GENERAL OBSERVATIONS. 



27 



Under normal circumstances of growth, the ratio of the curves is 

 practically constant * and in noting the wonderful accuracy with 

 which these high divergences are illustrated, one naturally con- 

 cludes that laws which hold with such mathematical exactness in 

 the higher series must also be true when lower divergences are 

 considered. Such an apparent confirmation of the Schimper-Braun 

 theory is very striking ; in fact, its very perfection leads one to 

 question the accuracy of the spiral hypothesis ; if a single growth 

 spiral can here work at an angle correct to minutes and seconds, 

 and the difference between |^ and ^^ is only 1' 41", why is the 

 mathematical precision not equal in simple cases, and the divergence 

 as readily measured as the angles of a crystal ? f 



* Cf. Weisse. Out of 140 plants 6 only were anomalous, the error being thus 

 only 4 per cent. 



t A primordium of the highest system described (Helianthus capitulum, fig. 

 13) subtends an angle of only 2° 5'. 



On the other hand the range of error in lower systems is very considerable. 

 As an example a seedling of Nymphaea alba may be taken, since the leaves arise 

 perfectly free from one another. There are thus no lateral pressures in the bud ; 

 the interstices are packed with hairs among which the leaf bases slide without 

 meeting any resistance (fig. 94). 



By making a careful camera-lucida drawing of such a plant 3 mm. in diameter, 

 magnified 70 diameters, restored to normal volume as far as possible by clearing 

 in Eau de Javelle, the centre of the vascular bundles may be taken as represent- 

 ing the centre of construction and the angular divergence measured on the 

 drawing with a considerable amount of accuracy. In such a specimen the angles 

 measured were— 



Three leaves form a complete cycle, and the error of observation diminishes 

 in taking the average of the three ; while the centre of each leaf may be fairly 

 accurately marked, the actual centre of the system is difficult to judge on an 

 apex which is bulging asymmetrically. 



Thus on another apex (fig. 94, corner) an obvious error of ^ mm. in judging 

 the centre thus empirically, introduced in the divergence of two particular leaves, 

 an error of 143° to 147° ; the angle to the next leaf not so much affected by this 

 error being 133° or 132°, In the Schimper-Braun system, the phyllotaxis of 



