PHYSIOLOGICAL BOTANY. 273 



of which this is adduced, and the example here calculated 

 out for beginners. 



Schimper's representation of the Arrangement of 

 Leaves is undoubtedly very ingenious, because it collects 

 the vague expressions of the spiral arrangement of leaves 

 into a comprehensive review. The formula given above 

 must be regarded as the fundamental formula, from 

 which the others may be deduced. Its application to 

 opposite and whorled leaves, the leaves of axillary bf anches, 

 even the development of the leaves in the buds, and the 

 parts of flowers, is no less ingenious. Schimper's ex- 

 planation is somewhat awkward, and it was therefore very 

 important that Al. Braun detailed this system more accu- 

 rately, more copiously, and more clearly. An excellent 

 memoir next appeared by MM. L. and A. Bravais, in the 

 'Ann. des Scienc. Natur./ 2d ser., t. vii, pp. 42-110. 

 The authors examined the spiral positions of the leaves 

 and foliar parts, the secondary spiral lines, as they are 

 represented on the developed surface of a primary cylinder, 

 where, namely, the spiral lines running from right to left, 

 and those from left to right intersect each other; and 

 point out as the foundation of the whole theory, that 

 when the numbers of the above two rows of spiral lines 

 are respectively primary numbers, there exists a spiral 

 line, which includes all the spots at which the leaves are 

 attached, a generating or including spiral, but if they have 

 a common divisor, verticillate positions occur. In the 

 first case, the angles both of the separate spirals (se- 

 condary spirals) and of the individual members in the 

 spirals will coincide with the horizontal line, and the 

 secondary divergences with the divergence of the pro- 

 ducing or general spiral. If we call the number of a 

 member in a secondary spiral line n, the divergence of 

 this spiral dn, the divergence of the general spiral d\ 9 

 and m the number of turns made by this spiral before it 

 arrives at n, we then have ndl = m . 360 + dn. This 

 formula serves for the calculation of the divergence of the 

 general spiral. It is then found, by direct observations, 



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