MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 337 



inches ; the curve thus approaches a circle in which the amount 

 of error is not greater than ;jigth of the radius. 



In all higher systems the approximation to the circle will be 

 successively closer, so that, as previously pointed out, the use of 

 circles in the quasi-squares of any system above (3 + 5) is beyond 

 any error of constructing a small diagram. The relation of the 

 curve to its " centre of construction " homologous with the centre 

 of the circle has also been previously indicated, since it is readily 

 noted on the geometrical diagram by taking the point of inter- 

 section of the intermediate spirals. 



The (2 + 3) curve, again, shows a marked distortion, and the 

 flattening on the side towards the origin is excessive, the general 

 outline obviously differing from a circle. This is still further ex- 

 aggerated in the (2 + 2) curve, iu which a distinct dimple begins to 

 appear at this point (fig. Ill, III.), and the " centre of construction " 

 shows still greater displacement. The (1 + 2) curve becomes dis- 

 tinctly kidney-shaped, with the centre of construction very close to 

 the depression (fig. Ill, IV., C"); and the limiting case is met with in 

 the (1 + 1) curve (fig. 112, A, the centre of construction being at C). 



It thus follows that lateral primordia may be represented in 

 theoretical construction diagrams as circles, within any error of 

 drawing, in any system from (3 + 5) upwards. In lower systems 

 the bilaterality of the ovoid is very marked, so much so, in fact, 

 that the occurrence of such a form at the apex of a plant-shoot 

 would not readily strike the observer as in any sense due to 

 the production of a primordium, the section of which would be 

 homologous with a circle, and within its sphere of growth 

 possessed of the same physical properties.* 

 ■ * Thfe same curves, or similar figures for any given ratio system whatever, 

 may be easily drawn within the error of drawing by a simple geometrical 

 method. For example, to draw the curve for the ratio 3 : 4, make a curve- 

 tracing for this ratio from the circular network of squares (p. 53), and with the 

 curve-tracing mark out a quasi-square mesh of the system. Divide this into 

 12 equal parts in either direction by describing 11 intermediate spirals in each 

 direction, and into these smaller squares transfer square by square the circle 

 inscribed in a true square similarly divided into 144 meshes, the points where 

 the circle cuts the meshes being judged by the eye with sufficient accuracy. 



Since the curves of the higher ratios so nearly approach a circle, the lower 

 ones figured are really the only ones which possess a special interest. 



