frequently takes place parallel to the long axis of the cell, when 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 exceptions that Berthold was led to qualify 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 it5. He was deeply and rightly impressed by the fact that other forces besides surface {358} tension, both external and internal to the cell, play their part in the determination of its partitions, and that the answer to our problem is not to be given in a word. How 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 return to a consideration of the simpler and more general phenomena. And in so doing, let us leave the case of the cubical, quadrangular or cylindrical cell, and examine the case of a spherical cell and of its successive divisions, or the still simpler case of a circular, discoidal cell.
When we attempt to inôÙvesôÙtiôÙgate mathôÙeôÙmatôÙiôÙcally the position and form of a partition of minimal area, it is plain that we shall be dealing with comparatively simple cases wherever even one dimension 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 at once obvious, then, that the partition must lie transversely to the long axis of the