X] ON GEODESICS 743 



points on the curve, the curve gives the shortest distance between 

 them. It often happens, in the geodesic systems which we meet 

 with in morphology, that two opposite spirals or rather helices 

 run separate and distinct from one another, as in Fig. 346, C; 

 and it is also common to find the two interfering with one another, 

 and forming a criss-cross or reticulated arrangement. This indeed 

 is a common source of reticulated patterns. 



The microscopic and even ultramicroscopic structure of the cell- 

 wall shews analogous configurations: as in the large cells of the 

 alga Valonia, where the wall consists of many lamellae, each com- 

 posed of parallel fibrillae running in spiral geodesies, and alternating 

 in direction from one lamella to another. Here, and not less clearly 

 in the young parenchyma of seedHng oats, it is the long-chain 

 cellulose molecules which follow a spiral course around the cell- wall, 

 right-handed or left-handed as the case may be, and inchned more 

 or less steeply according to the elongation of the cell. But these 

 highly interesting questions of molecular, or micellar, structure 

 lie beyond our scope*. 



Among the ciliated infusoria, we have a variety of beautiful 

 geodesic curves in the spiral patterns in which their ciha are 

 arranged; though it is probable enough that in some comphcated 

 cases these are not simple geodesies, but developments of curves other 

 than a straight hne upon the surface of the organism. In other words, 

 they seem to be instances of "geodesic curvature." 



Lastly, an instructive case is furnished by the arrangement of 

 the muscular fibres on the surface of a hollow organ, such as the 

 heart or the stomach. Here we may consider the phenomenon 

 from the point of view of mechanical efficiency, as well as from 

 that of descriptive anatomy. In fact we have a right to expect 

 that the muscular fibres covering such hollow organs will coincide 

 with geodesic lines, in the sense in which we are using the term. 

 For if we imagine a contractile fibre, .or an elastic band, to be fixed 

 by its two ends upon a curved surface, it is obvious that its first 

 effort of contraction will tend to expend itself in accommodating 



* Cf. (e.g.) C. Correns, Innere Struktur einiger Algenmembranen, Beitr. zur 

 Morphol. u. Physiol, d. Pflanzenzelle, 1893, p. 260; W. T. Astbury and others, 

 Proc, E.S. (B), cix, p. 443, 1932, and other papers; G. van Iterson, jr.. Nature, 

 cxxxviii, p. 364, 1936; R. D. Preston, Proc. U.S. (B), cxxv, p. 772, 1938; etc. 



