Fincke Principle Least Action
By Bernhardt FINCKE, M.D., BROOKLYN, N.Y.
Presented by Sylvain Cazalet
"Lorsqu'il arrive quelque changement dans la Nature, la quantité d'action nécessaire pour ce changement est la plus petite qu'il soit possible" (Oeuvres de M. de Maupertuis Lyon 1756 Tome IV p 36) i.e. when a change occurs in nature, the quantity of action necessary for the change is the least possible (Fincke High Pot. and Hom. Phila 1865 p 18).
The principle of the least quantity of action has a history which promises to be an important element in the history of culture. For our present purpose of showing the necessity of such a principle since the introduction of potentiation in Homoeopathics, it may suffice to give a short sketch, perusing Euler "sur le principe de la moindre actio" in the histoire de l'Academy Royale des sciences et belles-lettres. Annee 1752 Tom VII. p. 199.
The lex parsimoniae, as this principle is called, is extremely old. Aristotle mentions it and many others do so after him, as e.g. Isocrates who said: "the small forces produce the motion of the large masses"-Ptolemy, Fermat, Malebranche, s'Gravesande, Leibnitz, Wolff and others, until Maupertuis determined the law for the first time in a general formula.
The ancients observed, that nature never does anything without design and for naught, and selects the nearest paths, but they did not prove it. Ptolemy said, the rays of light come to us in straight lines, because that is the shortest path, and he deduced from the reflexion of light, that light passes from any point in its course before incidence, to any other in its reflected course, by the shortest paths, and in the least time, its velocity being uniform and equal before and after reflexion. (s. Arago Biographies translated by Smyth. Powell & Grant, Boston. Ticknor & Fields 1859 Sec. II. p. 189. Note).
Others assumed the circle to be the shortest line perhaps, because they knew from the geometers, that in the surface of the sphere, the arcs of the great circles were the shortest lines from two points. This they transferred to the heavenly bodies which at that time were thought to move in circles. Since they move however in the most transcendent curves, the opinion that nature affectates straight or circular lines is condemned, and the proposition, that nature everywhere wants a minimum, turns out quite the reverse. This no dought has caused Descartes and his followers to reject the doctrine of final causes in philosophy and they contended, that in all phenomena of nature much more an extreme inconstancy is to be discerned, than a certain and universal law.
With all that opposition the principle lived, supported by certain cases e.g. in the reflexion of light, but it did not hold good in the refraction of light.
Though, therefore, it is clear that in the direct and reflected motion of light nature really takes the shortest route, the mere computation, however, makes it apparent that the law could not consist in the selection of the shortest path, if not an infinity of other phenomena should be contrary to it. Another minimum then, the length of the path must be adopted, just so in the motion of direct as of reflected light, which in this case is merged into the shortest path, a minimum which would also find application at the refraction of light. After such considerations, Fermat determined, that the light in its motion selects not so much the shortest route, as that one by which it would travel in the shortest time from one point to the other. Or, he assumed that the light in the same medium moves with uniform velocity, so that in one medium the time were proportioned to the paths described, and that in direct or reflected motion the shortest route must necessarily be that one which was described in the shortest time: but that in transparent mediums such as air, water, glass, the velocity of the light were also different, much greater in the thinner medium such as the air, and less in the denser medium, such as glass: a supposition which seened to be in sufficient accordance with nature. And by this hypothesis which was attacked fiercely by Descartes, after overcoming the greatest difficulties in the calculation, he succeeded in explaining the phenomena of refraction and he found that the sines of the angles of coincidence and refraction are proportioned to each other in a definite eay, that is, that the sum of the times or of the spaces divided by the velocities is a minimum.
But Descartes, proscribing the final causes, explained the refraction of light by the laws of the shock of the bodies, comparing the rays of light to a continued series of fine globules, and he arrived at the same law of refraction, as ecperience shows, in a different way. But he differed from Fermat in that the light moves in the denser medium quicker, than in a thinner, quite the reverse of Fermat's velocity in glass, than in air, be owing to the lesser resistance the priciples of his philosophy. Considering, however, through the greatest distances, this theory is obviously inconsistent, because such a notion is not in accordance with the idea of velocity.
Though Fermat's proposition was adopted by most philosophers and mathematicians who did not adhere to Descartes' opinion, Fermat could not be considered to be the discoverer of a universal law which was pursued everywhere by nature. He had only noticed, that the principle of the least time extend upon the motion of light and no farther.
Gottfried Wilhelm von Leibniz (1646-1716) likewise has tried to subvert Fermat's explanation. In order to explain the refraction of light he has proposed to recall the final causes rejected by Descartes and to give again the explanation which Descartes, contrary to Fermat, had derived from the shock of the bodies. He commenced denying that nature select the shortest route of the paths of the least time, but he maintained that it select the easiest way, which should not be confounded with each other. The resistance serves to measure this easiest way, the resistance with which the light passes through the transparent mediums and he supposes that this resistance is different in different mediums. He even lays down that in dense mediums like water and glass the resistance is greater than in the air and in the thinner mediums, which seems to favor Fermat's opinion. In this presupposition Leibnitz considers the difficulty which light finds on passing through a medium, and he computes this difficulty by the path multiplied by the resistance. The ray always pursues that route in which the sum of the computed difficulties is the least; and according to this method de maximis et minimis he finds the rule which is confirmed by the experience. But though at first sight this explanation agrees well enough with Fermat's, yet afterward it is interpreted with such a singular subtlety that it becomes diametrically opposed to it and confirms the one advanced by Descartes. For though Leibnitz has taken the resistance of glass as being greater as that of the air, yet he contends that the light moves quicker in the glass than in the air and that the resistance of the glass is the greater one, which is certainly a paradox. The explanation of Leibnitz concurs with the one of Descartes in as much as both attribute to the light a greater velocity in the denser medium, but is differs much by the cause which each philosopher assigns to account for the greater velocity, because Descartes believed the resistance in the denser medium being lesser, while Leibnitz conceived it to be greater. Be that as it may, Leibnitz has never applied his principle of the easiest way to any other case, nor has he taught how this difficulty of which he had to make a minimum should be computed.
Leibnitz' great disciple Wolff, in the explanation of the refraction of light, renders the explanation of Fermat word for word in his Elements of Dioptrics. For in his 2. problem $35 he, supposing that the velocity of light in different media be different, greater in the thinner, lesser in the thicker one, seeks the time which a ray wants to pass through a path from one point to another in another medium. From this he concludes that, since nature always acts in the shortest way, this time must be the least possible.
Newton in his Optics, has a principle of the least resistance and in his Principia 2. book, he determines what must be the meridian curve of a solid of revolution in order that the resistance experienced in that body in the direction of its axis may be the least possible.
Franklin touched upon the principle of the least action in his happy common sense way when he said: if two suns were hung up in space and if upon one of them would alight a fly, the suns would be moved.
The discovery of s'Gravesande consists in that, if two inelastic bodies meet in such a manner that they are at rest after the shock, the sum of the living force before the shock is the least one, if it is assumed that the relative velocity remains the same.
This is about all that was known until the time when Maupertuis pronounced the Law of the Least Action as a universal principle from which all other principles naturally flow, and next to it is the Principle of Rest or Equilibrium as we shall see hereafter.
Maupertuis was a well educated, elegant French nobleman, who was first musketeer, then captain of the dragoons in France, and already at the age of twenty-five years in 1723 he was received into the Royal Academy, of sciences in Paris. He then went to London where he was received as member of the Royal Society, and was among the first who raised his voice in favor of the Newtonian philosophy against Descartes. He then, attracted by the celebrity of John Bernoulli, went to him in Basil, in company with Clairaut and there studied the mysteries of the new analysis. After his return he associated himself with La Condamine and Voltaire, who under his auspices studied the Newtonian philosophy in order, to treat of it in a proper and competent manner in his "Elements de la philosophie de Newton" a treatise which though of inferior scientific value has exerted a great and wholesome influence upon the acceptation of Newton's opinions on the continent. It was at that time that Maupertuis made the acquaintance of Koenig, who taught Mathematics to Madame Du Chatelet on the recommendation of Voltaire.
In 1736 Maupertuis was sent by the French Government to Lapland in order to measure a degree of the meridian for the purpose of ascertaining the figure of the earth. He was accompanied by Clairaut, Camus, Monnier, Outhier and Celsius. It was a daring enterprise as may be judged from the history of the expedition. The cold was at one time so extreme that the thermometer fell 37 degrees below zero. Nothing but brandy remained liquid, and in drinking it the lips would stick to the vessel containing it. Yet Maupertuis and his associates did their task very creditably. Maupertuis was celebrated through all Europe and became a member of the great Academies of Sciences in Europe.
Voltaire placed under his portrait the lines: "Le globe mal connu, qu'il a su mesurer, Devient un monument ou sa gloire se fonde; Son sort est de fixer la figure du monde, De lui plaire et de l'eclairer." i.e. the globe little known which he knew how to measure, becomes a monument of his fame. His destiny is to determine the figure of the earth, to be its favorite and to enlighten it.
The flattening of the poles suggested by Newton was now experimentally proved by Maupertuis' expedition.
In 1740 Maupertuis, invited by Frederic the Great, went to Berlin and thence to the field with the king in the seven-years war. At the battle of Mollwitz, Maupertuis was captured by the Austrian huzzars who plundered him, and among other valuables, took a watch of the celebrated Graham of London from him; a companion of his arctic voyage. Maupertuis was well received by the Emperor and the Empress Maria Theresia who returned to him another similar watch of Graham set with diamonds with the remark that the huzzars in plundering him only meant a joke, and that they send him his watch back again. He soon was exchanged and went back to Berlin.
In 1742 Maupertuis was received as a member in the Academy of Sciences of Paris.
In 1743 Maupertuis was received as a member in the Academy of France, the first instance of one person being a member of both academies of Paris at the same time. He was present at the siege of Fribourg, and was ordered to bring the news of victory to the French king.
In 1744 Maupertuis returned to Berlin and married an amiable young lady, a relative of the Minister of State, von Bork. In this year, April 15th he announced in the public session of the Academy of France, the Law of the Least Action as a universal principle. Shortlyafter this Euler wrote his: "Methodus inveniendi lineas curvas maximi minimive proprietat gaudentes" which contained a verification of this principle. In the memoir on the subject Maupertuis gavve the rigorous demonstration, deducing from this principle the Law of Motion and Rest and applying it to the refraction of light. The papers were printed in the memoirs of the Academy of France and in those of Berlin.
In 1746 Maupertuis was installed as President of the Royal Academy of Sciences in Berlin and adorned with the order of merit. The French king Louis XV made him pensionnaire veteran of the Academy of Paris with a pension of 4,000 liv.
Though fortunate in his enterprises, of studious habits, loaded with favors of kings and savans, and happily married, still, being of a hypochondriac disposition, M. felt miserable on the following account.
In 1751 Professor Samuel König of Franeker (1712-1757), the former pupil of Maupertuis, published in the Acta eruditorum of Leipzig, a letter from Leibnitz to Hermann, which was said to contain already the principle of the least action. Maupertuis considered this publication as an imputation of plagiarism and arraigned König as a member of the Berlin Academy, before this learned body. A commission of five was appointed and Koenig was called to produce the letter. On examining it it was found that the passage relating to the matter was forged. (See Memoires of the Royal Academy of Sciences 1752 p. 52 in the "expose concernant l'examen de la lettre de Mr. de Leibnitz, allegue par Mr. le Professeur Koening dans le mois de March 1751 des actes de Leipzig a l'occasion de la moindre action)."
Koenig then was expelled from the Academy and due justice done to Maupertuis. Leonhard Euler (1707-1783) who independently of Maupertuis, as it seems, had arrived at the same principle in his "Methodus" and therefore should have had some claim, if he had not come a little later and if he had at the sme time pronounced the universality of the principle, which he did not, defended Maupertuis and wrote several interesting lucid style. So Frederic the king, also wrote in his behalf. But Voltaire, another former friend and pupil of Maupertuis attacked him recklessly, with libellous papers, among which the "Diatribe du docteur Akakia medicin du Pape" was the most cutting. Frederic ordered the whole edition of this libel to be brought into his room. There he burned it with his own hands in the chimney. But unfortunately one copy found its way to Holland, and was there reprinted. Frederic then ordered this new edition to be burned by the hangman in the public places of Berlin, which was actually done on December 24, 1752. This was too much for Voltaire. He sends in his key and cross and resigns his pension. The king does not accept it and returns the insignia. There is a temporary lull of apparent reconciliation. But Voltaire wants to go to Plombieres, and after many dubious refusals and delays the king gives him the permission to go, but on the condition of his returning. Voltaire, arrived in Leipzig, receives from Maupertuis a cartel reidicule which is responded to in trenchant sarcasms. Voltaire goes to Frankfort on the Main. On the point of leaving this city, three persons, in the name of the king, detain him and ask for a volume of poetry of Frederic, given to him as a token of friendship. It is not present, but lays with other effects in Leipzig. Voltaire is forced to sign a paper that he will not leave Frankfort till the book is procured from Leipzig. He, with his niece, Madame Denis and his secretary, is lodged in a miserable tavern, his trunks are searched, they must even empty their pockets openly. The three victims are separated and watched by soldiers with bayonets. After a few days the order comes to release the prisoners. Their baggage is nearly all returned and Voltaire must pay the expenses of the whole fray. (See Carlyle, Frederic the Great). He travelled to and fro for fibe years after, till he settled down at Ferney, where he lived a useful life, full of splendor too, for twenty years longer.
Not so Maupertuis. From his fatigues on his arctic voyage his health had been greatly impaired and he was spitting blood twelve years before he died. But after this scandal which had hit him in his most vulnerable part, his honor, he never fairly rallied. He went several times to France and St. Malo, travelling for his health. Finally he came to his old friend Daniel Bernoulli in Basil, in whose house he expired July 27, 1759, 61 years old, attended by La Condamine. His works hae appeared in Lyon in four volume, quarto in 1752, and a translation of his Essay de Cosmologie into German has been published by Mylius in Berlin 1751.
Such were the throes of the birth of the Principle of the Least Action. What had moved Koenig and Voltaire to act so ignominiously toward their former friend, associate and teacher, is not difficult to say. It was probably nothing but the "invidia pessima" of which scholars, savans and artists are no less free than doctors of medicine of which it is proverbially predicated as of people even of lesser attainments. Maupertuis was a fine gentleman of nable birth, of much influence, the daily companion of the great king, somewhat sensitive, and somewhat vain and ambitious, and subject to hypochondria, but "of generous mind and nable intentions" according to Daniel Bernoulli's evidence. Voltaire had been previously the favorite of the king and very likely felt his influence decrease. Being ever of a satirical and malicious disposition, he growing older, took offence at the growing splendor of the president of the Academy, a post of honor which possibly might have been the option of himself.
So the unfortunate calumniation was concocted which had such a sad effect upon them all, offenders and offended. As to Koenig, a passage in a letter of Daniel Bernoulli to Euler, June 13, 1744, may throw some light upon the character of this forger. It appears that Koenig was banished from his native land Berne on account of some "mutineries" imputed to him. Bernoulli now recommends him to Euler for the Academy of Berlin "a tout prix," nay Bernoulli says, Euler would do a work of charity if he would employ Koenig some way or other. This is the same Bernoulli in whose arms Maupertuis expired.
It must be considered that upon Maupertuis' side stood such men as Frederic the Great, Euler, Lagrange, Daniel Bernoulli, and all the other Academicians. They all respected and loved him and have shown as much by their deeds and testimony. -No doubt the quarrel terminating so fatally has done injury to the promulgation and acceptance of the principle in question. Everybody was disgusted with the matter which was a disgrace to a world-renowned scholar, and many wounds were inflicted which needed time to heal up. When this time came, the persons concerned had either died, or grown old, or were forgotten, and the principle nver fairly came to a proper valuation notwithstanding its having been sustained by the most eminent and competent minds.
In the meantime the rise of the physical sciences, and especially the birth of chemistry had, to be sure, shown the necessity of guiding principles, but full of the new developments and discoveries, a theory was sufficient which construed matter out of ready made indivisible atoms which were movved by forces made to order mathematically, and so produced the experimental and experiential phenomena which was all that was needed for the present. Now after the experiments and experiences have accumulated and increased to such a mass that a new deduction of proper principles and classification of the facts to be registered under them is redered possible, the pure phoronomic laws assume their right and authority, and point to a Universal Law of Motion contained in the Principle of the Least Action of Maupertuis.
With the so-called Laws of Motion of Sir Isaac Newton (1643-1727), motion is inconceivable, because they are strictly the Laws of Rest. The first law is the Law of Inertia, as it is improperly called, but really is that of self-preservation based upon the principium identitatis.
The second law is the Law of equivalence of motion depending upon the third law, which expresses the mutuality of action, both, therefore, being exponents of the proportionality of motion, all three lead to the expression of the equilibrium rather than to that of motion.
The principle of the Virtual Velocities of Lagrange presupposing an infinitesimal motion = o, in order to demonstrate the equilibrium, is a mode of rendering part of Maupertuis' principle, but cannot likewise be considered a principle of motion. It is an infintiely small motion which causes Lagrange to construe the equilibrium by itself, but improperly. The difference between virtual and real is, that the former is only thought, but this is actual and it is that part of the overpoise which occurs in the first minimal moment of space and time, and with the minimal force, for whatever exceeds it, is already called real. Now, they say, an infinitesimal quantity is in comparison with a finite = o. Therefore all infinitesimal quantities which compose the virtual velocity = o in comparison with the real velocity which actually disturbs the equilibrium. Therefore the principle does not constue the equilibrium out of itself, but out of the motion which is opposed to it, for it borrows forces from the dynamics and makes them = o. This is a contradiction in itself. In other words: two bodies are equal to each other if their irtual movements are equal to zero. Or, two bodies are equal to each other if an infinitesimal force would moe them through infinitesimal sapace and time in inverse ratio. Voluntarily a difference is added and presumed that, if it be taken away again, it is as it was before. Therefore the principle of the Virtual Velocities is a principle which only hides the uniersal principle of the least action, being merely an application of it to the equilibrium.
The conservation of forces is another principle of the equilibrium from another point of view. It says forces can not be destroyed or created as little as matter, they only can neutalize, equalize each other. It shows the equilibrium between the forces gained and the forces lost, between the body moing and the body moved. It is a logical, and not a physical principle, as Faraday lretends to say. It says nothing about the motion itself which causes the equilibrium. It walks oer the first step and is content with the result expressed in the analytic formula of equation.
Principles are all logical and therefore metaphysical. Metaphysics is nothing more nor less than the science of the comprehensibility of physics, and logic is the mental instrument which mediates the process of cogitation. So Faraday is right in that he does not see a difference in Metaphysics and Physics. They are both essentially the same only Physics renders the facts to build up Metaphysics which in its turn helps on Physics in its investigations and observations. Metaphysics is by no means Mystics, nor fancy, nor anything which allows philosophizing without due ground of correct experience fortified by experiment and observation. Therefore so-called physical principles are of necessity metaphysical, but the conservation of forces is neither physical nor metaphysical, it is only a logical expression of physical phenomena which may also be differently expressed e.g. as equation in mathematics and if you please the very phenomena of conversion of forces into one another and of matter into one another are such other expressions. All of these expressions, however, do not make the principle of conservation of force important on account of the conservation. That nothing is lost in this world, that neither matter nor force can be destroyed or created, that forces can be reproduced by similar forces, are observations from experience but not, properly speaking, warrenting the principle of conservation. In this term is lurking the conception of teleology which is said to be foreign to genuine science, and we do not better the matter by endowing it with the name of a principle. Nay, the very principle of conservation of forces itself is only another impersonation of the principle of the least action, for its equivalent nature shows clearly that the conseration is as in all equivalence the least possible action in the given case. Such facts as mentioned above may eventually lead and they actually do lead to a general principle which we have found in the Universal Assimilation, and so we must consider the conservation of forces, and the correlation of forces, and the conversion of forces as stepping stones to the higher generalization of Homoeosis.
The so-called principle of D'Alembert (Jean Le Rond d'Alembert (1717-1783)): all the motions that have been lost or gained by the different bodies of a system by their reaction, necessarily balance each other under the condition of the connection which characterize the proposed system (Comie, positive philosophy) is likewise no dynamical principle but a statical one, as it relates to the equilibrium of various equilibria.
So among all the hitherto accepted dynamical principles we have really no true dynamical principle, if we do not adopt Maupertuis' Least Action.
It is difficult to understand that this principle should have met on one part with such oppostion and on the other part with such neglect, if we consider how lucidly and plainly it was at once demonstrated by its discoverer. Had Fermat introduced the element of time, Maupertuis brought in the element of velocity and reached thereby a perfection which makes it applicable to all cases of motion, and allows to constuct from it the Law of Rest or Equilibrium which Lagrange very aptly defines as "the result of the destruction of the several forces which combat each other, and which destroy reciprocally the action which they exercise upon each other (Mechan. analyt p. 2). By these means all statical questions are reduced to dynamics which concurs with the truth, because there is no absolute rest for anything, as there is nothing absolute in anything.
Perhaps the bery simplicity of the demonstration of our principle prevented its general acceptation. Motion and rest follow equally from this general principle, and the motion of hard and elastic bodies as well as all the rest of bodies, become so many problems to be solved by it.
From time to time only the principle of the least action has been subjected to new treatment, and has been acknowledged to be true and useful. So we find in Fischer's Geschichte der Kunste und Wissenschaften, Goettingen 1803 Vol. IV p. 95:
"The proposition in itself is true. If Leibnitz indeed knew it, yet he adopted quite a different principle in explaining the law of refraction of light. Maupertuis, therfore, has always the merit, to have discovered this truth, and to have developed it from natural laws with much acumen."
In the Dictionnaire des Sciences mathematiques, Paris 1838, we find: "Lagrange with the aid of the calculus of variations which he has discovered, has demonstrated in the most rigorous and elegant manner, that the principle" (of Maupertuis) "extended to every system of bodies under the laws of attraction, and acting otherwise upon each other in some certain way. It is especially to that beautiful proposition of lagrange, that the name of the principle of the least action has been attched to Mechanics."
Joseph-Louis Lagrange (1736-1813) in his "essai d'une nouvelle methode pour determiner les maxima et minima des formales integrales indefinies" laid the foundation to the calculus of variations which was afterward perfected and dilated by other analyticians. This calculus, then, was an offspring of Maupertuis' Principle of the Least Action. He also called it so and it is contained in the formula: in a system of moving bodies the sum of the products of the masses of the bodies by the integral of the products of the velocities, and the elements of the spaces passed over, is constantly a maxium or minimum.
Shortly after the demonstration of the principle of the Least Action by Maupertuis. Euler wrote his "Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes." In the supplement attached to it, this illustious geometer demonstrated, that in the trajectories which the bodies described about central forces the velocites multiplied into the element of the curve, is always a minimum. Euler himself says, that the product as he considers it, presents the action itself as Maupertuis defines it, and that this discovery has been made after the appearance of the Maupertuisian principle. He adds to this very modestly, that he had not believed to find a more extended principle, content to have detected this beautiful property in the movements about centres of forces.
Euler, in a letter to Goldbach 1752 Aug. 5, gives to Maupertuis his full due when he says: "What your honor please to ask about the formulas given by M. de Maupertuis on the leges motus no doubt will concern those by which he determines the regulas communicationis motus in conflictn corporum tam elasticorum quam non elasticorum; because they are the same as those long known before, they also agree with the Leibnitzians. But as the principium itself is concerned, from which M. de Maupertuis derives these regulas, such indeed is entirely new. For, though it has been maintained before, that nature act via facillima, yet neither Leibnitz not anybody else has shoen which were that very quantity which is a minimum in the operationibu naturae. M. de Maupertuis calls this quantity the quantitatem actionis, and determines the same by the product of the mass of the velocity and of the spatium, and derives there from very beautifully not only the regulas motus, but also other things.
"I also long before demonstrated, that in motibus corporum coelestium always the formula SMv ds be a minimum; where M signifies the massam, v the celeritatem and ds the spatium percursum. Therefore M v ds is the quantitas elementaris and S M v ds the totalis which consequently according to M. de Maupertuis must be a minimum. (Fuss. Corresp. St. Pet. 1843 v. I, p. 580)."
"The high opinion which the celebrated Daniel Bernoulli (1700-1782) entertained of Maupertuis appears from his letter to Euler d. d. July 7, 1745 (Fuss u. s. v. I, p. 577). M. Mauperuis according to his last letters is going to Berlin within three of four weeks, in order to enter upon the office of President of the Academy. This gives me the hope, that everything will go well with the Academy, because M. Maupertuis is the favorite of the whole court and will certainly make it a point of honor, to make the Academy prosper; he has a generous mind and noble intentions."
The principle of the Least Action, therefore, as we have seen led under the analytical power of Lagrange to the foundation of the calculus of Variations, afterward perfected by other analyticians.
Professor Peirce, the greatest American mathematician, fully acknowledges the grandeur and universality of the principle of the Least Action inhis Analytical Mechanics (Physic. and Celestial Mechanics, Boston. Little, Brown & Co., 1855 p. 316):
"When in the case of the fixed forces of nature, the initial and final positions of the system are given, as well as the intial power with which the system is moving, the variation of the characteristic function vanishes, and, therefore, the function is generally a maximum or a minimum. The action expended by the system, which is measured by this function, is also a maximum or a minimum; or in other words, the course by which the total expenditure of action is a macimum or a minimum. But it is obvious, that in most cases and always when the paths in which the various bodies move, cannot correspond to the macimum of expended action, and, therefore, in most cases the system moves from its given initial to its final position with the least possible expenditure of action."
"Many examples can, however, be given, in which the expended action is, in some of its elements a maximum, although, even in those cases, the expenditure is a minimum at each instant or for any sufficient short portion of the paths of the bodies."
"This principle of the least action was first deduced by maupertuis throught an a priori argument from the general attributes of Deity, which he thought to demand the utmost economy in the use of the powers of nature, and to permit no needless expenditure or any waste of action. This grand proposition which was announced by its illustrious author with the seriousness and reverence of a true philosopher, is the more remarkable that, deried from purely metaphysical doctrines, and taken in combination with the law of power, which likewise reposes directly upon a metaphysical basis, it leads at once to the usual form of the dynamical equations."
In Knight's Encyclopaedia, likewise, we find a vindication of the principle and the adhortation, that the student might look for further explanations in full treatises. "The principle of Least Action is the equivalent of the expression, that the integral of the product of the vis viva of a system by the element of time is in general a minimum."
Strange it is that Knight considers the principle in question as originating with Maupertuis in a limited sense, whilst the Law of Rest of Maupertuis himself and the Virtual Velocities and the Variations of Lagrange, are afterward merely derived from it, giving the very universal principle in a limited sense, all of which are merely applications upon the equilibrium. This, however, is only a kind of relative motion under distinct limitation. We infer from that circumstance, that the universal character of the principle of the Least Action, given to it by its discoverer, is not yet properly understood.
This is confirmed by another modern demonstration of this principle by Dienger (Archiv d Math. and Phyik v-Grunert 1864 Vol. 41, p 299), who simply falls back upon the rules of the calculus of Variations and flatters himself to have deprivved it of metaphysical subtleties by making it a mere sequela of the general propositions, excluding thereby every obscurity.
Euler, in his letters to a German princess (Leipzig 1769 Vol. 1, p. 263), gives a very good, clear and popular account of the principle in question. "If two bodies meet each other, so that without penetrating one another, they cannot remain in their state, the penetrability of both in like manner resists the permanence of this state, and by both in common the force is generated which hinders the penetration and the change of the state. In this case we say that both bodies act upon each other, and the force generated by their impenetrability is the cause of this mutual action. This force, therefore, acts also upon both bodies simultaneously, for since they should penetrate themselbes mutually, it repels them both and prevents in such a manner the penetration. It is, consequently, certain that the bodies can act upon each other, and it is said so much of the action of the bodies, e.g. if two billiard-balls shock each other, that this expression can not be unknown to your Highness. It must be remarked that this action extends no farther, than as far as their impenetrability suffers, and from that grows just such a force, as is necessary, to preent the penetration; in other words: such a force that every lesser one would no more suffice for this intention. A greater force of course would prevent also the penetration, but as soon as the bodies are no more in danger to penetrate themselves, so soon their impenetrability ceases to work; and the force, springing therefrom, consequently, must be the least possible which is just sufficient to prevent the penetration. If then the force is the least, its action, that is the change of the state produced thereby, must also be the least among all which are able to prevent the penetration, and if, therefore in the shock of two bodies, the continuation of its state becomes impossible, and from it a mutual action originates, this action is the least possible if the penetration is to be prevented. Here, Your Highness will find quite unexpectedly the foundation of the system of the Least Action of Maupertuis so much exalted and contested. He understands by it, that in all changes which take place in nature, the action produced by it is always the least possible. In the manner in which I have demonstrated this principle to Your Highness it is evidently founded in the nature of bodies, and all those are exceedingly wrong who deny it. But those do still more wrong who ridicule it. Your Highness will have seen, that certain persons who are not friends of Maupertuis seize every opportunity to make merry about the principium actionis minimce as also about the hole going as far as the center of the earth. But fortunately truth does not lose anything by it."
Ferdinand Jakob Redtenbacher (1809-1863) (Dynamiden-system, Mannheim 1857 p. 24) expresses the Law of the least Action, as follows, though evidently he is not aware of it.
"Very remarkable are these processes, which I will call dynamic metamorphoses or transmutations of motion."- He could habe called them just as well equivalents of motion analogous to Mayer-"Just as, namely, in the machines by the geometrically mechanical organization of its constituents a directly linear passage to and fro into a continual revolution, and the reverse, just so by proper influences the free motion of atoms in the bodies can be transferred to one another. From oscillations of ether and from oscillations of ether of a certain kind, ethereal oscillations of another kind, or by purely mechanical inactions (Einwiskungen) heat storm will furnish a striking evidence. I must candidly confess that it appears to me as if by these processes a remarkable mystery of nature was uncovered, and indicated how admirably simple the means are which nature uses, in order to attain its great universal purpose."
The eminent savant here touches the Universal Principle of Motion proclaimed by Maupertuis 113 years before in his Principle of the Least Action.
In the foregoing collection of the opinions of the most prominent scientific men, it is seen how the simple principle of Least Action needed the efforts of many centuries to lead finally to its clear enunciation by Maupertuis. But strange to say, the clearness of the conception is today as much obscured as a hundred years ago. Nay, even Maupertuis himself, who formulated it, failed to convey the characteristic universality which renders it the essential law of motion. We cannot follow the mathematical reasoning about this principle, which seems more to confuse it than to clear it up, but we utilize it for Homoeopathics by deducing from it the Principle of the Least Plus as the quantity of action necessary to produce any change in anture added on the positive or negative side - Additulum. If this Least Plus is acknowledged as the moving principle in the Universe in inanimate things (so-called), how much more is it applicable in animate beings, endowed with a sensitivity which calls for more refined medicine than the common old school offers. The remedies which in the course of sixty years have been developed from crude materials used for medicine, and from the dynamides, habve reached a fineness for which the term infinitesimals is only a compromise for our ignorance, since it recedes into the depth of minuteness which no man can fathom. But the action is there and the result of the action upon healthy and sick people, shows that the action is specific for each source from which it has been derived. The simple mechanical least action supplies force for labor to be performed in moving masses, from one place to another, and transferring forces geometrically in machinery to answer that purpose. The least action in natural processes produces the phenomena which are the objects of Physics, and the least action depending upon the assimilability of substances within infinitesimal limits, belongs to the department of chemics. The least action in organic bodies by which their organs carry on life is the prerogative of Biology. But in all these actions the least quantity is sufficient to turn the scale and induce the action and reaction without which no motion can take place, because action and reaction themselves are mediated by this Least Plus or Additulum which in itself is of no account, as it vanishes in the transference of it through the systems to which it is applied. Thus it is a pure metaphysical quantity which acts all things without ever being fixed as a real thing itself.
Thus the Homoeopathic potency, the Least Plus or Additulum of a medicine applied to the organism, either on its positive state of health or on its negative state of sickness, works the proving in the first instance, or the healing in the second, if selected according to the Homoeopathic law.
Thus the Similia of symptoms in the sick are equalized by the similia of the medicinal dose, if correctly selected, which is always a minimum, and there is no other way of healing, because in every case it is the least quantity of action which works the cure under the Law of the Similars. Ceterum censeo macrodosiam esse delendam.
INTERNATIONAL HAHNEMANNIAN ASSOCIATION 1897.
B. FINCKE, M.D.
Dr. Boger - I have only heard the latter part of that paper but I know the general purport pretty well from having read a similar paper by Dr. Fincke some time ago. It behooves us in all cases to be able to meet our allopathic friends with a foil to their arguments, and when they come with their multitudinous explanantions of the action of different remedies, it becomes us to be able to say something for ourselves. The explanation which I have found to be founded on the Organon, as well as to be unanswerable, is that our curative remedies depend upon a force acting in a similar direction to the disease force, and that no force moving in the universe is capable of any deflection in any direction by a force of equal magnitude and power acting in exactly the opposite direction; therefore, any force capable of changing a force already moving in one direction, necessarily moves in a similar direction. That is a fundamental principle in physics and in philosophy, which does not admit of any chang whatsoever. Therefore, everycure made, which is really a cure, is made along the line of potash, of the cm potency, or something else, every cure is along the line of similia, and that is an argument which no allopath will be able to refute. Every cure that has ever been made, or every cure that ever can be made, will necessarily be made, along that line. The method is a deflection of the disease force, moving it back into its normal channel, through a similar force which is found in the remedy.
Dr. Stow - I would like to ask what becomes of this power which we have been taught is somewhat antidotal. For instance, the vital force is disturbed by some particular force in a certain direction. It becomes necessary, therefore, in order to change this condition of the human economy, to annihilate this siease producing force, disturbing force, not to deflect it, because the mere act of deflecting turns it into another direction and leaves it in existence. That is the point I wish to have discussed here, at least to my satisfaction. We do not want to differ in regard to these question; we need to be a unit in describing the modus operandi of our remedies when we come to a discussion, either on paper, or verbally, with an allopathist. This paper is an extremely interesting one to me, but the trouble with me is this, that I need to take the paper and read it, and re-read it inorder to understand just what Dr. Fincke means. It is almost impossible here to follow out the thread of his thought, by simply listening to the reading of the article; hence I think it well to give this paper a conspicuous place in some conspicuous manner, so that we and others may take time to digest it, and there shall be no question about the real understanging of it, from beginning to end. When we get right down to the bottom of the question, it is this: Is it true that drugs tested upon the human economy, produce in certain potencies certain trains of symptoms? That we know to be true. Is it also true that when we find a certain train of symptoms in the human organism, not produced by any drug, but produced by some other force, that the selection of a remedy which will produce the greatest number of symptoms, corresponding with those presented in the case, will cure it? We absolutely know it does. That is true, and we look not so much to theory that may be offered, as we do to the fact brought out by the result. I would give more for those facts that are brought out in a case of pure homoeopathic practice, than for any amount of theory; yet it becomes necessary for us to place ourselves in such a position that we can meet the arguments of the scientific opponents of our school.
Dr. McLaren - That is quite true about deflecting the force; that is what the allopaths are doing all the time; they are always trying to deflect that disturbing force, and make more trouble by covering it up. My own impression is somewhat different from that of Dr. Boger, and it is this: that the disturbed vital force is moving in a certain direction, and you have got to get an exactly similar force, and the very mildest possible, the weakest possible, to move in exactly the opposite direction. When two express trains come together there is a terrible smash, but it takes only a very slight dynamic disturbance to make a man feel sick. That is something we cannot appreciate. The least bit of a fright, the least bit of a disturbance about how the man is going to meet his note tomorrow, may cause a sleepless night, and the man is sick. Such things are really imponderable, and bery slight in their force, and yet the results are great. We need the slightest possible force to counteract them, amd yet my impression is that it must be in the opposite direction. We have illustrations of it in nature. Off the coast of Norway, at a certain point where a cape juts out, the waves are exactly similar in height and number of vibrations, just opposite that point there is a perfectly dead calm. Oppposite forces of exact similarity, exact size and strength and wave height coalesce. It is the coalescing of the two opposing forces that produces the cure. That is my own interpretation of it and I give it for what it is worth.
Dr. Boger - I think the sole difference between the gentleman who has spoken and myself, is merely a difference in the apprehension of the term. The resultant of the two forces of equal magnitude and power, forces moving in opposite directions, is stasis, and stasis is death in every case, physically, mentally, or in any other condition, and the use of the term deflection, was perhaps unfortunate. A disease is, in itself, a deflection of the vital force. Perhaps it would be better to say that you are turning back again into the original channel, inflecting it, if you please-the dictionary perhaps would not sanction that way of using the word - but the only force capable of turning the vital force back into the normal channel, is one which moves in a direction similar to the disease force. That thought is carried through all nature, through physics and everywhere. That cannot be controverted, never can, never will.
Dr. James - I think there might probably be a misapprehension with regard to deflection, and I will merely suggest the idea. We have the parallelogram of forces, with which you are all familiar, where a force coming in one direction, striking an object that is situated there (illustrationg by the border of the blotter on the table), will send it in that line (along one border), and another force coming in another direction, at right angles, striking the same object, would send it over there (indicating), to the other border, in a line 90 degrees to the previous line. If both these forces are equal, and they come together at once, then the object takes a line between them, in this direction, which is 45 degrees to either of the previous lines, that is the resultant, which would be the diagonal of a square. If one of the forces be greater than the other, then it will be a parallelogram like this blotter. I have seen on the plains in the West a herd of cattle being driven, and one steer determined to leave the herd. One of the herders on horseback would chase him. He did not come opposite to him and stop him off suddenly; that was impossible. It meant death, of course, to the man who would attempt it. But with an instinctive understanding of the parallelogram of forces, the steer going in a direction away from the herd, the man went with him, and headed him around in a direction as nearly in line with the proper direction of the herd as possible, and he kept going around with him. This caused a deflection, which if it be analyzed, both directions taken together would be found to be a parallelogram of forces, and the steer's path a series of these reultants that finally produced the arc of a circle, and finally the steer came back to the herd.
This word resultant is the word that might be used as a means of understanding the application of the law of similars in the cure of a disease. The absolute collision between the horseman and the steer was impossible without death, but by following him around in the way I have described, he went around a series of resultants which finally became a circle, and the circle is a series of straight lines joined end to end.
So in the treatment of disease, positive opposition to the disease action causes disaster, as in the case of the herdsman and the steer. The law of similars enables us to travel with the disease, establishing a series of resultants which form the arc of a circle, and so the disease action is overcome, and there is a return to health.
Dr. Stow - That is all very interesting; it is a good geometrical proposition so far as the bodies of solids are concerned; we understand that. That is the geometry of force as developed by the contact od two or more bodies, coming together on different lines. We are not dealing with absolute matter; we are dealing with that quality of matter we call force. What is it? Have you any comprehension of it? I must say I have none. We simply know that there is force in it. Here I take a grain of dynamite, a little grain that I can hold on my finger. I place it on an anvil in a blacksmith's shop and take a hammer in my hand. We will suppose it is globular. It seems to be harmless, and is harmless unless some force be brought into operation against it to produce something else. I strike it with the hammer, and if I am not careful the hammer will be thrown from my hand by the reaction. What is done? A force is liberated that is sufficient to produce a shock. It is sufficient also, to throw or force the hammer from my hand. That is exactly what we want to get at; we wish to know absolutely how it comes to pass that forces acting in the human organism, similar forces, annihilate disease. Are we able to do it? I want to have that idea brought out by some of these thinkers.