September 2006
Vol. 1 No. 1
www.seattlelym.com/dynamis
EDITORS IN CHIEF
Peter Martinson
Riana St. Clasis
ASSISTANT EDITOR
Jason Ross
2
From the Editors
ART DIRECTOR
Chris Jadatz
4
Actually Relive History!
by Lyndon H. LaRouche, Jr.
LaRouche Youth Movement
Offices:
6
A Proposal to the World LaRouche Youth Movement on Taking the
Necessary Measurements With Which to Recreate the Discoveries of
Aristarchus and Eratosthenes
by Dennis Mason
8
Experimental Metaphysics: Leibniz’s Infinitesimal Captive
by Michael Kirsch and Aaron Yule
Boston, MA
617-350-0040
Detroit, MI
313-592-3945
Houston, TX
713-541-2907
33
The Inertia of Descartes’ Mind
by Jason Ross
35
A Very Useful Discovery Using Leibniz’s Calculus
by Peter Martinson
40
The New Biology
by Cecilia Quiroga and Thomas McGrath
Los Angeles, CA
323-259-1860
Oakland, CA
510-251-2518
Seattle, WA
206-417-2363
Washington, D.C.
202-232-6004
For submissions, questions, or
comments, please email
[email protected]
- or [email protected]
On the Cover
Albrecht Dürer,
Melencholia (1514).
Behind the shadows of
scientific investigation.
“…God, like one of our own architects, approached the task of constructing
the universe with order and pattern, and laid out the individual parts
accordingly, as if it were not art which imitated Nature, but God himself had
looked to the mode of building of Man who was to be.”
Johannes Kepler
Mysterium Cosmographicum
2
From the Editors
We face a dark moment, as if we were poised on a cliff,
overlooking this ghastly scene, and we felt ourselves slipping, as
if we were losing the privileged perspective of the viewer of
Peter Breugel the Elder’s The Triumph of Death, and felt
ourselves, instead, merging with the happenings on the ground.
Yet, were it not for the Moment of this moment, would we have
been impelled to launch a renaissance journal?
Most members of the LaRouche Youth Movement were not
alive when Lyndon LaRouche first ran for president in 1976.
Few of the younger members of this organization have personal
experience with the political interventions LaRouche had
conducted prior to the 2000 presidential campaign, interventions
that, in fact, date back to the late 1960s.
These political
mobilizations frequently determined world events, events that
most people only heard about as second hand gossip drooled out
by the local television network, but the motivations for these
events would have remained hidden to us, if we were dependant
solely on experience for our knowledge.
No one in our generation, the children of the Baby- Boomers,
has ever experienced a society (economy) that was not in
decline. None of us has had the opportunity to empirically
determine that we were in a dark age. Like the proverbial frog
∆υναµις Vol. 1, No. 1
sitting in the sauce pan, the contents of which are slowly heating
to a boil, we could have remained in the current cultural soup
until it were too late. We could have served as a stew for the
Olympian Gods, the “leading men and women” of society, who
would devour humanity’s future as avidly as Francisco Goya’s
famous image of Saturn consuming his children.
When LaRouche made his
fundamental discovery between
1948 and 1952, that discovery
carried
with
it
a
great
responsibility. If you were to
know something that is vital for
the survival of Man into the
future, would you be morally
accountable to convey that
knowledge to society, even if that
society
attacked
you
for
contradicting its closely held, but
erroneous, beliefs? LaRouche’s
role in world politics, his role as
statesman and scientist, has been
informed by his discovery.
October 2006
3
When we began to look closely at his record, at his amazing
success in economic forecasting, we were struck with awe, for
how could LaRouche know what so many “leading economists”
did not? He, and the organization he has formed and has fought
to maintain under the most horrendous circumstances, are
completely anomalous, when viewed from the paradigm of
today’s culture.
As Kepler states in the New Astronomy,
anomalies “lead men to wonder,” and “look into causes.”
So those of us who have come to realize how atypical LaRouche
actually is, have been provoked to investigate why LaRouche
has been able to do what he has. We know we must replicate
LaRouche’s discovery, if we are to guide our society safely into
the future. In so doing, we have been confronted with the
realization that our society has lost knowledge it once had, that
it has descended into the first phases of a dark age. This journal,
then, is a true renaissance undertaking. We intend that it
become a forum, in the tradition of the Acta Eruditorium and
Crelle’s Journal, for those minds who would rediscover the
great ideas of humanity’s past, as these are prerequisites for a
full comprehension of LaRouche’s discovery, and for those who
would work to extend LaRouche’s work, laboring to make new,
fundamental discoveries in the future.
We have been inspired in this enterprise by the work of our
Ibero-American counterparts, who have launched an on-line
journal, called Prometeo,1 which has served as an important
vehicle to convey LaRouche’s ideas and the work of his youth
movement to the Spanish speaking population of the Americas.
This has been a significant medium for delivering profound
republican ideas into the ferment around the contested Mexican
presidential election, for example.
We were also moved by our opportunity this last Christmas to
work with LaRouche on his paper The Principle of Power. In
this article, LaRouche had members of the LaRouche Youth
Movement from around the country help to provide pedagogical
examples for his text. The experience of the intense work and
collaboration that went into this project gave us a taste of what
the LaRouche Youth Movement were capable. We want to
expand this capability as we expand the opportunity for people
not yet familiar with LaRouche to also become struck with
wonder.
This first issue is truly dynamic. It opens, appropriately, with a
challenge, from the world’s leading economist, Lyndon H.
LaRouche, Jr., to all those who would wish to drag science out
of the muck of Sophistry, and transform this looming Dark Age
into a permanent Solar Renaissance. The second submission is
a proposal, by Dennis Mason, to initiate an international
observation experiment. As LaRouche stresses, the science
education curriculum of the LYM lays a heavy focus on
astrophysics, from the standpoint of Classical Greek Sphaerics,
through Gauss and Riemann’s work on curvature. Mason
proposes a collaborative effort to recreate the measurements of
Eratosthenes, to determine the curvature of the Earth. Third is
the centerpiece - a dialogue by Michael Kirsch and Aaron Yule,
∆υναµις Vol. 1, No. 1
on Leibniz’s discovery of the Catenary-cued principle of
Physical Least action. The reader is strongly urged to work
through this one with the necessary pedagogical equipment.
Fourth, as a supplementary article, is Jason Ross’ demonstration
that Leibniz’s vis viva was already fully relativistic, and, that
Descartes was never really necessary to physical science. Fifth,
Peter Martinson reports a slick application of the Calculus to
trigonometric functions, and attacks the current brainwashing in
the typical universities. And, finally, Cecilia Quiroga and
Thomas McGrath present a discussion of Vladimir Vernadsky’s
breakthroughs in Biogeochemistry. As LaRouche has been
recently saying, Vernadsky applied Leibniz’ s method of
dynamics to his discoveries, which makes his work crucial for
understanding the principles needed to rebuild the world’s
economy today.
Future issues will feature such work as translations of Bernhard
Riemann’s Theory of Abelian Functions, and Carl Gauss’ work
on the Pentagramma Mirificum. There will be special issues
devoted to pedagogicals for specific aspects of work the LYM is
doing, like Gauss’ 1799 dissertation. Other articles filling out
some historical research necessary to understand the
significance of LaRouche’s work will also be published. Also,
we will publish pedagogicals and insights into the LYM work
on Bach’s well-tempered system, Bel Canto choral work, and
the Pythagorean Comma. And, of course, several articles are
now in the works on the Economic Animation project.
This journal is not meant to be read all by your lonesome. The
future of civilization requires the current generation not only to
master the deep principles of Physical Economics, as advanced
by the application of the LaRouche-Riemann method, but to
then communicate these discoveries to others and change
national economic policies for the better. This journal is, thus,
as much a political intervention as a scientific-classical artistic
forum. It is the proceedings of the international LaRouche
Academy. Print out copies and circulate them! Hurl the
weapon of creativity in the faces of Aristotle and Descartes, and
the fascist Synarchist International, the “leading men and
women,” who would use their reductionist method to deprive us
of scientific and technological progress, and drive our
civilization into a Dark Age!
Riana St. Clasis
Peter Martinson
editors
1
http://www.wlym.com/~spanish/Prometeo/Prometeo.htm
October 2006
Actually Relive History!
LaRouche
4
Actually Relive History!
Lyndon H. LaRouche, Jr.
September 29, 2006
The birth of this publication reflects a significant moment in an
ongoing process of the reliving of the act of original discovery
in the case of Pythagoreans, Plato, and Plato's Academy, and,
now, the foundations of modern physical science in the work of
the Johannes Kepler as the avowed follower of Nicholas of
Cusa. So, the LYM has reached the point of actually launching
what will become, hopefully, a rebirth of that specific current of
science upon which the greatest achievements of European
civilization had been premised, heretofore.
I read the reports contained within this first edition of Dynamis
as marking the beginning of the true rebirth of the university
from a long, downward journey into that Sophistry of a thusself-doomed Athens which has become the characteristic
contemporary mood within popular opinion and governments in
Europe and the U.S.A. today. The opportunity to rejuvenate the
universities has thus arrived.
I take this opportunity to restate the crucial matter of scientific
principle placed at issue so.
I recall vividly the boisterous eruption among many of the
leading scientists associated then with the Fusion Energy
Foundation (FEF), when I presented the necessity of returning
modern physical science to its founding by the great follower of
Nicholas of Cusa, Johannes Kepler. Some of those who typified
the best living scientists of the 1970s and 1980s howled in
protest, in defense of the Isaac Newton whose chest of papers
has revealed him as a half- witted hoaxster and specialist in
witchcraft.
Grudging admissions on some part of the work of Kepler were
forthcoming in an FEF meeting a year later, but, as in the later,
last occasion of a personal meeting with those scientists, it was
conceded by some, that the fearful protests against my
insistence on study of Kepler reflected the fact that science in
the Americas and Europe today is dominated by an echo of the
ancient Babylonian priesthood, which is ruled over by
contemporary-banker- controlled "peer review committees."
Some of the finest achievements of modern physical science
which have been demonstrated crucial- experimental discoveries
in the laboratories, are suppressed by a priesthood of the type of
the Sixteenth-Century Paris-ites ridiculed by the great Francois
Rabelais, and, more recently, Jonathan Swift's portrayal of the
academic authorities of the legendary Island of Laputa.
This has been the prevalent state of affairs in European science
since the death of England's Queen Anne, and, in the extreme,
since such apostles of the hoaxster Bertrand Russell as: 1.) the
Josiah Macy, Jr,-based faker Professor Norbert Wiener, who
∆υναµις Vol. 1, No. 1
made a pagan religion of "information theory," and, 2.) his
crony, the virtually (or, actually) autistic John von Neumann,
who classed the human mind, including his own, as an "it." Both
of these particular dogmas of a modern Babylonian priesthood,
have found their typical official residences in such prominent
locations as the Massachusetts Institute of Technology's RLE.
The problem in science today, is the widespread presumption
that an experimentally demonstrable universal physical
principle, such as universal gravitation, is, virtually, merely an
opinion of formal-mathematical deduction, whose ontological
actuality is, at its least worst, that mere, deformed shadow of
experimental reality: therefore, ontologically, a mere
mathematical formulation.
All science (and true Classical artistic composition, too) pertains
to the discovery of experimentally validated universal physical
principles: that is to say, principles whose existence lies only in
their function as controlling powers intrinsic to the universe as a
whole. Therefore, such universal principles are expressed, by
reflection, within a formal mathematics, only in the form of
certain mathematically infinitesimal influences which, in their
reality, as universals, shape the action within the universe as a
whole.
Hence, the only form of the calculus which was ever discovered
was by Leibniz. Virtually, even Newton's existence was almost
a hoax, his claims to scientific orignality, entirely so. The
reality of the physical meaning of the Leibniz calculus, is
expressed for today in the implicit argument for an antiEuclidean geometry, which Abraham Kaestner student Carl F.
Gauss presented as a devastating refutation of de Moivre,
D'Alembert, Euler, Lagrange, and others, in 1799 doctoral
dissertation on the subject of The Fundamental Theorem of
Algebra.
Kepler's uniquely original discovery of both gravitation and its
expression as a principle of harmonic organization of the Solar
system, as this discovery was inspired by Nicholas of Cusa, are
the original modern root and typification of physical science. It
is typical of competent science, that gravitation is an object
which is, as Albert Einstein argued, coextensive with a finite
object, an object known as a finite, self-bounded universe.
Einstein traced that conception to, chiefly, the successive work
of both Kepler and Bernhard Riemann's concept of a
specifically dynamic organization of the physical universe as a
whole, as expressed by Riemann's presentation of the tensor as,
ontologically, a thoroughly anti-Euclidean, physical concept of a
universal dynamic process, rather than, as sometimes
misrepresented, a mere mathematical schema.
October 2006
Actually Relive History!
LaRouche
5
This view from the standpoint of the best leaders of modern
science, was prefigured, and made possible by the foundations
of that modern approach to a specifically anti- Euclidean,
physical geometry, which is found in European civilization as a
whole, found there only in the continuing influence of the work
of the Pythagoreans and Plato.
It is by reliving those great discoveries made available to us
from the past of the entirety of ancient and modern European
civilization, that the spirit and substance of science can be
brought to life, again, today.
It is the young adult mind which has captured the sense of those
Classical acts of discovery, who is enabled, thereby, to define
the nature of the sovereign individual mind of man or woman in
the universe. It is the individual who knows science in that way,
as an experience of the definition of the human individual which
makes man distinct from the beasts, who also has access to a
comprehension of a return to Classical artistic composition as a
way of thinking about mankind, a way which uplifts
contemporary society above the bestiality which prevails in
what is called popular culture and the prevalent Sophistry of
politics today.
It is eliminating that lunatic dichotomy, which separates what is
called science from what is called art, which is the indispensable
basis for the promotion of sanity in the populations, and also
their electorates, today.
The mission of the best souls from the generation of young
adults today, is restore consciousness of participation in real
humanity once again today, away from the stinking mass of
Sophistry inherited from the self- doomed Athens of Pericles,
the same Sophistry into which trans-Atlantic civilization, and its
political institutions, has so deeply descended today.
Thus, I see the great tradition of the university since Plato's
Academy at Athens, as being, once again, reborn. The promise
of a new Renaissance is here; let us continue the steps toward
making it an actuality.
∆υναµις Vol. 1, No. 1
October 2006
A Proposal to the World LaRouche Youth Movement
Mason
6
A Proposal to the World LaRouche Youth Movement on
Taking the Necessary Measurements With Which to Recreate
the Discoveries of Aristarchus and Eratosthenes
Dennis Mason
The Idea is this: on a given day to measure the angular
relationship of the shadow cast by the sun, relative to the North
Star; and the difference between the height of the object casting
the shadow and the length of the shadow itself. This would be
done, simultaneously, at various locations on the surface of the
Earth, the distances between which would be calculated with the
aid of GPS. If any two positions on a sphere are connected by a
segment of a shortest line, then these measurements could be
used to find the circumference of the Earth, were it a sphere.
Using different positions for the calculations will result in
different circumferences if the Earth were not a sphere, and the
differences would be useful in investigations of the true
curvature of the Earth.
Also, the measurements of the lengths of the shadow and
plumb-bob would aid, with data from three locations, in
triangulating the position of the sun, and in determining its
distance from the Earth.
As would be the case in the event of different circumferences
generated by different pairs of positions, a different set of three
positions used for triangulation would generate different
distances to, and positions of, the sun, were the Earth not a
sphere.
∆υναµις Vol. 1, No. 1
October 2006
A Proposal to the World LaRouche Youth Movement
Mason
7
The measurements would be taken with a plumb-bob, a ‘flat’
surface, and some paper… a level, and a measuring tape were
also necessary. The length of the plumb would be one meter,
and though calculation should, for rigor’s sake, be done in both
meters (Km) and inches (miles), meters would be the primary
unit.
We’d be using the North Star as our reference for determining
the angle of the shadow cast by our plumb, and so it were
necessary to take measurements at the sight beforehand,
calculating the right ascension of the star from some local point
of reference; this will ensure a unified point of measure, in
terms of the northern hemisphere. Those below the equator
would need to discover how to determine the angular position of
the North Star relative to their respective sights; all involved
would work on this problem with them. Dependant on how
long that takes, we could either solve this riddle before taking
measurements, or take the measurements in the northern
hemisphere, for something to start with, and then, later, take
them again with everybody.
This is just a starting point for an investigation; we will
probably run into lots of problems aside from the obvious (i.e.
weather), in trying to recreate these discoveries.
It were mandatory for all involved to study On Eratosthenes,
Maui’s Voyage of Discovery, and Reviving the Principle of
Discovery Today, by Lyndon LaRouche in the Spring 1999
issue of 21st Century Science and Technology.
If you would like to participate, contact Dennis Mason,
[email protected], in the Seattle Office.
∆υναµις Vol. 1, No. 1
October 2006
Experimental Metaphysics
Kirsch and Yule
8
Experimental Metaphysics:
On the Subject of Leibniz’s Captive
Michael Kirsch and Aaron Yule1
“Creativity, as I identify it here is the difference between you
and a monkey…..First, the member of the human species can
increase the potential relative population-density of his or her
species by the willful use of creativity, as no form of animal life
could do this. Second, progress of society over successive
generations, depends on the reenacting of the creative discovery
of those kinds of universal physical principles, by successive
generations. Taken together, these two expressions of
creativity(as I define it) provide the basis for what we might call
natural human morality, the kind of difference which separates
human morality and the culture of monkeydom.”2
Presently, as the world is being run by an elite order of apes,
there is little hope of the existence of mankind, unless the
principle of human creativity is asserted as the dominant
characteristic among nations. The study of making money for
money’s sake, with disregard for the reality of the living
conditions affected, has crushed living standards internationally
and brought the U.S. economy to its present near death
experience. Unless we gear up our industrial scientific
capabilities and return to an economy oriented toward scientific
progress, the challenges facing the U.S. and the World of now
more than 6 billion people, will not be solved. To achieve the
policy change required, the respect for any economics that
considers the belief ‘buy cheap and sell dear’ as a fundamental
truth, must promptly cease. Economics must now be studied as
the assimilation and application of the creative powers of the
human mind. However, the challenge is more complicated.
The technology/infrastructure deficit caused by the tragic
destruction of living standards worldwide will cause any honest
person to realize the interacting challenges involved in securing
a pathway out of the crisis before time runs out. The desperately
needed technologies for all nations, including water sanitation,
water desalinization, canal and port management, rail
transportation, nuclear power, etc, must be applied to the
productive sector and infrastructure grid of each nation with the
intention of accomplishing the most benefit possible considering
the present standing of each nation. The benefit to each nation is
measured in the ability to assimilate most effectively the
technology applied in such a way as to create the preconditions
for subsequent technological breakthroughs.
In other words, since the true substance of economy is creative
ideas, how can the projects organized by governments utilize
technology in ways that reorder the economy to exclusively
express creative ideas?3
These factors require a knowledge of the relationship between
applications of technology and the effected process of change in
the productivity of nations far beyond any former period in
history.
∆υναµις Vol. 1, No. 1
What will be the change in the relationship of the infrastructure
of a nation with its productive sector when that nation applies a
new scientific principle to its economy as a whole? Will the
results be in correlation with the challenges facing present world
population levels and resource availability? If we honestly care
about the future of the human species, we will not leave these
questions up to chance. If we are truly driven to develop the
human species we cannot afford to commit errors in these
matters; therefore, if your economics can not answer these
questions- the typical response of nearly every college student
or professor today- then it must be thrown out, and replaced
with one which can.
Taking these challenges into consideration, the prerequisite of
all future Eurasian economists, serious about making the
required shift in economic policy world wide, is rediscovering
U.S. Statesman LaRouche’s discovery in economic science: the
ability to measure the relationship of creativity, expressed as an
increase in the productive powers of labor, with the increase in
productivity of human populations.
Presently, the LaRouche Youth Movement, intent on turning
economics into a science able to sufficiently answer the present
challenge described above, is investigating the qualitative shifts
occurring from the introduction of new technologies to
economies, ranging from: Lincoln’s transcontinental railroad,
the application of electricity to production at the turn of the
century via the electric motor; FDR’s REA and four corner’s
projects; and JFK’s space program. In the cases where serious
research has been done, it has been demonstrated that only
LaRouche’s discovery of the universal characteristic of action in
economics has allowed these qualitative changes that occur in
the relationship of the power applied to the work output, the
qualitative effects of the productive powers of labors to
productivity, and so on, to be measurable.
To understand the unfolding of the economic processes more
fully it is required to dig deeper into the principles underlying
LaRouche’s Discovery in economic science. Keeping the above
overview of the current challenge in mind, we now proceed to
the specific aspect, relevant to this economic study, which this
scientific inquiry seeks to investigate.
The Challenge of a New Science: Measuring the
Action of Physical Principles
LaRouche in his text on mathematical economics, So You Wish
to Learn All About Economics?, states that, the measure for the
unfolding process of the developing relation between, on the one
side, creative discoveries applied as technology to basic
October 2006
Experimental Metaphysics
9
Kirsch and Yule
economic infrastructure, and on the other, those discoveries
applied to production, which is what we were seeking above, is
defined as potential relative population density. This
measurement of the physical action of the relations between
physical principles, LaRouche continues, can be expressed
through the mathematical physics of the Gauss-Riemman
complex domain. As Gauss demonstrated with his work on
magnetism, the physical relationships of magnetic potential are
measurable with complex functions. LaRouche states, that these
very same functions of a complex variable are also the
mathematical language for measuring the changing physical
relationship of economic potential.
Of crucial significance for this present inquiry is: one, the
unfolding process of an economy, potential population density,
must be conceptualized as a Leibnizian infinitesmal; two, the
establishment of the ability to measure the action of the relations
between physical principles, is historically rooted in Leibniz’
discovery of the Catenary cued principle of physical least
action.4 This discovery by Leibniz defined all universal
physical principles to be measured in an anti-Euclidean, antiCartesian domain of physical curvature, laying the basis for
what was to become the Gauss-Riemann complex domain.
calculus as a means to solve the challenge presented to us
above. We proceed with the added caution. As described,
LaRouche’s method is the most successful science developed
for economics; however, in the 1980’s when it was taught from
the standpoint of reductionist calculus, it was rendered impotent
in its ability to successfully measure potential population
density.5 On the contrary, the mathematical physics used to
measure LaRouche’s science of physical economy must be
firmly established on a Leibnizian metaphysical infinitesmal,
rather than a mere mathematical shadow.
Noetic Archeology
In Spring 2005, Lyndon LaRouche called on the LYM to relive
the discovery of the Leibniz Calculus from the true method of
Leibniz rather than the reductionist version taught in
Universities today.6 Taking up this challenge a couple of months
later, the Boston Office of the LYM decided that it would be a
good place to start by working through a recent translation of
Bernoulli’s Lectures on the Calculus.7 In working through that
piece, we found the concept of the dx/dy triangle very difficult
to conceptualize. There seemed to be no reason to memorize the
rules of the Calculus, creating a contradiction with Leibniz’
principle of sufficient reason. Therefore, with much
encouragement from the translator of Bernoulli's lectures, Bill
Ferguson, a couple of us decided to physically construct the
diagrams in Bernoulli's lectures. In the following pictures, the
results can be seen (Figure 1). After a period of physically
demonstrating some of the properties of the Catenary from the
lectures, we decided to investigate the concept of the
differential. The results were remarkable. The calculus, before a
mysterious language of mathematicians, was now
communicable through the language of physics.
After demonstrating the physical differential for these 20 points
of a paper clip Catenary, we attempted to demonstrate the
physical differential of the Catenary continuously with a
wooden curve.8
The magnitude of Leibniz’ breakthrough for mathematical
physics and economy is outlined by LaRouche:
“No actually fundamental, axiomatic advance in the subsuming,
essential mathematical principles of physical science has been
reported in the open literature, since the elaboration, as by
Gauss, Dirichlet, Riemann, and their collaborators, of the
implications of Leibniz’s discovery of the role of the
Catenary function in defining natural logarithms and as
expressed by Leibniz’ universal physical principle of
universal least action. It was this legacy, chiefly mediated
through the work of Leibniz, which has provided the foundation
for valid modern science since Leibniz’ death, and provided me
the indispensable foundations for my original, supplementary
contributions to the field of Leibniz’ original creation of the
science of physical economy.” [emphasis added]
Therefore, let us now set out to rediscover Leibniz’ infinitesmal
∆υναµις Vol. 1, No. 1
Although the experiment was not fully successful, the
implications of that design contained a bigger challenge:
demonstrating the true infinitesmal of the Catenary.
LaRouche put out this challenge at a Berlin Cadre school at the
end of December 2005 shortly after the publication of his
Powers Paper:
“Generate a Catenary, by some means other than a hanging
chain. Construct it! The way a machine-tool designer would
construct something.
“So now, you don’t show the principle, as such. But what you
do, is, you show how the principle works,, by generating a curve
which corresponds to the Catenary. And by generating it, by
your willful action, you show that your understanding of the
principle, is correct. It is now discussable, it is now
communicable.
October 2006
Experimental Metaphysics
10
Kirsch and Yule
Figure 1
The physical constructions made in
Boston, to demonstrate the
conceptions in Bernoulli’s lectures on
the Differential Calculus.
After
demonstrating the
physical differential
for these 20 points
of a paper clip
Catenary, we
attempted to
demonstrate the
physical differential
of the Catenary
continuously with a
wooden curve.
∆υναµις Vol. 1, No. 1
October 2006
Experimental Metaphysics
11
Kirsch and Yule
“What you do with any discovery, is, you make a discovery of
principle: The very fact that it's a principle means you can't see
it! It is not sense-perceptible. Its effect is sense-perceptible, but
it is not sense-perceptible. Now, you have to find out, to
demonstrate this principle, to demonstrate you have willful
control over the use of the principle. So therefore, you do
something that demonstrates, that you have willful control over
what you contend to be a principle.”
Later at a Jan. 11th Webcast LaRouche continued the challenge:
“Now, how can you see an object that fills up the universe as a
whole? Where are its boundary conditions? You can't.
“Now, in physical science, all discoveries, like the principle of
gravitation, can not be seen as objects of the senses. What you
can do, is that you can generate, as you do with machine-tool
design—if you understand the concept of a principle, you find a
way to express that, as a design. Now you demonstrate the
effectiveness of the idea, by a machine-tool design—as we did
with a number of these things in that edition, in which some of
the young people did that. Like the case of—instead of trying to
draw a Catenary, based on doing a parallel to a hanging chain,
actually construct and generate a Catenary. The Catenary
principle is not something you can see. It's a transcendental
function. And, you can not see transcendental functions. They
have the form of being zero, or everything. But they are a
something.
“So, once you have the idea and you demonstrate by
construction that you can generate that effect—which is what a
machine-tool designer does, if they are really good at it.
Particularly in research, test-of-principle work: You actually
say, "Does this principle work?" "Okay. How can you generate
an effect, that shows that this principle works?" Now, you've
proven it. That's called a proof of principle, a unique
experiment.”
For a couple months after LaRouche posed this challenge, we
authors read through Kepler’s New Astronomy. Although our
reading through of Kepler’s work did not make us masters of his
discovery, his concept of gravity gave us a new look at what
should be found in the Catenary. Thus, taking up the challenge
posed above by LaRouche, we returned to
Leibniz’
construction of the Catenary, intending to re-discover the true
infinitesmal, correlative with the machine-tool principle.9 From
the standpoint of the physical principle we had been
investigating, the veil over his paper was lifted and was seen
now, by us, in hindsight, as arising from a lack of physical
experimentation. What we had re-discovered as a singularity,
the key to the Catenary, betrayed the scent and put us on the
trail to re-discovering the true historical basis for Leibniz’
universal physical principle of least action, with implications
beyond our knowledge at that time. Much more will be
illustrated concerning Leibniz’ discovery; however, the
pedagogical following this introduction seeks to illustrate that
specific archeological expedition. But first, let the following
∆υναµις Vol. 1, No. 1
brief historical sketch serve as both a demonstratin of the central
role of the Catenary, and a basis for further research in the role
of the Catenary as the organizing principle of the LeibnizBernoulli calculus as a whole.
Why, Historically, the Catenary?
Leibniz’ revival of the method of Greek Sphaerics, Platonic
Ideas, and Keplerian Physics, can be rediscovered through
1) Analysis Situs: an anti-Euclidean constructive
geometry;
2) his Calculus: a language of the infinite able to
measure the principles of motion guiding physical
pathways of action;
3) and his science of Dynamics: application of his
metaphysics to a true concept of substance, the
force or the power of action in physical processes
defining the mechanics of extension, size, figure,
motion etc as effects.
As the following pedagogical demonstrates, not only does the
Catenary ontologically defined the entire domain in which the
principles of motion were investigated, itself opening up a new
method of science; but also, in what will be shown here, was, in
fact, the Catenary challenge that organized the Leibniz calculus
project into a movement of scientists committed to establishing
an anti-Cartesian physical science throughout Europe.
Leibniz knew that in order to inspire the scientists, to break
from the Cartesian dogmas of empiricism, the bane of scientific
progress, it was necessary to demonstrate the accepted
underlying axioms of Descartes as absurd. Just prior to writing
his 1686 Discourse on Metaphysics, Leibniz struck at the
foundation of Cartesian physics, the principle of the
conservation of motion, in writing a short piece on the error of
Descartes who maintained that all bodies in collision maintain
their quantity of motion, mass times velocity.10 The example
Leibniz gives, is where one body of mass 1 and velocity of 2,
falling from 4 feet, will have the power to move a body of mass
4 up one foot 1 foot. But he argues that the quantity of motion
for one is 2 and for the second is 4 and yet they both have the
same power. This demonstration showed simply that the force
was maintained even when the quantity of motion wasn’t.
Leibniz discusses that Descartes law is only true for the simple
machines of the time. “We need not wonder that in common
machines, the lever, the windlass, pulley, wedge, screw, and the
like, there exists an equilibrium, since the mass of one body is
compensated for by the velocity of the other; the nature of the
machine here makes the magnitude of the bodies-assuming that
they are of the same kind- reciprocally proportional to their
velocities, so that the same quantity of motion is produced on
either side.”11
A Cartesian, Abbot Catelan, shot back in objection to Leibniz’
coup on the established Rules of Motion. He defended that
Descartes so-called principle was true if the objects fell from
equal heights, corresponding to the simple machines of the time.
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Kirsch and Yule
Leibniz responded back in 1687 with a challenge in the Acta
using the issue of falling bodies. He writes of it in a letter to
Antoine Arnauld in 1688:
“I have only taken the opportunity of this argumentation to put
forward a very curious geometrico-mechanical problem which I
have just solved. It is to find what I called an isochronous curve
in which a body shall descend uniformly and approach equal
distances to the horizon in equal times, not with standing the
acceleration it under goes. This latter I off set by continually
changing the inclination. I did this in order to bring out
something useful and to show M. l’Abbe that the ordinary
analysis is to limited for difficult problems.”12
Here Leibniz’ genius can be seen. Since he challenged this
reaction of Catelan by posing a problem in the Acta which was
unsolvable with Cartesian principles of motion, it allowed all
the scientists of the time to realize themselves: one, that they
had to give up their underlying assumptions to solve it, and two,
that a method of truth existed. The ability, not simply to publish
scientific papers, but to overthrow axioms, inspired the
Bernoulli’s and others to join Leibniz’ attack on the Cartesians.
Indeed, he organized a movement, leading to the Catenary that
sparked the essential role of the Bernoulli's in both formulating
the Calculus into a comprehensive mathematical physics, and
advancing Leibniz’ Calculus further.
Leibniz recalls this process in his 1697 thoughts on Bernoulli’s
Brachistochrone in the Acta:
“There in lies, in my sense, the reason for the success of the
method of infinitesmals that I have initiated with respect to
differences and summations (and which became known as the
differential calculus), and of its adoption by a number of
imminent individuals: it turned out to be the most appropriate
method for solving problems. Indeed, I began to validate that
method, when, in response to M. Abbot Catelan in the News of
the Republic of Letters, where he had realized some objections
to my work on dynamics, and thus lending too much credibility
to the Cartesian methods, I got the idea of responding to him, as
well as to anyone who had the same sentiment, by showing that
I had solved the relatively easy problem of the isochronic curve.
“But, since there is always a certain continuity in everything,
my demonstration had the effect of suddenly inspiring Jacques
Bernoulli, who, up until that time, only had an occasional
flirting with the differential calculus that I had published in the
Acta, and without getting anything out of it. But, since he
grasped the importance of this method for questions of
mathematical-physics, he then submitted to me the problem of
the Catenary Curve that Galilieo had tackled without success.
logarithms. This resounding success provided the Bernoulli
brothers with a wonderful opportunity which enabled them to
later accomplish marvels with this calculus, so much so that,
from now on, this method is as much theirs, as it is mine.”
[emphasis added]
Hanging with a Modern Layman
Before beginning this dialogue, you’ll need: a heavy chain, a
thin chain which can be cut to different lengths, a compass,
poster board, straight edge, tape, a pulley, string, pencil.
Existentialists,
lacking
rigor,
may
avoid
physical
experimentation.
A young man leaving the math department walks up to another
man holding a chain in deep contemplation.
LAYMAN: Hey dork, whatcha doin?
WISEMAN: A fool always serves the wise.
LAYMAN: Uh, so why you out here looking like a fool?
WISEMAN: Wisdom doesn’t come easy.
LAYMAN: Well, you’re making it more difficult for yourself. If
you’d pay the tuition you could become wise like me.
WISEMAN: Wisdom is not paid for, its only harvested.
LAYMAN: Well, I pay for what was harvested and collected,
and its being served to me…. I skipped that step, why do the
work when someone’s already done it for you?
WISEMAN: How do you know the soil, from which the harvest
came, has not been poisoned?
LAYMAN: They wouldn’t let that happen! Who would want to
do that? Besides, who are you, a fool here, to tell me what
wisdom is? What can you figure out from this chain that I
haven’t already learned in my math class?
WISEMAN: Do you want to know? How could one, who’s idea
of truth is that which the professors condone, the mere popular
opinion of the lecture hall, be interested in the wisdom this
chain yields?
LAYMAN: Ok fine, what then, what is it you are asking here?
WISEMAN: Tell me, is that an extra ordinary curve?
“It was the study of my calculus that led M. John Bernoulli to
the right answer, after he had made the connection with the area
of the hyperbola, as I had done myself, but with the only
difference that he found the construction by means of the
rectification of the parabolic curve, which I made use of
∆υναµις Vol. 1, No. 1
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LAYMAN: Well, I feel tension in my arms, due to the weight of
the chain.
WISEMAN: Is that tension you feel simply the weight of the
chain? If you let go with one hand, and hold the chain hanging
from the other, is the tension on that arm equal to the tension on
both arms before?
LAYMAN: Honestly, I do not know.
WISEMAN: If you hold the chain between your hands, and
now move your hands apart, what changes?
LAYMAN: I feel an increase in the tension in my arms.
LAYMAN: Why, no it’s, a simple chain, like any I’ve seen.
WISEMAN: Is that because the weight increased?
WISEMAN: Layman, tell me why it hangs the way it does,
since it is simple, like any you’ve seen.
LAYMAN: Ha ha ha ha, you ask such silly questions, isn’t it
obvious? Of course not!
LAYMAN: Well, the weight of the chain is pulling the chain
down.
WISEMAN: Then how do you explain that as you move your
arms, since the weight has not increased, you feel a greater
tension?
WISEMAN: Pulling, what do mean pulling?
LAYMAN: The links, I mean, are being pulled by their own
weight.
WISEMAN: Ah, but this idea Layman, holds no weight, for the
cause of the links motion, could not be the links themselves.
How are you distinguishing between, the links and the weight?
Do not the links of chain contain the weight you just asserted as
the cause of the shape?
LAYMAN: Well then, if the weight and links are inseparable, as
you say, then this force that I attributed to the weight, must be
Gravity. That is the force pulling on the chain. Ah, yes, now I
have your answer, and I got to leave or else I’ll be late for my
next class on statistical economics.
WISEMAN: But hold, you have not answered the question I
asked you, and assumptions surely hold no weight, but only pull
tight the chains of the mind.
LAYMAN: Well, then what is it?
LAYMAN: There must be an added tension between the links
of this chain, from my pulling them apart.
WISEMAN: But yet the weight of the chain doesn’t change,
does it?
LAYMAN: Of course not!
WISEMAN: Ok, so the weight is constant, if we have the same
chain… and the tension increases as you pull your hands apart,
but what happens if you do not move your hands apart, but you
move your hands up and down.
LAYMAN: One feels heavier than the other, and one feels
lighter. This is relative to which one is above the other.
WISEMAN: How much of what you feel is due to the weight of
the chain, and how much is due to the added tension?
LAYMAN: What do you mean?
LAYMAN: I feel I’m gonna look like a fool!
WISEMAN: If your left hand is holding up a small portion of
the chain and your right is holding up more, what portion of the
tension in your right hand is due to the weight and what portion
of the tension is due to the your pulling?
WISEMAN: Is experimentation fools play?
LAYMAN: I can’t tell
LAYMAN: ok, fine.
WISEMAN: Do you think the hand with less chain is holding
less of the added tension, or the hand with more chain?
WISEMAN: Take this chain and tell me what you feel?
WISEMAN: tell me now, what you feel? What can you observe,
about this chain hanging between your hands?
∆υναµις Vol. 1, No. 1
LAYMAN: Again, to be quite frank, I know not.
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14
L: No
W: What if you move one hand down the chain, what is the
relation of tension caused by the weight, and the added tension
as you hold it in different spots?
W: Then what are you feeling there?
L: Something
L: The tension from the weight changes as there is less chain,
but still I can not tell how the added tension is changing.
W: Well, let us take this route: as you continue to move your
hand down, successively holding less and less weight, will there
be a spot in which you hold no weight?
L: How could you be holding no weight if you are holding a
chain?
W: As you pull it farther from where you hung it from, you feel
a greater increase of tension in your hands?
L: yes
W: and the tension from the weight is not increasing because the
chain is not increasing?
L: right, but there is an added tension.
W: So, what you are saying is, you are still holding up the
chain?
L: Yes, I am.
W: Well now, hold the chain in one hand letting it hang freely.
Now with your other hand grab the last link at the bottom of the
chain. In other words, move the chain as you like, but keep the
bottom link as the bottom of the chain. Is that hand holding up
any chain?
W: So then, what you felt at the last link as you pulled it
sideways was the added tension?
L: Yes, so the added tension in the chain is all I feel at the
bottom.
W: Now keep your hand on the bottom link that you are pulling.
I will hold the other side and while I hold up less and less by
moving my hand down, what is changing at the lowest link?
L: I feel nothing change at the lowest link. Wait, do this again,
and lets double check.(The wise man repeats this process again,
holding the chain lower and lower.)That’s a pretty good trick. A
constant tension at the bottom, where did you learn that one?
W: Now as I hold it at higher and higher points do you feel any
change?
L: no
W: And as I’m holding it higher and higher, does the added
tension caused by your pull effect the increase of tension in my
hand?
L: no, other wise it would be moving side to side.
W: So layman, let us return to the question we asked earlier,
having now established that a) if the chain length is constant,
tension is added when moving the position of the chain, and b)
if the chains position is fixed the tension from the pull is
constant while the length changes, how can you know the
relation of the tension from the weight and the added tension at
any given spot on the chain?
L: Well, lets see: when I hold the chain at top and I move my
hand down, the only thing that’s changing is the weight of the
chain. Therefore, I can know that when I’m at higher point on
the chain the portion of tension due to the weight is greater, but
the added tension is the same; when I’m lower in the chain the
added tension would be the dominant part of the portion.
W: but how could you know their relationship precisely?
∆υναµις Vol. 1, No. 1
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15
L: Yes, there is relation I know, but how precisely?
W: Observe again the effects that you perceive as the chain
changes position. What else changes that you can measure as the
chain changes, besides the quantity of tension from weight and
added tension, which you could use to relate to the relation of
the two tensions?
L: Ahhhh…. Yes, I see the X and Y coordinates of the chain, I
can’t believe I hadn’t thought about the coordinate system of
Descartes. Of course I’ve never really looked at something
outside of class. If we make a horizontal line and a vertical line,
I can see that as I move the chain, those coordinates will change
as well.
W: How is the relation between the X coordinate and the Y
coordinate related to the two tensions?
L: Well, the Y coordinate corresponds to the tension from the
weight of the chain, and the X coordinate corresponds to the
added tension in the chain.
W: It seems we must return from this crooked path. Answer the
question now, without relying simply on what your senses tell
you. Rather than putting an assumed X and Y coordinate system
upon the chain, let the chain define itself. Again, what else is
changing as the added tension and tension of the weight change?
W: So the longer the X coordinate the greater the portion of
added tension in the chain in relation to the tension from
weight, and if the Y coordinate is dominant the tension of the
weight will be dominant in the relation?
L: Well, as I move the chain around, the direction of the chain
changes.
L: Yes, that must be so… it looks like it to me.
W: How would you relate the change in direction to the change
in two tensions?
W: So if the X coordinate is 2 feet and the Y coordinate is 4
feet, then the added tension will be one third of the portion of
what someone feels holding the chain?
L: Hmmm… I don’t know, but I would think that the direction
could be measured as having a vertical and horizontal
component, I remember that from class, studying vectors.
L: yes, that is so..
W: Ok, but, the curve of the chain constantly changes direction
as you move along the chain, right?
W: so would you also say, that when the relation of the X
coordinate and the Y coordinate are the same, that the tension
from the weight and the added tension will be the same as well?
L: yes
L: Yes that is true.
W: Well, let us inspect this hypothesis. We hold this chain so it
has the same X and Y coordinate lengths. Feel the added tension
in the chain at the bottom, and then keeping that in mind, feel
the tension at the top link. If at the top link you feeling the effect
of holding up weight of the chain as well as feeling the added
tension, then if added tension and tension of the weight are
equal with these equal X and Y coordinates, then surely what
you should feel at the top link will be twice the added tension?
L: Why did you let me fall into this trap? On my calculation,
even without any more precision than this, I can say, beyond
any doubt that what I feel at the top link here, is much more that
twice the tension. Trying to make me look like a fool, eh?
W: Then how would you measure the direction in terms of
horizontal and vertical components at any one moment of the
curve, if the curve changes direction at every moment?
L: Is this what you were saying about going beyond the senses?
Hmmmmmm…you got me stumped.
W: At any one moment where’s the chain’s impetus?
L: Hmmmm ok, at any moment, the chain is tending in a
particular direction…. so you could extend the direction beyond
and see the relation of horizontal and vertical component which
could otherwise not be seen.
W: ah, yes. Another name for what you just figured out is called
a tangent.
L: Are you saying black boards are not necessary for tangents?
∆υναµις Vol. 1, No. 1
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Kirsch and Yule
W: Yes. But now further, tell me next… if you extend the
direction of motion at one moment, does the relation of
horizontal and vertical components of the direction correspond
to the relation of the two tensions.
W: Would you have the direction vertical, without the chain
hanging?
L: Sure, vertical.
L: Well, how would I do that. How can you relate the lengths of
these components to the two tensions?
W: Would vertical be the same, anywhere you go on the Earth?
W: Think back just now: what determines the direction?
L: No, I mean, well yes, its down. Anywhere I walk its down.
L: Ok, the direction changed when the portions of the two
tensions changed.
W: And how do you know what is down?
L: That’s where the chain hung
W: What direction was the chain in when there was no added
tension in the chain?
L: Uhhh, it wanted to go down, where all things tend without
impediment.
W: So what determines vertical?
L: Where chains hang.
W: And what is horizontal?
W: So, if we seek to compare the lengths of the horizontal and
vertical components of the direction with the two tensions, how
long is that length of chain, when it was hanging?
L: Perpendicular to the hanging chain
W: Does the chain become horizontal at the bottom?
L: Obviously the length of the chain itself.
L: It seems to. for a moment it would
W: Ok, so the vertical direction corresponds with the tension
solely due to the weight of the chain?
W: And also, remember what you said, when you pulled the
bottom away from a simply hanging chain?
L: Yes.
W: Now, think back to when you held it at lower and lower
points on the chain. The portion of the added tension became
more and more dominant in the relation of the two tensions.
What direction was it tending as you held it closer and closer to
the bottom?
L: Yes, I said that I felt a pull, but reason informed that what I
felt was not due to any tension of weight, but only the added
tension in the chain caused by my pulling.
W: So if all that was felt at the moment at the bottom of the
chain was the added tension, and the direction at the bottom is
horizontal, what does that tell us?
L: Horizontal.
L: Ah, we can relate the horizontal component of the direction
taken with our tangent to the added tension at the bottom of the
chain.
W: What do you mean horizontal?
L: Perpendicular to vertical
W: So, you see that you can not assume directions, they are not
a priori, but are defined by the impetus of force; that is,
direction is defined by what is causing the curve to take its
shape, not the other way around. In Euclid’s brain, shape is
described by a priori directions.
W: How do know what vertical is?
L: Perpendicular to horizontal
W: Now, didn’t you say a chain hanging freely from one hand
hangs down?
L: Its good Euclid was so smart, he probably used more than ten
percent of his brain.
L: Yes
W: And by down you said the direction which all things tend?
W: And for our sakes, we could call the added tension that
corresponds with the one horizontal moment on the chain,
horizontal tension that is constant?
L: Yeah, I said that.
L: That would be right
W: Would you call down, vertical?
W: And the tension of the hanging chain, vertical?
L:
Yeah,
the
∆υναµις Vol. 1, No. 1
chain
hangs
vertically,
sure
October 2006
Experimental Metaphysics
Kirsch and Yule
L: Yes
W: But now we face another problem. If we intend to relate the
horizontal and vertical tension with the lengths of the
components of direction, we must now confront this challenge:
how can we measure constant tension, as a length, just as we
took the length of the vertical tension as the length of chain
itself?
L: Hmmmm A length of chain equal to the constant tension? I
don’t have an answer off hand.
17
W: What will be the direction of the chain when those relations
are equal? Remember directions were defined by what was
causing the shape of the chain.
L: The components of direction being equal…ok….the tangent
of direction would be at 45 degrees.
W: …. Proceeding on with this hypothesis, into why this curve
is hanging the way it is, why don’t you hang a chain here
against the wall, and find the length of constant tension.
L: Yes, ok.
W: Think of this, remember before when you held the chain at
different points? As you held it at high points, the dominant
portion was the vertical tension, and as you held at low points
the dominant portion was the horizontal tension?
L: Yes, ok, so if the portion of what I felt became more and
more dominant in regards to the horizontal tension as I held the
chain lower and lower, there must be a point, where that length
of chain is equal to the tension.
W: When is it when the horizontal tension becomes more
dominant thanL:
-Wait, give me a second here…….ok, as I said
before, as we go lower the horizontal tension becomes the
dominant force… so…. Arg…I don’t know…..there is a
moment though, but how do I know it?
W: What else were we looking at, other then horizontal and
vertical tension?
L: Umm. We were looking at lengths of chain.
W: Anything else?
L: Ah, yes, direction!
W: And how was this useful to our inquiry?
L: We related the horizontal tension to the horizontal direction.
W: And?
L: We related the vertical tension to the vertical direction.
W: So therefore?
L: Now I think I got the idea. Since we are determined to find
the length equal to the constant horizontal tension to compare a)
the components of the tangent to b) the lengths of the horizontal
and vertical tension, then, assuming that these two relations are
proportional, when the horizontal and vertical components of
the tangent representing the direction are equal, then the
horizontal and vertical tension must be as well, unless this
relation we are investigating proves false.
∆υναµις Vol. 1, No. 1
Walking to the nearest
wall, the Layman strung
up a chain. Then, being a
geometer, he produced a
compass and a ruler.
Using circular action he
found the tangent to the
curve at 45 degrees.
(sound easy? Try this
yourself!)
W: Now you have your unity. Test it to see if that length is
equal to the constant tension. Take this pulley here and attach a
string to the bottom link. Then tie a length equal to the unity,
which you just found from the 45’, to the other end of the string
and place the mid section of the string over the pulley so as to
have the unit length hang vertical transferring the weight into a
horizontal pull. Does it pull the chain perpendicular to the pull
to the earth?
After doing these things (which the reader should also do!),
the Layman exclaimed:
L: “Remarkably so!” What does this all mean? Does that happen
for every chain? Are the unities different for each curve? What
does that mean?
W: What does unity make possible? How does constancy relate
to variability? Now that this chain has a number one, how will
the rest of the chain relate to it?
L: hmmmmmm Now that I have unity, the constant horizontal
tension as a length, I can compare it with what’s changing,
mainly the growing vertical tension, the growing length of
chain.
W: how should we do this?
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L: If I have this unity hanging here, I can rotate the chain down as
a length from the bottom point and compare it with the unity. As I
do this I make a triangle with one side constant, and the other side
growing, while rotating more chain down. Now I have the
relationship between the vertical and horizontal tension expressed
as a proportion here in these triangles.
W: Yes, that is what Leibniz and Bernoulli called the differential
relationship. You seem to be finding your way in this study.
W: And, before, what were we interested in comparing this
relationship with?
W: We know what is going on inside the chain. But, what is the
cause that creates these effects that we have measured? What is
the cause of the shape of the space that the chain is in?
L: The relationship of the vertical and horizontal components of
the direction of the impetus.
W: How do we compare these?
L: With the tangent triangles.
The Layman (and the reader) chose many moments on the
chain and extended the direction, drawing many tangent
triangles on the wall. He then took the length of chain at those
moments, and swinging those lengths of chain down, related
them to the constant horizontal tension. He found that the
two triangles were similar.
L: Now we know why chains hang the way they do. I guess I did
learn something. I never thought of the mathematics I study in
this way.
L: Excuse me? The changing relation isn’t an efficient enough
cause for you?
W: It’s an efficient cause, but we’re looking for something more.
L: What sort of cause are we looking for?
W: A cause defining the space expressed by the chain’s curve.
You said before the chain is being pulled to the earth. Who
realized this?
L: Well that’s Newton
W: Right; he got hit by a coconut! And the light turned into
darkness for him. But the shadow world made him go nuts and he
was forever doomed in the infernos of Dante.13 Yet his legacy
was a byproduct of a fight between Leibniz and the Cartesians
who hated the true discoverers of our civilization. Have you ever
heard of Johannes Kepler?
L: Yes, he had those three laws of motion.
W: Actually, that was Newton’s shadow of Kepler’s true
hypothesis. For Kepler, the immaterial idea of the sun generates a
magnetic whirlpool of harmonic characteristics; the laws of equal
area/ equal time becoming its effects. Kepler took the motions of
the planets as effects seen as reflections of a necessary physical
cause, itself the result of an immaterial ‘idea’ produced by the
sun. And in turn, this idea had a higher cause, the true substance
investigated by Kepler- that is, Reason itself.14
L: So this means that the relationship of the constant horizontal
tension to the growing vertical tension is everywhere proportional
to every moment of direction. Now I can precisely know which
portion I feel is due to the horizontal tension and which to the
vertical tension.
W: What more does this tell us about the relation between what is
constant and what changes? What else is constant about this
curve besides the horizontal tension?
L: Although the sides of the tangent triangle constantly change
their proportion, at every moment of the curve, they express the
physical relationship of the two tensions, as we constructed it. So,
not only is the horizontal tension constant, but the relationship of
the two triangles is always constant.
∆υναµις Vol. 1, No. 1
Therefore, Leibniz, being a student of Kepler, knew that the
curve the chain takes is an effect of a larger process of change in
the motions of the heavenly bodies15. Leibniz, while he was
looking at the characteristics of the physical hanging chain, was
chiefly concerned with the minds investigation of reason. Seeking
the cause in Reason, he turned the problem of physics into
discovering the dynamic determining the effects, where the
predicates thus become a clear expression of the substance.
Leibniz’ concept of individual substance, as the true cause of
action was like Kepler’s concept of ‘idea’; the immaterial force
causing the motions of the planets. As Leibniz says:
“ When a number of predicates are attributed to a single subject
while this subject is not attributed to any other, it is called an
individual substance. The subject term must always include the
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predicate term in such a way that anyone who understands
perfectly the concept of the subject will also know that the
predicate pertains to it. It is in the nature of an individual
substance…to have a concept so complete that it is sufficient to
make us understand and deduce from it all the predicates of the
subject to which the concept is attributed.”16
This was the difference between the Cartesians and Leibniz. Like
those whom Kepler refuted, [Ptolemy, Copernicus, and Brahe]
the Cartesians explained the universe through the mechanical
actions of extension, i.e., its predicates: size, figure, body,
motion, etc. Though causes were not denied, but located, without
reason, in mathematical laws, such as Descartes rules for
motion17.
Hence, the Catenary, cannot be the cause of itself. But when the
mind obtains a perfect concept of the ‘substance’, the curve is
known, not as the continuous function of the physical differential,
but as an expression of the dynamic. Since it was Leibniz’ intent
to demonstrate this principle of perfection, he found the “best of
all possible constructions.”
The physical relationship between the two tensions we discovered
just now seems to reflect the mechanics of the curve, but how do
we come to a geometry of change reflecting Kepler’s hypothesis?
L: Well then, I think you might be turning me into a fool. I can
honestly say that my idea of this physical chain has been
challenged just now. If this curve is an expression of gravity,
defined from Kepler’s idea of the solar system, I guess the
question I have now is, how did Leibniz discover this ‘substance’
of the Catenary?
W: Lets look at the effects again and see what we can derive.
L: Ok.
W: To what can you relate the relations of the growing chain and
constant to, besides the direction?
L: There is also the height of the curve arising from the physical
constant, where the height is taken from the abscissa created from
the bottom of the constant
W: What if we take first the 45`` moment and examine the
height? What’s the relation of the height at the unit, as the
diagonal of our triangle, with the length of chain and the
constant?
L: Hmmm, that’s incredible, it’s the diagonal of the triangle
whose sides are one!!
W: This here gives you a square where the sides are equal to the
constant pull that is perpendicular to the direction of a falling
object, and the diagonal is the height of the unit of the Catenary.
It seems we have transcended Meno!!
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L: Who’s Meno?
W: You’ll have to read Plato to do justice to that!!
L: Is that true then for every height on that curve?
W: Test it.
The Layman (and the reader) picked many moments on the
chain and constructed a growing series of triangles as before,
but this time paying heed to the diagonals instead of the sides.
Then he checked these by setting them under the moment of
the curve to see if they were the right height. After much
labor he concluded that that they seemed to be true for every
height.
21
L: Yes, we just take the diagonals of these triangles and stack
them up and the tops of all of them will be the curve.
W: But how would you know where to stack those heights if you
didn’t have a chain to stack them under? You could stack each
individually, but where? How far apart? And even if you knew
this, you could never stack every single one; therefore, to know
this curve and all of its predicates, is not to describe it for each
point with different numbers for the heights, as you may with
your graphing calculator, but, to generate this curve continuously.
You must know that the growth of the height is the product of a
continuous function that, unlike the continuous physical
relationship of the tensions, unveils the reason of the curve.
W: Good, now investigate these lengths. Relating the unity with
the chain and heights you found, what numbers arise?
Therefore, these heights are also effects, merely shadows, of the
physical pathway, and can never be explained by themselves, as
the Cartesians attempted to do. To know how to generate the
Catenary without the chain, one must form an idea of the
substance within the mind as a metaphor for the true substance.
L: The diagonal of the height of one was the square root of
two….
L: I’m not sure what you’re talking about with all of that, but
what continuous function was this curve only an effect?
W: Another name for that is the geometric mean between one and
two.
W: Leibniz after consulting Theatetus about his new idea, and
finding no objections, concluded that all incommensurables, and
indeed all linear magnitudes were from thence better known and
redefined as the arithmetic means between two extremes; thus
redefining all sense perceptible extension.
L: Yes. Ok, well then, if the length of chain is two, then the
diagonal is the geometric mean between one and five, or the
square root of five, and then if the length of the chain is three the
diagonal is the mean between one and ten or the square root of
ten… and, well, I can’t go any further.
W: Yes, there is a physical limitation here. But could you
generate the geometric means in between those?
L: How so?
W: He compared the length of chain to the height of chain. In
what way could you relate these two lengths?
L: You could add or subtract the length of chain from the height.
L: Hmmmm, yes I could construct the gm between one and three
using the gm of one and two, and construct the gm between one
and six using the gm between one and five, both of which I just
constructed, …but the others would be harder.
W: What does that create?
L: Two lengths, for the length of chain equal to the constant, the
square root of two minus the constant and plus the constant.
W: Yes, but do you think with some time we could find all the
geometric means between all numbers, including whole numbers,
fractions, and incommensurables?
W: And for the other lengths?
L: I suppose we could. Oh, so you are saying there are heights
that are not diagonals of whole number sides?
W: Right, the triangle growing continuously would generate
diagonals expressing all numbers.
L: Yes, if the long side is the gm between one and three then the
diagonal would be two. Or if the long side were four thirds then
the diagonal would be five thirds. Ok, yes so the heights can be
given a number, rational and incommensurable.
L: The square root of five minus two and plus two, and the square
root of ten plus three and minus three….
W: How do these lengths relate to the diagonal?
L: The square root of two minus the constant added to the square
root of two plus the constant will give me twice the square root of
two…divide that in two and I get back to the square root of
two… the diagonal is therefore the arithmetic mean between to
two extremes.
W: Does the ability to number these heights individually allow us
to know this curve continuously for every moment?
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W: And how do those extremes relate to the
constant?
L: The square root of two minus the constant is to
the constant as the constant is to the square root
of two plus the constant…so the constant is the
geometric mean between the two extremes..…
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W: A geometric mean derived from the physical
force. Indeed! And would these relations be true
for the diagonal and long side of the square root
of five minus two and the square root of ten
minus three?
L: Yes…
W: And what about the diagonals whose sides
would not grow arithmetically…..would they too
have this relation?
L: Ummmm, yes they all would.
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W: How do you generate all the lengths in between those?
L: Well, I could just keep doing what I’m doing.
W: Now, continue to compare the long side of the triangle to the
hypotenuse as it grows and tell me what you generate..
W: And do the spirals, after crossing the bottom point, ever come
back around?
L: Uhh. I see. No.
The Layman (and the reader!), constructed many extremes of
the growing length of chain compared to the growing diagonal.
After much speculation he exclaimed:
L: Huh, these comparisons yield many lengths which form a
curve…. two curves formed by these marks of addition and
subtraction of the length of chain!
W: What are these curves?
L: I don’t know
W: How are they changing?
L: Well, the two curves tend toward the point at the bottom.
W: Do you think they stop there? Now, do the other side.
The Layman(and the reader) did the same for the other side.
L: It appears that the moment at the bottom of the chain, is not
where the spirals stop, but where they cross. For now I see two
spirals which start at the bottom and crossing the bottom of the
chain, they continue out and curve back towards…… I don’t
know.
W: Do they start at the bottom? I don’t see that you constructed
the spirals that far down.
L: Uhhhhhhhh…. Yer right….
∆υναµις Vol. 1, No. 1
W: So if you were at the bottom of the spiral, where would the
other extreme of the spiral be?
L: Hmmmmm….. well, first, So, that moment is to one as one
would be to…. hmmmm…. Ok, to make a mark at the bottom
point, the chain would have to be long enough to get a triangle that
would compare an infinite side to an infinite diagonal.
W: For who dare measure the infinite, but One?
L: Ah, this is profound, one is the mean between the infinitely
small and the infinite and everything in between!
W: Yes, but there’s a point you’re missing. One is more
meaningful.
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L: It has more meaning than the infinite?
the heights?
W: Keep unity in mind. But, for now, lets move on. What is the
arithmetic mean of the spirals?
L: Can’t we relate the length of chain to
the height as before?
L: It’s the diagonal.
W: try it….
25
W: And what does the arithmetic mean, taken continuously,
generate?
L: Wouldn’t that be a straight line?
W: Right, and what does the geometric mean taken continuously,
generate?
L: A curve!
W: So the arithmetic mean relates to the straight, and the
geometric mean to the curved?
L: This is true.
W: So what do these spirals tell us about the diagonals?
L: They are always the arithmetic mean between the growth of
the two spirals.
W: How does this redefine the knowability of the heights?
L: I’m not sure
W: Remember this crucial point. Going beyond the visible path of
the chain, we related the physical forces to the physical directions
at every movement. This path always expressed the relationship of
the forces, leading us to investigate the diagonals that were given
reason from the spirals.
Layman (and reader) picked points on the curve
drawn on the wall. Finding the length of chain
at those points, he rotated the length down to
subtract it from the heights and rotated it up to
add it to the heights.
Continue now passing through this window of the unseen
investigated tangent to the visible domain, and proceed as Leibniz
did, to the visible curve of the chain itself. How can we find a
clear concept of the substance as Leibniz said a “Concept so
complete that it is sufficient to make us understand and deduce
from it all the predicates of the subject to which the concept is
attributed.”18
This brings us back to what we had discussed before, concerning
the knowability of the heights which turned out to be the diagonals
of the triangles.. How do you really know that those diagonals are
the true heights? What is the continuous function that will stack all
the heights correctly?
L: Well, I do remember that Jungius proved it wasn’t a parabola.
What would the equation be that could express this kind of
arithmetic geometric relations we’ve been running into? I wonder
if we can find the rate at which x grows as y grows. Let me see…..
W: What can we relate to the sense object of the curve drawn by
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L: Subtracting the weight of chain from the height gives me many
lengths….but what are they?
W: At one, the tangent to the curve at 45 is similar to the diagonal
in your triangle, right?
26
The Layman (and the industrious reader!) swung the
chain, and made many lengths by the subtraction and
addition to the height. The Layman noticed that the
marks formed on the wall by this action formed two
curves.
L: yes
W: so check to see if in the triangle the square root of two minus
one is the same length as the length you get when you swing the
chain of one down on your height at one.
L: it is the same
W: ok, but how do you really know…
L: I will do more……
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L: Would you look at that!
W: You are learned in geometry, now what are they equal to?
W: How would those curves grow?
L: Wow! They’re equal to the unit! What is constant about these
curves is the same as what is constant about the Catenary.
L: ….. I’m at a loss.
W: As we did for the Catenary before, ... if we find what is
constant, then we can discover the nature of the curves, by relating
what is constant to what is changing. In the Catenary we found the
constant horizontal tension and then unfolded the relations to the
growing length. What is constant about these curves?
L: I don’t know
W: Take some tangents.
Layman(and reader) takes tangents to the curves formed by
the marks.
W: Yes, and something else. Notice how first, each curve’s
subtangent could be the subtangent of the other curve at the same
time, but of tangents taken at different moments.
L: Yes?
W: At the bottom moment, which revealed the constant tension,
the pull is in both directions, right?
L: Yes, that’s astonishing! These curves reflect the physical
action.
W: Notice secondly, if you took the tangent to each curve such
that one tangent made a subtangent in one direction, and the other
a subtangent in the opposite direction, then the geometric mean
between the positive and negative subtangents would be the
constant tension, the one touching the bottom. L: And the
geometric mean between one and negative one is the square root
of negative one… hmmm this is getting more interesting all the
time. Is this why imaginary numbers come in pairs?
L: Oh! The subtangents are always equal.
W: Good question. Ok, now what does the constant subtangent tell
us about the growth of these curves?
L: They grow arithmetically in one direction…
W: And, what about the other direction??
L: Well, the logarithmic spirals had equal angles, and geometric
growth, are these vertical lengths growing geometrically?
W: How would you know that?
L: Didn’t the heights equal the diagonals as I had shown?
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W: Yes
L: So, these geometric extremes on the spirals will be the same
geometric lengths on the curves!
W: These are called natural logarithmic curves.
28
we’d generate the two spirals each growing opposite to the other
with the arithmetic mean of a straight line.
W: Yes. Now what kind of continuous action could you apply to
that above that would transform the spirals into the natural
logarithmic curves which would in turn transform the straight line
into a Catenary?
L: Natural logarithmic curves?
W: Yes, for the constant subtangents, from which we derived the
arithmetic growth, were generated from the constant horizontal
tension in the Catenary, not simply geometrically constructed. So,
how else does the constant tension relate to these logarithmic
curves?
L: It is the geometric mean!
W: And what would this geometric mean form, taken continuously
between the curves….
L: Good question…. Well, the equal angular growth would be
transformed into some kind of arithmetic growth along the
abscissa…
W: If you took every moment of the spiral between the square root
of two minus the constant to where the triangle disappears into the
constant, in what length of the abscissa would that infinite number
of lengths stack?
L: Well, we put the square root of two minus the constant
underneath the unit Catenary. I guess we’d have to find that
length.
L: hmmm.. a tangent at the weightless moment of the Catenary…
So this tangent, is physically derived!
W: And how could you find that length without the Catenary?
W: yes,
L: I’m stumped.
L: So, what you called before, the impetus of action at the moment
of the chain, is here expressed geometrically as this tangent,
everywhere the geometric mean of these curves.
W: Well, for now we can put that question aside. Let us return to
what we’ve been seeking. What do you say now? What is the
substance of the Catenary?
W: What is the arithmetic mean between the curves?
L: The Catenary!
W: Now, think back through the process of this construction.
Remember we asked how we could generate this curve without the
hanging chain?
L: Yes, I remember that.
W: Ok, So what is the constant unfolding relationship that
generates the Catenary curve?
L: Hmmmm, lets see, I know by now it must involve the unit of
the Curve, the length of chain equal to the horizontal tension.
W: Good thinking… and?
L: I don’t know, it definitely has something to do with the
relationship of the curves.
W: Ok, tell me what you make of the following: Take any
arbitrary right triangle and keep one side constant to represent the
unit of the Catenary. While the other side grows, if you could
constantly add and subtract the growing side from the height, what
would you get?
L: Oh, right, if we could construct a continuous growing triangle
with an action of continuous addition and subtraction in that way,
∆υναµις Vol. 1, No. 1
L: It is the arithmetic mean between two logarithmic curves. Now
I can construct every point without the curve. I think we have now
truly found the continuous function of the Catenary.
W: True, but this is not the substance. Have we not been seeking
for a cause?
L: Yes, the geometry explains it
W: But the cause is not in the geometry.
How did this geometry arise?
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L: Oh, right, first we found the physical differential.
W: And only one moment expressed this physical differential
relationship?
L: Right, only the bottom could give us the length of chain equal
to the constant, which is how we found the differential. All these
curves we constructed equal zero at that bottom moment.
W: How did you know the point as zero?
L: What do you mean how do you know the point?
W: Like the other moments of the Catenary, we only know this
moment as an effect of the growth of the other curves.
Furthermore, the whole domain of these complex curves is the
reflection of a continuous physical action, of which the curve is an
effect. How does that redefine this zero?
29
W: Now you remind me of those poor Cartesians, who spent so
many hours gazing at the motions of bodies in extension, never
knowing what their eyes observed. This weightless moment may
seem a paradox in the geometry, but is in truth a singularity
reflecting the substance; it is the primary predicate, the mirror or
gateway to the substance.
As Leibniz says:
“It is only atoms of substance, that is to say, real unities
absolutely destitute of parts, which are the sources of action and
the absolute first principles out of which things are compounded,
and as it were, the ultimate elements in the analysis of substance.
One could call them metaphysical points. They have something
vital … and mathematical points are the points of view from
which they express the universe….
It is only metaphysical points, or points of substance…
which are exact and real, and without them there would be nothing
real, since there could be no multitude without true unities.”19
We couldn’t be farther from Euclid. For our point is not a point,
but, indeed, the most perfect demonstration of what Leibniz called
the infinitesmal, the relationship guiding the unfolding of the
curve at every moment. From this mathematical point, the true
metaphor for the idea, the substance, the monad, comes to light.
Now further, do not all points in space contain this metaphysical
infinitesmal? All matter is subject to the substance expressed by
Catenary. So all points reflect what is only known in this one
point, thus making the infinitesmal, truly infinite and universal.
L: I think we can agree, that this is a better form of geometry, but
what you just said has nothing to do with it!
W: Layman, you are a geometer, but what is the cause of
geometry, what is the purpose?
L: Geometry serves its own purpose!
W: Yet, Leibniz created this new domain of geometry as a
language to investigate the true substance of the universe: Reason.
L: What does Reason have to do with it?
W: Wasn’t it your reason that led you to a knowledge of the
geometry?
L: Yeah, but my reason has nothing to do with the geometry.
L: Hold on, wait a second, Euclid defined the point for us; a point
is that which has no part, a zero!
W: Ahh, but this moment has a part to play! Indeed the leading
role!
L: How can zero play a part, you are making no sense!
∆υναµις Vol. 1, No. 1
W: Unlike geometers, using the effects of an already made
discovery as tricks to describe things, Leibniz knew that
investigating the universe is an investigation of ideas, and the
Reason for them.
L: I care not! Why should I worry about where the discovery came
from?
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W: There is a problem in the way your assumptions are causing
you to think here, Layman. For the axioms to which you have
submitted in order to receive your degree have put you in Euclid’s
box, in which reason does not reside.
30
principle of ‘perfection’ embodies the metaphysical intent of
Leibniz’ work
5
L: Well then, you tell me. What’s outside the box?
Lyndon LaRouche, Current Status of the LaRouche Riemann
Method, ICLC internal memo from 1985
W: Doesn’t Leibniz’ construction express perfection?
6
L: How can you seriously try to relate perfection to geometry?
Lyndon LaRouche, Cauchy’s Infamous Fraud, EIR April 1st
2005
7
W: For as Plato and Theatetus’ concept of ideas was
incommensurables, Leibniz advanced the science of ideas, to a
science of infinitesmal, opening the way to investigate the mind’s
relation with the physical action of unseen universal physical
principles.
He demonstrated, as Theatetus began to with the issue of the
incommensurable, that space is not linear, not Euclidean!
In discovering the infinitesmal, Leibniz, following Kepler,
succeeded in capturing the paradox of physical action, and opened
the road to a physics beyond geometry. The language he used to
pronounce the name of his captive expressed, ironically, the
principle of perfection, and thus redefined all physical action as
thenceforth being known only through what would later be called
the Complex Domain.
Thus, Leibniz, intending to demonstrate the principle of
perfection, did so with the best of all possible constructions for the
Catenary, validating his conception of the universe being “at the
same time the simplest in its hypotheses and the richest in
phenomena, as might be a geometric line whose construction
would be easy but whose properties and effects would be very
remarkable and of a wide reach.”20
L: Well, this has all been very interesting, but I’ve got a degree
I’ve got to worry about. I think I’ll stick to geometry. See you
later.
Johann Bernoulli Lectures on the Integral Calculus, translated
by Bill Ferguson in 21st Century Science and Technology,
Spring 2004
8
Lyndon LaRouche, The Principle of Power, EIR, Dec. 23 2005
Box 17, p.68.
9
Gottfried Leibniz, The String Whose Curve Is Described by
Bending Under Its Own Weight, and the Remarkable Resources
That Can Be Discovered from It by However Many Proportional
Means and Logarithms, translated by Pierre Beaudry in Fidelio
Spring 2001
10
It is worth noting here that for the 7 years prior to writing this
attack, Leibniz spent a great deal of time in the Harz mountains
advancing the science of mining, producing the following
inventions: a minimal friction pumping system, which required
less power, a water system for the mine that created a
continuous stream of power, and windmills able to work on less
wind than ever before. Although his inventions proved to work
magnificently, the project was eventually sabotaged. One could
ask, did the eventual sabotage of Leibniz’ mining project cause
him to find it necessary to launch his political attack against the
Cartesian dogma holding back science? Did his work with
simple machines give him the insight into the Cartesian fraud?
11
Gottfried Leibniz, A Brief Demonstration of a Notable Error
of Descartes and others concerning a Natural Law,
Philosophical Papers and Letters, Edited by Loemeker p.296
12
References
1
2
Boston Office: [email protected] and [email protected]
Lyndon LaRouche, Earth’s Next 50 Years
3
Gottfried Leibniz called such an expression, ‘perfection’; thus,
another way to pose the question is, how can the technology be
applied such that the whole economy’s expression is in
coherence with this principle of ‘perfection’?
4
Euler and Mapuertuis came up with the term ‘least action’ to
claim authorship over a principle Leibniz had already
discovered concerning the characteristic action of the universe.
[see the university text on Maupertuis principle of Least Action]
Although the term ‘least action’ is common, the terminology of
‘perfection’ will be used in this present inquiry instead, as the
∆υναµις Vol. 1, No. 1
Leibniz, Discourse on Metaphysics and Correspondence with
Arnauld, translated by Montgomery, Open court publishing
1902, p. 239
13
John Maynard Keynes, Essays in Biography (W. W. Norton
& Co 1963) See the biography on Isaac Newton
14
For a full working through of Kepler’s discovery of Universal
Gravitation see the LYM Animations website:
wlym.com/~jross/kepler
15
In 1689-90, the same period he discovered the Catenary
principle, Leibniz had been working through Kepler’s discovery
of Universal Gravitation. Quoting from his 1689 Essay on the
Causes of Celestial Motion (soon available from David
Dobrodt):
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“It was … brought about by divine providence that [Tycho's]
observations and efforts come into the hands of an incomparable
man, for whom it had been preordained that they would be
preserved, so that he might first make known to mortals, the
laws of the axis of the heavens and related matters and the faith
and laws of the gods. This man accordingly discovered that each
of the primary planets describes an elliptical orbit, in one of the
foci of which would be the sun, moved according to the law that
the areas swept out by the radii drawn from the sun to the
planet, are always proportional to the times.”
31
motion or rest; rest can be attributed to any one of them you
may choose, and yet the same phenomena will be produced. It
follows therefore(Descartes did not notice this) that the
equivalence of hypotheses is not changed by the impact of
bodies upon each other and that such rules of motion must be
set up that the relative nature of motion is saved, that is, so that
the phenomena resulting from the collision provide no basis for
determining where there was rest or determinate absolute
motion before the collision.”
In his Explanation of the system of nature 1695:
The influence of Kepler upon Leibniz can be seen most
prominently in the maturation of his science of Dynamics.
Kepler demonstrated in Part I of his New Astronomy, On the
Equivalence of Hypotheses, that without the physical cause of
the motions being discovered all the different hypotheses of
Ptolemy, Copernicus, and Tycho were all geometrically
equivalent. This thesis by Kepler of the failed statistical method
shines bright in the following locations by Leibniz. In Leibniz’
paper, Critical Thoughts on the General Part of the Principles
of Descartes, 1692 he says:
As for absolute motion, nothing can determine it
mathematically, since everything ends in relations. The results is
always a perfect equivalence in hypotheses, as in astronomy, so
that no matter how many bodies one takes, one may arbitrarily
assign rest or some degree of velocity to any one of them we
wish, without possibly being refuted by the phenomena of
straight, circular, or composite motion.
16
Leibniz, Discourse on Metaphysics, Loemker, p.307
17
“If motion is nothing but the change of contact or of immediate
vicinity, if follows that we can never define which thing is
moved. For just as the same phenomena may be interpreted by
different hypotheses in astronomy, so it will always be possible
to attribute the real motion to either one or the other of the two
bodies which change their mutual vicinity or position. Hence,
since one of them is arbitrarily chosen to be at rest or moving at
a given rate in a given line, we may define geometrically what
motion or rest is to be inscribed to the other, so as to produce the
given phenomena. Hence if there is nothing more in motion
than this reciprocal change, it follows that there is no reason in
nature to ascribe motion to one thing rather than to others. The
consequence of this will be that there is no real motion. Thus, in
order to say that something is moving, we will require not only
that it change its position with respect to other things but also
that there be within itself a cause of change, a force, an
action.”[emphasis added]
Leibniz, Critical Thoughts on the General Part of the
Principles of Descartes, Loemker, p. 393
In his Specimen Dynamicum of 1695
For example, Rule One, if two equal bodies A and B, with equal
velocities collide, both will be deflected with the velocities
equal their approach, maintaining the Q of M. If body A had a
mass of 4 and B a mass of 4, both with velocity 2, then the Q of
M of each is 8 and the total 16. After the collision it is still 16.
“Force is something absolutely real even in created substances
but that space, time, and motion have something akin to a
mental construction and are not true and real per se but only
insofar as they involve immensity, eternity, and activity or the
force of created substances. Hence it follows at once that there
is not vacuum in space and time; that motion apart from force is
in fact nothing but change of situation; and thus that motion
insofar as it is phenomenal consists in a mere relationship.
Descartes, too, acknowledge this when he defined it as
translation from the position of one body to the position of
another. But he forgot his definition when he deduced its
consequences and set up rules of motion as if motion were
something real and absolute. Therefore, we must hold that if
any number of bodies are in motion, we cannot determine from
the phenomena which of them are in absolute determinate
∆υναµις Vol. 1, No. 1
In 1692 after working through Kepler’s Discovery of Universal
Gravitation and Discovering the related Catenary Principle,
Leibniz completed a more thorough refutation of Descartes
including an exhaustive refutation of his rules of motion. What
Leibniz showed was that the mathematical law of the
preservation of quantity of motion contained jumps from rule to
rule, breaking from the continuity of motion, and the continuity
of reason! Understanding the dynamic as necessary for a
determination of motion, he saw the quantity of motion as an
effect of force. See the Jason Ross’ article in this edition of
Dynamis on the derivation of the quantity of motion from force.
As Leibniz demonstrates in his refutation, Descartes
mathematical rule is that all bodies in collision maintain the
same quantity of motion(Q of M), that is, mass times velocity.
Rule Two is that if A and B collide but A has a greater mass
then B is deflected and A continues; both with their earlier
velocities in the direction of B. In this case, if body A has a
mass of 6 and velocity 2, Q of M 12, and B has a mass of 4 and
velocity 2, Q of M 8, for a total Q of M of 20, then after the
collision the total mass now 10 must be moving at velocity 2 in
order to maintain the Q of M.
Here Leibniz objects. If here in Rule Two body A the larger is
reduced to come almost equal, that is, only an infinitely small
amount larger, than the idea that for body A to collide with body
October 2006
Experimental Metaphysics
Kirsch and Yule
32
B and yet maintain the same direction and take B along with it,
without being slowed down in any way by body B is absurd. But
further and more devastating is that Leibniz points out the
greater absurdity in the contradiction it creates with Rule One.
For when the two bodies were the same mass moving at same
velocity, they deflected in opposite directions, but now in Rule
Two if one is an infinitely small amount greater in mass than the
other, the lesser now changes direction. Where does the change
in motion take place? Must there not be a transition in direction
of motion? Mustn’t there be a slowing down for an object to
change directions? Never mind these questions- says a
Cartesian- the mathematics works out!
Rule 5, which states that A is greater than B, with A moving and
B at rest, then A carries B with it with the previous velocity. So,
if A had mass of 4 with velocity 2, for a total Q of M of 8, while
B was mass of 2 at rest, for a total Q of M of 0, then if A carries
B with it for a new mass of 6, the velocity must now be two
thirds less than it was before. Descartes gives no reason for this.
And in Rule 6, if A and B are equal mass, while A moves with
at rest, then A is deflected with three-fourths of its former
velocity and B is moved in the opposite direction of A with oneforth of A’s velocity. If A had mass of 4 and velocity 4, for a Q
of M of 16, and B with mass of 4 and velocity 0 for Q of M of 0,
then to maintain the Q of M, A mass 4 moves at velocity 3, Q of
M 12, and B moves with velocity 1, Q of M of 4, for a total of
16, which is what he meant to maintain. Here, Leibniz objects
again, that the rules contradict each other revealing absurd gaps
in the reasoning process of Descartes, who clearly meant only to
maintain a mathematical formula, placing the reason for the
motion in the motion itself.
For if in Rule 5 when A is greater than B, then A carried B
along with itself, but in Rule 6, if A and B were equal then A
deflected with three-forths of its original velocity. So if A were
an infinitely small amount larger than B, then it carries B with it
losing no velocity, but if it loses than infinitely small amount
and becomes equal than instantaneously changes directions with
no process of change! This creates further contradictions with
Rules One and Four, which the reader is encouraged to discover.
18
Leibniz, op cit.
19
Leibniz, A New System of the Nature and Communication of
Substances, as well as the Union Between the Soul and Body,
1695, Loemker, p. 456
20
Leibniz, Discourse on Metaphysics, Loemker, p.306
*** We’d like to thank Dan Yule for his crucial role in the
development of the pedagogy for this article as well as his
contribution and help with the dialogue***
∆υναµις Vol. 1, No. 1
October 2006
The Inertia of Descartes’ Mind
33
Ross
The Inertia of Descartes’ Mind
Jason Ross1
After working through Leibniz’s disproof of Descartes’s
quantity of motion with a member of the LaRouche Youth
Movement,2 the university physics student opined:
y = 4 – 2x
And substituting into energy, we have
“I admit that Leibniz was right about the error of Descartes. His
use of ‘quantity of motion’ or momentum as a measure for the
power of a body was wrong, and Leibniz’s vis viva, or ‘kinetic
energy,’ as I call it, is right. But still, both ideas are important.
Descartes may have made a mistake, but the Law of the
Conservation of Momentum is used all the time – in fact, where
would the universe be without it?
“Think back to your physics class,” the student continued:
“collisions of elastic bodies obeying two different, independent
laws, both of which are true: the law of conservation of
momentum, and the law of conservation of energy. (“Energy”
as used here, is a deadened, flattened form of Leibniz’s concept
vis viva.) You need both of them to solve problems! Here, look
at my physics textbook!”
½(2x2 + (4 – 2x)2) = 4
(2x2 + 16 – 16x + 4x2) = 8
6x2 – 16x + 8 = 0
3x2 – 8x + 4 = 0
Which, by applying the quadratic equation, we arrive at
x = (8 ± √(64 – 48))/6
x = (8 ± 4) / 6
x = 2 or ⅔
Problem:
Two perfectly elastic bodies, A and B are free to move along a
common line. A has a mass of 2, and B a mass of 1. Body A is
moving to the right with a velocity of 2, while body B is
stationary. What are the velocities of the bodies after they
collide? (Hint: elastic collisions conserve kinetic energy)
Solution:
Using m for mass and v for velocity (speed), the kinetic energy
of the bodies before the collision is:
∑½mv2 = ½(2·22 + 1·02) = 4
The momentum before the collision is:
∑mv = 2·2 + 1·0 = 4
Let’s call by x the velocity of A after the collision, and by y the
velocity of body B after the collision. Since by the laws of
conservation of momentum and energy, momentum and energy
must be the same after the collision as they were before the
collision. So we can say that:
∑½mv2 = ½(2·x2 + 1·y2) = 4
and
∑mv = 2·x + 1·y = 4
∑mv = 2x + y = 4
Solving the momentum formula for y, we have
∆υναµις Vol. 1, No. 1
The first solution would be if the bodies didn’t hit each other, so
we will use x = ⅔, and then we determine that y = 2⅔.
So, after the collision, body A has a velocity of ⅔, and B a
velocity of 2⅔.
“See!” exclaimed the student, “We are able to determine the one
outcome that would have both the same momentum and energy
as the bodies before they collided. Using these two formulas
together, we can predict the results of a collision! Using only
one of the two formulas leads to an ambiguous result.
Conservation of energy, your vis viva, by itself points to the
infinitude of possible motions which make this equation true:
½(2·x2 + 1·y2) = 4. So you could have (x=2, y=0) or (x=1,
y=√6), or (x=0, y=√8), or as many others as I’d like to list off.
Vis viva alone can’t tell you what will happen without quantity
of motion.
“Surely, this points to something real inhering in Descartes
concept of quantity of motion, doesn’t it? It has allowed us to
solve a problem which your vis viva, by itself, could not.”
“Or has it?” asked the LYMer: “Let’s take Leibniz’s idea of vis
viva in light of his argument in the Leibniz-Clarke Controversy
on the non-existence of an absolute space existing
independently of body:
Allow two observers to watch the collision described in your
textbook, the one as presented in your book, and another
observer, who is moving to the right with a constant velocity of
1. The first observer, as above, perceives A to have a motion of
2, and B to be standing still. The second observer, however,
October 2006
The Inertia of Descartes’ Mind
Ross
perceives A to have a velocity of only 1, and B to be moving
backwards with a velocity of –1.
Just as the first observer determines a vis viva of 4, the second
determines a vis viva of
34
References
1
Oakland office: [email protected]
2
Lyndon LaRouche, The Principle of Power, EIR, Dec. 23 2005
½(2·12 + 1·(-1)2) = ½(2 + 1) = 1½.
3
Now, allow the two Leibnizian observers to compare their
knowledge. The moving observer tells the first that, from his
point of view, the bodies have a vis viva of 1½, and that,
therefore, their vis viva after they collide must similarly be 1½.
The first observer, knowing that the vis viva from his vantage
point will be 4, is able to combine these two facts about the
post-collision motion of the bodies, thus:
A similar derivation, from Kepler’s actually physical
principles, of the mathematical form of Newton’s “Law of
Universal Gravitation” can be found in Appendix V to Lyndon
LaRouche’s The Science of Christian Economy
½(2·x2 + 1·y2) = 4
½(2·(x-1)2 + 1·(y-1)2) = 1½
Here, the second equation is the first observer’s restatement,
from his point of view, of the second observer’s knowledge.
The first observer knows that the second observer sees the same
speeds as decreased by 1 because of his motion. Now, restating
the equations for the two observers (while multiplying both
sides by 2):
2x2 + y2 = 8
2x2 – 4x + 2 + y2 – 2y + 1 = 3
And subtracting the one equation from the other, we arrive at:
4x – 2 +2y – 1 = 5
4x + 2y = 8
2x + y = 4
But wait! This is nothing other than the equation from your
textbook for the conservation of momentum:
∑mv = 2x + y = 4
Since we can derive your formula of conservation of momentum
entirely from Leibniz’s vis viva and relativity of observers, this
means that we can determine the outcome using the single
principle of vis viva!
So, what has Descartes’ idea of quantity of motion contributed?
Not a thing! How could it be a principle, if the universe doesn’t
need it to exist?3
∆υναµις Vol. 1, No. 1
October 2006
A Very Useful Discovery Using Leibniz’s Calculus
Martinson
35
A Very Useful Discovery Using Leibniz’s Calculus
Peter Martinson1
In April 2005, LaRouche called on the LaRouche Youth
Movement to develop a curriculum for teaching Calculus, which
would effectively nuke the fraud committed by the likes of
Augustin Cauchy2. In response, many of the LYM offices had
assembled groups to study the same physical problems
confronting Gottfried Leibniz when he made his discovery.
We’re now in the process of mastering the core principles at a
rapid rate, and taking the results onto the world’s streets and
university campuses, to uplift humanity, and stop the collapse
into a world dark age.
I was lucky enough to pop into Los Angeles in the midst of the
initial Calculus ferment last year. Sky Shields had just issued a
list of Calculus problems, and the LA LYM was busy trying to
solve them. Adrian Yule, Chase Jordan, and I succeeded in
cracking open the problem of finding the derivative of the Sine
function. This paper is an exposition of our Very Useful
method of solution, written in the form of an attack on the
lingering fraud of Cauchy.
1. Leibniz’s Calculus is Physical
The human body is equipped with biological sensory organs that
cannot see what causes the effects which are registered within
those organs. But this doesn’t mean you cannot “see” the
causes with your mind’s eye. The subject of Leibniz’s Calculus,
is Man’s ability to discover and master Universal Physical
Principles, which order the sensory domain.
As LaRouche states it, “What bounds the universe is the
dynamically interacting array of universal physical principles.
Taking that into account, how might we expect to find a
universal physical principle as an object of experience, an object
recognized as such in the circumstance in which its effect is
relevant to the situation we are considering? What form, as an
object, does that principle assume in that setting?
that you will never get to a circle, by increasing the number of
sides of a polygon ad infinitum. The circle is not one of the
polygons.
Cusa called the circle, the Maximum polygon. The circle
represents the boundary condition to the polygons. One effect
of this is, that the straight side of the polygon cannot measure
the circle’s circumference. The circle is generated by a higher
principle, which subsumes the lower ordered world of polygons.
Let’s look more closely into the circle, and see if we can locate
this principle. We will look at how a change in circular action
interacts with a change in linear action, these changes being
such tiny actions, that we cannot really see them. What Leibniz
developed, as the Calculus, was a method for projecting these
infinitesimally small changes into the visible. If a variable
references an infinitesimal, Leibniz prefixed a “d” to the
variable, such as “dx.”5
This is a somewhat modified version of what Yule, Jordan and
myself originally constructed last year. The reader will find this
construction very useful for further investigations into Gauss’
work on curvature,6 and Riemann’s breakthroughs in AntiEuclidean geometry.7
First, we will see what happens when we change the radius of
the circle. Let a circle AB be described about center O with
radius r, and let another radius be drawn from the center,
describing an angle ϕ , which meets the circumference at B. If
a line is dropped from B perpendicularly to the radius, at C, we
have created what is called the sine of the angle:
r sinϕ = BC .
“The answer? Try a point.
“At that point, how can we determine which universal principle,
such as universal gravitation, is operating? The principle is, as
Kepler emphasizes, acting efficiently at every imaginably small
interval, and yet smaller. It is expressed, thus, as a true
principle, a highly efficient apparent nothing, which we
recognize as a perfect singularity.”3
Gottfried Wilhelm Leibniz (1646-1716) was immersed in an
environment shaped by Johannes Kepler’s demand for a new
mathematics, for further investigating what Nicholas of Cusa
(1401-1463) discovered as transcendentals.4 The first known
transcendental relationship, was that between the curved side of
a circle and the rectilinear sides of a polygon. Cusa had shown
∆υναµις Vol. 1, No. 1
October 2006
A Very Useful Discovery Using Leibniz’s Calculus
36
Martinson
The
length
r cos ϕ
OC
= OC .
is
called
the
cosine
of
the
angle:
Extend the radius by length dr , to form a new circle ED, with
sine FE. Thus, FE = r sin ϕ + EG . Call EG, d ( r sin ϕ ) .
This is the length we want to find – the change of r sin ϕ .
More challenging – and fun! – is the change of r sin ϕ when
ϕ changes. Return to our original circle AB of radius r,
ϕ , and r sin ϕ = BC . Increase the angle by
dϕ . Now, d (r sin ϕ ) is EF − FG . Since the arc swept out
is proportional to the radius, EB = r ⋅ dϕ . Also, since GB is
parallel to FC, GBO = BOC = ϕ . Where is our similar
inscribed angle
triangle?
To find EG, observe that angle EBG = ϕ . By the similarity
of triangles, EG:BE::BC:OB, or,
d (r sin ϕ ) r sin ϕ
.
=
dr
r
Therefore, the derivative of r sin ϕ with respect to r is
d (r sin ϕ )
= sin ϕ .
dr
In other words, if the original sine were small (small angle),
then the increase or decrease of the sine with the radius would
be small.
∆υναµις Vol. 1, No. 1
Connect EB with a straight line. To be excruciatingly precise,
though EGB is a right angle, GEB is not exactly ϕ , but a little
less.
Since
OE
and OB are
separated by
dϕ , we see
GHE = ϕ + dϕ in Figure 5. But, since OEB is a little less
than a right angle, GEB is not quite ϕ + dϕ . But, GEB is
definitely a little bit {more} than ϕ , since OBE is also not a
right angle. Therefore, GEB is more than ϕ , less than
ϕ + dϕ !
Maybe it is
ϕ + 1 2 dϕ !
October 2006
A Very Useful Discovery Using Leibniz’s Calculus
Martinson
This is where the empiricists among us usually would exhibit a
psychotic reaction, throw up their hands, and refer to the nearest
calculus textbook for the formula. But, recalling the quote by
LaRouche earlier, we should remember that we’re not really
interested in calculating this tiny, almost nonexistent angle.
dϕ is very small. It’s so small, that you couldn’t measure it
with any known, or even any possible, instruments. It is beyond
the senses. This is LaRouche’s point. We are thus looking for
the principle involved, which keeps pushing the precise
measurement just out of our reach.
So, let’s stop splitting infinitesimals, and just let GEB = ϕ .
Now, we have our similar triangle. Hence,
and GE:EB::OC:OB, or
GE = d (r sin ϕ )
d (r sin ϕ ) r cos ϕ
=
.
r × dϕ
r
Thus, the derivative of r sin ϕ with respect to ϕ {phi} is
r cos ϕ . In other words, if the cosine is small, then the growth
of the sine with respect to the angle will be small, and vice
versa.
Here, we’ve located something. The growth of the linear sine,
as you change the curved angle, depends on the orthogonal
cosine. The change of the sine must, itself, change, depending
on what the cosine is doing. How does the sine know what the
cosine is doing? Either they’re discussing things between
themselves, or there must be a higher principle, which bounds
the interaction between the two orthogonal, transcendental
magnitudes, such that they appear to cooperate to produce the
circle.
Let’s look at what has replaced this geometrical analysis.
2. The Attack on Knowledge
In several of LaRouche’s recent works,8 he demonstrates that
the scientific works produced by Euler, d’Alembert, Cauchy,
and many of their associates during and after the so-called
“Enlightenment,” were not only scientifically invalid, but were
not {intended} to be scientific. Perhaps, there were some poor,
good-hearted folks who were swindled into believing and
mimicking these non-scientific frauds, but the overall effect was
similar to a slanderous propaganda attack.9 These frauds infect
today’s science education heavily, and the importance of
understanding them as frauds, and knowing the antidote, can be
seen by the woeful state of the world economy, and the content
of most present-day “scientific” journals.
Leibniz, possibly the most productive scientist ever, has been
virtually eliminated from university and other academic
literature. It is only recently that the true scope of not only his
genius, but his political revolutionary work, has been uncovered.
The ongoing intelligence work being done by Lyndon
LaRouche and his associates, continues to develop our
∆υναµις Vol. 1, No. 1
37
understanding of his role in the Great Conspiracy to end
reign of the Venetian, financier oligarchical enemies
humanity, who spawned the Synarchist International, and
right now aiming to destroy civilization, and, emphatically,
United States.10
the
of
are
the
Leibniz was the great forefather of the American System of
Political Economy. In the mid to late 1600s, he intersected a
fight between the followers of Descartes, on the one side, and
those of Kepler, Fermat, Pascal, et al. on the other. Leibniz
often provided crushing death blows to the Cartesians,11 though
they never admitted defeat. As Leibniz became the champion
anti-Cartesian revolutionary, he inspired and informed the
networks which would produce Abraham Kästner, Moses
Mendelssohn, Benjamin Franklin, George Washington, and
Alexander Hamilton. Thus, he also became the key target of
attacks by the same Venetian networks that sponsored Tomas de
Torquemada, Paolo Sarpi, Napoleon Bonaparte, and later,
Adolph Hitler and Benito Mussolini.
The culmination of the battle was the dispute between Leibniz
and that puny Cartesian, the Sun Myung Moon12 of science,
Isaac Newton, over, “who discovered Calculus first.” In 1712,
Newton wrote a conclusion to the definitive study on the issue,
conducted by the London Royal Society, which declared
Newton the discoverer.13 In truth, Newton never really
discovered anything, much less, the Calculus.
Over one hundred years later, in the midst of a renewed fight
over Calculus, a period today called the “Great Rigorization,”
the pig Augustin Cauchy pulled his fraud. Sitting on his perch
at the head of a destroyed Ecole Polytechnique in Paris, he
published his “limit theorem,” which hereditarily follows from
these definitions:
When the values successively attributed to the
same variable approach indefinitely a fixed
value, eventually differing from it by as little
as one could wish, that fixed value is called
the limit of all the others.
When the successive absolute values of a
variable decrease indefinitely, in such a way
as to become less than any given quantity, that
variable becomes what is called an
infinitesimal. Such a variable has zero for its
limit.14
Notice that there is no geometry involved. The argument for the
“rigorization” was that, Leibniz’s geometry was not accurate
enough. If geometry could be eliminated from the Calculus, to
be replaced by formal mathematical equations, then we
wouldn’t need to worry about more pesky constructions! We
could take the equations, and logically deduce the fundamental
axioms of the Calculus, which Leibniz obviously missed, since
he was so concerned with all his talk of discoveries.
The “limit” is included in the sequence of converging values of
October 2006
A Very Useful Discovery Using Leibniz’s Calculus
38
Martinson
the variable. In other words, the circle is really a polygon with
an infinite number of sides. Of course, to construct a circle,
nobody bothers to draw an infinite number of sides.
A = lim An
n →∞
The Greeks themselves did not use limits
explicitly. However, by indirect reasoning,
Eudoxus (fifth century B.C.) used exhaustion
to prove the familiar formula for the area of a
2
circle: A = π r .16
3. Free the Mind from Slavery!
Today, Mathematics, and Calculus most emphatically, is
mystified beyond belief. Here’s how the mystification is carried
out:
Sit the student in a huge
lecture hall with hundreds of
other soon-to-be debt slaves.
Have a crazed, unwashed,
and usually foreign man with
an unintelligible accent stand
at the front of the lecture
hall, scribbling miles of
incomprehensible formulas,
rules, and incantations on the
blackboard,
periodically
scaring everyone by pointing
out which incantations must
be memorized for the test.
Ted Kaczynski, a former
mathematics professor at
UC-Berkeley, resorted to
lobbing bombs at his
students.
The student leaves the
lecture hall, dazed, half
panicked and half zombified,
wondering how he or she
will memorize all of these
formulas and rituals in time.
This poor student will now typically spend his or her meager
free time, dealing with the anxiety through exciting nights of
binge drinking and dodging venereal diseases.15
What he or she is being packed full of, is nothing more than the
fraud of Cauchy. For an example, read this quote from the most
popular Calculus textbook on US campuses, James Stewart’s
Calculus:
The origins of calculus go back at least 2500
years to the ancient Greeks, who found areas
using the ‘method of exhaustion.’ They knew
how to find the area A of any polygon by
dividing it into triangles … and adding the
areas of these triangles.
It is a much more difficult problem to find the
area of a curved figure. The Greek method of
exhaustion was to inscribe polygons in the
figure and circumscribe polygons about the
figure and then let the number of sides of the
polygons increase … Let An be the area of the
inscribed polygon with n sides.
As n
increases, it appears that An becomes closer
and closer to the area of the circle. We say
that the area of the circle is the limit of the
areas of the inscribed polygons, and we write
∆υναµις Vol. 1, No. 1
Those equal signs are for real! Here, Stewart tacitly ignores the
discovery of Cusa, that the circle is not a polygon! This might
seem a fine point, but it happens to be quite crucial for the
future of humanity. Amassing sides of a polygon is not a
process which will achieve a circle, because the circle has no
straight sides. The circle represents a higher principle than
polygons. In the same way, no discovery is ever made, by
accumulating force fed formalisms and rituals. No matter how
well you master mathematics, in the form it is taught in current
university classrooms, you will never be taught how to make a
discovery.
Unfortunately, this ordeal is not intended to produce educated
humans. A mystical dogma has been used to replace the
discoveries made by true scientists like Leibniz. The student
will never know what Leibniz actually discovered through this
method of “learning.” He or she will be packed full of
equations, and a creeping suspicion that Aristotle was right there is no new knowledge, only periodic accidents which
produce new equations.
LaRouche has made it his mission, through his writings, and
through his development of the LaRouche Youth Movement, to
teach the method of making discoveries. Hopefully, this report
will prompt the young reader to make the healthy decision, to
drop everything and fight for the future of humanity. Join the
LYM’s fight to relive these discoveries of the past, to give to the
future generations.
References
1
Seattle Office: [email protected]
2
Lyndon H. LaRouche, Jr., Powers are Always Universals:
Cauchy’s Infamous Fraud, EIR, April 1, 2005
3
Lyndon H. LaRouche, Jr., The Principle of ‘Power,’ EIR,
December 23, 2005
4
Nicholas of Cusa, De Docta Ignorantia, translated by Jasper
Hopkins, cla.umn.edu/sites/jhopkins/, 1985
5
Gottfried Wilhelm Leibniz, Nova methodus pro maximis et
minimis, itemque tangentibus, quae nec fractas nec irrationales
quantitates moratur, et singulare pro illi calculi genus (A New
Method for Maxima and Minima as Well as Tangents, Which is
Neither Impeded by Fractional nor Irrational Quantities, and a
October 2006
A Very Useful Discovery Using Leibniz’s Calculus
Martinson
Remarkable Type of Calculus for Them), submitted 1676, and
published 1684 in Acta Eruditorum 3. English translation by J.
M. Child, The Early Mathematical Manuscripts of Leibniz,
(Open Court, Chicago, London, 1920)
39
16
James Stewart, Calculus, third edition, p. 39. (Brooks/Cole
Publishing Company, Pacific Grove)
6
Karl Friedrich Gauss, Disquisitiones generales circa
superficies curvas (General Investigations of Curved Surfaces),
p. 7. Published 1827, translated by Adam Hiltebeitel and James
Morehead (Raven Press Books, USA, 1965). Also, Gauss’s
work on Conformal Representation, 1824, as translated in
Smith, A Source Book on Mathematics.
7
Bernhard Riemann, Über die Hypothesen, welche der
Geometrie zu Grunde liegen, 1854 habilitation dissertation.
8
Op cit.
9
For example, recall the recent, mentally crippling effect on the
Baby Boomer generation, resulting from the Rohatyn-ShultzTrain sponsored “Get LaRouche” operations from the late
1980s. On John Train, see LaRouche PAC’s Spring 2005
pamphlet Bush’s Social Security Fraud – Stop George Shultz’s
Drive Toward Fascism!, p. 26.
10
There has been much written on the Synarchist International
recently, including the LaRouche PAC pamphlets Lyndon
LaRouche’s June 9 Webcast: Felix Rohatyn and the Nazis, and
LaRouche in Berlin Exposes Synarchist Enemies of the United
States. On the extended European conspiracy around Leibniz,
see H. Graham Lowry, How the Nation was Won: America’s
Untold Story (Executive Intelligence Review, Virginia, 1987)
11
Gottfried Wilhelm Leibniz, Discourse on Metaphysics, and
Specimen Dynamicum.
12
Laurence Hecht, Moonification of the Sciences: The RussellWells ‘No-Soul’ Gang Behind the Moonie Freak Show, 21st
Century Science and Technology, Winter 2002-2003. The
Reverend Sun Myung Moon owns more financial enterprises in
the world, and runs an international religious sex-cult, which has
massive influence in most major national governments. Your
congressman might wear a watch given as a gift by Moon, for
certain favors, possibly sexual. In reality, Moon is a “synthetic
personality” in the shell of a brainwashed Korean man, who is a
useful tool of international Synarchy. And, it is possible that he
united your parents in one of his “mass marriages.”
13
Lowry, p. 145
14
A. L. Cauchy, Analyse algébrique, 1821, pp. 19, 43. As
translated in Garrett Birkhoff, ed., A Source Book in Classical
Analysis, p. 2. (Harvard University Press, Cambridge, MA,
1973)
15
Or, very common today, through indulging in hard drugs,
such as heroin or crack cocaine.
∆υναµις Vol. 1, No. 1
October 2006
The New Biology
40
Quiroga and McGrath
The New Biology
Cecilia Quiroga and Thomas McGrath1
Man: Extending the Biosphere
Consider the way in which the living processes of this planet
have changed the morphological and physiological characteristics
of the planet as a whole relative to the surface of the moon, which
as far as we know had not been affected by living processes until
man arrived not more than 40 years ago. How does the change
that takes place on a planet void of living processes differ from
the change that takes place on a planet invaded by “living
matter?” If you can imagine the surface of the moon over a
milliard of years in a time-lapse video, you might see the
formation of craters, perhaps some moon dust slowly shifting
across the surface. Change is so slow that the footprints from the
first moon landing are still intact. The most significant geological
change on the moon is the impact of meteoroids or comets, which
are a rare occurrence. Frank Borman, commander of the Apollo 8
mission, commented of the moon “I know my own impression is
that it's a vast, lonely, forbidding expanse of nothing”. If you
were to then imagine looking down on the planet Earth and
witness it’s development over a milliard you would see the lush
forests develop, bodies of water forming, the extinction and
evolution of different animal species, ice ages, the re-emergence
of greenery, and the formation of deserts.
We are at a point now in the development of the planet where
mankind has reached physical and mental boundaries. As we
approach a human population of 7 billion there is increasing
concern over how the utilization of resources today will affect
future generations. Some look at the current wars, pandemics,
famine, energy crises, and other such pestilences as a
consequence of overpopulation, “Mother Earth’s immune
system” taking care of the virus which mankind has become. This
view, which asserts an underlying belief that man’s application of
his knowledge to change the biosphere is “unnatural”, has been
the one of the greatest hindrances to the types of development
∆υναµις Vol. 1, No. 1
Human Population Chart
What causes this?
which will be required for mankind to survive and progress. In
fact, with the present state of the global economy, if not
immediately addressed, this thinking will lead the world into a
new 14th century style dark age crisis. According to physical
economist and statesman Lyndon LaRouche, technological
progress, which induces an increase in population density, is not
only natural, but necessary for the development of the universe.
LaRouche’s discovery of relative potential population density and
pioneering work in the science of physical economics has made
him the most successful long range economic forecaster in
modern history. In order to have a successful reorganization of
the presently bankrupt international economic system,
LaRouche’s discovery in physical economics must be
rediscovered by policy makers. LaRouche has cited Russian
biogeochemist Vladmir I. Vernadsky’s work on how the
biogeochemical composition of the biosphere developed before
and after the advent of man. Vernadsky’s works are a good
pedagogical for understanding the role that mankind plays in the
development of the biosphere over geological time.
Vernadsky approached crucial questions that had once been
reserved to philosophical and religious inquiry from a
scientifically rigorous standpoint. He states “...one may make a
conclusion that biology cannot decidedly answer the question
October 2006
The New Biology
Quiroga and McGrath
41
its subsistence. In the development from single celled organisms
to organisms with highly complex central nervous systems, the
evolution of the biosphere as a whole has been shown to have a
general trajectory to increase the intensity and speed of the
“biogenic migration of atoms.”
Why does Animal Population Growth level off?
whether there is an impassable gap between the living and inert
bodies of the biosphere. I mean the biology based upon the now
available scientific facts and empirical generalizations. An
analysis shows that this question remains essentially unanswered
by biologists.”2 In asking the question “what is life?” Vernadsky
based his investigations on observing what life does and how it’s
actions change the biosphere. Think of Johannes Kepler, the 17th
century astrophysicist who demonstrated the idea of a universal
physical principle of organization, with his discovery of universal
gravitation. This discovery was the development of an
understanding that what we see are like the shadows in Plato’s
cave, a reflection of a certain unseen causes. Kepler laid the
foundation for modern physics, with many followers over the
centuries investigating light and magnetism. Vernadsky,
continuing in this tradition refused to evade the question of “what
is life” because he understood that the progress of science and
society depends upon gaining a better understanding of the
harmonies of the universe.
Instead of looking merely at the pair wise relationships between
individual organisms Vernadsky’s investigation launched a new
field of scientific research, biogeochemistry, in which he studied
the relationships between the biotic, abiotic and cognitive
processes as a whole over geological time. “A number of most
characteristic and important geological phenomena establish such
a character of the biosphere with certainty. It’s chemical
composition, as well as all the other features of its structure, is
not casual and is most intimately related to the structure and time
of the planet and determines the form of life observed... life is
continuously and immutably connected with the biosphere, it is
inseparable from the latter materially and energetically. The
living organisms are connected with the biosphere through their
nutrition, breathing, reproduction, and metabolism. This
connection may be precisely and fully expressed quantitatively by
the migration of atoms from the biosphere to the living organism
and back again-the biogenic migration of atoms.” 3 The “living
matter” although it is a small fraction of the mass of the planet as
a whole has dramatically changed the landscape in a way that
non-living matter is incapable of doing. In fact, the mass of the
biosphere makes up only 4.0384 of the entire mass of the planet
but the impact on the earth’s structure is extremely powerful. Life
will spread to any place that it might find the nutrients needed for
∆υναµις Vol. 1, No. 1
So, what is life? Life cannot exist separately from the abiotic, yet
as far as we know no other planet has expressed the harmonic
principle of life. The chemical elements found on earth are not
specific to this planet, so what could be the reason for a set of
elements to start interacting in a unified way as typified by a
living organism? What characteristics of the “primordial soup”
allowed for it to start changing the surface of the earth? Is it the
case that life is just a more complex form of chemical
compounds? How did the biosphere become structured in such a
way that it would create the conditions in which a creature that
would ask these questions emerged? Cognitive processes have
been able to create new chemical compounds and make
qualitative changes in the physiochemical composition. Human
beings have taken minerals and elements and have been able to
sculpt ideas into them and have formed physical economies.
Although ideas don’t have material attributes the generation and
transmission of them through generations has also drastically
changed the composition of the biosphere. Vernadsky called this
new phase space of human thought the “noosphere” taking the
Greek prefix “nooes” which means mind. A hierarchy of ordering
principles is most clearly evidenced by the fact that the
boundaries of life have been extended beyond the mere biosphere
with mankind’s exploration of the solar system the expression of
a new principle; cognition! The idea of cognition as a higher
ordering physical principle, or power as in the Greek word
dynamis is key for understanding the profound implications of
Vernadsky’s revolutionary work. Although not rigorously proven
until the development of the LaRouche-Riemann method,
Vernadsky had an intuitive sense that life was a universal
physical principle.
Like Kepler's revolutionary discovery of elliptical orbits in the
New Astronomy, Vernadsky’s life work elevated science beyond
that of the sophistries predating him. Vernadsky’s work implicitly
challenges the accepted second law of thermodynamics, the
Boltzmanniac dogma based on the cultish belief that the universe
functions as a closed mechanical system with a fixed amount of
energy and is therefore “winding down” towards entropy.
Vernadsky understood that a truthful scientific method requires
one to seek causes, and not simply deduce effects from localized
phenomena. In his studying the relationships between the living
and non living he found that “the whole work of the laboratory is
based on such a structure of the biosphere, on the existence of an
impassable sharp, materially energetical boundary between the
living and the inert substance.”5 The avoidance of the question of
where this energy comes has consequently retarded all fields of
science. As Lyndon LaRouche has quipped on the subject of
energy “The fact that we can measure the height of dogs, cows,
and people by the same yardstick, does not allow us to class all as
species of yardsticks.” You can measure the amount of energy
that is produced by any given process but understanding the cause
October 2006
The New Biology
Quiroga and McGrath
of the process is primary in Vernadsky’s work. The expression of
universal physical principles of organization like gravitation,
magnetism, life, and cognition are evidence of this unfolding
process of development and organization. So what has more
potential to organize the universe, cow society or human society?
In order to understand these processes physically, new
approaches to measurement were developed by Lyndon
LaRouche. The materialist and reductionist might ask; How much
does an idea weigh? How many kilowatts does it take to produce
an idea? LaRouche asks; How do you measure the power of an
idea? The attempt to understand mankind’s relationship to nature
from a reductionist standpoint becomes futile once you discover
that there exists a non-linear relationship between creativity and
the universe.
The study of the expressions of cognitive processes in the
biosphere over geological time has been taken up by Lyndon
LaRouche and his youth movement. The science of physical
economy is an investigation of the dynamic relationships between
and within these three “phase spaces”: abiotic, biotic and
cognitive. Think about the discovery of atomic energy. A piece of
matter, the atom, has never been seen, but has nevertheless been
conceptualized and mastered by the human mind and is now able
to produce great densities and quantities of heat-energy. What is
the new field of potential that has been created by this discovery?
Would the energy produced from the combustion of oil or the
fission of Uranium have existed if the scientific thought of
mankind had not been developed? Human scientific thought has
allowed for the biosphere to propagate in a way which could not
have been possible on its own. Mankind has shown resilience in
the creation of new resources. For example: fresh water, which
civilization is in desperate need of can be generated through
nuclear power desalination plants. This would increase
manufacturing potential, provide an economic framework in
which many conflicts may be resolved in underdeveloped areas,
and allows us to further develop the biosphere by greening the
deserts. These development projects would commence
immediately if governments were to craft policy and interact with
a conscious scientific sense of the role that humans play in the
universe. What does the ability to constantly develop forms of
technology that are ever more dense in energy such as the leap
from burning wood, to coal, to fossil fuels, nuclear fission and
fusion and matter- antimatter say about the physical boundaries
of humanity?
There is no need to be fooled into adopting a dead end
“population control” dogma.6 The idea that people compete over
a fixed amount of resources, like animals, which is the basis of
Globalization is the true cause of many of our economic
problems. The building of infrastructure and emergence of cities
are a stark example of how the development of society is
coherent with Vernadsky’s discovery of the increase of density
and speed of the “biogenic migration of atoms” over geological
time. The current system must be replaced by a new one based on
cooperative development and production of higher levels of
technology so that human society may flourish and master the
solar system. Human beings, unlike lemmings, have forethought.
∆υναµις Vol. 1, No. 1
42
Mankind has the capacity to exercise free will, but this freedom is
only true if accompanied with knowledge. The LaRouche Youth
Movement is organizing for an epistemological renaissance in
science and society in which the work being done on Vernadsky
is a key part. LaRouche’s development of Vernadsky’s work is
crucial if humanity is to avert the onrushing calamity of the
breakdown of physical economies worldwide. The implications
of LaRouche and Vernadsky’s work make it necessary for society
to have a shift in economic thinking. As Lyndon LaRouche has
said on the subject of economics,” Creativity does not exist to
make some men rich. Society needs riches to secure goals of
creative progress in the human condition. As President Franklin
Roosevelt showed in his practice: creativity is not a servant of
making money; money must be a regulated slave and instrument
of the mission of progress through creativity. If you agree, and if
enough of us agree on that, then our republic will survive this
crisis, and civilization will go forward.”7
References
1
Boston Office: [email protected]
2
V.I. Vernadsky: Scientific Thought As A Planetary
Phenomenon, B.A. Starostin, trans. (Moscow: Nongovernmental
Ecological V.I. Vernadsky Foundation, 1997). Vladmir I.
Vernadsky, Scientific Thought as a Geological Force (Russia:
1946)
3
Vladmir I. Vernadsky, On Some Fundamental Problems of
Biogeochemistry (Moscow: 1935)
Translation secured through the Columbia University files
contributed by V.I. Vernadsky's son, Professor George
Vernadsky, New Haven, Connecticut, U.S.A.
4
V.I. Vernadsky: Scientific Thought As A Planetary
Phenomenon, B.A. Starostin, trans. (Moscow: Nongovernmental
Ecological V.I. Vernadsky Foundation, 1997). Vladmir I.
Vernadsky, Scientific Thought as a Geological Force (Russia:
1946)
5
V.I. Vernadsky: Scientific Thought As A Planetary
Phenomenon, B.A. Starostin, trans. (Moscow: Nongovernmental
Ecological V.I. Vernadsky Foundation, 1997). Vladmir I.
Vernadsky, Scientific Thought as a Geological Force (Russia:
1946)
6
See “There Are No Limits to Growth,” New York: New
Benjamin Franklin House, 1983
7
LaRouche, Lyndon. Economy and Ideas,” Earth’s Next 50
Years.” Leesburg, VA: LarouchePAC, March 2005.
October 2006