we may deal with the case as though we had now to do with four equal cells, each one of them a perfect quadrant. And so our problem is, to find the shortest line which shall divide the quadrant of a circle into two halves of equal area. A radial partition (Fig. 145, A), starting from the apex of the quadrant, is at once excluded, for a reason similar to that just referred to; our choice must lie therefore between two modes of division such as are illustrated in Fig. 145, where the partition is either (as in B) {360} concentric with the outer border of the cell, or else (as in C) cuts that outer border; in other words, our partition may (B) cut both radial walls, or (C) may cut one radial wall and the periphery. These are the two methods of division which Sachs called, respectively, (B) periclinal, and (C) anticlinalÿ£¢6. We may either treat the walls of the dividing quadrant as already solidified, or at least as having a tension compared with which that of the incipient partition film is inconsiderable. In either case the partition must meet the cell-wall, on either side, at right angles, and (its own tension and curvature being everywhere uniform) it must take the form of a circular arc.
Now we find that a flattened cell which is apôÙproxôÙiôÙmateôÙly a quadrant of a circle invariably divides after the manner of Fig. 145, C, that is to say, by an apôÙproxôÙiôÙmateôÙly circular, anticlinal wall, such as we now recognise in the eight-celled stage of Erythrotrichia (Fig. 144); let us then consider that Nature has solved our problem for us, and let us work out the actual geometric conditions.
Let the quadrant OAB (in Fig. 146) be divided into two parts of equal area, by the circular arc MP. It is required to determine (1)