The Shapes of Birds' Eggs

usual way of describing a bird's egg is to report its length and maximum diameter. latter is not the diameter at the mid-point of the length, as a rule, because most eggs are bigger at one end than the other. A statement of length and maximum diameter is not a complete description therefore, and the question arises whether a complete description can be given, and, if so, how many measurements are needed to define it. Obviously, it requires at least three, but it may take more. In fact, it will be shown in what follows that most avian eggs require four measurements or constants (that is, two in addition to length and breadth), and that some require five. problem then becomes one of finding a general equation suitable for all eggs, of expressing the facts in the simplest, most logical, and most convenient way, and of devising apparatus for measuring the eggs and deducing the constants. This investigation was not undertaken primarily as a mathematical amusement. It seems likely that it may throw some light on several biological and ecological problems, but the present paper concerns itself merely with the broad question of what is the shape of a bird's egg. mathematics may conceivably show something of the physiology and mechanics of egg-laying, since the shape of the egg is a response to the forces exerted by the oviduct during shell-formation (Mallock, 1925; D'Arcy Thompson, 1943). These biological problems are perhaps more interesting than the purely geometrical one of defining egg shape. Thompson (op. cit: 936, footnote) seems to throw up his hands in the belief that egg shape is indescribable, particularly if it happens to be a guillemot's ( = murre's). Romanoff and Romanoff (1949: 88) are more explicit: The numerous variations in the contour of individual eggs obviously cannot be expressed in mathematical terms. It seems to me that, on the contrary, nothing can be more obvious than that, as a matter of theory, any such shape should be readily described; and, as a matter of fact, they take very little describing, and the results appear interesting. present paper will, therefore, be confined to a logical development of the mathematical aspects, leaving the biological and other problems, for which they may provide a solution, for later papers. Since this paper was completed and accepted by the A.O.U., I have received through the kindness of Professor Bartels and Dr. Storer, both of Ann Arbor, Michigan, a reprint of a paper by Jun-ichi Okabe (On the Forms of Hens' Eggs Reports of Research Institute