PBAGTIGAL BOTANY. 



163. The arc interposed between the insertion of two 

 ccessive leaves is called the angle of divergence. 'Wliile 

 e fraction i expresses the angle of divergence of tristi- 

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

 il. 



As regards distichous leaves, the term angle can not 

 iply to their divergence, since the vertical lines are sepa- 

 ted by half the circumference of the axis ; it is, liow- 

 er, represented by the fraction ,J. 



The fractions, ^, i, f , -f, ^, etc., severally represent, not 

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

 odes. They have for their numerator the number of 

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

 '.nominator the number of leaves in the cycle, or rather 

 .6 number of intervals between the points of insertion of 

 ese leaves or their verticals. 



It was Bormet, who lived in the middle of the last cen- 

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

 ive recently been extended and generalized by Schiraper, 

 raun, and others. 



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

 V^a/iiked, etc. The 21-ranked arrangement has 21 

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

 ire ^Y for an angle of divergence. The fractions, ^f, 

 b li' !%■> represent angles of divergence in cycles, con- 

 sting of 34, 55, 89, 144 leaves, and completed by 13, 21, 

 t, 55 revolutions or turns of the spiral. 

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

 on, thus, -1-, i, f, I, J^, ^, il, fi, li, -jfi^j, we shall 

 ladily perceive the relation that they bear to each other, 

 eginning with the third fraction, we notice, on compar- 

 ig through the successive fractions, the numerators and 

 jnominators, that each fraction has for its numerator the 



