HARDIFICRE'S SCIENCE-GOSSIP. 



the subject of what we called " secondary spirals." 

 It is obvious that difficulties are met with in deter- 

 mining the phyllotaxis of fir-cones, and in other 

 cases where, by the shortening of the internodes and 

 consequent approximation of the nodes, the leaves 

 or scales are so closely situated that at first sight it 

 seems hopeless to unravel the mode in which these 

 leaves or scales are arranged. But this is accom- 

 plished by having recourse to the system of secondary 

 spirals. If a cone be attentively examined it will 

 be seen that, starting from any scale at the base, 

 spirals wind round the cone to the right and left 

 from the starting-point. Now it is obvious that the 

 spirals running from right to left embrace between 

 them all the scales of the cone. And the same 

 applies to the spirals running from left to right. 

 The least reflection will prove that this is the case. 

 Hence, if we count the number of spirals running 

 (say) from left to right, we know that in every one 

 of these spirals the number of scales embraced in 

 that spiral will be represented by the fraction having 

 I for its numerator and the number of spirals for its 

 denominator. For example, let the cone of the 

 white pine (Finns Strobus) be taken. In this cone 

 we find eight secondary spirals passing from left 

 to right and five from right to left. Thus we see 

 that in each spiral running from left to right there is 

 contained one-eighth of the total number of scales, 

 and in each spiral running from right to left there is 

 contained one-fifth of the total number of scales. Hence 

 it becomes possible to number every scale of the 

 cone ; for starting from the lowest scale, which we 

 call I, the next scale on the spiral running from left 

 to right will be numbered 9, the next 17, and so on, 

 the common difference being eight. 



In the same manner, the spiral originating in scale 1, 

 but running from right to left, will have its second 

 scale numbered 6, its third numbered II, and so on, 

 the common difference being five. Thus we have all 

 the materials for determining the generating spiral 

 as it is termed, or that spiral which passes through 

 every scale before arriving at the one vertically above 

 the one from which we started, and this is effected 

 by numbering all the scales. It will, of course, be 

 observed that it is only the generating spiral which 

 passes through the numbers o, 1, 2, 3, 4, 5, &c, 

 consecutively, whereas the secondary spirals pass 

 through the numbers already given. In the white 

 pine, then, the 14th scale is immediately over the 

 first, and there are five turns round the axis. Hence 

 the arrangement is represented by the fraction T 5 j, 

 which, by the way, is a " curviserial " arrangement, 

 as will be explained shortly. It affords excellent 

 practice to the student of phyllotaxis to unravel the 

 arrangement of fir-cones, and, more than anything 

 else, tends to give definite ideas on the subject of 

 phyllotaxis. The ^ plan is not uncommon. It is 

 found in the house-leek and wormwood. Although 

 usually uniform in the same species, still this is not 



always the case. For in some plants there may be 

 one arrangement at the base and another at the summit. 

 But in most cases this is observed only in the young 

 state of the plant, and disappears with growth. Some 

 species of the genus Sedum furnish examples. Should 

 the generating spiral follow a similar course in both 

 stem and branches, the arrangement is homodro?nous, 

 but if this is not the case then it is called hetero- 

 dromous (o/u-os, like, 'irtpos, different, 8p6/j.os, course). 

 Other series are 5 8 r (Pinus sylvestris, the Scotch fir,) 

 other species of Pinus $, |^, &c. The expression 

 "curviserial arrangement" was used above. By 

 this it is meant that no leaf is exactly over the leaf 

 from which we start in situation, but it is placed a 

 little to the right or left of that position. For if the 

 fraction f 5 be calculated in degrees, it will be found 

 that the result is not a whole number, but that some 

 odd minutes are present. Thus we see that in this 

 case the circle is not equally divided by the scales, 

 and hence it is impossible for them to be situated in 

 vertical rows. The fact of the leaves being thus 

 arranged in an infinite curve suggested to Bravais the 

 term curvise?-ial, which is used in contradistinction to 

 the rectiserial arrangement. In this the fractions give 

 whole numbers, and the circle is divided into cases 

 of equality ; thus in this case the leaves or scales are 

 actually arranged in perfectly vertical rows in the 

 axis, and the vertical position of the leaves forming 

 the commencement and termination of the cycle is 

 maintained. 



Before dismissing the subject of alternate phyllotaxis 

 it is well to notice that in Dicotyledons the first leaves 

 (cotyledon) have an opposite arrangement, and it is 

 by the lengthening of the axis that this becomes 

 alternate. In Monocotyledons, as already mentioned, 

 alternate phyllotaxis is of necessity the rule. 



Opposite phyllotaxis is not quite so commonly met 

 with as alternate. In it two leaves are given off at a 

 node on opposite sides of the axis, and very frequently 

 the successive pairs of leaves are arranged at right 

 angles to each other. When this occurs the arrange- 

 ment is said to be decussate. In some cases, however, 

 the leaves do not exactly cut one another at right 

 angles, but deviate slightly from this decussation. In 

 this case there is a more or less spiral arrangement, 

 and a tendency to alternation. Opposite phyllotaxis 

 is particularly characteristic of orders, for example the 

 Labiatae, and an instance may be cited in the common 

 white dead-nettle (Lamium album). Again, in Caryo- 

 phyllacese, or the chickweed order, opposite phyllotaxis 

 is very frequent. In purging flax (Linum catharticum) 

 the arrangement is seen not to be strictly decussate, 

 so that the second pair of leaves does not exactly 

 cut the first at right angles. 



Examples of the whorled or verticillate phyllotaxis 

 may be found in the order Rubiaceae, which includes 

 the common "bed-straw" (Galium). It is worthy of 

 notice that in this case also the leaves frequently 

 decussate with one another in successive whorls. 



