Feb. 1, 1S69.] 



HARDWICKE'S SCIENCE-GOSSIP. 



35 



spicuous object during the spring and summer, as 

 every breeze exposed their brilliant under-surfaces. 

 Proceeding as before, we see in this case that the 

 fourth bud is directly over the first, and our 

 imaginary spiral drawn through them will make one 

 revolution round the branch. It is evident that the 

 leaves in this case are differently arranged from 

 those of our former example. Their arrangement 

 may be expressed by the fraction £ (see fig. 27), 

 which shows that in one complete circle round the 

 stem occur three leaves, separated from each other by 

 i the circumference, thus making with each other 

 an angle of 120°— i.e., they are situated in three rotes. 

 The alder {Alnus glutinosa) is another example of 

 this arrangement. 



The neighbouring hawthorn will furnish us with 

 another example for study. Let us fix our attention on 

 a given bud, and, running our eye along the branch, 

 until we arrive at another bud in a direct line with the 

 former, let us count the buds forming the series. 

 Five is the number, the sixth being seated im- 

 mediately in a line with the first. Let us now trace 

 out a spiral passing through each bud, from the 

 first to the sixth : it makes, we find, two complete 

 circles round the branch. This arrangement of five 

 leaves in two revolutions round the stem is ex- 

 pressed by the fraction f . This expression indicates 

 that the leaves are separated from each other by f 

 the circumference, and make with each other an 

 angle of 1M°, and that the cycle of five leaves is 

 completed in two revolutions. The oak, the chest- 

 nut, and others have a similar vernation. The 

 leaves in all these cases arc arranged in fee rotes. 



In a species of willow we are furnished with yet 

 another arrangement. Here we find the series to 

 consist of eight buds seated along a spiral making 

 three revolutions round the stem. The expression 

 for this mode of disposition is §. It shows, in the 

 same manner as before, that the leaves make with 

 each other an angle of 135°, and that they are 

 separated by % the circumference. It is clear, there- 

 fore, that they are arranged in eight rotes. Other 

 plants will be found to have their leaves arranged 

 in one or other of the plans expressed by the 

 fractions \, ± f, f, -£,-, 5 8 T , j$, &c. 1 am now speak- 

 ing exclusively of plauts with non-whorled leaves. 

 This series of fractions, then, may be looked upon 

 as indicating the phyllotaxis of non-whorled plans— 

 i.e., the law regulating their disposition of leaves. 



A singular relation is found to subsist between 

 the different terms of this series. If we add 

 together the respective numerators and denominators 

 of any two consecutive terms, we thereby obtain the 

 succeeding one. Thus ^ and £, the first and second 

 terms of the series, gives us j^ = g the third term ; 

 if now we take the second term, £, and the third 

 term, f, and treat them in a similar manner, we 



Similarly g=| 9 



obtain ^=g,the fourth term 



3+5 _ 8_ 5+_8_ _ 13 p 

 8+13 21' 13+21 — 3i' (EC - 



4 - «-v. Again, obtaining the deno- 

 minators by the method given, we may set down 

 the numerators by inspection ; for each successive 

 denominator becomes the numerator of the next term 

 but one succeeding. This will readily be seen by 

 examining the series of fractions. The curious 

 manner in which these numbers may be derived 

 from each other not only renders the series of terms 

 expressing the phyllotaxis of non-whorled plants 

 easy of acquisition and retention, but, if forgotten, 

 of reproduction. Like Bode's celebrated law of 

 interplanetary distances, while indicating, with re- 

 markable clearness and precision, a great group of 

 important facts otherwise difficult of remembrance, 

 it is altogether empirical, and could not be deduced 

 in any other way than by multiplied observations. 



The disposition of leaves confers upon trees much 

 of their individuality as species. As branches 

 spring from the self-same nodes as leaves, it follows 

 that their arrangement will be the same. Thus in 

 plants whose phyllotaxis is half, the twigs arising 

 from the branches will be arranged alternately in 

 the opposite rows, and the branches will necessarily 

 have a flattened appearance. The branches of trees 

 having the arrangement of leaves represented by 

 i, f, &c, have a very different appearance, wholly 

 attributable to this difference in disposition of the 

 leaves and branches. Trees and shrubs having 

 decussate leaves, the simplest case of whorl 

 structures, as the ash, the horse-chestnut, &c, differ 

 widely from the foregoing cases; each pair of 

 secondary branches and twigs being perpendicularly 

 opposite to the succeeding, and parallel to the 

 alternate pair. It is unnecessary to state that the 

 appearance of leaf-covered twigs and branchlcts is 

 greatly affected by the mode of arrangement of 

 their leaves. 



In addition to its influence on the physiognomy 

 of trees, the disposition of leaves is of considerable 

 physiological importance. As leaves are functionally 

 devoted to the elaboration of the sap by respiration, 

 exhalation, &c, and as free access of light is essen- 

 tial to the perfect performance of this function, it is 

 of some consequence that they should not crowd too 

 much upon each other. This crowding is pre- 

 vented by the method in which they are arranged. 

 In no case is a leaf immediately in front of the one 

 preceding it. In those plants, like the elm, the 

 hazel, the hornbeam, &c, and in whorl-leaved plants, 

 which are most obnoxious to crowding, the leaves 

 are separated by twice the interval between one 

 leaf and another, for the third leaf is over the first, 

 the fourth over the second, &c. In other cases 

 they are more widely separated. Thus, in the next 

 case the fourth is over the first, in the next the 

 sixth, then the ninth, then the fourteenth, &c. 



The phyllotaxis of plants with whorled leaves is 

 more difficult and complex. It can only be reduced 



