Fig. 11.—Diagram to show the Characteristics of an “Harmonious-equipotential System.”
The element X forms part of the systems a b or a1 b1 or a2 b2; its prospective value is different in each case.
But the formula is not yet complete: s and l are what the mathematicians call variables: they may have any actual value and there will always be a definite value of p.v., i.e. of the actual fate which is being considered; to every value of s and l, which as we know are independent of each other, there corresponds a definite value of the actual prospectivity. Now, of course, there is also a certain factor at work in every actual case of experimental or normal development, which is not a variable, but which is the same in all cases. This factor is a something embraced in the prospective potency of our system, though not properly identical with it.
The prospective potency of our system, that is to say of each of its elements, is the sum total of what can be done by all; but the fact that a typically proportionate development occurs in every possible case, proves that this sum comes into account, not merely as a sum, but as a sort of order: we may call this order the “relation of localities in the absolutely normal case.” If we keep in mind that the term “prospective potency” is always to contain this order, or, as we may also call it, this “relative proportionality,” which, indeed, was the reason for calling our systems “harmonious,” then we may apply it without further explanation in order to signify the non-variable factor on which the prospective value of any element of our systems depends, and, if we denote the prospective potency, embracing order, by the letter E, we are now able to complete our formula by saying p.v. (X) = f(s, l, E).
So far the merely analytical study of the differentiation of harmonious-equipotential systems.[59]
Instances of “Harmonious-Equipotential Systems”
We must try at first to learn a few more positive facts about our systems, in order that we may know how important is the part which they play in the whole animal kingdom, and in order that our rather abstract analysis may become a little more familiar to us. We know already that many of the elementary morphogenetic organs have been really proved to be harmonious-equipotential systems, and that the same probably is true of many others; we also know that the immature egg of almost all animals belongs to this type, even if a fixed determination of its parts may be established just after maturation. Moreover, we said, when speaking about some new discoveries on form-restitution, that there are many cases in which the processes of restitution do not proceed from single localities, the seat of complex potencies in the organism, but in which each single part of the truncated organism left by the operation has to perform one single act of restoration, the full restitution being the result of the totality of all. These cases must now be submitted to a full analysis.
All of you have seen common sea-anemones or sea-roses, and many of you will also be familiar with the so-called hydroid polyps. Tubularia is one genus of them: it looks like a sea-anemone in miniature placed on the top of a stem like a flower. It was known already to Allman that Tubularia is able to restore its flower-like head when that is lost, but this process was taken to be an ordinary regeneration, until an American zoologist, Miss Bickford, succeeded in showing that there was no regeneration process at all, in the proper sense of the word, no budding of the missing part from the wound, but that the new tubularian head was restored by the combined work of many parts of the stem. Further analysis then taught us that Tubularia indeed is to be regarded as the perfect type of an harmonious-equipotential system: you may cut the stem at whatever level you like: a certain length of the stem will always restore the new head by the co-operation of its parts. As the point of section is of course absolutely at our choice, it is clear, without any further discussion, that the prospective value of each part of the restoring stem is a “function of its position,” that it varies with its distance from the end of the stem; and so at once we discover one of the chief characteristics of our systems. But also the second point which enters into our formula can be demonstrated in Tubularia: the dependence of the fate of every element on the actual size of the system. You would not be able to demonstrate this on very long stems, but if you cut out of a Tubularia stem pieces which are less than ten millimetres in length, you will find the absolute size of the head restored to be in close relation to the length of the stem piece, and this dependence, of course, includes the second sort of dependence expressed in our formula.