INTRODUCTIOK. 7 



This clearly formed the weakest point of the theory. It is quite 

 useless to take angular measurements as the basis of a theory when 

 they cannot be checked. 



Again, in considering the common quincuncial (f ) type, it is quite 

 easy to suppose that if five members developed in spiral series were 

 left isolated on a stem, they would space themselves out at equal 

 angles of 72° if they developed symmetrically: but it does not 

 follow that they were produced at exact successive angles of 144°, 

 although this number may have been approximated. 



It is, in fact, a matter of ready observation, as Bonnet noticed, 

 that in none of the cases usually described as |, and continued for 

 several members, does the sixth member come exactly over the first, 

 but rather falls a little earlier in the gap between 1 and 3. The 

 longer the internodes, the nearer it appears to so come, but the 

 range of error may clearly be very large : thus, to form the 6th 

 leaf of a § cycle the spiral should have rotated 5 x 144 = 720° ; the 

 nearest 6th leaf of any other cycle is that of the -pg, to form which 

 the spiral rotates 692°. In a given case, therefore, when it becomes 

 necessary to decide whether the cycle stops at f , or is continued on 

 to ^g-, a range of error as great as -^=14° requires to be negotiated. 

 Such a range in a system which in higher values comes down to 

 miiiutes and seconds does not tend to render the original spiral 

 theory very acceptable. 



The determination of the fractional value depends, therefore, 

 since angular measurements are out of the question, on the deter- 

 mination of a member vertically superposed, to one taken as a 

 starting point. The theory of Sehimper and Braun really stands 

 or falls, then, with the observation of " orthostichies," that is to say, 

 according as a leaf which appears to stand vertically above any 

 given one is actually so. Of this, again, proof is impossible : the 

 very fact that in going up the series to count the divergence on a 

 specimen, a nearer and nearer vertical point is obtained at every 

 rise, suggests that the one ultimately selected is only an approxima- 

 tion, the eye being as incapable of judging a mathematically straight 

 line as it is of measuring an angle to fractions of a degree. 



That orthostichies tend to become cuniserial in the higher 

 divergences was more fully recognised by Bravais, and very in- 



