230 



KNOWLEDGE 



[October 1, 1890. 



sum of tlie numerators of any two adjacent ft-actions is 

 the numerator of the succeeding fi-action ; and the same 

 law holds good for the denominators. 



This series of fi-actions is very commonly found in 

 dealing with phyllotaxis, and has been termed by botanists 

 the Ordinary system. But other fractions are met with 

 in phyllotaxis which eaunot be referred to the above 

 series. These form the two series j, f , ^, yV' • • • ^^^ i' 

 i, |, 3^. . . . They have been called the Secondary and 

 Tertiary series respectively. As in the Primary so in these 

 the sum of two adjacent numerators gives the succeeding 

 numerator, and the sum of two adjacent denominators the 

 succeeding denominator ; but they ditfer from the first 

 inasmuch as in it the numerator of any one fraction is 

 the denominator of the next but one preceding. Still, 

 the three series are connected with one another, for the 

 numerator and denominator of any one fraction in the 

 Primary series furnishes us with the denominator of 

 the corresponding fi-action of the Secondary series, and 

 the numerator and denominator of any one fraction of the 

 Secondary gives the denominator of that fiaction in the 

 Tertiary series. 



In the case of the phyllotaxis represented by the fraction 

 ^, the leaves are disposed in two vertical rows on the stem ; 

 in the ^ arrangement there are three vertical rows ; in the 

 I five rows, and so on. The denominator of the fraction 

 gives the numbers of the rows in each case. This must 

 needs be so, as it indicates the number of leaves which are 

 passed in the genetic spiral from the leaf with which the 

 spiral may be supposed to commence to that which is 

 placed vertically above it. These vertical rows are called 

 orthoatichiex (opBo's straight, and o-nf a row). They may 

 be clearly seen on reference to Fig. II. {a, h, c, d, e), in 

 which the bark of the stem is supposed to be unrolled, and 

 placed flat on the paper. 



It will also be noticed that spirals other than the 

 genetic spiral — secondary xpimls — can be traced on the 

 diagrams. These are not so evident when the leaves 

 are placed Eit a distance from one another, but when 

 they are developed close together they are very easily 

 noticed, and in fact they often obscure the genetic 

 spiral. The lines which trace them out are termed 

 pdidsti'hies. The number of parallel parastichies in one 

 direction can be found by subtracting the number of one 

 leaf from that of the one next it on the same parastichy. 

 Thus in the case of the f spiral there are thi-ee parallel para- 



stichies to the right. In one of these we find the figures 

 1, 4, 7 ; subtracting one from four we get three, which is 

 the number of parasticliies to the right ; in one of those 

 to the left we meet with the figures 2, 4 ; two from four 

 gives two, the number of parallel parastichies to the left. 



It is often a matter of extreme difficulty to detennine 

 the genetic spiral, but by taking any two secondary spirals 

 wliich cut one another, that is one to the left and another 

 to the right, it may be made out with comparative ease. 

 Take the number of parallel parastichies in one direction, 

 and number the leaves as stated above ; for instance, in 

 the case just cited, as there were three parallel para- 

 stichies to the right, the leaves of one of these would be 

 numbered 1, 4, 7, and so on, and the others would also 

 be numbered after the same fashion. Then pick out the 

 parallel parastichies to the left, which cut the former ; 

 presuming there were two, the leaves of the one which 

 arose nearest the one formerly selected would be num- 

 bered 2, 4, 6, and so on. The whole of the leaves could 

 thus get their correct numbers, after which the tracing 

 of the genetic spiral would be a matter of no difliculty. 

 This method is followed in complicated cases where the 

 leaves are very close to one another, such as in the 

 Pine-cones, and in the capitula of the Composita; ; for 

 example, in tracing the order of the arrangement of the 

 florets of the Simflower. 



We have already stated that the arrangement of leaves 

 may be opposite, whorled, or scattered. In the first case 

 it generally happens that the adjacent pairs stand at right 



angles to one another on the stem, giving what is termed 

 the dixitsxak' arrangement, ex. Andiiallis arvensis (Fig. III. 

 (rt) ). The whorled arrangement is seen in many of the 

 .Rnbiaciic ; ex. Qulum critcittta, or Crosswort, one of the Bed- 

 straws. It has been suggested that the alternate arrange- 

 ment of leaves is the normal one, and that the whorled 

 arrangement has arisen by suppression of the internodes. 

 In support of this view, Mr. G. E. Massee writes in 

 Natvre of 12th July 1877, stating that in Lysiuiacliid 

 nt'morum, a small plant with yellow flowers, which is found 

 in abimdance carpeting our woods in early summer, the 

 leaves are opposite, but that the flowers springing in the 

 axils of the opposite leaves are never both equally deve- 

 loped at the same time, the one boing expanded while the 

 other is in bud ; and also that the oldest or most fully 

 developed flower appears alternately on opposite sides of 

 the stem." 



The axis or stem on which the flowers arise is the 

 foml a.i-is, and its mode of branching is termed the 



* In reality in Lysimachia nemorum, according to Sir. G. E. 

 JIassee, the first-formed flower alone belongs to tbe stem of the 

 plant ; the others are all that is left of aborted branches which have 

 developed from one another in such a way as to give the appearance 

 of a single stem. 



