78 BOTANICAL GAZETTE [july 



variation curves fall upon the numbers of the Fibonacci series, 2, 3, 5, 8, 13, 

 etc., and low multiples of them, or on the similarly constituted "Trientalis- 

 series," 3, 4, 7, n, etc., and their low multiples. The latter series was dis- 

 covered by Ludwig in Trientalis, whence its name. Vogler 10 had found 

 earlier that the modes in the number of umbellary rays of Astrantia major fall 

 upon members of the Fibonacci series when only the primary umbels are 

 included, but in the secondary umbels the modes are on the Trientalis series. 

 More recently the same author 11 has reported on the number of ray flowers in 

 Arnica montana, Buphthalmum solid folium, Eupatorium molle, Aster novi- 

 belgii, Senecio erucifolius, and Chrysanthemum Parthenium. Several of these 

 species gave well-marked modes on the Fibonacci numbers, but in other collec- 

 tions of data from the same species the mode occurred not infrequently on some 

 quite unrelated number. For example, in Arnica montana a collection from 

 Rigi, Switzerland, in 1908, showed modes on 13, 16, and 21, while heads of the 

 same species collected the next year at Klosters presented a well-developed 

 mode on 1 1 and only a slight indication of a mode on 1 3 . Later analysis of this 

 case showed that the terminal heads give modes on 13 and 16, while secondary 

 heads give modes on 11 and 14, the latter numbers bearing the same relation 

 to the Trientalis series that the former do to the Fibonacci series. This whole 

 problem as to the position of the modes in variation curves of ray flowers, and 

 other organs which are related more or less definitely to the phyllotactic 

 spiral, is still unsolved, though it is evident that the Fibonacci series supplies 

 the modal numbers in many cases, and that other equally definite series are 

 followed in other cases. It is very rare that the number of variates used by 

 investigators is sufficiently great to establish with any considerable degree of 

 probable correctness these relatively superficial features of the curves. Ritter 

 has gone to the length of asserting that non-phyllotactic variates among plant 

 organs have their modal numbers also related to the Fibonacci or Trientalis 

 series. He even contends that this is true of graduated variates. In support 

 of this view he tabulates 12 a rather meager series of measurements of 

 width and length of leaves and leaflets of Stellaria media, Oxalis Acetosella, 

 Lysimachia nummularia, Hypericum perforatum, Caragana arborescens, Rosa 

 canina, Medicago saliva, Symphoricarpus racemosus, Fragaria vesca, and Cytisus 

 Laburnum, and the width and length of fruits of Alnus glandulosa, Rosa canina, 

 Quercus Robur, and Q. sessiliflora. He believes that the measurements of 

 surfaces, such as leaf blades, give modes related to the square roots of the 

 Fibonacci numbers, namely, on io|/T, ioi/"2, 101/J, ioi/"s, 101/8, etc., 



10 Vogler, P., Variationstatistische Untersuchungen an den Dolden von Astrantia 

 major. Beih. Bot. Centralbl. 24: 1-9. figs. 6. 1908. 



"Vogler, P., Neue variationstatistische Untersuchungen an Compositen. 



Naturwis 



12 Ritter, G., Uber d; 

 Centralbl. 25:1-29. 1909. 



Beih. Bot. 



