XIV] OR PHYLLOTAXIS 641 



consequence a series of symmetrical patterns, whose nature will 

 depend upon the form of the entire surface. If the surface be 

 that of a cylinder we shall have a system, or systems, of spiral 

 helices : if it be a plane, with an infinitely distant focus, such as 

 we obtain by "unwrapping" our cylindrical surface, we shall 

 have straight lines; if it be a plane containing the focus within 

 itself, or if it be any other symmetrical surface containing the 

 focus, then we shall have a system of logarithmic spirals. The 

 appearance of these spirals is sometimes spoken of as a " subjective " 

 phenomenon, but the description is inaccurate : it is a purely 

 mathematical phenomenon, an inseparable secondary result of 

 other arrangements which we, for the time being, regard as primary. 

 When the bricklayer builds a factory chimney, he lays his bricks 

 in a certain steady, orderly way, with no thought of the spiral 

 patterns to which this orderly sequence inevitably leads, and which 

 spiral patterns are by no means "subjective" The designer of 

 a wall-paper not only has no intention of producing a pattern 

 of criss-cross lines, but on the contrary he does his best to avoid 

 them ; nevertheless, so long as his design is a symmetrical one, 

 the criss-cross intersections inevitably come. 



Let us, however, leave this discussion, and return to the facts 

 of the case. 



Our second question, which relates to the numerical coincidences 

 so familiar to all students of phyllotaxis, is not to be set and 

 answered in a word. 



Let us, for simplicity's sake, avoid consideration of simultaneous 

 or whorled leaf origins, and consider only the more frequent 

 cases where a single "genetic spiral" can be traced throughout 

 the entire system. 



It is seldom that this primary, genetic spiral catches the eye, 

 for the leaves which immediately succeed one another in this 

 genetic order are usually far apart on the circumference of the 

 stem, and it is only in close-packed arrangements that the eye 

 readily apprehends the continuous series. Accordingly in such 

 a case as a fir-cone, for instance, it is certain of the secondary 

 spirals or " parastichies " which catch the eye; and among 

 fir-cones, we can easily count these, and we find them to be 



T. G. 41 



