578 THE FORMS OF TISSUES [ch. 



As a matter of fact, while we have abundant simple illustrations 

 of the principles which we have now begun to study, apparent 

 exceptions to this simplicity, due to an asymmetry of the cell itself 

 or of the system of which the single cell is a part, are by no means 

 rare. We know that in cambium-cells division often takes place 

 parallel to the long axis of the cell, though a partition of much less 

 area would suffice if it were set cross- ways: and it is only when a 

 considerable disproportion has been set up between the length and 

 breadth of the cell that the balance is in part redressed by the- 

 appearance of a transverse partition. It was owing to such excep- 

 tions that Berthold was led to quahfy and even to depreciate the 

 importance of the law of minimal areas as a factor in cell-division, 

 after he himself had done so much to demonstrate and elucidate 

 it*. He was deeply and rightly impressed by the fact that other 

 forces besides surface tension, both external and internal to the cell, 

 play their part in determining its partitions, and that the answer 

 to our problem is not to be given in a word. Hpw fundamentally 

 important it is, however, in spite of all conflicting tendencies and 

 apparent exceptions, we shall see better and better as we proceed. 



But let us leave the exceptions and consider the simpler and 

 more general phenomena. And let us leave the case of the cubical, 

 quadrangular or cylindrical cell, and examine that of a spherical 

 cell and of its successive divisions, or the still simpler case of a 

 circular, discoidal cell. 



When we attempt to investigate mathematically the place and 

 form of a partition of minimal area, it is plain that we shall be 

 dealing with comparatively simple cases wherever even one dimen- 

 sion of the cell is much less than the other two. Where two 

 dimensions are.small compared with the third, as in a thin cylindrical 

 filament like that of Spirogyra, we have the problem at its simplest ; 

 for it is obvious, then, that the partition must lie transversely to 

 the long axis of the thread. But even where one dimension only 

 is relatively small, as for instance in a flattened plate, our problem 



* Cf. Protoplasmamechanik, p. 229: "Insofern liegen also die Verhaltnisse hier 

 wesentlich anders als bei der Zertheilung hohler Korperformen durclj fliissige 

 Lamellen. Wenn die Membran bei der Zelltheilung die von dem Prinzip der 

 kleinsten Flachen geforderte Lage und Kriimmung annimmt, so werden wit den 

 Grund dafiir in andrer Weise abzuleiten haben." 



