GENERAL OBSERVATIONS. 19 



the same sense with it, and by zigzagging from scale to scale the 

 whole cone is numbered up. From a consideration of the para- 

 stichies of other systems, Braun* tabulated a series of fractions ^, f , 

 h l> ^S' si' ^^-i complementary to those of Schimper's series of 

 orthostichies. The method is strictly accurate, and clearly affords 

 the only way of numbering up the members of a complicated spiral 

 system when the lateral members retain their original close contact 

 on a condensed axis. 



However, as soon as attempts are made to bring the system of 

 parastichies into line with that of orthostichies, difficulties arise. 

 The system of (8 + 13) parastichies corresponds in the Schimper- 

 Braun theory to a phyllotaxis of ^j-, but it is at once clear that 22 

 is not superposed to 1. The scale marked 35 is practically over it, 

 and hence the phyllotaxis would be usually given as one stage 

 higher, i.e. ^, to fit which divergence it would be necessary to 

 assume that the correct parastichies are those passing through 

 1, 14, 27, and 1, 22, 43 respectively. In other words, the steepest 

 parastichies are to be taken as a guide. As in the preceding case 

 of Sempervivum, however, there is no evidence that 35 is vertically 

 superposed to 1 ; the figure, in fact, shows that it is only approxi- 

 mately so, and that if 56 were normally placed it might be nearer the 

 vertical line. Owing to the sloping off of the cone, 56 is clearly 

 well off the line, and 35 remains the nearest for practical purposes. 

 There is no proof of its accuracy ; but by comparison with Semper- 

 vivum, the strongest presumption in favour of its being only an 

 approximation, owing to the limited number of members on a 

 cylindrical portion of the axis. There is certainly no clear justifi- 

 cation for assuming any secondary displacements in order to save 

 the theory. 



In fact, there is only one mathematically accurate statement 

 which can be made about such a construction, and that is, that 

 taking four scales in contact, or making use of a rhomb of rhombs 

 {e.g., fig. 7: 1, 9, 22, 14), the cone is composed of (8+13) inter- 

 secting spirals, of which eight are longer and thirteen shorter. 



Adopting the convention that the right-hand direction is that 

 marked by the hand of a watch at 12 o'clock, the cone figured 

 * Flora, 1835, p. 157. 



