84 RELATION OP PHYLLOTAXIS TO MECHANICAL LAWS. 



positions will hold for the asymmetrical condition, and since, again, 

 uniform growth in a plane circular system may be indicated by a 

 series of " circles '' enclosed in a "square" meshwork, in which 

 successive " squares " and " circles " are in geometrical progression, it 

 follows that growth is equally uniform throughout the spiral 

 system, and successive members along any spiral path are also in 

 geometrical progression. These relations, following from the 

 system of construction, are indicated for a (3 + 5) system in 

 fig. 28. 



Although circles have been inscribed in the orthogonal areas, it 

 is clear that the proper figure which only becomes a circle as the 

 orthogonal area becomes a square, is of an " ovoid " form,* while a 

 point which may be found for each area by drawing twice the 

 number of curves in either direction, and therefore represents the 

 intersection of the intermediate spirals, will at the same limit 

 become the centre of the circle. It may be now termed the " Centre 

 of CoThstruction " of the ovoid. These centres of construction fall on 

 circles the radii of which are again in geometrical progression, and 

 growth in each lateral member is uniform with that of the circle 

 which represents the main axis ; and in such an expanding system, 

 circles drawn through the centres of construction of the lateral 

 members will indicate the relative size of the axis when the 

 member was laid down. Since growth is uniform throughout the 

 system, this ratio is a constant and may be used to define the 

 system ; and for the same reason, the same diagram which ex- 

 presses the relative size of the developing primordia will also 

 represent the relative position of the areas in which the first 

 impulses originated as possibly mathematical points corresponding 

 to the centres of construction. The ovoid figures approach circles 

 so closely when the angle they subtend is small, that the error is 

 almost beyond the error of drawing; below 60° it is practically 

 unnoticeable, and the inscribed curves may be thus represented by 

 actual circles. At the same time it must be remembered that, 

 though these figures are in a spiral system ovoid curves in their 

 relation to the parent axis, they represent growth-centres with a 

 partially individualised activity, and may therefore by their own 

 * Gf. Mathematical Notes, for the equation and construction of the true curve. 



