56 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



Thus taking the parastichies noted, 34 + 55 



and 21 + 89, 

 the same points may also (with others) be mapped out by 



13 + 144, 

 or 8 + 233, 

 5 + 377, 

 3 + 610, 

 2 + 987, 

 and ultimately 1 + 1597, the least con- 

 centrated system. 



Thus, on reducing the whole system to one grand spiral only, 

 which is, in fact, the genetic spiral, passing through 1, 2, 3, etc., 

 the nearest approach so far to a true orthostichy line is the log. 

 spiral cutting the genetic spiral orthogonally, which passes through 

 / and 1598, a point too near the centre to mark clearly, though 

 378, 611, 988, are indicated. 



The error of attempting to define such a system by the Schimper-' 

 Braun system of fractions, in which it might pass as a ff type, is 

 at once apparent : the 89 curve bears no obvious relation to the 

 genetic spiral, though it bisects the angle between the 34 and 55 

 curves. The system of intersections may be defined in terms of 

 any pair of these orthogonally intersecting spirals but not in terms 

 of one of each of two different sets. The only true expression for 

 the Sunflower head lies in taking the nearest numerical ratio, 

 which is that in which the squares, or the circular primordia 

 contained in them, are in actual lateral contact. There is in the 

 whole system figured no member radially superposed to number 

 one, and the construction is thus in agreement with Hofmeister's 

 law ; while other important deductions are — 



(1) The genetic spiral follows the direction of the short curves (55). 



(2) One coil of the " genetic spiral " approximates a circle, almost 

 within the error of drawing the system. 



(3) The false "orthostichy" line (89) is very nearly a radius, or 

 within the expression of hypothetical " torsion " in a specimen. 



(4) The nearest approach to an orthostichy line (1597) is as 

 near a radius as the genetic spiral is to a circle.* 



* For the true orthostichy line, cf. Mathematical notes. 



