NATURAL SCIENCES OF PHILADELPHIA. 527 



Jon by the study of organic forms, to the endeavor to decipher the principle 

 or purpose of a complicated machine by the contemplation of its shadow.* He 

 discourages the notion that the shape of the egg is susceptible of a mathemat- 

 ical explanation. The form of the egg, he considers, is not the immediate 

 product of a formative tendency, but the mechanical result of a twisting action 

 of the oviduct, and gives as little hope of an explanation of the forming forces 

 as, for example, the shape of a top does of comprehending the law of formation 

 of the person who turned it. f 



Meckel}: accounts for the form of the egg in a similar manner. He cites 

 Thienemann to show that when the egg is forced rapidly through the oviduct, 

 in consequence of persistently chasing the hen, the egg is then deformed, 

 being greatly elongated and without a hard shell. He also alludes to the ex- 

 periments of M. St. Hilaire in proof of the fact that hen's eggs placed verti- 

 cally during incubation either do not come to development or else produce 

 monsters. On the whole, he appears to be of opinion that the form of the 

 egg may not only have a mechanical origin, but may be important as a me- 

 chanical means in determining the form of the embryo. 



OF MATHEMATICAL OVOIDS. 



Fechner adopts the oval of Descartes, proposed by Steiner, as the true rep- 

 resentative of the form of the egg. The elliptic spheroid he considers to be 

 a rough approximation to the true form ; but M. St. Hilaire states that out of 

 six eggs of the Epiornis, sent to France, five were nearly true ellipsoids. || 

 The other had a large and a small end. We shall now consider particularly 

 the curve proposed by ourself to represent the longitudinal section of an 

 egg. This curve belongs under a general formula which includes the ellipse. 

 We shall principally consider a curve having an obtuse and an acute end, and 

 which may be called the hyper-ellipse, and the solid generated by its revolu- 

 tion, the hyper-ellipsoid.^ 



Construction of the hyper-ellipse. Measure the length and thickness of the 

 egg. Draw (Fig. 1, Plate 1) A B, H D, each equal to the length of the egg, 

 and bisecting each other at right angles in C. Make D K equal to the half- 

 thickness of the egg, and on H K describe a semicircle cutting A B in F. 

 Then A B is the axis of the hyper-ellipse, and F is the focus. 



Construct an ellipse (Fig. 2) with the semi-axes F A, F B equal respectively 

 to the same distances in Fig. 1, and draw any radius vector F P. 



In Fig. 3 draw B F, F A, as in Fig. 1, and make the angle AFP equal to 

 twice the angle A F P of Fig. 2 ; also make F P equal to the same in Fig. 2. 

 The point P is then a point of the hyper-ellipse. In a similar manner any re- 

 quired number of points may be found, and the curve traced through them 

 by hand. Instead of beginning the construction at A, we may commence at B, 

 making the angle B F P equal to twice the same of Fig. 2, and the radius 

 F P the same. 



*AUgemeine Physiologic des Koerpjr lichen Libens. Leipzig. 1S51, p. 328. 



t Ibid. p. 335. The labors of Ilanstein and Wright in investigating tne law of phyllotaxis al- 

 though they do not prove mathematically the cause of phyllotaxis, but rather pertaiu to its teleo- 

 logical significance appear to me to contain remarkable applications of mathematics to the 

 study of Organic Morphology, and to take much from the general force of Lotze's criticism. Ilan- 

 ,-ttein ueber den Zusamrnenhang der Blattstelluug mit dem Bau des c.ikotylen Holzringes. Mo- 

 aatsber. d. Koenl. L'reuss. Ak.d. Wiss., Berlin, 1S57, p. 105. Wright on the most thorough, uniforai 

 distribution of points about an Axis. Mathematical Monthly, April, 1809. 

 % Die Bildung der fuer partielle Fnrchung bestimmten Kier der Voegel, Ac. Zeitschr. f. Wiss. 

 Zoologie, vol. 3, 1851, p. 432. 



g We may refer the reader to Mr. Hay's Principles of Symmetrical Beauty, and to Purdie on 

 Form and Sound (Edinburgh, 1859), for information concerning the composite ellipse a figure 

 which seems to offer or to suggest means for closely imitating the forms of various eggs. 



(| Note sur de3 ossemeate et des oeufs trouve* a Madagascar, dans des alluvions moderns, ct 

 provenant d'un oiseau gigantesque ; par M. Isidore Geoffroy-Saiut-Hilaire. Comptes Kendus, 

 vols, xxxii. p. 101 ; xxxix. p. 833; xlii. p. 315, and xliii. p. 518. 



fi This curve may be termed the hyper-ellipse, because its radius vector is a powar of the radius 

 vector of an ellipse, taken from the fjcus, or bocause its radius is derived from the ellipse as in 

 the following construction. 



1862.] 



