THE BILATERALITY OF APPENDAGES. 303 



such theoretical diagrams; thus, as each leaf-rhomb is extended 

 laterally, over an older primordium and under a younger member, 

 the shorter curves assume their characteristic serrated appearance, 

 and the long curves remain with an unaltered clean contact edge. 

 The recognition of the phyllotaxis ratio of such a growing shoot 

 then becomes increasingly obvious ; the smooth edges of the lines 

 of contact of one set of curves directly indicating that they must 

 be the longer curves of the construction. 



Again, the lower the parastichy ratio in the Fibonacci or any 

 other series, the more marked is the "spiral of dorsiventrality " 

 as an obliquely horizontal path ; while with higher ratios this so 

 nearly approaches a circle that the flattening and position of the 

 leaf-insertions along a spiral path are not readily observed ; nor 

 will it be noticed when dorsiventrality becomes excessive, or again 

 is correlated with phenomena of sliding growth. 



Thus, in the case of the sporophylls of Stangeria (fig. 29), the 

 rhombs are obviously extended along an obliquely horizontal 

 spiral, and the same fact may be noticed in Pine-cones ; while 

 in the lowest cases of asymmetrical phyllotaxis, the oblique 

 insertion of the members becomes very marked {cf. Gasteria, fig. 

 58a). Also, comparison of the diagram (fig. 110) shows how 

 little the amount of sliding may be that is sufficient to bring the 

 transverse diagonal of each rhomb very approximately parallel 

 with the circle, i.e. truly horizontal. 



It is also obvious that the " spiral of dorsiventrality " does not 

 bear any necessary relation to the " ontogenetic spiral " in a given 

 shoot: it is the spiral which is orthogonal to the curve previously 

 termed Schimper's orthostichy line, and since this latter, which 

 bisects the angle of the intersection of the construction spirals, 

 winds in the direction of the longer curves, the " spiral of dorsiven- 

 trality" always follows the direction of the shorter spirals, and 

 bears the same relation to the genetic-spiral as do these (cf. Table, 

 fig. 34). With a right-hand genetic-spiral, that is to say, the 

 foliage-leaves on a (2 + 3) and a (3-1-5) shoot are obliquely inserted 

 in converse directions, and their bases remain obliquely inserted 

 even after the shoot has passed through a zone of elongation 

 which renders Schimper's orthostichy line sufficiently straight 



