000- .309 147 1.000 3.236 2.618 .236 .618 1.000+ .309 148 1.000 3.4142 2.4714 .4714 1.0572 .4714 149 1.000 3.4142 2.5606 .2677 1.0606 .4268 150 1.000 3.4142 2.5858 1.000 .4142 151 1.000 3.4142 2.7071 .2929 ? .3535 152 1.000 3.4142 2.7071 .7071 1.0606 .3535 153 1.000 ? ? ? ? ? ? 154 1.000 (3.4142) 2.7071 .5460 1.000 (.3535) 155 1.000 3.382 2.618 .309 .545 1.000 .382 156 1.000 3.382 2.764 .236 .764 1.000 .309 157 1.000 3.528 2.764 .764 1.090? .382 158 1.000 3.618 2.764 .666 1.236 .427 159 1.000 3.618 2.854 1.146 .382 160 1.000 3.618 2.854 .7236 1.146 .382 161 1.000 3.854 2.854 1.545 .500 162 1.000 4.666 3.500 2.000 .5833 [179] GEOMETRY OF GREEK VASES 135 Kylix. Inv. 00.336. Off-set lip. Beazley, V. A., -p. 22. Interior, Youth testing arrow. Early archaic style. Height, 0.07975 m. Width, including handles, 0.238 m. Diameter of bowl, 0.173 m. Diameter of foot, 0.079 m. The enclosing rectangle is obviously composed of three squares, and the foot is exactly contained in the central square. Without the handles the kylix is placed in a 2.191 rectangle, the projection of the handles being .4045, or four whiriing square rectangles. The diameter of the stem is .4472; its height, including the ring, .250. The height of the hp is .236. 136 Kylix. Inv. 10.179. Beazley, V. A., p. 86, no. 19. Interior, Seilen seated on a pithos. " Panaitios Painter." Height, 0.10635 m. Width, including handles, 0.318 m. Diameter of bowl, 0.248 m. Diameter of foot, 0.11 m. The containing area is composed of three squares, but the analysis of details is unusually comphcated. If a whirling rectangle be applied at the bottom of the central square, the intersection of the diagonal of half this rectangle with the diagonal of the reciprocal fixes the diameter of the ring on the foot, as is shown by the perpendicular dropped from the point of the intersection. The length of ^^' w - w 5 V -r-''^'" "/S- -^ :>■ y w w — this perpendicular is .3504 (the reciprocal of 2.854). The diameter of the foot is .3504X3, or 1.0512. The diameter of the bowl is, 2.3504, i. e., two squares plus the ratio .3504. If a hoiizontal line is drawn at the height .3504, the diagonals of each half of the containing shape cut the line so as to fix the diameter of the foot. The rectangle with the diameter of the foot as width and the .3504 line as height is thus shown to be made up, like the over-all rectangle, of three squares. In the drawing the diagonal of the small square at the right is continued, and cuts the top of the over-all rectangle so as to define the diameter of the bowl. The pro- jection of the handles beyond the bowl is expressed by the ratio .3248, which is the reciprocal of 3.0787 — a rectangle made up of two squares and a 1.0787 (.927) rectangle composed of one and a half whirling square rectangles. It is noteworthy that the top of the 1.0787 rectangle coincides with the line at the level .3504. The scheme is more clearly shown in the small diagram. It is also noteworthy that if the diameter of the foot be taken as unity the total width and the diameter of the bowl are expressible in simple ratios, viz., 2.236 = V5- •3168 „ „., , .2483 = 2.854 and .111 .111 C 180] KYLIX € 181 GEOMETRY OF GREEK VASES 137 Kylix. Inv. 89.272. Beazley, V. A., p. 105, no. 69. Interior, Man and boy. Exterior, Men and youths. Attributed to Makron. Height, 0.14 m. Width, including handles, 0.422 m. Diameter of bowl, 0.332 m. Diameter of foot, 0.124 m. The ratios are: With handles 3.000. 0.14 X 3 = .420. Error, 0.002 m. Without handles 2.382. 0.14 X 2.382 = .33348. Error, 0.0015 m. Dikmeter of foot 8944. 0.14 X .8944 = .126216. Error, 0.0022 m. Diameter of ring on foot 691 Diameter of stem 1708? Projection of handles 309 138 Kylix. Inv. 01.8074. Off -set lip. Interior, Crouching archer. Height, 0.0736 m. Width, including handles, 0.228 m. Diameter of bowl, 0.1645 m. Diameter of foot, 0.075 m. If the height is made 0.074 m. (0.0006 m., more than the average; 0.001 m., less than the diameter of the foot) the ratios are : With handles 3.09. 0.074 X 3.09 = .22866. Error, 0.00066 m. Without handles 2.236. 0.074 X 2.236 = .16.5464. Error, 0.000964 m. Foot 1.000. 0.074 X 1.000 = .074. Error, 0.001 m. Diameter of stem 309 Height of lip 236 Projection of handles 427 C 182] KYLIX [ 183 ] GEOMETRY OF GREEK VASES 139 Kylix. Inv. 13.84. Beazley, V. A., p. 132. Interior, Youth and woman. Exterior (A) and (B), Seilens and maenads. "Penthesilea Painter." Height, 0.1146 m. Width, including handles, 0.351 m. Diameter of bowl, 0.271 m. Diameter of foot, 0.105 m. Assuming the height to be 0.114 m., the ratios are: With handles 3.090. 0.114 X 3.090 = .35226. Error, 0.001226 m. Without handles 2.382. 0.114 X 2.382 = .271548. Error, 0.000548 m. Diameter of foot 927? Diameter of ring on foot 5609? Projection of handles 354 140 Kylix. Inv. 95.35. Beazley, V. A., p. 24. Hambidge, p. 119. Interior, Athlete with jumping weights and javelins. Exterior (A) and (JB), Seilens and maenads. The paintings are placed by Beazley in the group headed by the works of the "Painter of the London Sleep and Death." The potter may well have been Pamphaios. Height, 0.1337 m. Width, including handles, 0.413 m. Diameter of bowl, 0.3315 m. Diameter of foot, 0.1195 m. The ratios are : With handles 3.090. 0.1337X3.090 =.413133. Error, 0.000133 m. Without handles. 2.472. 0.1337X2.472 = .3394964. Error, 0.001094 m. Diameter of foot 8944 0.1337 X .8944= .1337. Error, 0.00008 m. Diameter of ring on foot 5652 Diameter of stem 236 Projection of handles 309 C 184 ] KYLIX C 185] GEOMETRY OF GREEK VASES 141 Kylix. Inv. 01.8029. Beazley, V. A., p. 98. Interior, Youth at laver. Unsigned; the painting is ascribed to Douris. Height, 0.07925 m. Width, including handles, 0.2615 m. Diameter of bowl, 0.198 m. Diameter of foot, 0.077 m. If the height be assumed to be 0.08 m., the ratios are: With handles 3.236. 0.08 X 3.236 = .25888. Error, 0.00262 m. Without handles 2.472. 0.08 X 2.472 = .19776. Error, 0.00024 m. Diameter of foot 1.000? 0.08 X 1.000 =0.08? Error, 0.003 m. Diameter of ring on foot 691 Projection of handles 382 142 Kylix. Inv. 00.338. Beazley, V. A., p. 97, no. 6. Interior, Diskoboles. Exterior (A) and (B), Battle scenes. Signed by Douris as painter. Height, 0.09325 m. Width, including handles, 0.3055 m. Diameter of bowl, 0.2395 m. Diameter of foot, 0.095 m. Assuming the height to be 0.094 m., the ratios are: With handles 3.236. 0.094 X 3.236 = .304184. Error, 0.00132 m. Without handles 2.528. 0.094 X 2.528 = .237632. Error, 0.00187 m. Diameter of foot 1.000. Error, 0.001 m. Diameter of ring on foot 764 Diameter of stem 236 Projection of handles 354 [ 186] KYLIX ^ ¥\ r\ ^\ [ 187 ] GEOMETRY OF GREEK VASES 143 Kylix. Inv. 01.8020. Beazley, V. A., p. 86, no. 8. Interior, Diskobolos. Exterior, Athlete. " Panaitios Painter." Height, 0.08893 m. Width, including handles, 0.291 m. Diameter of bowl, 0.224 m. Diameter of foot, 0.10175 m. If the height is taken as 0.0895 m. (0.00057 m., greater than the average). The ratios are : With handles 3.236. 0.0895 X 3.236 = .289622. Error, 0.00138 m. Without handles 2.528. 0.0895 X 2.528 = .226256. Error, 0.0022 m. Foot 1.146. 0.0895 X 1.146 = .102567. Error, 0.00081 m. Smallest diameter of stem 236 Diameter of ring on foot. ... .691 Projection of handles 354 144 Kylix. Inv. 01.8034. Beazley, V. A., p. 93. Interior, Symposion. Ex- terior, Symposion. "Berlin Foundry Painter." Height, 0.1172 m. Width, including handles, 0.3755 m. Diameter of bowl, 0.298 m. Diameter of foot, 0.1175 m. Assuming the height to be 0.117 m., the ratios are: With handles 3.236. 0.117 X 3.236 = .378612. Error, 0.003112 m. Without handles 2.545. 0.117 X 2.545 = .297765. Error, 0.00023 m. Diameter of foot 1.000. Error, 0.0005 m. Diameter of ring on foot 691 Diameter of stem 236 — Projection of handles 3455 = .691 -i- 2 C 188] KYLIX C 189] GEOMETRY OF GREEK VASES 145 Kylix. Inv. 01.8022. Beazley, V. A., p. 105, no. 70. Hambidge, p. 121. Interior, Woman and man. Exterior, Youths, men and women. Attributed to Makron. Height, 0.126 m. Width, including handles, 0.413 m. Diameter of bowl, 0.332 m. Diameter of foot, 0.1175 m. Assuming the height of the kylix to be 0.127 m., the ratios are: With handles 3.236. 0.127 X 3.236 = .410972. Error, 0.002028 m. Without handles 2.618. 0.127 X 2.618 = .332486. Error, 0.000486 m. Diameter of foot 927. 0.127 X .927 = .117729. Error, 0.000229 m. Diameter of ring on foot . . . .764 Projection of handles 309 146 Kylix, Inv. 95.32. Beazley, V.A.,p. 23. Interior, Seilen. Exterior (A) Arming scene. (B) Combat. Signed by Pamphaios as potter. Height, 0.1324 m. Width, including handles, 0.425 m. Diameter of bowl, 0.3435 m. Diameter of foot, 0.129 m. If the height be assumed to be 0.1315 m., the ratios are: With handles 3.236. 0.1315 X 3.236 = .425534. Error, 0.000534 m. Without handles 2.618. 0.1315 X 2.618 = .343267. Error, 0.000233 m. Diameter of foot 1.000. Error, 0.0025 m. Diameter of ring on foot 691 Height of top of painted band on exterior, beneath figures 618 Diameter of top of painted band 1.618 Diameter of bottom of painted band 1.309 Diameter of stem 236 Projection of handles 309 [ 190 ] KYLIX [191] GEOMETRY OF GREEK VASES u ai O •4J 03 • ■ ;h a a bl) CD i i o .2 >» . S3 s-T o ■i-i 02 i .g a .3 ^ a O ►il :S o 00 CO t M s: o a ^ (O o CO +=, d a § -s fe ^ •SPd « o 5a T-H CO O iH[ao tn fH tH O O O h^ a h e ^ 1^ •« «> o eq CO TjH rA ' ' ' lO . . 00 CD CO s a a (X) t-H CO (N h- o> o o o o o o o o o o o (M ■"ti O rH 00 00 »0 00 l>- (N rH O (N CO CO TfH O O 1—1 CO CO -* ko o CO c O O 03 •G P-i '-' ^ K o3 03 pq _g CO o ^ o >^ . i o CO o o CO c5 &0 a •rH • 1-1 • • <1> .- t4 OS'S ^ I ■s o M d as ^- to CO &0' •rH w c* t^ C 05 CO CO c« CO ei -a c3 S XX § M :2 O .23 "c^ 0) S +3 (» £ ^ f-i 0) a ^ C 197] GEOMETRY OF GREEK VASES 152 Kylix. Inv. 00.344. Beazley, V. A., p. 189. Interior, Herakles, Nessos, Deianeira. Exterior, Centauromachy. Signed by Erginos as potter, and by Aristophanes as painter. 153 Kylix. Inv. 00.345. Beazley, V. A., p. 189. The paintings are a replica of those on the above kyhx, but are unsigned. These two cups are a pair. Though only one is signed, both are decorated by the same hand, and both must have been made in the same factory. One might expect that they would conform to the same scheme. This, however, is not the case. The unsigned example fits fairly accurately a simple scheme. The signed piece varies widely in the height, the total width, and the diameter of the bowl. The discrepancy in the diameter of the foot and the diameter of the ring on the foot, though much smaller, is still appreciable. Attempts to fit the signed vase into another scheme have remained unsuccessful. The dimensions of the two cups are as follows: No. 152 No. 153 Height 0.1329 m. 0.1315 m. Width including handles 0.442 0.452 Diameter of bowl 0.34925 0.3559 Diameter of foot 0.142 0.1405 Diameter of ring on foot 0.096 0.0925 Projection of handles 0.046375 0.04855 Diameter of maeander band on exterior, A 0.1345 0.1335 Diameter of maeander band on exterior, B 0.159 0.159 Diameter of maeander band on interior, A 0.163 0.177 Diameter of maeander band on interior, B 0.192 0.2085 The proportions of the unsigned kylix are as follows: With handles 3.4142. 0.1315 X 3.4142 = .4489673. Error, 0.003 m. Without handles 2.7071. 0.1315 X 2.7071 = .355984. Error, 0.0000984 m. Diameter of foot 1.0606. 0.1315 X 1.0606 = .1394689. Error, 0.001 m. Diameter of ring on foot .7071 Projection of handles 3535 C 198] KYLIX [199] GEOMETRY OF GREEK VASES -x \ ■■'■■ f \^ / f 1 CO < o • r-t f-t K c _-.£ri _^ ; / [207] GEOMETRY OF GREEK VASES Vm a a 03 o w oq T3 03 O o O o Q 1—1 ^^^ ~ CO .2 ^ 01 u a a .3i w ^ S S P^ C 208 J LEKYTHOS LEKYTHOS BLACK-FIGURED PEEIOD 163 Lekythos of Corinthian Shape. Inv. 08.291. Ann. Rep. 1908, p. 61. Pictures in three zones. Height, 0.2095 m. Diameter, 0.093 m. This vase is contained, with re- markable exactness, within a \/5 rec- tangle, and its details, including most of the painted bands are clearly de- finable in terms of this rectangle. A \/5 rectangle may be divided into five reciprocals, in this case five •\/5 rectangles placed horizontally. The top of the lowest of these \/5 rectangles coincides with the band on which the lowest of the three zones of decoration rests. The diameter of the bottom of the body equals the height of this \/5 rectangle (.4472), or the side of its central square. Simple sub- divisions of the adjoining whirling square rectangles give the diameter of the foot (.7888). If the diagonals of these whirling square rectangles are produced they meet at the level of the band on which the second zone of decoration rests, i. e., at the height of two whirling square rectangles placed side by side (.809). The reciprocal rectangle applied to the top of the containing shape ac- counts for all the details of the lip and neck, and its base coincides with a painted band on the shoulder. The diameter of the lip is seen to equal the side of 163 the central square (.4472), and all the details of lip and neck can be simply expressed in terms of this square. The diameter of the neck equals the side of a \/5 rectangle placed in the middle of the square. This \/5 rectangle (.200) is a reciprocal of the reciprocal of the over-all shape. The upper of the two bands on which the highest of the three zones of painted decoration rests is placed at the level at C 209 ] GEOMETRY OF GREEK VASES 164 which the diagonal of the whole shape intersects the diagonal of a whirling square rectangle. The area above this line is thus seen to be the familiar rectangle, 1.0652. 164 Lekythos. In v. 95.15. Hambidge, p. 132. Achilles and Ajax playing draughts in the presence of Athena. Height, 0.2825 m. Diameter, 0.142 m. The containing rectangle is composed of two squares; and all the proportions except the height to the top of the shoulder, the height of the neck, and its lower diameter, are expressible in terms of squares, or in simple fractions, using the diameter as unity. The ratios are: Height 2.000 = 2 Height to shoulder . 1.400 = l' Height of lip, neck and shoulder 600 = | Height of lip 1666 = | Diameter 1.000 = 1 Diameter of lip 500 = J Smallest diameter of neck 200 = ] Diameter of bottom of body 4286 = f Diameter of foot . . . .600 = | [210 ] LEKYTHOS 165 165 Lekythos. Inv. 98.922. Female figure mounting a quadriga; behind her, Hermes. Height, 0.1105 m. Diameter, 0.0582 m. The containing area has the ratio, 1.8944; it is composed of a square plus two \/5 rectangles (1.000 +.4472 +.4472). The proportions of details fall into simple ratios, as follows : Height 1.8944 Height to shoulder 1.2764- Height of lip, neck, and shoulder 618+ Height of Up and neck 500 Height of painted band below figures .500 Diameter 1.000 Diameter of hp 500 Diameter of top of neck 250 Diameter of bottom of body . .4472 Diameter of foot 6584 [211 ] GEOMETRY OF GREEK VASES 166 Lekythos. Inv. 99.528. Warrior lead- ing horse. Height, 0.2615 m. Diameter, 0.09 m. The containing rectangle has the ratio, 2.8944, or 1.4472X2, i. e., it is composed of two squares plus two y/h rectangles. The ratios are : Height 2.8944 Height to neck 2.0854 Height of neck and lip 809 Height of lip 382 Height of neck 427 Diameter 1.000 Diameter of hp 5669+ DiiE^neter of top of neck 236 Diameter of bottom of neck 382 Diameter of bottom of body 382 Diameter of foot 764 166 [ 212 ] LEKYTHOS RED-FIGURED PERIOD 167 Lekythos. Inv. 13.195. Beazley, V.A.,Tp. 26. Hambidge, p. 131. Cows led to the sacrifice. Signed by Gales as potter. Height, 0.310 m. Diameter, 0.124 m. The containing area is made up of two squares and a half. All the proportions can be expressed in simple fractions. The lekythos is a good example of what is called by Mr. Hambidge " static symmetry." The ratios are : Height 2i Height to bottom of shoulder . . If Height of lip, neck and shoulder . | Height of lip ^ Height of neck ^ Height of shoulder ^ Diameter 1 Diameter of hp | Diameter of bottom of body J Diameter of foot f 167 C 213 1 GEOMETRY OF GREEK VASES 168 169 168 Lekjrthos. Inv. 13.199. Bearded man in Ionic dress reciting and playing lyre. Height, 0.3925 m. Diameter, 0.14 m. 169 Lekythos. Inv. 13.198. Beazley., V. A., p. 114. Young hunter with dog. " Pan Painter." Height, 0.3875 m. Diameter, 0.138 m. These two lekythoi are said to have been found together, and resemble one another closely. Both are contained in the rectangle, 2.809. The proportions of [214 ] LEKYTHOS the foot are the same in both, but those of the lip, neck and shoulder are different. The ratios are: No. 168 No. 169 Height 2.809 2.809 Height to shoulder 2.000 1.941? Height of Up, neck and shoulder .809 .868? Height of shoulder 2236 .250? Height of Up and neck 5854 .618 Diameter l.OOQ 1.000 Diameter of Up 5854 .5669 Diameter of top of shoulder . . . .236 .236 Diameter of bottom of body . . . .382 .382 Diameter of foot 691 .691 170 Lekythos. Inv. 95.44. Beazley,F. A., p. 76. Menelaos and Helen. "Providence Painter." Height, 0.411 m. Diameter, 0.142 m. The enclosing rectangle has the ratio, 2.8944. The height to the shoulder equals twice the diameter. The diameters of the lip and of the top of the shoulder are obtained from a whirling square rectangle applied at the top of the enclosing area. The diameter of the bottom of the body and of the foot are expressed in terms of a whirling rectangle apphed to half of the lower large square. Height 2.8944 Height to shoulder 2.000 Height of Up 3455 Diameter 1.000 Diameter of Up 528 Diameter of top of shoulder 236 Diameter of bottom of body 382 Diameter of boot 691 [215 ] GEOMETRY OP GREEK VASES 171 Lekythos. Inv. 13.189. Beazley, V. A., p. 91. Woman taking skein of wool out of a basket. " Brygos Painter." Height, 0.332 m. Diameter, 0.1125 m. The containing rectangle has the ratio 2.944 = 1.472X2. The vase up to the shoulder is enclosed in two squares. Most of the details can be expressed in simple ratios, as follows: Height 2.944 Diameter 1.000 Height to shoulder 2.000 Diameter of hp 6669 Height of lip 382 Diameter of top of neck 236 Height of neck 309 Diameter of bottom of body 382 Height of shoulder 253 Diameter of foot 666 172 Lekythos. Inv. 95.41. Beazley, V. A., p. 98. Athlete with jumping weights. By Douris. Height, 0.300 m. Diameter, 0.101 m. The containing rectangle is composed of three squares (3.000). The diameter of the bottom of the body has the ratio .400, that of the foot, .800, suggesting that the proportions belong to the static class. No other simple ratios have been noted. C 216 ] LEKYTHOS ■ / \ X : A /.,.. \ ■■•L- i '■ / %:1'i---y V : X: Xh 171 172 C 217 ] GEOMETRY OF GREEK VASES 173 Lekythos. Inv. 95.45. Beazley, V. A., p. 76. Apollo with phiale and lyre. " Providence Painter." Height, 0.391 m. Diameter, 0.1305 m. The enclosing area, composed of three squares, is divided " dynamically." The height of the vase to the neck is expressible in a simple ratio. The height of the lip (.3618 = 2.764) makes with the over-all diameter an area composed of two 1.382 rectangle (1.382X2 = 2.764). The intersection of the diagonal of one of these rectangles with the diagonal of a square applied to it fixes the diameter of the lip (.5854). The diameters of the bottom of the body (.382) and of the foot (.691) are found often in lekythoi. Height 3.000 Height to neck 2.309 Height to shoulder 2.059 Height of lip 3618 Height of lip and neck 691 Height of shoulder 250 Diameter 1.000 Diameter of Hp 5854 Diameter of top of neck 236 Diameter of top of shoulder 2764 Diameter of bottom of body 382 Diameter of foot 691 174 Lekythos. Inv. 98.885. Seilen pursuing a maenad. The outlines of the figures incised; the woman painted white. Height, 0.221 m. Diameter, 0.0745 m. The containing rectangle is composed of three squares. The body occupies two squares; the lip, neck, and shoulder are enclosed in the remaining square. The details are expressible in simple ratios as follows : Height 3.000 Height to shoulder 2.000 Height of lip and shoulder 1.000 Height of lip 382 Height of neck .472 Height of shoulder '. . .146 Height of foot 191 Diameter 1.000 Diameter of lip 618 Diameter of top of neck . . .236 Diameter of top of shoulder 382 Diameter of bottom of body 382 Diameter of foot 764 [ 218 ] LEKYTHOS ■■^^^■•J 174 [ 219 ] GEOMETRY OF GREEK VASES 175 Lekythos. Inv. 93.103. Youth and woman at stele. Height, 0.4215 m. Diameter, 0.1275 m. 176 Lekythos. Inv. 93.104. Woman with perfume vase and youth with spear at stele. Height, 0.4215 m. Diameter, 0.123 m. These lekythoi are undoubtedly a pair, made in the same factory. The paint- ings are by a single artist, "a florid imitator of the Achilles Painter " (Beazley, Vases in America, p. 166). They are of the same height, the heights of all the members are the same; the diameters are also the same except for the la,rgest diameter of the body, which is 0.0035 m. , less in the second than in the first. ' The first can be simply analyzed in terms of the rectangle, 3.309, as appears from the drawing and the table of ratios. The over-all ratio of the second is 3.427; but at- tempts to express the proportions of details by means of subdivisions of thi$ rec- tangle have been unsuccessful. If a geometrical scheme was used by the potter, it seems likely that he made the lekythos, no. 175, conform to it exactly, and changed the diameter of no. 176, either purposely or accidentally. The ratios of no. 175 are as follows: Height 3.309 Height to top of neck : 3.000 Height to bottom of neck 2.500 Height to shoulder 2.191 Height of Hp, neck and shoulder 1.118 Height of Hp 309 Height of neck 500 Height of shoulder l 309 Height of foot 1708 Diameter 1.000 Largest diameter of Hp 5528 Smallest diameter of Hp 2764 = .5528 4- 2 Diameter of bottom of body 309 Diameter of foot ; . .6584 [ 220 ] LEKYTHOS C 221 ] GEOMETRY OF GREEK VASES -'-•/. '• - • ' :V': .:..-: .. / /-•\\ ; 1 /■■••>- 177 177 Lekythos. Inv. 95.49. Beazley, V. A., p. 186. Nurse bringing baby to mother. School of the Meidias Painter. Height, 0.1375 m. Diameter, 0.082 m. The containing area has the ratio, 1.691. The proportions are as follows: Height 1.691 Height to shoulder 1.073 Height of lip and neck 618 Height of lip 309 Height of neck 309 Height of band below figures, top .309 Height of band below figures, bottom .236 Diameter 1.000 Diameter of hp 4472 Diameter of shoulder 382 Diameter of bottom of body 764 Diameter of foot 7888 [ 222 ] LEKYTHOS \ ' . ' '.' 7*7 ^"^^TT — ; >. / ••• V • • • • * • I t^ * 178 178 Lekythos. Inv. 01.8119. Two youths seated, Eros and a woman dancing. Free style. Height, 0.132 m. Diameter, 0.077 m. The enclosing area has the ratio, 1.7071, and all the details can be expressed in terms of the -s/2 rectangle, as follows: Height 1.7071 Height to shoulder 1.0606 Height of hp 2929 Height of neck and shoulder 3535 • Diameter 1.000 Diameter of hp and shoulder 500— Diameter of bottom of body 7071 Diameter of foot 7677 = .3535 + .4142 [ 223 ] GEOMETRY OF GREEK VASES 179 179 Lekythos. Inv. 98.884. Nike cutting off the forelock of a bull in the pres- ence of Athena and a youth. Free style. Height, 0.1515 m. Diameter, 0.079 m. The enclosing rectangle has nearly the ratio, 1.8944. The height to the shoulder is 1.118, the height of the lip, .382, the height of the neck, .3944. The diameter of the lip is .500, that of the shoulder somewhat less than .382, that of the bottom of the body, .7236. C 224 ] LEKYTHOS 180 180 Lekythos. Inv. 95.1402. Beazley, V. A., p. 186. Acorn-shaped. Eros and woman. Meidias Painter. Height, 0.1585 m. Diameter, 0.0595 m. The containing rectangle has the ratio, 2f . The height of the acorn-cup is two-thirds of the diameter. C 225 ] GEOMETRY OF GREEK VASES PYXIS 181 Red-figured Pj^is. Inv. 93.108. Hambidge, p. 77. (A) Domestic scene. (B) Nike crowning a youth. Free style. Height to top of knob, 0.178 m. Height to top of lid, 0.1365 m. Height to junction of vase and lid, 0.1205 m. Diameter of lid and of pyxis, 0.15 m. The over-all rectangle has the ratio, 1.191. This can be divided in various ways, one of the simplest being .809 +-382. The Ud exactly fits a .382 (2.618) rectangle, and the visible portion of the pyxis therefore is enclosed in the familiar shape .809 or 1.236, i. e., two whirling square rectangles. The area in which the base of the pyxis is enclosed is seen to be composed of two -\/5 rectangles. The other details are obtained in an equally simple way. The ratios are : Height to top of knob on lid 1.191 Height of knob with its neck 2764 Height to top of lid 9146 Height of lid 382 Height to junction of pyxis and lid 809 Height of base 2236 Greatest diameter of lid and pyxis 1.000 Diameter of knob 2764 Smallest diameter of pyxis 750 Diameter of foot 8292 = .2764 X 3 C 226 ] PYXIS [227] GEOMETRY OP GREEK VASES 182 Polychrome Pyxis. Inv. 98.887. A neatherd, a cow, and six Muses, painted in colors on a white ground. Beazley, V. A.,p. 128. Hambidge, p. 49; photograph opp. p. 50. Height to top of knob, 0.1765 m.; to top of hd, 0.125 m.; to junction of pyxis with hd, 0.107 m. Greatest diameter of pyxis and of hd, 0.146 m. The interesting relation which this pyxis bears to a white pyxis in the Metro- politan Museum, New York, has been brought out by Mr. Hambidge, op. cit. Both are enclosed in the rectangle, 1.2071, i. e., .7071-|-.500, or a a/2 rectangle placed horizontally above two squares. But the proportions of details are en- tirely different, though definable in both cases in simple areas based on the \/2 rectangle. In the present example the lid with its knob is enclosed in an area made up of three ■\/2 rectangles placed vertically, side by side (cf., the black- figured kylix by Tleson, no. 124). The visible portion of the pyxis occupies an area composed of three ■\/2 rectangles placed horizontally above two squares. The vases with the lid, but omitting the knob, is placed in a rectangle composed of two horizontal \/2 rectangles above two squares. The knob with its neck is placed in a rectangle made up of a ■\/2 rectangle and a square. The knob alone is contained in area similar to the over-all rectangle (1.2071), as is shown in the drawing by the diagonals of half the shape. It is noteworthy also that the top of the lid is at half the height of the \/2 rec- tangle of the containing shape, and that a square inscribed in the centre of this -\/2 rectangle defines the smallest diameter of the pyxis. If a square be applied at the top of the over-all shape, its base coincides with the bottom of the body of the pyxis; and the indented line near the top of the vase is at half the height of this square. The intersections of the diagonals of half the over-all rectangle with the base of this square determine the diameter of the base of the pyxis. The ratios are : Height to top of knob 1.2071 = "^^ "^ ^ Height to top of Hd 8535 = ^^ + ^ 4 Height to junction of pyxis and lid 7357 = — — i-- 6 Height of lid with knob 4714 = ^ o Height of knob with its neck -.3535 = ^^ 4 Height of knob without its neck 250 = - 4 Height of base 2071 = "^^ ~ ^ 2 Height, omitting base ". 1.000 = 1 [ 228 ] PYXIS L.i^ .•■ A". \ •■■■ •■■-. / i tma \ '■■■ •■■■■ 182 Greatest diameter of pyxis and of lid 1.000 = 1 Diameter of knob 2071. = — Smallest diameter of pyxis 7071 = V2 Diameter of base 8284 = 2V2 - 2 [ 229"] GEOMETRY OF GREEK VASES PERFUME VASE 183 Perfume Vase. Inv. 97.367. Covered with black varnish. A broad band of decoration in the black-figured technique round the mouth ; a tongue pattern round the lid. Height, with cover, 0.173 m.; without cover, 0.1285 m. Diameter, 0.2395 m. Diameter of foot, 0.1475 m. The containing area is a 1.382 rectangle. This divides into two .691 rec- tangles, each made up of a square plus a \/5 rectangle. The vase with the cover, but without the foot, is contained in the two squares; the foot is contained in the two \/5 rectangles. Without the cover the vase is enclosed in a 1.854 rectangle, i. e., three whirling square rectangles placed side by side. The area up to the ridge of the bowl is composed of four vertical whirling square rectangles placed side by side. The ratios are given in two forms, A , with the height taken as unity, B, with the diameter taken as unity. A B Height with lid 1.000 .7236 Height without lid 7453 .5394 Height to ridge of bowl 559 .4045 Height of foot 309 .2236 Diameter 1.382 1.000 Diameter of ring at bottom of knob 1459 .0927 Diameter of lid 618- .4472 Diameter of bottom of bowl 472 .3416 Diameter of foot 854 .618 [ 230 ] PERFUME VASE 183 A 183 B [ 231 ] GEOMETRY OF GREEK VASES 184 A 184 Perfume Vase. Inv. 99.530. Described in Ann. Rep., 1899, p. 73, no. 29. Hambidge, p. 85. No figure decoration. The surface is left in the color of the clay, or painted black, as shown in the drawing, in which added red color is repre- sented by hatching. Height, with cover, 0.2455 m. ; without cover, 0.1825 m. Diameter, 0.2685 m. The bowl of this vase resembles an archaic Doric capital while the foot recalls Ionic column bases. The proportions are heavy, but from the point of view of execution the vase is a masterpiece of the potter's art, to be classed with the amphorae painted by Amasis (nos. 4 and 7) and the kylix made by Tleson (no. 124). It is probably to be dated in the same period. Study of the proportions shows clearly that they are derived from the whirling square rectangle, but many of them cannot be expressed in a simple geometric construction or in familiar ratios. The vase without its cover is contained in the rectangle 1.472 (.6793). The diameter of the moulding at the top of the stem is .472 ; the height of the bowl including this same moulding is also .472; the height of the stem is .528. These proportions are shown in the second drawing. The centre of a square ap- [ 232 J PERFUME VASE 184 B plied at the right end of the 1.472 rectangle fixes the diameter of the moulding; and a square applied to the top of the .472 shape at the left fixes the height of the bowl. Whirling square rectangles applied at the top and bottom of the .472 square determine the height of the raised fillet on the exterior of the bowl and the level of the bottom of the incurved rim; the intersection of these two horizontal lines with diagonals of the square determine (with a slight inaccuracy) the diam- eter of the foot. The area of which the diameter of the foot is the long side and the height of the stem, including the moulding at the top, is the short side is a whirling square rectangle. If a whirling square rectangle is apphed to the top of the whole containing area including the cover, its base coincides with the uppermost of the three red rings on the stem. Its diagonals cut the line of the top of the vase so as to divide the area above this line into a square and two whirling square rectangles (4.236 = .236). The proportions of the details of the knob can be simply expressed in terms of this .236 square. [ 233 ] GEOMETRY OF GREEK VASES ..■■■■■r-'::X I..-:-.. 3. 185 A 185 Alabaster Perfume Vase. Inv. 81.355. An inaccurate drawing is pub- lished in Journal of Hellenic Studies XXXI, 1911, p. 87, Fig. 15. Provenance unknown. Height, with cover, 0.256 m.; without cover, .1955 m. Diameter, 0.1585 m. The stem and foot are in one piece. The bowl is in two pieces, the upper part, with the incurved rim, resting in a rebate cut at the inner edge of the lower part. These two pieces show no signs of having been cemented together. The lid and its knob are in two pieces. The vase, with its cover, is contained in a whirling square rectangle. Without the cover it fits the rectangle, 1.236, or two whirling square rectangles placed horizontally. It is noteworthy that the most important horizontal line in the composition — the ridge of the bowl — is at half the height of the applied square. The drawing of the vase in section shows that the junction of the stem and bowl is C 234 ] PERFUME VASE 185 B at half the height of the whole vase with the foot omitted. This drawing also shows that the lid and knob, considered separately, are enclosed in a square, and that the joint between the lid and the knob is at the level .191 in this square. The knob, therefore, is contained in the rectangle .809 appUed to the small square. The ratios are : Height with cover 1.618 Height without cover 1.236 Height to ridge of bowl 1 .118 Height to top of stem 882 Height of bowl alone 354 Height of foot 146 Diameter 1.000 Diameter of rim of bowl 5528 Diameter of top of stem 264 Diameter of foot 528 Diameter of lid 4236 C 235 ] PRINTED AT THE HARVARD TJNIVERSITy PRESS CAMBRIDGE, MASS., IT. S.A.