The “Harmonious-Equipotential System”
We have an ectoderm of the gastrula of a starfish here before us; we know that we may cut off any part of it in any direction, and that nevertheless the differentiation of the ectoderm may go on perfectly well and result in a typical little embryo, which is only smaller in its size than it would normally be. It is by studying the formation of the highly complicated ciliary band, that these phenomena can be most clearly understood.
Now let us imagine our ectoderm to be a cylinder instead of being approximately a sphere, and let us imagine the surface of this cylinder unrolled. It will give us a plane of two definite dimensions, a and b. And now we have all the means necessary for the analytical study of the differentiation of an harmonious-equipotential system.
Our plane of the dimensions a and b is the basis of the normal, undisturbed development; taking the sides of the plane as fixed localities for orientation, we can say that the actual fate, the “prospective value” of every element of the plane stands in a fixed and definite correlation to the length of two lines, drawn at right angles to the bordering lines of the plane; or, to speak analytically, there is a definite actual fate corresponding to each possible value of x and of y. Now, we have been able to state by our experimental work, that the prospective value of the elements of our embryonic organ is not identical with their “prospective potency,” or their possible fate, this potency being very much richer in content than is shown by a single case of ontogeny. What will be the analytical expression of such a relation?
Let us put the question in the following way: on what factors does the fate of any element of our system depend in all possible cases of development obtainable by means of operations? We may express our results in the form of an equation:—
p.v. (X) = f( . . . )
i.e. “the prospective value of the element X is a function of . . .”—of what?
We know that we may take off any part of the whole, as to quantity, and that a proportionate embryo will result, unless the part removed is of a very large size. This means that the prospective value of any element certainly depends on, certainly is a function of, the absolute size of the actually existing part of our system in the particular case. Let s be the absolute size of the system in any actual experimental case of morphogenesis: then we may write p.v. (X) = f(s . . . ). But we shall have to add still some other letter to this s.
The operation of section was without restriction either as to the amount of the material removed from the germ, or as to the direction of the cut. Of course, in almost every actual case there will be both a definite size of the actual system and a definite direction of the cut going hand-in-hand. But in order to study independently the importance of the variable direction alone, let us imagine that we have isolated at one time that part of our system which is bounded by the lines a1 b1, and at another time an equal amount of it which has the lines a2 b2 as its boundaries. Now since in both cases a typical small organism may result on development, we see that, in spite of their equal size the prospective value of every element of the two pieces cut out of the germ may vary even in relation to the direction of the cut itself. Our element, X, may belong to both of these pieces of the same size: its actual fate nevertheless will be different. Analytically, it may be said to change in correspondence to the actual position of the actual boundary lines of the piece itself with regard to the fundamental lines of orientation, a and b; let this actual position be expressed by the letter l, l marking the distance of one[58] of the actual boundary lines of our piece from a or b: then we are entitled to improve our formula by writing p.v. (X) = f(s, l . . . ) (Fig. 11).