98 HISTORY OF BOTANY 



of 120 degrees, a condition represented by the fraction J. 

 The next condition was when two spiral turns had to be 

 made, starting from the initial leaf and touching each 

 successively higher leaf, before the leaf immediately above 

 the initial leaf in the same vertical plane was reached, 

 while five leaves had to be passed in the two circuits of 

 the stem. This was expressed by the fraction f. It 

 was then seen that by adding the numerators and de- 

 nominators of any two successive fractions the next 

 higher fraction in the series could be arrived at, thus 

 1 + 1 = 2. 1 + 2 = 3. 2 + 3 ^ _6^^ and so on. A vast amount 

 of ingenuity and skill was expended, both by Schimper 

 and later by Alexander Braun, in working out this 

 idea, and it was not long before the whole subject, 

 with its cacophonous terminology of orthostichies, 

 parastichies, and what not, became the new fad of 

 the mathematically minded botanist. The relationship 

 between the point of origin of the leaf, its form, and, 

 most important of all, its functions passed into the 

 background, and all efforts seemed to be focussed 

 on the attempt to establish this pseudo-mathematical 

 idea as one of the fundamental laws of nature. You 

 can scarcely open one of the older textbooks of botany 

 without meeting with pages of calculations and drawings 

 of transparent stems, ideahsed pine cones, and machine- 

 turned figures of capitula, etc., with various Unes drawn 

 upon them, illustrating the principles on which this new 

 subject of phyllotaxis was supposed to be based. I have 

 no desire to inflict on you the miseries I and my fellow- 

 students endured in our endeavours to master what was 

 regarded as one of the articles of a botanist's faith in 

 the later years of last century. 



I ought at this point to sketch for you the very 

 important advances made in anatomy and histology 

 during the period with which we are at present concerned, 

 but I prefer to delay consideration of that aspect of the 

 subject till the next lecture, so that I may present the 



