84 PRACTICAL BOTANY. 



163. The arc interposed between tlie insertion of two 

 successive leaves is called the angle of divergence. While 

 the fraction ^ expresses the angle of divergence of tristi- 

 chons leaves, the fraction f designates that of the quincun- 

 cial. 



As regards distichous leaves, the term angle can not 

 apply to their divergence, since tlie vertical lines are sepa- 

 rated bj half the circumference of the axis ; it is, how- 

 ever, represented by the fraction ^. 



The fractions, |, i, f , f , y^, etc., severally represent, not 

 only the angle of divergence, but the whole plan of these 

 modes. They have for their numerator the number of 

 the spiral turns of which the cycle is composed, and for 

 denominator the number of leaves in the cycle, or rather 

 the number of intervals between the i)oint3 of insertion of 

 these leaves or their verticals. 



It was Bonnet^ who lived in the middle of the last cen- 

 tury, that pointed out these modes of phyllotaxy, but they 

 have recently been extended and generalized by Schimper, 

 Braun, and others. 



The 8-ranked arrangement is followed by the IZ-ranked, 

 21-ran7ced, etc. The 21-ra?i7ced arrangement has 21 

 leaves in one cycle, with 8 turns of the spiral, there- 

 fore -^j for an angle of divergence. The fractions, ^, 

 ih li' tWj represent angles of divergence in cycles, con- 

 sisting of Si, 55, 89, 144 leaves, and completed by 13, 21, 

 34, 55 revolutions or turns of the spiral. 



ISTow if we arrange this series of fractions in a progres- 

 sion,^ thus, -1-, J, f , I, ^\, ^, -If, f i l-f, ^j5^, we shall 

 readily perceive the relation that they bear to each other. 

 Beginning with the third fraction, w^e notice, on compar- 

 ing through the successive fractions, the numerators and 

 denominators, that each fraction has for its numerator the 



