CHAPTER X 



A PARENTHETIC NOTE ON GEODETICS 



We have made use in the last chapter of the mathematical 

 principle of Geodetics (or Geodesies) in order to explain the con- 

 formation of a certain class of sponge-spicules ; but the principle 

 is of much wider apphcation in morphology, and would seem to 

 deserve attention which it has not yet received. 



Defining, meanwhile, our geodetic line (as we have already 

 done) as the shortest distance between two points on the surface 

 of a solid of revolution, we find that the geodetics of the cylinder 



give us one of the simplest of 

 cases. Here it is plain that the 

 geodetics are of three kinds: (1) 

 a series of annuh around the 

 cyhnder, that is to say, a system 

 of circles, in planes parallel to 

 one another and at right angles 

 to the axis of the cyhnder (Fig. 

 236, a); (2) a series of straight 

 lines parallel to the axis; and 

 (3) a series of spiral curves wind- 

 ing round the wall of the cyhnder 

 (6, c). These three systems are 

 all of frequent occurrence, and 

 are all illustrated in the local 

 thickenings of the wall of the 

 cyhndrical cells or vessels of plants. 



The spiral, or rather helicoid, geodetic is particularly common 

 in cylindrical structures, and is beautifully shewn for instance in 

 the spiral coil which stiffens the tracheal tubes of an insect, or 

 the so-called "tracheides" of a woody stem. A similar pheno- 



B C 



Fig. 236. Annular and .spiral thick- 

 enings in the walls of plant-cells. 



