246 BRAUN'S VIEWS. [BOOK * 



metical relations, which are the expression of the various 

 combinations of a certain number of elements, disposed in a 

 regular manner. All the spires depend upon the position of 

 a fundamental series, from which the others are deviations. 

 The nature of the fundamental series is expressed by a frac- 

 tion, of which the numerator indicates the whole number of 

 turns required to complete one spire, and the denominator 

 the number of scales or parts that constitute it. Thus -^ 

 indicates that eight turns are made round the axis before any 

 scale or part is exactly vertical to that which was first formed, 

 and the number of scales or parts that intervene before this 

 coincidence takes place is 21. 



The following are some of the results thus obtained by 

 Braun, in studying the composition of the spires of different 

 plants : 



4- in Asarum, Aristolochia, Lime tree, Vetch, Pea, the 

 spikes of all grasses. 



i is rare in Dicotyledons, and generally changes into more 

 complicated spires. It exists in Cactus speciosus, and some 

 others. 



-f- is the most common of all, and represents the quincunx. 

 Mezereum, Lapsana communis, Polemonium coeruleum, 

 Potato, are examples. 



f is also common, as in the Bay-tree, the Holly, common 

 Aconite, and the tuft of radical leaves of Plantago media. 



-i^ exists where the leaves are numerous and their intervals 

 small. Wormwood, common Arbutus, dwarf Convolvulus, 

 and the tufts of leaves in London Pride and Dandelion, are 

 instances. 



-r in Woad, Plantago lanceolata, the bracts of Digitalis 

 lanata. 



i n Sempervivum arboreum, the bracts of Plantago 

 media, and of Protea argentea. 



f-i- was found on an old trunk of Zamia horrida, and two 

 species of Cactus (coronarius and difformis). 



It does not, however, appear that this inquiry has led to 

 any thing beyond the establishment of the fact, that, 

 beginning from the cotyledons, the whole of the appendages 



