GEOMETRICAL REPRESENTATION OF GROWTH. 31 



includes an apical region in which the embryonic protoplasm is 

 engaged in the formation of new cell-units, without regard to any 

 marked longitudinal extension of the member as a whole, and may 

 be taken to imply the general increase of the protoplasmic mass, 

 equally in all directions in space, and uniformly throughout its 

 substance; the actual subdivision into units of an approximately 

 equal size being a secondary specialization. The mass of protoplasm 

 may be thus considered as a whole, without reference to the 

 cell-membranes of the component units, and this in its structure, 

 and in being supplied with new material along an axial-con- 

 ducting portion, presents many analogies with a jet of semi-fluid 

 substance. 



Sachs * also pointed out the remarkable similarity of the shape of 

 the growing apex of a plant to a paraboloid of revolution, and that 

 in a radial longitudinal section of a typical apex the periclinal 

 walls formed a series of confocal parabolas crossed by a coaxial 

 system of confocal parabolas which formed the anticlinal walls. The 

 mathematical fact that two such sets of confocal parabolas intersect 

 at all points orthogonally was of the utmost importance in enabling 

 him to formulate his theory of the orthogonal intersection of cell- 

 walls. Sachs, however, left the matter entirely a theory of 

 geometrical construction, although, as he himself states in deciding 

 against the spiral theory, geometrical constructions tell nothing of 

 causes but only express facts ^ still it is clear that these phenomena 

 must be based on some definite laws, probably mechanical. 



Thus Errera f has shown that the cell-wall at the moment of its 

 formation has the properties of a weightless fluid film, and that the 

 direction taken by such cell-membranes is identical with that taken 

 by soap-bubble films which impinge orthogonally on previously- 

 formed films. A large number of cases proposed as presenting an 

 apparent contradiction of Sachs' generalization have been shown 



* Sacts, Lectures on the Physiohgrj of Plants, Eng. trans., 1887, p. 448 : " Many- 

 hundreds of median longitudinal sections through growing points of shoots and 

 roots, drawn by very different observers without even the most distant per- 

 ception of the fundamental principle, accord with the construction I have given, 

 and demonstrate its accuracy'' 



+ Errera, "Ueber Zellenformen und Seifenblasen," Biol. Gentralb., IBS'?, 

 1888, p. 728. 



