ON THE THEORY OF VEGETABLE 

 SPIRALS*.t 



THE circumstances under which the following remarks have 

 been composed would be a sufficient excuse for any defects, if 

 they were not rather a reason why no such task should have 

 been attempted. But the error, if it be one, will not be re- 

 peated, and, like Socrates, ^^KaXv^ra^evo^ p& \. 



The following remarks on the Theory of Vegetable Spirals, 

 appear to possess some interest, though it is very possible that 

 it is only my imperfect acquaintance with the history of the 

 subject which makes me think they contain anything not already 

 noticed. On the chance however that they are new I shall en- 

 deavour to put them down, though my state of health obliges 

 me to do so in a hurried and imperfect manner. 



The fundamental principle of what follows is the resolution 

 of the symmetrical spiral into portions actually unsymmetrical 

 yet capable of becoming regular without essential alteration. 

 Symmetry is therefore regarded as something superadded to the 

 essential principle of the spiral, and we are thus led to trace 

 the existence of this principle in cases in which it exists apart 

 from any appearance of symmetry. 



A spiral may be divided in two ways : by two azimuths or 

 by a horizontal circle. We may either obliterate all the leaves 



* Now first published. 



t Though the nature of the subject and my own ignorance make the whole of 

 these remarks unsatisfactory, no part of them seems to me more so than that which 

 relates to the way in which simpler forms are included in the more complex. I 

 have not expressed my own idea, imperfect as that is. The following statement 

 would be nearer it : " Every form includes all simpler forms, modifying them alter- 

 nately in opposite directions, i. e. by expansion and contraction." 



t Plat. Phcedr. 



