The Shapes of Birds' Eggs: Mathematical Aspects

Abstract

In the Handbook o! North American birds the description of egg sizes, and more particularly of egg shapes, proceeds on somewhat different lines from those of previous works. The present paper explains the methods of measuring and methods of "reducing" the observations. The elements actually measured are the conventional average length and breadth, plus the (unconventional) radii of the blunt and pointed ends. The standard de•Sations of each of these are estimated from the sample. From these measurements, all of which in effect are measurements of size, certain dimensionless quantities are derived hereinafter referred to as asymmetry, bicone, and elongation. These are specifications of shape independent of size. This paper sets forth the theory, modifies the theory to achieve maximum practical utility, and gives reasons for believing the results valid and adequate. In addition methods of sampling are discussed, and a new procedure of sampling is given for getting somewhat more representative results than heretofore. TaE only figures usually given for the size of an egg are its length and its maximum diameter, hereinafter called its breadth. The older texts, such as the Catalog of eggs in the British Museum (1901) or Seebohm (1896), give the extreme values that have been encountered for these figures in some survey or other, and this specifies the "range" of lengths or breadths within which the dimensions of another egg of the same species may reasonably be expected to fall. In the Handbook of British Birds (Witherby et al., 1938), in Bent (1919), and in some other more recent texts the average value of length and breadth is given, together with the extreme values. Neither method is entirely satisfactory, for even if a hundred eggs were examined to obtain these extremes, it is obvious that if we examine another hundred, half the extremes may be exceeded. Still more recently, some authors have adopted the more useful procedure of giving the averages and the standard deviations rather than the average and the extreme values. Then, provided that the "sample" is drawn from a "normal" or "Gaussian" population of eggs-and Van Bree (1957) has shown that at least for the population he examined this assumption is sound-we can estimate the probability that an egg will depart by any given amount from the average. There is one precaution that has not normally been observed. Measurements sh'ould be strictly independent and on unbiased samples. As it has been shown (Preston and Preston, 1953) that parentage has a significant effect on egg size, a valid sample, if it is to represent a species, should have equal numbers of eggs from each parent, and indeed it would be best to have as many parents represented as possible. In the case of a human population, for example, it would not be wise, in estimating the average 454 The Auk, 85: 454-463. July, 1968