69 



pected that on the continuation of the construction, with successive 

 circles decreasing in the same ratio, an arrangement of logarithmic 

 spirals should result. This should perhaps enable one to put Church's 

 constructions on another basis. Van Iterson succeeded in proving 

 mathematically that Beijerinxk's expectation was right, and this 

 question became the starting point for van Iterson 's thesis i). It was 

 very difficult to make Beijerinck agree with this work and its con- 

 struction, especially to a complete separation of the mathematical and 

 the morphological sides in the presentation, but in later years he 

 stated spontaneously that this separation was correct. Af ter Beije- 

 rinck had been established for a number of years in Gorssel, he said 

 at one time that of all his reading this thesis was the work he studied 

 most intensely. Evidence that this was really the case is seen in many 

 computations found after his death, and also in a short publication 

 entitled "Verband tusschen de bladstellingen van de hoofdreeks en de 

 natuurlijke logarithmen" (Relation between natural logarithms and 

 phyllotaxis of the Fibonacci series), which appeared in 1927 2). 



Beijerinck's opinion stated therein has never been completely 

 clear to the writer. In the main it is as follows. 



If one draws two helices in opposite directions on the surface of a 

 cylinder placed vertically, in such a way that the one helix makes an 

 angle of inclination whose tangent equals a/»/2 ( — 1 + V5), while the 

 other helix is perpendicular to the first one, then it may be proved 

 that consecutive points of intersection of the helices on the surface 

 of the cylinder are placed, with respect to each other, at angles of 

 divergence equal to the limiting angle of the Fibonacci-series (137°30' 

 28"). It may also be expressed as follows : the surface of the cylinder is 

 divided by these two helices into rectangular areas whose centres are 

 placed at the said angle of divergence to each other. If one considers 

 the cylinder's surface capped by a hemisphere of the same radius, 

 and constructs thereon the helices at the same inclination, then near 

 the top of the sphere these helices approximate to logarithmic spi- 

 rals drawn on a plane. These spirals will divide the plane into areas 

 of gradually-diminishing size, which wdll still have the above-men- 

 tioned angle of divergence with each other. Beijerinck has given 

 to an area delimited by two logarithmic spirals with these angles of 

 inclination the name of "Folium logarithmicum aureum". 



Beijerinck supposes that in the ideal case with higher plants the 

 meristematic cell-substance at the surface of the growing-point is 

 distributed in areas such as are indicated above for the top of the 

 hemisphere; each area being a "Folium logarithmicum aureum" but 



1) G. VAX Iterson Jr., Mathematische und mikroskopisch -anatomische Studiën 

 über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen, Jena 

 1907. 



2) Verslagen Afdeeling Natuurkunde Koninklijke .\kademie van Wetenschappen 

 Amsterdam 36, 585-604, 1927 {Verzamelde Geschrifte tub, 28-45). 



