Permutation

“ Before Galois an Italian named Ruffini and a Norwegian named Abel had employed permutation groups to prove that the general equation of the fifth degree can not be solved by successive extraction of roots. In doing this the former studied a number of properties of permutation groups and is therefore generally regarded as the founder of this theory.
The definition of a permutation group is very simple. It is merely the totality of distinct permutations which do not change the formal value of a given expression. Such a totality of permutations has many remarkable properties. ”

“ As to bell-ringing, the adepts insist upon it that this is a science; and they give it the name of campanology. We all know, ever since we learnt about permutation and combination at school, that if there are six, eight, ten, or any number of distinct things, we may arrange them in an enormous number of ways, each way differing from every other. ”

“ Moreover every change in digit arrangement effects a new complexion. The number of the possible complexions or the probability of the given state is therefore equal, under the conditions specified, to the possible "permutations with repetition." [21] In the simple example chosen we have for such permutation, according to a known formula, Consequently in the general case, we have where the sign signifies the product extended over all the elementary regions.
Result contained in (16) is equally true for any other physical system, say, one involving radiant energy. ”

“ In evolution heterogeneity is a constant measure of progress, hence the laws stating the variety of effects producible from given elements have a direct interest and value. These are the laws of combination and permutation. Combinations, mathematically, are groups where the presence and not the position of an element counts for difference—thus B C A and B A C are the same combination but different permutations. As additions are made to the elements, combinations increase in geometrical progression with 2 as constant factor. ”

“ Here two living organisms are made out of one, without death intervening, and by permutation alone may one calculate, through how few generations cells need pass in order to be numbered by millions, without a death necessary to the process. ”

“ Lastly, we at once perceive that the ratios which happen to exist between these various factors, the ratio for instance between the growth-factor and the rate of angular revolution, will give us endless possibilities of permutation of form. For instance (7) with a given velocity of vectorial rotation, a certain rate of growth in the generating curve will give us a spiral shell of which each successive whorl will just touch its predecessor and no more; with a slower growth-factor, the whorls will stand asunder, as in a ram’s horn ”

“ We have to find the number of permutations such that exactly m of the letters are in places they originally occupied. We have the particular redundant product
(ax1 + x2 + ... + xn) ξ1 (x1 + ax2 + ... + xn) ξ2 ... (x1 + x2 + ... + axn) ξn
in which the sought number is the coefficient of am xξ11 xξ22 ... xξnn. The true generating function is derived from the determinant
and has the form
It is clear that a large class of problems in permutations can be solved in a similar manner, viz. by giving special values to the elements of the determinant of the matrix. ”

“ Time is the condition of all our thoughts, space of all our actions, and therefore both in art and science we are to a great extent concerned with permutations. Language, for instance, treats different permutations of letters as having different meanings.
Permutations of things are far more numerous than combinations of those things, for the obvious reason that each distinct thing is regarded differently according to its place. ”

“ The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.
The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives special importance to such groups. ”

“ In some questions the number of permutations may be restricted and reduced by various conditions. Some things in a group may be undistinguishable from others, so that change of order will produce no difference. ”

“ As to that which is moved, we have shown that it is primary and simple and ungenerated and indestructible and generally unchanging; and the mover has an even better right to these attributes. It is the primary that moves the primary, the simple the simple, the indestructible and ungenerated that which is indestructible and ungenerated. Since then that which is moved, being a body, is nevertheless unchanging, how should the mover, which is incorporeal, be changed? ”

“ Do they constitute other kinds of movement than those already noticed, generation and destruction, growth and decrease, movement of displacement and alteration? Shall composition and decomposition be reduced to some one of these kinds of motion, or shall we look at this process inversely? If composition consist in approximating one thing to another, and in joining them together; and if, on the other hand, decomposition consist in separating the things which were joined, we have here only two movements of displacement, a uniting, and a separating one. ”

“ A perusal of the arguments against Perpetual Motion made by thinking men with scientific minds even though long before the thorough establishment of the doctrine of transmutation and Conservation of Energy, discloses the fact that those arguments in fact depend finally on the principle now known and designated Conservation of Energy. ”

“ If transmutation, when thus tested, has decidedly the advantage over progression and yet is comparatively in disfavour, we may reasonably suspect that its reception is retarded, not so much by its own inherent demerits, as by some apprehended consequences which it is supposed to involve and which run counter to our preconceived opinions. ”

“ But with the next Element added to our resources, in the next Round, Permeability will become so manifest a characteristic of matter, that the densest forms of this Round will seem to man's perceptions as obstructive to him as a thick fog, and no more. ”

“ In this case we may distinguish two stages or degrees in the transposition: first, before we think of our own pulling, we say the object itself pulls; in the first transposition we say it pulls against us, its pull is the counterpart or rival of ours but it is still conceived in the same direct terms of effort; and in the second transposition this intermittent effort is made potential or slumbering in what we call strength or force. ”

“ Hitherto, no rival hypothesis has been proposed as a substitute for the doctrine of transmutation; for 'independent creation,' as it is often termed, or the direct intervention of the Supreme Cause, must simply be considered as an avowal that we deem the question to lie beyond the domain of science. ”

“ There is, therefore, every reason to believe,—since repulsion, as the fact of forcible compression shews, takes place while the particles of bodies are still at a certain distance,—that the motion produced in one body by another, and ascribed to immediate impulse, is produced, without actual contact, by this mutual repulsion, as it is called, of the bodies when brought within a certain invisible degree of vicinity to each other ”

“ One of the most common fallacies by which the superficial projectors of machines for obtaining a perpetual motion have been deluded, has arisen from imagining that any number of weights ascending by a certain path on one side of the center of motion, and descending on the other at a greater distance, must cause a constant preponderance on the side of the descent ”

“ We maintain, therefore, that the presentation of sense, the continuum or manifold, or what you will, consists in the transmutations of a real substance itself unextended and unperceived; that the laws of these transmutations are what constitute the geometric all-containing Space; that at a point in this real energetic system organically related to the intelligent self, the transmutations occurring there constitute the individual's sensible experience ”

“ It may be thought almost paradoxical that writers who are most in favour of transmutation (Mr. C. Darwin and Dr. J. Hooker, for example) are nevertheless among those who are most cautious, and one would say timid, in their mode of espousing the doctrine of progression; while, on the other hand, the most zealous advocates of progression are oftener than not very vehement opponents of transmutation. We might have anticipated a contrary leaning on the part of both, for to what does the theory of progression point? ”

“ It is precisely under this aspect that transmutation is of such inestimable value to morphology; it enables us to rise from the bare empirical knowledge of numberless isolated facts to a philosophical conception of their efficient causes. ”

“ Transmutation is a great natural Mystery, Metallick, and not contrary to the Course of Nature, nor repugnant to the Order of GOD, as many men of it do falsly judge. For imperfect Metals, are changed neither into Gold, nor into Silver, without this Stone of Philosophers. ”

“ The transmutation of one sort of atom into another is therefore not a mere visionary fancy, but an established fact; and although our laboratory experiments in this direction may not as yet have gone very far, they have gone far enough to show that a Law of Transmutation does exist in Nature. ”

“ The transmutation theory insists that they owe their origin to gradual transformations, and are not, therefore, the result of distinct creative acts. All agree, however, that we arrive at a lower stratum where no trace of life is to be found. ”

“ It is complete in itself, and has no need to stretch out indefinitely into an empty space that does not exist. Hence, too, it is spherical. It is equally real in every direction, and the sphere is the only form that meets this condition. Any other would be in one direction more than in another.
89.Parmenides, the father of materialism. To sum up. What is, is a finite, spherical, motionless corporeal plenum, and there is nothing beyond it. The appearances of multiplicity and motion, empty space and time, are illusions. ”

“ It may be demonstrated, indeed, to all those capable of reasoning in a sound manner on those sciences, that a perpetual motion is impossible: for, to be possible, it is necessary that the effect should become alternately the cause, and the cause the effect. It would be necessary, for example, that a weight raised to a certain height by another weight, should in its turn raise the second weight to the height from which it descended. ”

“ The perpetual motion has been the quicksand of mechanicians, as the quadrature of the circle, the trisection of an angle, etc., have been that of geometricians: and as those who pretend to have discovered the solution of the latter problems are in general persons scarcely acquainted with the principles of geometry, those who search for, or imagine they have found, the perpetual motion, are always men to whom the most certain and invariable truths of mechanics are unknown. ”

“ CHAP. XIV.—The seeming probability of effecting a Continual Motion by Solid Weights in a Hollow Wheel or Sphere.
The third way whereby the making of a perpetual motion has been attempted is by the Natural Affection of Gravity; when the heaviness of several bodies is so contrived, that the same motion which they give in their descent, may be able to carry them up again. ”

“ Within these twelve years past the mathematicians who deny the possibility of a perpetual motion have been repeatedly and publicly called upon, both in the foreign and English prints, to produce a single instance of these demonstrations. They have not done it. They might have produced, indeed, the demonstrations of Huygens, De la Hire, and others to prove, as Desaguliers very properly expresses it the fallacy of the schemes of most of the pretenders to the perpetual motion. ”

“ Notwithstanding the fact that a countless number of devices for the attainment of Perpetual Motion have been proclaimed and exhibited, it is to be supposed that those actually proclaimed and brought to light constitute but an infinitesimally small proportion of those actually made. ”

“ In other words, these men in reasoning on specific problems presented to them, and on the problem of Perpetual Motion in particular, appear to have appreciated and applied in their reasonings, the principle of Conservation of Energy. ”

“ But in these things Nature hath her singular Privilege granted by GOD, for the sake of Man. I speak this of Transmutation, that you may understand, that the foolish Philosopher Aristotle, was not well grounded in his Philosophy, but was meerly fatuate. Now I will give you the Receipt, that you may know how, in all places of Germany, to make Copper of Iron; whence we may easily gather, what the power of Transmutation is, and that many other Transmutations may be made ”

“ And in Book viii. we have such propositions as this (c. 10) : “Demonstration that a finite mover (movens) cannot move anything in an infinite time.” This is illustrated by a diagram in which two hands are represented as engaged in moving a whole sphere, and one hand in moving a hemisphere.
This mode of representing force, in diagrams illustrative of mechanical reasonings, by human hands pushing, pulling, and the like, is still employed in elementary books. Probably this is the first example of such a mode of representation.
Roger Bacon. ”