THE USES AND 



ES IN I'LAin. 391 



e: are 



origin of spiral arrangements in general, and of the whorl, is significantly in ac- 

 cordance with the observation that the forms peculiar to the two lo* 

 more frequent among fossil plants than among surviving ones. 



But waiving these theoretical considerations for the pi ent. 1 will now examine, 

 quite independently of theory, the properties in the spiral arrangements of all 



the fractions between j and |, or rather between 1 and 1, and of a less de- 



nomination than 14ths. I adopt these limits because the character of all fraction! 

 greater than £, or less than J, will be sufficiently shown by this limit, and because 

 fractions less simple than 13ths cannot be distinguished in nature from simpler 



ones. The fractions between J and . being complements of those great r than 



J, need not, of course, be separately studied; since they express the one ar- 

 rangements, only counted in the opposite direction an mud ti > st< n I have 

 chosen to represent these possible arrangements by the larger fractions rather 

 than by the smaller (their complements), for reasons I have given. It hough 

 theoretical considerations on the origin of spiral arrangements in general sug- 

 gest the latter and more usual mode as the proper one. Tins may be given 

 as an additional reason for the choice, since \ l shall not thus be led to con 

 found a conventional mode of representation with a law of nature, or have 

 any undue bias in consequence, but shall be able to judge the hypothesis on 

 its own merits. The best reason, howev.r, for the choice is, that by representing 

 the cycle by the larger number of turns, or by counting the longer i iy mnd. 

 we are able to spread out into greater detail in the accompanying diagram 

 the steps of the distribution, and see more clearly its char; ten 



In the following table the first column contains all the fractions 1 have 



denned 



d 



in the order of their denominations, with their d imal eqim nts to 

 thousandths, and at the end the irrational quantity ft, with its .pproxim te decimal 

 value. The second column contains the same fractions, arranged in fee ord r of 

 their magnitudes, with their decimal values and the differences between sueeeastve 

 ones. The third contains the complement, of these decimal , In, They represent 

 the smaller of the two parts or angles of divergence into whieh two success- 

 leaves would divide the circumference, or represent these angles reck ned the 

 shorter way round. The fourth contains the differences between the de.ma s of 

 the second and third, together with the ratios of these to those of the th.rd 

 (These differences 



th 



gles 



llv corrected by a unit, to allow for the 



pproximate character of scne of these decimal,) They are the ^nten , 



of divergence introduced by the third leaf of a cycle between ,t and -he fir 



