THE BILATERALITY OP APPENDAGES. 313 



presents a striking picture of the familiar quincunoial calyx of the 

 great majority of pentamerous Dicotyledonous flower-types ; and 

 given an axis producing such a system of lateral members, the 

 reason for retaining five with such constancy to constitute a pro- 

 tective investment becomes increasingly obvious.* These relations 

 are necessary consequences of the utilisation of lines of equal 

 distribution in spiral series, and are of especial interest in connec- 

 tion with the Fibonacci system, in that they give the clue to the 

 number of members requisite to give the best and most equal 

 arrangement, whether in the form of a single cycle or many, so 

 that in dealing with the numerical relations of the parts of asym- 

 metrically constructed flowers it becomes possible to deduce a 

 normal or average type of construction. 



The (3 -J- 5) system, in an adult condition, and represented by 

 spirals of Archimedes, from the standpoint of the Schimper-Braun 

 formulae, becomes a | type, and not a f as it is usually reckoned ; 

 the error being introduced with the assumption that a specialisa- 

 tion of Jive members implies a cycle of f , although, as previously 

 noted (p. 15), there was no possible criterion for such an assump- 

 tion. That is to say, in dealing with a spirally constructed 

 pentamerous flower, the (3 + 5) system brings the first petal to 

 the front in the gap between sepals 1 and 3, while by regarding 



* The apparent mimicry of pentamerous flower mechanism observed in the 

 inflorescence of many Composites, in which a calyx-like involucre of five 

 segments is succeeded by a corolla-like series of ray-florets, 5 or 8, and a 

 series of disk-members resembling the spirally arranged sporophylls of a 

 Ranunculaoeous type, is thus solely due to the working out of corresponding 

 phyllotaxis rules in the two cases ; one full cycle of protective members being 

 succeeded by one full cycle of decorative ones and one or more cycles of 

 reproductive members. If the phyllotaxis system is low, (2-)- 3) or (34-5), the 

 retention of 5 or 8 members in one full cycle of contact is as normal as the re- 

 tention of full cycles of higher- terms of the series in Sunflower capitula. There 

 is no proof that this so-called mimicry is intentional, or even biologically 

 advantageous ; it is the necessary outcome of a similar low phyllotaxis system 

 combined with an attempt to reduce the members of each kind to a minimum ; 

 cf. the 5-star flower-like capitula of Ohrysogonum virginianum, and Tagetes 

 signata ; as also Dahlia coccinea — 5 involucral segments, 8 rays, and (8-t-13) 

 disk-florets— with Aconitum iiapellus—b sepals, 8 petals, and (8 + 13) sporo- 

 phylls. 



