THE SPIRAL THEORV 



497 



which spnng from the growing-point, appear in a sequence and order which excludes 

 the employment of the spiral. Payer, who has done such excellent service with 

 respect to the developmental history of flowers, showed that the flower of the 

 common Mignonette (Fig. 334) is a strictly bilateral structure. From its growing- 

 point the sepals, petals, stamens, and carpels are put forth in such a manner that 

 their development begins at one point of the circumference, and then proceeds 

 right and left on both sides of the young flower towards the opposite point. 



Closely connected with the spiral theory, was also the view that lateral shoots 

 from a main shoot can only arise in the axils of the leaves. But even this so-called 

 principle of axillary branching presents itself on unprejudiced observation only as 

 a special case, occurring chiefly among the Phanerogams, and even there only 

 when lateral shoots arise on erect and radially constructed shoots. Where the 

 dorsi-ventrality of the mother-shoot is sharply expressed, however, as in the cases 

 mentioned above, the lateral shoots may arise on the flanks of the mother-axis 

 while the leaves are situated dorsally, or vice versa. In Ferns and Mosses, the most 

 careful investigations into the development have shown that normal lateral shoots 

 may arise on the back or on the 

 flank of the base of the leaf, or 

 even quite independently of any 

 relation to the leaf. Even in the 

 Phanerogams, from which the older 

 theory of phyllotaxis started, it is 

 not always possible to apply the so- 

 called principle of axillary branching: 

 the floral region especially aff"ords nu- 

 merous examples of this. Not rarely, F,c. 334.-Developn,ent of, he flower of /..W..^«,-..., after Payer). 



n« in tViP Prilrifpr!*» mnnv Ponilinn '^° "^^ ^'^"- ^ y°"''S. to the right an older bud; the anterior sepals s 



as m tne «^rUClIerae, many rapillOn- ^,^ ^^, ^„^y f,„„ ^l,^ l^,,^, t,,^ posterior ones remain, pp petals i 



arpn= and T^ono-inpcc. in Ai-nirlpc*! j/ stamens, the posterior ones already advanced in size, the 



acese ana UOragmeae, m AlOiaeae, ones not yet developed ; ^carpel-rudimentary fruit. 



and others, the flowers arise on the 



mother-axis without being preceded by leaves, in the axils of which they might arise. 

 They are leafless branch-systems ; and in some Boragineae and Crassulacese, 

 where it is true leaves are formed on the dorsi-ventral axis of the inflorescence, 

 these are situated with no obvious relation to the flowers, and even the number of 

 these leaves does not always correspond to the number of the flowers. 



The theory of phyllotaxis, with its assumption of the spiral as a fundamental 

 law of growth, has, to the great injury of all deeper insight into the growth of the 

 plant, established itself so firmly that even now it is not superfluous to show up its 

 errors point by point. Thus, among the errors of this theory is the one that the 

 spiral arrangement of all organs on a common axis must necessarily follow from its 

 so-called parasiichies. By the term parastichy is understood the serial arrangement 

 of lateral organs in two or more directions which cross one another ; this appears 

 very clearly in cases where numerous organs are situated close beside one another on 

 a common axis. If in such a case the organs have approximately the same form, they 

 involuntarily present rows to the eye, which it can follow from right to left or from left 

 to right. The two dotted lines a and I) in Fig. 335 will at once show what is meant. 

 Now it is of course possible, with certain premisses which the founders of the spiral 



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