HELICES AND SPIRALS OP ARCHIMEDES. 63 



spiral curve may be, must necessarily pass into a parallel screw 

 thread type when all the members become and remain equal. 



If the helices are secondary productions, it is very possible that 

 the Archimedean spirals which would represent them on a plane 

 system are equally secondary. The fact that a stem may go on 

 producing leaves to infinity, without producing a terminal member, 

 and that the leaves develop as similar primordia, is alone sufficient 

 to suggest that the genetic spiral is a log. spiral, rather than a 

 spiral of Archimedes which winds directly to the centre of the 

 system and allows for no further development. 



From the equation to the spiral of Archimedes {r = a6), by 

 taking a as different values of the 2, 3, 5, 8 series, while r and 6 

 are constant, it is easy to construct a series of spirals to correspond 

 to these ratios (fig. 33). 



In such a series the intersections of successive members of the 

 series, drawn in the opposite direction, are seen to be, in accordance 

 with the closeness of the ratios 3 : 5 : 8 : 13, etc., practically identical 

 within the limit of construction error. 



A tracing from such a pair may therefore be used to map a 

 system corresponding to the data observed in the given plant, either 

 as a symmetrical or asymmetrical construction.* 



In such a diagram it is at once observed that the intersections are 

 not orthogonal, and therefore afford no clue to the distribution of 

 pressures; while the rhombs are relatively much flatter at the 

 circumference, but become very steep towards the centre : so steep 

 do they become as all the spirals fall into the centre, that not only 

 cannot they be adequately represented in the diagram, but it is at 

 once obvious that it is impossible that such rhombs can in any way 

 indicate the structure of the actual primordia arising on a growing 

 apex, which are either isodiametric or elongated tangentially. 



" A familiar example of the former is seen in the chasing on a watch-case, 

 and will serve to illustrate the weak points of the system. 



These curves also present a beautiful example of a subjective effect produced 

 by an indirect method of construction. 



Engraved as wavy circles which have radii differing by a constant increment, 

 the sloping curves fall into series as spirals of Archimedes ; the number of waves 

 being constant in each circle, the construction is symmetrical and the spirals 

 thus appear equal in member in either direction. 



