402 DISTRIBUTION OF THE GREEN LEAVES ON THE STEM. 



There are only few plants on whose branches or axes several stories occur with 

 twenty-one or thirty-four successive leaves in each. On the other hand, it happens 

 that on many shoots, not even one story is completely formed, or in other words, 

 that in more than a hundred leaves which project from the axis, no two are to be 

 found situated quite vertically above one another, and consequently, in these cases, 

 rectilineal orthostichies are out of the question. In many fir-cones, for example, 

 rectilineal lines are sought for in vain, and it is impossible, even approximately, to 

 estimate how many leaves are included in one story. It has been also conjectured 

 that in such cases the leaves of a story are innumerable, and if so, the fraction by 

 which such a system of leaf-insertion would be represented would be an absurd 

 figure. 



In such shoots it is anything but easy to establish the successive ages of the 

 leaves, that is, to number them in their proper order of development, especial!}' 

 when the leaves are thickly crowded together. This becomes the more difiicult 

 when the leaves on such very crowded axes arrange themselves in spiral series, or 

 lines which are much more apparent to the eye than the lines of development or 

 genetic spirals. These spiral series, which are seen on shoots of many succulent 

 plants (Sedum, SeTnpervivum), on species of Pandanus and Yucca, on the branches 

 of lycopodiums and conifers, and especially also in the inflorescence of crucifers 

 and the cones of many firs, of which a pine-cone, represented in fig. 101, may be 

 taken as an example — these series are called paras^icA-ies. They may be utilized in 

 order to ascertain which leaves succeed one another in age, thus — by first of 

 all ascertaining how many such jiarallel spiral lines ascend to the right, and how 

 many to the left on the axis examined. In a pine-cone, for example (see illustration 

 below), eight such lines or parastichies are seen to ascend in a somewhat sharply 

 oblique direction to the left, and five to the right in a rather less sharply oblique 

 direction. In order to find out which leaves succeed one another in age, the lowest 

 leaf is called 1, and the numbers 8 and 5 are used in the following manner. The 

 leaves of those steep parastichies, on the left adjoining 1, are numbered by additions 

 of 8 respectively, 9, 17, 25, 33, 41, &c. The leaves of the less steep parastichies on 

 the right, which adjoin 1, are numbered, on the other hand, li}' additions of 5 

 respectively, 6, 11, 16, 21, 26, &c. The numbering of the other parastichies is then 

 easily completed by subtractions and additions of the numbers 8 and 5, and the 

 numbers so obtained represent the successive ages of the leaves on the cone. This 

 somewhat complicated arrangement may be best demonstrated by imagining the 

 surface of a leafy, almost cylindrical axis, e.g. of a pine-cone, to be slit up longi- 

 tudinally, rolled out flat, and extended so that all the leaf -scales lie in one plane, aa 

 represented in the plan illustrated in the right-hand figure opposite. 



Naturally the most lively interest has been aroused at all times by the geo- 

 metrical ratios of phyllotaxis here generally reviewed, and it could not fail to follow 

 that the most diverse speculations should have been connected with them. This is 

 not the place to consider these in detail, but in so far as the remarkable and 

 actually existing conditions of the geometric arrangement of the leaves have a 



