The more complicated cases, when t, T and Tÿ£¢ãý are all unequal, are already sufficiently explained.
The biological facts which the foregoing conôÙsiôÙdeôÙraôÙtions go a long way to explain and account for have been the subject of much argument and discussion, especially on the part of the botanists. Let me recapitulate, in a very few words, the history of this long discussion.
Some fifty years ago, Hofmeister laid it down as a general law that ãThe partition-wall stands always perpendicular to what was previously the principal direction of growth in the cell,ããor, in other words, perpendicular to the long axis of the cell3. Ten {305} years later, Sachs formulated his rule, or principle, of ãrectangular section,ã declaring that in all tissues, however complex, the cell-walls cut one another (at the time of their formation) at right angles4. Years before, Schwendener had found, in the final results of cell-division, a universal system of ãorthogonal trajectories5ã; and this idea Sachs further developed, introducing complicated systems of confocal ellipses and hyperbolûÎ, and distinguishing between periclinal walls, whose curves apôÙproxôÙiôÙmate to the peripheral contours, radial partitions, which cut these at an angle of 90ô¯, and finally anticlines, which stand at right angles to the other two.
Reinke, in 1880, was the first to throw some doubt upon this explanation. He pointed out various cases where the angle was not a right angle, but was very definitely an acute one; and he saw, apparently, in the more common rectangular symmetry merely what he calls a necessary, but secondary, result of growth6.
Within the next few years, a number of botanical writers were content to point out further exceptions to Sachsãs Rule7; and in some cases to show that the curvatures of the partition-walls, especially such cases of lenticular curvature as we have described, were by no means accounted for by either Hofmeister or Sachs; while within the same period, Sachs himself, and also Rauber, attempted to extend the main generalisation to animal tissues8.
While these writers regarded the form and arrangement of the cell-walls as a biological phenomenon, with little if any direct relation to ordinary physical laws, or with but a vague reference to ãmechanical conditions,ã the physical side of the case was soon urged by others, with more or less force and cogency. Indeed the general resemblance between a cellular tissue and a ãfrothã {306} had been pointed out long before, by Melsens, who had made an ãartificial tissueã by blowing into a solution of white of egg9.
In 1886, Berthold published his Protoplasmamechanik, in which he definitely adopted the principle of ãminimal areas,ã and, following on the lines of Plateau, compared the forms of many cell-surfaces and the arrangement of their partitions with those assumed under surface tension by a system of ãweightless films.ã But, as Klebs10 points out in reviewing Bertholdãs book, Berthold was careful to stop short of attributing the biological phenomena to a definite mechanical cause. They remained for him, as they had done for Sachs, so many ãphenomena of growth,ã or ãproperties of protoplasm.ã
In the same year, but while still apparently unacquainted with Bertholdãs work, Errera11 published a short but very lucid article, in which he definitely ascribed to the cell-wall (as Hofmeister had already done) the properties of a semi-liquid film and drew from this as a logical consequence the deduction that it must assume the various conôÙfiôÙgurôÙaôÙtions which the law of minimal areas imposes on the soap-bubble. So what we may call Erreraãs Law is formulated as follows: A cellular membrane, at the moment of its formation, tends to assume the form which would be assumed, under the same conditions, by a liquid film destitute of weight.