84 TRANSACTIONS, NATURAL HISTORY SOCIETY OF GLASGOW. 



The fact that such leaves as, e.g.^ Yillarsia have a nearly 

 circular shape, is to my mind simply explained by the considera- 

 tion that, as they are nearly symmetrically arranged with regard 

 to all external, agencies such as light and air, itc, there is no 

 reason why they should elongate further in one direction than in 

 another, and therefore must be circular. These leaves are not 

 exposed to interference from their neighbours, which would 

 prevent this equal development. As further examples of this 

 circular type, Nuphar, Nymphsea, Hydrocotyle vulgaris, Linn., 

 Ranunculus peltatics, Schrank, and Viola palustris, Linn., may 

 be mentioned. 



Turning now to the diamond type of leaf, there are many oval, 

 elliptic, and even linear leaves which can be placed here. The 

 leaves of Veronica hedercefolia, Linn., Polygonum Boberti, Loisel, 

 Dentaria hulbifera, Linn., Parietaria officinalis. Linn., Epilobium 

 roseum, Schreb., can be fitted together in a nearly perfect manner. 

 Amongst others may be mentioned Arenaria trinervia, Linn., 

 Epilobium alpinum, Linn., Galeobdolon luteum, Huds,, Polygala 

 vulgaris, Linn., Polycarpon, and Hieracium murorum, Linn. 



Some of these leaves give, to my mind, patterns of great beauty, 

 probably in consequence of the perfect manner in which they fit 

 one another. 



So far the ovate type of leaf, except when it is broadly ovate, 

 and cordate at base, has not been alluded to. I shall just men- 

 tion a curious point with regard to the leaves of Saponaria offici- 

 nalis, Linn, and Mentha sativa, Linn. These plants have their 

 leaves opposite and in four rows. The angle of the wedge- 

 shaped base is in both a right angle, whilst the angle at the apex 

 is 60°. The bases of the four leaves thus fit exactly, but the 

 exact geometrical idea of this shape is not very clear. 



The point which I wished to discover was whether the question 

 could be simplified by the finding of a geometrical idea or plan, 

 which a leaf will, of course, vary to suit its own individual needs. 

 These variations in individuals are sometimes distinctly connected 

 with variations in the environment (as I tried to show in a paper 

 read before the Linnean Society, December 18th, 1890; see 

 Journal of the Linnean Society, vol. xxviii.) ; but without some 

 clear plan or geometrical idea, it is obvious that the fitting 

 could not be as perfect as it now is, 



