THE RELATIONS OF SURFACE TO MASS. 37 



The Relations of Surface to Mass. 



However much the weight of an animal increases during its development, 

 the ratio of the free surface to the mass alters but slightly from the ratio estab- 

 lished when the embryo begins to take food from outside. It is only for conve- 

 nience that I express this law in this precise form ; in reality, about it our knowl- 

 edge is scanty and our conceptions vague. According to a geometrical principle, 

 when the bulk of a body bounded by a simple surface increases, the surface 

 enlarges less than the mass — in the simplest case of a cube, the surface increases 

 as the square, the mass as the cube, of the diameter. If in a cube of unit diam- 

 eter one unit of surface bounds one unit of mass, then in a cube of three units 

 diameter nine units of surface will bound twenty-seven units of mass, the pro- 

 portion in the first cube is i : i, in the second 1:3. To maintain the proper 

 proportion in the embryo, simple enlargement is insufficient, therefore the sur- 

 face increases by becoming more and more irregular. The irregularities are 

 characteristic of each organ and part, and may be either large or microscopic. 

 They may be conveniently grouped under two main heads — projections and 

 invaginations. 



Projections are illustrated by the limbs, filaments of the gills in fishes, the 

 villi of the intestine, folds of the stomach in ruminants, etc. In every case the 

 projection is covered by an epithelium and has a core of mesodermic tissue. 



Invaginations exist in much more varied form and play a principal part in 

 the differentiation of the animal body. They may be classified under four 

 principal heads: (1) Dilatations; (2) diverticula; (3) glands; (4) vesicles. Di- 

 latations have considerable importance in embryology; the stomach, lungs, 

 bladder, and uterus arise as gradual dilatations of canals or tubes of originally 

 nearly uniform diameters. Diverticula, in the sense of relatively large blind 

 pouches, also form important organs, such as the caecum and appendix vermi- 

 formis, or the gall-bladder; these structures arise, each as a blind outgrowth of a 

 canal, the walls of which at a certain point rapidly grow to form the pouch. 

 Glands are, as first shown by Johannes Midler's classic researches, only small 

 diverticula, which end blindly and appear in an immense variety of modifica- 

 tions ; the manifold types of glands are discussed below in a separate paragraph ; 

 they constitute the largest class of organs with which we have to deal. The 

 glands are developed from epithelium and push their way into the mesoderm 

 upon which the epithelium rests, while in dilatations, and in diverticula, the 

 epithelium and mesoderm expand together. Vesicles we call those epithelial 

 sacs which develop somewhat like glands by growing into the mesoderm, but the 

 mouth of the invagination closes by the coalescence of the epithelium, thus 

 shutting the cavity. The closed sac separates from the epithelium from which 



