TWENTY-SIXTH ANNUAL MEETING. 35 



03 = ( 0X6)+ 0= 

 (Add) _0 (Add) _J. = ( X 6 ) + 1 



13=,( 0X6)+ 1= i(AddM 

 (Add) _^ (Add) _7 =. ( 1 X 6 ) + 1 



23 = ( 1X6)+ 2= s 



(Add) 2 



_3 (Add) _19 = ( 3X6) + 1 



33 = ( 4X6)+ 3= 27 'At^^^iJ 



_6 _37 = ( GX6)+1 



43= ( 10X6)+ 4= 64 — 



JO _G1 = (10X6) + 1 



53 = ( 20X6)+ 5= 125 — ? 



_15 _91 = (15X6) + 1 



63= ( 35X6)+ 6= 216 — 



_21 127 = (21X6) + 1 



73 =( 56X6)+ 7= 343 -2 



28 1G9 = (28X6) + 1 



83= ( 84X6)+ 8= 512 —^ 



36 217 = ( 36 X 6 ) + 1 



93 = (120X6)+ 9= 729 _ 



_45 _271 = ( 45 X 6 ) + 1 



103 = ( 165 X 6 ) + 10 = 1000 — 



55 331 = ( 55 X 6 ) + 1 



113 = (220X6) + 11 = 1331 _il 



66 397 = ( 66 X 6 ) + 1 



123 = ( 286 X 6 ) + 12 = 1728 



Thus we find that the cubes are equal to the successive terms of the tetrahe- 

 dral series multiplied by the constant factor 6, plus the root. The differences 

 between the cubes are equal to the successive terms of the trigonal series multi- 

 plied by 6, plus the constant 1. The differences themselves are the successive 

 terms of the hexagonal series. 



1 cannot do better in closing this paper than to produce the beautiful 



PHYLLOTACTIC SERIES. 



I have carried this to the twelfth term; and that is about as high and com- 

 plex as our observations on phyllotaxy have gone. A more frequent method of 

 arranging these numbers is to place them in the form of fractions, thus: 

 112 3 5 8 13 21 34 55 89 



1 2 3 5 8 13 21 34 55 89 144 233 



In which either term of any fraction is obtained by addition of the two preceding 

 terms of the same denomination. This may not be considered exactly a figurate 

 series; but in nature it expresses the most beautiful and symmetrical arrange- 

 ment of leaves, scales, bracts, seeds, etc., on stem, branch, or fi-uit, as the leaves 

 on a house-leek, the florets on a spike of petalostemon, the scales on the cup of 

 an acorn or a cone of pine or spruce, the seeds of a magnolia cone or a straw- 

 berry, the florets and seeds on the head of a sunflower, etc. These numbers 

 show two things — how leaves, etc., have the greatest possible distribution to 

 sunlight and air, and how the greatest number of leaves, seeds, etc., may be 

 packed into the smallest and most economical space, in bud and fruit. The 

 fractions express the distance around a stem, cone, or capitulum, from one leaf 

 or seed to the next above. 



