GEOMETRICAL REPRESENTATION OF GROWTH. 37 



more perfect and uniform production of a radially symmetrical 

 axis. 



In other words, the " circular-vortex " construction of the sym- 

 metrical apex is secondary, and must be regarded as a special case 

 of a more primitive "spiral-vortex" construction, which is not, 

 however, necessarily a peculiar property of protoplasm, as assumed 

 in the original conception of the spiral theory, but the mere ex- 

 pression of asymmetrical growth. 



In dealing with the spiral development of lateral members, it is 

 therefore necessary to take as a starting point the more general 

 case of a Spiral Vortex, rather than the circular one implied by 

 the typical Angiosperm apex of Sachs. 



In such a spiral vortex, the stream lines are logarithmic or equi- 

 arigular spirals,* which only reach their pole at infinity, and lines 

 of equal pressure and flow will be marked out by the paths of 

 orthogonally intersecting log. spirals. In other words, each circle 

 becomes a coil of a log. spiral, and the radius is represented by a 

 portion of that log. spiral, which cuts the other orthogonally. 



Just as the circular-vortex construction is that of an ideal apex, 

 and is usually masked in any given specimen by secondary 

 phenomena of unequal growth and pressure of the component cells, 

 and possibly even at the theoretically initial group by subsidiary 

 vortices in the main stream, so the spiral vortex structure will also 

 be masked and almost obUterated. Beautiful examples of circular- 

 vortex construction persist in the loose and undifferentiated endo- 

 cortex of many roots (c/. Zea, Philodendron), while a more typical 

 root shows parenchyma more or less hexagonally packed. 



The apex of the root of a Fern f affords a convenient example 



* The logarithmic spiral is the curve whose polar equation iar=a^, where a 

 is constant. It is called logarithmic because another form of the equation is 

 log. r = e log. a. The log. spiral has the property that the tangent at any point 

 makes a constant angle with the radius vector. 



+ The tetrahedral cell of the Fern-root is here selected as an iUuetration, owing 

 to the fact that it is easily ohservable, fairly large, and in the large apex of a 

 healthy root a considerable number of segments can be obtained in a fairly level 

 series. 



At the same time it must be clearly imderstood that the ceU in question cuts 

 off a fourth segment in the sequence to form the root-cap. The exact series is 



