This investigation we may approach in two ways: by considering, namely, the partitioning off from some given space or area of one-half (or some other fraction) of its content; or again, by dealing simultaneously with the partitions necessary for the breaking up of a given space into a definite number of compartments.
If we take, to begin with, the simple case of a cubical cell, it is obvious that, to divide it into two halves, the smallest possible partition-wall is one which runs parallel to, and midway between, two of its opposite sides. If we call a the length of one of the edges of the cube, then a2 is the area, alike of one of its sides, and of the partition which we have interposed parallel, or normal, thereto. But if we now consider the bisected cube, and wish to divide the one-half of it again, it is obvious that another partition parallel to the first, so far from being the smallest possible, is precisely twice the size of a cross-partition perpendicular to it; {347} for the area of this new partition is a × a ⁄ 2. And again, for a third bisection, our next partition must be perpendicular to the other two, and it is obviously a little square, with an area of (½ a)2 = ¼ a2 .
From this we may draw the simple rule that, for a rectangular body or parallelopiped to be divided equally by means of a partition of minimal area, (1) the partition must cut across the longest axis of the figure; and (2) in the event of successive bisections, each partition must run at right angles to its immediate predecessor.
Fig. 136. (After Berthold.)
We have already spoken of “Sachs’s Rules,” which are an empirical statement of the method of cell-division in plant-tissues; and we may now set them forth in full.
- (1) The cell typically tends to divide into two co-equal parts.
- (2) Each new plane of division tends to intersect at right angles the preceding plane of division.
The first of these rules is a statement of physiological fact, not without its exceptions, but so generally true that it will justify us in limiting our enquiry, for the most part, to cases of equal subdivision. That it is by no means universally true for cells generally is shewn, for instance, by such well-known cases {348} as the unequal segmentation of the frog’s egg. It is true when the dividing cell is homogeneous, and under the influence of symmetrical forces; but it ceases to be true when the field is no longer dynamically symmetrical, for instance, when the parts differ in surface tension or internal pressure. This latter condition, of asymmetry of field, is frequent in segmenting eggs[386], and is then equivalent to the principle upon which Balfour laid stress, as leading to “unequal” or to “partial” segmentation of the egg,—viz. the unequal or asymmetrical distribution of protoplasm and of food-yolk.
The second rule, which also has its exceptions, is true in a large number of cases; and it owes its validity, as we may judge from the illustration of the repeatedly bisected cube, solely to the guiding principle of minimal areas. It is in short subordinate to, and covers certain cases included under, a much more important and fundamental rule, due not to Sachs but to Errera; that (3) the incipient partition-wall of a dividing cell tends to be such that its area is the least possible by which the given space-content can be enclosed.