MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 339 



The angle subtended by a primordium belonging to any system 

 of orthogonally intersecting log. spirals is given by the following 

 formula : — 



27r 360° 



The bulk-ratio for circular primordia was represented by the 

 sine of half the angle subtended at the origin ; the same expression 

 may be regarded as representing the approximate bulk-ratio in 

 the case of the ovoid curves which so nearly approach circles. 



From the above table it will be seen that the difference between 

 the angles subtended by the ovoid curves and those subtended by 

 circles having the respectively simple bulk-ratios obtained approxi- 

 mately by geometrical construction is a very small one. Such an 

 error is quite within any limit of construction error in small dia- 

 grams, and is far within the error of checking systems in the case of 

 the plant, in which, as previously noted, the construction adjustments 

 that must be made in the bulk-ratio before a new spiral path is 

 introduced must be necessarily often very considerable; so that 

 for practical purposes the integral bulk-ratios of the respective 

 systems may be taken as sufficiently accurate statements of the 

 phenomena observed. The empirical geometrical method of 

 estimating the bulk-ratio of any given system is therefore suffi- 

 ciently reliable, if the convention can be of any assistance, and 

 does not involve any special mathematical knowledge. 



Note V. — The Oscillation Angle. 



Taking the construction of a constant asymmetrical spiral 

 phyllotaxis system as the result of adding members at a constant 

 divergence angle, or as a phenomenon of growth oscillation, — a 

 convention which only holds, however, as has been previously 

 made clear, for integral ratios only divisible by unity as a common 

 factor, — the measurement of the true angular divergence of the 

 members of the systems constituting the Fibonacci series becomes 

 of special interest from the standpoint of comparison with the 

 helical divergences of the Schimper-Braun-Bravais conveutiou. 



