THEIR ARRANGEMENT. 149 



those beyond, it is 137 plus a variable number of minutes, which 

 approaches nearer and nearer to 30'. Hence M. Bravais considers 

 all these as mere alterations of one typical arrangement, namely, 

 with the angle of divergence 137 30' 28", which is irrational to 

 the circumference, that is, not capable of dividing it an exact num- 

 ber of times, and consequently never bringing any leaf precisely in 

 a right line over any preceding leaf, but placing the leaves of what 

 we take for vertical ranks alternately on both sides of this line and 

 very near it, approaching it more and more, without ever exactly 

 reaching it. These forms of arrangement he therefore distin- 

 guishes as curviserial, because the leaves are thus disposed on an 

 infinite curve, and are never brought into exactly straight ranks. 

 The others are correspondingly termed rectiserial, because, as the 

 divergence is an integral part of the circumference, the leaves are 

 necessarily brought into rectilineal ranks for the whole length of 

 the stem. Organic forms and arrangements, it may be observed, 

 always have a degree of plasticity and power of adaptation, even 

 in their numerical relations, which approximate, but are never en- 

 tirely restricted to mathematical exactness. 



249. A different series of spirals sometimes occurs in alternate 

 leaves, viz. , |, f , T 3 ; and still others have been met with ; 

 but these are all rare or exceptional cases, and do^ not require to 

 be noticed here. 



250. Opposite Leayes (236). The arrangement of opposite leaves 

 usually follows very simple laws. Almost without exception, the 

 second pair is placed over the intervals of the first, the third over 

 the intervals of the second, and so on. More commonly, as in 

 plants of the Labiate or Mint Family, the successive pairs cross 

 each other exactly at right angles, so that the third pair stands di- 

 rectly over the first, the fourth over the second, &c., forming four 

 equidistant vertical ranks for the whole length of the stem. In 

 this case the leaves are said to be decussate. In other cases, as in 

 the Pink Family, it may often be observed that the successive 

 pairs deviate a little from this line, so that we have to pass several 

 pairs before we find one exactly superposed over the pair we start 

 with. This indicates a spiral arrangement, which falls into some 

 one of the modes already illustrated in alternate leaves, only that 

 here each node bears a pair of leaves. 



251. Yerticillate or Whorled Leaves (236) follow the same modes of 

 arrangement as opposite leaves. Sometimes they decussate, or the 



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