10 THE STRUCTURE OF FLOWERS. 



leaves in a "cycle," and tlie numerator the number of times 

 a spiral line, passing through the position of the leaves, coils 

 round the stem in forming a cycle ; thus, with the f arrange- 

 ment, any leaf being taken as number 1, the sixth leaf will 

 be first that falls in the same vertical line Avith number l,the* 

 leaves 1 to 5 constituting the cycle. The portion of the 

 spiral line which passes through the leaves 1 to 6 coils twice 

 round the stem, and if projected on a plane forms two circles. 

 The ano-ular distance, measured from the centre of the stem 

 or circles, between any t\v'o successive leaves is always found 

 by multiplying 360° by the fraction : thus f X 360° = 144°. 



The interpretation, therefore, of the prevailing numbers 

 3 and 5 in floral whorls is that they are, in most cases, cycles 

 of the ^- or f types respectively; while 4 is primarily due 

 to the union of two pairs of opposite and decussate parts. 



6, 8, 10 are merely the doubles of the preceding, and mostly 

 represent two pairs of whorls or cycles blended together, 

 thus forming one whorl, or so closely appi'oxiraated as 

 scarcely recognizable as two ; though the rare number 8, in 

 some cases, such as Nigella, and Helleborus fcetidus, may repre- 

 sent a cycle of the f type. Similai'ly, the still rarer numbers 



7, 9, and 11 in flowers correspond to the absence of these 

 numbers as denominatois of any fractions of the above 

 prevailing series. 



With the exception of dimerous and tetramerous whorls, 

 all the rest are presumably due to alternate arrangements. 

 Now, opposite leaves present a more primitive type than 

 alternate ; that this is so, is not only reasonable from the 

 primordial condition of the cotyledons of Dicotyledons, but 

 the transition from an opposite to an alternate condition 

 may be often witnessed on rapidly growing stems, such as 

 of the Jerusalem Ai'tichoke. Whenever this plant bears 

 opposite leaves below, and alternate leaves above, it will be 



