CLOSING EXCERPT FROM
THE THREE-FOLD NUMBER
BENJAMIN PIERCE, LOUIS AGASSIZ, FIBONACCI, AND THE SOLAR
It may well be that the extension of the "Three-fold Number" beyond terrestrial boundaries is simply the logical continuation of Ovid's initial observation that "The three-fold number is present in all things whatsoever" --an observation that in view of the nature of spiral galaxies need not remain with planetary systems per se. Nevertheless, the linking of natural growth to the structure of planetary systems was undoubtedly a bold and momentous step even though it also reflects the second part of the quotation from Ovid: "Nor did we ourselves discover this number, but rather natures teaches it to us." At least this seems applicable in the case of Benjamin Peirce,26 who integrated both to successfully apply the Fibonacci series to the structure of the Solar System. The latter's work was originally published in the Proceedings of the AAAS in 1850 and given additional permanence with a further airing in Louis Agassiz's Essay on Classification in 185727. All to little or no avail, it would seem, for in spite of the details and the implications the work it still remains in relative obscurity to the present day. In some respects this may be understandable, though the subsequent lack of attention or acceptance can hardly be blamed on the quality of the work or the means of presentation. All too easily dismissed as "speculative biology" (Lurie 1962:128) 28 it would seem, it is likely that it was also one of the first victims of "Bode's "Law" which first surfaced less than a decade later (1866-1871) despite its fatal mathematical flaws and ad hoc origins. Indeed, if longevity and popularity alone provide the guidelines, then "Bode's Law" would win hands-down in any comparison between the two planetary frameworks. If, however, the standard by which such matters are judged depends not on popularity or elementary mathematics, but on human progress and increased understanding, then one can only wonder what else might have been accomplished since Agassiz's time and sadly lament the loss.
The complete description of Benjamin Pierce's application of the Fibonacci series to the structure of the Solar System as published by Louis Aggassiz is provided below; perhaps significantly, the words "Fibonacci" and/or the "Golden Section" (and the like) are noticeably absent--such words perhaps already unacceptable to the powers that be and also a perceived threat to the status quo. Nevertheless, there can be no mistaking the sequence applied or the major premise, called here perhaps fittingly enough (for the moderns, at least) "the law of phyllotaxis". One may also note that Peirce had already considered the practical differences between his theoretical treatment and the Solar System itself and subsequently considered not only the position of Earth, but also discepancies encountered for the positions of Mars, Uranus and Neptune. Initially Pierce also applied a double form of Fibonacci series but subsequently reduced the set to arrive are a situation similar to that involving the synodic difference cycle between adjacent planets.
ESSAY ON CLASSIFICATION
Louis Agassiz 1857
FUNDAMENTAL RELATIONS OF ANIMALS
COMBINATIONS IN TIME AND SPACE OF VARIOUS KINDS OF RELATIONS AMONG ANIMALS
It must occur to every reflecting mind, that the mutual relation and respective parallelism of so many structural, embryonic, geological, and geographical characteristics of the animal kingdom are the most conclusive proof that they were ordained by a reflective mind, while they present at the same time the side of nature most accessible to our intelligence, when seeking to penetrate the relations between finite beings and the cause of their existence.
The phenomena of the inorganic world are all simple, when compared to those of the organic world. There is not one of the great physical agents, electricity, magnetism, heat, light, or chemical affinity, which exhibits in its sphere as complicated phenomena as the simplest organized beings; and we need not look for the highest among the latter to find them presenting the same physical phenomena as are manifested in the material world, besides those which are exclusively peculiar to them. When then organized beings include everything the material world contains and a great deal more that is peculiarly their own, how could they be produced by physical causes, and how can the physicists, acquainted with the laws of the material world and who acknowledge that these laws must have been established at the beginning, overlook that à fortiori the more complicated laws which regulate the organic world, of the existence of which there is no trace for a long period upon the surface of the earth, must have been established later and successively at the time of the creation of the successive types of animals and plants?
Thus far we have been considering chiefly the contrasts existing between the organic and inorganic worlds. At this stage of our investigation it may not be out of place to take a glance at some of the coincidences which may be traced between them, especially as they afford direct evidence that the physical world has been ordained in conformity with laws which obtain also among living beings, and disclose in both spheres equally plainly the workings of a reflective mind. It is well known that the arrangement of the leaves in plants148 may be expressed by very simple series of fractions, all of which are gradual approximations to, or the natural means between 1/2 or 1/3, which two fractions are themselves the maximum and the minimum divergence between two single successive leaves. The normal series of fractions which expresses the various combinations most frequently observed among the leaves of plants is as follows: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55, etc. Now upon comparing this arrangement of the leaves in plants with the revolutions of the members of our solar system, Peirce has discovered the most perfect identity between the fundamental laws which regulate both, as may be at once seen by the following diagram, in which the first column gives the names of the planets, the second column indicates the actual time of revolution of the successive planets, expressed in days; the third column, the successive times of revolution of the planets, which are derived from the hypothesis that each time of revolution should have a ratio to those upon each side of it, which shall be one of the ratios of the law of phyllotaxis; and the fourth column, finally, gives the normal series of fractions expressing the law of the phyllotaxis.149
Table I (Agassiz-Pierce 1857)
In this series the Earth forms a break; but this apparent irregularity admits of an easy explanation. The fractions: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, etc., as expressing the position of successive leaves upon an axis, by the short way of ascent along the spiral, are identical as far as their meaning is concerned with the fractions expressing these same positions by the long way, namely, 1/2,2/3, 3/5, 8/13, 13/21, 21/34, etc.
Let us therefore repeat our diagram in another form, the third column giving the theoretical time of revolution.
Table II (Agassiz-Pierce 1857)
It appears from this table that two intervals usually elapse between two successive planets, so that the normal order of actual fractions, 1/2, 1/3, 2/5, 3/8, 5/13, etc.,or the fractions by the short way in phyllotaxis, from which, however, the Earth is excluded, while it forms a member of the series by the long way. The explanation of this, suggested by Peirce, is that although the tendency to set off a planet is not sufficient at the end of a single interval, it becomes so strong near the end of the second interval that the planet is found exterior to the limit of this second interval. Thus, Uranus is rather too far from the Sun relatively to Neptune, Saturn relatively to Uranus, and Jupiter relatively to Saturn; and the planets thus formed engross too large a proportionate share of material, and this is especially the case with Jupiter. Hence, when we come to the Asteroids, the disposition is so strong at the end of a single interval, that the outer Asteroid is but just within this interval, and the whole material of the Asteroids is dispersed in separate masses over a wide space, instead of being concentrated into a single planet. A consequence of this dispersion of the forming agents is that a small proportionate material is absorbed into the Asteroids. Hence, Mars is ready for formation so far exterior to its true place, that when the next interval elapses the residual force becomes strong enough to form the Earth, after which the normal law is resumed without any further disturbance. Under this law there can be no planet exterior to Neptune, but there may be one interior to Mercury.
Let us now look back upon some of the leading features alluded to before, omitting the simpler relations of organized beings to the world around, or those of individuals to individuals, to consider only the different parallel series we have been comparing when showing that in their respective great types the phenomena of animal life correspond to one another, whether we compare their rank as determined by structural complication with the phases of their growth, or with their succession in past geological ages; whether we compare this succession with their embryonic growth, or all these different relations with each other and with the geographical distribution of animals upon earth. The same series everywhere! These facts are true of all the great divisions of the animal kingdom, so far as we have pursued the investigation; and though, for want of materials, the train of evidence is incomplete in some instances, yet we have proof enough for the establishment of this law of a universal correspondence in all the leading features which binds all organized beings of all times into one great system, intellectually and intelligibly linked together, even where some links of the chain are missing. It requires considerable familiarity with the subject even to keep in mind the evidence, for, though yet imperfectly understood, it is the most brilliant result of the combined intellectual efforts of hundreds of investigators during half a century. The connection, however, between the facts, it is easily seen, is only intellectual; and implies therefore the agency of Intellect as its first cause.150
And if the power of thinking connectedly is the privilege of cultivated minds only; if the power of combining different thoughts and of drawing from them new thoughts is a still rarer privilege of a few superior minds; if the ability to trace simultaneously several trains of thought is such an extraordinary gift, that the few cases in which evidence of this kind has been presented have become a [p.131] matter of historical record (Caesar dictating several letters at the time), though they exhibit only the capacity of passing rapidly, in quick succession, from one topic to another, while keeping the connecting thread of several parallel thoughts: if all this is only possible for the highest intellectual powers, shall we by any false argumentation allow ourselves to deny the intervention of a Supreme Intellect in calling into existence combinations in nature, by the side of which all human conceptions are child's play?
If I have succeeded, even very imperfectly, in showing that the various relations observed between animals and the physical world, as well as between themselves, exhibit thought, it follows that the whole has an Intelligent Author; and it may not be out of place to attempt to point out, as far as possible, the difference there may be between Divine thinking and human thought. Taking nature as exhibiting thought for my guide, it appears to me that while human thought is consecutive, Divine thought is simultaneous, embracing at the same time and forever, in the past, the present, and the future, the most diversified relations among hundreds of thousands of organized beings, each of which may present complications again, which, to study and understand even imperfectly, as for instance, Man himself, Mankind has already spent thousands of years. And yet, all this has been done by one Mind, must be the work of one Mind only, of Him before whom Man can only bow in grateful acknowledgment of the prerogatives he is allowed to enjoy in this world, not to speak of the promises of a future life.
I have intentionally dismissed many points in my argument with mere questions, in order not to extend unduly a discussion which is after all only accessory to the plan of my work. I have felt justified in doing so because, from the point of view under which my subject is treated, those questions find a natural solution which must present itself to every reader. We know what the intellect of Man may originate, we know its creative power, its power of combination, of foresight, of analysis, of concentration; we are, therefore, prepared to recognize a similar action emanating from a Supreme Intelligence to a boundless extent. We need therefore not even attempt to show that such an Intellect may have originated all the Universe contains; it is enough to demonstrate that the constitution of the physical world and, more particularly, the organization of living beings in their connection with the physical world, prove in general the existence of a Supreme Being as the Author of all things. The task of science is rather to investigate what has been done, to inquire if possible how it has been done, than to ask what is possible for the Deity, as we can know that only by what actually exists. To attack such a position, those who would deny the intervention in nature of a creative mind must show that the cause to which they refer the origin of finite beings is by its nature a possible cause, which cannot be denied of a being endowed with the attributes we recognize in God. Our task is therefore completed as soon as we have proved His existence. It would nevertheless be highly desirable that every naturalist who has arrived at similar conclusions should go over the subject anew from his point of view and with particular reference to the special field of his investigations; for so only can the whole evidence be brought out. I foresee already that some of the most striking illustrations may be drawn from the morphology of the vegetable kingdom, especially from the characteristic succession and systematical combination of different kinds of leaves in the formation of the foliage and the flowers of so many plants, all of which end their development by the production of an endless variety of fruits. The inorganic world, "considered in the same light, would not fail to exhibit also unexpected evidence of thought, in the character of the laws regulating the chemical combinations, the action of physical forces, the universal attraction, etc., etc. Even the history of human culture ought to be investigated from this point of view. But I must leave it to abler hands to discuss such topics.
Last Section (31st)
31st. The combination in time and space of all these thoughtful conceptions exhibits not only thought, it shows also premeditation, power, wisdom, greatness, prescience, omniscience, providence. In one word, all these facts in their natural connection proclaim aloud the One God, whom man may know, adore, and love; and Natural History must in good time become the analysis of the thoughts of the Creator of the Universe, as manifested in the animal and vegetable kingdoms, as well as in the inorganic world.
It may appear strange that I should have included the preceding disquisition under the title of an "Essay on Classification." Yet it has been done deliberately. In the beginning of this chapter I have already stated that Classification seems to me to rest upon too narrow a foundation when it is chiefly based upon structure. Animals are linked together as closely by their mode of development, by their relative standing in their respective classes, by the order in which they have made their appearance upon earth, by their geographical distribution, and generally by their connection with the world in which they live, as by their anatomy. All these relations should therefore be fully expressed in a natural classification; and though structure furnishes the most direct indication of some of these relations, always appreciable under every circumstance, other considerations should not be neglected which may complete our insight into the general plan of creation. (Louis Agassiz, ESSAY ON CLASSIFICATION, Ed. E. Lurie, Belknap Press, Cambridge, 1962:127-128)
As far as Pierce's still largely unheralded contribution and its attempted furtherance by Louis Agassiz are concerned one might note that although the latter gives due prominence to the subject in his Essay on Classification this highly significant issue is still rarely mentioned in abstracts or notes on the latter's work itself. Yet the understanding inherent in the basic premise was hardly likely to have been entirely isolated, as Agassiz himself stresses in the following passage, which also carries with it familiar yet ancient echoes of unity and applied intellect:29
These facts are true of all the great divisions of the animal kingdom, so far as we have pursued the investigation; and though, for want of materials, the train of evidence is incomplete in some instances, yet we have proof enough for the establishment of this law of a universal correspondence in all the leading features which binds all organized beings of all times into one great system, intellectually and intelligibly linked together, even where some links of the chain are missing. It requires considerable familiarity with the subject even to keep in mind the evidence, for, though yet imperfectly understood, it is the most brilliant result of the combined intellectual efforts of hundreds of investigators during half a century. The connection, however, between the facts, it is easily seen, is only intellectual; and implies therefore the agency of Intellect as its first cause
One major difference between this approach and others lies in the direction used by Peirce; i.e., the latter commenced from the outermost regions and applied Fibonacci-related divisions while moving inwards towards the center. Here the location of Neptune was perhaps a key (or a hindrance) in that the secondary position (i.e., the synodic location of the exponential framework) happens to be similar to that of Neptune itself. On the other hand, however, the 1 : 1 occurrence was perhaps--rightly or wrongly--also an alerting factor for the Fibonacci series itself. Nor should this necessarily matter, for the premise itself was already absorbed and applied.
In retrospect it is hard to say how far this line of inquiry might have been taken, or what might ultimately have resulted, but it must surely have been a far more useful endevour than the circular, simplistic and ad hoc diversions introduced and perpetuated by "Bode's Law." And how could something so momentous and far-reaching have been so easily driven into obscurity? According to the modern editor of Agassiz' Essay on Classification, (E. Lurie) it was partly the work of Asa Gray and Chauncey Wright, as explained in the following footnote (the latter's No.149):30
Agassiz tried to interest Americans in this concept, an idea typical of German speculative biology and one that he had been much impressed with since his student days at the University of Munich. See Asa Gray, "On the Composition of the Plant by Phytons, and Some Applications of Phyllotaxis," Proceedings, AAAS, II (1850), 438-444, and Benjamin Peirce, "Mathematical Investigations of the Fractions Which Occur in Phyllotaxis," in ibid., 444-447. Gray was never entirely convinced of the validity of this ideal conception. He subsequently encouraged Chauncey Wright to examine the problem of leaf arrangement, with the result that such facts were shown to be understandable in terms of the principle of natural selection.
but it is still incredible
that it should have been driven down so swiftly, except, perhaps that
it was undoubtedly heliocentric as well as a major departure away from
the views perpetuated by organized religion.
Thus it may have come too late, a century after Linneaus' classifications, a little less with respect to Cook's voyages, and half a century or more of continued activity that was simply too much for those who wished to maintain the status quo. But what else took place during this period of hopeful enlightenment only to fade from view?
For that we turn next to
the perhaps unexpected subject of spiral formations in shells.
Copyright © 2002. John N. Harris,
updated on May 25, 2003.
RETURN TO THE THREE-FOLD NUMBER
RETURN TO THE MAIN PAGE