THE USES AND 





S85 



theory of types, that the typical properties of them are not fully determined 



by what we have yet observed respect in 



? 



them 



For, although these is 



I 



great variety of such arrangements, these do not include all the possible ones, 

 nor even all the simplest. There must still be soother principle of choice 

 besides what determines the rational fraction and the spiral arrangement What 



ie 



result 



this is, is the problem of the mathematical theory of Phyllotaxy. II 

 of this investigation was a classification of all the fractions that occur in 

 natural arrangements under the general form of the continued fraction 



1 



a + 1 



1+1 



1 -J- &c, 



in which a may have the values 1, 2, 3, or 4. The successive approximations 

 of these four continued fractions give four series of proper fractions, which 

 include all the arrangements that occur in nature. These series are for 



a 



a 



a 



a 



1 

 2 

 3 

 4 



5, h t. 



i 



t, fa 



• v *» *» *' A» &c 



ii !• h A» t 6 ?> &c - 



• h h h l\> A> &c - 



d series, and 



The first series is not usually 

 fractions of the 

 opposite direct 

 line coi 



given, since they 



are the complements of th 



express the same 



gements, but in an 



ound the circumfe 



or by supposing that the 



piral 



necting the leaves is drawn from leaf to leaf the longer way round. 



•st series, we shall still have in the others, as they stand, 



Omitting then the first 

 developed to five terms, many more 



in actual plants 



fract 



th 



have 



been observed 



could be observed 



I 



propose m 



what follows to subject the mathemat 



indue ion expressed 



by these series to careful 



actual observation from what is 



precisio 



the amount of inductiv 



angle 



rests. 



tical examination, to distinguish what is matter of 



deduced from theory, and to ascertain with 



hich the theory of the typical 



itrictly inductive investiga- 



idence on w 



Pursuing the subject afterwards by a 



tion, I shall estimate what there is of 

 think, to the rejection of the theory 



typical ai 

 nugatory. 



uth in the theory 



1 



will lead, 1 



it stands, or under the form of the 



g 



but will not render the obsen 



on which it depends wholly 



On the 



contrary, it will show that this observation really leads 



th 



true 



planat 



of the occurrence of 



certain fractions in the sp 





