54 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



give equally correct results; the ratio 34:55 may be taken as 

 33 : 55 within the error of drawing in small squares, or 3 : 5. The 

 curves (3 : 5), taken in a circle divided into 90 parts, will give 

 results well within the limit of drawing the square meshwork 

 correctly. So close are the ratios of the stages of the continuous 



1 

 fraction 1 + 1 that within the error of drawing any one will 



T + etc., 



give satisfactory results ; the error being considerable only in the 

 centre of the system, where the difficulty of measuring the squares 

 is also greatest. The true curve for 34 : 55, and in fact that for 

 the ideal angle of the continued fraction, lies between (3 : 5) and 

 (5 : 8), and may be closely approximated from the ratios 3,: 5 in a 

 circle with 90 radii, or 5 : 8 in one of 80. 



Such a pair of curves is, then, well within the error of drawing, 

 accurate for a (34 + 55) system, and may be used to map out a 

 spiral orthogonal construction; for practical purposes a pair of 

 curves may be cut out in card, fixed to the paper by a pin through 

 the centre of the circle, and used as a rule. By taking a circle of 

 radius equal to that of the curve pattern, and dividing it into 55 

 and also into 34 equal parts, so that one point may be common to 

 the two sets, and using the curves as a rule to mark 55 short 

 curves and 34 long ones, the whole circle will be plotted out into 

 a spiral meshwork of squares in orthogonal series, corresponding 

 to the parastichies of the Sunflower capitulum taken as a type, and 

 the plan may now be used as a check on the actual phyllotaxis 

 (fig. 25). 



It is obvious that either the points of intersection may be re- 

 garded as the centres of construction of the lateral members, or 

 the square areas themselves as the actual members, if packing 

 is so close that no interspaces are left; and the appearance 

 of circular flower - primordia may be indicated by describing 

 'circles in the approximately square areas. Eegarding the point 

 on the circumference common to both sets as No. 1, the 

 whole system may be numbered up by Braun's method, the 

 meshes along the short spirals differing by 55, those along the 

 long by 34. 



