on the other exterior side he is nude, but the other standing figure on this side is nude. The two standing figures here (PLATE III) face in the same direction, whereas on the other side they face toward one another, with the nude one to the right, another case of variety in the midst of symmetry. We see it

again in the cylices and baskets which hang on the wall. Three baskets slightly differentiated and part of one cylix and another complete cylix are preserved in the drawing. In the centre of the foot of the cylix to the right there is a solid black circle. within a ring instead of a small open circle within a larger circle, as on the other side. The black relief lines also vary.

It has seemed worth while in the case of this important vase to go into some detail, as an analysis brings out many interesting features of Greek art in general, and of Greek vases of the severe red-figured style in particular, especially that of variety in the midst of symmetry and similar motives. I hope I have also demonstrated that the cylix is in the style of Duris of which there are two others in Baltimore (Hoppin, op. cit. I, p. 277; Beazley, op. cit. p. 99), the cylix with top-spinning assigned by Hartwig to the "Master with the Spray," being also Durian. Even if some one argument is doubted, the cumulative evidence is conclusive. In the case of Duris there is less doubt than with other vasepainters about an unsigned vase, since we have forty or more signed vases and his style is clearly marked. The resemblance of our cylix in style to the signed vases and to unsigned vases which have been attributed to Duris with general consent is very great. The subject, the meander and star border, the palmette pattern, the shape of the heads, the hair and beard with the relief lines, the anatomical details, the slender arms and sharp elbows, the noses and ears, the eyes drawn with circle and dot (cf. J. H.S. XXXIV, 1914, p. 189), the drapery, the recurrence of similar gestures, the general proportions of the figures, especially the elongated standing youths whose small heads are on a level with those of the reclining figures, the love of variety in detail in the midst of symmetrically arranged groups, but above all the resemblance in style to signed vases of Duris, make it almost certain that we have a cylix painted by Duris himself or by one of his best pupils in his second period (hardly senile as Beazley calls it) when he painted his other cylices with banquet scenes and when in the midst of a certain stiffness he exhibited not only care but a greater power of facile execution (Fowler-Wheeler, Greek Arch

aeology, p. 495). Jacobsthal is inclined to date some of these vases as late as 452, but in view of the lack of true rendering of the anatomy in many cases I am not inclined to date our cylix later than 470. On the other hand I hesitate after the remarks of Hauser (text to Furtwängler-Reichhold, op. cit. II, p. 232) and Jacobsthal (op. cit. p. 63 and passim) to date Duris's second period as early as 480.

THE JOHNS HOPKINS UNIVERSITY,
BALTIMORE, MD.

DAVID M. ROBINSON.

Institute of America

DYNAMIC SYMMETRY: A CRITICISM

I

THOUGH published under university auspices and enlisting for its preparation the interest and seemingly the sympathy of the classical specialists of two of our great American museums, Mr. Hambidge's treatise on dynamic symmetry has thus far elicited very little archaeological comment. Yet his theory, if true, is of fundamental importance for Greek esthetic theory and for our understanding of the relations between mathematics and artistic practice in antiquity. In certain modern circles, among lay theorists and practical designers, Mr. Hambidge is said to have succeeded in gaining a considerable following for his methods; but it is not as a working formula for artistic production today, so much as a brilliant and novel explanation of the structural formulae of sixth and fifth century Greek vases that Mr. Hambidge's treatise commands the attention of classical archaeologists, who owe its author their gratitude for turning his labor and their attention so searchingly to the fundamentals and minutiae of the formal structure of many of the most beautiful shapes of ancient pottery.

Those who had not the fortune of initiation through personal instruction had for long heard distantly of Mr. Hambidge's process of "Dynamic Symmetry" as of some great thaumaturgy through which the theory of ancient design had been fundamentally affected. When at last Mr. Hambidge published his book on structural design, based on accurate and very detailed measurements of Greek vases in the Boston, New York, and New Haven museums, his expectant public among classical archaeologists felt, with disappointment, that the subject had not been made as accessible as had been hoped. Very remarkable properties were apparently inherent in ancient vases; but there was no simple and direct statement of what dynamic symmetry was, or how it was to be detected, or what artistic properties it imparted. The shadow of a geometric mysticism seemed to obscure the issues. Before Jay Hambidge, Dynamic Symmetry: the Greek Vase. Yale University Press. 1920.

1

American Journal of Archaeology, Second Series. Journal of the
Archaeological Institute of America, Vol. XXV (1921), No. 1.

18

one can appreciate, one must understand; but the understanding of dynamic symmetry had been left extremely difficult.`

From Mr. Hambidge's treatise it is apparent that dynamic symmetry has to do with the relation of surface areas in a design. Static symmetry" (which is treated as its antithesis) depends upon simple commensurability of lengths, of linear measurements. But dynamic symmetry, apparently, is not mere commensurability of area. An ellipse and a circle having twice the area of the ellipse are apparently not an instance of dynamic symmetry. Mr. Hambidge confines his instances to rectangles. The computation of areas of curvilinear outline would be distressingly difficult. Accordingly, in the case of Greek vases, it is not the area of the vases which is computed. The actual area of the vase surfaces, I may say, is nowhere computed, and is a matter of indifference. For the curvilinear area of the vase a simple rectangle is substituted. This is the containing rectangle, of which the sides are parallel to the vertical axis and the base-line of the vase. It is, as it were, the smallest rectangular frame into which the whole vase will fit. To this rectangle the analysis for dynamic symmetry is applied. Not its size, but its shape, is important. If this rectangle can be split up into rectangles of similar and related shapes, and if these smaller rectangles can be used to determine recognizable elements of the vase, the occurrence of dynamic symmetry is held to be established.

In a sense, the first condition can always be fulfilled geometrically, since within any rectangle an infinite number of rectangles of similar and related shapes can be constructed. But dynamic symmetry apparently requires not merely that similar rectangles shall be discoverable, but that the whole rectangle may be completely subdivided into squares and rectangles similar to the original rectangle or of closely related shape. This process of subdivision is the chief geometrical element in dynamic analysis. Such analysis of a rectangle is usually accomplished (1) by division into halves, thirds, etc., or (2) by laying off the shorter side on the longer side (so as to form a square) and then treating the remainder of the original rectangle as a new rectangle subject to similar analysis. The completed process will thus show a disintegration of the original rectangle into squares and rectangles. These rectangles will be similar to the original rectangle only under certain conditions. Rectangles fulfilling these conditions are the only ones used for dynamic symmetry. Mathematically

it is perfectly easy to formulate these conditions and discover what rectangles will satisfy them.1

1 As this is nowhere clearly performed in Mr. Hambidge's book, a brief notice may be of service:

(1) When the short side is laid off on the long side of a rectangle so as to form a square, the requirement that the remainder of the rectangle shall be similar to the original rectangle may be stated thus,

where x is the long side, and the short side is 1. This equation will be satisfied √5+1

2

if x= (i.e. very nearly 1.618). A rectangle with its sides in the ratio of 1.618 to 1 will, therefore, satisfy the condition. This particular 1.618 shape is called by Mr. Hambidge the "Whirling Square Rectangle” and is, next to the square, the most frequent form in dynamic analysis.

(2) Since

√x

= it follows that rectangles of which the sides are to each other as the square root of an integer is to 1, will have a special property of subdivision into shapes similar to the whole. Substituting the value 2 for x, it follows that when a "Root-Two Rectangle" (i.e. one of which the longer side is to the shorter as √2 is to 1) is cut into 2 parts (sc. at the mid point of the longer side) each part will be a "Root-Two Rectangle"; similarly (substituting 3 for x) when a "Root-Three Rectangle" (i.e. with sides as √3 to 1) is cut into 3 parts, each part will be a “Root-Three Rectangle" (and so on for higher values of x). Because of this property of subdivision, the "Root-Rectangles" are peculiarly suitable for "dynamic" analysis.

(3) The "Root-Five Rectangle" (i.e. with sides as √5 to 1) is related to √5-1 √5+1 the "Whirling-Square Rectangle." Since is the reciprocal of 2 (by equation 1) it may be simply stated that a "Root-Five Rectangle" is made up of a square plus two "Whirling-Square Rectangles," because

2

The geometric analysis of "Dynamic Symmetry" is therefore based on these three equations:

square and "Whirling-Square Rectangles").

The remarkable subdivisibility of the rectangles used in "Dynamic Symmetry" is thus due to certain simple inherent mathematical properties of the particular rectangles selected. It is not due to the potter nor to the construction of the vase.