MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 333 



shoots of higher plants should have evolved such a growth- 

 construction as a means of increasing the body surface remains 

 unsolved ; but so long as a shoot possesses a persistent embryonic 

 growing apex, some such construction would appear necessary : 

 in fact, if the growth-centre is to remain a point, dichotomy would 

 be the only simple alternative. Why, again, certain curves should 

 be selected, their number, ratio, inequality or approximate or 

 actual equality, still constitute further problems apparently 

 hopeless of any immediate solution, although teleological sug- 

 gestions may be put forward to explain the frequency of simpler 

 forms of symmetry, e.ff, (2 + 2) and (1-|-1). But granted the 

 initiation of such construction systems by the plant itself as the 

 elaborated response to some general co-operation of external 

 agencies, these fundamental characters are mathematical conse- 

 quences, however much or little the subsidiary action of special 

 influences may tend to subsequently mould the growth-forms 

 thus initiated by the shoot-apex. The influence of external 

 environment, of which so much is expected in these days by 

 enthusiastic materialists, must have something to act upon: the 

 use of the favourite expression adaptation implies the pre-existence 

 of a certain something which can be modified ; and just what this 

 something is, and how far it goes, is thus defined in mathematical 

 terms by such a generalisation as that of the growing system of 

 growth-centres. 



Leaf - appendages in centric growth - systems are therefore 

 hilateral, dorsiventral, and isophyllous, not from direct relation to 

 the action of any such agencies as gravity or vertical light, but 

 from the mechanical laws controlling the distribution of the 

 material substance of which they are composed. Further, a 

 plant shoot builds such primary appendages of one kind only; 

 whether they are all to be classed under the general term leaf- 

 members, or whether this term is to be restricted to the specialised 

 assimilating organs, is a matter of little consequence. The abstract 

 Urblatt of Goethe is now exchanged for an actual concrete and 

 mathematically defined appendage, the quasi-circle primordium, 

 and the futility of any discussion as to the priority of foliage-leaf 

 or sporophyll becomes obvious. 



