36 JOURNAL OF THE ROYAL HORTICULTURAL SOCIETY. 



PHYLLOTAXIS; OR, THE ARRANGEMENTS OF LEAVES IN 

 ACCORDANCE WITH MATHEMATICAL LAWS. 



By the Rev. Professor G. Henslow, M.A., F.L.S., V.M.H., &c. 



[Lecture given on June 25, 1907.] 



We all know that the motions of the heavenly bodies are subject to 

 strict mathematical laws ; that the formation of crystals in the inorganic 

 world is likewise in accordance with laws which can be represented 

 mathematically ; but when we turn to the organised world we do not, 

 somehow, expect to find structures of animals and plants reducible to 

 any representation by mathematical formula?. l 7 et this is very often quite 

 possible. The animal frame is composed of bones and muscles, which 

 form levers of various kinds. The flight of birds is strictly in accordance 

 with certain muscular actions, which counteract gravity and can be 

 represented mathematically ; while the bee has practically solved the 

 problem of making cells with the least amount of material, but combined 

 with the greatest capacity and strength. When we turn to the vegetable 

 kingdom we are again amongst organic forces, and we look about almost 

 in vain for results which can be tested by mathematics or which can be 

 represented by their formula?. The most remarkable instance is probably 

 the arrangement of leaves, and which forms the subject of the present 

 lecture. 



If several leafy shoots from different plants be taken, it will be 

 observed that many, probably the majority, have their leaves placed one 

 at a time on the stem, or, as botanists say, alternately ; e.g. the Garden 

 Flag, a Sedge, the Oak, and the Holly. Others will almost always have 

 two leaves at the same position (or node), but situated on opposite sides of 

 the stem ; e.g. Lilac, Privet, and Horse-chestnut. Of the latter it will be 

 also noticed that each pair of leaves stands at right angles to those above 

 and below it. Such series of pairs of opposite leaves constitute what has 

 been called the decussate arrangement. Extended observations will only 

 strengthen the conclusion that leaves are for the most part alternate or 

 opposite* 



Alternate Leaves. — If I take a branch of the May or Oak, and hold it 

 vertically with any selected leaf before me, and then pass my finger 

 upwards along the stem from that leaf to the next, and thence to the 

 third, fourth, fifth, and sixth leaf in succession, I find that the one last 

 reached (sixth) is exactly over, or in the same vertical line with, the first ; 

 and if I proceed further I shall find the seventh is vertically over the 

 second, the eighth over the third, and so on, the eleventh being, therefore, 

 over both the sixth and first. 



The following observations will result from this examination : — 

 Obs. 1. All the leaves on the branch are arranged in five vertical rows : 



* Leaves will occasionally be found grouped in threes or some higher number ; 

 they are then said to be whurled or vcrticiUate. 



