52 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



cannot be traced, and the orthostichies are equally incapable of 

 observation, the whole system is plotted out by taking the observed 

 number of parastichies and expressing them as mutually intersecting 

 log. spirals. 



A method is thus attained which gives perfect mathematical 

 expression to such a system as that deduced from observation of the 

 Pine cone or Sunflower capitula, so perfect that any deviations 

 from it in the actual plant must be due to the influence of some 

 extraneous force not yet considered. 



The following construction which expresses a (34 + 55) Sunflower 

 head may be taken as a type : — 



Since the whole construction hinges on the diagrammatic 

 representation by intersecting log. spirals, a simple method is 

 required for drawing these curves with a degree of accuracy which 

 will at least cover the error of observation; and, as the graphic 

 constructions will be found to afford sufficient geometrical evidence 

 of the truth of the method, it will not be necessary to include any 

 strict mathematical proof. A simple way of obtaining very accurate 

 results is as follows: — Describe a large circle and divide it into 

 a conveniently large number of parts (50-100); draw the same 

 number of radii through these points, and then, proceeding from the 

 circumference inwards, draw, with the same centre, a series of 

 concentric circles,, making with the radii a meshwork of squares, 

 as near as can be judged by the eye. In such a circular network 

 of squares, arranged in radial series in geometrical progression, 

 all lines which are drawn through the points of intersection in 

 any constant manner are logarithmic spirals, and when drawn in 

 reciprocal fashion intersect at all points orthogonally, the simplest 

 case being that in which symmetrical diagonals are drawn across 

 the meshes, which gives, in fact, the preceding case for the structure 

 of alternating whorls (fig. 24). 



An unequal pair of curves may be selected by taking a diagonal 

 across two squares in one direction on one side, and across two in 

 a converse way on the other; by continuing these, two asym- 

 metrical log. spirals will be obtained having by construction the 

 ratio 1:2. By filling in such curves all round the figure, it may 

 be proved experimentally that by using the full number of short 



