The Principle of the
least action,
the universal law of motion and posology.
By Bernhardt
FINCKE, M.D., BROOKLYN, N.Y.
Presented by Sylvain
Cazalet
Pierre-Louis Moreau
de Maupertuis (1698-1759)“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.
René Descartes (1596-1650)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)Leibnitz 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.
Moreau de MaupertuisThis 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.
Maupertuis in Lapland
Gravure J.Ansseau, Source “Vie des savants illustrés”, Louis
Figuier, 1882.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.
Moreau de MaupertuisIn 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.
Samuel König (1712-1757)In 1751 Professor Koenig of Franeker, 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 Koenig 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).”
Leonhard Euler (1707-1783)Koenig then was
expelled from the Academy and due justice done to Maupertuis. Euler 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.
Sir Isaac Newton (1643-1727)With the so-called Laws
of Motion of Newton, 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.
Jean Le Rond d’Alembert
(1717-1783)The so-called principle
of D’Alembert: 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)Lagrange 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).”
Daniel Bernoulli (1700-1782)“The high opinion
which the celebrated Daniel Bernoulli
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.”
Redtenbacher Ferdinand Jakob
(1809-1863)Redtenbacher (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.DISCUSSION
Dr Bernhardt Fincke
(1821-1906)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 Cyrus Maxwell Boger
(1861-1935)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.Copyright ©
Sylvain Cazalet 2001
