Insight - World - LaRouchePAC

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Th e Jo u r n a l o f T h e laro u c h e –ri e m a n n me T h o d o f Ph y s i c a l ec o n om i c s
december 2008 Vol 3 No. 3

December 2008
Vol. 3 No. 3
www.seattlelym.com/dynamis
EDITORS
Peter Martinson
Riana St. Classis
Jason Ross
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On the Cover
Raphael’s
Alba Madonna.
An instant, which
encompasses all eternity.
ΔΥΝΑΜΙΣ
Th e Jo u r n a l o f T h e laro u c h e -ri e m a n n me T h o d o f Ph y s i c a l ec o n om i c s
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3
26
38
41
44
48
From the Editors
The Calling of Elliptical Functions
By Michael Kirsch
On the Subject of ‘Insight’
By Lyndon H. LaRouche, Jr.
Third Demonstration of the Theorem
Concerning the Decomposition of Integral Algebraic
Functions into Real Factors
By Carl Friedrich Gauss
Letter from Carl Gauss to Wilhelm Bessel
December 18, 1811
The First Integral Calculus
Johann Bernoulli
Exclusive Interview: René Descartes
What’s the Matter with Descartes?
By Timothy Vance
“...God, like one of our own architects, approached the task of constructing
the universe with order and pattern, and laid out the indivitual parts accordingly,
as if it were not art which imitated Nature, but God himself had looked
to the mode of building Man who was to be.”
Johannes Kepler
Mysterium Cosmographicum

Editorial
At the moment this issue of Δυναμις is being put to
print, there is a vast, immediate battle being waged for the future
of the human race. On the one side, Lyndon LaRouche is mobilizing
a team of national governments to erect a new international
credit system, based on the best aspects of the American System
of Economics. On the other side, an old oligarchical empire, centered
in London, England, is unleashing the terrorist hordes it has
been fostering through the international drug trade for years, to
preserve its power and create chaos, potentially leading to the assassination
of America’s president elect.
To think clearly in a time of crisis such as this, one needs
to be creative.
August of this year past marked an historic turning
point in the development of the scientific capabilities of humanity.
Concluding a month long seminar series on the conceptions
associated with Carl Gauss’s discovery of the orbit of Ceres, the
Gauss “Basement” team passed the baton to the next team, who
was tasked with blazing the trail into Riemann’s conception of
higher hypergeometries. Simultaneously, a selection of former
Basement dwellers produced an hour long Δυναμις video, taking
Harvard University, and the rest of so-called academia, to task,
for graduating uneducated boobs (such as George W. Bush) into
positions of influence in our society. This video, The Harvard
Yard (http://www.larouchepac.com), presents a real history of
Kepler’s discovery of our harmonic solar system, as opposed to
the Newtonian witch-quackery of “action at a distance” heaped
out in its place today.
The new Riemann team has already made history, by
making several new discoveries about Riemann’s early development.
In looking back at Riemann’s own written draft manuscripts
on Geistesmassen, or thought-objects, they found that
several key sections had been kept out of the published version
in his collected works, edited (selectively) by Heinrich Weber. In
re-translating the entirety of Riemann’s writing on this subject,
they put together an initial picture of how Riemann’s work on the
mind informed his conceptions of related physical phenomena
such as gravitation, light, heat, and magnetism, through to the
end of his life.
A crucial insight was made by the team leader, Sky
Shields, who had begun investigating Wolfgang Köhler and his
school of Gestalt Psychology. Köhler found that the mind does
not assemble its ideas from small parts and sense impressions, but
that all thoughts are derived from whole gestalts. In attempting
to find a mapping between this activity of the mind and the human
nervous system, he ran into a conceptual block which came
from physics. Assuming that thoughts are just epiphenomena of
mechanical or chemical (i.e. nonliving) processes in the brain,
implicitly eliminates the possibility of a higher mental principle
that can order lower thought phenomena. Köhler had thought that
the most appropriate physical model, where a higher principle or-
ders the lower, observed phenomena, was what Carl Gauss, and
later Riemann, called potential. The 20th Century introduction
and promotion of the science of little hard particles, called atomic
science, had virtually replaced the study of potential.
Köhler saw that the observed relationship between the
mind and the senses can only exist in a universe composed of
gestalts – which can, in cross section, be expressed as what we
might call individual thoughts, much like a potential field expresses
itself, in cross section, as the motion of the bodies in that
field. This notion was in exactly the same spirit as Riemann, who
declared that all of what we explain as “forces” are expressions
of the motion of a space-filling matter, not a Newtonian war of
nut-tugging.
Nobody who thinks, really believes in the particles-acting-at-a-distance
hoax, anyhow. It is merely a form of social control.
To help break this control, the Riemann team has produced
a video, called The Matter of Mind (http://www.larouchepac.
com), laying out the argument that, only the science from Kepler
through Gauss, Riemann, and Köhler is real – the other stuff is
what makes slaves.
During the transition to the Riemann team, one of the
outgoing members of the Gauss team, Michael Kirsch, prepared
a report about the conceptual history of elliptical functions, from
Nicholas of Cusa up through Gauss and Riemann. His report
forms the centerpiece of this issue of Δυναμις, and is supplemented
by translations of Carl Gauss’s third proof of the Fundamental
Theorem of Algebra and a letter he wrote to his student Bessel
on the true nature of physical functions, and also a translation
of Johann Bernoulli’s first lecture on the integral calculus. Also
included is a report written by Lyndon LaRouche, to aid in clarifying
a crucial issue in how humans generate discoveries. Last,
Δυναμις has obtained an exclusive interview with none other
than René Descartes.
The reader should be reminded, that the reports contained
inside Δυναμις represent the rebirth of true science, which
has virtually died out over the past century. The true mission of
a young person today, must be to become a true scientist (which
also means learning how to perform classical choral counterpoint).
The next real scientific breakthroughs must be made in
this generation, and not wait for some distant point in the future.
The survival of the human race depends on breakthroughs in our
understanding of the universe, so don’t waste your time. Anybody
can read a popular science magazine, but it was only a few
driven individuals who went further than anybody else ever had.
Peter Martinson
Riana Nordquist
Jason Ross
Δυναμις Vol. 3 No. 3 December 2008
2

The Calling of Elliptical Functions
Kirsch
The Calling of Elliptical Functions
How a Lemniscate is Not Other than a Riemann Surface
Michael Kirsch
Hear now from those who sought to tame the principles
of transcendental physical pathways, bringing into
mankind’s understanding the principles which were
uncertain.
Gauss’s discovery of the characteristics of functions,
related to those processes which Johann Bernoulli defined as elliptical,
and Riemann’s later explicit reworking of his concept of
an elliptical function, led to a new degree of conceptual freedom
for mankind—an ability to understand that the reality of a process
can be understood by the internal unseen characteristics of the
process itself, rather than by any predicates.
The importance of this realization, is that it defines the
historical arc of investigating transcendental processes of nature,
as a continous development of demonstrating the human mind’s
ability to conceive of invisible principles in their own domain,
without depending on false shadows and images. This is not to
say that the invisible principles are now seen, sensed, or even understood
directly, but rather, the principles are come to be known
not in terms of what effects they produce, but how those effects
are produced.
The specific principle here goes far beyond the subject
matter involved; it is as Kepler’s captive, the physical pathway
of the Mars orbit, which, when brought into the understanding of
the human mind, is applied to increase mankind’s power and effectiveness.
Just as certain principles, when applied to society as
a whole, have the effect of increasing the power of labor through
technological advancement, this principle of understanding how
processes can be known and conceptualized, is as important as it
is extensive.
This important realization defines a continuity through
the foundation of modern science to its maturity. It leads one to
understand something fundamental about the human mind, why
it must be defended, and why civilization’s main object is the
pursuit of its development.
The Foundation of Modern Science
Cardinal Nicolas of Cusa defined the method of investigating
processes in his dialogue On Not-Other. The following
reasoning from his dialogue serves to introduce his concept:
What causes us to know, what most gives us knowledge
of the world? Definition. How is definition known? The definition
is expressed from the defining of a thing---definitions define
all things. And if definition defines everything, does it not define
itself? Then the definition defining everything is not other than
the defined.
What Cusa presents to the reader of On Not-other, is
that principles of nature, such as light, or heat, define themselves.
From this characteristic, principles do not have pieces from which
they are made; they are indivisibles, units. The distinguishing
character of men then becomes whether they, in investigating the
effects of light, for example, keep their mind on the principle of
light which shares in this Not-other characteristic, that of being
self-defined. Only those who ask, “What is the cause of this process,”
can come to discover this characteristic, a realization only
obtained by having a clear concept of the cause.
The history of modern science is founded upon this
method.
The lack of this approach is due to the fact that it has
been replaced by a priori assumptions about physics, geometry,
and mathematics; their presence acts in such a way, that although
there have been many discoveries which have overthrown these
assumptions, the shackles are continually re-fastenend, and new
discoveries are re-stated in terms of the old ones, re-explained by
what was known before.
Like the old flame that keeps translating you back into
the person which their fantasy first thought up when you just met,
for the reductionists, nothing is really new.
The Error of Archimedes
This method of approach was founded in Cusa’s correction
of the error of Archimedes in his attempt to square the
circle.
Cusa shows in his paper, On the Quadrature of the Circle,
that the curvatures of the angles inside and outside the circle
have an invisible difference which nonetheless exists, an essential
characteristic which is the result of the kind of action which
generates a circle, as the action which Kepler discovered generates
the ellipse corresponded to a physical relationship present at
every infinitesimal moment of the orbit. This defines the curved
and straight lines partaking in those angles, as incomparable with
one another.
After noting this ‘species’ difference, he shows that from
straight lines, it is impossible to determine the radius of a circle
which would have an equal perimeter to a triangle, but, that one
can come seemingly very close. This error of precision is pointed
out by him as fundamental to the way in which the human mind,
seeking to measure truth, must approach principles. 1
1 “The measure with which man strives for the inquiry of truth has no
rational proportion to Truth itself, and consequently, the person who
Δυναμις Vol. 3 No. 3 December 2008
3

Cusa wrote later in his
Theological Complement, that
the defining difference of his approach
with Archimedes’ is based
on the fundamental distinction
between the rational and intellectual
parts of the mind. He states
in that location, that the gravest
mistake of Archimedes was his
reliance on his rationality to measure
a principle only graspable
in his intellect; the problem was
not that in his mind he sought to
measure the circle with a straight line, but that he endeavored to
manifest this rationally. 2
His point is best expressed by two different responses to
the following question: how do you find the perimeter of a circle,
whose measure is a straight line?
Archimedes’ reply was to use an exhaustive method of
approximation through bisection of arcs in comparing the straight
to the curved. Cusa, however, answered that the real circle whose
area is measured by a straight line exists only in the infinite.
“The ancients sought after the squaring of a
circle. 3 If they had sought after the circularizing of
a square, they might have succeeded…a circle is not
measured but measures… Given a finite straight-line,
a finite circular-line will be its measure.
“Thus, given an infinite circular line, an infinite
straight-line will be the measure of the infinite
circular-line… Because the infinite circular line is
straight, the infinite straight-line is the true measure
that measures the infinite circular-line…
“Therefore, the coincidence of opposites is as
is contented on this side of precision does not perceive the error.
And therein do men differentiate themselves: these boast to have
advanced to the complete precision, whose unattainability the wise
recognize, so that those are the wiser, who know of their ignorance.”
Nicholas of Cusa, from Toward a New Council of Florence, translated
by William F. Wertz, Jr. (Schiller Institute: Wasthington, DC
1993)
2 Cusa writes: “But the coincidence of those features in every polygon
in terms of a circle, ought to have been sought intellectually; and
[then those inquirers] would have arrived at their goal.” The rationality
determines the properties of a subject, such as a radius of a
sphere, or the geometrical properties of a curve. The intellect, can
conceive of the concept of an infinite sphere, where the center and
circumference coincide, or, as the infinite circle, whose measure is a
straight line. Cusa, ibid.
3 Such as Archimedes
The Calling of Elliptical Functions
Kirsch
the circumference of an infinite circle; and the difference
between opposites is as the circumference of a
finite polygon.” [emphasis added]
Cusa’s solution is outside the comprehensibility of the
rational, but exists in the intellect; in conceiving of the essence
of a thing, the intellect brings the relations between different species
to clarity; bringing the boundaries of a species into the understanding,
it thereby illuminates the concept of a generating
principle. 4
In other words, a process such as a circle is the projection
of an unseen principle, and thus the essence determining its
uniqueness will be unseen. This process is known to the mind, not
by describing the process (the arc of the circle), with its effects
(the straight lines of the polygon), but only through the paradox
of the infinite. Cusa’s concept of the infinite is that the mind must
ascend to the generating principle, the principle of the not-other,
to see that the principle of the circle defines itself.
1. Descartes’ Fraud
Holding this approach of Cusa in the mind, travel forward
to beyond Kepler’s rigorous demonstration of Cusa’s method
for the elliptical orbit, his own paradox in measuring such a
curve, and his discovery of universal gravitation bounding the
system of our sun. 5
In his New Astronomy, Cusa’s follower Johannes Kepler
had uncovered the physical paradox of constant change in the
universe. This paradox he captured by determining the constant
physical relationship governing the relation between the physical
cause of the sun, and the sense perceptible observations of Tycho
Brahe. As this relationship was found to take the form of an ellipse,
it was accompanied by a problem he called upon future
geometers to solve.
In the aftermath of Kepler’s mortal death the networks
of Descartes had attempted to remove the method of investigating
physical paradoxes. By means of his limited powers of mind,
Descartes announced a ban on such physical paradoxes as Kepler
left for geometers, or, what he termed, ‘mechanical’ problems.
4 Cusa had made this point in De Docta Ignorantia when he
brought the infinite to mathematics. Cusa used the example of the
infinite line to demonstrate that the maximum is in all things and all
things are in the maximum. Each finite line could be divided endlessly
and yet, a line would always remain. Thus the essence of the
infinite line was in a finite line. Likewise each line, when extended
infinitely, became equal, whether it was 4 feet or 2 feet. Thus the essence
of each finite line was in the infinite line, although participated
in by each finite line in different degrees. Here, the circle is in every
polygon, in such a way that each polygon is in the circle. “The one
is in the other, and there is one infinite perimeter of all.” http://cla.
umn.edu/sites/jhopkins/DI-I-12-2000.pdf
5 See http://www.wlym.com, and the LaRouche PAC video The
Harvard Yard
Δυναμις Vol. 3 No. 3 December 2008
4

“Probably the real
explanation of the refusal of
ancient geometers to accept
curves more complex than
the conic sections lies in
the fact that the first curves
to which their attention was
attracted happened to be the
spiral, the quadratrix, and
similar curves, which really
do belong only to mechanics,
and are not among the
curves that I think should be
included here [in geometry],
since they must be conceived
of as described by two separate
movements whose relation
does not admit of exact
determination.” 6 [emphasis
added]
This means that geometry
was not to be used in determining
physical paradoxes. But, to say you can’t use geometry is
to say you can’t know it, since geometry is the means by which
we come to measure what something is and is not.
Gottfried Leibniz, who like Kepler before him, had been
in training to be a priest before devoting himself to the study
of mathematical physics, exposed this fraud. The ‘mechanical’
curves, as Descartes defined them, were indeed constructible, but
only by rising above the assumptions of Descartes.
Leibniz’s reply was the following:
“If you wished to trace geometrically (that is
by a constant and regulated motion) the Archimedean
spiral, or the Quadratrix of the Ancients, you could do
it without any difficulty by adjusting a straight line to a
curve, in such a way that the rectilinear motion would
be regulated from the circular motion. And that is why,
contrary to what Descartes has done, I will not exclude
such curves from geometry, because the lines which
are so described are exact, and they involve properties
which are very useful, and are adapted to transcendental
magnitudes.”[emphasis added]
In an earlier letter on the subject to Antoine Arnauld in
1686, Leibniz further elaborates and clarifies this sentiment, saying
that his new method provides the means of reducing transcendental
“curves to calculation, and I hold that they must be
received into geometry, whatever M. Descartes may say.
“My reason is that there are analytical prob-
6 From The Geometry of Rene Descartes Translated by Smith
and Latham
The Calling of Elliptical Functions
Kirsch
lems which are of no degree…e.g. to cut an angle in
the incommensurable ratio of one straight line to another
straight line….” 7 “… I show that the lines which
Descartes would exclude from geometry depend upon
equations which transcend algebraic degrees but are
yet not beyond analysis, nor geometry. I therefore call
the lines, which Descartes accepts, algebraic, because
they are of a certain degree in an algebraic equation.
The others I call transcendental.” [emphasis added]
Circular and Exponential Transcendentals
What made Leibniz’s method of constructing transcendentals
with geometry possible, was, that unlike Cartesian geometry,
which treats a geometrical image as a self-evident fixed object,
Leibniz discovered the characteristic of change of a process
to be its most essential nature. With this conception, combined
with experimentation by means of moments of change in the geometry,
the mind can discover patterns and lawfulness that lead
to a realization about the constant relationship which is present at
every moment of change, one that exists in each infinitesimally
small interval, guiding the process of the sense perceptible effect.
Inversely, the curve could then be constructed as an expression of
that relationship acting as though infinite to each moment of generation
of the curve, as from above. From this, the curve is known
not as an a priori sense perceptible object, but as an expression of
the unfolded relationship maintained at every moment. 8
7 See Box 1
8 See Box 2. A basic demonstration of the application of infinitesmal
calculus can be easily be understood by means of the simple
Δυναμις Vol. 3 No. 3 December 2008
Box 1
Bernoulli shows an example of Leibniz’s description of this transcendental relation
in his lectures on the integral calculus. Since the expression for the circumference of
a circle is 2 times Pi times the radius of the circle, then two circles, having radii which
are in proportion to each other as two lines which are incommensurable, such as 1 and
square root of 2, will have perimeters which are likewise incommensurable.
Imagine a cycloidal curve, produced
by two circles in such a relationship
described, one rolling about the other.
In such a case there will be no number
of times the rolling circle will come back
around to the same position, such that it
would trace out a finite number of cycloidal
curves; rather an infinite number of
curves will be traced. The consequence of
this is that an equation produced by the
intersection of a line with these infinite
number of curves will have an infinite
number of roots, be of infinite degree, i.e.
transcendental, and non-algebraic.
5

This geometrical differential
of change blasts the empiricism
of object-fixated analysts 9 , and defines
the calculus as a language of
change, capable of communicating
ideas about indivisible principles
which lay invisible to processes,
and yet exist as their cause.
As a corollary of this general
method, Leibniz showed how
such ‘transcendental’ and other
curves could be constructed by different
means, for example, not directly,
but by the quadrature of geometric
curves.
geometric parabolic and hyperbolic curves.
9 Anal-ists
Box 2a
The equation for the parabola
is ob = cb 2 .
ob = oa
oa' = ob'
If cb = w and oa = z, then w:2
z::dw:dz
Since z = w 2 , then dz = 2 w
dw, which is the expression
for d(z) =d(w 2 ) . The rela-
tionship cb
2oa
is maintained for
any tangent to the parabola.
Inversely, the constant expression
of that relationship
is the parabola.
The Calling of Elliptical Functions
Kirsch
Box 2b
Take the hyperbola (top diagram). What
is the equation for finding the simple com-
mensurable points of the curve?
Y = X−1 then 1
X
or Y = 1
X
is to 1 as 1 is to X, or 1
X
So if Y = 1
X
Δυναμις Vol. 3 No. 3 December 2008
1
1 = X
. For
example, if X = 4, then ¼ is to 1 as 1 is to
4
Think, what is the geometric relation
of this curve?
It is such that it always has 1 or a
constant as the geometric mean between
X and Y.
Therefore, how do you create a
geometric mean?
Have you ever doubled a
square?
If yes, then you know what a
geometric mean consists of and therefore
will know where the constant would be
between X and Y on the hyperbola.
Did you figure it out yet?
Ahh, well, I’ll give you the answer.
It’s at half the angle between X and
Y (in the top diagram, it is 45°). If you
don’t know why, double a square, until
you create a spiral. Then look at the properties
of the spiral.
This then remarkably gives us
not only any X and Y relation, but also
a tangent to the hyperbola of ½ X and Y,
meaning that the point of the curve which
is the tangent is at the coordinate ½ X
and ½ Y (see the top diagram, in this case
it is 2 for X and ⅛ for Y)
Lets translate this now into mathematical
mumbo jumbo and see why this
works out according to Leibniz’s calculus
of differential. Supposedly, the curve we found is the tangent, which means that it is
the characteristic triangle of the curve or the dy
dx = in X−2 terms of mathematics.
Now, by way of the mathematics of the calculus, we know that the differential
of Y = 1
X is dy = X−2 dx or dy
dx = X−2 . We also see that the triangle we find
geometrically as the tangent has a length of X and a height of X−1 ; which means
that the proportion is 1
X 1
dy
1 = . But our calculation says that X dx = X−2
That’s ok, because its actually X−2 over 1, which we find is a similar triangle
(bottom diagram) to 1
X 1
1 = X .
1
1
6

Leibniz’s notation
for the integration
of curves showed
that the expression for
both the quadrature of
the hyperbola and the
arc length of a circle,
as Cusa had indicated
earlier, was indeed
transcendental, and
not able to be integrated
algebraically 10 ;
however, although
transcendental, the
integrals of the circle
and hyperbola could
be related to the physical
function to which
they correspond, and
used to construct other
transcendentals related
to them. Therefore,
although they
themselves were not
algebraically solvable
with the Leibniz
method of integration,
the functions were
known, and thus the
concept of the construction
was clear.
In this way
curves, such as conchoids,
could be
constructed from an
equivalent area made
up of a hyperbola and
a circle, or, many oth-
er transcendental curves, such as the exponential curve, could be
constructed with two simultaneous actions.
Maintaining a relationship between a constantly grow-
10 The concept of integration depended on the relationship between
the function and its change at every moment. Without such
a relationship, the integral could not be found, e.g., in Box 4, the
differential of the quantity under the square root, –2 x dx, is not the
differential part of the expression, that is dx, and thus there is no
means to integrate the expression, since there is not a direct relation
between a quantity and its differential. (See Bernoulli’s Lectures on
the Integral Calculus, Translated By Bill Ferguson, in this issue of
Δυναμις.) For certain algebraic functions, many methods were used
to solve integrals; however, for the transcendental curves, the only
recourse was the ability to relate the differential to the actual function
to which it corresponded.
The Calling of Elliptical Functions
Kirsch
Box 3
Bernoulli shows that the exponential curve,
a transcendental of no degree, can be constructed by
means of the quadrature of the hyperbola.
Starting with the equation for the curve of constant
subtangent(the characteristic of the exponential
curve), dw:dz :: w:a, one sets up the equality a dz = a2 dw: w.
Bernoulli then shows that when the equality of
this relationship is maintained through all possible values
of z and w, the exponential curve will be constructed.
The integral � a2dw : w = az . To construct
the left side of the equation, the z axis of the
hyperbola has the lengths, a2
w . This multiplied by
the infinitesimally small change in w, gives us the area under the hyperbola. The area of the
hyperbola grows arithmetically in proportion to the geometric growth of the w axis (as investigated
by St. Vincent de Gregoire, 1 to
whom Leibniz gives credit as one of his
three major inspirations.) The right side
of the equation, a dz, is a simple rectangular
growth, whose sum would be az, a
constant times the z azis.
This area az maintaining its
equality to the area of the quadrature
of the hyperbola sum of a2 dw : w, will
therefore, in arithmetic rectangles along
the z axis, project geometric growth in
the w axis, which is the exponential
curve.
So, the sum of a2 dw : w, the
quadrature of the hyperbola, is the logarithm
of w, that is, the logarithm of the
geometric growth, w.
1 See Bill Ferguson’s article, in the Upcoming Issue of Dynamis for more on St. Vincent.
ing quadrature of one geometric curve on one axis, and the
equality of a rectangular area on the other axis, produces two
simultaneous actions which construct the relation on which the
transcendental curve depends. An example of such a construction
of a transcendental curve is the method of constructing the exponential
curve by means of the quadrature of the hyperbola (Box
3).Within this method, for calculation, the arclengths closely related
to the quadrature of the circle could be converted to Arcsine
and Arccosine, while the quadrature of the hyperbola could be
converted to logarithms. This method, related to a whole class of
transcendentals, was able to solve many problems not otherwise
solvable using the algebraic rules of the integral calculus. 11
11 See Box 4 and 5
Δυναμις Vol. 3 No. 3 December 2008
7

Physical Transcendentals
With this understanding
of the infinitesimal
calculus, the understanding
of how to conceptualize
processes, such as Cusa had
indicated much earlier, was
brought to a higher degree
of maturity; however, the
real breakthrough for science
came with Leibniz’s
investigation of physical
transcendentals. The
boundary of this domain of
transcendentals was defined
by the investigation of the
physical principle existing
ontologically outside the
domain of the ‘geometric’
transcendentals.
This superior
campaign waged by Leibniz
in the method of conceptualizing
processes, and
capturing them for mankind,
necessilarily leads
us to introduce what may
seem to some, an unrelated
subject.
Leibniz had taken
Box 4
CONVERTING ARC LENGTH OF CIRCLE TO
RADIANS
�
adx
vπ
s = √ =Arc Cosine x =
a2 − x2 360◦ 1
as = Quadrature of the Circle
2
1
as = Quadrature of the Circle
2
s = Arc length ofCircle
s = Arc length ofCircle
AB = a, a =1
AC = x
CD = y = √ 1 − x 2
it upon himself personally, since leaving Paris in 1676, to wipe
out the interrelated Cartesian fraud which pertains to the domain
of physics. It is Descartes’ religious belief, that there is nothing in
The Calling of Elliptical Functions
Kirsch
ABD = 1
�
2
The expression for the arc length of the circle cannot be
algebraically integrated using the rules of the integral
calculus; this underscores its transcendental nature,
as there is no way to capture the true meaning of the
arc length by means of the sine and cosine expressed as
the algebraic magnitude, x. What is shown here, is that
the expression for the arc length can be simply turned
into a certain amount of radians which are equal to
the arc length. This is not ‘solving’ the integral from
within the functions, but rather, simply stating what
the arc length equals in terms of the circle itself. It is,
in a sense a tautology: the arc length equals the arc
length. This is easily seen if one substitutes Cosine v for
x in the integral expression for s; the ratio of the value
for d(cos υ)a : √ a 2 − cos υ 2 becomes simply dv, thus,
υ = � dυ . As an example, if the angle is 60°, then the arc
length in radians is π
3.
dx
√ x 2 − 1
Box 5a 1
Now, for an Equilateral Hyperbola AC = CH.
AH =(HC) √ 2 AF = x+√x2 √ −1
2
But, for this axis, since AB is taken as the constant
log 1 √ 2 must be subtracted.
Thus: 3
FH = (IH)
√ 2
AG = 1 √ 2
Since AF*IF = AG*BG, then ABG = ADE and
thus, ABD = IFBG. 2
IFBG =
By Box 3:
1
� �
dw 1 dx
=
2 w 2 x2 = ABD
− 1
�
1 dw 1
= log w
2 w 2
1 For more on the relation between definite integrals
and logarithms, see Bill Ferguson’s article, which
will appear in a future issue of Δυναμις, for a more
generalized study of definite intergrals involving the
hyperbola and logarithms.
2 Here, log w means the natural logarithm.
3 See Box 5b
bodies which is not contained in extension, for which he is falsely
praised. With Descartes’ science of ‘extension’, substance (in
the sense of Plato, Cusa, and Kepler) was banned from thought.
Δυναμις Vol. 3 No. 3 December 2008
1
1
1
2
� w
1
dw
w
�
1
= log w − log
2
1
�
√ =
2
1
2 log (w√2) Therefore:
log (x + � x2 �
− 1) =
dx
√ x 2 − 1
8

Box 5b
Here, it is quite easy to see, that log w + log ww = log
www. What is more, it is generally true that log b +
log d = log (bd)
e a = b, a =log b
e c = d, c =log d
log b +log d
=log e a +log e c
=log e (a+c)
=log [(e a )(e c )]
=log (bd)
He held that all phenomena are only modifications of extension
and should be explained by their properties, such as form, position,
and motion---explaining all phenomena in terms of sense
perceptible quantities. This “salvation from mysticism” 12 must
be understood, as having been done with a full consciousness of
Kepler’s already established method for defining causes beyond
extension.
Descartes doctrine of extension thereby reduced the
study of bodies in motion to purely geometric analysis, as Ptolemy
had done before him. Causes were, not outrightly ‘denied’, but,
in end effect, located, without reason, in the bodies themselves;
‘causes’ explained by mathematical laws such as Descartes rules
for motion or Ptolemy’s epicycles. 13 In this way, no experiments
12 It is common jargon, that Descartes modernized physics to
have replaced what was seen as the ancient Greek, mystical explanations
of the ‘substance’ of bodies.
13 This is an entirely different meaning of ‘self-defined’. From
Descartes’ reasoning one might be led to believe that after mankind
was freed from his body, they would be free from his ideas. If only
this were so in the case of Descartes! (See Δυναμις exclusive interview
with Descartes, this issue) For a concise exposition of the
absurdity of Descartes’ rules of motion based on his maxim, the
The Calling of Elliptical Functions
Kirsch
are necessary in physics, but rather, everything is deduced from a
priori axioms about geometry.
This is the battle field upon which Leibniz must be understood.
Taking this into account, we will look at two physical
transcendentals, one which ontologically defined the whole domain
of transcendentals just discussed, and a second, posed by
Leibniz, which leads us into the discoveries of Gauss and Riemann.
2. Experimental Metaphysics
It is important to make the point here at the outset, that
the following detailed elaboration, seeming to some as an extended
tangent to our current arc of thought, is in fact a crucial
element defining the entire pathway of our methodological discussion.
Leibniz’s method, described above, of finding the relationship
maintained throughout the curve – the differential relationship
– took on a different challenge in application to the
catenary, the curve formed by a hanging chain. 14 As with curves
such as the parabola, a constant relationship is sought which is
present throughout the curve; however, in the case of the hanging
chain, the constant relation is a physical one, not geometric.
The catenary is formed by the tension between two tensions:
the weight of the chain tending in the vertical direction 15 ,
and the horizontal tension amongst the links themselves. The
distance between the end links that hang the catenary defines the
amount of horizontal tension that exists throughout the chain,
and thus the differentiation between catenaries for any particular
length of chain.
Through physical experiment, the proportion of vertical
and horizontal tension at a given point in the chain is found to be
directly related to the direction which the chain is tending at that
point. What is added to this derived fact, is the most essential
property of the catenary: the horizontal tension is constant for
any one particular catenary hung between two points, regardless
of the amount of chain that is removed.
These experimentally derived relations lead the physicist
to traverse from the seen to the unseen, in order to investigate
quantity of motion, see the October 2006 of Dynamis: Experimental
Metaphysics, Kirsch, Yule, ft: 17. Also, in the same Issue, Inertia of
Descartes Mind, Ross. See Fermat Book on Light, Ross.
14 Jacob Bernoulli, posed this challenge at the end of his response
to Leibniz’s Isochrone challenge of 1689, which was put forward
to expose the Cartesian fraud in physics and geometry.
15 There are assumptions brought to investigating physical
curves from geometry with which we must dispense. First of all,
directions are not arbitrary, lying on an infinite space of many directions,
rather, they are physically defined. What is down? Down
corresponds to the direction which a weight falls, and horizontal is
that which is perpendicular to a falling weight. Any other definition
is arbitrary.
Δυναμις Vol. 3 No. 3 December 2008
9

the changing relation at every moment of change. If the direction
at two moments of the chain are extended by means of tangents,
and the weight of the chain is placed where the tangents meet, the
problem of measuring the relation, of horizontal and vertical tension
in the chain, can be turned into another relation proportional
to trigonometric functions created by the tangents. 16
Employing this, a unique singular moment where the direction
of one of the tangents is completely horizontal, turns the
just mentioned relation into one relating the vertical tension of
16 See Box 6
Box 6a
Box 6b
The Calling of Elliptical Functions
Kirsch
the whole chain, to the horizontal tension. 17
The physical relationship of tensions guiding
the change in direction at every moment of the
curve is discovered: for any given length of chain,
the proportion which represents the vertical and horizontal
slope of the curve at that point, is equal to the
proportion of the weight of the chain to the constant
horizontal tension. 18
Once the physical differential was discovered,
both Leibniz and Johann Bernoulli saw its relation
to the quadrature of the Hyperbola. From there
however, their methods were entirely different.
The most essential characteristic of the
curve, the constant horizontal tension, makes the
catenary the best expression of how nature performs
the least action pathway in traversing space, thus defining,
in its expression, the curvature of gravity and
tension in bodies moving around the sun. This characteristic
plays the main role in unfolding the unseen,
physical paradox of this curve of least action. Here
we come
to the moment
of
inflection
which leads to
a completely
new, ontologically
defined
geometry.
H o w
do you define a
process? With
what axioms do
you approach
an unknown
process in order
to measure it?
The discovery
of the catenary
allows none.
What is the algebraic
relationship
which defines the
coordinates? Here, there
are none to be found. As
Carl Gauss would later
demonstrate the method
17 See Box 6b
18 See Box 7. The constant horizontal tension represented by the
letter a, can be derived experimentally with a pulley attached to the
bottom link of the chain, or, as will be shown below, geometrically,
once the physical principle is clear.
Δυναμις Vol. 3 No. 3 December 2008
10

The Calling of Elliptical Functions
Kirsch
of ‘intrinsic
curvature’ in
his method of
d e t e r m i n i n g
the curvature
of a surface,
the principle
of the catenary
can only be
d i s c o v e r e d
through utilizing
the physical
relations of the
curve, dumping
all axiomatic
systems in the garbage.
Going to the immediate point: how would one measure
the height of a catenary? In order to measure the coordinates
of the curve, where does one place the abscissa, the horizontal
line beneath the curve? Should it be tangent to the bottom point?
Somewhere below? If so, where?
There is only one non-arbitrary distance below the curve
to construct and measure the changing heights of the catenary:
the length, one. But what length is one? The relation of vertical
and horizontal tensions in the chain, to the vertical and horizontal
direction, leads to a singularity at a tangent of 45°, where the proportion
of vertical and horizontal direction is equal to one. 19 This
length of chain is thus equal to the constant horizontal tension,
the unit length, defined by the physical principle of the catenary
itself.
Only this intrinsic measurement, constructed under the
catenary as the height of the abscissa, led Leibniz to discover the
relation of the catenary to the quadrature of the hyperbola.
As soon as he investigated, experimentally, the growing
lengths of chain laid down straight as lengths at the bottom of the
catenary the physical differential of the tensions was revealed in
a new light. Investigating the diagonals of the triangle whose two
sides are 1) the horizontal constant, which is set as the height
at the bottom of the curve, and 2), the length of chain laid down
straight, he then discovered these diagonals to be equal in length
to the heights corresponding to that given length of chain.
This points to the most essential corollary of these relations.
In utilizing the square Pythagorean theorem, for these
three sides—the diagonals (heights), constant tension (constant
side), and length of chain (long side)—the physical differential
relation 20 was transformed into one expressing the differential in
terms of the quadrature of the hyperbola. 21
S dx
S
19 Here, our realationship a = dy = 1, thus by our above relation a
=1, or in otherwords S (length of chain) = a (horizontal tension).
20 S dx
a = dy
21 See Box 7a
Box 6c
S = Length of Chain
a = Constant Horizontal Tension
S
a
=Tan w = dx
dy
= dx
dy
However,
unlike Bernoulli’s
construction, Leibniz,
employing his knowledge
of the relation of
the exponential curve
to the quadrature of
the hyperbola, 22 inverted
and translated
this particular relation
for y, in the above
diagram, into one for
x, showing the height
of the catenary to be
the arithmetic mean
between two exponential
curves. 23
In other words, Bernoulli showed the catenary could be
drawn with the quadrature of the hyperbola, but Leibniz showed
that the inverse function for the quadrature of the hyperbola is the
catenary. 24
By this means, the substance of the catenary defined its
own predicates, by its physical principle alone. Every effect of
the physical differential is given definition, as a function of two
transcendental functions. 25
22 See above Box on Relation of Quadrature of Hyperbola and
Exponential Curve.
23 The idea for the last step is credited to Bill Ferguson. See box 7b
24 Johann realized, which his brother Jacob failed to do, that it
was indeed a transcendental curve, and not algebraic, a discovery
which recruited him to Leibniz’s method of seeking for the ironies in
nature, as for example, his Brachristicone; however, his construction
was inferior to Leibniz’s and did not leave the domain of the earlier,
“geometric” transcendentals discussed, thus failing to capture a sub-
stantial irony involved in the paradox of physical least action.
1
1
25 It is highly of note that the exponential curves which are de-
Δυναμις Vol. 3 No. 3 December 2008
1
Box 7a
S = Length of Chain
a = Length of Chain equal to Constant Horizontal Tension
Height = � S 2 + a 2 = x
S
a
dy = adx
S
= dx
dy
= adx
√ x 2 − a 2
Since a is equal to 1,
dx
dy = √
x2 − 1
Box 7b
From Box 7a
�
y =
dx
√
x2 − 1
From Box 5
From Box 3
Thus
y =log (x + � x 2 − 1)
e y = x + � x 2 − 1
e y − x = � x 2 − 1
x = ey + e −y
2
1
1
11

In the fantasy of his own mind,
Descartes imagined a physics based only on
extension, one which required only a priori
geometrical deduction without experimentation,
i.e. an infinitely boring universe in which
nothing new ever happens. On the contrary,
in physics, a concept of the individual substance,
or monad, is possible, a concept which
is “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” 26 —provided one exists in the real
universe of experimental metaphysics.
The Power of Leibniz’s Construction
Shattering the domain of Cartesian
geometry and space, Leibniz’s construction
captures, in the most accurate
paradoxical metaphor, the unique irony
between geometry and physics.
All curves which could be described
by one ‘coordinate system,’ an
action defining a geometric space, are
now seen to be inferior to those like the
catenary which require two independent
geometries. His construction points one
in the direction of asking: doesn’t the
catenary substantiate and define the existence
of the exponential curve? That is, since the catenary is
generated so easily by nature, and from it is derived two exponential
curves, doesn’t only the physical least action of the catenary
define the domain in which this relation of exponential curves
exists?
And further, doesn’t the creation of two exponential
curves in opposite directions mean that one exponential curve is
only a special case of the two, making the two together a single
higher function, and in fact primary, defined ontologically by their
generation by least action in the field of gravity and tension?
In this sense, one is not putting geometric curves together
in Cartesian space, but rather, a single unified physical principle
is generating a geometric space—a different geometry of
physical space altogether. It is not that the catenary can be drawn
by the hyperbola or exponential, but rather, the catenary, is what
draws the other curves. The physical principle thus defines geometry
in a paradoxical manner, a geometry which goes beyond
rived from the catenary are those whose logarithms are natural, i.e.
the subtangents of the exponential curves are equal to one, or better
said, whose subtangents are equal to the constant tension in the catenary
itself.
26 Gottfried Leibniz, Discourse on Metaphysics, Section VIII
The Calling of Elliptical Functions
Kirsch
Descartes.
To elaborate this point further, think back
to Johann Bernoulli’s demonstration of constructing
the exponential curve by means of the quadrature
of the equilateral hyperbola. The right cone, to
which this equilateral hyperbola would belong as a
conic section, would, in order to create the double
exponential construction of Leibniz, have to be
joined with another right cone. Therefore, whereas
in sense perceptible, descriptive geometry there exists
one cone, in physics, there exist two cones set
at 90 degrees to one another.
Looking back upon geometry and ‘geometric’
transcendentals from the standpoint
of Leibniz’s catenary, one asks: doesn’t this
mean that all prior analysis that didn’t involve
physics was only a special case of sense perceptible
geometry, not the kind which subsumes
the true nature and characteristics of
the functions?
Even more specifically: Leibniz’s
discovery redefines ontologically, the whole
domain of the quadrature of the circle and
hyperbola, and all the lower geometric curves
which could be constructed by quadrature. 27
The physical action is primary and generates,
as a lower domain, the predicates of geometrically
related quadrature. In this way, the circular
and logarithmic are in a sense geometric
transcendentals; they are not related to physical transcendentals
directly, but only indirectly, existing as their effects and projected
shadows.
The catenary’s relation with the quadrature of the hyperbola
shows that the transcendental quadratures were sufficient for
describing certain processes, but Leibniz shows their limitation,
by bringing the concept of the transcendental to a higher domain,
a domain which corresponds to least action, expressing principles
organizing the space of gravity and tension.
Above all, Leibniz’s method of discovery demonstrated
the Not-other characteristic of the catenary---a discovery possible
only with the intrinsic physical geometry defined by the process
27 This is also demonstrated in the case of Leibniz’s quadrature
machine, where the tractrix, the evolute of the catenary, is used
as a generalized quadrature principle. See Extension of geometric
measurements using an absolutely universal method of realizing all
quadratures by way of motion: accompanied by different procedures
of construction of a curve from a given property of its tangents. (A.
E. September 1693, M. S. V p. 294-301). Latin Title: SUPPLE-
MENTUM GEOMETRIAE DIMENSORIAE SEU GENERALIS-
SIMA OMNIUM TETRAGONISMORUM EFFECTIO PER MO-
TUM: SIMILITERQUE MULTIPLEX CONSTRUCTIO LINEAE
EX DATA TANGENTIUM CONDITIONE. Translated by Pierre
Beaudry, ftp.ljcentral.net/unpublished/Pierre_Beaudry/.
Δυναμις Vol. 3 No. 3 December 2008
12

itself. He demonstrated the way in which the mind can be elevated
to an understanding of this characteristic, by discovering
the geometrical principle which expresses ironically the physical
principle at every moment. The truest expression of reality is the
irony with which the relation between the predicates, points to the
truth that the process defines the predicates.
And thereby, it is this type of domain in which intellectual
ideas truly exist.
It is the transformation between domains, which gives
a power to man, and pushes his ability to new degrees. In this
sense, there is not an object which one can see as the transcendental,
only an effect is seen. The transcendental is only present
in the mind, seen as an anomaly between what is yet undiscovered
and what is known and yet transformed ontologically by the fact
of the existence of the transcendental.
In this sense, for those who do not ask “why?” when
confronted with a paradox to their preconceived notions, the
transcendental does not exist.
3. Descartes’ Fraud, Again
We saw above that the differential expressions for the
quadrature of the circle and hyperbola could not be integrated
but could be related to their known physical functions. Yes, these
transcendental expressions were a rigorously true definition for
the area of the curve, but not indicative of how they were generated,
i.e. of the domain to which they actually corresponded.
However, there were other physical transcendentals
which couldn’t be related in any way to known functions, not
even in the way that the catenary is ‘related’ to quadrature of the
hyperbola.
Accordingly, Leibniz had his own challenge for scientists
which, in his lifetime, did not become solved to a sufficient
degree. He posed a challenge of constructing a curve which
couldn’t be related in any way to the quadrature of the circle and
hyperbola, and thus posed a completely new test for physicists.
Like Kepler’s elliptical orbit, what Johann Bernoulli called the
integrals of “elliptical” curves, left a challenge. A true solution
to them would not be solved in his life time by him nor the Bernoullis
and would lead to the paradox of what Johann Bernoulli
called the ‘elliptical’ integral. As Carl Gauss himself would later
say, it is this domain beyond the shadows of the quadrature of the
circle and hyperbola that is of essential interest to those who seek
to open up new domains of thought. 28
28 “In the computation of integrals I have always had little interest
in matters that simply follow from substitutions, transformations,
etc.--in short, making use of a certain mechanism in an appropriate
way to transform integrals into algebraic, logarithmic, or circular
functions; instead, my real interest has been a more careful and deep
consideration of transcendental functions that cannot be transformed
into those named above. We can now deal with logarithmic and circular
functions as we can with 1 times 1, but that lovely goldmine
that contains the higher functions is still almost completely terra
incognita. …One stands in awe before the overflowing treasure of
The Calling of Elliptical Functions
Kirsch
However, the gateway to this new domain of study involves
an often missed, but crucial, historical irony.
Among most scholars of science and mathematics today,
there is little understanding that higher analysis in geometry originated
from the overturning of Descartes’ ban on transcendentals.
But what is even less understood today, is that the development
of the analysis of what came to be known as ‘elliptical’ integrals,
was born directly out of the instance of Leibniz’s incorporating
his method of transcendental physical curves into his fight against
the Cartesian ‘physicists’. The weight of this historical truth is
necessary for comprehending the tensions within the body of
knowledge discovered by Carl Gauss and Bernhard Riemann.
Leibniz’s Dynamics
In 1686, Leibniz exposed the fraud of Descartes’ quantity
of motion, causing an irrational freak out by a well known
Cartesian, Abbott Catalan 29 ; despite further correspondence on
the subject, his cult-like belief in Descartes would not be challenged.
30 Understanding the steps one takes to determine a new
paradoxical proposition, Leibniz consciously recruited others to
his method by posing a challenge whose solution required the
discovery of a new principle or the application of a new method.
In 1689, Leibniz challenged those infected by Cartesian
methods to solve both the simple isochrone, (the curve a body
would take traveling equal vertical distances in equal times) 31 ,
and the paracentric isochrone (the curve a body would take receding
equal distances from a given point in equal times), knowing
that only those who surpassed the dogma of Descartes could approach
the problem.
The political guts of Leibniz, and the realization of the
uniqueness of his method, inspired the Bernoullis to take up the
challenge of these isochrones, and led in end effect to a movement
of scientists who could demonstrate, simultaneously, the fallacy
of both Descartes’ physics and his ban on applying geometry to
mechanical curves. 32
new and highly interesting truths and relations which these functions
offer.” –Carl Gauss, Letter to Schumacher, as translated in Carl
Friedrich Gauss, A Biography, Tord Hall, p. 135
29 Gottfried Leibniz, Brief Demonstration of the Error of Descartes,
Leroy Loemker (Kluwer Academic Publishers 1989)
30 See Leibniz’s correspondence with Arnauld, translated by
Montgomery (Open Court Publiching 1901)
31 This curve led to the cubic parabola, which Jacob Bernoulli
divided in such a way as to play a unique role in inspiring Fagnano’s
Discoveries.
32 Leibniz recounts in the Acta Eruditorum of 1697, that it was
his isochrone challenge which led Jacob Bernoulli at the conclusion
of his solution published in the Acta Eruditorum to pose the Catenary
challenge; Leibniz comments that this was the first true application
of the power of his method, allowing them “to later accomplish marvels
with this calculus, so much so that, from now on, this method is
Δυναμις Vol. 3 No. 3 December 2008
13

Box 8a
The Paracentric Isochrone
The Calling of Elliptical Functions
Kirsch
The paracentric isochrone is the curve
formed by a body which moves away from a
given point in equal distances corresponding to
equal times. To construct such a curve requires
a method of inversion. Rather than having a
curve, and determining its physical properties,
as with the catenary or elastic curve, this curve
is found as the solution to the aforesaid physical
properties.
The body is dropped from a height i,
the radius of the circle in the diagram. Since the
property is traveling equal distances in equal
times away from a point, once the body reaches
the start of the curve at A, it will fall the same
distance away from A at the first moment of the
curve, as it will in a moment along any other
part of the curve. This important fact is combined
with two other known laws of motion, to
lead to transcendental relation which is to be
constructed.
First, the relationship between the speed
at the first moment to the speed at the second moment is proportional to the proportion of the two distances traveled at
those moments. Secondly, the squares of the speeds are proportional to the vertical distances fallen. This means that the
squares of the proportion of the distances traveled at two moments, are directly proportional to vertical distances from
which they have fallen. Obtaining a representation of this proportion is our first step.
In our diagram, call Aw the infinitely small distance fallen at the first moment of the curve, δα distance traveled
in a moment at some other point δ of the curve. The distance βα, is the change in the distance of the body from the
given point A, in the time it travels form from point δ to an infinitely small distance away point α. From what was said,
this distance is equal to Aw. Therefore, as Aw
βα
equals the speed of the body at A to the speed at δ, so equals the same
βαβα Ai =
δα
relation. And, from what was also stated above, βαβα Ai
δαδα Ai+αγ
δαδα = . Since βαβα + δβδβ = δαδα , this can be coverted to
Ai+αγ
βαβα Ai
βαβα Ai
δαδα = . To express this we take into consideration a
αγ
δαδα =
simple geometric application of the differential calculus.
αγ
Let Aγ = x, γα = y, ζ� = z, A� = a, Aδ = t, βα = dt, y
By Box 4, o� = adz
√ aa−zz , thus, βδ = tdz
√ aa−zz .
Therefore, our relation above βαβα
δβδβ
Rearranged and reduced, this becomes dt
√ t =
Δυναμις Vol. 3 No. 3 December 2008
t
= z
a
At
tt dz dz tz
= , becomes, dt dt :
αγ aa−zz = a : a .
adz √ . Integrating, we have 2
aaz−zzz √ t = �
δα
adz
√ aaz−zzz . How to con-
struct this relation?
In Bernoulli’s 1694 paper on the elastic curve, he compares the relation of the tension in an elastic band, to its
length, and the width of the band to its stretching. He also compares the length of the band to a small element, and the
radius of curvature to its stretching. By comparison of these two relations, he determines a transcendental relation,
which he proceeds to construct by an elaborate method of quadrature, similar to that in Box 3, but much more complicated.
Through this, he is able to construct the elastic curve, the arc of which AQ, is found to be equal to �
In comparison of his two constructions, one for the paracentric, and one for the elastic curve, he finds that the
mean z is the third proportional of a and u, the abscissa of the elastic curve, or in other words, z = uu
a . By substituting
this in the equation above for the parcentric isochrone, we are left with √ at = �
aaaa du
√ aaaa−uuuu .
aaaa du
√ aaaa−uuuu .
14

Box 8b
By the acceptance of this challenge, physical transcendental
curves were now opened to be studied. What this signified
was a decisive defeat for the Cartesian method, and also an opening
to a new domain.
Jacob’s solution to “Leibniz’s Curve”
The differential relationship of the paracentric isochrone
led to a form similar to the differential expression for the quadrature
of the circle and hyperbola, i.e., a transcendental one. 33 However,
unlike those, and related others, there was no way to relate
the paracentric isochrone to any known, constructible functions.
In the construction of such a curve, Jacob tried to uncover
the identity of this curve by enticing relatives of the curve, and
seeing if they would, as though spilling the family dirt, help in
identifying its true character. In this way, he found the integral of
the paracentric was directly related to the elastic curve, the curve
of a bent flexible rod, whose integral rises to the fourth degree. 34
as much theirs, as it is mine.” (ftp.ljcentral.net/unpublished/Pierre_
Beaudry/)
33 See Box 8a. Jacob Bernoulli, Complete Works 1740, Volume I, No.
LIX, p. 601 Jacobi Bernoulli Solutio Problematis Leibnitiani: De
Curva Accessus and Recessus aequibalis a puncto dato, mediante
rectificatione Curva Elasticae
34 Jacob had begun studying this curve upon working on the
Catenary chain. His paper was published in 1694, the same year as
his ‘solution’ to Leibniz curve. See H. Linsenbarth’s 1910 German
translation Abhandlungen ueber das gleichgewicht und die Schwin-
The Calling of Elliptical Functions
Kirsch
Jacob Bernoulli’s construction
of the Paracentric Isochrone1 OA OA = x
AB AB = y
OA = OA' OA
OM = a
OD = a
�
OM =1
OD = √ 2
Arclength of Lemniscate OA′
�
a2 dx
√
a4 − x4 Applicata of Elastic Curve AB
�
x2 dx
√
a4 − x4 Arc Length of Ellipse CD
� (a 2 + x 2 )dx
√ a 4 − x 4
1 Bernoulli’s original diagram included a hyperbola on either side of the lemniscate.
1
However, due to the non-constant ‘law
of tensions’ which the elastic curve obeys, another
curve would have to be used. 35 Jacob found
a saving grace. He writes “as if from a prayer, a
curve of four dimensions presents itself” shaped
like “the bow of a French ribbon.” This curve
was the lemniscate, which, by employing it instead
of the elastic curve, since the expression
for their arclengths are identical, he could turn a
quadrature into a rectification. 36 In relating these
curves to one another, Jacob came upon an interesting
relation between the ellipse, lemniscate,
and elastic curve. 37
But, as the elastic curve and lemniscate
were themselves only described in terms of the
functions they produced, relating such curves
to the paracentric isochrone did not reveal the
identity of such an integral. The problem was,
all the relatives shared in the family secret; they
only posed new paradoxes, leaving as a mystery
the principle involved in these higher transcendentals.
One is left asking: what was the principle
which created the process of the paracentric
isochrone, and its relatives? What was the
process to which these predicates correspond?
Such was the boundary of this higher transcendental.
But, just as the quadrature of the hyperbola needed to
be ontologically defined by a higher transcendental of a different
species, the catenary, so the ‘elliptical’ integrals of the lemniscate
and elastic curve, related to the paracentric isochrone, demanded
to be defined by a higher domain. What seemingly was limited
to physics was in fact an impassable barrier into the realm of the
principles of space.
Looking back from the origin which led to the study of
‘elliptical’ integrals, as Bernoulli referred to them, it is seen that
Descartes’ absurd physics had an ironic effect of rendering the
investigation of transcendental physical processes, and the functions
related to them, impossible. Leibniz’s method of dynamics
gungen der ebenen elastischen Kurven Von Jakob Bernoulli (1691,
1694, 1695) und Leonh. Euler (1744) (Leipzig: Verlag von Wilhelm
Engelmann)
35 Jacob Bernoulli, Complete Works 1740, Volume I no. LXIV,
p. 627, G.G.L Construction Propia Problematis De Curva Isochrona
Paracentrica
36 The elastic, like other transcendentals noted in Bernoulli’s lectures,
required for its construction, the double action of maintaining
an equality of area between a quadrature of a geometric curve, and
a growing rectangle equal to it. Rectification is simply a geometric
curve being drawn, as in the case of the lemniscate.
37 See Box 8b
Δυναμις Vol. 3 No. 3 December 2008
1
15

Multiple Arc Principle
Arc OZ = 2 Arc OW
when
OZ = 2 OW√ 1 − OW 4
1+OW 4
The Calling of Elliptical Functions
Kirsch
Box 10a
Comparison between Euler’s and Gauss’s view of the Lemniscate
A =
Euler Gauss
� 1 − aa
1+aa
Corda Arcus dupli = 2aA
1−aaAA
Corda complementi dupli = AA−aa
1+aaAA
corda arcus (n +1)cupli =
corda complementi = AB−ab
1−abAB
aB + bA
1 − abAB
Box 9
Fagnano’s Discoveries
Complement Principle
Arc OA = – Arc BM
When
√
1 − OA2 OB = √
1+OA2 Complement Arc Principle
Double Arc Principle
Multiple Arc Principle
coslemn p =
The Complement Principle discovery
may very well be connected with the
following:
Let OA = z, AE = y, OE = x
Since (xx + yy) 2 = xx − yy
and zz = xx + yy
If OM = 1, then CM = OB !
which means that DM is also = OA
�
1 − sinlemn pp
1+sinlemn pp
2sc
Sehne des doppelten Bogens 1−sscc
cc−ss
Cosehne des doppelten Bogens 1+ccss
(sinlemn p)(coslemn q) ± (sinlemn q)(coslemn p)
sinlemn sinlem (p + ± q)
=
1 ∓ (sinlemn p)(sinlemn q)(coslemn p)(coslemn q)
(coslemn p)(coslemn q) ∓ (sinlemn q)(sinlemn p)
coslemn coslem (p + ± q)
=
1 ± (sinlemn p)(sinlemn q)(coslemn p)(coslemn q)
Δυναμις Vol. 3 No. 3 December 2008
1
1
x = z√1+zz 2
y
x =
1
1
y = z√ 1 − zz
2
�
1 − zz CM
=
1+zz OM
16

thus opened the door to the confrontation with the essential, and
aggravating problem of dealing with physically related transcendental
pathways of four dimensions.
A Great Frustration
The one crucial, actual discovery in the study of elliptical
integrals occurring before the time of Gauss, was made by the
Italian, Giulio Carlo Fagnano, who had been studying philosophy
and theology before devoting himself to the study of geometry,
particularly, the investigations of the Lemniscate.
He begins his most famous work Metodo per Misuare
La Lemniscata, thus:
“The two greatest geometers, the brothers 1
Giacomo (Jacob) and Giovanni (John) Bernoulli have
made the lemniscate famous, using its arcs to construct
the paracentric isochrone”.
The Calling of Elliptical Functions
Kirsch
Box 10b
p =Arc OM =Arc om
Euler q =Arc ON =Arc on Gauss
OM = a
ON = b
Om = A
On = B
Box 10c
OM =sinlemn p
Gauss compared ON arcs =sinlemn on the lemniscate, q with arcs on the circle. As
Gauss noted in his Om notebook, =coslemn the ptangent
of an angle from the center
of the lemniscate, On =coslemn equals the qlemniscatic
cosine at that angle.
So inspired by
‘Leibniz’s curve’, Fagnano
discovered fundamental
principle’s.
First, what is the algebraic
relation between
the functions related to
complementary (opposite)
arcs of the curve?
This involves a geometrical
relation with
the tangent taken at a
unique singularity of
the curve, allowing one
to compare the angles of
the relevant functions.
Second, as multiple arcs of the lemniscate are traversed,
what is the algebraic relation between the functions of the
curve? 38
The origins of Fagnano’s discovery are still to
be uncovered by intrepid discoverers; only passing clues
are left in his two part paper, Metodo per Misuare La
Lemniscata. In his first work, he presents a relation between
the equilateral hyperbola, ellipse, and lemniscate,
in a diagram very similar to Jacob’s construction for the
paracentric isochrone. Fagnano states that his complementary
arcs principle was discovered by him in relation
to rectifiying the arcs of a certain parabola. In connection
with his second principle of multiple arcs, he states that
the measure of the lemniscate depends on the extension of
the equilateral hyperbola and a type of ellipse, while the
measure of the cubic parabola depends on the extension
of the lemiscate. This echoes Jacob’s construction noted
above. 39
With the combination of these two principles,
Fagnano showed how to divided the Lemniscate into equal parts,
such as 2,3,5 parts; and generally divisions which fall under 2
times 2 to the m, 3 times 2 to the m, 5 times 2 to the m, where the
exponent m represents any positive whole number.
These discoveries by Fagnano were crucial, as they made
clear certain relations between the curve itself, and the functions
produced by the curve. This at least allowed the shadows, which
the higher transcendental generated, to form a pattern with which
to work. However, although Leonard Euler generalized the relation
of multiple arcs to their functions, which became known
as the ‘addition theorem’, the principle of organization was still
38 See Box 11.
39 Fagnano then references its relation to a paper in the Acta Eruditorum
of 1695, concering the division of the Cubic Parabola. The
cubic parabola was the solution to the first isochrone challenge given
by Leibniz in 1 1689, and Jacob Bernoulli had investigated dividing it
into equal arcs.
Δυναμις Vol. 3 No. 3 December 2008
1
OM = a
ON = b
Om = A
On = B
OM =sinlemn p
ON =sinlemn q
Om =coslemn p
On =coslemn q
17

The Calling of Elliptical Functions
Kirsch
Series approximations of the Lemniscate integral
by Stirling and Euler, as noted in Carl Gauss’s notebooks
completely unknown 40 ; the cause which generated the process,
created a great frustration looming over the cognizant minds who
attempted the search.
4. Removing the Training Wheels
Carl Gauss’s daily log and notes from January 1797
through fall of 1798, maps out the discovery of a new field of
science involving the nature of the Lemniscate. 41
Gauss began with the methods of his predecessors, but
looked at the same function with completely different eyes. His
notebooks show that he read everything by Euler on the Lemniscate
related to his elaborations of Fagnano’s discoveries, such
as his Observationes de Comparatione Arcuum Curuarum Irrectificabilium.
He also tackled the many works on determining
values of transcendentals with infinite series. For example, John
Stirling’s De summatione et interpolatione serierum and Leonard
Euler’s, De Miris Proprietatibus Curvae Elasticae, the latter
written specifically on finding infinite series for the lemniscate
integral.
However, he wasn’t interested in these methods because
they were only good for finding numerical values. Unlike Euler,
who treated the expression for the lemniscate as a complicated
algebraic expression, which merely begged a numerical approximation
by series, Gauss had something else in mind when consid-
40 When Fagnano was nominated to the Berlin Academy in 1750, he
sent the Academy a copy of his Produzione Matematiche which
reached Euler’s hands on 23rd December 1751, a day described by
Jacobi as “the birthday of elliptical functions”. However, in the papers
he would present to the academy in the following years on the
subject of the lemniscate addition theorem, Euler’s work was nothing
more than an elaboration of the implications of Fagnano’s discoveries;
Euler himself made no original discoveries of his own. Euler,
who did nothing more than take the insights of Fagnano and fill them
out, was led to great fame, and is today considered one of the great
geniuses in mathematics. Fagnano, an actual genius, is held in obscurity,
barely known by the recipients of his original discoveries.
41 Werke, Volume III p. 404-480, Volume X pages 145-206, 509-
543
ering this function, and simply
utilized the methods of series for
a different task. For Gauss, the
paradox of elliptical integrals
served as a tool to open up a new
domain of truths, a new instrument
for experimentation, not a
paradox to be reduced back into
something relatable to so-called
known domains, as Euler had
exhaustively attempted to do.
In this spirit, Gauss
deviated from the prior method
of Fagnano and treated the lemniscate
problem, not as an algebraic relation of functions whose
complementary arcs are arithmetically related, but as a periodic
function (what he called the sine and cosine of the lemniscate). 42
With this understanding of the periodic Sine and Cosine
of the lemniscate, Gauss proceeded along a new path, untread.
Gauss, unlike all his earlier predecessors, investigated not how
the ‘elliptical integrals’ could be expressed in terms of their functions,
but rather, what is the process such that it defines these
particular predicates.
To clarify the point: just as the sine and cosine of a circle
are the trigonometric functions of the arc of the circle, so the sine
and cosine of the lemniscate, an ‘elliptical’ integral, are the ‘elliptical’
functions of the arc of the lemniscate.
Therefore, the question restated in another way is: rather
than investigating how the elliptical integral could be defined
as a function of its functions, Gauss asked, how can the elliptical
function be defined as a function of elliptical integrals? The
process is thus investigated not by what it generates, but how it
generates it, i.e., by looking at the way the function of elliptical
integrals can express the elliptical function, he asked the question:
how does the process, define itself?
Carl Gauss’s experimentation with the Lemniscate
function, expressed as quotients of infinite products and
series, from his notebooks
42 See Box 10 on Lemniscatic Sine
Δυναμις Vol. 3 No. 3 December 2008
18

The Calling of Elliptical Functions
Kirsch
Box 11
Zeroes and Infinities
Gauss was more interested in finding how the function
itself acts, rather than finding infinite series approximations.
In other words, Gauss was searching for singular
characteristics of the function, to distinguish it from others.
Take an average, everyday transcendental function, such
as the sine of an angle. What are some singular properties
of the sine function? The function continuously changes,
as its radius moves around the arc of a circle, but it has a
maximum and a minimum – positive 1 and negative 1, respectively.
It also traverses zero twice in its period around
the circle. Its change is non-constant, so it is impossible
to determine its exact length, in terms of the diameter of
its circle, except at those singular points and a few others.
We can begin to develop a mathematical expression of
those singular points, in hopes of defining our function
with them.
In 1799, Gauss proved beyond a doubt that, since
algebra is merely a description of a real physics, an algebraic
equation was always decomposable into a number
of factors equal to the highest degree of the equation.
Each of these factors represent where the equation is
zero. So, the equation x2 − 4=0
sin x = x(1 − x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 (1 − x4
π4 has two zeroes: at positive
and negative 2. Therefore, it can also be written as
(x − 2)(x +2)=0 .
Every algebraic equation has a finite number of
zeroes, and thus factors, but our sine function is transcendental,
thus having an infinite number of zeroes. Therefore,
we can attempt to approximate our sine, using an algebraic
equation with an infinite number of factors. Since the
zeroes are all at integral numbers of half-circumferences
(0, π, 2π, 3π,..., −π, −2π, −3π,... ), we can write:
Now, whenever the arc x is equal to a denominator,
that factor will become zero, and we have a zero of
the function. Try it out for some
)
non-singular point, such
as x =1 radian (2π radians = 360◦ ). Our function gives
1(1 − 1
every time it traverses one lobe of the lemniscate. Therefore,
when x is 0, Π, 2Π, 3Π,..., −Π, −2Π, −3Π,... ,
the lemniscate function is zero. Our function
should, therefore, look much like the sine function:
Let’s see what Gauss really said:
OK, now this looks a bit different than what we
found. Let’s analyze it. First, instead of (1 −
1 1
1 1 1
π )(1 − 2π )(1 − 3π ) ···(1 + π )(1 + 2π )(1 + 3π ) ···
etc., which will be, approximately, 0.866054044. The actual
sine of one radian is more like 0.8414709848, which
is pretty close.
Gauss applies this reasoning to the lemniscate
1
function. If the lemniscate function is the radius of the
lemniscate, and its variable is the arclength of the lemniscate
x, then we can find its zeroes. The radius is zero
x
Π ) , Gauss
has (1 − x4
Π4 x
) . Second, if you look closer, Gauss actually
has written down a ratio, with denominator terms.
Still, if the arc is equal to some whole number of lobes, Gauss’s
function equals zero. But, what if any of the factors in the denominator
equal zero? Then, this function expresses where
the lemniscate function is infinity! Where on the lemniscate,
is its radius infinite? Nowhere! What is Gauss doing here?
2 − 4=0
sin x = x(1 − x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 (1 − x4
π4 ) when x4
Π4 = −1 , or when the arc is equal to the
fourth root of negative 1 times Π. What is the fourth root of
negative 1? (In other words, the biquadratic root of negative
one.) Gauss would also write, for the good old circular
sine,
In this function, which actually gives a much better
approximation to the sine function, the first factor (1 + x2
π2 − 4=0
sin x = x(1 −
)
equals zero when x equals positive or negative π times
the square root of negative one. Where is that zero on the
circle? Notice, also, that the circle has no infinity points,
or poles, but the lemniscate does. Where are those poles?
x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 − x4
π4 x
)
2 − 4=0
sin x = x(1 − x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 (1 − x4
π4 x
)
2 − 4=0
sin x = x(1 − x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 (1 − x4
π4 x
)
1
2 − 4=0
sin x = x(1 − x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 (1 − x4
π4 )
x
1
2 − 4=0
sin x = x(1 − x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 (1 − x4
π4 x
)
2 − 4=0
sin x = x(1 − x x x
)(1 − )(1 −
π 2π 3π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
π 2π 3π
sinlem x = x(1 − x x x
)(1 − )(1 −
Π 2Π 3Π )
···(1 + x x x
)(1 + )(1 + ) ···etc.
Π 2Π 3Π
sin x = x(1 − x2 x2 x2
)(1 − )(1 − )
π2 4π2 9π2 ···(1 + x2 x2 x2
)(1 + )(1 + ) ···etc.
π2 4π2 9π2 (1 − x4
π4 19
)
Δυναμις Vol. 3 No. 3 December 2008

The Calling of Elliptical Functions
Kirsch
Box 12
Lagrange’s Inverse Function
Continuing to utilize the mathematical
apparatus of his predecessors,
Gauss capitalized on the Lagrange inverse
theorem to get an expression for
the inverse function for the lemniscate,
i.e. if by some variable y, we denote the
arc length of the lemniscate which is a
function of some variable x, then how
can we express x, in terms of a function
of y?
At left is an algebraic infinite series
approximation of the inverse function,
revealing nothing too profound
about the nature of x.
Where Lagrange used
his inverse function to simply
get a numerical expression for
the function he was looking at,
Gauss was using the inverse
function as a means, a stepping
stone for a more elaborate investigation
into the nature of
the function, using calculations
to determine what the inverse
function does.
Since Gauss looked upon the
sine and cosine of the lemniscate as real
functions, not simply numerical quantities,
he experimented with them, and
looked at how they change taking their
derivatives, and reciprocals, to determine
the properties of the function. This
stands in contrast to Euler, who looked
at the function as a magical box, where
one puts something in, and gets something
out.
The uniqueness of Gauss’s investigation
here can also be seen as he
finds, in an analogy to the circle, the tangent
and the derivative of the tangent.
But wait – on the visible lemniscate, as
we saw above in Box 9 on Fagnano’s
use of the tangent, there is no ‘tangent’
of the lemniscate, because it equals the
sine and cosine of the lemniscate!
Δυναμις Vol. 3 No. 3 December 2008
20

One asks, “But how
can these functions be defined
by the process, if the process
is the principle you do not yet
know and for which you are
searching?” That is exactly the
difficulty. In this respect, one might exclaim, looking back at the
earlier calculus, “Ah, training wheels!”
To accomplish the fullfillment of this method, Gauss
generalized analysis for the first time. He took into consideration,
the arc of the lemniscate which was not visible on the geometrical
lemniscate: a ‘complex’ one. 43 Gauss, knowing the crime which
Euler perpetrated in this regard, wrote that, by the neglect of
imaginary magnitudes, the field of analysis “forfeits enormously
43 Gauss never breathed a word about how he incorporated complex
numbers into his analysis of higher functions, such as the lemniscate,
except to his collaborator Bessel in 1811.[see letter in this
issue]
The Calling of Elliptical Functions
Kirsch
“On the lemniscate we have found out the most elegant
things exceeding all expectations and that by methods
which open up to us a whole new field ahead.”
– Gauss 1798
in beauty and roundness and
in a moment all truths, which
otherwise would be universally
valid, are necessarily yoked to
the most cumbersome limitations.”
[emphasis added] 44
44 This lack of universality arising as a consequence of dismissing
‘imaginary’ values, is not limited to higher analysis. Gauss demonstrates
two years later in his 1799 doctoral dissertation, that equations
have imaginary roots, and likewise, that no general theorem
can be stated about algebraic magnitudes, without their physical
significance understood. This statement is also reflected in one who
looks back upon the study of quadratic residues from the standpoint
of biquadratic residues;what may seem as a paradox, such as -1 being
a quadratic residue of all prime numbers of the form 4n+1 and
not 4n+3, is clear as day to one who incorporates ‘imaginary’ numbers
into the field of arithmetic, and contemplates the geometry of
the complex modulus. http://www.wlym.com/~animations/ceres/
index.html
Δυναμις Vol. 3 No. 3 December 2008
21

So this was true indeed for the higher functions of the
4th degree. Without this, their nature, and a new field, was not
possible.
Gauss examined the quotient of periodic functions from
this standpoint and determined which values of the periodic functions
make the numerator equal to zero, giving a value of zero for
the elliptical function , and which values of the functions make
the value of zero for the denominator, giving a value of infinity
for the elliptical function.
Without expressing the sine of a real arc plus an ‘imaginary’
arc, the infinty points were not representable, indeed they
could not be found; nor was the characteristic that the elliptical
function is found to be periodic in two different ways. 45
Once Gauss obtained the knowledge of the zero and
infinity values of the of the elliptical function, he could then
(employing some of his earlier inductive researches related to
Lagranges algebraic method of inversion 46 ) represent them, first,
by means of a quotient of infinite products, and, subsequently,
by a quotient of power series, leading to a basis for investigating
further into the nature of the lemniscatic function. 47 For example,
Gauss finds that the logarithm of the numerical value he approxi-
mates for the denominator is equal to π
2 , which he says “is most
remarkable, and a proof of which promises the most serious increase
in analysis” (See Figure 1)
45 See Appendix
46 See Box 12
Box 13
The value of the angle which produces
a given sine on a circle, is ambiguous
from the standpoint of the sine.
Another way to think of it:
Two values for x, for each value of y
y = xx
√
y = x
47 For a related expression of infinite products see Box 11.
The Calling of Elliptical Functions
Kirsch
A Simple example of Multi-Valued
functions
Derivative of the Arc Length
ds
dz =
1
√
1 − xx
x ∞ 0 1 −1 2 −2
ds
dz 0 1 ∞ ∞ √ 3i √ 3i
Two values of x for every one value
of ds
dz
1
By this, Gauss discovered a whole
new set of relations for the numerator and
denominator of the sine of the lemniscate,
relating the arcs of the lemniscate, the circle,
and the exponential functions. Using
this, he continued to investigate the nature
of the lemniscatic function as expressable
through the infinity and zero points, but in
ways which revealed deeper truths.
Applying this new found relation,
he investigated which values of the sine of
the circle make the sine of the lemniscate
equal to zero, and which make it equal to
infinity, representing the lemniscate now by
a quotient of infinite products whose variables
were sines of the circle; later on, he
converted this to a trigonometric series of
multiple angles of circular sines.
About the implications of this, he
writes of this in late July 1798 in his day
book: “On the lemniscate we have found
out the most elegant things exceeding all expectations
and that by methods which open
up to us a whole new field ahead.”
What was involved with this, was his study of the interelated
properties of the numerator and denominator themselves. 48
A Priest’s Calling
Gauss investigated the domain in which the nature of the
process generating elliptical functions exists. He investigated the
characteristics which uniquely define elliptical functions. However,
Gauss never made clear in his lifetime how he conceptualized
the principle involved in these higher functions, and thereby,
did not bring their potential fully into access for use by the human
mind in conceptualizing the universe, and its processes.
Bernhard Riemann, who like Kepler and Leibniz before
him, had been in training to be a priest, inverts the problem altogether,
defining ontologically what Gauss never stated.
In Riemann’s lectures on Elliptical Functions, he begins
with the characteristic that elliptical functions are doubly peri-
48 This study of the numerator and denominator of the quotient
expression of the elliptical function led to various paths. In 1799,
Gauss’s first entries about his use of the arithmetic-geometric mean
arise out of this research of the relations between the numerator and
denominator of the elliptical function. Also, the development of his
study of the numerator and denominator of the elliptical function
becomes the basis for Gauss’s 4th proof of quadratic reciprocity. Jacobi,
in his own studies of elliptical functions, references this paper
by Gauss, when employing the same series, which he called the Theta
Function. Later in Riemann’s lectures on elliptical functions, he
discusses the Theta Function, seen here in a unique way in Gauss’s
notebooks, as one of many expressions for elliptical functions.
Δυναμις Vol. 3 No. 3 December 2008
1
etc ...
22

odic. From this fact, he then derives what the consequent characteristics
would have to be, representing the zero and infinity
points on a complex plane tiled with parallograms, whose sides
represent the two periods which were originally set forth. From
these characteristics alone, the total of all the possible values
of the function to which a single parallelogram on the complex
plane corresponds, is found to be equivalent to an elliptical
integral. Thus, the elliptical integral is shown to be derived
simply from the double periodicity.
He then shows that since for doubly periodic functions
of the second order, there are two values of the differential
of the elliptical integral for every value of the inverse
function, therefore, in order to conceptualize this inverse
function, the two values have to be made distinguishable
and unique. To achieve this, Riemann imagines two separate
sheets, one for each value, and once more employs the
characteristic zero and infinity points, but, this time, as discontinuities
of the inverse function, to reunite the sheets in
such a way as to obtain a connected surface, where one can
traverse the whole range of possible values of the function
continuously. However, the concept of the way in which
these sheets are connected, representing the different values
of the function, is not a concept which allows itself to be visually
represented in sense-perceptible, three-dimensional space;
this defines the nature of this class of higher transcendentals, to
which the lemniscate and all of its related functions belong as
special cases, as existing in such a domain, outside of sense perceptible
physical space, conceivable in the intellect alone. 49
The Calling of Elliptical Functions
Kirsch
The doubly-periodic function, from Bernhard Riemann’s
lectures on Elliptical Functions
49 It is unclear if Gauss had come to this conception but simply
failed to present it. Two years after Gauss’s introduction of complex
magnitudes into the study of higher transcendental functions, he
demonstrates complex functions as a multiply connected geometry
of two dimensions, in his 1799 Fundamental Theorem of Algebra.
In letter to Bessel in 1811, Gauss writes of a proof for path independence
for integration in the complex plane, that this path independence
holds as long as integration is not through zero points, and
Thus, as Cusa defined the characteristic of Notother,
which all true principles in the universe share, their
nature is that they, like the definition defining everything
as being not other than the defined, define themselves.
So, man is freed from the domain of the shadows,
depending on predicates to express curves. Rather,
the process which generates the sense perceptible curve is
conceived in the mind; a concept of the substance so clear
that, by the interconnected leaves of the Riemann surface,
every value, every predicate is defined, as a result of the
action of the substance itself.
From the standpoint of this conception of elliptical
functions---realizing that these are not sense perceptible
curves, but have their nature and are derived from
the characteristic of double periodicity---it is then clear
that the former method of dealing with geometric curves,
which for the most part looked at the characteristics of the
effects of the process, was entirely inferior.
Looking back from Riemann’s conceptual discovery,
it becomes clear that with the correct understanding of higher
physical functions, all of sense perceptible geometry exists as a
special case, a projection of some higher function which can be
conceptualized but not sense perceptibly represented.
Branch Cuts, from Riemann’s lectures on Elliptical Functions
Ah! But the question remains:
If the catenary was the necessary physical process which
defined the lower geometric transcendentals ontologically, and
the multiply-connected Riemann surface is the concept which defines
the geometric lemniscate, to what physical process does this
concept correspond?
that integrating through infinity points leads to multiple values of the
function. Five years later, in 1816, in his third proof of the Fundamental
Theorem of Algebra, he again discusses integration of areas
on a plane, and makes an almost identical point that he earlier made
to Bessel concerning the ambiguity of infinity points, with respect to
values of an integral.
Δυναμις Vol. 3 No. 3 December 2008
23

Appendix: Imaginary Arcs
In 1839, C. G. J. Jacobi read a paper at the Berlin Academy
of Sciences devoted to the use of complex magnitudes in
arithmetic. As introduction, he said this:
“Gauss, in his investigations of biquadratic
residues, introduced the complex numbers of the form
a + b √ −1 as moduli or divisors... But, however simple
such an introduction of complex numbers as moduli
may now seem, it belongs nonetheless to the most
profound notions of science; indeed, I don’t believe
that arithmetic alone led the way to such concealed
notions, but rather that it was derived from the study
of elliptical transcendentals, and indeed the particular
type that are given by the rectification of the arcs of
the lemniscate... [J]ust as the arcs of the circle can be
divided into n parts through the solution of an equation
of the nth degree, the arcs of the lemniscate can
likewise be divided into a + b √ −1 by the solution of
an equation of degree aa + bb .”
Jacobi was on to something, but he did not have the insight
of Bernhard Riemann, or his teacher Carl Gauss. How did
Gauss find the poles of the lemniscate function? Nowhere on its
arc, does the lemniscate attain an infinite radius, although those
infinite radii are represented in the denominator of the quotient
Gauss has in his notebooks. Soon after this entry, Gauss demonstrates
more of how he constructed a notion of functions:
Here, we see Gauss determining the lemniscatic sine of
an arc t plus an imaginary arc u √ −1 . How long is an imaginary
arc? Use your imagination! Up until Gauss’s time, the predominant
word on complex magnitudes was that, though they are impossible,
and therefore imaginary, we need to use them to make
the math come out alright. But, don’t go around thinking that
they are actually real – we mathematicians can just make up anything
we want, so our systems work out.
Gauss thought that idea was not just perverted and lazy,
but downright damaging to the progress of human science. In
1799, he blasted the academics on this point, and showed that
imaginary magnitudes were, in fact, more real than so-called real
magnitudes. Gauss represented those complex magnitudes on a
surface, where one direction was an increase in the real part of
the number, and the perpendicular direction was an increase in the
imaginary part of the number. So, an arc of a+bi must have two
perpendicular components – but that doesn’t make any sense.
The Calling of Elliptical Functions
Kirsch
First, let us see how this imaginary arc will represent
itself through our sinlem equation. Using Gauss’s series approximation
for the sinlem, let us see what happens when we make the
arc imaginary.
sinlem iφ = iφ − i 1
10 φ5 + i 1
120 φ9 − i 11
15600 φ13 + ···
= i(φ − 1
10 φ5 + 1
120 φ9 − 11
15600 φ13 + ···)
= i sinlem φ
How about for the coslem?
coslem iφ =1− i 2 φ 2 4 1
+ i
2 φ4 6 3
− i
10 φ6 8 7
+ i
40 φ8 − ···
=1+ φ 2 + 1
2 φ4 + 3
10 φ6 + 7
40 φ8 + ··· =?
Well, we know how long the coslem is in terms of the
sinlem, so,
�
1 − (sinlem iφ)
coslem iφ =
2
1+(sinlem iφ) 2
�
1+(sinlem φ)
=
2
1 − (sinlem φ) 2
This should seem to not quite correspond to observed
physics. While the physical lemniscate has only real arcs, our
math equations are perfectly happy with imaginary arcs. But,
where are they? One gets the sneaking suspicion, that when we
answer this question, we will no longer be looking at a lemniscate
curve.
Now, let us force the addition formula to cough up some
answers about this lemniscate function. First, we have the regular
addition formula:
(sl a)(cl b)+(sl b)(cl a)
sinlem (a + b) =
1 − (sl a)(sl b)(cl a)(cl b)
Let us smoothly introduce our “imaginary arc” which, using the
relations we just found,
Now, our lemniscatic sine of a complex arc is expressed
as the lemniscatic sines and cosines of purely real arcs. Let’s put
in some values, to see what happens. 1
We know the minimum and maximum values of the lem-
1
Δυναμις Vol. 3 No. 3 December 2008
1
1
=
1
coslem φ
(sl t)(cl ui)+(sl ui)(cl t)
sinlem (t + ui) =
1 − (sl t)(sl ui)(cl t)(cl ui)
sl t
cl u + i(sl u)(cl t)
=
1 − (sl t)i(sl u)(cl t) 1
cl u
= (sl t)+i(sl u)(cl t)(cl u)
(cl u) − i(sl t)(sl u)(cl t)
1
1
1
24

niscatic sine and cosine of real arcs. If Π represents one half of an
arc (or one lobe), then we can calculate several values. Thus, at
the origin, the sinlem is zero and the coslem is 1. At an arc equal
to one half Π, the sinlem is one, but the coslem is 0. At an arc
of Π, the sinlem again equals zero, but the coslem has continued
to the other side, to equal –1. At three-halves Π, the sinlem has
passed over to the other side, and equals –1, but the coslem has
returned to the center, and equals zero. Finally, at two Π, the
sinlem is again zero, while the coslem is one.
Now, we can see what kinds of values our “imaginary
arcs” formula will give us. Let’s set u equal to one half Π, and let
t cycle through our extreme points:
for t = 0
for t = one half Π
for t = Π
for t = three halves Π
for t = 2Π
0+i · 1 · 0 · 1
1 − i · 0 · 1 · 1 =0
1+i · 0 · 0 · 1
= ∞
0 − i · 1 · 1 · 0
0+i ·−1 · 0 · 1 0
=
0 − i · 0 · 1 ·−1 0 =?
−1+i · 0 · 0 · 1
= −∞
0 − i ·−1 ·−1 · 0
0+i · 1 · 0 · 1
1 − i · 0 · 1 · 1 =0
This is getting interesting. We have found some infinite
lemniscatic sines! The lemniscatic sine for t = Π is a bit ambiguous,
but that is OK, since we are just experimenting. Here is a
table, with the values of sinlem for all values of u and t between
zero and 2Π:
u
1 0 2
t
Π Π 3
2Π 2Π
0 0 0 0 0 0
1
2Π 1 ∞ −1 ∞ 1
0
Π 0 0 ? 0 0
0 ? 0
3
2Π −1 −∞ 1 −∞ −1
2Π 0 0 0 0 0
The industrious reader will recognize that, if this table is
continued to higher angles than 2 Π, it will simply repeat, in both
the t and the u directions. Gauss recognized that, since the characteristics
of repeating is different for the two directions, what
we have here is a function that is doubly–periodic. It has two
periods: one real, the other “imaginary.”
One final step we can take, imagining what Gauss must
have thought, a well-behaved function must be smooth everywhere,
with the only exceptions at its poles. Imagine a sphere
with infinite radius. Imagine that on that sphere is a set of perpendicular
lines, one representing the growth of t, the other repre-
The Calling of Elliptical Functions
Kirsch
senting the growth of u. Let a surface of varying height be placed
upon this sphere, whose heights correspond with those values of
the lemniscatic sine in our table. What would this look like?
Where is the lemniscate, now? Perhaps it is only an effect
of a higher, unseen function!
Δυναμις Vol. 3 No. 3 December 2008
1
25

On the Subject of ‘Insight’
LaRouche
Science in its Essence:
On the Subject of ‘Insight’
Lyndon H. LaRouche, Jr.
This article first appeared in the May 9, 2008 issue of Executive
Intelligence Review
In my Sir Cedric Cesspool’s Empire, 1 I emphasized the
importance of the concept of “insight” as key for, among other
things, understanding the mechanisms of evil which characterized
the most notable writings of the leading Fabian Society figure
H.G. Wells. Here, I return to that notion of insight for conceptualizing
the root-causes of the present plunge of world civilization,
into the prospect of an immediate new dark age of mankind,
a prospect caused by the role of the same standpoint of Wells in
his threatening the planet as a whole, with what has now become
its currently accelerating plunge toward an abyss.
In real life, one never really knows what has
been done, until one knows not only why and how it
was done, but is capable of replicating the formation
of the concept.
As I have indicated within written and oral reports published
earlier: looking back from today, the most crucial
event in my life, has been my surefooted rejection of the
concept of Euclidean geometry on the first day of my encounter
with it in my secondary classroom. The most crucial implication
of that for my later life, has been, that, in rejecting Euclidean geometry
as intrinsically incompetent, as I did that day, I had actually
made a decision which was to shape the essential features of
my life over the seventy years which have followed that event.
To repeat what I have said repeatedly on the subject of
that event, over the intervening years, the following should be
noted as an entry-point into the discussion to follow here.
My fascination with the Boston, Massachusetts Charlestown
Navy Yard, had been centered in the ongoing construction-work
there. This had forced my attention to the fact of the
challenge of understanding the geometric principle of construction
through which the ratio of mass and weight of supporting
structures to the support of the total structure, is ordered. This
repeated experience, on both my several relevant visits there, and
my haunting possession of the fact of that experience, had already
established the meaning of “geometry,” as physical geometry, for
me, that already prior to my first encounter with secondary school
geometry. 2
The continuing importance of my flat rejection of socalled
Euclidean geometry at first classroom encounter with it, is
typified by considering the way in which this reverberating experience
led, a decade and more later, to my flat rejection of the
sophistry of Professor Norbert Wiener’s presentation of so-called
“information theory,” of the still wilder insanity of John von Neumann’s
notions of “economics,” and von Neumann’s matching,
pervert’s view of the principle of the human mind. These latter
goads, and related experiences, prompted me, in 1953, to discover
and adopt the appropriate consequence of Leibniz’s work,
as the standpoint of Bernhard Riemann’s 1854 habilitation dissertation.
In that light, this adolescent experience, with its outcome,
is the best illustration from my experience of the proper technical
meaning of the term “insight.” 3 In fact, it was an integral feature
of the process which had led me, during adolescence, to adoption
of the work of Gottfried Leibniz as the chief reference-point of
my intellectual life, then, and, implicitly, to the present day.
From that point in my youth, onwards, the chief philosophical
reference-points in my intellectual development, were
wrestling against the sophistry of Immanuel Kant’s series of “Critiques,”
and the systemic sophistry of both Aristotle and his follower
Euclid. It was against that background—those rejections,
which had been fully established already for me during the course
of my adolescence, that I came to recognize, and to rely upon the
concept of insight per se: Insight as being the Platonic domain of
hypothesizing the higher hypothesis, a concept of the nature of
the human species and its individual member, which is central to
all of the discoveries of principle by Plato.
The LYM Science Project
Presently, three relevant, major projects by the LaRouche
2 This development was associated, during that same period of
my life, with my father’s principal intention in selecting those visits,
the ritual tour of the U.S.S. Constitution; my own attention was
focused on the mysteries of the construction in other parts of that
yard.
1 See LaRouche, H. G. Wells’ ‘Mein Kampf’: Sir Cedric
Cesspool’s Empire, 2008 http://www.larouchepac.com/
node/10610
3 Wolfgang Köhler: please forgive me; it was necessary!
Δυναμις Vol. 3 No. 3 December 2008
26

On the Subject of ‘Insight’
LaRouche
a discovery which is maliciously denied
to exist, as such, in conventional
academic and related programs today.
This is the aspect of Kepler’s work
which was strongly upheld by Albert
Einstein, against those Twentieth-
Century Max Planck-hating thugs of
the modern positivist tribes associated
with the pathetic Ernst Mach, and with
the worse Bertrand Russell of Principia
Mathematica notoriety.
In the third case-study, the
work of Carl F. Gauss, I had proposed
to the incoming team, from the outset,
that Gauss rarely presents the history
of his actual processes of discovery,
but, rather, presents the results, and
also provides a plausible approach to
study of the way in which he might
have effected the relevant discovery.
The mission assigned to the incoming
team was, therefore, to discover how
Gauss’s mind actually worked in his
making his key discoveries. Obviously,
that assignment for the incoming team
had been crafted by me as a challenge
within the realm of epistemology, the
Investigating the shape of space with Kepler
domain of insight properly defined.
This frankly original approach
to the study of Gauss’s work,
Youth Movement (LYM) have preceded that association’s pres- has produced some uniquely useful findings, findings which proently
approaching treatment of the implications of Riemann’s vide a uniquely original approach to taking up the unique revolu-
1854 dissertation.
tion effected by Bernhard Riemann, from the point of his 1854
The first of those three had been based on a West Coast habilitation dissertation, the change which launched the Riemann
team, which had worked through some crucial features of the an- revolution in science, through those challenges which Riemann
cient origins of modern European science, as located in the re- posed to such among his successors as the Italy school of Betti
lated work of the Pythagoreans, Plato, and the modern reflection and Beltrami.
of this treatment of dynamics in the work of Leibniz.
To explain the significance of those listed, four initial
A second team had worked through the main features of stages of work for understanding human scientific creativity in
the founding of modern European science by Cardinal Nicholas general, I proceed now with reference to the relevant implica-
of Cusa’s and by Leonardo da Vinci’s follower, Johannes Kepler. tions of what I define, once more, ontologically, as the principle
The LYM’s thorough-going, published report on the Kepler proj- of insight.
ect, is a uniquely competent treatment, as similarly expressed in
This will clear the pathway for the study of the uncom-
the work of Albert Einstein, as by relevant others, but is not completed projects of Riemann, as the case is only illustrated by the
petently taught in known university programs otherwise available work of Betti and Beltrami, as by the challenges posed by V.I.
today.
Vernadsky and Albert Einstein, later. Here, comprehension de-
In the second study, that of the uniquely original dismands the more precise treatment of the notion of insight which
covery of gravitation, by Kepler, the difficulty, highly relevant to is included in the following pages.
the matter of insight, is that secondary sources on Kepler’s work
The importance of treating that subject in this fashion
have been (see http://www.wlym.com/~animations), chiefly, vi- here, is to be located, in significant part, in the fact that the third
ciously fraudulent evasions of the actual development of Kepler’s in a continuing series of science projects conducted by teams
original and crucial discovery of a principle of Solar gravitation, from the LYM is nearing the point at which the team’s study of
Δυναμις Vol. 3 No. 3 December 2008
27

the mystery of Carl F. Gauss’s career is now entering
its completion, a point at which a comprehensive treatment
of the work of Bernhard Riemann will be undertaken
by a new team, the essential contributions to
advancing the frontiers of modern science to be found
in the work of Bernhard Riemann and his immediate
associates and other collaborators.
1. Man as Man, or Beast?
The quality of insight, as I define it, again, here,
is a specific potentiality which is fairly defined
as being unique to all those individual human
beings who are not victims of relevant physical or psychological
damage.
The present definition of human, as distinct
from beasts, is the specific power of the human species
to alter its behavior, as a species, to the effect that the
potential relative population-density of the members
of a culture is increased willfully, as this is illustrated
not only by a human culture’s ability to increase its
potential relative population-density willfully, but by
the manifest transmission of such specific qualitative
changes from one, to other members of the human species,
as, for example, through stimulation of discovery
of a physical principle by individuals presented with
the appropriate intellectual stimulus.
This quality is demonstrated, crucially, by the
willful increase of the relative population-density of
the human species, as expressed in the quality of antientropic
increase of the mass of the Earth’s Noösphere,
that relative, functionally, to the specific masses of the
Biosphere and the mass of matter originally generated as part of
the abiotic domain.
Thus, there is no species of ape, or other beast, which is
capable of meeting the standard of this test.
On this account, there is only one human race, and no
essential human differences in species, or variety, within the
ranks of humanity so defined. 4 This functional distinction in the
potentials of human behavior, whether expressed by individuals,
or by societies as a whole, is properly approached for examination
from the vantage-point established by Plato, both respecting
Plato’s refined definition of the concept of hypothesis, and the
systemically related subject of the quality of the individual human
soul, as that subject was treated by Plato and Plato’s follower
Moses Mendelssohn. 5
4 Any deviation from that rule is “racism, per se,” which is, in
itself, the expression of an impulse tantamount, under natural law, to
crimes against humanity.
5 I.e., both Plato’s Phaedo and the treatment of Phaedo by
On the Subject of ‘Insight’
LaRouche
Bernhard Riemann (1826-1866)
In general, the Classical term hypothesis, when employed
in any approximation of a meaningful, Platonic way, is
already a reflection of specifically human potential for creativity.
The simplest expression of that distinction is the difference
between reason and Sophistry. For the purposes of our discussion
here, Sophistry is typified by the reductionist method, opposed to
reason, which was shared among Aristotle, Euclid, and the hoaxster
Claudius Ptolemy, as typical of the Aristotelean form of the
method of lying called “Sophistry,” or, in current argot, “spin.”
The typical expression of corruption of the human mind
in contemporary, globally extended European culture, is Anglo-
Dutch Liberalism, otherwise known as the legacy of the New
Venice faction of Paolo Sarpi. The extremely degenerate expressions
of Liberalism (e.g., empiricism) today, are extreme expressions
of Liberalism’s intellectual degeneracy such as positivism
Mendelssohn. This is also the method of Nicholas of Cusa, as in De
Docta Ignorantia, his follower Leonardo da Vinci, Johannes Kepler,
Pierre de Fermat, Gottfried Leibniz, and Bernhard Riemann.
Δυναμις Vol. 3 No. 3 December 2008
28

and existentialism. 6
Therefore, we shall proceed with our exposition here by
taking up the case of Aristotle’s follower Euclid, as in the case of
the work titled Euclid’s Elements.
Minds Blinded by Sight
The Aristotelean form of Sophistry represented by the
Euclid of Euclid’s Elements, is premised upon so-called a-priori
presumptions, assumptions which are associated with reliance
upon the believed absurdity that “seeing is believing.”
For example, it would be impossible to discover the universal
principle of gravitation, as characteristic of the organization
of the Solar System, except by relying, as Johannes Kepler
did, upon the clear evidence of a systemic contradiction between
the Solar System viewed from the standpoint of an assumed paradigm
of sight, rather than the fruitfully paradoxical solution provided
by contrasting the characteristic of hearing, as Johannes
Kepler did, with the characteristic, linear presumption usually associated
with a naive notion of the characteristic of sight. 7
The entirety of the purely arbitrary presumptions underlying
Euclid’s Elements, was located in a naive presumption
respecting the assumed ontological elementarity of the characteristic
of vision.
Thus, true insight sees vision as such as representing the
primitive level, sees that one’s opinions on this level, are products
of a foolish belief in the reality of simple sense-experience.
The lowest level of actual human intelligence, the level of actual
insight, is the recognition of the fact that one’s opinions respecting
sight alone, are being formed in the grip of a kind of form
of mass-insanity such as “sense-certainty,” which is to be recognized
as a mind blinded, thus, by blind faith in sight.
For matters of science, and also history, naive seeing as
such must be superseded by insight. 8
6 Typically, mathematical formulations, such as mere statistics,
are substituted for actual physical principles, and even for simple
truth.
7 Kepler’s reflection on the apparent role of the series of Platonic
solids in locating the organization of the planetary orbits, led
him, by aid of reflections on the preceding work of Nicholas of Cusa,
Luca Pacioli, and Leonardo da Vinci, to recognize the composition
of those Solar bodies then known to him as being an harmonic ordering.
It was this recognition that led Kepler to his principled discovery,
through recognition of the paradoxical juxtaposition of the
assumptions of sight and the assumptions of harmonically ordered
hearing.
8 As in the distinction of Max Planck’s actual discovery from
that positivists’ perversion (e.g., Ernst Mach, et al.) known as “quantum
mechanics.”
On the Subject of ‘Insight’
LaRouche
Kepler’s discovery of the principle of general gravitation,
provides a typical kind of crucial proof of the fallacy of
sense-certainty. In his Harmony of the World, the discovery of
general gravitation within the Solar System required the juxtaposition
of two notions of senses, those of sight and hearing (i.e.,
harmony), for the derivation of a general principle of gravitation
among the planets. This leads to the recognition that our powers
of sense-perception are to be regarded as the natural experimental
instruments which “come in the box of accessories”: when the
infant is delivered from “the manufacturer.”
A similar insight into the fallacy of “sense-certainty”
was expressed by the ancient Pythagoreans and Plato, as this was
typified then in a crucial way by the construction of the doubling
of the cube by Plato’s friend from Italy, the Pythagorean Archytas.
Similarly, the significance of Eratosthenes’ praising that
construction, was shown afresh through Europe’s Eighteenth-
Century conflict between the work of Gottfried Leibniz and the
Anglo-Dutch Liberals (a.k.a. empiricists) Voltaire, Abraham de
Moivre, D’Alembert, Leonhard Euler, and Euler’s dupe, Joseph
Lagrange. 9 The modern history of that conflict begins with the
Eighteenth-Century algebra of Ferro, Cardan, Ferrari, and Tartaglia,
on the subject of quadratic, cubic, and biquadratic geometries,
and continues through, and beyond, the work of Carl F.
Gauss in such matters as the evolution of his treatment of his
Fundamental Theorem of Algebra and related matters.
Gauss’s Personal Situation
Carl Gauss suffered the misfortune of having come to
maturity in the aftermath of the French Revolution, a time which
Friedrich Schiller identified as expressing a lost, great moment
of opportunity in history (the American Revolution and the great
work of Abraham Kästner, Gotthold Lessing, Moses Mendelssohn,
Gaspard Monge, Lazare Carnot, et al. as a moment which had
fallen prey to “a little people.” Thus, although Gauss’s achievements
themselves were to be essentially a continuation of the
legacy of Cusa, Leonardo, Kepler, Fermat, and Leibniz, Gauss’s
professional career depended upon his avoiding the appearance
of support for all things which might suggest indifference to the
alleged genius of the hoaxster Galileo, Sir Isaac Newton, and of
such Eighteenth-Century enemies of Leibniz and Leibniz’s follower
Abraham Kästner as Voltaire, de Moivre, D’Alembert, Euler,
Lagrange, and their Nineteenth-Century successors such as
Laplace, Cauchy, Clausius, Grassmann, and Kelvin.
Thus, once more, the early Nineteenth Century had
brought on a period in which the minds of most were blinded by
9 Lagrange, in the last years of his life, edified the tyrant Napoleon
Bonaparte, an effort used by Napoleon to disperse the leaders of
the Ecole Polytechnique into technical duties in the tyrant’s military
service. It was Laplace and Cauchy who destroyed the educational
program of the Ecole, on orders from London.
Δυναμις Vol. 3 No. 3 December 2008
29

sight.
Thus, when I first introduced the LYM’s current “basement
team” to the challenge of their present work (presently nearing
completion) on the work of Gauss, I forewarned them, that,
whereas Gauss’s work is brilliant, and his post facto account of
the discoveries plausible; such was the nature of his time, that his
actual method of discovery was tucked, as in the case of his personal
preference for non-Euclidean geometry, behind a protective
screen of intellectual camouflage.
The implied duty laid upon him, or his successors, on
account of that carefully crafted, protective screen, included the
complementary obligation to uncover what lay, awaiting today’s
attention, behind the camouflage imposed by those hoaxsters who
represented the reputed embodiment of the alien, Newtonian tyrant.
However, today, the present result of adopting that implied
mission, is, that, to the degree Gauss’s discoveries are now being
presented as finished reports from the standpoint of Bernhard Ri-
On the Subject of ‘Insight’
LaRouche
Carl Friedrich Gauss (1777-1855)
emann’s frankness in this matter, the results, thus far, are, increasingly,
most agreeable.
Thus, the true genius of Carl Gauss could be recognized
by students today, only when the fact is considered, that much of
what Bernhard Riemann said and wrote, was indebted to what
Gauss, in his adult years, rarely dared to say publicly. Therefore,
to really understand Gauss, it is necessary to know Riemann, and
then to see how much of Riemann’s wonderful work, his habilitation
dissertation and beyond, had been made possible by what
Riemann recognized as having been lurking within the shadows
of what Gauss had permitted himself to say.
Gauss’s repeated treatments of the subject of his doctoral
dissertation, on the subject of The Fundamental Theorem
of Algebra (as complemented by the related paper on the law of
quadratic reciprocity), are to be recognized as a recurring theme
in much of the span of Riemann’s work. 10
10 Gauss’s Fundamental Theorem was first presented in 1799,
Δυναμις Vol. 3 No. 3 December 2008
30

2. The Infinitesimal
That much said thus far: shift the choice of subsuming topic,
back from the account of Gauss’s role as such, to the
ontological implications of insight per se—the point of
reference, the ontological standpoint, at which Gauss’s published
accounts of his discoveries, are, for reasons noted above, often
met at their relatively weakest expression. Gauss’s recurring,
fresh treatment of the subject of his first three statements of what
he would come to call his “Fundamental Theorem of Algebra,”
and the intimately related, higher subject of “the law of quadratic
reciprocity,” is typical.
Nonetheless, Gauss’s intention, however bounded by
the ugly peer-review pressures of his time and place as a young
adult, onward, is nevertheless to be seen as persistent in his effort
to provide his more sensible readers crucial evidence leading
them, hopefully, toward the relevant conclusions which Gauss
dares not state explicitly. 11 Once Riemann’s 1854 habilitation dissertation
and his treatment of Abelian functions are taken into
account, and the preceding writings of Gauss viewed from this
standpoint, the debated matter of Gauss’s ontological intention,
contrary to D’Alembert, Leonhard Euler, and the crooked British
imperial assets Laplace and Cauchy, et al., should be clear to any
qualified student of such matters. 12
Gauss’s treatments of the subject of the Fundamental
uttered as a direct rebuttal of Euler’s 1760 publication on that subject
and the closely related matter of the law of quadratic reciprocity. In
all of his published work on this subject, the underlying theme which
Gauss references, but does not state explicitly, is the Leibniz notion
of the ontologically infinitesimal, a connection made implicitly
clear in Gauss’s work.
11 See Bernhard Riemann, Über die Hypothesen. Welche
der Geometrie zu Grunde liegen (New York: Dover reprint edition,
1953): Sections numbered I. (Begriff einer nfach ausgedehnten
Grösse), p. 273, and II. Massverhältnisse, deren eine Mannigfaltigkeit
von n Dimensionen fähig ist ...), p.276.
12 With the defeat of the Emperor Napoleon Bonaparte, the
French intention of electing Lazare Carnot President of a French
Republic was defeated by action of the relevant British occupation
authority, the Duke of Wellington, sticking a wretched Bourbon on a
London-controlled French throne. Under this British reign over occupied
France, the scoundrels Laplace and Cauchy were installed to
uproot the educational program of the Ecole Polytechnique’s Gaspard
Monge. Monge was dumped, and his associate Lazare Carnot
went to die as an exiled hero, in Magdeburg. The mental disease
called positivism, thus grabbed control, but for a relatively few stubborn
heroes, of the official French scientific intellect. Cauchy’s role
as a hoaxster, and plagiarist of the work of Abel, was finally exposed
by examining Cauchy’s post-mortem files. Carnot was a fellow
member, with Alexander von Humboldt, of the Ecole.
On the Subject of ‘Insight’
LaRouche
Theorem of Algebra and its crucial, correlated reflection of that
“Theorem,” as reflected in what he defines as a “law of quadratic
reciprocity,” point the alert student toward the ontological issue
which he wishes to argue, but, considering the auspices, he dares
not do that too explicitly. The often referenced parallel, related
case of what is actually anti-Euclidean geometry, is to be considered
in this light, as being a correlative of that view of the
Fundamental Theorem.
The relevant argument to that effect, is as follows.
Once we acknowledge, as the Pythagoreans and Plato
already knew, that the objects of sense-certainty are never better
than shadows cast by an unsensed, but nonetheless efficient reality,
and, when the same matter is then reviewed from the standpoint
of Riemann’s work, the issues are much clearer.
The crucial point, as I have repeatedly emphasized in
earlier locations, is the fact that the enemy of Leibniz, of Gauss,
of Riemann, et al., in science, has been the pack of hoaxsters
typified by the Eighteenth-Century Liberals such as Antonio
Conti, Voltaire, de Moivre, D’Alembert, Leonhard Euler, and
Euler’s dupe Joseph Lagrange. With that British victory over
France which Britain secured through, successively, the siege of
the Bastille, the French Terror, Napoleon Bonaparte’s reign, and
the British monarchy’s triumph at the Congress of Vienna, young
Gauss had now entered the Nineteenth Century, entering a world
in which official science was oppressed by the top-down enforcement
of that moral, intellectual corruption known as the Liberalism
of Euler and Euler’s followers.
If we, then, take into account the specific issues of scientific
method posed, still today, by that same Liberal political
corruption, of the reigning official opinion in science of that time,
and ours, too, we are enabled to distinguish what Gauss clearly
intended, from what the same fear of reactions by powerful adversaries
prevented him from stating clearly, as was the case in
his suppression of reports of his own discoveries in anti-Euclidean
geometry. To present this case, it is necessary to restate here
the related point made in locations published by me earlier.
The Roots of Science
When we trace the history of European science from its
roots, in Sphaerics, from the ancient maritime culture which settled
Egyptian civilization (including that, notably, of Cyrenaica),
we must recognize what can be competently termed “science” as
being rooted essentially in the development of the navigational
systems of the ancient, seafaring maritime cultures of the great
periods of glaciation, rather than such silly, but popular academic
myths as attempting to trace civilization from “riparian” cultures
as such. It was the observation of both seemingly regular and
anti-entropic cycles in the planetary-stellar system, which is the
only supportable basis for the notion of “universal,” as that term
could be properly employed for grounding the notion of science
per se today.
Δυναμις Vol. 3 No. 3 December 2008
31

The case of the settlement
of Sumer and its culture, from the
sea, by a non-Semitic people’s
sea-going, Indian Ocean culture’s
colonizing of southerly Mesopotamia,
is indicative. 13 In any case,
the very idea of science would have
no secured basis in knowledge unless
very long spans of ocean-going
maritime cultures were taken into
account for crucially relevant features
of ancient calendars.
In short, the notion of
universal, which does not exist as
a functional conception in Liberalism,
is the essence of any competent
effort at developing actual scientific
knowledge. Only long-ranging
ancient maritime cultures could
have been impelled to produce the
elementary considerations underlying
the Sphaerics from which all of
competent strains in European, or
other science has been derived. The
idea of a universal physical principle,
on which all competent science
is premised, could not come into
existence for mankind in any other
way, unless we were to presume
the source of this opinion to be,
arbitrarily, colonists arriving from
“outer space.” I emphasize, that the
true concept of universal, does not
actually exist as a scientific conception
within the bounds of empiricism or its spin-offs.
What we know with certainty, respecting contrary views
on the possibility of the existence of a practice of science, is that
the contrary views are all either implicitly “malthusian,” or are
products of a type of culture, such as the typical “oligarchical
model,” congruent with malthusianism. I emphasize, that all
such latter types known to us generally now, belong to a category
known to ancient through modern European cultures as “the oligarchical
model,” a model to be recognized as being congruent
with Aeschylus’ representation of the Satanic-like figure of that
Delphic Olympian Zeus. This was the Zeus, who, in Aeschylus’
13 Suspected to have been an offshoot of a maritime culture of
the Dravidian, or closely related language-group. Herodotus indicates
a kindred maritime-cultural origin for Ethiopia. So, Bal Gangadhar
Tilak back-traced the origins of Sanskrit to a colonization,
across land, from the north coast of Siberia, through mid-Asia, into
Iran and northern India (Orion, and Arctic Home in the Vedas).
On the Subject of ‘Insight’
LaRouche
Johannes Kepler (1571-1630)
discoverer of Universal Gravitation
account, banned the knowledge
of science (e.g., “fire”) from the
minds of those mortal men and
women such as Lycurgan Sparta’s
helots, the lower, subjugated social
classes.
It is to be emphasized now,
as we contemplate the global wave
of mass-starvation which has been
caused by the spread of the massmurderous,
neo-Malthusian model
of that British lackey otherwise
known as former Vice-President
Al Gore, that virtually all of the
great crises of known civilizations
have been the result of those same
policies of practice which are fairly
identified as pro-Satanic attempts
to ban scientific knowledge and its
practice from the great majority of
the world’s human populations. 14
Such has been the accelerating
decline of the physical economy
of the U.S.A., per capita and per
square kilometer, since the terrible
developments and aftermath of
1968. 15
The upshot of that line of
inquiry, is that we exist within a
stellar universe which is governed
by what Albert Einstein, for example,
emphasized as being universal
physical principles of change.
14 Former U.S. Vice-President Al Gore, a British agent against
the U.S.A.’s American System of political-economy, who walks in
the footsteps of the de facto traitor to the U.S.A., and sometime U.S.
Vice-President Aaron Burr, is a typical advocate of the “oligarchical
model.” President Andrew Jackson of “Trail of Tears” notoriety, had
been an accomplice of Burr’s anti-U.S. conspiracy, and had served
as U.S. President as a lackey and accomplice of Land-Bank swindler
and later U.S. President Martin van Buren.
15 It is not merely the actions of the trans-Atlantic “sixty-eighters”
and the U.S. Richard Nixon Administration which have caused
the pattern of accelerating physical decline of the economies of the
Americas and Europe since 1968. Trends do not perpetuate themselves,
except as the relevant trend takes life, as a form of “tradition,”
within the culture of those who are shaping the policy-making
proclivities of the society. To free the U.S.A., in particular, from the
grip of forty years of self-destruction, we must free control over our
society’s policy-shaping from the hands and minds of those who embody
the “68ers” tradition.
Δυναμις Vol. 3 No. 3 December 2008
32

These principles are presented to us in this capacity,
as they were to long-ranging ancient maritime cultures,
presented so in their astronomical expression,
as a combination of both ostensibly regular and antientropic
universal physical principles of change. Some
cycles, such as the equinoctial cycle, are long-ranging,
and may appear to be fixed. However, contrary to the
neo-Aristotelean fraudster Claudius Ptolemy, and to
Clausius, Grassmann, and Kelvin, the universe is not,
ontologically, a domain of cycles of repeatedly fixed
no-change: the universe is essentially anti-entropic.
In the latter case, that universe of change, the
universe is finite, but anti-entropic, in the respect that
nothing exists outside it. Thus, rather than the foolishness
of a ignorant believer’s assumption of an Euclidean
or Cartesian, limitless space, the universe is not Euclidean,
nor Cartesian, but a dynamic system in the sense
of dynamic employed by the ancient Pythagoreans and
Plato, or such as Leibniz, Riemann, Max Planck, and
Einstein, in modern science. This notion of a physically
efficient universality which I have just presented here
so, is, as Albert Einstein emphasized, indispensable for
modern universal science; without this notion, no competent
notion of the work of Kepler, Fermat, Leibniz,
Gauss, or Riemann can be reached.
This notion which I have just so emphasized, is
crucial for understanding the great Nineteenth-Century
crisis in science which Gauss and Riemann addressed.
The interwoven conceptions of a “Fundamental Theorem
of Algebra” and “law of quadratic reciprocity”
in the work of Gauss, are typical of this. Riemann’s
remedy for what is lacking in the work of Gauss, addresses
precisely this conceptual problem, a problem
which continues to underlie not only the ongoing essential
work of all modern science, but the systemically
dynamic form of social crisis menacing the very existence
of world society today.
Our Universe
That aspect of the efficiently existing universe which is
accessible to our sense-perceptual powers, is the passing footprints
of those powers which generate such shadows themselves. As Albert
Einstein made this point in his own fashion, it is through the
relevant power of insight, like that of Kepler’s uniquely original
discovery of universal gravitation, which is, manifestly, uniquely
specific to the human species, that we are enabled to adduce the
eternal motion of that great unseen entity which has left those
footprints upon our heavens. Such is the implication of Riemannian
dynamics, as also that of Leibniz before him.
As emphasized here earlier, the fact that the organization
of the Solar System is fairly regarded as in conformity with
On the Subject of ‘Insight’
LaRouche
Cupola of Santa Maria del Fiore, in Florence, Italy
Kepler’s harmonic approximation, as Albert Einstein emphasized
the principle involved, defines a universe which is ontologically
finite. That is to say, that principles, such as the principle of
gravitation as discovered by Kepler, principles which envelop our
universe, are discoverable, and provable, only through the kind
of method of dynamics which Gottfried Leibniz revived from
the earlier discoveries of the Pythagoreans and Plato. We owe
comprehension of the implications of that fact, as Albert Einstein
emphasized, chiefly to the work of Johannes Kepler and Bernhard
Riemann. However, that discovery had already been made
implicitly by Cardinal Nicholas of Cusa, in such among his works
as the seminal De Docta Ignorantia, but it had also been known,
earlier, by the Pythagoreans and Plato.
To restate this same point: the principled form of ac-
Δυναμις Vol. 3 No. 3 December 2008
33

tion which is expressed to our senses as a predicate of universal
principles, is the universal principle on which all manifest forms
of apparently principled actions depend for their expression. The
universe of experience is defined, thus, as Einstein defined it, as
self-bounded. Thus, it is a finite universe in that sense, but without
any external boundary but the principle of anti-entropic, creative
powers associated with the notion of a Universal Creator.
The human faculty upon which such higher-ranking
knowledge of that higher, efficiently necessary existence depends,
is the object of insight in the fullest sense of Plato’s presentation
of that notion. Thus, all competent modern science depends upon
the view of this matter by Nicholas of Cusa.
To summarize that point: the notion of an ontologically
existing universe, as opposed to some Euclidean or kindred sort
of Sophist’s fantasy, depends upon the notion of universal lawfulness,
as Einstein’s view of Kepler’s work illustrates the crucial
point of all this present discussion.
To illustrate that point, take the case of the history of the
modern European discussion which led into Gauss’s first statement
of what was to become known as his view of the challenge
of the Fundamental Theorem of Algebra. Go back to the previously
referenced, Sixteenth-Century treatment of the subject of
the relations among quadratic, cubic, and biquadratic residues, as
by Cardan et al.
The ontological implications of this Sixteenth-Century
treatment of those matters must be considered against the background
of Archytas’ duplication of the cube. Against that historical
background of Sphaerics, the principled nature of the systemic
fallacy of the method employed by Cardan et al. should have
been obvious. What should have been the obvious remedy for
that had been supplied, during the Fifteenth Century by the work
of Filippo Brunelleschi, 16 Nicholas of Cusa, and Luca Pacioli, as
also by the surviving known fragments of the work of Leonardo
da Vinci. In brief, the necessary approach would have been the
same concept of physical geometry on which I had insisted during
my adolescence, or, much more appropriately, Riemannian
physical geometry, rather than the ivory-tower formalities of an
implicitly pro-Euclidean algebra.
In other words, when the empiricist followers of Descartes
and Antonio Conti employed the fallacy of the hoaxsters de
Moivre and D’Alembert, in crafting the hoax of so-called “imaginary
numbers” for the fraudulent attack on Leibniz by themselves,
Leonhard Euler, et al., they were not merely constructing
a fraud against physical science. They were behaving as a-priorist
incompetents in refusing to grasp the readily accessible, physicalgeometry
implications of the uniqueness of Archytas’ method for
constructing a process of duplication of the cube, rather than the
intrinsically incompetent, Sophist method of Aristotle, Euclid,
16 As in Brunelleschi’s employment of the catenary as a principle
of physical geometry which had been the required principle of
design for the construction of the cupola of Santa Maria del Fiore.
On the Subject of ‘Insight’
LaRouche
and Claudius Ptolemy.
Admittedly, this erroneous presumption reflected a crucial
oversight which had been made by the Sixteenth-Century
set of Cardan et al., prior to the experimentally crucial discovery
of least action by Pierre de Fermat. However, the discoveries by
Kepler and Fermat were an integral feature of both the uniquely
original discovery of the calculus (ca. 1676) by Leibniz, but,
more emphatically, Leibniz’s taking into account the crucial principle
of Fermat in Leibniz’s own crafting, in collaboration with
Jean Bernouilli, of the concept of a universal physical principle
of least action.
This “imaginary number” fraud by de Moivre,
D’Alembert, Euler, et al., was not merely a reflection of their apparent
ignorance of elementary principles of physical geometry
known since no later than Archytas and Eratosthenes. It was to be
seen as an echo of the “malthusian” oligarchical-model hoax expressed
by the Olympian Zeus of Aeschylus’ Prometheus Trilogy.
When that aspect of the matter is taken into account,
the difficulty which threatened Carl Gauss in the matter of the
Fundamental Theorem of Algebra, ought to become transparent.
Gauss’s third statement of that case ought to have made it clear,
retrospectively, to all modern mathematical physicists re-considering
Gauss’s proof, once the publication of Riemann’s habilitation
dissertation had made clear the essential issue lurking in the
shadows of Gauss’s own argument.
From the appearance of Riemann’s habilitation dissertation
and his Theory of Abelian Functions, onward, the deeper
implications of the history of modern science since Nicholas of
Cusa’s De Docta Ignorantia should have been clear, as Albert
Einstein located the root of competent modern physical science
in those methods which Kepler had attributed to Cusa’s work, the
work which, chiefly, founded competent forms of modern European
science.
Such is the nature of true insight.
3. Insight Reviewed
At the close of July 2007, the world as a whole entered
a phase-shift into chronic hyperinflation, into what has
been, ever since that date, a general breakdown-crisis of
the present world system as a whole. Since that time, the entire
world’s presently existing, post-August 1971 monetary-financial
system, has been doomed to its extinction, in one way, or another.
There are alternatives, but these mean abandoning what
has become the 1971-2008 world monetary-financial system.
It means putting the present system under a juridical system of
reorganization-in-bankruptcy, and replacing it with an echo of
the principles and intentions of President Franklin Roosevelt’s
policy for a Bretton Woods world monetary system free of those
vestiges of British imperialism which, unfortunately, reign, and
ruin us all, still today.
Δυναμις Vol. 3 No. 3 December 2008
34

It is important
to recognize
that we are
obliged to use that
term, “British Imperialism,”
because
that is the name by
which it goes. The
content of what that
term connotes, is an
international financial
tyranny whose
appropriate technical
term of description
is Anglo-Dutch
Liberalism, which
means the present
form of organization
of a network
of financier and
closely associated
interests which was
built up in northern
maritime Europe by
Venice’s Paolo Sarpi
and his followers.
“British” in “British
Imperialism” marks
that empire-in-fact,
the leading single
imperial power in
the world today
(since the 1971-
1972 betrayal of the U.S.A. by the Administration of President
Richard Nixon), which had first been established as the imperial
power of a private company, the British East India Company
through the implications of the Paris Peace of February 1763.
Such is the great challenge to the creative powers of the
members of mankind today.
Thus, on July 25th, I spoke: “... this occurs at a time
when the world monetary system is now currently in the process
of disintegrating. There’s nothing mysterious about this; I’ve
talked about it for some time; it’s been in progress, it’s not abating.
What’s listed as stock values and market values in the financial
markets internationally is bunk! These are purely fictitious
beliefs. There is no truth to it; the fakery is enormous. There is
no possibility of a non-collapse of the present financial system—
none! It’s finished, now! The present financial system cannot continue
to exist under any circumstances, under any Presidency,
under any leadership, or under any leadership of nations. Only a
fundamental and sudden change in the world monetary-financial
system will prevent a general, immediate, chain-reaction type of
On the Subject of ‘Insight’
LaRouche
The author, presenting an international webcast on July 25, 2007
collapse. At what speed we do not know, but it will go on, and it
will be unstoppable! And the longer it goes on before coming to
an end, the worse things will get. And there is no one in the present
institutions of government who is competent to deal with this.
The Congress—the Senate and the House of Representatives—is
not currently competent to deal with this. And if the Congress
goes on recess, and leaves Cheney free, then you might be kissing
the United States and much more good-bye by September.
“This is the month of August; it’s the anniversary of
August 1914. It’s the anniversary of August 1939. The condition
now is worse,objectively, than on either of those two occasions.
Either we can make a fundamental change in the policies of the
United States now, or you may be kissing civilization good-bye
for some time to come....” 17
17 From the original transcript of my remarks on that occasion.
(For the complete transcript of LaRouche’s July 25, 2007 webcast,
see EIR, Aug. 3, 2007.)
Δυναμις Vol. 3 No. 3 December 2008
35

The Individual in History
As I have said repeatedly, of late, the history of mankind
is not event-driven; it is man-driven. The most essential decisions
which drive the actually crucial changes in the course of history
have often been what was deemed impossible by conventional
opinion-makers earlier. It is not what happened in yesterday’s
usually fraudulent leading press reports which drives history;
it is men or women of a special kind of influence, such as our
Benjamin Franklin, or the great historian and dramatist Friedrich
Schiller, who choose to lead nations in one direction or another.
It is rarely a matter of choosing from among multiple choices
on the table; the most momentous turns in history have been the
changes, changes made by the initiative of a seemingly tiny minority,
changes like the founding of our Constitutional republic
which had seemed, in July 1776, to the world at large, not merely
impossible, but an ill-fated conceit of a few.
The greatest decisions in history are made by men or
women, as individuals, decisions which have seemed virtually
impossible to conventional institutions and public opinion even
a relatively short time before. All great turns in history of that
quality come as the unique innovation in thought and will by relatively
rare individuals. So, President Abraham Lincoln saved our
republic, virtually despite itself; so, the greatest poets and scientists
did what no one else had dreamed before.
The greatest of all such deeds occur in such times as those
of which the great English Classical poet, Percy Bysshe Shelley
wrote in his In Defence of Poetry. There are times when much
of a people is overcome by a marvelous increase in the power of
imparting and receiving profound and impassioned conceptions
of man and nature, as by the inspiration of the then already deceased
Friedrich Schiller in calling forth the great initiative of the
German people led by Scharnhorst in organizing, according to the
principle of strategy defined by Schiller’s studies of the religious
wars in the Netherlands and the Thirty Years War, to accomplish
the otherwise seemingly impossible defeat of the tyrant Napoleon
Bonaparte in Russia and in that tyrant’s desperate effort to return
to France to raise a new army and a new general war.
So, a Genoese sea-captain working in the service of Portugal,
the greatly talented and inspired Christopher Columbus,
was led by his continuing study of the testament of the founder
of modern science, Cardinal Nicholas of Cusa, one of the greatest
geniuses of all modern history, to devise a plan for realizing Cusa’s
program, for great strategic voyages across the great oceans,
to rescue a corrupted European culture by extending its reach to
distant lands. This was Cusa’s intention, as actually adopted, with
full consciousness of that intention, by Columbus from about
1480 onward, which created the Americas, and brought about
that subsequent colonization of New England which gave birth to
what became our United States.
This was the object of the actual founding of our republic,
the U.S.A., whose morality was defined, first, by the crucial
On the Subject of ‘Insight’
LaRouche
Christopher Columbus (1451-1506)
passage of a work denouncing the evil slaver John Locke, the passage,
“the pursuit of happiness,” from Gottfried Leibniz’s New
Essays on Human Understanding, which is the core principle
of our Declaration of Independence and the root of the principle
of moral law of our republic which is elaborated, as in the spirit
of the Peace of Westphalia, as also reflected in the great Platonic
and Christian principle of agape, in the Preamble of our Federal
Constitution.
Thus, the true history of mankind is only that which is
defined by the actuality of the perfectly sovereign creative powers
which can be expressed only by the individual creative personality.
These are the same creative powers, unique to sovereign
individual minds, which are expressed by uniquely great discoveries
of scientific principle, as by the Pythagoreans, Plato, Cusa,
Kepler, and Leibniz, or Classical qualities of artistic principle,
such as those of Friedrich Schiller, or the combination of initiatives
rooted in a concurrence of scientific craft and moral inspiration
in the achievement of Christopher Columbus.
The contrary implication to be considered, against that
Δυναμις Vol. 3 No. 3 December 2008
36

ackground, is that the chief source of the ugliest failures of
humanity is a certain kind of popularized stupidity of the type
demanded by the Olympian Zeus of Aeschylus’ Prometheus
Bound, as demanded by the creature of the British Foreign Office’s
Jeremy Bentham, Thomas Malthus, or as the lame-brained
perversions uttered by that pathetic puppet known as the incumbent
President of our U.S.A. Popular opinion, such as that induced
by our presently, inherently corrupt and lying major news
media, is the deadliest of the Trojan Horses inserted into the domains
of mankind today.
In that sense, the issue of the development of the creative
powers of the individual young member of society is, in
the final analysis, the most crucial political, and also moral issue
of the existing cultures of this planet, most notably our presently
dumbed-down, Boomer-ridden U.S.A. Our present educational
systems have assisted greatly in making our people stupid enough
to be influenced by the opinions uttered by the proverbial “paid
prostitutes” of our presently popular “yellow” press.
The Relevant Paradox
The power of creativity, as I have presented the case
summarily in the preceding chapters here, is, as I have already
emphasized, not only a built-in natural potential of the human
individual, a potential absent in all animal species; it is unique
to all persons who are not victims of relevant damage to their
potential range of human powers. In broad terms, therefore, every
individual should be developed as a truly creative personality.
As the case may be, as cows do not make for intelligent
citizens, it is wrong to attempt to train people to become cows, as
the latter has been done, in effect, to most of the human population
in most known cultures to present date. The subject, therefore,
is, once more, the case of the suppression of knowledge of
“fire” by order of the archetypical Malthusian (or, present-day
Malthusian and lying former Vice-President Al Gore). Only under
artificial conditions such as those prescribed by Britain’s leading
anti-humanist, the World Wildlife Fund’s Prince Philip, is the
natural, human intellectual potential of the person suppressed in
ways—pro-Malthusian ways—which turn children into the virtually
half-witted cattle of today’s neo-Malthusian movements.
Consider what caused the legendary Olympian Zeus to
cook up this anti-human role of “environmentalism.” There are
two, complementary motives.
First, actually creative and brave people will not willingly
submit to either a legendary Olympian Zeus, or a Prince
Philip or Al Gore. Second, since mankind’s creativity is typically
expressed through its realization as scientific and related progress
in developing prevalent human conditions, the continuation of
the progress which man’s true nature demands, “uses up natural
resources” in ways which only the natural advances in the science-driven
and related creative productive powers of mankind
could remedy.
On the Subject of ‘Insight’
LaRouche
On the latter account, of the Earth’s total mass, the portion
corresponding of pre-biotic masses is shrinking as a percentile
relative to the product of living processes, while the rate of
increase of the portion of the mass generated by human activity is
increasing, relative to both abiotic residues and residues of other
kinds of living processes.
Thus, to keep large populations sufficiently stupefied to
be reigned over by the tyrannical likes of the Olympian Zeus, it
is necessary (for the sake of that tyranny) to keep subject populations
as stupid as possible, and, therefore, to prevent actual increases
in the productive powers of human labor, or, even, as has
been done in the U.S.A., and in western and central Europe since
1989, to reverse previous economic progress absolutely. 18
For that reason, nominal American citizens such as former
Vice-President (and traitor) Aaron Burr and former Vice-
President-turned-British-lackey Al Gore do not like honest patriots
of our U.S.A. very much.
However, on the opposite side of that matter, the potential
for developing true scientific creativity, and also artistic creativity
in the individual member of society, is there. It exists, and
can be promoted, if we come to understand this subject-matter,
and are willing to make its achievement the essential goal for the
development of our future individual citizen.
My own dedication to that mission is multifarious; but,
my most essential, relevant skill is in the field of those expressions
of physical-scientific creativity which are coincident with
my special competence in the domain of physical economy. To
this end, I have promoted an approach to the students’ replication
of the development of the principal valid currents of physical science,
ranging, explicitly, and most typically, from the Pythagoreans
and Plato through Cusa, Leonardo, Kepler, Fermat, Leibniz,
Gauss, the Monge-Carnot phase of the Ecole Polytechnique,
Dirichlet, and Riemann. Those who work in relevant forms of
teams, to relive the acts of discovery which are most relevant
for re-experiencing first-hand knowledge of the most-relevant
discoveries, can generally succeed in one significant degree or
another.
With great science and great Classical art, combined, we
can generate among us new generations sharing the quality of
temperament we should require for those generations of our new
citizens. The benefit would be, not only skills, but the fostering of
the truly creative powers of the human mind, upon which progress
depends.
Best of all, once one knows that expressed quality of
potential in oneself, which distinguishes one from an ape, or brutalized
slave, insight comes naturally, because it is natural, for as
long as people are developed for what the human individual is,
and is intended to become.
18 As in the pattern set by the predatory, dictatorial, Thatcher-
Mitterrand “conditionalities” imposed upon Germany.
Δυναμις Vol. 3 No. 3 December 2008
37

Third Demonstration
Gauss
Th i r d de m o n s T r aT i o n o f T h e Th e o r e m Co n C e r n i n g T h e
Decomposition of Integral Algebraic
Functions into Real Factors
Carl F. Gauss
Δυναμις Vol. 3 No. 3 December 2008
This, the third of Carl Gauss’s four proofs of the Fundamental
Theorem of Algebra, appeared in 1816. Here, Gauss makes a rare inter-
ventiaon for a physical basis of mathematics. It was translated by Mervyn
Fansler, who offers this disclaimer: “Since my German is much further along
than my Latin, this translation derives from a German translation of the
original published by E. NE t t E in Ostwald’s Klassiker der exakten
Wissenschaften, B. 14. The original Latin version, which appeared in the
Commentationes recentiores... and is the version published in Gauss’s
Gesammelte Werke, was consulted as a reference.
After the previous treatise was already printed, continued
meditations upon the same subject led me to a new proof
of the theorem, which, similar to the preceding, is purely
analytical, yet is based upon entirely different principles, and,
with respect to simplicity, appears by far to be superior to the for-
mer. To this third proof are the following pages now devoted.
1.
Let the following function of the indeterminate x be
given:
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
whose coefficients A, B, C,... are determined real magnitudes.
Considering r, ϕ to be other indeterminate magnitudes and set-
ting
The factor r can evidently be canceled out of the de-
nominator of the last formula, since t’, u’, t’’, u’’ are divisible
by it. Finally, let r be a determined positive magnitude, which,
though indeed arbitrary, should still exceed the highest of the
magnitudes
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =
for these magnitudes the signs of A, B, C, D, ... should be ne-
glected; that is, any negative signs occurring should be changed
into positive. Following this preparation, I claim that
+
TT � + UU �
r = R
tt � + uu �
assuredly obtains a positive value if we set r = R, which also gives
a (real) value to ϕ.
Demonstration. Setting
it follows:
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1
1 + Bx m−2 + Cx m−3 + ... + Lx + M,
− 1)ϕ + Br m−2 cos(m − 2)ϕ
− 3)ϕ + ... + Lr cos ϕ + M = t
− 1)ϕ + Br m−2 sin(m − 2)ϕ
− 3)ϕ + ... + Lr sin ϕ = u
−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
1 + Bx m−2 + Cx m−3 + ... + Lx + M,
− 1)ϕ + Br m−2 cos(m − 2)ϕ
− 3)ϕ + ... + Lr cos ϕ + M = t
− 1)ϕ + Br m−2 sin(m − 2)ϕ
− 3)ϕ + ... + Lr sin ϕ = u
−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos(45 ◦ + mϕ) =T,
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin(45 ◦ + mϕ) =U,
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ) =T � ,
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =U � ,
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(4
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(4
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + L
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + L
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos(45 ◦ + mϕ) =T,
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin(45 ◦ + mϕ) =U,
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ) =T � ,
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =U � ,
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos(45 ◦ + mϕ) =T,
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin(45 ◦ + mϕ) =U,
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ) =T � ,
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =U � ,
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin(
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ) =
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos(45 ◦ + mϕ) =T,
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin(45 ◦ + mϕ) =U,
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ) =T � ,
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =U � ,
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos(45 ◦ + mϕ) =T,
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin(45 ◦ + mϕ) =U,
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ) =T � ,
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =U � ,
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(4
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(4
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... +
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + L
mA √ 2, 2
�
mB √ 2, 3
�
mC √ 2, 4
�
mD √ 2, ...;
R m cos45 ◦ + AR m−1 cos(45 ◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ)+M cos(45 ◦ + mϕ) =T,
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ)+M sin(45 ◦ + mϕ) =U,
mR m cos45 ◦ +(m − 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
+(m − 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m − 1)ϕ) =T � ,
mR m sin45 ◦ +(m − 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
+(m − 3)CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =U � ,
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
X = x m + Ax m−1 + Bx m−2 + Cx m−3 + ... + Lx + M,
ϕ
r m cos mϕ + Ar m−1 cos(m − 1)ϕ + Br m−2 cos(m − 2)ϕ
+ Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ + M = t
r m sin mϕ + Ar m−1 sin(m − 1)ϕ + Br m−2 sin(m − 2)ϕ
+ Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u
mr m cos mϕ +(m − 1)Ar m−1 cos(m − 1)ϕ +(m − 2)Br m−2 cos(m − 2)ϕ
+(m − 3)Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t �
mr m sin mϕ +(m − 1)Ar m−1 sin(m − 1)ϕ +(m − 2)Br m−2 sin(m − 2)ϕ
+(m − 3)Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �
m 2 r m cos mϕ +(m − 1) 2 Ar m−1 cos(m − 1)ϕ +(m − 2) 2 Br m−2 cos(m − 2)ϕ
+(m − 3) 2 Cr m−3 cos(m − 3)ϕ + ... + Lr cos ϕ = t ��
m 2 r m sin mϕ +(m − 1) 2 Ar m−1 sin(m − 1)ϕ +(m − 2) 2 Br m−2 sin(m − 2)ϕ
+(m − 3) 2 Cr m−3 sin(m − 3)ϕ + ... + Lr sin ϕ = u �� ,
(t 2 + u 2 )(tt �� + uu �� )+(tu � − ut � ) 2 − (tt � + uu � ) 2
r(t 2 + u 2 ) 2
= y.
38

+ Rϕ)+BR cos45 + AR cos(45 cos(45 +2ϕ) + CR + ϕ)+BR cos(45 +3ϕ)+... cos(45 + +2ϕ) LR cos(45 +(m− 1)ϕ)+M cos(45 + mϕ) =T,
R m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ +3ϕ)+... + LR cos(45 +2ϕ)
◦ +(m− 1)ϕ)+M cos(45 ◦ + CR + mϕ) =T,
m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m− 1)ϕ)+M cos(45 ◦ cos(45 + mϕ) =T,
◦ + ϕ)+BR m−2 cos(45 ◦ +2ϕ)
+ CR m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m− 1)ϕ)+M sin(45 ◦ + ϕ)+BR + mϕ) =U,
m−2 sin(45 ◦ R +2ϕ)
m sin45 ◦ + AR m−1 sin(45 ◦ + ϕ)+BR m−2 sin(45 ◦ cos(45 +2ϕ)
◦ +3ϕ)+... + LR cos(45 ◦ +(m− 1)ϕ)+M cos(45 ◦ + mϕ) =T,
Third Demonstration
Gauss
mR m cos45 ◦ +(m− 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +3ϕ)+... + LR sin(45 +2ϕ)
◦ +(m− 1)ϕ)+M sin(45 ◦ + CR + mϕ) =U,
m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m− 1)ϕ)+M sin(45 ◦ sin(45 + mϕ) =U,
◦ + ϕ)+BR m−2 sin(45 ◦ +2ϕ)
+(m− 3)CR m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m− 1)ϕ) =T � cos(45 ,
◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ mR +2ϕ)
m cos45 ◦ +(m− 1)AR m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ sin(45 +2ϕ)
◦ +3ϕ)+... + LR sin(45 ◦ +(m− 1)ϕ)+M sin(45 ◦ + mϕ) =U,
mR m sin45 ◦ +(m− 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ cos(45 +2ϕ)
◦ +3ϕ)+... + LR cos(45 ◦ +(m− 1)ϕ) =T � +(m− 3)CR ,
m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m− 1)ϕ) =T � R ,
m−1 cos(45 ◦ + ϕ)+(m − 2)BR m−2 cos(45 ◦ +2ϕ)
I. t is composed of the terms
R
Δυναμις Vol. 3 No. 3 December 2008
m−1
m √ 2 (R + mA√2cos (45◦ + ϕ))
+ Rm−2
m √ 2 (R2 + mB √ 2cos (45◦ +2ϕ))
+ Rm−3
m √ 2 (R3 + mC √ 2cos (45◦ +3ϕ))
+ Rm−4
m √ 2 (R4 + mD √ 2cos (45◦ R
+4ϕ))...,
which, for each determined real value of ϕ,
as is easily seen,
has a single positive value; consequently, t must be taken as a
positive value. In a similar manner it will be proven that u, t’, u’
also become positive, such that
1
m−1
m √ 2 (R + mA√2cos (45 ◦ + ϕ))
+ Rm−2
m √ 2 (R2 + mB √ 2cos (45 ◦ +2ϕ))
+ Rm−3
m √ 2 (R3 + mC √ 2cos (45 ◦ +3ϕ))
+ Rm−4
m √ 2 (R4 + mD √ 2cos (45 ◦ +4ϕ))...,
TT� + UU� r = R
T cos(45 ◦ + mϕ)+U sin(45 ◦ + mϕ),
T sin(45 ◦ + mϕ) − U cos(45 ◦ + mϕ),
T � cos(45 ◦ + mϕ)+U � sin(45 ◦ + mϕ),
T � sin(45 ◦ + mϕ) − U � cos(45 ◦ + mϕ),
tt� + uu� must also become a
positive magnitude.
II. For r = R, the functions t, u, t’, u’ change over into
T cos(45
1
◦ + mϕ)+U sin(45◦ + mϕ),
T sin(45◦ + mϕ) − U cos(45◦ + mϕ),
T cos(45◦ + mϕ)+U sin(45◦ + mϕ),
T sin(45◦ + mϕ) − U cos(45◦ R
+ mϕ),
respectively, as will be easily shown by the actual development.
Consequently, the value of the function
m−1
m √ 2 (R + mA√2cos (45 ◦ + ϕ))
+ Rm−2
m √ 2 (R2 + mB √ 2cos (45 ◦ +2ϕ))
+ Rm−3
m √ 2 (R3 + mC √ 2cos (45 ◦ +3ϕ))
+ Rm−4
m √ 2 (R4 + mD √ 2cos (45 ◦ +4ϕ))...,
TT� + UU� r = R
T cos(45 ◦ + mϕ)+U sin(45 ◦ + mϕ),
T sin(45 ◦ + mϕ) − U cos(45 ◦ + mϕ),
T � cos(45 ◦ + mϕ)+U � sin(45 ◦ + mϕ),
T � sin(45 ◦ + mϕ) − U � cos(45 ◦ + mϕ),
tt� + uu� R
, for r = R, will
equal
1
m−1
m √ 2 (R + mA√2cos (45 ◦ + ϕ))
+ Rm−2
m √ 2 (R2 + mB √ 2cos (45 ◦ +2ϕ))
+ Rm−3
m √ 2 (R3 + mC √ 2cos (45 ◦ +3ϕ))
+ Rm−4
m √ 2 (R4 + mD √ 2cos (45 ◦ +4ϕ))...,
TT� + UU� r = R
T cos(45 ◦ + mϕ)+U sin(45 ◦ + mϕ),
T sin(45 ◦ + mϕ) − U cos(45 ◦ + mϕ),
T � cos(45 ◦ + mϕ)+U � sin(45 ◦ + mϕ),
T � sin(45 ◦ + mϕ) − U � cos(45 ◦ + mϕ),
tt� + uu� and thus will be a positive magnitude.
Incidentally we conclude from the same formula that
the value of the function t2 + u2 T 2 + U 2
mA √ �
2, mB √ 2, 3�
mC √ t
, for r = R, equals
2
2 + u2 T 2 + U 2
mA √ �
2, mB √ 2, 3�
mC √ t
and will thus be positive, and thence it follows that for no
value of r which is greater than the individual magnitudes
2
2 + u2 T 2 + U 2
mA √ �
2, mB √ 2, 3� mC √ 2 ,..., can t = 0, u = 0 at the same
time.
2.
Theorem. Within the boundaries r = 0 and r = R, as
well as ϕ = 0 and ϕ = 360°, there exists such values of the
indeterminates r, ϕ, for which t = 0 and u = 0 at the same
time.
Demonstration. We will suppose that the proposition is
not true; it is evident that the value of t2 + u2 T 2 + U 2
mA √ �
2, mB √ 2, 3�
mC √ extending from r = 0 to r = R and from ϕ = 0 to ϕ = 360°,
which thus acquires a finite, completely determined value. This
value, which we will signify by Ω,
must be obtained regardless
of whether the integration is performed first with respect to ϕ and
then with respect to r, or the inverse order. We have, however, the
indefinite, if we consider r as a constant,
�
tu − ut
ydϕ =
r(t
1
for all values of
the indeterminates within the assigned limits must be a positive
magnitude, such that the value of y remains finite. Let us consider
the double integral � �
2
ydrdϕ
2 + u2 ) ,
�
�
ydϕ
as is easily confirmed by differentiating with respect to ϕ. A
constant is not added, if we presume that the integration begins
at ϕ = 0, since for ϕ
tu
= 0 one obtains
1
1
1
� −ut �
r(t2 +u2 ) =0
�
ydϕ
�
ydr
TT � +UU �
T 2 +U 2
�
TT
T
= mϕ +45◦ − arctan U
T
arctan U
� �
ydrdϕ
�
ydϕ =
. Now, since
T
tu� − ut� r(t2 + u2 ) ,
tu � −ut �
r(t2 +u2 ) =0
�
ydϕ
�
ydr = tt� + uu� t2 ,
+ u2 TT � +UU �
T 2 +U 2
� � � TT + UU
T 2 dϕ
+ U 2
= mϕ +45◦ − arctan U
T
arctan U
� evidently also vanishes for ϕ = 360°, then the integral
ydϕ from ϕ = 0 to ϕ = 360° becomes = 0, while r remains
indefinite. But from here follows Ω = 0.
Likewise we have the indefinite integral, in which we
consider ϕ as constant, �
tt + uu
ydr =
t
T
1
2 � �
ydrdϕ
�
ydϕ =
,
+ u2 as is likewise easily confirmed by differentiating with respect to
r: again no constant is needed here, supposing that we begin
the integration with r = 0. Hence, according to the proofs in the
preceding paragraphs, the integral, extending from r = 0 to r = R,
equals
tu� − ut� r(t2 + u2 ) ,
tu � −ut �
r(t2 +u2 ) =0
�
ydϕ
�
ydr = tt� + uu� t2 ,
+ u2 TT � +UU �
T 2 +U 2
� � � TT + UU
T 2 dϕ
+ U 2
= mϕ +45◦ − arctan U
T
arctan U
, and will, consequently, according to the theorems
of the preceding paragraphs, always be a positive magnitude
for each real value of
T
1
ϕ � �
ydrdϕ
�
ydϕ =
. Therefore, Ω will also necessarily
be a positive magnitude, that is, the value of the integral
tu� − ut� r(t2 + u2 ) ,
tu � −ut �
r(t2 +u2 ) =0
�
ydϕ
�
ydr = tt� + uu� t2 ,
+ u2 TT � +UU �
T 2 +U 2
� � � TT + UU
T 2 dϕ
+ U 2
= mϕ +45◦ − arctan U
T
arctan U
from
T
1
ϕ = 0 to ϕ = 360°. 1 This is absurd, since just before we had
found the same magnitude = 0. Our presumption, thus, can not be
true, and with it the validity of the theorem is proven.
3.
The function X transforms into t + u √ −1
x = r(cos ϕ +sin ϕ √ −1)
t − u √ −1
x = r(cos ϕ − sin ϕ √ −1)
x = g(cos G +sin G √
t + u
by the substitution
√ −1
x = r(cos ϕ +sin ϕ √ −1)
t − u √ −1
x = r(cos ϕ − sin ϕ √ −1)
x = g(cos G +sin G √ −1), x = g(cos G − sin G √ −
x − g(cos G +sin G √ t + u
and into
−1)
√ −1
x = r(cos ϕ +sin ϕ √ −1)
t − u √ −1
x = r(cos ϕ − sin ϕ √ t + u
by the
substitution
−1)
x = g(cos G +si
and simil
√ −1
x = r(cos ϕ +sin ϕ √ −1)
t − u √ −1
x = r(cos ϕ − sin ϕ √ −1)
x = g(cos G +sin G √ −1), x = g(cos G − sin G √
x − g(cos G +sin G √ −1)
and similarly x − g(cos G − sin G √ . If for determined values
of r, ϕ, say for r = g, ϕ=G, results that t = 0, u = 0 simulta-
1 This is clear in and of itself. Moreover, one will easily ascertain
the indefinite integral = m ϕ + 45° – arctan
−1).
U
T , and can be proven
in different ways (indeed in itself it is not yet obvious which
of the infinitely many values of the multi-valued function arctan U
T ,
+(m− 3)CR
which correspond to ϕ =360°, one must adopt), that the value which
one obtains for the integration for ϕ = 360°, must be set = m • 360°
or = 2 m π. However, this is not necessary for our purpose.
m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m− 1)ϕ) =U � ,
+(m− 3)CR
1
1
m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m− 1)ϕ) =U � ,
1
1
sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ mR +2ϕ)
m sin45 ◦ +(m− 1)AR m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ R +2ϕ)
m−3 cos(45 ◦ +3ϕ)+... + LR cos(45 ◦ +(m− 1)ϕ) =T � ,
sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m− 1)ϕ) =U � R ,
m−1 sin(45 ◦ + ϕ)+(m − 2)BR m−2 sin(45 ◦ +2ϕ)
R m−3 sin(45 ◦ +3ϕ)+... + LR sin(45 ◦ +(m − 1)ϕ) =U � ,
1
1
and similarly 1 x − g(cos G1 − sin G1 and similarly
1
√ −1).
1
x 2 − 2g cos Gx + g 2 ;
1
39

t + u
Third Demonstration
Gauss
neously, (and that there are such values became verified in the
previous paragraphs), then X obtains the value 0 for each of the
substitutions
√ −1
x = r(cos ϕ +sin ϕ √ −1)
t − u √ −1
x = r(cos ϕ − sin ϕ √ −1)
x = g(cos G +sin G √ −1), x = g(cos G − sin G √ −1),
x − g(cos G +sin G √ −1)
and similarly x − g(cos G − sin G √ −1).
x 2 − 2g cos Gx + g 2 t + u
and will consequently be divisible by the indefinite
;
sin G =0
cos G = ±1
= x ∓ g
1
√ −1
x = r(cos ϕ +sin ϕ √ −1)
t − u √ −1
x = r(cos ϕ − sin ϕ √ −1)
x = g(cos G +sin G √ −1), x = g(cos G − sin G √ −1),
x − g(cos G +sin G √ −1)
and similarly x − g(cos G − sin G √ −1).
x 2 − 2g cos Gx + g 2 t + u
So long as G is not = 0, nor g = 0, these divisors are unequal,
and X will consequently also be divisible by their product
;
sin G =0
cos G = ±1
= x ∓ g
1
√ −1
x = r(cos ϕ +sin ϕ √ −1)
t − u √ −1
x = r(cos ϕ − sin ϕ √ −1)
x = g(cos G +sin G √ −1), x = g(cos G − sin G √ −1),
x − g(cos G +sin G √ −1)
and similarly x − g(cos G − sin G √ −1).
x 2 − 2g cos Gx + g 2 ;
however, if either sin G = 0 and thus cos G = ±1 or g = 0, each
sin G =0 of the factors will be identical, namely, = x∓ g. It thus holds
true that the function X possesses a real divisor of the second or
cos G = first ±1 degree, and since the same conclusion will again hold for the
quotient, X is thus completely dissolvable into such factors.
= x ∓ g
4.
Although we have brought the task we had undertaken
completely to an end with the preceding, still it would not be superfluous
to add something further concerning the inferences of
§ 2. By the assumption that t and u vanish at the same time for
no values of the variables r, ϕ within the boundaries given there,
we happen upon an inevitable contradiction, whereupon we have
concluded the incorrectness of the assumption itself. This contradiction
must therefore cease, if there is an actual value of r, ϕ
for which t and u become = 0 at the same time. In order to further
clarify this, we observe that for such values t
1
2 + u2 T 2 + U 2
mA √ �
2, mB √ 2, 3�
mC √ will = 0 and
thus y become infinite, so that it no longer is permitted to treat the
double integral
2
�� ydrdϕ as an assignable magnitude. Given
in general, if ξ, η , ζ signify the coordinates of spatial points, the
integral �� ��
ydrdϕ ydrdϕ signifies the volume of a body, which is
bounded by the five planes whose equations are
ξ =0, η =0, ζ =0, ξ = R, η = 360 ◦ ,
� tu
ydϕ = � − ut� r(t2 + u2 )
�
ηdξ
η = 1
ξ2 C − 1
and by a surface whose equation is ζ = y, if one considers those
parts of the body as negative whose coordinates ζ are negative.
However, here it will be tacitly assumed that the sixth surface will
be continuous; if the latter condition comes to pass in such a way
that y will be infinite, it can very well happen that the former conception
no longer makes sense. In this case, there can be no talk
of the evaluation of the integral �� The integration
�
tu − ut
ydϕ =
r(t
ydrdϕ , and for that reason
it is not incomprehensible that analytical operations applied in
blind reckoning to meaningless things lead to nonsense.
2 + u2 )
is only an actual integration---that is, summation---so long as y is
everywhere a finite magnitude between the boundaries in which
one integrates; on the contrary, it will be absurd if y is infinite
anywhere � between those bounds. If we specify such an integral
ηdξ , which generally indicates the surface between the axis of
the abscissa and the curve developed according to the customary
rule in which the ordinate η corresponds to the abscissa ξ, and
are withal negligent of the continuity, then we could very often
find ourselves entangled in contradictions. If we set, for example,
η =
1
1
1
1
ξ2 ��
ydrdϕ
ξ =0, η =
� tu
ydϕ =
, then analysis will yield the integral
� − ut� r(t2 + u2 )
�
ηdξ
η = 1
ξ2 C − 1
ξ , which will
correctly indicate the surface so long as the curve maintains its
u
continuity; since this is interrupted at ξ = 0, tthen,
if anyone of
unreasonable manner should ask for the magnitude of the surface
from a negative abscissa to a positive, the formula
for
would
r = e,
deliver
ϕ = E by
the absurd reply that it would be negative. However,
r
what
= e,
these
ϕ = F by
and similar analytical paradoxes mean shall be followed
r = f,
up
ϕ
in
= E by
more detail on another occasion.
r = f, ϕ = F by
Here it is only possible to affix but a single remark.
Were questions proposed without restriction, which in certain arctan Θ − arc
cases could become absurd, then analysis thereupon aids itself
very frequently by delivering a response in part variable. If we,
for example, signify the integral �� ��
ydrdϕ
ξ =0, η
� tu
ydϕ =
ydrdϕ extending from r =
e to r = f and ϕ = E to ϕ = F and the value of
� − ut� r(t2 + u2 )
�
ηdξ
η = 1
ξ2 C − 1
��
ydrdϕ
ξ =0, η =0, ζ =0, ξ = R, η = 360
ξ
u
t
for r = e, ϕ = E b
r = e, ϕ = F b
r = f, ϕ = E b
r = f, ϕ = F b
arctan Θ − a
◦ ,
� tu
ydϕ = � − ut� r(t2 + u2 )
�
ηdξ
η = 1
ξ2 C − 1
ξ
u
t
for r = e, ϕ = E by Θ
r = e, ϕ = F by Θ� r = f, ϕ = E by Θ�� r = f, ϕ = F by Θ��� arctan Θ − arctan Θ � − arctan Θ �� ��
ydrdϕ
ξ =0, η =0, ζ =0, ξ = R, η = 360
then by analytical operations one easily obtains the integral val- +arctan
ue
◦ ,
� tu
ydϕ = � − ut� r(t2 + u2 )
�
ηdξ
η = 1
ξ2 C − 1
ξ
u
t
for r = e, ϕ = E by Θ
r = e, ϕ = F by Θ� r = f, ϕ = E by Θ�� r = f, ϕ = F by Θ��� arctan Θ − arctan Θ � − arctan Θ �� +arctanΘ ��� .
The integral in actuality can then only have a determined value,
if y remains finite between the specified boundaries. This value
is itself contained in the specified formula, however it is not fully
determined by the same, since indeed the arctan is a multivalued
function, and it must further become decided which functional
values are preferable in a determined case, by means of other
not-so-difficult considerations. If, on the contrary, y is infinite
anywhere between the specified boundaries, then the question respecting
the value of the integral �� 40
ydrdϕ is absurd. This
does not hinder, that, if one will obstinately extort a response
from analysis, this might soon give different methods, whereby
the individual values of the foregoing general formula are obtained.
u
t
ξ
Δυναμις Vol. 3 No. 3 December 2008
for r = e, ϕ = E by Θ
r = e, ϕ = F by Θ �
r = f, ϕ = E by Θ ��
r = f, ϕ = F by Θ ���
1
1
1
1

Gauss to Bessel
December 18, 1811
Correspondence
Carl Gauss to Wilhelm Bessel
December 18, 1811
In this letter, Gauss responds to an assertion made by his student
Bessel about the theory of numbers. Gauss used this opportunity to intervene
on the way mathematicians had been thinking about their science, making clear
that mathematics must represent an investigation of physics, not merely mathematical
theorems.
A
few days ago, I finally recieved the Königsberger Archiv
that I had ordered. With great interest, dear Bessel, I read
through your first treatise, 1 and for now, have at least
skimmed the other. That and the wonderful assistance, which it
offers in the determination of li for large numbers, is now all the
more agreeable to me, since I have recently obtained a beautiful
table of factors from Ch e r n a C up to 1,020,000, appearing in this
year of Deventer, with which I want to count up, little by little, the
prime numbers from myriad to myriad, in order to compare them
with the value of the integral
� e x−1
x dx
a + b √reckoned from x = 0 on. You have made known to me the desire,
that −1=a I report + bi this theorem in our GGA: before I do it, dear Bessel,
our friendship obligates me to converse in writing with you
li(a + about bi) one or another point, where my outlook is not entirely in
� accord with yours. Therefore, kindly accept the following re-
ϕ(x) marks · dx and, in the meantime, impart to me your thoughts just as
frankly and candidly, as I mine. I think I can infer from one of
α + βiyour
casual remarks, that a basic principle is common to both of
us: “In mathematics, there are no controversial truths,” and thus I
x = a do + bi not doubt, that through the mutual exchange of our ideas, we
will already be in agreement.
ϕ(x) · dx First and foremost, from someone who wants to introduce
a new function into analysis, I would ask for an explanation,
a + biwhether
he will understand them as applied to merely real mag-
� nitudes (real values of the arguments of the function), and regard
ϕ · dx the imaginary values of the arguments as if they were only quasioutgrowths;
or whether he would accede to my principle, that, in
ϕ(x) the realm of magnitudes, the imaginaries a + b
ϕ(x) =∞
√ −1=a + bi
must enjoy equal rights with the reals. The discussion here is not
of practical utility, rather, to me, analysis is li(a an independent + bi) science,
which, by the neglect of those imaginary � magnitudes, loses
ϕ(x) · dx
1 Investigation of expressible transcendental functions by the
integral α + βi
� dx
lx , Königsberger Archive of Natural Science and Medicine
1, 1812, S.1, FR. W. Be s s e l’s Werke II, Leipzig 1876, S.330
� x−1 e
x dx
enormously in beauty and roundness and, in a moment all truths,
which otherwise would be universally valid, are obliged to add
the most weighty limitations. I must employ the assumption, that
you are essentially in accordance with me about this point, since
your elucidation2 a + b
in art. 18 already indicates, that you by no means
intend to block the way to investigations about
√ � x−1 e
x
a + b
−1=a + bi
li(a + bi) . What
should be imagined then with
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x)
ϕ(x) =∞
√ a + b
−1=a + bi
li(a + bi)
�
ϕ(x) · dx,
where
α + βi
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x)
ϕ(x) =∞
1
√ � −1=a + bi
x−1 e
li(a + x bi)
�
ϕ(x) · dx
α + βi
x = a + bi?
Evidently, if one wants to proceed from clear notions, it must
be accepted that x proceeds, through infinitely small ϕ(x) increments · dx
(each of the form
a + bi
�
ϕ · dx
ϕ(x)
ϕ(x) =∞
dx
a + b √ � x−1 e
x
−1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi),
from that value for which the integral
should be 0, up to
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x)
ϕ(x) =∞
1
dx
a + b √ a + b
−1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi,
and then add all
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x)
ϕ(x) =∞
1
√ � −1=a + b
x−1 e
x li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
. Thus the significance is fully established. Now, however, the
pathway can be constructed in infinitely many ways: thus a + as bi the
entire realm of all real magnitudes can be thought of as an � infinite
straight line, so the entire realm of all magnitudes, real and ϕ imag- · dx
inary, can be made physical with an infinite plane, where each
point, determined by abscissa = a, ordinate = b, can represent ϕ(x) the
magnitude
ϕ(x) =∞
dx
a + b √ � x−1 e
x
−1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
a + bi.
Accordingly, the continuous path from a value
of x to another �
ϕ · dx
ϕ(x)
ϕ(x) =∞
1
dx
a + b √ −1=a + bi a + b
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
a + bi is represented by a line, and is consequently
possible in infinitely
�
many ways.
I maintain ϕ · dx then, that the integral
ϕ(x)
ϕ(x) =∞
1
√ � x−1 e
x
−1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx always obtains
the same value along to two different pathways, as long as
ϕ(x)
ϕ(x) =∞
dx
a + b √ −1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x) never = 0 within the area enclosed between the two lines
representing the pathways. This is a beautiful theorem,
ϕ(x) =∞
1
3 � x−1 e
x
whose
not difficult proof I will give on an appropriate occasion. It is
connected with other beautiful truths relating to the expansion of
series. A pathway to each point can always be found, which never
contacts such a place where
dx
a + b √ −1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
� x−1
a + bi
e
x
�
ϕ · dx
ϕ(x)
ϕ(x) =∞.
Hence I insist, that one
must avoid such points, where the original, fundamental principle
of
dx
a + b √ � x−1 e
x
−1=a + bi
li(a + bi)
�
ϕ(x) · dx evidently loses its clarity and easily leads to inconsistencies.
Furthermore, it is at the same time clear from this,
how α a + function βi produced by
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x)
ϕ(x) =∞
dx
a + b √ � x−1 e
x
−1=a + bi
li(a + bi)
�
ϕ(x) · dx can always have many
values for the same value of x, when namely, the paths can go
around such a point where α + βi
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x)
dx
a + b √ −1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x)
ϕ(x) =∞,
either never, or once, or
multiple times. 4 a + b
If one defines, e.g., log x by
2 See Be s s e l’s Werke II, Leipzig 1876, S. 341, 1. Spalte.
3 Strictly speaking, it is still yet supposed, that
√ −1=a + bi
li(a + bi)
�
ϕ(x) · dx
α + βi
x = a + bi
ϕ(x) · dx
a + bi
�
ϕ · dx
ϕ(x) is itself
a single-valued function of x, or at least only one system of values,
without interruption in continuity, is assumed for whose ϕ(x) value =∞ within
each whole surface-space.
Δυναμις Vol. 3 No. 3 x = a + bi
December 2008
ϕ(x) · dx
a + bi
� ϕ · dx
4 It appears in the manuscript as x = ∞
ϕ(x) =∞
41
1
1

� ϕ(x) · dx
Gauss to Bessel
December 18, 1811
�
1
· dx,
x
+2πi
−2πi
ϕ(x) = ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
�
1
· dx,
x
where x starts off = 1, then log x is arrived at either without including
the point x = 0 or by rotating around it one or more times;
each time, the constants +2πi
−2πi
ϕ(x) =
� x e − 1
dx
x
� x e dx
x
1
ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
�
1
· dx,
x
+2πi
or −2πi
ϕ(x) =
� x e − 1
dx
x
� x e dx
x
1
ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
α + βi
x = a + bi
ϕ(x) · dx
�
1
· dx,
x a + bi
�
ϕ · dx
are added; thus the multi-
+2πi
ple logarithms of each number is clear. If ϕ(x) can never become
infinite for a finite value of x, then the integral is always only a
−2πi
single valued function. This is e.g. the case ϕ(x) for=∞
ϕ(x) =
1
� x e − 1
dx
x
� x e dx
x
1
ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
�
1
· dx,
x
+2πi
−2πi
ϕ(x) =
so that
� x e − 1
dx
x
� x e dx
x
1
ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
�
1
· dx,
x
+2πi
−2πi
ϕ(x) =
is certainly a single-valued function of x, whose value is always
converging, and will always be represented by one and only one
sensible series
� x e − 1
dx
x
� x e dx
x
1
ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
I would that Herr so l d n e r had chosen this one, since he
would introduce a simple new function, instead of his
�
dx
li x −
log x
� x e − 1
dx
x
� x e dx
x
1
,
�
1
· dx,
x
+2πi
−2πi
ϕ(x) =
since a single valued function is always regarded as simpler and
more classical than a multi-valued one, especially since log x is
itself already a multi-valued function. It would perhaps also be
advantageous, for
ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
�
1
· dx,
x
+2πi
−2πi
ϕ(x) =
� x e − 1
dx
x
or at least for
� x e dx
x
1
ex − 1
,
x
� x e − 1
dx
x
x + 1 1
xx +
4 18 x3 + 1
96 x4 +etc.
�
dx
lix −
logx ,
� x e − 1
dx
x
� x e dx
x
to introduce a suitable symbol and name, so much the more since,
with the problems from physics derived from li x, x itself is commonly
an exponential magnitude. If the truths for so l d n e r’s li
are carried over to my 1
� x e dx
x ,
negative, infinite magnitude.
I have mentioned all of this before hand, in order to establish
my point of view, that I must join Euler, when he said, 5
� x e dx
x
that
,
�
dx
logx
is taken as real for the case where
�
x x e < dx1,
and for values > 1, necessarily
obtains imaginary values.
li(0.5+0.001i), li(0.6+0.001i), x Its difference from the real,
which
li(0.7+0.001i), li(0.8+0.001i),
Ma s C h e r o n i, so l d n e r and you add to it, is =πi or 3πi or 5πi
li(0.9+0.001i),
etc. I do not
li(1.0+0.001i),
examine the path
li(1.1+0.001i),
through x = 1; one
etc.,
would
until
be
li(1.5+0.001i)
able to
prove in an entirely similar way, that log – x = log + x (a theorem
which can be accepted, if limited to real magnitudes, but which
must be immediately ommitted, when my above principle of two
dimensions is bestowed upon the realm of all magnitudes.) Make
li x real for any single value of x between 0 and 1! But which
value will then be assigned?
1
,
� x e dx
x
�
dx
logx
li(0.5+0.001i), li(0.6+0.001i), li(0.7+0.001i), li(0.8+0.001
li(0.9+0.001i), li(1.0+0.001i), li(1.1+0.001i), etc., until li(1.5+0.00
1
,
� x e dx
x
�
dx
logx
li(0.5+0.001i), li(0.6+0.001i), li(0.7+0.001i), li(0.8+0.001i),
li(0.9+0.001i), li(1.0+0.001i), li(1.1+0.001i), etc., until li(1.5+0.001i)
1
,
� x e dx
x
�
dx
logx
li(0.5+0.001i), li(0.6+0.001i), li(
li(0.9+0.001i), li(1.0+0.001i), li(1.1+0.001
1
,
42
�
dx
logx
li(0.5+0.001i), li(0.6+0.001i), li(0.7+0.001i), li(0.8+0.001i),
li(0.9+0.001i), li(1.0+0.001i), li(1.1+0.001i), etc., until li(1.5+0.001i)
Without doubt imaginary, but it should follow the law of the continuity,
being nowhere a break ex abrupto? If you then go from
li(1.5+0.001i), while allowing the imaginary part 0.001i to decrease
to 0, [to li 1.5], it absolutely does not come to a real value
of li 1.5, but to one which depends upon -\pi i. With what you
furnish for proof against eu l e r and myself, I find to criticize 1)
that you say, if li x must become real in the whole circumference,
then one must etc., but of course the onus lies not on us, if the
continuity should not be lifted without cause, and it should even
be proved, of course, that it should be taken as real in the whole
circumference. 2) To be sure,
�
dy
y
is log y as well as log –y, but never both simultaneously, rather the
former, if the integral � can begin from y – 1, the latter, if it begins
dx
from y = –1; the second integral = C + is l(±lx)+etc., just as understood as the first,
in general log y + C,
logx
if the first time C is set = 0, the second time
C is set =±πi or ±3πi etc. However, it is very true that eu l e r’s
remarks require a correction in so far as, if the integral should
start from z = 0 , in no case � will x e −
C
1
be infinite. – Thus, according
to my opinion, one may not set dx
x
�
dx
= C + l(±lx)+etc.,
logx
�
dx
1
logx
which for brevity I will signify with Ei x, Exponential Logarithm,
then the integration is taken, � so that Ei disappears for any real,
x +
dx
Δυναμις Vol. 3 No. 3 December 2008
1 1
xx +
4 18 x3 5 Institutiones calculi integralis I, 1768, Sections 228, L. Euler
Opera omnia, ser. I, vol. 11, S.128 +etc.
logx
li(0.5+0.001i), li(0.6+0.001i), li(0.7+0.001i), li(0.8+0.001i),
(1 +2αx + βxx)
i(0.9+0.001i), li(1.0+0.001i), li(1.1+0.001i), etc., until li(1.5+0.001i)
x + 1 1
xx +
4 18 x3 + etc.

�
dy
� y
dy
y
rather must be determined either with l(+lx) or with l(–lx), but
only one decided on. –
�
Moreover, I dx believe the extension of the investigation to
= C + l(±lx)+etc.,
imaginary arguments logx will provide grounds for supremely interesting
results. Nevertheless, from the foundations derived above,
I would rather chose the function
� x e − 1
dx
x
�
dx
logx
x + 1 1
xx +
4 18 x3 �
dx
= C + l(±lx)+etc.,
logx
� x e − 1
dx
x
than
�
dx
logx
x +
+ etc.
(1 +2αx + βxx)
1 1
xx +
4 18 x3 �
dx
= C + l(±lx)+etc.,
because I surmise, that logx the first will give a coinciding result. Thus
for example, I would love to know, whether this function, or what
is the same, the series
+ etc.
x +
(1 +2αx + βxx)
1 1
xx +
4 18 x3 �
dy
y
�
dx
= C + l(±lx)+etc.,
logx
� x e − 1
dx
x
�
dx
logx
+etc.
can become 0, for certain finite values of x of the form a + bi. I
can not yet maintain with certainty, x + whether it is even very probable
to me. If there is such a value (then most certainly infinite),
these magnitudes will be very remarkable decomposed and the
entire series can be decomposed into infinite factors of the form
1 1
xx +
4 18 x3 + etc.
(1 +2αx + βxx) .
From a few other details, I can only add a few words at
this time...
1
1
1
1
Gauss to Bessel
December 18, 1811
Δυναμις Vol. 3 No. 3 December 2008
43

The First Integral Calculus
Bernoulli
The First Integral Calculus
A Selection from Johann Bernoulli’s Mathematical Lectures on the Method of Integrals and Other Matters
Johann Bernoulli
Translated by William A. Ferguson, Jr., from the German translation
of Dr. Gerhardt Kowalewski.
We have seen previously how to find the differentials
of quantities. Now we will, inversely, show how the
integrals of differentials are to be found, i.e., those
quantities, from which the differentials are derived. Now it is already
known from previous statements, that dx is the differential
of 1
2x2 x2dx 1
3x3 1
4x4 or 1
2x2 x2dx 1
3x3 1
4x4 1
2
+ or – a constant quantity,
x2
x2dx 1
3x3 1
4x4 1
2
the differential
of
x2
x2dx 1
3x3 1
4x4 1
2
or
x2
x2dx 1
3x3 1
4x4 + or – etc. and x3 1
2
dx the differential of
x2
x2dx 1
3x3 1
4x4 1
2
or
x2
x2dx 1
3x3 1
4x4 + or – a constant quantity, likewise also
From this the following general rule can be formed:
axp dx
a
p +1 xp+1
ax
is the differential of
p dx
a
p +1 xp+1 .
Therefore if the integral of any differential quantity is
to be taken, then one must first of all consider, whether the given
quantity is the product of any differential quantity multiplied by
its “absolute quantity” raised to a certain power. This is then an
indication that one can find the integral by the rule given above.
If, for example, the integral of the quantity dy √ dy
a + y is to be
found, then I see first, that dy is multiplied by a multiple of its
absolute quantity a + y, raised to the power a + y
1
2
√ a + y
a + y
1
2 ; next I seek its
integral by the above rule, namely,
1
1
1 + y) 2
2 +1(a +1
1
1
1 + y) 2
2 +1(a
, i.e.
+1
2
3 (a + y)√ It should be noted, that sometimes quantities present
themselves, whose integrals at first glance, it seems, cannot be
found by this rule. Nonetheless, the dxintegral may easily be found
after a certain transformation, as in the following cases.
1. If one writes
a + y
.
√ a2x2 + x4 xdx √ a2 + x2 ( 1
3a2 + 1
3x2 ) √ a2 + x2 instead of
dx √ a2x2 + x4 xdx √ a2 + x2 ( 1
3a2 + 1
3x2 ) √ a2 + x2 dx
, then one finds the integral of the latter, namely
√ a2x2 + x4 xdx √ a2 + x2 ( 1
3a2 + 1
3x2 ) √ a2 + x2 dx
. And if one writes
√ a3 +3a2x +3ax2 + x3 (a dx + xdx) √ a + x
2
5 (a2 +
(3ax 3 for dx
dx
√ a3 +3a2x +3ax2 + x3 (a dx + xdx) √ a + x
2
5 (a2 +2ax + x 2 ) √ a + x
(3ax 3 dx +4x 4 dx) � ax + x2 dx
, then one finds the integral
√ a3 +3a2x +3ax2 + x3 (a dx + xdx) √ a + x
2
5 (a2 +2ax + x 2 ) √ a + x
(3ax 3 dx +4x 4 dx) � ax + x2 dx
.
2. Also conversely it can occur, that one must pull one
or more variables under the root sign, before the integral can be
taken, as with the following example
√ a3 +3a2x +3ax2 + x3 (a dx + xdx) √ a + x
2
5 (a2 +2ax + x 2 ) √ a + x
(3ax 3 dx +4x 4 dx) � ax + x2 .
The integral of this cannot be taken by our rule as it
appears. However, if one pulls in an x (under the root sign), the
result is
(3ax 2 dx +4x 3 dx) � ax3 + x4 = 2
3 (ax3 + x 4 ) � ax3 + x4 xdx
a4 +2a2x2 + x4 (3ax
,
whose integral one finds by the rule
2 dx +4x 3 dx) � ax3 + x4 = 2
3 (ax3 + x 4 ) � ax3 + x4 xdx
a4 +2a2x2 + x4 (3ax
.
3. If a fraction presents itself, whose denominator is a
square, cube, or other power, then one must choose its root as the
absolute quantity. Therefore, for
2 dx +4x 3 dx) � ax3 + x4 = 2
3 (ax3 + x 4 ) � ax3 + x4 (3ax
xdx
2 dx +4x 3 dx) � ax3 + x4 = 2
3 (ax3 + x 4 ) � ax3 + x4 xdx
a4 +2a2x2 + x4 (3ax 2 dx +4x 3 dx) � ax3 + x4 = 2
3 (ax3 + x 4 ) � ax3 + x4 xdx
a4 +2a2x2 + x4 (3ax 2 dx +4x
= 2
3 (ax3 +
a4 adx is the differential of etc.
ax dx "
"
ax "
"
"
"
etc.
+2
2 dx
ax3 ax
1
2
dx
ax2
1
3ax3 1
4ax4 2
3
Δυναμις Vol. 3 No. 3 December 2008
(a + y)√ Likewise one finds the integral of xdx
a + y
3√ a2 + x2 ,
which is the following
1
2
1
3 +1(a2 + x 2 ) 1
3 +1 = 3
8 (a2 + x 2 ) 3� a2 + x2 ;
the integral of dy : √ a + y
2 √ a + y
dx : x
1
0 x0 = 1
dy :
equals
× 1=∞
0 √ a + y
2 √ a + y
dx : x
1
0 x0 = 1
dy :
, the integral of
× 1=∞
0 √ a + y
2 √ a + y
dx : x
1
0 x0 = 1
dy :
equals
× 1=∞
0 √ a + y
2 √ a + y
dx : x
1
0 x0 = 1
0 × 1=∞ a
.
2 + x2 −1 :(2a2 +2x2 )
a4 +2a2x2 + x4 a2 + x2 −1 :(2a2 +2x2 )
a4 +2a2x2 + x4 a2 + x2 −1 :(2a2 +2x2 )
a4 +2a2x2 + x4 a2 + x2 −1 :(2a2 +2x2 )
a4 +2a2x2 + x4 a
is to be chosen as the absolute quantity, and one obtains
then
2 + x2 −1 :(2a2 +2x2 )
a4 +2a2x2 + x4 −1 :(2a
. If one chose
2 +2x2 )
a4 +2a2x2 + x4 as the absolute
quantity, then the integral of this fraction would not be
obtainable by this rule.
4. If the integrals of two quantities cannot be found individually,
then sometimes it will be the case, that one can find the
integral of their combination. Example:
1
a4 +2a2x2 + x4 a2 + x2 1
1
1
44
1

The First Integral Calculus
Bernoulli
adx
√
2ax + x2 Δυναμις Vol. 3 No. 3 December 2008
+
xdx
√
2ax + x2 adx + xdx
√
2ax + x2 √
2ax + x2 adx + xdx
√
3a +2x
ax dx + x2 adx
√
2ax + x2 The integral of either quantity is not known. However, the integral
of their sum,
dx
√
3ax2 +2x3 √
1
3 3ax2 +2x3 1
+
xdx
√
2ax + x2 adx + xdx
√
2ax + x2 √
2ax + x2 adx + xdx
√
3a +2x
ax dx + x2 adx
√
2ax + x2 is
dx
√
3ax2 +2x3 √
1
3 3ax2 +2x3 1
+
xdx
√
2ax + x2 adx + xdx
√
2ax + x2 √
2ax + x2 adx + xdx
√
3a +2x
ax dx + x2 adx
√
2ax + x2 .
5. Sometimes, a fraction may seem not to have an integral,
but if one multiplies its numerator and denominator by
the same quantity, then its integral can be obtained easily. So it
is with
dx
√
3ax2 +2x3 √
1
3 3ax2 +2x3 1
+
xdx
√
2ax + x2 adx + xdx
√
2ax + x2 √
2ax + x2 adx + xdx
√
3a +2x
ax dx + x2 adx
√
2ax + x2 .
Multiply the numerator and denominator by x. Then one obtains
dx
√
3ax2 +2x3 √
1
3 3ax2 +2x3 1
+
xdx
√
2ax + x2 adx + xdx
√
2ax + x2 √
2ax + x2 adx + xdx
√
3a +2x
ax dx + x2 adx
√
2ax + x2 dx
√
3ax2 +2x3 ,
√ whose integral is
1
3 3ax2 +2x3 1
+
xdx
√
2ax + x2 adx + xdx
√
2ax + x2 √
2ax + x2 adx + xdx
√
3a +2x
ax dx + x2 dx
√
3ax2 +2x3 √
1
3 3ax2 +2x3. 6. Conversely, sometimes the numerator and denominator
are to be divided by the same quantity, in order to obtain its
integral. Example:
ax
1
2dx √
a2x2 + x4 ax dx
√
a2 + x2 a √ a2 + x2 ax .
Divide every term by x. Then one obtains
2dx √
a2x2 + x4 ax dx
√
a2 + x2 a √ a2 + x2 ax
.
Its integral follows by the rule:
2dx √
a2x2 + x4 ax dx
√
a2 + x2 a √ a2 + x2 .
7. It also occurs sometimes, that the integral of a given
quantity is not obtainable by the rule. If one however adds another
quantity to it, whose integral one knows, something may be
produced whose integral can be taken. Next, one subtracts from
that integral the added quantity, therefore the remainder is the
desired integral. Example: xdx √ a + x
adx √ a + x
(a dx + xdx) √ a + x
= 2
5 (a + x)2√a + x
adx √ xdx
.
Because its integral cannot be taken by a simple
method, add
a + x
√ a + x
adx √ a + x
(a dx + xdx) √ a + x
= 2
5 (a + x)2√a + x
adx √ a + x
2
3 a(a + x)√ xdx
to the given quantity. The result is
a + x
√ a + x
(a dx + xdx) √ a + x
= 2
5 (a + x)2√a + x
2
3 a(a + x)√ xdx
,
whose integral is found by the rule to be equal to
a + x
√ a + x
adx √ a + x
(a dx + xdx) √ a + x
= 2
5 (a + x)2√a + x
adx √ a + x
2
3 a(a + x)√ xdx
.
If one subtracts from this the integral of
a + x
√ a + x
adx √ a + x
(a dx + xdx) √ a + x
= 2
5 (a + x)2√a + x
adx √ a + x
2
3 a(a + x)√a + x
2 2
(a + x) √ a + x − 2 a(a + x) √ =
which is
,
a + x
2
5 (a + x)2√a + x
adx √ a + x
2
3 a(a + x)√a + x
2
5 (a + x)2√a + x − 2
3 a(a + x)√a + x
xdx √ a + x
x 2 dx √ a + x
(a 2 +2ax + x 2 )dx √ a + x
a 2 dx √ a + x
2ax dx √ a + x
x2 dx √ =
,
therefore
a + x
1
2
5 (a + x)2√a + x
adx √ a + x
2
3 a(a + x)√a + x
2
5 (a + x)2√a + x − 2
3 a(a + x)√a + x
xdx √ a + x
x 2 dx √ a + x
(a 2 +2ax + x 2 )dx √ a + x
a 2 dx √ a + x
2ax dx √ a + x
x2 dx √ =
adx
remains as the integral of the given quantity
a + x
1
√ a + x
2
(a + x)2
5
xdx √ a + x
(a 2 +
x2 dx √ =
. In this
same manner one finds the integral of
a + x
2
5 (a + x)2√a + x
adx √ a + x
2
3 a(a + x)√a + x
2
5 (a + x)2√a + x − 2
3 a(a + x)√a + x
xdx √ a + x
x 2 dx √ a + x
(a 2 +2ax + x 2 )dx √ a + x
a 2 dx √ a + x
2ax dx √ a + x
x2 dx √ adx
. In other
words, the integral of
a + x
1
√ a + x
2
3 a(a + x)√a + x
2
5 (a + x)2√a + x − 2
3 a(a + x)√a + x
xdx √ a + x
x 2 dx √ a + x
(a 2 +2ax + x 2 )dx √ a + x
a 2 dx √ a + x
2ax dx √ a + x
x2 dx √ 2
3
is known, as
is that of
a + x
1
a(a + x)√a + x
2
5 (a + x)2√a + x − 2
3 a(a + x)√a + x
xdx √ a + x
x 2 dx √ a + x
(a 2 +2ax + x 2 )dx √ a + x
a 2 dx √ a + x
2ax dx √ a + x
x2 dx √ 2
5
, and afterwards was found immediately
above, that of
a + x
1
(a + x)2√a + x − 2
3 a(a + x)√a + x
xdx √ a + x
x 2 dx √ a + x
(a 2 +2ax + x 2 )dx √ a + x
a 2 dx √ a + x
2ax dx √ a + x
x2 dx √ 2
5
.
Therefore one has the integral of the remaining term,
a + x
1
(a + x)2√a + x − 2
3 a(a + x)√a + x
xdx √ a + x
x 2 dx √ a + x
(a 2 +2ax + x 2 )dx √ a + x
a 2 dx √ a + x
2ax dx √ a + x
x2 dx √ a + x.
In this same manner one will find the integrals of
the quantities x
1
3 dx √ a + x
x4 dx √ a + x
xp dx √ a + x
(2ax 3 + x 4 )dx √ a + x
2ax3 dx √ a + x
x4 dx √ x
or
a + x
3 dx √ a + x
x4 dx √ a + x
xp dx √ a + x
(2ax 3 +
2ax3 dx √ a + x
x4 dx √ x
and even
a + x
3 dx √ a + x
x4 dx √ a + x
xp dx √ a + x
(2ax 3 + x 4 )dx √ a + x
2ax3 dx √ a + x
x4 dx √ x
. Thus also, if a quantity consisting of several terms
is given, its integral will be found by parts. One such quantity is
a + x
3 dx √ a + x
x4 dx √ a + x
xp dx √ a + x
(2ax 3 + x 4 )dx √ a + x
2ax3 dx √ a + x
x4 dx √ x
. First I seek the integral of the first part
a + x
3 dx √ a + x
x4 dx √ a + x
xp dx √ a + x
(2ax 3 + x 4 )dx √ a + x
2ax3 dx √ a + x
x4 dx √ x
, then that of the second,
a + x
1
3 dx √ a + x
x4 dx √ a + x
xp dx √ a + x
(2a
2ax3 dx √ a + x
x4 dx √ 45
a + x. Their
sum gives the integral of the whole.
Admonition
These are the most important cases in which integrals
can be formed. Indeed several, even infinitely many others yet remain,
with the help of which integrations are possible. However
they do not all come to mind, and furthermore, most of them can
be reduced to those cited here, so that with the help of these, the
desired ones can be achieved; ultimately a thousand methods of
solution and manifold cases according to the nature of the given
quantities present themselves to the attentive observer. For this
reason it were no less impossible than useless, were we to provide
yet several others aside from those offered here.
Let the one remark suffice, that important mathematical
problems and theorems directly depend on the finding of integrals,
both those already found as well as such as are yet desired to be
found, as for example the quadrature of plane surfaces, the rectification
of curves, the cubature of bodies, the method of inverse
tangents, or the finding of the nature of a curve from given properties
of its tangents, as well as that which belongs to mechanics,
like the methods of finding the center of mass, of impulses, of oscillations,
and so forth. Through the finding of integrals, one also
obtains the involutions of curves, and the method by which to
determine their nature, and with the help of the involute to rectify
the curves themselves, as Ts c h i r n h a u s did with his caustics.
The ease of finding the differential of any given quantity
is matched, conversely, by the difficulty of finding the integral
of any given differential, so that we at times cannot even confidently
assert, whether the integral of the given quantity can be

The First Integral Calculus
Bernoulli
formed or not. I venture to assert at least, that every whole and
rational quantity, which is multiplied or divided by x p� a2 − x2 , x p� ax − x2 , x p� a2 + x2 (a 3 + ax 2 − x 3 �
a + x
)dx
x
(a 3 + ax 2 − x 3 �
a + x
)dx
x =
(a4 + a3x + a2x2 − x4 2 − x2 p
, x
)dx
√
ax + x2 1
� ax − x2 , x p� a2 + x2 (a 3 + ax 2 − x 3 �
a + x
)dx
x
(a 3 + ax 2 − x 3 �
a + x
)dx
x =
(a4 + a3x + a2x2 − x4 , is either integrable or reducible to
the quadrature of the circle or the hyperbola. This we will show
in what follows. Therefore, above all, it is to be carefully considered
whether the given quantity, which one would integrate, can
be reduced by multiplication, division, or through the extraction
of roots to a quantity which has one of these root terms, multiplied
by a whole and rational quantity. If it is possible, then it is
immediately a serious proposition, that the given quantity can be
integrated; xif not, it is dependent on and reducible to the quadrature
of the circle or the hyperbola.
If, for example, the following quantity is given,
)dx
√
ax + x2 1
p� a2 − x2 , x p� ax − x2 , x p� a2 + x2 (a 3 + ax 2 − x 3 �
a + x
)dx
x
(a 3 + ax 2 − x 3 �
a + x
)dx
x =
(a4 + a3x + a2x2 − x4 ,
and its integral is to be taken, then it appears at first glance neither
to be integrable nor to have a relationship to the quadrature of the
circle. In other words, if one assumes the absolute quantity to be
that under the root sign, the fraction
)dx
√
ax + x2 (a+x)
sumed variable.
This will be better clarified through an example. Let the
quantity whose integral is desired be (ax + x
, then its differential
x
will also be a fraction, so from that nothing can be concluded by
the rule. Therefore I multiply the numerator and denominator of
this irrational fraction by the numerator, and unite the product of
the numerator with itself to the rational component of the quantity,
so that a fraction results whose numerator is purely rational
and whose denominator is irrational. Namely,
This quantity now shows, that either it has an integral,
or is reducible to the quadrature of the hyperbola. Rules will be
given below, as to how this can be recognized and done.
There remains yet something else, before we come to
the use and application of the integral calculus. To wit, we will
explain another procedure for the formation of integrals, which
condenses the general method by more than a little. Because
sometimes, due to the complexity of the given quantity, it is not
immediately clear, whether it is of a kind which is reducible to
one of the cases that we have presented before, and even, whether
it has an integral or not. This procedure, however, reduces the
quantity to fewer terms, so that one may find the desired integral
without difficulty. This is done however, by taking the quantity
under the root sign as the absolute quantity and setting it equal
to some variable, and transforming the quantity to be integrated
accordingly into another consisting only of terms of the assumed
helping variable. One takes the integral of this quantity, which for
the most part appears much simpler, which can be transformed
back to the desired integral, by reinserting the value of the as-
2 )dx √ a + x
√
a + x = y
x = y2 − a
dx =2ydy
(ax + x 2 )dx √ a + x =2y 6 dy − 2ay 4 dy
= 2
7y7 − 2
5ay5 2
7 (x + a)3√x + a − 2
5 a(x + a)2√ (ax + x
.
For that purpose, I set
x + a
1
2 )dx √ a + x
√
a + x = y
x = y2 − a
dx =2ydy
(ax + x 2 )dx √ a + x =2y 6 dy −
= 2
7y7 − 2
5ay5 2
7 (x + a)3√x + a − 2
(
√
a + x = y
. Then x = y
a(x + a)
5
1
2 − a
dx =2ydy
(ax + x 2 )dx
= 2
7y7 − 2
5ay5 (ax + x
and further
2
(x + a)3√
7 2 )dx √ a + x
√
a + x = y
x = y2 − a
dx =2ydy
(ax + x 2 )dx √ a + x =2y 6 dy − 2ay 4 dy
= 2
7y7 − 2
5ay5 2
7 (x + a)3√x + a − 2
5 a(x + a)2√ (ax + x
. Therefore in the whole quantity
x + a
1
2 )dx √ a + x
√
a + x = y
x = y2 − a
dx =2ydy
(ax + x 2 )dx √ a + x =2y 6 dy − 2ay 4 dy
= 2
7y7 − 2
5ay5 2
7 (x + a)3√x + a − 2
5 a(x + a)2√ (ax + x
.
The integral of this is found immediately, easily, without
further ado, and indeed is
x + a
1
2 )dx √ a +
√
a + x = y
x = y2 − a
dx =2ydy
(ax + x 2 )dx √ a + x =2y 6 d
= 2
7y7 − 2
5ay5 2
7 (x + a)3√x + a − 2
(ax + x
. Now if one inserts the
value of y, one obtains
a(x +
5
1
2 )dx √ a + x
√
a + x = y
x = y2 − a
dx =2ydy
(ax + x 2 )dx √ a + x =2y 6 dy − 2ay 4 dy
= 2
7y7 − 2
5ay5 2
7 (x + a)3√x + a − 2
5 a(x + a)2√x + a.
In the same manner the integral of the quantity
(a
1
2 +2x2 )dx
√
a2 + x2 √
a2 + x2 = y
x = � y2 − a2 dx = ydy : � y2 − a2 (a 2 +2x 2 )dx : � a2 + x2 =
(2y 3 − a 2 y)dy : � y4 − a2y2 will be found by setting y = √ a2 + x2 √
a2 + x2 = y
. Then will x = � y2 − a2 dx = ydy : � y2 − a2 (a
and
(
2 +2x2 )dx
√
a2 + x2 √
a2 + x2 = y
x = � y2 − a2 dx = ydy : � y2 − a2 (a 2 +2x 2 )dx : � a2 + x2 =
(2y 3 − a 2 y)dy : � y4 − a2y2 . Further we get for the quantity itself
The integral of this is � y4 − a2y2 (a − x)dx
√
2ax − x2 √
2ax − x2 = y
x = a ± � a2 − y2 dx = ∓y dy : � a2 − y2 (a − x)dx
√ = dy
2ax − x2 = y = √ 2ax − x2 . Likewise, if one has
�
y4 − a2y2 (a − x)dx
√
2ax − x2 √
2ax − x2 = y
x = a ± � a2 − y2 dx = ∓y dy : � a2 − y2 (a − x)dx
√ = dy
2ax − x2 = y = √ 2ax − x2 ,
set y = √ a2 + x2 �
y4 − a2y2 (a − x)dx
√
2ax − x2 √
2ax − x2 = y
x = a ± � a2 − y2 dx = ∓y dy : � a2 − y2 (a − x)dx
√ = dy
2ax − x2 = y = √ 2ax − x2 �
y4 − a2y2 (a −
√
2ax
√
2ax − x2 = y
. Then x = a ± � a2 − y2 dx = ∓y dy : � a2 − y2 (a − x)
√
2ax −
= y = √ 2ax − x2 �
y4 − a2y2 √
2ax − x2 =
x = a ±
,
� a2 dx = ∓y dy :
= y = √ �
y4 − a2y2 (a − x)dx
√
2ax − x2 √
2ax − x2 = y
x = a ±
2ax −
� a2 − y2 dx = ∓y dy : � a2 − y2 (a − x)dx
√ = dy
2ax − x2 = y = √ 2ax − x2 �
y4 − a2y2 (a − x)dx
√
2ax − x2 √
2ax − x2 = y
x = a ±
and
� a2 − y2 dx = ∓y dy : � a2 − y2 (a − x)dx
√ = dy
2ax − x2 = y = √ 2ax − x2 �
y4 − a2y2 (a − x)dx
√
2ax − x2 √
2ax − x2 = y
x = a ±
.
The integral of this is
� a2 − y2 dx = ∓y dy : � a2 − y2 (a − x)dx
√ = dy
2ax − x2 = y = √ 2ax − x2 x
.
This rule can henceforth be applied in infinitely many
cases, even in those which seem almost hopeless because of their
complexity. Because, aside from the fact that this rule sometimes
makes the quantity in question much briefer, it also offers the
advantage, that it stands immediately before one’s eyes, whether
the transformed quantity can be integrated.
To all these methods of finding integrals one can add the
following, which because of its usefulness and easiness, is almost
preferable to all the rest. This method is however only relevant
to those quantities which are combined with irrationals. Its whole
application, accordingly, consists of transforming irrational quantities
into rational ones, so that the given quantity assumes a com-
p� a2 − x2 , x p� ax − x2 , x p� a2 + x2 (a 3 + ax 2 − x 3 �
a + x
)dx
x
(a 3 + ax 2 − x 3 �
a + x
)dx
x =
(a4 + a3x + a2x2 − x4 x
)dx
√
ax + x2 p� a2 − x2 , x p� ax − x2 , x p� a2 + x2 (a 3 + ax 2 − x 3 �
a + x
)dx
x
(a 3 + ax 2 − x 3 �
a + x
)dx
x =
(a4 + a3x + a2x2 − x4 (a
)dx
√
ax + x2 2 +2x2 )dx
√
a2 + x2 √
a2 + x2 = y
x = � y2 − a2 dx = ydy : � y2 − a2 (a 2 +2x 2 )dx : � a2 + x2 =
(2y 3 − a 2 y)dy : � y4 − a2y2 (a2 +2x2 )dx
√
a2 + x2 √
a2 + x2 = y
x = � y2 − a2 dx = ydy : � y2 − a2 (a 2 +2x 2 )dx : � a2 + x2 =
(2y 3 − a 2 y)dy : � y4 − a2y2 46
Δυναμις Vol. 3 No. 3 December 2008
1

The First Integral Calculus
Bernoulli
pletely rational character, after which, when it is possible, its integral
is easy to form. This leads therefore into the Diophantine
problem, which provides excellent assistance on such occasions,
as will become more clearly evident by examples.
For example, let
a3dx x √ ax − x2 √
ax − x2 ax − x2 ax − x 2 = a2x2 m2 x = am2 :(a2 + m2 )
√
ax − x2 2 2 2 = a m :(a + m )
dx =2a3mdm :(m2 + a2 ) 2
a
be the quantity to be integrated, which is not feasible by the previous
methods, but which can be done in the following manner.
Since
1
3dx x √ ax − x2 √
ax − x2 ax − x2 ax − x 2 = a2x2 m2 x = am2 :(a2 + m2 )
√
ax − x2 2 2 2 = a m :(a + m )
dx =2a3mdm :(m2 + a2 ) 2
a
is an irrational quantity, then to make it rational,
1
3dx x √ ax − x2 √
ax − x2 ax − x2 ax − x 2 = a2x2 m2 x = am2 :(a2 + m2 )
√
ax − x2 2 2 2 = a m :(a + m )
dx =2a3mdm :(m2 + a2 ) 2
a
must become a square. Let therefore
1
3dx x √ ax − x2 √
ax − x2 ax − x2 ax − x 2 = a2x2 m2 x = am2 :(a2 + m2 )
√
ax − x2 2 2 2 = a m :(a + m )
dx =2a3mdm :(m2 + a2 ) 2
a
.
From this assumption
1
3dx x √ ax − x2 √
ax − x2 ax − x2 ax − x 2 = a2x2 m2 x = am2 :(a2 + m2 )
√
ax − x2 2 2 2 = a m :(a + m )
dx =2a3mdm :(m2 + a2 ) 2
a
and consequently
1
3dx x √ ax − x2 √
ax − x2 ax − x2 ax − x 2 = a2x2 m2 x = am2 :(a2 + m2 )
√
ax − x2 2 2 2 = a m :(a + m )
dx =2a3mdm :(m2 + a2 ) 2
a
,
1
3dx x √ ax − x2 √
ax − x2 ax − x2 ax − x 2 = a2x2 m2 x = am2 :(a2 + m2 )
√
ax − x2 2 2 2 = a m :(a + m )
dx =2a3mdm :(m2 + a2 ) 2
,
therefore the whole given quantity
a
1
3 dx
x √ ax − x2 = 2a3 dm
m2 −2a3 : m
m = � a2x :(a− x)
�
4a5 − 4a4x −
= −2a
x
2
�
a − x
x
a3 dx : x √ ax − x2 dx 3√ x2 +2ax + a2 x
x2 +2ax + a2 a
,
whose integral is easy to find, and indeed is
3 dx
x √ ax − x2 = 2a3 dm
m2 −2a3 : m
m = � a2x :(a− x)
�
4a5 − 4a4x −
= −2a
x
2
�
a − x
x
a3 dx : x √ ax − x2 dx 3√ x2 +2ax + a2 x
x2 +2ax + a2 a
.
If one now inserts the value
3 dx
x √ ax − x2 = 2a3 dm
m2 −2a3 : m
m = � a2x :(a− x)
�
4a5 − 4a4x −
= −2a
x
2
�
a − x
x
a3 dx : x √ ax − x2 dx 3√ x2 +2ax + a2 x
x2 +2ax + a2 a
, the
result is
3 dx
x √ ax − x2 = 2a3 dm
m2 −2a3 : m
m = � a2x :(a− x)
�
4a5 − 4a4x −
= −2a
x
2
�
a − x
x
a3 dx : x √ ax − x2 dx 3√ x2 +2ax + a2 x
x2 +2ax + a2 a
as the integral of
3 dx
x √ ax − x2 = 2a3 dm
m2 −2a3 : m
m = � a2x :(a− x)
�
4a5 − 4a4x −
= −2a
x
2
�
a − x
x
a3 dx : x √ ax − x2 dx 3√ x2 +2ax + a2 x
x2 +2ax + a2 a
.
Likewise to integrate
3 dx
x √ ax − x2 = 2a3 dm
m2 −2a3 : m
m = � a2x :(a− x)
�
4a5 − 4a4x −
= −2a
x
2
�
a − x
x
a3 dx : x √ ax − x2 dx 3√ x2 +2ax + a2 x
x2 +2ax + a2 a
,
3 dx
x √ ax − x2 = 2a3 dm
m2 −2a3 : m
m = � a2x :(a− x)
�
4a5 − 4a4x −
= −2a
x
2
�
a − x
x
a3 dx : x √ ax − x2 dx 3√ x2 +2ax + a2 x
x2 +2ax + a2 must become a cube. Let therefore x + a = y3
x = y3 − a
dx =3y2 dy
3√
x2 +2ax + a2 2 = y
dx 3√ x2 +2ax + a2 =
x
3y4 dy
y3 x + a = y
.
then will
− s
3
x = y3 − a
dx =3y2 dy
3√
x2 +2ax + a2 2 = y
dx 3√ x2 +2ax + a2 =
x
3y4 dy
y3 x + a = y
,
− s
3
x = y3 − a
dx =3y2 dy
3√
x2 +2ax + a2 2 = y
dx 3√ x2 +2ax + a2 =
x
3y4 dy
y3 x + a = y
and
− s
3
x = y3 − a
dx =3y2 dy
3√
x2 +2ax + a2 2 = y
dx 3√ x2 +2ax + a2 =
x
3y4 dy
y3 Therefore the whole quantity
dx
− s
3√ x2 +2ax + a2 =
x
3y4 dy
y3 − a .
If one can obtain the integral of this, then one also has the integral
of the given quantity.
Δυναμις Vol. 3 No. 3 December 2008
47

What’s the Matter with Descartes
Vance
Exclusive Interview: René Descartes
What’s the Matter with Descartes?
Timothy Vance
There’s no denying it, folks.
No matter how you slice it,
there’s just no amount of
geometric extension in the world
that could bail out Descartes’ utterly
bankrupt notion of moving
bodies. You may keep trying to
extend this matter of his if you like,
but it still won’t move. “It’s impenetrable,”
the experts may say.
I don’t care, my friend. It’ll take
a miracle from God to get those
bodies of his moving again. Why?
Because Descartes’ argument for
motion lacks force ... and not just
the kind necessary for convincing
you he’s right, which, by the way,
he would enjoy very much. This
may seem a little suprising at first
because René, as I like to call him,
is a man whose books are still purchased
every year by undergraduates
enrolled in philosophy courses
they can ill afford to take, and
shouldn’t. He is a man who is credited with making mathematics
“modern” by freeing Geometry from the tyranny of ... well, Geometry;
And finally, Descartes is a man whom my loving friend
Wilhelm Gottfried Leibniz would go out of his way on just about
every occasion to refute. Naturally, such a reputation led me to
expect more gravitas from Monsieur Descartes; instead, I found a
man led astray (from making any damn sense) by too great a faith
in his own genius and inflated sense of self–importance.
What follows is a hastily arranged interview I conducted
with Monsieur Descartes himself on Sepember 28 of this year:
VANCE – Monsieur Descartes, thank you so much for
taking the time to be with us today.
DESCARTES – It really is my pleasure, and I’m very
lucky to have actually made it here this evening. I had some
trouble earlier today ...
Tim Vance interviews René Descartes
V – Trouble with your flight over the Atlantic?
D – No, trouble getting out of bed. It’s where I draw the
vast majority of my philosophical conclusions, you know; and
it’s so very hard to find a nice quiet place these days for meditative
contemplation (much more so if you have friends like mine).
I’ve also discovered that one’s bed provides the perfect environment
for thinking through life’s most challenging questions like
“What can I know?” and “How can I know it?” – especially if
doe in the late morning and early afternoon when everybody else
is at work.
V – You’ve been described by leading academics as “one
of the most influential thinkers in human history.” That’s quite a
remarkable statement! How did you come by all that knowledge
which made you so famous?
D – That’s a great question Tim, thanks so much for asking
me. It’s really simple. “All that I have, up to this moment, ac-
Δυναμις Vol. 3 No. 3 December 2008
48

cepted as possessed of the highest truth and certainty, I received
either from or through the senses.” 1 The idea for it came to me in
a dream a few winters back. It was then that I discovered something
very important which “got the ball rolling” so to speak–
V – What did you discover?
D – MYSELF! You know: Cogito ...
V – Ergo sum?
D – Yeah! Isn’t that great?
V – Excusez–moi, Monsieur. Would you mind if I were
to ask you a few simple questions concerning motion? It appears
that my friend Leibniz could not have disagreed more with
you on this very subject. He accuses you of preferring applause
over certainty, or rather a fan–base over fact; and attributes your
failure to properly describe the laws of nature–
D – I FAILED?
V – Well, before I answer you, René, let me ask: What
is motion when it’s properly understood?
D – Well, I am most sure of myself when I say “it is the
translation of one part of matter, or of one body, from the vicinity
of those bodies that are in direct contact with it and are viewed as
at rest to the vicinity of others. Where by ‘one body’ or ‘one part
of matter’ I understand everything that is transferred at the same
time, even if this itself might again consist of many parts which
have other motions in themselves. And I say that translation is not
the force or action that transfers, as I shall show that this [motion]
is always in the mobile, not in the mover.” 2
V – Très bien, Monsieur. However, I became a little
confused when you assigned the name “motion” to what is in fact
only an effect of motion (namely “translation,” or a change from
one place to another in a rectilinear fashion I presume), only to
separate that effect immediately afterwards from its cause (that is,
the “force or action that transfers”). You make little distinction
between something which is moved, i.e. the mobile as you say,
and that which moves, i.e. the mover.
D – “It is not a matter of that action which is understood
to be in the mover, or in that which arrests motion, but of translation
alone and of the absence of translation, or rest.” 3
1 René Descartes Meditations – No. 1
2 René Descartes Principles of Philosophy Part II – ‘On Motion’
Translated by M.S. Mahoney, 1977
3 Mahoney, ibid.
What’s the Matter with Descartes
Vance
V – But that would imply that–
D – “[M]otion and rest ... are nothing other than two
different modes–” 4
V – Of the same thing?
D – Yes, all corporeal substance.
V – Substance made of what?
D – Extension which is impenetrable.
V – But how are things extended?
D – How else are things extended through Euclidean
space? By merely adding or subtracting individual points in the
case of a line, or by adding and subtracting a series of points in
the case of a surface, and so on.
V – Euclid tells me a point has no parts. So what part of
a point would make it impenetrable?
D – I don’t know.
V – I think you’re making it a lot harder for yourself
by applying the same generic metric of “translation,” or the lack
thereof, to both a body in motion and one at rest, and similarly
applying it to any moving body regardless of whether it moves or
is moved. Once again, why lump the two together?
D – “[Because] no more action is required for motion
than for rest.”
V – But René, if no more action is required for motion
than for rest, what has a body formerly at rest ever done in order
to move? And if a moving object isn’t doing anything either, who
or what has moved it?
D – “God.” 5
V – I see, très bien! And let me guess: “God is the primary
cause of motion and always conserves the same quantity of
motion in the universe.”
D – “Mais oui.”
V – Uh–huh ... Right.
4 Mahoney, ibid.
5 Mahoney, ibid.
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D – “[Well], it seems clear to me that it is nothing other
than God Himself, who in the beginning created matter together
with motion and rest and now conserves just as much motion and
rest as a whole as He then posited. Now, although this motion in
moved matter is nothing other than its mode, nevertheless it has
a certain and determinate quantity, which we easily understand
to be able to be always the same in the whole universe of things,
even though it be changed in its individual parts.” 6
V – For example ...?
D – “[I]t is evident [...] when one part of matter is moved
twice as fast as another, and this second [part of matter] is twice
as large as the first, there is as much motion in the smaller as in
the larger; and by as much as the motion of one part is made slower,
the motion of some other equal to it is made faster. We also
understand perfection to be in God, not only that He is immutable
in Himself, but that he works in a most constant and immutable
way, such that, save those changes that clear experience or divine
revelation renders certain and that we believe or perceive to be
made without any change in the Creator, we should suppose no
other [changes] in His works, lest one then argue an inconstancy
in Him. Whence it follows that it is most wholly in accord with
reason that we think on this basis alone that God moved the parts
of matter in various ways when He first created them and that He
now conserves all of this matter clearly in the same way and for
the same reason that He formerly created, and that He also conserves
the same amount [tantundem] of motion in it always.” 7
V – C’est magnifique! Except ...
D – Except what?
V – Except that doesn’t happen.
D – Whatever ... C’est la vie.
V – Descartes!
D – “I will continue resolutely fixed in this belief, and
if indeed by this means it be not in my power to arrive at the
knowledge of truth, I shall at least do what is in my power, viz.,
[suspend my judgment.]” Explain! 8
V – Your straightforward example of two bodies which
possess the same “quantity of motion” – or what you define as the
product of mass times velocity – certainly pleases the senses. It
6 Mahoney, ibid.
7 Mahoney, ibid.
8 Meditations – “I. Of the Things of Which We May Doubt”
What’s the Matter with Descartes
Vance
even works if we limit our investigation of motion to the common
machines of antiquity, e.g. the lever, windlass, pulley, wedge, and
screw – what is often referred to as “mechanics,” or “statics” to
be more precise. But, as the great Monsieur Leibniz would point
out, in all of those machines “there exists an equilibrium, since
the mass of one body is compensated for by the velocity of the
other” and that “the nature of the machine here makes the magnitudes
of the bodies – assuming they are of the same kind – reciprocally
proportional to their velocities, so that the same quantity
of motion is produced on either side.” 9 Therefore, your law of
motion only works by accident.
D – It’s no accident. I didn’t look into any other kind
of machine!
V – You know, René, it wasn’t a pulley that put man on
the moon ...
D – The moon?
V – Anyways, shall I now proceed to show an exception
to your rule using a demonstration from Leibniz?
D – I suppose if you must.
V – Thank you ... First, assume that a body falling from
a given height acquires the same force which is necessary to lift
it back to its original height if its direction were to allow it, and
there was no interference with its motion through friction, air–
resistance, and the like.
D – As for example with an ideal pendulum?
V – Yes, exactly. Second, assume that the same amount
of effort is consumed in raising a four–pound body to a height of
one foot, as is consumed in raising a one–pound body to a height
of four feet.
D – I will generously grant Monsieur Leibniz these two
assumptions, as will any of my pupils.
V – Wonderful! Then a four–pound body falling one
foot has acquired the same amount of force as a one–pound body
falling four feet.
D – Mais oui, for what you say is to me both “clear and
distinct.”
9 Gottfried Leibniz A Brief Demonstration of a Notable Error
of Descartes and Others Concerning a Natural Law, Philosophical
Papers and Letters Translated by Leroy E. Loemker, Kluwer Academic
Publishers, Boston, 1956.
Δυναμις Vol. 3 No. 3 December 2008
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V – Except, the one–pound body acquires only two units
of velocity.
D – What of it?
V – Descartes! Do the math ... shouldn’t it have acquired
by your reasoning four units of velocity? Remember how
evidently, you implied, a body twice as large as another would
travel half as fast, thus always conserving the same quantity of
motion? 10
D – Aidez–moi! Do I have to show my work?
V – Don’t worry about it. I’ll just assume you’ve actually
calculated the results of this or any other physical experiment
for that matter, ok?
D – What a relief! Merci.
V – But remember: Galileo’s proposition, and subsequent
demonstrations show that a height is proportional to the
square of the velocity of an object having fallen from it. Which
means that, by your definition of ‘quantity of motion’, were we
to construct a machine which would allow us to transfer all of the
motion of the four–pound body falling one foot (previously in
motion but now at rest) to the one–pound body (previously at rest
but now set into motion), we would miraculously discover that
such a one–pound body would be able to raise itself to a height
of sixteen feet!
D – What’s wrong with that?
V – Well, earlier you had agreed with Leibniz when
he asserted that as much effort was consumed in raising a four–
pound body to a height of one foot, as was consumed in raising a
one–pound body to height of four feet. Thus, with as much effort
as was consumed in raising a one–pound body to a height of sixteen
feet, we could have raised the four–pound body to a height
of at least four feet. But we previously stated that the four–pound
body in question originally fell from a height of one foot (before
transferring its motion to the one–pound body), which should
have only been enough to raise itself to a similar foot. Where did
all that extra motion come from?
D – Je ne sais pas. But I do know that such results could
be used to produce some pretty interesting machines–
V – Don’t even say it! Perpetual mechanical motion
10 If Descartes’ total “quantity of motion” (mass times velocity)
is to be preserved in this example, then m1 · v1 = m2 · v2 (here:
4 times 1 = 1 times 4), i.e. a body of four pounds with one unit of
velocity would impart four units of velocity to a one pound body.
What’s the Matter with Descartes
Vance
doesn’t exist. This is because, as Leibniz would say, there would
be no reason for it to exist. For if effects could at any time exceed
their causes, of what use would those causes be if not for producing
all of their subsequent effects!
D – How was I supposed to see that?!
V – You weren’t! You don’t see causes, you know them
by their power. And so the very things by which you correctly
understand motion or change (powers), happens to be the name
of a new science of motion founded by Leibniz 11 in opposition to
yours and Newton’s. And so, while you sought to preserve the
same quantity of motion, Monsieur Leibniz sought to preserve
the same quantity of power, i.e. the equality of cause and effect.
D – I see.
V – René, all this “seeing” with your eyes got you into
this trouble in the first place. You unfortunately assumed that
there was nothing to be found in motion, or in corporeal bodies
for that matter, which could not be measured through your senses.
If indeed there was nothing more to it, then quantity of motion
would consist of the only two things one could sense about a
moving body, namely mass and velocity ... I mean, have you ever
seen with your own two eyes actual squares on a velocity?
D – [Non cogito–]
[Note: Having over–extended himself at this point in
the interview, Descartes was no more.]
V – Ergo, it appears you no longer exist, and neither
does the credibility of your methods.
11 i.e. Dynamics
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