NA TURE 



145 



THURSDAY. DECEMBER 19, 1907. 



MATHEMATICS IN BOTANY. 

 Matl}cmatisc!:e und mikroskopisch-anatomisclie 

 Sliulicn iiber BlattsteUuitgen. By Dr. G. van 

 Iterson, Jun. Pp. .\ii +331 + plates. (Jena: G. 

 Fischer, 1907.) Price 20 marks. 



THE subject of Phyllotaxis, which formerly involved 

 the study of the arrangement, though now more 

 l).irticularly the mode of origin, of the lateral members 

 of a plant-shoot, since it was first placed on a scientific 

 footing by Bonnet in 1754, has afforded one of the 

 most fascinating branches of botany, and, it must be 

 frankly admitted, one which is very inadequately 

 treated in text-books ; this being again the expression 

 of the fact that the more that is known with regard 

 to it, the more complex do its problems appear, and 

 the more hopeless of final solution. The subject, 

 again, possesses possibly a special interest in that it 

 IS strictly non-utilitarian, and remains a field of ab- 

 stract scientific work dealing with some of the most 

 fundamental questions of protoplasmic life, which 

 attain but little emphasis in the animal kingdom, 

 owing to the restricted output of systems of ramifi- 

 cation and appendage-production in the more con- 

 densed type of animal-organisation. The literature of 

 the subject is, however, very voluminous, and slowly 

 increases, the present volume of Iterson being the 

 only important contribution since the loss of Schumann, 

 and the publication of some of the work of Church 

 (1904). 



In that the arrangement of lateral members in the 

 case of shoot-construction usually involves phenomena 

 of periodicity or rhythm, phyllotaxis becomes capable 

 of a certain amount of mathematical treatment; and 

 il is to this fact, perhaps, more than to any other 

 that the subject is often viewed with a vague distrust 

 by the majority of botanists ; mathematical results 

 can only follow from given premises, which must first 

 be interpolated into the question; for example, 

 Church pointed out in 1901 that the accepted usage 

 for seventy years of mathematical expressions based 

 on systems of regular helices rendered all discussions 

 of the arrangement of members on growing apices 

 purely nonsensical, and vitiated all deductions based 

 on the apparent imperfection of such constructions. 

 Some working hypothesis is clearly necessary to start 

 with, and the assumption of different data may involve 

 a completely different mathematical presentation. 

 Iterson 's volume, including 300 pages on the botanical 

 aspect of the question, devotes igo to mathematical 

 speculations, the greater part of which will therefore 

 not appeal to the average botanist at all. The 

 mathematician necessarily starts with a severe handi- 

 cap, since the data of the actual appearances pre- 

 sented at the apex of a vegetative shoot are so illusive. 

 New primordia rise up as rounded pimples, wholly 

 independent of the segmentation of the apex into con- 

 stituent cells, and often without any visible connection 

 with each other, yet falling along the paths of what 

 is often a very elaborate pattern, most readily defined 

 as a meshwork of intersecting radiating curves; and 



NO. 1990, VOL. yjl 



it is admittedly impossible to measure any lines or 

 angles, or even to plot the form of the actual primor- 

 dium with any such degree of accuracy as a precise 

 mathematical presentation would appear to demand ; 

 hence observers who are more familiar with the con- 

 ditions obtaining at a growing-point become naturally 

 suspicious of mathematical speculations which are 

 incapable of verification. 



In all the speculations which have been introduced 

 into the subject, the difficulty is to find anything what- 

 ever which can be established as a reasonable proof 

 of the working hypothesis selected ; thus in Dr. 

 Iterson's volume, models of spheres in helicoid or 

 conical arrangements may edify the beginner, but 

 they have no particular relation to the origin of leaf- 

 primordia on the surface of a shoot-apex ; systems of 

 circles in tangential contact must remain unsatisfac- 

 tory while there is no evidence that primordia can be 

 treated as circles, or that the tangential contact is 

 absolute ; the mathematics of a conical surface which 

 can be unrolled has little reference to the curved 

 dome of a plant-shoot, the curve of which is beyond 

 present calculation, while any unrolling of the systems 

 destroys the only essential feature of the system of 

 intersecting curves; again, the projection of a spherical 

 primordium to a " folioid " curve gives suggestive 

 imitative results, when these folioid curves are con- 

 tinued in a log. spiral system ; but there is no evidence 

 to connect the folioid with the shape of a leaf-primor- 

 dium ; and it does not, as a matter of fact, fulfil the 

 normal demands of a phyllota,\is-system. 



Writers on phyllotaxis in the past may be divided 

 into three categories : first, those who seek merely 

 for a method of simply enumerating and cataloguing 

 the phenomena observed in the beautifully rhythmic 

 patterns, usually expressed as spiral curves, in plant- 

 shoots and buds, familiar examples of which are ob- 

 served in the Pine-cone and the arrangement of the 

 disk-florets of Composites. For these the empirical 

 helical formulae of Schimper and Braun (1835) still 

 afford a sufficient basis, so long as observations are 

 restricted to adult structures, and no very rigid 

 accuracy is required. A second class of writers start 

 off on the attempt to imitate the appearances, hoping 

 thereby to explain them ; the most unscientific line of 

 approach conceivable, the physiological fallacy of 

 such mimetic methods having been fully exposed by 

 Sachs. To this line of argument botany is indebted lor 

 numerous theories of " torsion," since torsion will 

 give a spiral effect ! Such imitative conceptions cul- 

 minate in the contact-pressure theory of Schwendener. 

 Lastly, there is a more modern class of investigators 

 who require something more fundamental, in the 

 nature of a physical cause for the phenomena of 

 rhythm, which clearly lies behind the first visible rise 

 of rounded primordia, these being but the expression 

 of more concealed growth-factors. 



The treatise of Dr. Iterson, who apparently remains 

 in the imitative line of approach, may be briefly de- 

 scribed as an attempt to harmonise the largely 

 accepted theories of Schwendener and older writers with 

 a corrected mathematical presentation which in itself 

 renders the difficulties of these writers largely illusory ; 

 as in the case of these observers, Iterson gets little 



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