332 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



II. The appendages further possess that peculiar attribute 

 called by Sachs their dorsiventrality — a term which, as previously 

 shown, requires to be taken in a purely metaphorical sense, and 

 which only holds its own in that it is as useful as any other 

 expression for indicating the fact that the members possess two 

 unlike surfaces, and that these are upper and lower surfaces, a 

 point not implied by the term bifacial. The form of the quasi- 

 circle shows that the peripheral portion of the appendage, as 

 seen in section, is larger than the interior part; so that, in 

 carrying these members over the slope of a dome-shaped apex, 

 the exaggerated side becomes the lower surface of the leaf : this 

 again being the mathematical consequence of the fact that the 

 growth-centre is transferred to a point nearer the inner side of 

 the curve. 



III. The term isophylly indicates still more concisely that 

 property of the members in which the bilaterality of the 

 appendage is expressed in the form of two equal sides about an 

 axis of the member in the tangential plane of the system, and is 

 usually applied to the shape of the lamina surface rather than to 

 its section: this again is equally a mathematical property of a 

 growing primordium possessing such a curve-section. 



The mathematical investigation thus shows that all primordia, 

 whatever value he given to m and n, present these properties as 

 fundamental and unavoidable features of construction. Every 

 appendage is mathematically bilateral, dorsiventral, and isophyllous, 

 with regard to the shape of the curve and the position of its centre 

 of construction. These are mathematical necessities of the type of 

 growth-system adopted by the shoot as a centric growth-centre 

 producing a rhythmic series of subsidiary centric growth-centres. 

 A growing system might evidently have one such centre or more 

 than one. One is the simplest case, and as a matter of observa- 

 tion is the general rule ; on the other hand, the case of multiple 

 growth-centres is included under the botanical title of fasciation 

 phenomena. Here at last is a definite foundation on which to 

 buUd the morphology of the shoot ; and it now becomes possible 

 to draw a distinction between the necessary properties and the 

 accidentia, or phenomena of subsequent adaptation. Why the 



