NOTATION. 217 



adopted of connecting the two numbers by the sign + , which may 

 be taken as meaning simply and, or more pictorially as a cross. 



The data are simply that so many curves cross so many, nothing 

 is added as to the angle of intersection, and such a formula 

 includes the simple facts of observation. 



Any attempt to indicate a ratio introduces a source of error ; 

 the formula (5 : 8) would mathematically imply a construction by 

 log. spirals in that ratio ; and, although it has been suggested that 

 such is actually the case at the growth-centre, the expression has 

 been avoided until the proof appears more satisfactory. Still 

 greater is the error of the old notation which states that 5 and 8 

 parastichies imply an ^ genetic spiral with orthostichies as 

 straight lines. Such a mathematical statement is alone possible 

 for the spirals of Archimedes and helices originally postulated by 

 Bonnet and Calandrini for adult constructions. 



It has been pointed out that in no growing system is any helical 

 construction possible, and that the retention of the old fractional 

 notation constitutes a hopeless state of confusion which still 

 vitiates much of the literature of the subject ; since it is clear that 

 no theory which implies unstated the mathematics of helical 

 construction, and which therefore deals with members of equal 

 bulk or points equally spaced, can ever afford any insight into 

 the construction of a growing system of gradated primordia. In 

 no instance is the unfortunate error of this gratuitous interpola- 

 tion of helical mathematics more conspicuous than in dealing 

 with the phenomena of contact-pressures ; the two things cannot 

 coexist in the plant. Contact-pressures must be growth-pressures, 

 and equal volume is only attained in the lateral members at the 

 close of their growth period, when growth-pressures cease with the 

 growth of the members : when they are mutually pressing one 

 another they are not equal in size, and the Archimedean notation 

 becomes so misleading that deductions involving this standpoint 

 are often quite unintelligible.* 



While, again, the genetic-spiral hypothesis only includes a certain 

 number of phyllotaxis constructions, all such rhythmic patterns 

 may be considered from the standpoint of the simplest method of 

 * Cf. Soliwendener, Berichte Deutsch. Bot, Gesell., 1902, p. 264. 



