december 2008 Vol 3 No. 3

Transcription

ΔΥΝΑΜΙΣ
The Journal of the LaRouche–Riemann Method of Physical Economics
december 2008 Vol 3 No. 3
December 2008
Vol. 3 No. 3
www.seattlelym.com/dynamis
ΔΥΝΑΜΙΣ
The Journal of the LaRouche-Riemann Method of Physical Economics
EDITORS
Peter Martinson
Riana St. Classis
Jason Ross
LAROUCHE YOUTH
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- or [email protected]
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2
From the Editors
3
The Calling of Elliptical Functions
By Michael Kirsch
26
On the Subject of ‘Insight’
By Lyndon H. LaRouche, Jr.
38
Third Demonstration of the Theorem
Concerning the Decomposition of Integral Algebraic
Functions into Real Factors
By Carl Friedrich Gauss
41
Letter from Carl Gauss to Wilhelm Bessel
December 18, 1811
44
The First Integral Calculus
Johann Bernoulli
48
Exclusive Interview: René Descartes
What’s the Matter with Descartes?
By Timothy Vance
On the Cover
Raphael’s
Alba Madonna.
An instant, which
encompasses all eternity.
“...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
2
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-
Δυναμις Vol. 3 No. 3
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
December 2008
3
Kirsch
How a Lemniscate is Not Other than a Riemann Surface
Michael Kirsch
H
ear 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
December 2008
4
Kirsch
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 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
December 2008
5
Kirsch
“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]
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.
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 prob6
From The Geometry of Rene Descartes Translated by Smith
and Latham
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
December 2008
6
Kirsch
Box 2a
Box 2b
Take the hyperbola (top diagram). What
is the equation for finding the simple commensurable points of the curve?
Y = X −1 or Y = X1 So if Y = X1
1
The equation for the parabola
is ob = cb2 .
ob = oa
oa' = ob'
If cb = w and oa = z, then w:2
z::dw:dz
Since z = w , then dz = 2 w
dw, which is the expression
for d(z) = d(w2 ) . The relacb
tionship 2oa
is maintained for
any tangent to the parabola.
Inversely, the constant expression of that relationship
is the parabola.
2
1
then X is to 1 as 1 is to X, or X1 = X1 . 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
dy
−2
in X
terms
of mathematics.
the characteristic triangle of the curve or the dx =
Now, by way of the mathematics of the calculus, we know that the differdy
= X −2 . We also see that the triangle we find
ential of Y = X1 is dy = X −2 dx or dx
geometrically as the tangent has a length of X and a height of X −1 ; which means
1
dy
−2
that the proportion is X1 = X1 . But our calculation says that dx = X
−2
That’s ok, because its actually X
over 1, which we find is a similar tri-
This geometrical differential of change blasts the empiricism
of object-fixated analysts9, 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 di- angle (bottom diagram) to
rectly, but by the quadrature of geometric curves.
1
X
1
=
1
X.
geometric parabolic and hyperbolic curves.
9
Anal-ists
1
December 2008
1
7
Kirsch
Leibniz’s noBox 3
tation for the integration of curves showed
Bernoulli shows that the exponential curve,
that the expression for a transcendental of no degree, can be constructed by
both the quadrature of means of the quadrature of the hyperbola.
the hyperbola and the
Starting with the equation for the curve of conarc length of a circle, stant subtangent(the characteristic of the exponential
as Cusa had indicated curve), dw:dz :: w:a, one sets up the equality a dz = a2
earlier, was indeed dw: w.
transcendental, and
Bernoulli then shows that when the equality of
not able to be inte- this relationship is maintained through all possible valgrated algebraically10; ues of z and w, the exponential curve will be constructhowever,
although ed.
 2
transcendental, the
a dw : w = az .
The integral
To conintegrals of the circle struct the left side of the equation, the z axis of the
2
and hyperbola could hyperbola has the lengths, a . This multiplied by
w
be related to the phys- the infinitesimally small change in w, gives us the area under the hyperbola. The area of the
ical function to which hyperbola grows arithmetically in proportion to the geometric growth of the w axis (as investhey correspond, and
tigated by St. Vincent de Gregoire,1 to
used to construct othwhom Leibniz gives credit as one of his
er transcendentals rethree major inspirations.) The right side
lated to them. Thereof the equation, a dz, is a simple rectanfore, although they
gular growth, whose sum would be az, a
themselves were not
constant times the z azis.
algebraically
solvThis area az maintaining its
able with the Leibniz
equality to the area of the quadrature
method of integration,
of the hyperbola sum of a2 dw : w, will
the functions were
therefore, in arithmetic rectangles along
known, and thus the
the z axis, project geometric growth in
concept of the conthe w axis, which is the exponential
struction was clear.
curve.
In this way
So, the sum of a2 dw : w, the
curves, such as conquadrature of the hyperbola, is the logachoids, could be
rithm of w, that is, the logarithm of the
constructed from an
geometric growth, w.
equivalent area made
up of a hyperbola and 1 See Bill Ferguson’s article, in the Upcoming Issue of Dynamis for more on St. Vincent.
a circle, or, many othing quadrature of one geometric curve on one axis, and the
er transcendental curves, such as the exponential curve, could be
equality of a rectangular area on the other axis, produces two
constructed with two simultaneous actions.
simultaneous actions which construct the relation on which the
Maintaining a relationship between a constantly growtranscendental curve depends. An example of such a construction
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.
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
December 2008
8
Kirsch
Box 4
CONVERTING ARC LENGTH OF CIRCLE TO
RADIANS
1
as = Quadrature of the Circle
2
1
as = Quadrature of the Circle
2
s = Arc length of Circle

vπ
a dx
length
of Circle
√s = Arc =
s=
Arc Cosine
x=
360◦
a2 − x2
Physical Transcendentals
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
υ)a : a2 − cos υ 2 becomes simply dv, thus,
for d(cos

υ = dυ . As an example, if the angle is 60°, then the arc
π
length in radians is 3 .
Box 5a1
Now, for an Equilateral
Hyperbola
√ AC = CH.
√
With this underx+ √x2 −1
AH
=
(HC)
2
AF
=
2
standing of the infinitesimal
√
√1
F H = (IH)
AG
=
calculus, the understanding
2
2

of how to conceptualize AB = a, a = 1
1
dx
Since AF*IF = AG*BG, then ABG = ADE and
√
ABD =
processes, such as Cusa had AC = x √
2−1
2
x
2
thus, ABD = IFBG.2
indicated much earlier, was CD = y = 1 − x


1
1
dw
dx
brought to a higher degree
=
= ABD
IF BG =
2
w
2
x2 − 1
of maturity; however, the

real breakthrough for sci1
1
dw
By Box 3:
= log w
ence came with Leibniz’s
2
w
2
investigation of physiBut, for this axis, since AB is taken as the concal transcendentals. The
stant log √12 must be subtracted.
boundary of this domain of
Thus:3
transcendentals was defined



√
1
1
1
1 w dw
by the investigation of the
=
log w − log √ = log (w 2)
2 1 w
2
2
2
physical principle existing
ontologically outside the
Therefore:
domain of the ‘geometric’


dx
transcendentals.
√
log (x + x2 − 1) =
x2 − 1
This superior
campaign waged by Leib1 For more on the relation between definite integrals
niz in the method of conand logarithms, see Bill Ferguson’s article, which
will appear in a future issue of Δυναμις, for a more
ceptualizing processes, and
generalized study of definite intergrals involving the
capturing them for manhyperbola and logarithms.
kind, necessilarily leads
us to introduce what may
2 Here, log w means the natural logarithm.
seem to some, an unrelated
3 See Box 5b
subject.
Leibniz had taken
bodies which is not contained in extension, for which he is falseit upon himself personally, since leaving Paris in 1676, to wipe
ly praised. With Descartes’ science of ‘extension’, substance (in
out the interrelated Cartesian fraud which pertains to the domain
the sense of Plato, Cusa, and Kepler) was banned from thought.
of physics. It is Descartes’ religious belief, that there is nothing in
December 2008
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9
Kirsch
Box 5b
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
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)
ea = b, a = log b
ec = d, c = log d
log b + log d
= log ea + log ec
= log e(a+c)
= log [(ea )(ec )]
= 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
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 direction15,
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.
December 2008
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Kirsch
Box 6a
Box 6b
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
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.
December 2008
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Kirsch
Box 6c
= Length of Chain
a = Constant Horizontal Tension
S
dx
S
= Tan w =
a
dy
of
‘intrinsic
curvature’ in
his method of
determining
the curvature
of a surface,
the
principle
of the catenary
can only be
discovered
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
relation20 was transformed into one expressing the differential in
terms of the quadrature of the hyperbola.21
S
dx
19
Here, our realationship
a = dy = 1, thus by our above relation
=1, or in otherwords S (length of chain) = a (horizontal tension).
=
20
S
a
dx
dy
21
See Box 7a
S
a
Box 7a
S = Length of Chain
a = Length of Chain equal to Constant Horizontal Tension

Height = S 2 + a2 = x
dx
S
=
a
dy
a dx
a dx
= √
dy =
S
x2 − a2
Since a is equal to 1,
dx
dy = √
x2 − 1
However,
Box 7b
unlike
Bernoulli’s

construction, Leibniz,
dx
employing his knowl√
y=
From Box 7a
x2 − 1
edge of the relation of
the exponential curve

to the quadrature of
y
=
log
(x
+
x2 − 1)
From Box 5
the hyperbola,22 inverted and translated

this particular relation
ey = x + x2 − 1
From Box 3
for y, in the above

diagram, into one for
ey − x = x2 − 1
x, showing the height
ey + e−y
x=
of the catenary to be
2
Thus
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,
= dx
dy
“geometric” transcendentals discussed, thus failing to capture a substantial irony involved in the paradox of physical least action.
25
1
1
It is highly of note that the exponential curves which are de-
December 2008
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Kirsch
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
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: SUPPLEMENTUM GEOMETRIAE DIMENSORIAE SEU GENERALISSIMA OMNIUM TETRAGONISMORUM EFFECTIO PER MOTUM: SIMILITERQUE MULTIPLEX CONSTRUCTIO LINEAE
EX DATA TANGENTIUM CONDITIONE. Translated by Pierre
Beaudry, ftp.ljcentral.net/unpublished/Pierre_Beaudry/.
December 2008
13
Kirsch
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
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 Catalan29; 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
December 2008
14
Kirsch
Box 8a
The Paracentric Isochrone
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,
βα
this distance is equal to Aw. Therefore, as Aw
δα equals the speed of the body at A to the speed at δ, so δα equals the same
βαβα
Ai
Ai
relation.
And, from what was also stated above, βαβα
. Since βαβα + δβδβ = δαδα , this can be coverted to
δαδα = Ai+αγ
δαδα = Ai+αγ
βαβα
Ai . To express this we take into consideration
βαβα
a
simple
geometric
application of the differential calculus.
Ai
=
δαδα = αγ
t, βα =αγdt, yt = az
Let Aγ = x, γα = y, ζ = z, A = a, Aδ =δαδα
t dz
a dz
By Box 4, o = √aa−zz
, thus, βδ = √aa−zz
.
Therefore, our relation above
βαβα
δβδβ
=
At
αγ
, becomes, dt dt :
tt dz dz
aa−zz
=a:
tz
a .
√

dt
√ a dz
√ a dz
2
t
=
=
Rearranged and reduced, this becomes √
.
Integrating,
we
have
. How to conaaz−zzz
aaz−zzz
t
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 compliaaaa du
cated. Through this, he is able to construct the elastic curve, the arc of which AQ, is found to be equal to √aaaa−uuuu
.
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,
words, z = uu
a . By substituting
√ orin other
aaaa
du
this in the equation above for the parcentric isochrone, we are left with at = √aaaa−uuuu .
December 2008
15
Kirsch
However, due to the non-constant ‘law
of tensions’ which the elastic curve obeys, anJacob Bernoulli’s construction other curve would have to be used.35 Jacob found
a saving grace. He writes “as if from a prayer, a
of the Paracentric Isochrone1
curve of four dimensions presents itself” shaped
OA==xx
OA
like “the bow of a French ribbon.” This curve
AB==yy
AB
was the lemniscate, which, by employing it inOA
OA
OA = OA'
stead of the elastic curve, since the expression
OM
1a
OM==√
for their arclengths are identical, he could turn a
OD
=
a
OD = 2
quadrature into a rectification.36 In relating these
Arclength
of
Lemniscate
OA′

2
curves to one another, Jacob came upon an intera dx
√
esting relation between the ellipse, lemniscate,
4
4
a −x
and elastic curve.37
But, as the elastic curve and lemniscate
Applicata
of Elastic Curve AB were themselves only described in terms of the

x2 dx
functions they produced, relating such curves
√
a4 − x4
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
Arc Length
of Ellipse CD

2
2
only posed new paradoxes, leaving as a mystery
(a + x )dx
√
the principle involved in these higher transcena4 − x4
dentals.
One is left asking: what was the prin1 Bernoulli’s original diagram included a hyperbola on either side of the lemniscate.
ciple which created the process of the paracenBy the acceptance of this challenge, physical transcentric isochrone, and its relatives? What was the
dental curves were now opened to be studied. What this signified
process to which these predicates correspond?
was a decisive defeat for the Cartesian method, and also an openSuch was the boundary of this higher transcendental.
But, just as the quadrature of the hyperbola needed to
ing to a new domain.
be ontologically defined by a higher transcendental of a different
Jacob’s solution to “Leibniz’s Curve”
species, the catenary, so the ‘elliptical’ integrals of the lemniscate
and elastic curve, related to the paracentric isochrone, demanded
The differential relationship of the paracentric isochrone
to be defined by a higher domain. What seemingly was limited
led to a form similar to the differential expression for the quadrato physics was in fact an impassable barrier into the realm of the
ture of the circle and hyperbola, i.e., a transcendental one.33 Howprinciples of space.
ever, unlike those, and related others, there was no way to relate
Looking back from the origin which led to the study of
the paracentric isochrone to any known, constructible functions.
‘elliptical’
integrals, as Bernoulli referred to them, it is seen that
In the construction of such a curve, Jacob tried to uncovDescartes’
absurd physics had an ironic effect of rendering the
er the identity of this curve by enticing relatives of the curve, and
investigation
of transcendental physical processes, and the funcseeing if they would, as though spilling the family dirt, help in
tions
related
to
them, impossible. Leibniz’s method of dynamics
identifying its true character. In this way, he found the integral of
Box 8b
the paracentric was directly related to the elastic curve, the curve
of a bent flexible rod, whose integral rises to the fourth degree.34
gungen der ebenen elastischen Kurven Von Jakob Bernoulli (1691,
1694, 1695) und Leonh. Euler (1744) (Leipzig: Verlag von Wilhelm
Engelmann)
as much theirs, as it is mine.” (ftp.ljcentral.net/unpublished/Pierre_
Beaudry/)
35
Jacob Bernoulli, Complete Works 1740, Volume I no. LXIV,
p. 627, G.G.L Construction Propia Problematis De Curva Isochrona
Paracentrica
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
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
34
Jacob had begun studying this curve upon working on the
a growing rectangle equal to it. Rectification is simply a geometric
Catenary chain. His paper was published in 1694, the same year as 1
curve being drawn, as in the case of the lemniscate.
his ‘solution’ to Leibniz curve. See H. Linsenbarth’s 1910 German
translation Abhandlungen ueber das gleichgewicht und die Schwin37
See Box 8b
December 2008
1
16
Kirsch
Box 9
Fagnano’s Discoveries
Complement Principle
Multiple Arc Principle
Arc OZ = 2 Arc OW
when
√
2 OW 1 − OW 4
OZ =
1 + OW 4
Arc OA = – Arc BM
When
√
1 − OA2
OB = √
1 + OA2
The Complement Principle discovery
may very well be connected with the
following:
Let OA = z, AE = y, OE = x
2
Since (xx + yy) = xx − yy
and zz = xx + yy
√
√
z 1 + zz
z 1 − zz
x=
y=
2
2

CM
y
1 − zz
=
=
x
1 + zz
OM
If OM = 1, then CM = OB !
which means that DM is also = OA
Box 10a
Comparison between Euler’s and Gauss’s view of the Lemniscate
Euler
A=

Gauss
Complement Arc Principle
1 − aa
1 + aa
coslemn p =

1 − sinlemn pp
1 + sinlemn pp
Double Arc Principle
Corda Arcus
dupli =
Corda complementi dupli =
Sehne des doppelten Bogens
Cosehne des doppelten Bogens
2aA
1−aaAA
AA−aa
1+aaAA
2sc
1−sscc
cc−ss
1+ccss
Multiple Arc Principle
corda arcus (n + 1) cupli =
corda complementi =
aB + bA
1 − abAB
AB−ab
1−abAB
sinlemn
sinlem (p +
± q) =
(sinlemn p)(coslemn q) ± (sinlemn q)(coslemn p)
1 ∓ (sinlemn p)(sinlemn q)(coslemn p)(coslemn q)
coslemn
coslem (p
(p +
± q)
q) =
(coslemn p)(coslemn q) ∓ (sinlemn q)(sinlemn p)
1 ± (sinlemn p)(sinlemn q)(coslemn p)(coslemn q)
1
December 2008
1
1
1
17
Kirsch
So inspired by
‘Leibniz’s curve’, Fagp = Arc OM = Arc om
nano discovered funq = Arc ON = Arc on
Euler
Gauss
damental principle’s.
First, what is the algebraic relation between
OM = a
the functions related to
ON = b
complementary (oppoOm = A
site) arcs of the curve?
On = B
This involves a geometrical relation with
OM = sinlemn p
the tangent taken at a
OM = a
ON = sinlemn q
unique singularity of
ON = b
Om = coslemn p
the curve, allowing one
Om = A
On = coslemn q
to compare the angles of
On = B
the relevant functions.
Box 10c
Second, as multiple arcs of the lemniscate are traversed,
OM = sinlemn p
what is the algebraic relation between the functions of the
Gauss comparedON
arcs=on
the lemniscate,
with arcs on the circle. As
sinlemn
q
curve? 38
Gauss noted in his
notebook,
the
tangent
of
an
angle
from
the
cenOm = coslemn p
The origins of Fagnano’s discovery are still to
ter of the lemniscate,
theq lemniscatic cosine at that angle.
On =equals
coslemn
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,
thus opened the door to the confrontation with the essential, and
Fagnano showed how to divided the Lemniscate into equal parts,
aggravating problem of dealing with physically related transcensuch as 2,3,5 parts; and generally divisions which fall under 2
dental pathways of four dimensions.
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.
A Great Frustration
These discoveries by Fagnano were crucial, as they made
The one crucial, actual discovery in the study of ellipticlear certain relations between the curve itself, and the functions
cal integrals occurring before the time of Gauss, was made by the
produced by the curve. This at least allowed the shadows, which
Italian, Giulio Carlo Fagnano, who had been studying philosophy
the higher transcendental generated, to form a pattern with which
and theology before devoting himself to the study of geometry,
to work. However, although Leonard Euler generalized the reparticularly, the investigations of the Lemniscate.
lation of multiple arcs to their functions, which became known
He begins his most famous work Metodo per Misuare
as the ‘addition theorem’, the principle of organization was still
La Lemniscata, thus:
Box 10b
“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”.
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
1 1689, and Jacob Bernoulli had investigated dividing it
by Leibniz in
into equal arcs.
December 2008
1
18
Kirsch
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.
Series approximations of the Lemniscate integral
In this spirit, Gauss
by Stirling and Euler, as noted in Carl Gauss’s notebooks
deviated from the prior method
of Fagnano and treated the lemcompletely unknown40; the cause which generated the process,
niscate
problem,
not
as
an
algebraic
relation of functions whose
created a great frustration looming over the cognizant minds who
complementary
arcs
are
arithmetically
related, but as a periodic
attempted the search.
function (what he called the sine and cosine of the lemniscate).42
4. Removing the Training Wheels
With this understanding of the periodic Sine and Cosine
of the lemniscate, Gauss proceeded along a new path, untread.
Carl Gauss’s daily log and notes from January 1797
Gauss, unlike all his earlier predecessors, investigated not how
through fall of 1798, maps out the discovery of a new field of
the ‘elliptical integrals’ could be expressed in terms of their funcscience involving the nature of the Lemniscate. 41
tions, but rather, what is the process such that it defines these
Gauss began with the methods of his predecessors, but
particular predicates.
looked at the same function with completely different eyes. His
To clarify the point: just as the sine and cosine of a circle
notebooks show that he read everything by Euler on the Lemare the trigonometric functions of the arc of the circle, so the sine
niscate related to his elaborations of Fagnano’s discoveries, such
and cosine of the lemniscate, an ‘elliptical’ integral, are the ‘elas his Observationes de Comparatione Arcuum Curuarum Irrecliptical’ functions of the arc of the lemniscate.
tificabilium. He also tackled the many works on determining
Therefore, the question restated in another way is: rathvalues of transcendentals with infinite series. For example, John
er than investigating how the elliptical integral could be defined
Stirling’s De summatione et interpolatione serierum and Leonas a function of its functions, Gauss asked, how can the elliptiard Euler’s, De Miris Proprietatibus Curvae Elasticae, the latter
cal function be defined as a function of elliptical integrals? The
written specifically on finding infinite series for the lemniscate
process is thus investigated not by what it generates, but how it
integral.
generates it, i.e., by looking at the way the function of elliptical
However, he wasn’t interested in these methods because
integrals can express the elliptical function, he asked the questhey were only good for finding numerical values. Unlike Euler,
tion: how does the process, define itself?
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 consid40 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, 509543
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
December 2008
The Calling of Elliptical
Functions
2
19
x −4=0
x2 −Kirsch
4=0
Box 11
Zeroes and Infinities
x
x
x
)
sin x =x x(1 − x )(1 − x )(1 −
3π There)
)(1the
− 2π
)(1 lobe
− π of
sinitxtraverses
= x(1 − one
every time
lemniscate.
3π
2π
π
fore, when x is 0, Π, 2Π,
x 3Π, . x. . , −Π, x−2Π, −3Π, . . . ,
) · · · etc.
)(1
+
)(1
+
·
·
·
(1
+
x
x
x
the lemniscate
function
is2π zero.
3πOur function
) · · · etc.
+ π )(1 +
· · · (1 + )(1
π look2πmuch 3π
should, therefore,
like the sine function:
x
x
x
)
sinlem x =x x(1 − x)(1 − x)(1 −
3Π
sinlem x = x(1 − )(1 − Π )(1 −2Π )
Π
2Π
3Π
x
x
x
·x· · (1 + x)(1 + x)(1 +
) · · · etc.
3Π
· · · (1 + )(1 + Π )(1 +2Π ) · · · etc.
3Π
2Π
Π
Let’s see what Gauss really said:
2
x
x2
x2
sin x =x2x(1 − x2 2)(1 − x2 2)(1 − 2 )
sin x = x(1 − 2 )(1 − π 2 )(1 −4π 2 ) 9π
π
4π
9π
2
2
x
x
x2
·x·2· (1 + x2 2)(1 + x2 2)(1 + 2 ) · · · etc.
· · · (1 + 2 )(1 + π 2 )(1 +4π 2 ) · · · 9π
etc.
π
4π
9π
x4
x4 (1 − π 4 )
π4 )
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
(1 −
its circle, except at those singular points and a few others.
We can begin to develop a mathematical expression of
x2 − 4 = 0
those singular points, in hopes of defining our function
with them.
x
x
x
)
)(1 −
sin x = x(1 − )(1 −
In 1799, Gauss proved beyond a doubt that, since
3π
2π
π
algebra is merely a description of a real physics, an alx
x
x
gebraic equation was always decomposable into a num) · · · etc.
)(1 +
· · · (1 + )(1 +
3π
2π
π
ber of factors equal to the highest degree of the equation.
Each of these factors represent where the equation is
x
x
x
zero. So, the equation x2 − 4 = 0 has two zeroes: at posi)
)(1 −
sinlem x = x(1 − )(1 −
3Π
2Π
Π
tive and negative 2. Therefore, it can also be writtenxas
OK,
x now this looks a bit different than what we
x
x
) analyze it. First,
)(1 − Let’s
sin x = x(1 − )(1 − found.
(x − 2)(x + 2) = 0 .
x−Π
x
x
(1
) , Gauss
instead
of
3π
2π
π
)
·
·
· etc.
)(1
+
)(1
+
·
·
·
(1
+
2
4
4=0 x
3Π
2Π
Π
Every algebraic equation has a finite number2 of x −
x −
x (1 − Π4 ) . Second, if you look closer, Gauss actux 4 = 0has
x
zeroes, and thus factors, but our sine function
is transcenx
x with xdenominator
· · · etc.
)(1 +ally )has
)(1 +
· · · (1 +
written down a ratio,
terms.
) x2
)(1
−
)(1
sin x = x(1
3π
2π
π
xx2 of
x2 −
x −whole
dental, thus having an infinite number of zeroes. Therefore,
x
Still, if thesin
arcxis=
equal
to
some
number
lobes,
Gauss’s
3π
2π
π
)(1 −
x(1
−= x(1
sin x
− 2π2 )(1
)(1 −
− 3π 2) )(1 − 2 )
π
we can attempt to approximate our sine, using an alge4π
function
equals
zero. But,
in the dex
x
x
x factors 9π
xany of the
x what ifπ
braic equation with an infinite number sinlem
of factors.
) · · x·zero?
)(1 − equal
x =Since
x(1 −the)(1 −nominator
)
·
·
·
etc.
)(1
+
)(1
+
(1 + Then,
this
function
expresses
where
x
x
2Π
Π
2π x2) · · ·3π
etc.x2
· ·3Π
· (1 + )(1 +π x2 )(1 +
the
lemniscate
function
is
infinity!
Where
on
the
lemniscate,
2 zeroes are all at integral numbers of half-circumferences
·
·
·
(1
+
)(1
+
)(1
+
)
·
·
·
etc.
3π 2
2π2
π
x −4=0
2
x
π
4π is Gauss
9π 2 doing here?
canxwrite: x
− 4 = 0( 0, π, 2π, 3π, . . . , −π, −2π, −3π, . . . ),· ·we
Nowhere!
What
x
x
x
·4 · · etc.infinite?
)(1 +is its )radius
· (1 + )(1 +
x Π
x
x
)(1
sinlem
= ,x(1
2Π
x4
x is−equal) to the
x − arc)(1
xor −when
(13Π
− πx4sinlem
) when
=x−1
x −
x −
x − )(1
)
)(1
sin x = x(1
4 x(1
3Π
2Π
Π the
Π
)
)(1
−
)(1
−
x
=
−
sin x = x(1 − )(1 −π )(1 −
2π ) 3π
Π
fourth root of negative 1 times
Π.2Π
What is 3Π
the fourth root of
π
2π
3π
2
2
x
x
x
x1?2 (In· other
x
x
xsin x = x(1 − x )(1 − xnegative
) · ·root
· etc.of nega)(1
+
)(1
+the biquadratic
·x
· (1 +words,
x
x
)(1
−
)
x +
x +
· ·x
· (1 + )(1
)(1
) · · · etc.
2Π
2
· 3Π
· etc.
)(1
+ for) ·the
)(1 +Π
·Gauss
·2(1 + would
π2
4π
tive
one.)·9π
also
write,
good old circular
etc.
· · · (1 + )(1 +π )(1 +
2π ) · · ·3π
3Π
2Π
Π
3π
2π
π
sine,2
2
2
Now, whenever the arc
x
to+a xdenominax2
x2 2
x2
x )(1 + x )(1 + x ) · · · etc. sin x = x(1
x· · · (1
x is equal
x have
x and
x −zero,
2
2
x2 − 2 )(1
)(1
− π 2a) zero 4π
)(1
− we
sinlem
xbecome
= x(1
x2 − 2 )(1
x − 2)
9π
tor, that
factor
will
of
π
) 3Π
−
sinlem x = x(1 − )(1 −Π )(1 2Π
sin x = x(1 − 2 )(1 −
)(14π
− 2 ) 9π
3Π
2Π
Π some
π
4π 2
9π
the function. Try it out(1for
non-singular
point,
such
x4
)x ◦
2
2
x
x − π4=
x
x
x2
2π
radians
360
).
Our
function
gives
as x =1 radian
(
x +
x +
) · · · etc.
)(1
· · ·x(1 + )(1
2 +
2 +
2 +
·
·
·
(1
)(1
)(1
) · · · etc.
x
x
x
2
2
2
)
·
·
·
etc.
)(1
+
)(1
+
1 · · · (1 +
1
1
1
1
1
3Π
Π ) · · · (12Π
π
1(1 − π )(1 − 2π )(1
+ 3Π
· · · (1 + 2 )(1 +
)(14π
+ 2 ) · ·9π
· etc.
Π − 3π2Π
π )(1 + 2π )(1 + 3π ) · · ·
2
π
4π
9π
1
etc., which will be, approximately, 0.866054044. The ac- (1 − x4 ) In this function,
1
which
actually
gives a much better
4
4
2
2
2
π
x 2like 0.8414709848,
x
x
2
tual sine of one
which
(1 − πx4 )
2
x2 is−more
x
x
sinradian
x = x(1
)(1
−
)(1
−
)
approximation
to
the
sine
function,
the
first
factor (1 + πx2 )
sinclose.
x = x(1 − 2 )(1 −
−2 2 ) 9π 2
π 2 2 )(14π
is pretty
π
4π
9π
equals zero when x equals positive or negative π times
Gauss applies2 this reasoning
to2 the lemniscate
2
1 the
the square root of negative one. Where is that zero on
x
x
x
2lemniscate
2 is the radius of the
function. If the
x2 +function
x
· ·x
· (1
+ 2 )(1
)(1
+
)
·
·
·
etc.
2
circle? Notice, also, that the circle has no infinity points,
· · · (1 +
)(1 +
)(14π
+2 2arclength
) · ·9π
· etc.
π
lemniscate,
and
of the lemπ 2its variable
4π 2 is the
9π
4
or poles, but the lemniscate does. Where are those poles?
x
4
) x, then we can find its zeroes. The radius is zero
(1 −niscate
1 − πx4 ) π4
December 2008
20
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!
December 2008
21
Kirsch
One asks, “But how “On the lemniscate we have found out the most elegant in beauty and roundness and
can these functions be defined things exceeding all expectations and that by methods in a moment all truths, which
by the process, if the process
otherwise would be universally
which open up to us a whole new field ahead.”
is the principle you do not yet
valid, are necessarily yoked to
– Gauss 1798
know and for which you are
the most cumbersome limitasearching?” That is exactly the
tions.” [emphasis added]44
difficulty. In this respect, one might exclaim, looking back at the
earlier calculus, “Ah, training wheels!”
44
This lack of universality arising as a consequence of dismissing ‘imaginary’ values, is not limited to higher analysis. Gauss demTo accomplish the fullfillment of this method, Gauss
onstrates two years later in his 1799 doctoral dissertation, that equageneralized analysis for the first time. He took into consideration,
tions have imaginary roots, and likewise, that no general theorem
the arc of the lemniscate which was not visible on the geometrical
can be stated about algebraic magnitudes, without their physical
lemniscate: a ‘complex’ one.43 Gauss, knowing the crime which
significance understood. This statement is also reflected in one who
Euler perpetrated in this regard, wrote that, by the neglect of
looks back upon the study of quadratic residues from the standpoint
imaginary magnitudes, the field of analysis “forfeits enormously
of biquadratic residues;what may seem as a paradox, such as -1 be43
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]
ing 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
December 2008
22
Kirsch
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 cirThe value of the angle which produces
cle, and the exponential functions. Using
a given sine on a circle, is ambiguous
Derivative of the Arc Length
this, he continued to investigate the nature
from the standpoint of the sine.
1
ds
of the lemniscatic function as expressable
=√
dz
1 − xx
through the infinity and zero points, but in
ways which revealed deeper truths.
x ∞ 0 1 −1 √2
−2
√
etc . . .
ds
Applying this new found relation,
0 1 ∞ ∞
3i
3i
dz
he investigated which values of the sine of
the circle make the sine of the lemniscate
Two values of x for every one value
ds
equal to zero, and which make it equal to
of dz
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.
Another way to think of it:
About the implications of this, he
Two values for x, for each value of y
writes
of
this in late July 1798 in his day
y = xx
√
book:
“On
the lemniscate we have found
y=x
out the most elegant things exceeding all expectations and that by methods which open
So this was true indeed for the higher functions of the
up to us a whole new field ahead.”
4th degree. Without this, their nature, and a new field, was not
What was involved with this, was his study of the interepossible.
lated properties of the numerator and denominator themselves.48
Gauss examined the quotient of periodic functions from
this standpoint and determined which values of the periodic funcA Priest’s Calling
tions make the numerator equal to zero, giving a value of zero for
Gauss investigated the domain in which the nature of the
the elliptical function , and which values of the functions make
process
generating
elliptical functions exists. He investigated the
the value of zero for the denominator, giving a value of infinity
characteristics
which
uniquely define elliptical functions. Howfor the elliptical function.
ever,
Gauss
never
made
clear in his lifetime how he conceptualWithout expressing the sine of a real arc plus an ‘imagiized
the
principle
involved
in these higher functions, and thereby,
nary’ arc, the infinty points were not representable, indeed they
did
not
bring
their
potential
fully into access for use by the human
could not be found; nor was the characteristic that the elliptical
mind
in
conceptualizing
the
universe, and its processes.
45
function is found to be periodic in two different ways.
Bernhard
Riemann,
who like Kepler and Leibniz before
Once Gauss obtained the knowledge of the zero and
him,
had
been
in
training
to
be
a priest, inverts the problem altoinfinity values of the of the elliptical function, he could then
gether,
defining
ontologically
what
Gauss never stated.
(employing some of his earlier inductive researches related to
In
Riemann’s
lectures
on
Elliptical
Functions, he begins
46
Lagranges algebraic method of inversion ) represent them, first,
with
the
characteristic
that
elliptical
functions
are doubly periby means of a quotient of infinite products, and, subsequently,
by a quotient of power series, leading to a basis for investigating
48
This study of the numerator and denominator of the quotient
further into the nature of the lemniscatic function.47 For example,
expression of the elliptical function led to various paths. In 1799,
Gauss finds that the logarithm of the numerical value he approxiGauss’s first entries about his use of the arithmetic-geometric mean
π
mates for the denominator is equal to 2 , which he says “is most
arise out of this research of the relations between the numerator and
remarkable, and a proof of which promises the most serious indenominator of the elliptical function. Also, the development of his
crease in analysis” (See Figure 1)
study of the numerator and denominator of the elliptical function
Box 13
A Simple example of Multi-Valued
functions
45
See Appendix
46
See Box 12
47
For a related expression of infinite products see Box 11.
1
1
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.
December 2008
23
Kirsch
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 doubly-periodic function, from Bernhard Riemann’s
the characteristic of double periodicity---it is then clear
that
the former method of dealing with geometric curves,
lectures on Elliptical Functions
which for the most part looked at the characteristics of the
odic. From this fact, he then derives what the consequent chareffects of the process, was entirely inferior.
acteristics would have to be, representing the zero and infinity
Looking back from Riemann’s conceptual discovery,
points on a complex plane tiled with parallograms, whose sides
it becomes clear that with the correct understanding of higher
represent the two periods which were originally set forth. From
physical functions, all of sense perceptible geometry exists as a
these characteristics alone, the total of all the possible values
special case, a projection of some higher function which can be
of the function to which a single parallelogram on the complex
conceptualized but not sense perceptibly represented.
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 Branch Cuts, from Riemann’s lectures on Elliptical Functions
of the function, is not a concept which allows itself to be visually represented in sense-perceptible, three-dimensional space;
Ah! But the question remains:
this defines the nature of this class of higher transcendentals, to
If the catenary was the necessary physical process which
which the lemniscate and all of its related functions belong as
defined the lower geometric transcendentals ontologically, and
special cases, as existing in such a domain, outside of sense perthe multiply-connected Riemann surface is the concept which deceptible physical space, conceivable in the intellect alone.49
fines the geometric lemniscate, to what physical process does this
concept correspond?
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
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.
December 2008
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Kirsch
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.
1
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.
1
1 9
11 13
φ −i
φ + ···
sinlem iφ = iφ − i φ5 + i
10
120
15600
1
1 9
11 13
φ −
φ + · · ·)
= i(φ − φ5 +
10
120
15600
= i sinlem φ
How about for the coslem?
1
3
7
coslem iφ = 1 − i2 φ2 + i4 φ4 − i6 φ6 + i8 φ8 − · · ·
2
10
40
1
3
7
= 1 + φ2 + φ4 + φ6 + φ8 + · · · = ?
2
10
40
Well, we know how long the coslem is in terms of the
sinlem, so,

1 − (sinlem iφ)2
coslem iφ =
1 + (sinlem iφ)2

1 + (sinlem φ)2
=
1 − (sinlem φ)2
=
1
coslem φ
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,
(sl t)(cl ui) + (sl ui)(cl t)
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) cl1u
sinlem (t + ui) =
=
(sl t) + i(sl u)(cl t)(cl u)
(cl u) − i(sl t)(sl u)(cl t)
Now, our lemniscatic sine of a complex arc is expressed
as the lemniscatic sines and cosines of purely real arcs. Let’s put
1
in some values, to see what happens.
We know the minimum and maximum values of the lem1
December 2008
1
1
1
1
25
Kirsch
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
0+i·1·0·1
=0
1−i·0·1·1
for t = one half Π
1+i·0·0·1
=∞
0−i·1·1·0
for t = Π
0
0 + i · −1 · 0 · 1
= =?
0 − i · 0 · 1 · −1
0
for t = three halves Π
−1 + i · 0 · 0 · 1
= −∞
0 − i · −1 · −1 · 0
for t = 2Π
0+i·1·0·1
=0
1−i·0·1·1
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!
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
3
0
Π
Π
2Π
2
2Π
0
0
0
0
0
0
1
Π
1
∞
−1
∞
1
2
t
0
0
Π
0
?
0
?
0
0
0
3
−1 −∞ 1 −∞ −1
2Π
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-
December 2008
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26
LaRouche
Science in its Essence:
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.
A
s 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
See LaRouche, H. G. Wells’ ‘Mein Kampf’: Sir Cedric Cesspool’s Empire, 2008 http://www.larouchepac.com/
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
node/10610
3
Wolfgang Köhler: please forgive me; it was necessary!
December 2008
27
LaRouche
Investigating the shape of space with
Youth Movement (LYM) have preceded that association’s presently approaching treatment of the implications of Riemann’s
1854 dissertation.
The first of those three had been based on a West Coast
team, which had worked through some crucial features of the ancient origins of modern European science, as located in the related work of the Pythagoreans, Plato, and the modern reflection
of this treatment of dynamics in the work of Leibniz.
A second team had worked through the main features of
the founding of modern European science by Cardinal Nicholas
of Cusa’s and by Leonardo da Vinci’s follower, Johannes Kepler.
The LYM’s thorough-going, published report on the Kepler project, is a uniquely competent treatment, as similarly expressed in
the work of Albert Einstein, as by relevant others, but is not competently taught in known university programs otherwise available
today.
In the second study, that of the uniquely original discovery of gravitation, by Kepler, the difficulty, highly relevant to
the matter of insight, is that secondary sources on Kepler’s work
have been (see http://www.wlym.com/~animations), chiefly, viciously fraudulent evasions of the actual development of Kepler’s
original and crucial discovery of a principle of Solar gravitation,
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 TwentiethCentury 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
domain of insight properly defined.
Kepler
This frankly original approach to the study of Gauss’s work,
has produced some uniquely useful findings, findings which provide a uniquely original approach to taking up the unique revolution effected by Bernhard Riemann, from the point of his 1854
habilitation dissertation, the change which launched the Riemann
revolution in science, through those challenges which Riemann
posed to such among his successors as the Italy school of Betti
and Beltrami.
To explain the significance of those listed, four initial
stages of work for understanding human scientific creativity in
general, I proceed now with reference to the relevant implications of what I define, once more, ontologically, as the principle
of insight.
This will clear the pathway for the study of the uncompleted projects of Riemann, as the case is only illustrated by the
work of Betti and Beltrami, as by the challenges posed by V.I.
Vernadsky and Albert Einstein, later. Here, comprehension demands the more precise treatment of the notion of insight which
is included in the following pages.
The importance of treating that subject in this fashion
here, is to be located, in significant part, in the fact that the third
in a continuing series of science projects conducted by teams
from the LYM is nearing the point at which the team’s study of
December 2008
28
LaRouche
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.
T
1. Man as Man, or Beast?
he 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
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 AngloDutch 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.
December 2008
29
LaRouche
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.”
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 EighteenthCentury 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.
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LaRouche
Carl Friedrich Gauss (1777-1855)
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-
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,
December 2008
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LaRouche
T
2. The Infinitesimal
hat 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.
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
11
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.
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.
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LaRouche
The case of the settlement
account, banned the knowledge
of Sumer and its culture, from the
of science (e.g., “fire”) from the
sea, by a non-Semitic people’s
minds of those mortal men and
sea-going, Indian Ocean culture’s
women such as Lycurgan Sparta’s
colonizing of southerly Mesopohelots, the lower, subjugated social
tamia, is indicative.13 In any case,
classes.
the very idea of science would have
It is to be emphasized now,
no secured basis in knowledge unas we contemplate the global wave
less very long spans of ocean-going
of mass-starvation which has been
maritime cultures were taken into
caused by the spread of the massaccount for crucially relevant feamurderous, neo-Malthusian model
tures of ancient calendars.
of that British lackey otherwise
In short, the notion of
known as former Vice-President
universal, which does not exist as
Al Gore, that virtually all of the
a functional conception in Liberalgreat crises of known civilizations
ism, is the essence of any compehave been the result of those same
tent effort at developing actual scipolicies of practice which are fairly
identified as pro-Satanic attempts
entific knowledge. Only long-rangto ban scientific knowledge and its
ing ancient maritime cultures could
practice from the great majority of
have been impelled to produce the
the world’s human populations.14
elementary considerations underlying the Sphaerics from which all of
Such has been the accelerating
competent strains in European, or
decline of the physical economy
other science has been derived. The
of the U.S.A., per capita and per
idea of a universal physical princisquare kilometer, since the terrible
ple, on which all competent science
developments and aftermath of
is premised, could not come into
1968.15
existence for mankind in any other
The upshot of that line of
way, unless we were to presume
inquiry, is that we exist within a
the source of this opinion to be,
stellar universe which is governed
arbitrarily, colonists arriving from
by what Albert Einstein, for examJohannes Kepler (1571-1630)
“outer space.” I emphasize, that the
ple, emphasized as being univerdiscoverer of Universal Gravitation
true concept of universal, does not
sal physical principles of change.
actually exist as a scientific concep14
Former U.S. Vice-President Al Gore, a British agent against
tion within the bounds of empiricism or its spin-offs.
the
U.S.A.’s
American System of political-economy, who walks in
What we know with certainty, respecting contrary views
the footsteps of the de facto traitor to the U.S.A., and sometime U.S.
on the possibility of the existence of a practice of science, is that
Vice-President Aaron Burr, is a typical advocate of the “oligarchical
the contrary views are all either implicitly “malthusian,” or are
model.” President Andrew Jackson of “Trail of Tears” notoriety, had
products of a type of culture, such as the typical “oligarchical
been an accomplice of Burr’s anti-U.S. conspiracy, and had served
model,” congruent with malthusianism. I emphasize, that all
as U.S. President as a lackey and accomplice of Land-Bank swindler
such latter types known to us generally now, belong to a category
and later U.S. President Martin van Buren.
known to ancient through modern European cultures as “the oligarchical model,” a model to be recognized as being congruent
15
It is not merely the actions of the trans-Atlantic “sixty-eightwith Aeschylus’ representation of the Satanic-like figure of that
ers” and the U.S. Richard Nixon Administration which have caused
Delphic Olympian Zeus. This was the Zeus, who, in Aeschylus’
the pattern of accelerating physical decline of the economies of the
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).
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.
December 2008
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LaRouche
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.
Cupola of Santa Maria del Fiore, in Florence, Italy
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
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-
December 2008
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LaRouche
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.
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.
A
3. Insight Reviewed
t 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.
December 2008
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LaRouche
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
The author, presenting an
imperial power in
the world today
(since the 19711972 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
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.)
December 2008
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LaRouche
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
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
December 2008
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LaRouche
background, 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 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 VicePresident-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, ThatcherMitterrand “conditionalities” imposed upon Germany.
December 2008
m−1
X m−3
= xm
+M,
Bxm−2
+ m−2
Cxm−3
+ m−3
... + +
Lx...++M,
m−1
X = xm + Axm−1 + Bxm−2 + Cx
++...
+
Lx
X
=Ax
xmm−1
++
Ax
+ Bx
+ Cx
Lx + M,
m
m−2
m−3
m
m−1
m−2
m−3
m
m−1
m−2
m−3
X
=
x
+
Ax
+
Bx
+
Cx
+
...
+
Lx
+ M,
x + Ax
+ Bx
+ ... + Lx + M,
X = x + Ax
+ Bx X+=Cx
+ ... + Lx
+ M, + Cx
ϕ
38
ϕ
ϕ
Third Demonstration
ϕ
Gauss
ϕ
ϕ
ϕ
rm cos mϕ + Arm−1 cos (m − 1)ϕ + Brm−2 cos (m − 2)ϕ
Third Demonstration of the Theorem Concerning the
m−3
m−2+ Ar
cos
(m cos
−cos
1)ϕ
cos
−cos
2)ϕ
r +cos
m−1
+
Cr
(m+
3)ϕ++Br
... m−2
+(m
Lrcos
ϕ+
Brmϕ
cos
(m+
−Ar
2)ϕ
rm cos mϕ + Arm−1 cos (m − 1)ϕ
mϕ
(m
−−Br
1)ϕ
(m
−M
2)ϕ= t
rm cos
m
m−1
m−2
m r cos mϕ
m−1
m−2
m−3 cos m−1
m
m−1
m−2
+
Ar
(m
−
1)ϕ
+
Br
cos
(m
−
2)ϕ
m
m−2
m−3
cos+
mϕ
−
1)ϕ
+
Br+
cos
−ϕLr
2)ϕ
r 1)ϕ
Cr
cos
−
3)ϕ
+
Lr
cos
+sin
M(m
m−3
Ar
cos (m −
Br
cos
(m
−(m
2)ϕ
r cos mϕ + Cr
sin
mϕ
+
Ar
sin
(m
−...
1)ϕ
+(m
Br
3)ϕ
... +
+rAr
Lr+
cos
ϕcos
+
M
=(m
t cos
+
Cr
(m
−
3)ϕ
+
...
+
cos
ϕ=+t−M2)ϕ
=t
m−3
m−3
m
m−1 cos m−3
m−2
m−3
+
Cr
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
+
M
==
t u
m
m−1
m−2
Cr
cos
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
+
M
=
t
r+
mϕ
+
Ar
sin
(m
−
1)ϕ
+
Br
sin
(m
−
2)ϕ
m
m−1
m−2
Cr
cos(m
(m −
− 1)ϕ
3)ϕ
+sin
... +
+
Lr
cos
ϕ
+
M
=
t
+
Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
r sin mϕ + Ar
sin
Br
sin
(m
−
2)ϕ
r sin mϕ
+ Ar
sin (m − 1)ϕ
+ Br
sin (m − 2)ϕ
m
m−1
m−2
m−1
m−2
m−3 sin (m − 1)ϕ
m−2
m
m−1
m−2
r
sin
mϕ
+
Ar
+
Br
sin
(m
−
m−3
rm1)ϕ
sin
mϕ
+
Ar
sin
(m
−
1)ϕ
+
Br
sin
(m
−
2)ϕ
+
Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u2)ϕ
m−3
rm sin mϕ + Cr
Arm−1
sin (m −
+
Br
sin
(m
−
2)ϕ
mr
cos
mϕ
+
(m
−
1)Ar
cos
(m
−
1)ϕ
+
(m
cos (m
3)ϕ ... Lr sin ϕ =
+ uCr
sin (m − 3)ϕ + ... + Lr sin
ϕ−=2)Br
u
m−3
m−3
m
m−1
m−2
m−3
m−3
+
Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
+
Cr
(m
−
+m−1
...
Lr
sin+
ϕ
=
u
mr(m+cos
mϕ
+sin
(m
−=
1)Ar
cos
(m+
−cos
1)ϕ
−++
2)Br
cosm−2
(m
−t2)ϕ
m
Cr − 1)Ar
sin m−1
(m −cos
3)ϕ
Lr
ϕ
u(m
+
(mm−2
−3)ϕ
3)Cr
(m
−(m
3)ϕ
... +
Lr
ϕ =cos
mrm cos mϕ + (m
−...mr
1)ϕ
+
(m
−sin
2)Br
cos
(m
−cos
2)ϕ
cos
mϕ
+
−
1)Ar
(m
−
1)ϕ
(m
−
2)Br
(m
m
m−1
m−2
mmr cos mϕ
m−1m−2
m−2
m−3 cos
 m−2
m+
m−1
(m
−
1)Ar
(m
−
1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
m−3
 m−3
mrcos
cos
+
(m
−
1)Ar
cos
(m
−
1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
+
(m
−
3)Cr
cos
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
=
t

mrm cos mϕ + (m − 3)Cr
1)Arm−1
(mmϕ
− 1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
mr sin
+cos
(mϕ
−=
1)Ar
sin (m
(m −
− 3)ϕ
1)ϕ +
+ ...
(m+−Lr
2)Br
3)ϕ
... mϕ
+ Lr
t
+
(m
−
3)Cr
cos
cos ϕ = sin
t (m
m−3
Carl F. Gauss
m−3
 = t
m−1 cos
m−2
m−3 mrm sin +
 −
+
(m
−
3)Cr
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
m−3

m
m−1
m−2
(m
−
3)Cr
cos
(m
3)ϕ
+
...
+
Lr
cos
ϕ
=
t
mϕ
+
(m
−
1)Ar
sin
(m
−
1)ϕ
+
(m
−
2)Br
sin
(m
−
m
m−1
m−2
+
(m
−
3)Cr
cos
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
=
t
+(m
(m−
−1)Ar
3)Cr
sin
(m−−1)ϕ
3)ϕ++(m
... +
Lr
u2)ϕ
mr sin mϕ
1)Ar
sin (m − mr
1)ϕ +sin
(mmϕ
− 2)Br
sin (m −sin
2)ϕ
+
(m
−
2)Br ϕ =sin
(m
m
m−1
m−2
m
m−1
m−2
m−3

m four proofs of the Fundamental
m−1
m−2
mr
sin
mϕ
+
(m
−
1)Ar
sin
(m
−
1)ϕ
+
(m
−
2)Br
sin
(m
−
2)ϕ
2
m
2 −
m−1
2−
m−2
This, the third of Carl Gauss’s
m−3
mr
sin
mϕ
+
(m
−
1)Ar
sin
(m
1)ϕ
+
(m
−
2)Br
sin
(m
2)ϕ
+
(m
−
3)Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
m−3

mr sin mϕ + (m − 3)Cr
1)Ar
sin (m −m1)ϕ
(m
2)Br
sin
(m −sin
2)ϕ
r +cos
+sin
(mϕ−
−=3)Cr
1)
cos
1)ϕ
− 2)
3)ϕ
... mϕ
+−Lr
u Ar
+
(m
(m(m
−−
3)ϕ
++
...(m
+ Lr
sin Br
ϕ = u cos
Theorem of Algebra, appeared in 1816.
Here, Gauss makes a2 rare
interm−3
2 m−3
m−1
2 ϕ =
m−2
m−3
2cos
m−3
+
(m
−
3)Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
u
2 m
m−1m2 rm cos2+
m−2
(m
−
3)Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
mϕ
+
(m
−
1)
Ar
(m
−
1)ϕ
+
(m
−
2)
Br
cos
−
3)Cr
sin
...mϕ
+ Lr
ϕ−
+
(m
−=
3)u2 Ar
Crm−1 cos
cos
(m−−1)ϕ
3)ϕ++(m
... +
Lr
ϕ(m
= cos
t 2)(
m r cos mϕ
+ (m
− 1)
ArMervyn
cos(m
(m−
−3)ϕ
1)ϕ
2)sin
Br
2)ϕ
m
rm+
cos
+
(m
1)
(m
−m−2
2)2cos
Brm−2
ventiaon for a physical basis of mathematics.
It was
translated
by
2 m
22
m−1
2
2
m
2
m−1
2
m−2
m−3

2 m
m−1
2 (m
2cos
m−1
2(m
m−2
r (m
cos
mϕ
1)
Ar
(m
−
1)ϕ
2)(m
Br−ϕcos
(m
− 2)+
2
m−3
 m−3
rm
cos
mϕ
−(m
1)+−
cos
−
1)ϕ
+
(m+
−(m
2)+−Br
+
(m
−Ar
3)
Cr
(m
−
3)ϕ
+
...
Lr
=cos
tcos
r cos
mϕ + (m
1)further
Arm
cos
−
1)ϕ
2)(m
Br
2)ϕ
Fansler, who offers this disclaimer:m
“Since
my German
is much
along
m+
rm +
sin
+
(m
−
1)
Ar
sin
(m
−
1)ϕ
+
2)
Brϕ−
(
− 3)
Cr
3)ϕ
...mϕ
Lr
cos
ϕm−2
=2(m
tcos
+
−
3)
Cr
cos
(m
−
3)ϕ
+
...cos
+ Lr
=2)ϕ
tsin
22
m−3

2
m−3

2
m
m−1
2
m−2
m−3
 −
than my Latin, this translation derives
of
the
2
m−3

+
(m
−
3)
Cr
cos
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
=
t
2 mfrom a German translation
2
m−1
2
m−2
+
(m
−
3)
Cr
cos
(m
3)ϕ
+
...
+
Lr
cos
ϕ
=
t
m
mϕ
+ (m
(m
−+
1)(m
Ar
1)ϕ
−++
2)(m
sinϕ
(m
m−2
3) Ar
Cr
cosr(m
(msin
−21)ϕ
3)ϕ
...
+
Lr
cos
ϕ1)
=2 Ar
tsin
+
(m
−
3)
Crm−1
sin
(m+
3)ϕ
...Br
+
Lr
=−
u
,(
m r sin mϕ + (m − 1)
sin
−
+
−
2)
Br
(m −
2)ϕ
m
rm sin
mϕ
−
sin
(m
−−(m
1)ϕ
−m−2
2)2sin
Br
sin2)
original published by E. Nette in Ostwald’s
Klassiker der2 exakten
22
m−1
2
2 m
m 2 rm sin mϕ + (m −
2 1)
m−1
2 2)
m−2
m−3
 (m
2 m
m−1
2
m−2
2
2



2

2
Ar
sin
(m
−
1)ϕ
+
(m
−
Br
sin
−
2)ϕ
m−3

sin
mϕ
(m+−(m
−
1)ϕ
(m
2)+
sin
(m
3)(m
Cr
3)ϕ
+
Lr
ϕLr
=+
uuuϕ
,−)=2)ϕ
m−3
+
uϕsin
)(tt
uu
)+
+(m
(tu
−
utBr
) ...sin
−+(tt
(t2)
m r Latin
sin mϕ
1) Cr
Armin rthesin
(m
−+
1)ϕ
Br
sin,+
(m
−
2)ϕ
+ (m
− appeared
3)
3)ϕ
...1)
+−Ar
Lr
sin
=2(m
uCr
Wissenschaften, B. 14. The original
version,
which
+
−
3)
sin
−−...
3)ϕ
+
sin
u ,
2  m−3

 = y.
2 ++
2 )2+ Lr
223)m−3
2(m −
r(t
 u
2...
 sinTϕT=
2 +
2  m−3

+
Cr
sin
(m
−
3)ϕ
u
,
2
2


2

2
U
U
+
u
)(tt
+
uu
)
+
(tu
−
ut
)
−
(tt
+
uu
)
(t
+
(m+
Cr
3)ϕ
Lrutsin
Commentationes recentiores... and is the version
in+Gauss’s
2+sin
 + ... +
 2ϕ = u ,
+ (m
3) Cr
3)ϕ
+
Lr
= (m
u +
,−uu
+published
u− )(tt
uu sin
) +(m
(tu−−
ut
) −...
−3)
(tt(t
))(tt
(t
+uu
uϕ2sin
) + (tu −
) − (tt +=uuy. )2
=
2
2
 
 y.2
 2 22  2 
  2  2
= y.
2 + u 2 )22 (t
Gesammelte Werke, was consulted as a reference. (t2 + u2 )(tt + uur(t
u
)(tt
+
)
+
(tu
−
ut
)
−
(tt
+)uu )
 u
2 2+
 + uu
 )uu
2+ (tu
r(t
+
u
)
2
2
2
+
)(tt
−
ut
)
−
(tt
+
uu
(t
) + (tu − ut ) − (tt + uu )
r(t + u )
= y.
r
=
R
=
y.
=
y.
2
2
2
r(t2 r(t
+ u2+
)2u )
r(t2 + u2 )2
fter the previous treatise was already printed, continued
meditations upon the same subject led me to a new proof
The factor r can evidently be canceled out of the deof the theorem, which, similar to the preceding, is purely
nominator of the last formula, since t’, u’, t’’, u’’ are divisible
analytical, yet is based upon entirely different principles, and,
by it. Finally, let r be a determined positive magnitude, which,
with respect to simplicity, appears by far to be superior to the forthough indeed arbitrary, should still exceed the highest of the
mer. To this third proof are the following pages now devoted.
magnitudes



√ 2
√ 3
√ 4
√
1.
mA 2, mB 2, mC 2, mD 2, ...;
m
m−1
m−2
Decomposition of Integral Algebraic
Functions into Real Factors
A
for these magnitudes the signs of A, B, C, D, ... should be neglected; that is, any negative signs occurring should be changed
given:
m−1m−1
m−2m−2
m−3m−3
m

m
m−1
m−2
m−3
X =Xxm
+
Ax
+m−1
Bx
+m−2
Cx
+m−3
...++...Lx
+
M,
+ uu 
m
=
x
+
Ax
+
Bx
+
Cx
+
Lx
+
M,
into positive. Following this
that tt
 
preparation,
 I claim
X = x + Ax
+ Bx
+ Cx
+ ... + Lx + M,
√
√ 4
√
√
√
√
√
2
3
2
3
4
m m
m−1
m−2
m−3
assuredlyobtains a positive
setmC
= R, 2,
which
also 2,
gives
m−1
X = x=m−3
+ Ax
+ Bx
+ Cx
+ ...
+
+ M,
1r2,
mB
2,
mC
mA
−1
mBif we
2,
mD
...; 2, mD
mA
2, value


+
Ax
+m−1
Bxm−2
+m−2
Cxm−3
+m−3
...√Lx
++Lx
+
M,



m
◦
m−1
◦
m−2
◦


+ Bxm−2 +X
Cx xX +
++
LxAx
+
M,



=...
xm
+ Bx
+ Cx
...
+
Lx
+
M,
√
√
√
ϕ .(45√√
cos
45
+
AR
cos
+ ϕ) + BR
cos (45
+4√
2ϕ) 3√ √ 4
2 √ R
3 √
4 √ to
a (real)
value
√ 3√
2
3
4
2√
m−2
−1
mB
2,
mC
mD
...; 2,2,...;2 mB1 mA
2,
xm +
+ ...
Ax+m−1
+ Cxm−3 mA
+ ... +
Lx
+2,M,
1 2, 2,
mB
mC
2, 2,
mD
mA
+ Bxm−2 +X
Cx=m−3
Lx + Bx
M,
mB
2, mC
2,
2,
mC
2,
mD
2, ...; 2, mD
mA
1
m−3
◦
◦




Demonstration.
Setting
...
+
CR4 √ cos
(45√+ 3ϕ)
+
+√
LR 
cos (45√ + 
(m − 1)ϕ)
ϕ whose
√
√
√
ϕ ϕ coefficients A, B, C,... are determined real √magnitudes.
2
3
4
2√
3√
1
√ + M cos
√
3
4
follows:
mB 1mC
2,◦ 2,
mCitm−1
2, mD
2, ...;2,12 mB
mA
2,set-m2,
mB
mD
2,
...;
mA 2,
Considering r, ϕ to be other indeterminate magnitudes
and
◦
m−2
◦
2,
mC
2,
mD
2, ...;
mA





+√ϕ) + BR √ sin (45√+ 2ϕ)
ϕ ϕ
√sin (45
√R 3sin 45√+ AR
√
√ 2
2
3
4
4
tingϕ
mB
2, +mC
2, sin
mD
mA
2, ...;
m−3 
◦
2, mC +
2, mD
2,
mA 2, mB 

(45
+ 3ϕ)
... +m−1
LR
(45◦2,+...;
(m −
1)ϕ)m−2
+ M sin
ϕ
√ m3 CR √◦ sin
√ 2
m√
◦ ◦
m−1
m−2◦
4
R
cos
45
+
AR
cos
(45 cos
+m−2
ϕ)
BR
R
cos
45
+
AR
cos
(45
+
ϕ)
+
BR
(45+◦ +
2ϕ)◦ cos (4
mB
2,
mC
2,
mD
2,
...;
mA
2,
m m
m−1m−1
m−2m−2
m
◦
m−1
◦
m
m−1
m−2
+ Ar+ Ar cos
(m
−
+ Br
cos
(m
−
r cos
mR
45 m−2
+ (m m−2
− 1)AR
cos (45 ◦+ ϕ) m−3
+ (m − 2)BR
cos (45 + 2ϕ)
cos
coscos
(m1)ϕ
−
1)ϕ
+ BrBr
coscos
(m2)ϕ
−−
2)ϕ
r rmϕ
m mϕ
m−2
m−
◦ 2)ϕ cos
◦
m−3
m ◦+
◦ m−1
m−1
cos mϕ
+ Arm−1
(m
1)ϕ
(m
m◦ +
◦ +
m−1+cos
◦◦ +
m−2
m
◦(45
m−1
◦ +
m−2
◦+◦ ...
R
cosR
45
+ 45
AR
cos
(45cos
+(45
ϕ)◦++BR
cos
+R
2ϕ)
CR
(45
3ϕ)
+
cos
+cos
CR
cos
(45
3ϕ)
... (45
+
LR
cos
(45
(mLR
− 1)ϕ)
cos
+ AR
ϕ)
+cos
BR
(45
2ϕ)
cos
45
+
AR
cos
+
ϕ)
+
cos
(45(
R
45
+
AR
cos
(45
+
ϕ)
+
BR
cos
(45
++
2ϕ)
m m
m−1
m−2
m−3
m−3
◦
◦BR
m−3
m−1
m−2
m−3
mϕmϕ
+
Ar
coscos
(m
− 1)ϕ
+
Br
cos
(m
−
2)ϕ
r rcoscos
Cr
3)ϕ
...
+
Lr
cos
ϕ
+
M
=
t
m−2
+
(m
−
3)CR
cos
(45
+
3ϕ)
+
...
+
LR
cos
(45
+
(m
− 1)ϕ)
+
Cr
cos
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
+
M
=
t
m−3
m
m−1
m−2
+
Ar
(m
−
1)ϕ
+
Br
cos
(m
−
2)ϕ
m−3
◦◦2)ϕ
◦ m−1
◦ ◦◦ ◦
m − 1)ϕ + Br
− 2)ϕ cos (mm− 1)ϕ
m
◦
m−1
m−2
m
◦
◦
m−2
◦
m−3
◦
Cr
3)ϕ
...
+
Lr
cos
ϕ
+
M
=
t
mϕ(m
+ Ar
+
Br
cos
(m
−
r cos cos
m◦
◦
m−1
◦
m−2
◦
m−1
m−2
◦
m−3
m−3
◦+
◦ ◦2ϕ)
+
CR
cos
(45
3ϕ)
+BR
...R+
LR
cos
(45
+
(m
−
1)ϕ)
(45
+
mϕ)
R
sin
45
+
AR
sin
(45
+cos
ϕ)
+
BR
(45
45
+
AR
sin
(45
+
ϕ)
+
sin
(45
+
m...
◦
m−1
◦M
m−2
+ CR
3ϕ)
+sin
+ 45
LR
cos
(45
+2ϕ)
(m
−
1)ϕ)
++BR
M
cos
(45
+=
mϕ)
T,
R45
cos
45
AR
ϕ)
+
BR
cos
(45
+
R+ ...
cos
+m−2
AR
cos
(45cos+
+(45
ϕ) ++
cos
(45
+m−1
2ϕ)
+
CR
cos
(45
+
3ϕ)
+T,
...+=
+(m
LR
cos
(4
+
CR
cos
3ϕ)cos
...BR
+
LR
(45
−sin
1)ϕ)
m−3
m m
m−1
m−2
m−2
m
◦(45
m−2
◦2ϕ)
R
+
AR
cos
cos
(45
+
m−2
mmϕ
m−1
m−2
+
Cr
cosm−1
(m
− 3)ϕ
+
Lr
cos
ϕm−2
+
=
t=
sin
Ar
sin
(m
−
+
Br
sin
(m
−
2)ϕ
cos
(m
−
1)ϕ
Br
cos
(m
−M
2)ϕ
rrm
mR
sin
45◦m−2
+
(mcos
− 1)AR
sin
(45
+ ϕ)+m−3
+ϕ)
(m+−
2)BR
sin
(45
+ 2ϕ)
r ...
mϕ
+m−3
Ar
sin
(m1)ϕ
−m−
1)ϕ
+
Br
sin
(m
−
2)ϕ
m
m−1
m−
− 3)ϕ
1)ϕ
+
Br
cos
(m
−
2)ϕ
+ mϕ
Cr
cos
3)ϕ
+ 1)ϕ
...
+◦+
Lr
cos
ϕM
+
t◦M
m−1
◦ ◦ m−3
m
+
+ sin
Lr
cos
ϕ
+
Mm−3
=
tcos
◦
◦
◦
m
◦
m−1
◦
m−2
◦
rsin
Ar
sin
(m
Br
sin
(m
−
2)ϕ
+
Cr
3)ϕ
...
+
Lr
cos
ϕ
+
=
t
m−3
◦
◦
◦
m−3
◦
◦
m
◦
m−1
◦
m−2
◦
◦ +
m−1
◦ +
m−2
◦ + ...
R sinm−2
+
AR
sin
(45
++(45
ϕ)
++
BR
sin
(45
+
2ϕ)
m
◦m
m−1
◦sin
m−2
◦
m
◦ sin
m−1
◦ sin
m−2
◦(45
+
sin
(45
+
3ϕ)
++
LR
sin
(4
CR
sin
(45
3ϕ)
+
...
+
LR
sin
(45
+
−
m−3
◦ϕ)
◦(m
R45
45
+ AR
ϕ)
+
BR
sin
(45
+
2ϕ)
CR
cos
(45
+
3ϕ)
...
+
LR
cos
(45
+
(m
−
1)ϕ)
+
M
cos
(45
+cos
=
T,
R
45
AR
sin
(45
+
ϕ)
+
sin1)ϕ)
(451)
+
CR
cos
(45
3ϕ)
+
...
LR
cos
+
(m
−
1)ϕ)
M
cos
(45
+
mϕ)
=mϕ)
T,
R
sin
45
+
AR
sin
(45
+CR
BR
sin
(45
+
2ϕ)
m−1
cos
45
++
AR
cos
(45
+
ϕ)cos
++
BR
cos
(45
+
2ϕ)
m−3
R3)ϕ
cos
45
+
AR
cos
(45
+R
ϕ)
+
BR
cos
(45
+
2ϕ)
m−3
◦(45
◦BR
+
CR
++
3ϕ)
++
...
+
LR
(45
(m
−
m−3
m mϕ +
m−1
m−2
m−3
rm3)ϕ
sin
Ar
sin
(m
−
1)ϕ
+
Br
sin
(m
−
2)ϕ
Cr
...
+
Lr
sin
ϕ
=
u
cos
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
+
M
=
t
m−2
+
(m
−
3)CR
sin
(45
+
3ϕ)
+
...
LR
sin
(45
+
(m
−
1)ϕ)
=
m−3
+
Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
m
m−1
m−2
m
−
+
...
+
Lr
cos
ϕ
+
M
=
t
r
sin
mϕ
+
Ar
sin
(m
−
1)ϕ
+
Br
sin
(m
−
2)ϕ
m−3
◦◦2)ϕ ◦
◦
m − 1)ϕ + Br
m
◦ ◦+ Mm−3
◦◦
m
◦ (45
m−1
◦ (45◦m−1
m−2
m−3
◦
◦ (45
Cr2)ϕ sin (m
3)ϕ
...CR
+m−3
sin
ϕ(45
=−
u
r sinsin
mϕ(m
+−
Ar
Br
(m
m+
◦ Lr
m−1
m−2
◦◦
m− 1)ϕ
◦+
m−1
m−2
◦
◦mϕ)
m−3
m−2
◦m−1
sin
+
3ϕ)
+
...
+
LR
sin
+
(m
−
1)ϕ)
sin
++
U,
m−3
◦(m
◦ sin
◦CR
mR
cos
45
+CR
(m
−
cos
+
ϕ)
+−cos
(m
−
mR
45
+sin
(m
−
1)AR
cos
(45
+
ϕ)
−=
2)BR
(45
m
◦
◦3ϕ)
m−2
◦
+ sin
CR
3ϕ)
+cos
...cos
+
LR
sin
(45
+1)ϕ)
(m
−
1)ϕ)
+1)AR
M
sin
(45
+
mϕ)
U,
R45
sin
45
AR
(45
ϕ)
BR
+
2ϕ)
R
sin
+
AR
sin
(45
+3ϕ)
ϕ)◦+
++
BR
sin
(45
+
2ϕ)
sin
(45
+
3ϕ)
+
+
LR
sin
(42
+
sin
(45+
+
+
...◦BR
+
LR
sin
(45
+=
(m
1)ϕ)
m−3
cos
cos
(45
+...
ϕ)
BR
2ϕ)
+
CR
cos
(45
+
3ϕ)
+
...
+
LR
cos
(45
+
(m
− ...
1)ϕ)
+
M
cos
m
m−1
m−2
m m
m−1
m−2
+
CR
cos
(45
+
+−+
LR
(45
+(45
(m
−
+
M
cos
(45
+
mϕ)
=
T,
R
sin
45
+
AR
sin
(45
+
ϕ)
+
sin
(45
+
2ϕ)
m−1
m−2
m−3
m−2
mmϕ
m−1
m−2
+
Cr
sin
(m
− 3)ϕ
+m
...
+
Lr
sin
ϕ
=
u
mr
cos
(m
−
1)Ar
cos
(m
−
1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
r
sin
Ar
1)ϕ
Br
sin
(m
−
2)ϕ
m
m−1
m−2
m−3
mr
cos
mϕ
+
(m
−
1)Ar
cos
(m
−
1)ϕ
+
(m
−
2)Br
cos
(m
2)ϕ
m
−
1)ϕ
+
Br
sin
(m
−
2)ϕ
+
Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
◦+−
m−1
◦
m−2
◦
m − 3)ϕ +mr
... + cos
Lr sin
u − 1)Ar
m−3
◦
m−3
◦
◦
m
◦
m−1
◦
m−2
◦
mϕϕ+=Cr
(m
cos
(m
1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
sin
(m
−
3)ϕ
...
+
Lr
sin
ϕ
=
u
m−3
◦◦cos
◦◦cosm−1
◦◦ +
m−3
◦cosm
◦ −−
◦ ◦ +cos
m
◦(45
m−1
◦ +
m+
◦+◦+
◦++2ϕ)
mR
45
+
(m
−
1)AR
(45
+
(m
2)BR
+
2ϕ)
◦ϕ)
m−1
◦1)AR
m−2
m cos
◦ cos
m−1
◦ sin
m−2
+
(m
−sin
3)CR
(45
3ϕ)
+
+
(m
3)CR
cos
(45
3ϕ)
...
LR
(45
+
(m
m−3
◦M
◦m−2
mR
45
+
(m
−
1)AR
(45
+...
ϕ)
(m
2)BR
cos
(45
CR
+
3ϕ)
+
LR
sin
(45
++
(m
−
1)ϕ)
+
M
sin
+
=
U,
+
CR
sin
(45
+
3ϕ)
...
++
LR
sin
(45
+
(m
1)ϕ)
(45
mϕ)
=mϕ)
U,
mR
cos
45
(m
−
1)AR
cos
(45
+cos
ϕ)
+
(m
2
mR
cos
45
+
−
cos
(45
+
ϕ)
+(45
(m
−
2)BR
cos...
(45
m m
m−2
cos
(45
+
3ϕ)
+
...
+−
LR
cos
(45
(m
−+
1)ϕ)
+
M
cos
(45
+
mϕ)
=
T,(m
R(45
sin
45
AR
sin
(45
+−
ϕ)
++(45
BR
sin
(45
+
2ϕ)
m−3
m−3 m−1
R
sin
45
+Lr
AR
sin
(45
+
+
sin
(45
+(m
2ϕ)
+
CR
sin
+
3ϕ)
+
...+
+
LR
sin
(45
+
−−
1)
m−3
BR
m−1
m−2
m−3
cos+
mϕ
+ Lr
(m
−+1)Ar
cos
(m
−+1)ϕ
+
(m
−++
2)Br
cos
(m
−
2)ϕ
+ (m
(m
−
3)Cr
cos
(m
−
3)ϕ
+ sin
...
Lr
ϕcos
=
tϕ)
Cr
sin
(m
−
3)ϕ
...1)ϕ
+
ϕ
=
ucos
m−1
m−2
(m
−
3)Cr
cos
(m
−
3)ϕ
...
+
Lr
ϕ
=
t
m−3
 2)ϕ
m
m−1
m−2
mmr
−cos
3)ϕ
+
sin
ϕ
=
u−
mr
+
−
1)Ar
cos
(m
−
+
(m
−
2)Br
cos
(m
−
m−3
◦
◦

(mcos
−...mϕ
1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
m
◦
m−1
◦
m
◦
m−1
◦
m−2
m−3
◦
◦

3)Cr
3)ϕ
...
+
Lr
cos
ϕ
=
t
mr
cos
mϕ
+
(m
1)Ar
cos
(m
−
1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
m
◦−
m−1
m−2
◦1)AR
m
◦ + (m
m−1
◦mR
m−2
◦sin
m−3
◦ + ϕ) +
m−3
◦=
◦ (45
m
m−1
◦
m−2
◦
3)CR
cos
(45
+
3ϕ)
+
...
+
LR
cos
(45
+
(m
−
1)ϕ)
T
,
m−3
◦
m−3
◦
◦
◦
mR
sin
45
+
(m
−
sin
(45
(m
−
2(◦
sin
45
(m
−
1)AR
(45
+
ϕ)
+
(m
−
2)BR
sin
m
m−1
m−2
+
(m
−
3)CR
cos
(45
+
3ϕ)
+
...
+
LR
cos
(45
+
(m
−
1)ϕ)
=
T
,
mR
cos
45
1)AR
ϕ)(45
+
(m
−
+sin
2ϕ)
mR
cos
+
(mLr
−cos
1)AR
cos
(45
ϕ)
+
(m
−
2)BR
cos
2ϕ)
+
(m
−
3)CR
(45
+
3ϕ)(45
...+cos
+(m
L
(m
−2)BR
3)CR
cos
3ϕ)
+cos
...=
+U,
LR
cos
m−3
m−2
R45+
sin
AR
sin
+...+
ϕ)
+
BR
sin
+1)ϕ)
2ϕ)
CR
3ϕ)(45
...
+
LR
1)ϕ)
++M
sin
m m
m−1
m−2
m
m−2
+
CR
sin
(45
+
3ϕ)
+−
+−
LR
sin
+(45
(m
+cos
M+(45
sin
(45
+(45
mϕ)
mR
cos
45
−
1)AR
cos
ϕ)
+
(m
−
2)BR
m−1
m−1 mr
m−2
m−3
(45
mmϕ
m−1
m−2
+
(m
−
3)Cr
cos
(m
−
3)ϕ
...
+
ϕ
=
t
mr
sin
+
(m
−
1)Ar
sin
(m
−
1)ϕ
+
(m
−
2)Br
sin
(m
2)ϕ
cos
cos
(m
−
1)ϕ
+
(m
−
2)Br
cos
(m
−
2)ϕ
m−3

mr
sin
mϕ
+
(m
−
1)Ar
sin
(m
−
1)ϕ
+
(m
−
2)Br
sin
(m
2)ϕ
m
m−1
m−2
m−3

3)Cr
...◦++Lr
ϕ2)Br
=
t m−1
cos (m
(m −
− 3)ϕ
1)ϕsin
+ mϕ
(m+−
2)Br
cos
(mcos
− 3)ϕ
2)ϕ
◦ − 1)ϕ
◦t◦ (m ◦
m−2
◦◦
cos
+
...
=mR
tm m
m−3
m−3
◦
◦ ◦◦
mr
1)Ar
sin
(m
(m
−m−1
2)ϕ
+Lr
(mcos
−ϕ3)Cr
3)ϕ
...cos
+
Lr
cos
ϕ
=◦◦sin
m−3
◦+
+
m
◦ ◦+
m−1
◦◦ +
m−3
◦m−2
 + sin
m−
◦++
m−1
m−2
m−3
◦ sin
◦m−2
sin
45
+
(m
−
1)AR
sin
(45
+
ϕ)
(m
2)BR
(45
+
2ϕ)
m
◦3ϕ)
m−1
◦+
◦m sin
m−1
m−2
(m
−
3)CR
3ϕ)
...
++
(m
−
3)CR
sin
(45
+
3ϕ)
LR
sin
(45
+(45
◦ϕ)
◦(m
mR
45
+
(m
−
1)AR
sin
(45
+
ϕ)
(m
−
2)BR
sin
(45
+
3)CR
cos
(45
+
3ϕ)
...
+
LR
cos
(45
+
(m
−
1)ϕ)
=
T...
,(45
+
(m
−
3)CR
cos
(45
+
+
...
+
LR
cos
(45
+
(m
−
1)ϕ)
=2ϕ)
Tsin
,+
mR
sin
45
+m−3
(m
−
sin
+LR
ϕ)
+
(m
−
2◦L
mR
45
+
(m
−
1)AR
sin
(45
+
(m
−
2)BR
sin
m m
m−1
m−2
CR
sin
3ϕ)
+sin
...
+−
LR
sin
(45
+
(m
−
1)ϕ)
+(45
M
(45
+(45
mϕ)
=
U,
mR
cos
45
1)AR
cos
(45
+
ϕ)
+1)AR
(m
−
2)BR
cos
(45
+
2ϕ)
m−3
(45
mR
cos
45
+
(m
−
1)AR
cos
+
ϕ)
+
(m
−
2)BR
2ϕ)
+
(m
−
3)CR
cos
+
3ϕ)
+
...
+
cos
(45
m−3
2)ϕ
m−3
 (m
m−1
m−2
m−3
mrcos
sin
mϕ
+
(m
−
1)Ar
sin
(m
−
1)ϕ
+
(m
−
2)Br
sin
(m
−
+
(m
−
3)Cr
sin
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
cos
(m
−
3)ϕ
+
...
+
Lr
cos
ϕ
=
t
m−1
m−2
+
(m
−
3)Cr
sin
(m
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
m−3

m +
m−1
m−2
(msin
−
3)ϕ
+
...
+
Lr
cos
ϕ
=
t
mr(m
mϕ
1)Ar
1)ϕ
(m
−
2)Br
sin
(m
−
2)ϕ
m−3
◦
◦

sin
−
1)ϕ
(m
−
2)Br
sin
(m
−
2)ϕ
m−3
◦
◦

3)Cr m sin
3)ϕ
... +
sin
ϕ(45
=◦sin
u◦ +
mr sin mϕ + (m − 1)Ar
sin 45
(m
−
1)ϕ
+3)CR
(m
−Lr
2)Br
(m
−++
2)ϕ
◦−
m−1
m−2
m−1
m−2
◦−
m−3
◦◦
m−3
◦=1)ϕ)
◦
m ◦sin
m−2
◦−
+
(m
sin
3ϕ)
+
...
+
sin
(45
+
(m
1)ϕ)
U3ϕ)
, =+sin
m−3
◦m−1
m−3
◦sin
+
m
◦...
◦2ϕ)
m−2
+
(m
3)CR
3ϕ)
+LR
+
LR
(45
+
(m
,(45
45
1)AR
sin
(45
ϕ)
+
(m
−
2)BR
sin
(45
+
+
(m
−
1)AR
(45
ϕ)
(m
2)BR
sin
(45
+
2ϕ)
+
(m
−◦3)CR
3ϕ)
+
+
L
(m
−(45
3)CR
sin
(45
...(m
+(45
LR
sin
(45
+sin
(m
m−3
 sin
mR
cos
cos
(45
+...
ϕ)
+
(m
−
2)BR
cos
(45
++
+◦+
(m
−
3)CR
cos
+
3ϕ)
+
...
cos
+
(m
− ...
1)ϕ)
2 m
2
m−1
2−
m m
m−1
m−2
+
(m
−
3)CR
(45
+ +
3ϕ)
+−
LR
cos
+(45
(m
−
1)ϕ)
=
T2ϕ)
, LR
mR
sin
45
+
(m
−
1)AR
sin
(45
+
ϕ) U
+
−+
2)BR
(=
22sinmR
m−1
m−1mmr
m−2
m−3
m−2
m−1
2cos
m−2
+ (m
−+3)Cr
(m
−(m
3)ϕ
+
...
Lr
sin
ϕ−
=
u2m−2
cos
mϕ
+ ...
(m
−
1)
Ar
cos
1)ϕ
++
(m
−
2)
Br
cos
(m
−
2)ϕ
sin
1)Ar
(m
−−
1)ϕ
+
(m
−
2)Br
sin
(m
−
2)ϕ
m−3
m−3
m
r−
cos
mϕ
(m
−
1)
Ar
cos
(m
−
1)ϕ
+
(m
2)
Br
cos
(m
−
2)ϕ
m
2 sin
m−1
2
m−2
sinr(m
(m22m
−2mr3)ϕ
1)ϕ
2)Br
2)ϕ
3)Cr
3)ϕ
...
+
Lr
sin
ϕ
=
u
sin
+
+
Lr
sin
ϕ
=
u
cos mϕ + (m − 3)Cr
1) Arm sin
cos(m
(m−
−
1)ϕ
+
(m
−
2)
Br
(m
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
m−3
◦◦ − 2)ϕ
◦◦
m−2
m−3
◦ cos
◦


m
◦
m−1
◦
◦
◦
m−1
◦
m−2
◦
m−3
◦
◦
+
(m
3)CR
sin
(45
+
+
...
LR
sin
+
(m
=
+
(m
−
3)CR
sin
(45
+
3ϕ)
...
sin
(45
+(45
(m+−
U+2)BR
,3ϕ)
m
2 22m−1
2 2−
m−2
cos
(45
+ 3ϕ)
3ϕ)
+LR
...
+
LR
cos
(45
+
(m
−=1)ϕ)
1)ϕ)
=UT+,,... +
mR
sin
+ (m
(m
−+
1)AR
ϕ)1)ϕ)
+−
(m
−
sinLR
(45sin+(45
2ϕ)+
m−3
m−3
 
mR
sin
45
+
(m
−
sin
(45
+
ϕ)
+
−
2)BR
sin
(45
2ϕ)
++
(m
−
3)CR
sin
(45
22m−2
m−3

m−3
 (m
2 cos
m−2
m−3
 45
m2m
rsin
mϕ
+ (m
−++
1)(m
Ar
cos
(m
−−
1)ϕ
+
(m
−+
2)
Br
cos
(m
2)ϕ
+ ...
(m
3)
Cr
cos
(m
−
3)ϕ
+...(m
...
+
cos
=
tcos
3)Cr
3)ϕ
Lr
sin
ϕϕ
=
m−1
2
m−3

−
3)
Cr
cos
(m
−+
3)ϕ
+1)AR
...Lr
+
Lr
cos
=
t−
2−mϕ
m
2 sin
m−1
2uϕ
m−2
(m
−
Lr
sin
ϕm−1
=
u
rm
cos
1)
Ar
1)ϕ
−
2)
Br
(m
− 2)ϕ
cos
(m
1)ϕ
(m−
2)
Br
cos
(m
−m
2)ϕ
3)
Cr
3)ϕ
...
+
cos
ϕ
=
tcos
m
r3)ϕ
cos+mϕ
+−
(m
−
1)
Ar
cos
(m
−
1)ϕ
−Lr
2)
Br
(m
◦ + (m
m−1
◦ − 2)ϕ
m−2◦
◦
m−3
◦
m−3
◦
◦

2 22 m−3
 m−2
mR
sin...45
+
(m
1)AR
sin
(45++(m
+...ϕ)
(m
2)BR
sin
(45 +
−+
3)CR
sin
+1)ϕ)
3ϕ)
+
m 22 m
m−1
22−
m−2
22 m
+
−+
3)CR
sin
(45
+−
3ϕ)
+
LR
sin−(45
+(45
(m −
=+...
U2ϕ)
, LR sin (45 + (m − 1)ϕ) =
22 m−2
m−1
2
 (m
m−1
2 m−3
mmϕ
2m−2
+
(m
−(m
Cr
cos
(m
−(m
3)ϕ
+
Lr
cos
ϕ
t=
sin
++
(m
−
1)
Ar
sin
(m
−
+
(m
2)
Br
m
rrm
cos
cos
(m
−
1)ϕ
++
(m
−
2)−
Br
cos
(m
−2)ϕ
2)ϕ
m−3

mcos
m
m−1
2 m−2
m−2
2=m−1
m−3
(m
r 2−
mϕ
+3)
(m
−Cr
1)
sin
(m
−(m
1)ϕ
(m
2)=
Br
sin
−−
2)ϕ
cos
(m
−rsin
1)ϕ
+mϕ
−3)
2)
Br
−1)ϕ
2)ϕ
3)ϕ
...
+−
Lr
ϕ
tsin
(m
3)ϕ
...
+
Lr
cos
ϕAr
m
sin
1)
Artcos
sin
(m
−
1)ϕ
+
(m
2)
Br
(m
2)ϕ
+
(m
−
3)
Cr
cos
3)ϕ
...cos
+−
Lr
cos
ϕ
=
tsin
m−3
◦
◦
2 22m−1
2
m−2
+
(m
−
3)CR
sin
(45 + 3ϕ) + ... + LR sin (45 + (m − 1)ϕ) = U  ,
m−3
 −
22m−2
m−3

m−3
 (m
2 sin
m−1
2sin
m−2
,2)ϕ
m2m
rm
mϕ
+
(m
−...
1)
Ar
sin
(m
−
1)ϕ
+
(m
−++
2)
Br
sin
++
(m
−
3)
Cr
sin
−
3)ϕ
+
...
Lr
ϕϕ
=
usin
, (m
cos
(m
−
3)ϕ
+(m
...
+
Lr
cos
=
t(m
m−1
2Ar
+
(m
−
3)
Cr
sin
(m
−
3)ϕ
...
+
Lr
sin
=
u
m−3
 2)ϕ
2−
m
2=m−3
m−1
2ϕ
m−2
cos
(m
3)ϕ
+
+
Lr
cos
ϕ
t
rm
sin
mϕ
+
(m
−
1)
sin
(m
−
1)ϕ
+
−
2)
Br
−
Δυναμις
Vol.
3
No.
3
December 2008
sin
(m
−
1)ϕ
(m
−
2)
Br
sin
(m
−
2)ϕ
3) Ar
Cr
3)ϕ + (m
... +−Lr
m r sin mϕ + (m − 1)
sin (m − 1)ϕ
2) sin
Brϕ = usin, (m − 2)ϕ
m−3
2 − 3)
22 Cr
 m−3
(m
 +
 2+ Lr
 2ϕ =m−2
2  , 
2 m
2222
m−1
22−

 ...
22−sin
22 (m − 2)ϕ
m−1
m−2
2


+
(m
sin
−
3)ϕ
u
+
u
)(tt
+
uu
)
+
(tu
−
ut
)
−
(tt
+
uu
)
(t
msin
r
sin
mϕ
+
(m
1)
Ar
sin
(m
−
1)ϕ
+
(m
2)
Br
sin
m−3

+2Lr
u sin
)(tt
uu
) +
(tu
ut
)++
uu
(t +
2 ϕ 2=
 u
2 (tt
, 2
m−3
3)
Cr
3)ϕ
...ut
Lr
ϕ +
=sin
u)ϕy.
sin (m
(m −
− 3)ϕ
1)ϕ +
+ ...
(m
2)
Br
sin
(m
2)ϕ
,uu
+
u
)(tt
+
)−
+
(tu−
)−
−sin
(tt+
uu
)=
(t
+−
(m
−
3) +
Cr
sin
(m
− −
3)ϕ
...
+
Lr
= y.
u ,
=
2 +
2 )2 222 22
2 2 2 2  2 m−3
 r(t
 u
 
 2 2
22 ut
= y.

m−3
 (m
)−
2 (tt
u
)(tt
+
uu
)
+
(tu
−
)
+
uu
)
(t
r(t
+
u
2
2
2
+
(m
−
3)
Cr
sin
−
3)ϕ
+
...
+
Lr
sin
ϕ
=
u
,


2


2
+)(tt
utu) −)−ut(tt
2
r(t− +
 2 + uu )= y.  2
...
Lr
ϕuu
u+uu
, (tu
+ uu sin
) +(m
(tu−−3)ϕ
ut(t+
)+
−u+
(tt2)(tt
+sin
) =2)+
+
uuu
) + (tu
) − (tt + =
uuy.)
(t
2 )22 2  2
y.
r(t
+
= y.

2u
 2
2 2 (t 22+ u2 )(tt
  + uu
2 =
r(t
+
u
)
)
+
(tu
−
ut
)
−
(tt
+ uu )2
2
2
2
+ uu r(t
)++
(tuu −
) ut ) − (tt + uu )
r(t + u )
=
y.
=
y.
r(t2 + u2 )2
r(t2 + u2 )2
Let the following function of the indeterminate x be
cosBR
45 + AR
(45 + ϕ)
BR + 3ϕ)
cos+(45
2ϕ)
+Rϕ) +
cos (45 cos
+ 2ϕ)
CR
cos+(45
... ++LR
cos (45 + (m − 1)ϕ) + M cos (45 + mϕ) = T,
◦
m−2◦ ◦
◦
m−3
◦
◦
◦ m−2
m
m−1
◦
cos
(45 ++...ϕ)
BR
cos
(45
+ 1)ϕ)
2ϕ)
CR
(45
+
3ϕ)
++...M+cos
LR(45
(45mϕ)
+sin
(m
− ◦1)ϕ)
+ M cos (45◦ + mϕ) = T,
+ 3ϕ)
++
LR
cos45
(45cos
(m −
+
= (45
T,
R
sin
++AR
sin (45
ϕ)
+cos
BR
+ 2ϕ)
◦
◦
◦
m ◦
◦
◦m−3
m−2
◦
m−2
◦
cos
(45
+453ϕ)
... m−1
+ LR
(45
+
(m
− ◦1)ϕ)
+
cos
(45
+ mϕ)
sinBR
++AR
sin
(45
+
ϕ)
+(45
BR
sinM
2ϕ)
+R
ϕ)
+
sin
(45
++cos
2ϕ)
CR
sin
+ 3ϕ)
+(45
...
++
LR
sin
(45◦=
+T,
(m − 1)ϕ) + M sin (45◦ + mϕ) = U,
m−2
m−3◦◦
◦
m
m−1
◦◦ +
sin
(45 ++...ϕ)++
BR
2ϕ) +
CR
+ 1)ϕ)
3ϕ)
LR
sin
(45
+
+ 3ϕ)
LR
sin
(45sin+(45
(m◦ −
+...
M+sin
(45
mϕ)
◦
Third Demonstration
m−2
39
(m
−2)BR
1)ϕ)Gauss
+ M cos (45 +
+ 2ϕ)
mϕ) = U,
= U,
mR cos 45 + (m − 1)AR
cos (45 + ϕ) + (m
−
◦m−2
◦
m−1
◦m−3
◦
m−2 ◦
◦
◦
◦
sin
3ϕ)
... −
+
LR
sin(m
(45
+(45
(m −
M−
sin
(45+ +
=cos
U,+
mR
+
ϕ)
(m
2)BR
(45
2ϕ)+ (m − 1)ϕ) = T  ,
cos(45
(45cos+
+45
ϕ) +
++(m
(m
− 1)AR
2)BR
cos
(45
+1)ϕ)
2ϕ)
+
−cos
3)CR
cos++
(45
+
3ϕ)
...mϕ)
+cos
LR
(45
◦ ◦ 
m−3 ◦ m−2
◦

m
◦(m
AR
(45
+...
ϕ)
+
− (45
2)BR
cos
(45(45
+◦+
(m
−
3)CR
+ 1)ϕ)
3ϕ)
+
...T2ϕ)
LR+cos
+ (mm−2
− 1)ϕ)
= ◦T +
, 2ϕ)
cosm−1
(45◦cos
+ 3ϕ)
+45
LR
+(45
(m◦m−1
−
=+
mR
sin
+cos
(m
−cos
1)AR
sin
+, ϕ)
(m(45
− 2)BR
sin (45
extending
from r = 0 to r = R and from ϕ = 0 to ϕ = 360°,
m−3
◦

m ◦
◦ ◦
m−1
◦◦m−3
m−2
◦
m−2
◦
◦
CR
(45
+ 3ϕ)
++
... (m
+ LR
cos
(m
− 1)ϕ)
=T
mR(45sincos
+ (m
− 1)AR
(45(45+ +
ϕ)
+(45
(m
−
(45
2ϕ)+ (m − 1)ϕ) = U  ,
sin
+45
ϕ)
2)BR
2ϕ)
−sin
3)CR
sin
+2)BR
3ϕ)
+ ,... +sin
LR
sin+
(45
which thus acquires a finite, completely determined value. This
m−3 ◦ m−2

AR
(45+◦+(m
+...ϕ)
(msin
−
2)BR
sin
(45◦+=+...U2ϕ)
−
3)CR
sin+(45
+ 1)ϕ)
3ϕ)
+
= which
U  , we will signify by Ω , must be obtained regardless
sin m−1
(45◦ sin
+ 3ϕ)
++
LR
(45
(m◦ −
, LR sin (45◦ + (m − 1)ϕ)
value,

of whether the integration is performed first with respect to ϕ and
CRm−3 sin (45◦ + 3ϕ) + ... + LR sin (45◦ + (m − 1)ϕ) = U  ,
then
with
respect
to
r,
or
the
inverse
order.
We
have,
however,
the


√
Rm−1
if we
consider
√indefinite,
(R + mA
2 cos
(45◦ +r as
ϕ))a constant,
ydrdϕ
I. t is composed of the terms
m 2

√
tu − ut
Rm−2 2
√
Rm−1
,
y dϕ
=
√
(R
+
mB
2 cos (45◦ +
2ϕ))
+
◦

2 + u2 ) 
√ (R + mA 2 cos (45 + ϕ))
m−1m 2
r(t
√
R
◦
m 2
ydϕ
√ (R

+ mA 2 cos (45 + ϕ))
m−3
ydrdϕ
√
Rm−2
tu − ut
m +2R √ (R3 + mC √2 cos (45◦ + 3ϕ))
+ √ (R2 + mB 2 cos (45◦ + 2ϕ))
ydϕ
=
,
ϕ


is easily confirmed by differentiating
to . A
r(t2with
+ u2respect
)
m 2
Rm−2 m2 2 as √
◦
2 cos
(45
+
2ϕ))
+ √ (R
ydrdϕ
√
constant
is
not
added,
if
we
presume
that
the
integration
begins
m−4+ mB
Rm−3 3
m +2R √ (R4 +ϕmD√2 cos (45◦ϕ+ 4ϕ))...,
+ √ (R + mC 2 cos (45◦ + 3ϕ))
tu −ut
=
0
at
=
0,
since
for
=
0
one
obtains
.
Now,
since
2
2
m 2
 r(t +u ) 
√

Rm−3m 32
tu −ut
◦
tu
−
ut
√
Rm−4 4
ϕ
0 + 3ϕ))
√
2 cos
(45
evidently
also vanishes for ydϕ
= 360°,
the ,integral
2) =
= then
r(t2 +u
+ √ (R + mD 2 cos (45◦ + 4ϕ))...,+ m 2 (R + mC
2 + u2 )
ydϕ= r(t
y dϕ from ϕ = 0 to ϕ = 360° becomes
0, while
r remains
m 2


√ ydϕ
Rm−4 4
tuΩ −= ut
◦But from here follows
indefinite.
0.


,
T T + U U+ √ (R + mD 2 cos (45 + 4ϕ))..., ydϕ =
2 + u2 ) integral, in which we 
2 seen,
Likewise we have the r(t
indefinite
which, for each determined real value of ϕ , as is m
easily
tu −ut
) =0
ydr
as2constant,
considerr(tϕ2 +u

has a single positive value; consequently,
must be taken as a

r =t R
tt + uu
tt +=uu

positive value. In a similar manner
it
will
be
proven
that
u,
t’,
u’
,
ydr
m−1
√
y dr = 2
R
2,
2
tu −ut
ydϕ
t + u2t + u
+ mA
cosbecome
(45◦ + ϕ))
UU
also become positive, such that T T√ +(R
must 2also
a 2 +u2 ) = 0
r(t
m 2
positive magnitude.
T T  +UU 

m−2
√
R
 T 2 +U 2 
◦
◦
ydϕ
2 t’, u’ change over◦into



r
=
R
II. For r = R, the functions
t,
u,
T cos
+Tismϕ)
+ U sin
(45 confirmed
+ mϕ), by differentiating
likewise
easily
with
to
T
+UU
2ϕ))(45 as
+ √ (R + mB 2 cos (45 +
tt +
uurespect
2 +U 2
,
ydr
=
m 2
◦ r:T again
◦
2+
2 we begin
no
constant
is
needed
here,
supposing
that
+
mϕ)
−
U
cos
(45
+
mϕ),
T
sin
(45
t
u

m−3
T cos (45◦ + mϕ)R+
U sin3(45◦ + mϕ),
√

TT
to the proofs in the
 according

◦ 
◦ the integration
 with r
◦ = 0. Hence,
tt
+
uu
◦
◦
√
+
mC
2
cos
(45
+
3ϕ))
(R
+
cos
(45
+
mϕ)
+
U
sin
(45
+
mϕ),
T

T sin (45 + mϕ)m− U2 cos (45 +1mϕ),


,
ydr
=
preceding paragraphs,
the integral,T2extending
T
T +2 U U from r = 0 to r = R,
◦ 
◦
  cos
+(45
mϕ)
U sin−
+ mϕ),
T cos (45◦ + mϕ)
+ U sin (45◦ √
+ mϕ),T cos (45
++mϕ)
(45◦ + mϕ), t +2 u 2 dϕ
T  ◦sin
T (45
TU
+UU
m−4
,
and
will,
consequently,
according
to
the
theoequals
T
+
U
1
R
1
2
2
4 (45◦ + mϕ),
◦
T +U
◦
T sin (45◦ ++mϕ)
U cos
√− (R
+ mD 2 cos (45
4ϕ))...,
+ mϕ) rems
− U cos
(45◦preceding
+ mϕ), paragraphs, always be a positive magniT sin+(45
of the
m 2
1
U

◦ T T  +UU  
◦
◦
Ω will
cos (45 + mϕ)
+
U for
sineach
(45 real
+ mϕ),
−
arctan
= mϕ
+ 45also
tude
value of ϕ . Therefore,
necessarily

2
2


respectively, as will be easily shown by the actualTdevelopment.
T +U
T
U
U
T T + Uintegral

◦
be
apositive
−
=Umϕ
+(45
45◦◦magnitude,
+ mϕ) −
cos
+ arctan
mϕ), that is, the value of2the
dϕ
Consequently, the value of the function tt + uu , T
forsin
r =(45
R, will
2
U
T
T +U
arctan

equal T T  + U U  and thus will be a positive magnitude.
t2 + u 2
T
U
T T  + UU
arctan
dϕ
t2 + u2 Incidentally we conclude from the same formula that
2 + U2
T
T
U
r = of
R the function t2 + u2 , for r = R, equals T 2 + U 2
the value
= mϕ + 45◦ − arctan
tt + uu
2
2
T
and
T +will
U thus be positive, and thence it follows that for
√ no
√ ϕ
√
3
0 to ϕ
= 360°.1 This is absurd, since just before we had
T 2 +than
U 2 the individual magnitudes
mA 2, mBfrom
2,◦ =mC
2UU
value of 
r which isgreater
√
√ 3
√
−arctan
arctan
= mϕ +
45 the
√
found
sameTmagnitude
= 0. Our presumption, thus, can not be
 t =√0, u
mA 2, mB 2, mC 2 √
,..., ◦can
= 0 at√the
t + uT −1
◦ same
3
√
TmA
cos (45
+
mϕ)
+
U
sin
(45
+
mϕ),
true,
and
with
it
the
validity
of the theorem is proven.
2, mB 2, mC 2
time.
t + u −1
U
◦
◦
√
arctan
T sin (45 + mϕ) − U cos (45 + mϕ),
x=
T
√ r(cos ϕ + sin ϕ
3. −1)
√
2. ◦ + mϕ) + U  sin (45◦ + mϕ),
t + u −1
T  cos (45
x√= r(cos ϕ + sin ϕ −1)
√
◦
◦
Thet function
into t + u −1 by the sub− u −1X transforms
+ mϕ) −
U 0cos
(45
T  sin
√
Theorem. Within
the(45
boundaries
r=
and
r =+R,mϕ),
as
√
−1)
x
=
r(cos
ϕ
+
sin
ϕ
stitution
and into t − u −1 by the
√
well as ϕ = 0 and ϕ = 360°, there exists such values of the
√
= r(cos
ϕ + sin
ϕ −1)
substitution x =
determined
values
√ r(cos ϕ − sin ϕ −1). Ifxfor
indeterminates r, ϕ , for which t = 0 and u = 0 at the same
√ 1
t−u
ϕ − sin ϕ −1)
of r, ϕ , 1say
for −1
r = g, ϕ =G, results that t = x√
0,=ur(cos
= 0 simultatime.
t − u −1
tt Demonstration.
+ uu
√
We will suppose that the proposition is
√
√
1
x
=
r(cos
ϕ
−
sin
ϕ
−1)
2
2
x =ofg(cos
+ sin G one
−1),
= g(cos
G − sin G
√
not true; it is evident that the value of t + u for all values of
1
This is clear in and
itself.GMoreover,
willx easily
ascerx = r(cos
ϕ − sin
−1)G + si
x =ϕg(cos
U
the indeterminates within the assigned limits must be a positive
tain the indefinite integral = m ϕ
1 + 45° – arctan T , and can be prov2
2
1
1 en in different ways (indeed in itself it√is not yet obvious which
√
T + finite.
U
magnitude, such that the value of y remains
Let us conx = g(cos G + sin G −1),
x = g(cos G −
U sin G −√
1
sider the double integral  
of the infinitely many values of the multi-valued function arctan T ,
√ 
√ 
√
3
x = g(cos G + sin G
mA 2, mB 2, mC 2 which correspond to ϕ =360°, one must adopt), that the value which
√
y dr dϕ
one obtains for the integration for
set =Gm+• 360°
x −be
g(cos
sin G −1)
1 ϕ = 360°, must
√
or = 2 m π . However, this is not necessary for our purpose.
and similarly x − g(cos G − sin G −1).
December
2008
√and simil
1
1
x − g(cos G + sin G −1)
√
and similarly
1 − sin G1 −1).
1 x − g(cos G
and similarly
1
2
x
−
2g
cos
Gx
+
g2;
1
1
1
1
m ◦◦
sin (45◦◦
√
√−1
t + u√
+u
u −1
−1
tt +
√
√−1)
x = r(cos ϕ + sin ϕ√
40
x=
= r(cos
r(cos ϕ
ϕ+
+ sin
sin ϕ
ϕ −1)
−1)
Third Demonstration
x
√
Gauss
√−1
t − u√
−u
u −1
−1
tt −
√
The integration
neously,
(and
that
√
x = r(cos ϕ − sin
ϕ√
−1)there are such values became verified in the

x=
= r(cos
r(cos
ϕ−
− sin
sin
ϕ −1)
−1) then X obtains the value 0 for each of the
ydrdϕ
previous
paragraphs),
x
ϕ
ϕ

tu − ut
substitutions
y dϕ =
√
√
r(t2 + u2 )
√−1), x = g(cos G − sin G√
√−1),
x = g(cos G + sin G√
x=
= g(cos
g(cos G
G+
+ sin
sin G
G −1),
−1), x
x=
= g(cos
g(cos G
G−
− sin
sin G
G −1),
−1),
ξ = 0, η =
x
is only an actual integration---that is, summation---so long as y is
everywhere a finite magnitude between the boundaries in which
and will consequently be divisible by the indefinite

one integrates; on the contrary, it will be absurd
if y istuinfinite

− ut
ydϕ
=
anywhere
between
those
bounds.
If
we
specify
such
an
integral
√

r(t2 + u2 )
√−1)
x − g(cos G + sin G√
η dξ , which generally indicates the surfacebetween the axis of
x
−
g(cos
G
+
sin
G
−1)
x − g(cos G + sin G√−1)
the abscissa and the curve developed accordingηdξ
to the customary
√−1).
and similarly x − g(cos G − sin G√
and similarly
similarly x
x−
− g(cos
g(cos G
G−
− sin
sin G
G −1).
−1).
rule in which the ordinate η corresponds to the abscissa ξ , and
and
1
  of the continuity, then we
are withal negligent
very often
η =could
2
So long as G is not = 0, nor g = 0, these divisors are unydrdϕ
ξydrdϕ
find
ourselves
entangled
in
contradictions.
If
we
set,
for
example,
equal, and X will consequently also be divisible by their product 
1
1
C
−
η = ξ2 , then analysis will yield the integral
ydrdϕ
ξ , which will
x22 − 2g cos Gx + g 22;
correctly
indicate
the
surface
so
long
as
the
curve
maintains its ξ = 0,◦ η
ξ = 0, η = 0,
x2 −
− 2g
2g cos
cos Gx
Gx +
+ gg2 ;;
u ζ = 0, ξ = R, η = 360 ,
x
continuity; since this is interrupted at ξ = 0,t then, if anyone of
unreasonableξ manner
should
theR,
magnitude
= 0, η =
0, ζ ask
= 0,for
ξ=
η = 360◦of
, the surface
for would
r = e,deliver
ϕ = E by
however, if either sin G = 0 and thus cos G = ±1 or g = 0, each
from
a
negative
abscissa
to
a
positive,
the
formula



tu −ϕut= F by
sin G = 0

tu − ut
r
=
e,
∓
of
the
factors
will
be
identical,
namely,
=
x
g.
It
thus
holds
ydϕ
=
sin G
G=
= 00
the absurd replyydϕ
that =
it would
be negative. However, what
2these 2
sin
r(t2 + u2 )
r =r(t
f , up+ϕinu=)E by
true that the function X possesses a real divisor of the second or


and
similar
analytical
paradoxes
mean
shall
be
followed


tu − ut 
cos G = ±1
first
degree, and since the same conclusion will again hold for the ydϕ more
= detail
ηdξ r = f , ϕ = F by
cos G
G=
=
±1
ηdξ occasion.
on
2 + u2
cos
±1
r(t
) another
quotient, X is thus completely dissolvable into such factors.
Here it is only possible to affix but a single
remark.

=x∓g
1
1
ηdξ Were questions
=x
x∓
∓ gg
η which
= 2 in certain
arctan Θ − arc
proposed
without
restriction,
=
η
=
4.
ξ
ξ 2 absurd, then analysis thereupon
cases
could
become
aids
itself
1
1
1
η = 2very frequentlyC by
C
−
delivering
a response
in
part
variable.
If
we,
Although we have brought the task we had undertaken
−
ξ

ξ
ξ
y dr dϕ extending from r =
for example, signify the integral
completely to an end with the preceding, still it would not be suC − 1ξ e to r = f and ϕu = E to ϕ = F and the value of ut
perfluous to add something further concerning the inferences of
t
§ 2. By the assumption that t and u vanish at the same time for
for r = e, ϕ = E by Θfor r = e, ϕ = E b
no values of the variables r, ϕ within the boundaries given there,ut
r = e, ϕ = F b
r = e, ϕ = F by Θ
we happen upon an inevitable contradiction, whereupon we have

for
r
=
e,
ϕ
=
E
by
Θ
r = f, ϕ = E b
r = f , ϕ = E by Θ
concluded the incorrectness of the assumption itself. This con

r
=
e,
ϕ
=
F
by
Θ
r = f, ϕ = F b
ϕ
r = f , ϕ = F by Θ
tradiction must therefore cease, if there is an actual value of r,

r
=
f
,
ϕ
=
E
by
Θ
for which t and u become = 0 at the same time. In order to further
r = f , ϕ = F by Θ
 integral arctan

clarify this, we observe that for such values t2 + u2 will = 0 and
Θ−a
then by analytical operationsarctan
one easily
obtainsΘthe
valΘ − arctan
− arctan Θ
+ arctan
1
thus y become infinite,
so
that
it
no
longer
is
permitted
to
treat
the
1
ue

1
y dr dϕ as an assignable
T 2magnitude.
+ U2
double integral
Given
arctan Θ − arctan Θ − arctan Θ + arctan Θ .
 if ξ , η , ζ signify the coordinates of spatial
  in general,
√ points,
√the
√
3
y dr dϕ signifies the volume mA
ydrdϕ
2, mB
integral
of a body,
which2,is mCThe2 integral in actuality can then only have a determined value,
bounded by the five planes whose equations are
if y remains finite between the specified boundaries. This value
is itself contained in the specified formula, however it is not fully
ξ = 0, η = 0, ζ = 0, ξ = R, η = 360◦ ,
determined by the same, since indeed
the arctan is a multivalued
1
function, and it must further become decided which functional
and by a surface whose equation is ζ = y, if one considers those
values are preferable in a determined case, by means of other

partstuof −the
as negative whose coordinates ζ are negative.

utbody
not-so-difficult considerations. If, on the contrary, y is infinite
ydϕ However,
=
anywhere between the specified boundaries,
then the question rer(t2 + here
u2 ) it will be tacitly assumed that the sixth surface will

be continuous; if the latter condition comes to pass in such a way

specting the value of the integral
y dr dϕ is absurd. This
ηdξ that y will be infinite, it can very well happen that the former condoes not hinder, that, if one will obstinately extort a response
from analysis, this might soon give different methods, whereby
1ception no longer makes sense. Inthis case, there can be no talk
η = 2of the evaluation of the integral
y dr dϕ , and for that reason
the individual values of the foregoing1 general formula are obξ
it is not incomprehensible that analytical operations applied in
tained.
1
C − 1ξ blind reckoning to meaningless things lead to nonsense.
1
u
t
for
r
r
r
r
= e,
= e,
= f,
= f,
ϕ=E
ϕ=F
ϕ=E
ϕ=F
by
by
by
by
1
Θ
Θ
Θ
Θ
1
1
December 2008
41
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 Chernac 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

ex−1
dx
x
√reckoned from x = 0 on. You have made known to me the desire,
a + b that
−1 I=report
a + bithis 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 + bi whether he will understand them as applied to merely real magnitudes (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 −1 = a + bi
must enjoy equal rights with the reals. The discussion here is not
ϕ(x) =of∞
+ bi)
practical utility, rather, to me, analysis isli(a
an independent
science, which, by the neglect of those imaginary
magnitudes,
loses

ϕ(x) · dx
1 Investigation
of expressible transcendental functions by the

α +Science
βi
integral ldxx , Königsberger Archive of Natural
and Medicine 1, 1812, S.1, FR. W. Bessel’s Werke II, Leipzig 1876, S.330
x = a + bi
 x−1 ex−1
√
e = a + bi
a + bx−1−1
dx x
e
x
√dx
enormously in beauty and roundness and, in a moment
aall+xtruths,
bbi)−1 = a + b
li(a
x−1
e
which otherwise would be universally
valid,
are
obliged
to
add


dx

x−1
√
e√
xbiex−1
a−1
+=
bemploy
=the
a+
the most weighty limitations.
assumption,
that
ϕ(x)
·
li(a
+
bi)
a√+ bI must
a−1
+dx
bi
√ dx dx
x −1 =
+ b point,
a + bi
you are essentially ain+accordance
since
b −1 = awith
+ bixme aboutathis

li(a
+ bi) that you by
your elucidation2 in art. 18
already
indicates,
noϕ(x)
means
α+
βi
·dx x−1
li(a
+ bi)
√
e
li(a
+
bi)
intend to block
the
way
toabi)
investigations
about
.
What
a
+
b
−1
=
+
bi
li(a
+
dx

√
√
√ 
ϕ(x) · dx ,awhere
x=
aa+
α=+
βi
a + b be
−1imagined
= aa+
+b −1
+bibi? x
should
ϕ(x)
dx
+bi
b then
−1 with
=
a +· bi
ϕ(x) · dx
li(a
+ bi)
Evidently, if
one
wants
from clear notions,
it must
ϕ(x)to· proceed
dx
α + βiinfinitely
x
=
a + bi
ϕ(x)
· dx
li(aaccepted
+ bi)  that
li(a
+
bi)
be
x proceeds,
through
small
increments
li(a
+ bi)α + βi
√
α which
+ βi the inteϕ(x)
·
dx
(each
of
the
form
),
from
that
value
for
a
+
b
−1
=
a
+
bi
α
+
βi



x bi
=, aand
+ bi
bi · dx
ϕ(x)
ϕ(x)
· dx be 0,ϕ(x)
· adx+
then ϕ(x)
add all
gral
should
up to· dx
x=a+
x =however,
a + bi the ex−1
+ βi x = a
bili(aestablished.
. Thus the α
significance
is +
fully
Now,
+ bi)

dx
ϕ(x) · dxmany
ϕ
a ·+dx
bi the x
α + βi can beα constructed
α +ways:
βi thus
pathway
as
dx
+ βi ϕ(x)in
· infinitely
ϕ(x)
a+
bi magnitudes
ϕ(x)
· dx ϕ(x)
entire realmxof=all
real
can· dx
be thought
of as· dx
an
 infinite
ϕ imag· dx
x = a +line,
bi soxthe
x = a real
+ϕ(x)
biand
straight
of+allbimagnitudes,
+ bi a
= entire
a + biarealm
√
ϕ(x)
· dx
aphysical
+ bi aαwith
inary, can be
made
++
−1infinite
= a + plane,
bia + biwhere each
b βian
· dx =ϕ(x)
ϕ(x)
ϕ(x)
=∞
ϕ(x) ·determined
dx
· dx
point,
by
b, can
represent
the
dxa,ϕordinate
ϕ(x)
dx ϕ · =
· abscissa
ϕ ·from
dx a value
x =the
+
bi
ϕ · dx li(a
magnitude a + bi. Accordingly,
continuous
path
+abi)
ϕ(x)by a line,aand
ϕ(x) = ∞
a+
bi another
+ biis consequently
of
x to
ϕ(x)
 a + bi is represented

ϕ(x)
ϕ
·
dx
ϕ(x)
ϕ(x)
·
dx
possible
in
infinitely
many
ways.
ϕ(x)
·
dx



ϕ(x)
ϕ · dx I maintain
ϕ · dx always ob∞integral
ϕ · dx
then,ϕ(x)
that=the
= ∞
x−1
e pathways,
ϕ(x)value
bidifferent
ϕ(x)
= to
∞
tains the same
along
αatwo
++βi
dxϕ(x) =as∞long as
x
ϕ(x) never = 0ϕ(x)
ϕ(x) the two lines
within the area
 x−1
 enclosed between
∞
representingϕ(x)
the =
pathways.
This
whose e
x
=ϕa·isdx
+abibeautiful theorem,3 √
dx
ϕ(x)difficult
= ∞ proof
ϕ(x) occasion.
= a∞+ b −1
ϕ(x)I =
∞give on an appropriate
not
will
It is= a x
+ bi
√
ϕ(x)
a + b with
−1 =other
a + beautiful
bi
connected
truths
ϕ(x)
· dxrelating to the expansion of
series. A pathway to each point can √
always be found, li(a
which
never
+ bi)
∞. =
li(a +such
bi) a place whereaϕ(x)
a++bib= −1
a + bi
contacts
Hence
I insist,
that
one

must
avoid
such
points,
where
the
original,
fundamental
principle
ϕ(x)
·
dx


li(a
+clarity
bi) and easily leads to inϕ its
· dx
of ϕ(x) · dx evidently loses
consistencies. Furthermore, itis at the same time clear
α +from
βi this,
βi
ϕ(x) · dx can always have many
howα a+function
produced byϕ(x)
x = acan
+ bigo
values for
same value of x, when namely, the paths
 the
x = such
a +ex−1
bia point where ϕ(x)
α +=
βi∞, either never, or once, or
around
dx
ϕ(x) · dx
multiple times.
x 4 If one defines, e.g., log x by
ϕ(x) · dx
x = a + bi
a + bi
a + bi
ϕ(x) · dx
1
1

ϕ
·
dx
2
See Bessel’s Werke II, Leipzig 1876, S. 341, 1. Spalte.1
ϕ · dx
a + bi
3
Strictly speaking, it is still yet supposed, that
1 ϕ(x) is itself
a single-valued function of x, or
least1 only one system
ϕ(x)
ϕ at
· dx
1 of values,
without interruption in continuity, is assumed for whoseϕ(x)
value=within
∞
each whole
ϕ(x)
= ∞ surface-space. ϕ(x)
4
It appears in the manuscript as x = ∞
ϕ(x) = ∞
1
December 2008
ϕ(x) · dx
a + bi

1
ϕ · dx

ϕ(x) · dx
α + βi
x = a + bi
+2πi
−2πi
+2πi
−2πi
−2πi
+2πi
+2πi
−2πi
−2πi
42
December 18, 1811

1

· dx, infinite magnitude. ex dx
negative,
1
x
,
· dx,
ϕ(x) · dx

I have mentioned all of xthis before hand, in order to es x
1
1
· dx, tablish my point of view, that I must join Euler, when he said,5

· dx,
x
a
+
bi
1
x · dx,
that
where x starts off = 1, then logx x+2πi
is arrived
 at either without in
dx or more times;
cluding the point x = 0 or by rotating aroundϕit·one
dx

each time, the constants +2πi 1or −2πi are added; thus the multilogx
dx, If ϕ(x) can never become
ple logarithms of each number
 xis ·clear.
infinite for a finite value−2πi
of x, 1then
the integral is always only a
 xx< 1, and for values > 1, nec· dx,
ex − 1as real for the case where
 x
is taken
e dx
x the case
single valued function. This isxe.g.
for= ∞
,
ϕ(x) =
e dx
e − 1 ϕ(x)
,
 x
essarily
imaginary
values.
Its
difference
from
the
real,
x obtains
,
,
ϕ(x) =
0.001i),
x li(0.7 + 0.001i), li(0.8 + 0.001i),
x
e dx
li(0.5
+
li(0.6
+
0.001i),
x
x
dx
e
e − 1which Mascheroni, Soldner and you add to it, is =πi or 3πi or 5πi x
x
,
e −1
,
li(0.9 + 0.001i),
li(1.0x+ ,0.001i),
li(1.1
+ 0.001i),
etc.,
until
x ϕ(x) =
ϕ(x) = ex − 1 ,
the path
through
x = 1; one
would
beli(1.5
able to+ 0.001i)
x etc. I do not examine
x
,
ϕ(x) =
in an entirely similar way, that log – x = log + x (a theorem
 prove
x
x

so that
− 1 can be accepted, if limited
ewhich

 x
todx
real magnitudes, but which
dx
dx
e −1

x
must
be
immediately
ommitted,
when
my
above
principle
of
two
dx

logx

x
dx
x
logx
 x xe − 1
e − 1 dimensions is dx
bestowed upon the realm of all magnitudes.) Make
,
ϕ(x) e=
dx
logx
 x −x1xdx
logx
x
li x real for any single value of x between 0 and 1! But which
e x−e 1− 1
, whose value is always
ϕ(x)
=
dx
is certainly a single-valued
function
of
x,
value will then be assigned?
x
x
1 3
1one
1 4
converging, and will1always 1be represented
by
one
and
only
li(0.6 + 0.001i),
+ 0.001i),
+ 0.001
x +li(0.5
xx
+
x+
x
+
+0.001i),
etc.
1 4
li(0.5 +li(0.7
0.001i),
li(0.6 +li(0.8
0.001i),
li(
3
18
4
96

xx + xx + x + etc.
sensible series x + li(0.5
li(0.9
+
0.001i),
li(1.0
+
0.001i),
li(1.1
+
0.001i),
etc.,
until
li(1.5
+0.001
0.00
+18
li(0.7
+
0.001i),
li(0.8
+
0.001i),
− 1 96li(0.6 + 0.001i),
e0.001i),
li(0.9
+
0.001i),
li(1.0
+
0.001i),
li(1.1
+
4
1 li(0.5 1+ 0.001i),
1 li(0.6
+ 0.001i), li(0.7 + 0.001i), li(0.8 + 0.001i),
1 3 +dx
1
1 4
xx0.001i),
+ x3 etc.,
x4 + etc.
x+ +
+ until
 li(1.0
li(0.9x++0.001i),
li(1.1
li(1.5
+ 0.001i)
xx
+
x
+
etc.
xxx + 0.001i),
li(0.9
+
0.001i),
li(1.0
+
0.001i),
li(1.1
+ 0.001i), etc., until li(1.5 + 0.001i)
18
4
96
1e −
1
1 4
18
4
3 1 96
dx
x + xx + x + x + etc.
Without
doubt imaginary, but it should follow the law of the con
4
18 x 96
dx

nowhere a break ex abrupto? If you then go from
lix −tinuity, being
,
dx
logx
lix −
,
li(1.5+0.001i),
while allowing the imaginary part 0.001i to de
1  logx
1 chosen this one, since he dx
I would that1 Herr Soldner
had
x + xx + x3 +dx x4 + etc.
crease
to
0,
[to
li
1.5], it absolutely does not come to a real value
lix
−
,

would introduce a simple
new
function,
18
41 lix
96 instead of his
−
logx of li 1.5, 1but to one which depends upon -\pi i. With what you
1 3 logx
dx1 , 4
x + ,x + etc.
+ −
x + xx lix
 furnish
18 logx
4
for proof against Euler and myself, I find to criticize 1)
dx96
ex − 1

,
li x −
x
that you
dx say, if li x must become real in the whole circumference,
1 x
e − log
x one must etc., but of course the onus lies not on us, if the

dx
 x
then
 x x dx
−
1
e
lix −e − 1 ,
dx continuity should not be lifted without cause, and it should even

logx
dx regarded as simpler and
x
since a single valued functionexisx−always
1dx
lix −
,
dx especially since log x is  be proved, of course, that it should be taken as real in the whole
more classical than a multi-valued
one,
x logx
circumference.
2) To be sure,
ex dx
itself already a multi-valued function.
ex dx It would perhaps also be

x
 x
advantageous, for
 x
dy
x1
e −
e dx
x dx
dx
e
y
  xxx
x
1
e e−xdx
dx
xx
is log y as well as log –y, but never both simultaneously, rather the
or at least for

former, if the integral
can begin from y – 1, the latter, if it begins
 x
dx
1 y = –1; the second
e dx
from
integral
just as+understood
as the first,
= C +1is
l(±lx)
etc.,
logx
 1xx
in
general
log
y
+
C,
if
the
first
time
C
is
set
=
0,
the
second
time
e dx
1
C
is
set
=±πi
or
±3πi
etc.
However,
it
is
very
true
that
E
uler
’s
1x
1
remarks
require
a
correction
in
so
far
as,
if
the
integral
should
to introduce a suitable symbol and name, so much the more since,
 will
start from z = 0 , in no case
C 1be infinite. – Thus, according
with the problems from physics derived from li x, x itself is comex −
dx
to
my
opinion,
one
may
not
set
monly an exponential magnitude. If the truths for Soldner’s li
x
are carried over to my
1

dx
= C + l(±lx) + etc.,
 1x
logx
e dx
,

1
1
x
dx
1
1
logx
which for brevity I will signify with Ei x, Exponential Logarithm,
5 Institutiones calculi
1 I, 1768, Sections 228, L. Euler
1 integralis
xx S.128
+ x3 + etc.
x
+
then the integration is taken, so that Ei disappears for any real,
Opera
omnia,
ser.
I,
vol.
11,
18
4
dx
Δυναμις Vol. 3 No. 3 logx
December 2008
1
1
x + xx + x3 + etc.
18
4

+2πi
Gauss to Bessel
li(0.5 + 0.001i), li(0.6 + 0.001i), li(0.7 + 0.001i), li(0.8 + 0.001i),
(1 + 2αx + βxx)
li(0.9 + 0.001i), li(1.0 + 0.001i), li(1.1 + 0.001i), etc., until li(1.5 + 0.001i)

dy
 y
dy
y
Gauss to Bessel
43
December 18, 1811

rather must be determined
either with l(+lx) or with l(–lx), but
dx
only one decided on. – = C + l(±lx)
 + etc.,

dy of the investigation to
Moreover, Ilogx
believe the extension
dx
C + l(±lx)y+ etc.,
imaginary argumentslogx
will=provide
grounds for supremely interesting results. Nevertheless, from the foundations derived above,
 x
I would rather chose the function
e −1
  x x dx
dxe − 1
= Cdx
+ l(±lx) + etc.,
logx x
than

dx
 x
 logx
dx e − 1 dx

dx
x + etc.,
= C +logx
l(±lx)
because I surmise, that
the first will give a coinciding result. Thus
logx
for example, I would love1to know,1 whether this function, or what
x + xx + x3 + etc.
is the same, the series
18
4
dx
1
1
x + 1 xx + 1 x3 + etc.
3 logx
18
4
x + xx + x + etc.
4
18
(1 + 2αx + βxx)
can+become
x of the form a + bi. I
(1 + 2αx
βxx) 0, for certain finite1values of
1 3 it is even very probcan not yet maintain with certainty,
whether
x + xx + x + etc.
18most certainly infinite),
4 (then
able to me. If there is such a value
these magnitudes will be very remarkable decomposed and the
entire series can be decomposed into infinite factors of the form
(1 + 2αx + βxx) .
From a few other details, I can only add a few words at
this time...
1
1
1
1
December 2008
44
Bernoulli
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.
It should be noted, that sometimes quantities present
themselves, whose integrals at first glance, it seems, cannot be
√
√
4
found
Nonetheless, the
may
dxintegral
a2 x2 +
x4easily be found
dx a2by
x2this
+ xrule.
after a certain transformation, as in the following
cases.
1 2
e have seen previously how to find the differentials
√ √ 3
2x
√ x dx√a21.+ xIf
2 x + 3ax
2 +a
2 + 3a
2
instead
of2 + x3
one
writes
dx
x
dx
a
x
1 2
of quantities. Now we will, inversely, show how the
dx a3 + √
3a22x +
3ax42 + x3
2
2x
1 2
1 2
integrals
of differentials are to be found, i.e.,
x2those
dx √ dx a x + x√, then one finds the integral of the
√ latter,√namely
2x
2x
1 2 (a
1 dx
1 2 2 √1 2
2 +x
2++xx
3 (+
2
3 2 . And if one(writes
1 2
dx)
a
+
x
)
a2 +
x2 a + x
a
+
x
)
a
dx
a
3a
x
+
3ax
2
quantities,
from which the differentials are derived.
Now it is al√√3a + x
3
3
3 dx)
(a dx + x
x dx
2x
2
2
x dx
1 3
for
dx a√a3++x3a2 x + 3ax2 + x3 , then one finds the integral
ready
known
x2 dx
x2from
dx previous statements, that dx is the differential
3x
1 2
1 2
2
(a
dx
+
x
dx) a +√
x √
1
3
ofx2 x or 2 x + or – a constant quantity, x dx the differential
2 2
1 2
1 2
2
3
√
2 ax+
1 4
(
a
+
x
)
a2 +
(a +
3
(a
dx
+
x
dx)
1
1
3
3
2 x
2
3
3
of 3 x or 3 x + or – etc. and x dx the differential of 4 x or
(a
+
2ax
+
x
)
a
+
x
.
5
1 3
2
5
1 x
4 2 dx
x dx
√
3x
2 2
4 x + or – a constant quantity, likewise also
1 4
1 4
(a + 2ax + x2 ) a + x √
x
x
2. Also conversely
it2can2 occur, that 2one must pull one
is the differential of 1ax4 etc.
5
4
1 3
14 3 a dx
(a + 2ax + x ) a + x
2 "
41x
3x
3 x ax dx
or
more
variables
under
the
root
"
ax
5 sign, before the integral can be
2
1
3 "
2
taken, as with the following
example
"
(3ax3 dx
1 4
1 4 ax dx
3 ax
(3ax3 dx + 4x4 dx) ax + x2
1
4
4x
4 x ax3 dx
"
"

4 ax
etc.
(3ax3 dx + 4x4 dx) ax + x2 
.
3
4
p
dx
+
4x
dx)
ax + x2
(3ax
ax dx
From this the following general rule can be formed:
The integral of this cannot be taken by our rule as it
a
p+1
p
x
appears.
However, if one pulls in an x (under the root sign), the 2
is
the
differential
of
.
ax dx
p+1
(3ax dx + 4x
result
is
a
p+1

x quantity is

Therefore if the integral of any differential
3
4

2
3 2 dx
p+1
(3ax
+ 4x3 3
dx)
(3ax
x4 ax + x
2 dx + 4x3 dx) ax3 +
to be taken, then one must first of all consider, whether the given
24x dx) 3ax + x4 ,3
(3ax dx
+
4
(3ax dx + 4x
 dx) ax + x
quantity is the product of any differential quantity
2
3
√ multiplied by
(3ax
dx
+
4x
dx)
ax3 + x4
2
dy This
a + is
y then an
its “absolute quantity” raised to a certain power.
whose integral one finds by the rule
= (ax3 +
3
indication that one can find the integral by the rule given above.
√

 3
2
a + ay + y is to be
4
2
If, for example, the integral of the quantity dy
=4 (ax
xx44). ax3 + x4
(ax33 +
= 2 (ax
+
ax343+
+
2 xx4 )3)3 ax
=
+
xax3 + x4
found, then I see first, that dy is multiplied
by a1 +1
multiple of its
1
3
(ax
=
+
x
)
32

1 2
(a +ay)
3
1
absolute quantity a + y, raised to the
power
;
next
I
seek
its
+
y
3
2
= (ax
+ x4 ) ax3 + x4
2 +1
3. If a fraction
3 presents itself, whose denominator is a
integral by the above rule, namely,
1
square, cube, or other power, then one must choose its root as the a4 + 2
2
x dx
√
absolute quantity. Therefore, for
1
2
dx
1
xx dx
(a + y) a + y
(a + y) 2 +1
4
x
dx
1
+4 2a2 x2 + x4
4 + 2a2 x2 a
, i.e. 3
.
+ xx
2 x2 +
4
2 +1
aa4 +
2a
4
2 2+
4
2
xadx+ 2a
a x
+ x2 x
√
3
4
2
2
4
a + 2a x + x
Likewise one finds the integral of x dx a2 + x2 ,
√
2
2
2
2
which is the2following
−1 : (2a2 + 2x2 )
a
+
x
(a + y) a + y
a2 +
+ xx2
a
2
2
3
a + x is to be chosen as the absolute quantity, and one obtains

2
2 2 −1 2: (2a2 + 2x2 ) . If one chose a4 + 2a2 x2 + x4 as the ab√dy : √a +12 y 2
3
1
then
a
x 2 + 2x2 )
3
−1+
(2a
dy : a + y 1
(a + x2 ) 3 +1 = (a2 + x2 ) √a2 + x2 ;
−1
:: (2a
2x
2)
−1+: (2a
+ 2x2then
)
solute
quantity,
the integral of this fraction would not be
dy : a + y
8
+
1
3
√
4
2 2
4
2
2
√
4
2
2
4
+
2a
x
+
x
a
obtainable
by
this
rule.
−1+: (2a
2xx4)
2a2 xx2++
+
√
2 a + 2y a + y
√
2a
aa4 +
x2If+the
x4integrals of two quantities cannot be found india4 + 2ax24.
the integral of dy : a + y equals 2 a + y , the integral of
2 2
x + x4then sometimes it will be the case, that one can find the
a4 + 2avidually,
dx : xdx : x equals √
integral
of their combination. Example:
dx
:
x
2 a+y
1 0 11x0 = 1 × 1 = ∞
1 0
1
x = 0 × 1 =0 ∞ .
x = ×1=∞
0
dx : x0
0
0
1 0
1
December 2008
x = ×1=∞
0
0
W
1
1
1
1
1
√
2
2
√√
2
a(a
+
x)
√ 2 a(a + x)√a(a++xx)2 a + x=−2 (a
+
x)
aa++xx
5 22
√
a dx 2 a +3x √
53
2√
√
2
2
(a
++x)
x)x) aaa+
=
+
√ + x)2 a + x=
(a
a(a+
+xxx
− 55(a
45
=
x Calculus
dx
5 a+x
3
The First Integral
√
2
√ Bernoulli
√
a(a + x) a + x
x dx a + x
a dx a + x
3√ 2 √
√
2√
2
√
2
x) a + x − a(a + x) ax+ dx
x a+x √
dx (a
+ xx
aa dx
aa +
5 which is
3 √
x dx
a dx
a dx a + x
x
dx
a
dx
√
√
2
√ +√
+ √2
√2
x2 dx a + x
2
2ax
+ x22ax +2ax
+
x
a(a
+ x) a + x
x
dx
a dx
2ax
x
x+
dxx a + x
√
22 √ √ 23
√
+√
2
2 +
a(a
+ax)
x)
+ xx
, + x)√a + x
a(a
2ax + x2 a dx
2ax + x2
(a +33x)
a(a
+ xaa−+
x dx
√
The integral of either aquantity
is not x
known.
However, +
the√inte√5
(a2 + 2ax + 3x2 )dx a + x
dx √2ax
dx
2
2
2
√
2ax + x therefore x dx a + x
+x
+√
gral of their sum, √
√
x dx a + x 2
2 + x dx
2ax
+
xx2dx
2ax +a xdx
a
dx
+
√
(a + 2ax + x2 )dx2 a + x 2 √
2
√
√2
(a + x) a + √
x − a(a + x) a + x
2ax++x xdx2ax + x2
√
a dx
√
√
2
2
2
2
5
3
√ +
√
(a +
+ x)
x)22 aa +
a(a
+ x) aa +
+ xx −
− a(a
+ xx
2
(a
√
x2 dx
a√+x)
x
5
3
(a + x)2
2ax + x2
3
5
2
a dx + x dx
√
x
dx
a
+
x
a
dx
a
+
x
5
2
2
√
√
√
+ 2ax + x )dx
a dx + x dx
√
√
√
√ a+x
√
dx aa +
+(a
xx
2ax + x2 xx dx
√
2
2ax +
x22ax + x2 .
dx aa++xx. In this
remains
as
the
integral
ofa the
a
dx
+ xgiven quantity xx3dx
2
is
2ax + x
√
√
2
same manner
one finds the integral
of x √dx a√+ x . In other
2ax + x2
5. Sometimes, a fraction may seem not to have an in√
√
√
4
3
2
2
2
2
dxx ais +
x
x dx the
a +integral
x
dx
+xxx√
+
2axaa+
)dx xa +
tegral, but if one √
multiplies
words,
of (axx +dx
known,
as
x adxdx + xand
2 its+numerator
dx denominator by
√
2ax +ax√dx
2ax
dx
a
+
x
√
2
√ be obtained easily. So it
the
same
quantity,
then
its
integral
can
2
a
dx
a
+
x
√ immediately
, and afterwards was pfound
is4that√
of
2ax + x
3a++x2x
a dx
dx3a + 2x
x dx a + x
x dx a + x 2ax dx√a + x
√
√
is with
.
above,
that
of
3
√
x dx a + x
3a + 2x
√
2
a dx + x dx
(a2 +
2ax
+ xremaining
)dx a +term,
x
√aTherefore
one has the integral
of √
the
xp3 dx√
+x
√
√
√
√
x
dx
a
+
x
a dx + x dx
√ 3a + 2x
2
22 + 2ax + 2x22 )dxx3adx
(a
+
x
a
+
x
x
dx
a
+
x
.
4
(a
+
2ax
+
x
)dx
a
+
x
a
dx
a
+
x
√
2
√
√
(a(2a
+
2
. x 2 dx a + x √
4
3a+
+xax
2x
ax dx
dx
In
this
same
manner
one
will
find
the
integrals
of
dx
a
+
x
x
dx
+
x
dx
2ax
dx
a
+
x
2
√
√
√
x
dx
a
+
x
√
4
√
3
√
2 3 2 p 3
x dx a + x and even
x 3 or
the quantities x dx a +
ax3ax
dx 2++x2x
dx +
3ax
x 2x
dx a + x
+ x41)dx a √
+x
(2ax
√
Multiply the numerator and
p
√ denominator by x. Then one obtains
2
x
dx
a
+
x
.
Thus
also,
if
a
quantity
consisting
of several terms
2
3
a
dx
√
√
2
3ax + 2x
√ √xp dx a 3+a x+√x
√
4
ax dx + x dx
2+
dx
a
x
x
2
is
given,
its
integral
will
be
found
by
parts.
One
such
2ax
dx quantity
a + x is
1 aa dx
√ √
dx
+ xxa + x
+
2axaadx
√
ax dx + x2 dx
√
2a +
√ 3
3
4
x2 dx
+ 2x
x3
3ax
1
2
√
+
x
)dx
a
+
x
(2ax
1
.
First
I
seek
the
integral
of
the
first part
√
2
3
2x
√
3
4
√
√
3 3ax +
p
3 3ax + 2x
3ax2 + 2x3 ,
4 +x
+
x
)dx
a
(2ax
3
√
x
dx
a
+
x
dx
a
+
x
x
2ax
dx
a
+
x
, then that of the second,
. Their
1
2
3
(2ax3 +
√
3 3ax + 2x
sum√gives
the integral
of the whole.
√
1
2ax
dx
a
+
x
√
1
4
√
√
x2xdx
x x
√
dxa +
a+
√ whose integral is 3 3ax2 + 2x3 .
√
2ax
dx aa +
+ xx
1
3
2ax
dx
2
3
(2ax3 + x4 )dx a + x
Admonition
2ax dx
+x
√
andadenomina3 3ax + 2x 6. Conversely, sometimes the numerator
2ax3 dx a + x
√
tor are to be divided by the same quantity, in order
to obtain its
3
√
These
cases
√are the most important
dxin awhich
+ x integrals
1 2ax
2
x4 dx a + x
√
integral. Example:
x
dx
a
+
x
4
can
be
formed.
Indeed
several,
even
infinitely
many
others yet re√x dx a + x
√
√
√
√
dx main,
+ xxwith the help
4 are possible.
xx22 dx
aa +
3 which integrations
of
2
+ x a +However
x dx x adx
2ax dx a + x
x
ax2 dx
they
do
not
all
come
to
mind,
and
furthermore,
most
of them can
√ 2
1
√
2 x2dx
2
aax
+ x4 .
axbe
dx
reduced to those
dx ahere,
+ x1so
x4 cited
√
1 that with the help of these, the
√
2
2
4
desired
ones
can
be
achieved;
ultimately
a thousand methods of
a2 x2 + x4
a x +x
Divide every term by x. Then one obtains
solution
and
manifold
cases
according
to
the nature of the given
ax dx
√
quantities
present
themselves
to
the
attentive
observer. For this
ax
a2 dx
+ x2
axreason
dx it were no less impossible than useless, were we to provide
√
√
√
a2 + x2 .
a2yet
+ several
x2
others aside from those offered here.
a a2 + x2
√
√
Let
the one remark suffice, that important mathematical
√
a a2 +Its
x2integral follows by the rule: a a2 + x2 .
x dx and
a +theorems
x
problems
directly depend on the finding of integrals,
7. It also occurs sometimes,
that the integral of a given
√
both
those
already
found
as well as such as are yet desired to be
dxthearule.
+ x If one however adds anquantity is√not obtainablexby
found,
as
for
example
the
quadrature of plane surfaces, the rectidx a +
otherxquantity
toxit, whose integral one knows,√something may be
fication
of
curves,
the
cubature
of bodies, the method of inverse
√
a dx one
a +subtracts
x
produced whose integral can be taken. Next,
from
tangents,
or
the
finding
of
the
nature
of a curve from given propx
dx
a
+
x
√ that integral the added quantity, therefore the remainder is the
√
a dx a
+x
erties of its tangents,
as
well
as
that
which
belongs to mechanics,
√
desired integral. Example: x dx a + x .
of xfinding the center of mass, of impulses, of os(alike
dx the
+ xmethods
dx) a +
Because √
its integral cannot
√ be taken by a simple
cillations, and so forth. Through the finding of integrals, one also
dx)given
a + quantity.
x
method, add a√dx a(a+dx
The result is
x +toxthe
obtains the involutions of curves, and the method by which to
1
√
(a dx + x dx) a + x ,
1
determine their nature, and with the help of the involute to rectify
a dx a + x
√
whose integral is found by the rule
1 to be equal to
√
the2 curves themselves,
as Tschirnhaus did with his caustics.
(a dx + x dx) a + x
= (a + x)2 a + x
The
ease
of
finding
the differential of any given quantity
5
1
√√
2 dx + x 2dx)
(a
a
+
x
is
matched,
conversely,
by
the
difficulty of finding the integral
= (a + 1x) a + x .
√
2
5
2
of
any
given
differential,
so
that
we at times cannot even confi= (a + x) a + x
√
5
dently
assert,
whether
the
integral
of the given quantity can be
If one subtracts from this the integral of a dx 2 a + x , 2 √
(a
+
x)
=
a
+
x
√
5
Δυναμις
Vol. 3 No. 32
December 2008
√
a dx a
+x
√
= (a + x)2 a + x
2
5
a(a + x) a + x
3
√ 2
√
a dx a + a(a
x + x) a + x
√
√ 2 a(a + x) a + x 3
a dx a3 + x
√
2
√
√
a(a + x) a +2x(a + x)2 a + x − 2 a(a + x) a + x
1
3
√
√
(ax + x2 )dx a +
(ax + x2 )dx a + x
√
46
(ax + x2 )dx a + x
(
Bernoulli
√
√
a
+
x
=
y
√
a + x = y√
(ax + x2 )dx a + x
formed or not. I venture to assert at least, that every
whole and sumed avariable.
+x=
y

√
= y2 − a
p
p will2 be better
2 − x2
2 x clarified
through ana example.
+√
x = y Let the
a− x2 This
x, =xpy 2 −
x
a
ax
,
x
+
x
a
rational
quantity,
which
is
multiplied
or
divided
by


2
2
(ax
+
x
)dx
a + x.
quantity
whose
integral
is
desired
be
p
p
−
a
x
=
y
√ dx = 2y dy
a2 − x2 , x ax − x2 , x a2 + x2 , is either integrable or reducible to
a + x = y . Then x = y 2 − a and furdx
=
2y
dy
For
that
purpose,
I
set
the quadrature of the circle or the hyperbola. This we will show
ther dx = 2y dy . Therefore in the whole quantity
in what follows. Therefore, above all, it is to be carefully consid
√
x = y2 − a
dx = 2y2dy √
6
ered whether the given quantity, which one would integrate,3can 2
a + x =2y √
6
4 x )dx a + x = 2y d
(ax
3

(ax + x )dx a + x = 2y
dy
−
2ay+
dy .
(a
+
ax
−
x
)dx
√
be reduced by multiplication,
division, or through the extraction
x
a+x
2
dx =
dy a + x = 2y 6 dy − 2ay 4 dy
(ax
+ 2y
x2 )dx
2
(a3 + ax
− x3 )dx
of roots
to a quantity
which
has
one
of
these
root
terms,
multi−
a
x
=
y
x
(ax + x2 )dx
The integral of this is found immediately, easily, without
plied by a whole and rational quantity. If it is possible, then it is
2 7
2
5
2
5ado,
further
and
inserts the

dx =
2yindeed
dy is = 7 y − 5 ay . Now if2 one√
immediatelyp 
a serious proposition,
that thep 
given quantity can be
= 27 y 7 −
5 ay
p
(ax + x )dx a + x = 2y 6 dy −
2
2
2
2
2
2
2
x
−
x
,
x
,
x
+
x
a
ax
−
x
a
7
5
value
of
y,
one
obtains
integrated; if not, it is dependent on and reducible to the quadra=7 y − 5 ay
5
a+x
ture of the 
circle or the hyperbola.
3
2
3
=2 27 y 7 − 25 ay
√
2
=
(a + ax − x )dx
√ (ax + 2x2 )dx√a +
√x(x=+2ya)
63 x + a −
2
3
2
x
dy − 2ay 4 dya(x +
If, fora example,
the following quantity is given,
+x
(x +2a) x2 + a −(a√
2a(x +
2 2a) 72x + a .
5
=
(a3 + ax2 − x3 )dx
√
+)dx
2x2 )dx
2 7
2
2 )dx
(a + 2x
74 )dx
5ay+5(a2x
2
(a4 + a3 x + a2 x2 − x
x
√ = (x
a(x
+
a)
a)35√
x2+√a 2−
x
+
a
7 y+ −

√
√
2
a + ax2 +5x2
a2 +7x2
2
a+x
(x + a)3
ax
+
x
(a4 + a3 x + a2 x2 − x4(a
)dx
3
2
3
In
the
same
manner
the
integral
of
the
quantity
+ ax − x )dx
7
√
x ,
= 27 y 7 − 25 ay 5
ax + x2
√
2
2
3
√ √
√
2
2
(x + a) x + a − a(x + a)
(a
+
2x
)dx
2
2
2
2
2
2
a + ax +
=xy = y
5
7
a neither
+x =y
√
and its integral is to be taken, then it appears at first glance
√
2 + x2 = y
2 + x2
a
a
to be 
integrable nor to
have
a
relationship
to
the
quadrature
of
the


√
√
 
p 

2
2
2 −2x2 , pxp 2ax −2 x2 , xp x =
+ a)2 x + a
(x + a)3 x + a − a(x
− a2 x = x y=2 − ya22 − a2
yx22be
a2 +to
xp In
− x2 ,words,
xpx ax
, x athe+absolute
x
a2 other
−
xassumes
√
circle.
if aone
quantity
5
7
2
2
2
2
by setting
(a+x)
√  will be found
a + x, then its differential
 y = a + x . Then will x = y − a
that under the root
2
2 +2 x2 =2 y
x
=
(a3 sign,
+ axthe
− fraction
x3 )dx
2
2
a
2
2


y dy
= y: dyy: − ya − a . Further we get for the quantity
and
itdx = y dy
−=adx
x can be concluded
will also be a fraction, so from that nothing
by: ydx
y 4 − a2 y 2
a2 ya22
 self
dx = y dy : y 4 y−2 −
4
3
2
2
4
4 − a22 y 2 2
the rule. Therefore I(amultiply
the
and denominator of

+ a x
+ anumerator
x − x )dx
y
x = 4 The
y a−

2integral
of this is y 4 − a2 y 2 .Likewise, if one has
ya2
this irrational fraction by the√numerator,
uniteathe
product of y −
+x
a+
+x2x2 and
3
y4 
− a2
y2
3
2
3 (a3ax


ax − component
x )dx
(a +with
ax itself
− x to
)dxthe+rational
the numerator
of
the
quan(a − x)dx
2
2
2
2
2
2
2
2
2
2
2
x
x
dx = y dy : (a
y 
−
=x2√
(a2x
=
+a2x )dx : (aa ++
x +)dx
=2x :)dxa: + ax22 +
(a − x)dx
tity, so that a fraction results whose numerator is purely rational
4
2
2




2ax
− x2
y
−
a
y
(a
−
x)dx
√
(a
− x)dx
3
2
3
2
3
2
4(2y
−−aa−2y)dy
: y:4 − ya42 y−2 a2√
−
y2
y 4 − a2 y 2 (2y − a y)dy : (2yy√
yx22a y)dy
and whose denominator is irrational. Namely,
(a −
2ax
2ax − x2
2ax − x2 √ (
This quantity now shows, that either it has an integral,
2ax

√
√
− x2: = ay2 + x2 (a
2ax − x2 =

(a2 + 2ax
2x2 )dx
= − x)dx

√
√
(a − x)dx
√
√ √
a+x

2ax − x2 2= y
a+x
3 2ax
2 −x
2: =√2y4 − a2 y 22ax − x2

=
(a3 + ax2 −=x3 )dx
2ax − x = y
(a3 + ax2 − x3 )dx
−
a
y)dy
(2y
2
2
2ax
−
x
x
, y − x2 = y
x = a ± a −2ax
x
x = a ± a2


x=a± 
a2 − y 2 2
+4 )dx
a3 x + a2 x2 − x4 )dx
(a4 + a3 x + a2 x2(a−4 x
y 2 
y
x = a ± a2 −√
√
x = a ± a2 −
√
√
2
2 dx
√
∓y dy x
: =aa2 ±
− y 2a2 − y 2 , dx = ∓y dy :
==
yThen
y
= 2ax
a2 +−xx
ax + x2
.
set
ax + x2
2
− xdy=: y 
2

dx2ax
= ∓y
a2 −
y
dx = ∓y dy : a2 − y 2 
dx = ∓y dy : a2 − y 2  2
and

or is reducible to the quadrature of the hyperbola. Rules will be
dx = ∓y dy : a − y 2
x2 = a ± a2 − y 2
2
−
y
x
=
a
±
a
(a − x)dx
given below, as to how this can be recognized and done.
√
= dy

(a
−
x)dx
2
2

There remains yet something else, before we come to
2ax
− x2
dx
= 2∓y dy√: (a a− x)dx
− y = dy
(a
−
x)dx
2
dx = ∓y dy : a − y
√
√
=(ady− x)
2ax − x2 2 = dy.
the use and application of the integral calculus. To wit, we will
2ax − x
2ax − x2 √
explain another procedure for the formation of integrals, which
√
√2ax −
The integral of this is = y = 2ax − x2 .(a − x)dx
= y = 2ax −
condenses the general method by more than a little. Because
√
√
(a − x)dx
= dy
√ be
=y= √
2ax − x2 2 rule can √
henceforth
many
= applied
dy2 √
sometimes, due to the complexity of the given quantity, it is not
2axin−infinitely
x2
= y = 2ax −This
x
= y2ax
= −2ax
2 −x
x
2
cases,
even
in
those
which
seem
almost
hopeless
because
of
their
=
y
=
2ax
−
x
immediately clear, whether it is of a kind which is reducible to
1 the fact that this rule sometimes 1
complexity.
Because,
aside
from
one of the cases that we have presented before, and even, whether
√
1
√
makes the2=
quantity
in question
y = 2ax
− x2 much briefer, it also offers the
it has an integral or not. This procedure, however, reduces the
= y = advantage,
2ax − x that it stands immediately before one’s eyes, whether
quantity to fewer terms, so that one may find the desired integral
the transformed quantity can be integrated.
without difficulty. This is done however, by taking the quantity
1
To all these methods of finding integrals one can add the
under the root sign as the absolute quantity and setting it equal
following,
which because of its usefulness and easiness, is almost
to some variable, and transforming the quantity to be integrated
1
preferable
to all the rest. This method is however
1 only relevant
accordingly
into another consisting only of terms of the assumed
1
to
those
quantities
which
are
combined
with
irrationals.
Its whole
helping variable. One takes the integral of this quantity, which for
application,
accordingly,
consists
of
transforming
irrational
quanthe most part appears much simpler, which can be transformed
tities
into
rational
ones,
so
that
the
given
quantity
assumes
a
comback to the desired integral, by reinserting the value of the as-
December 2008
1
47
Bernoulli
pletely rational character, after which, when it is possible, its integral is easy to form. This leads therefore into the Diophantine
a3 dx
√
problem, which provides excellent assistance on such occasions,
x ax − x2
as will become more clearly evident by
examples.
a3 dx
√
For example, let
a3 dx
x√ ax − x2
√
a3 dx
2
x ax − x2
√ 3 dx ax
3− x
x √a
ax −√
xa2 dx 3
√
a2 dx
2 −
2x
xaxxax
x ax −
− x√
ax − x2
√
x
ax − x2
ax − x2
√
2the quantity
be
to
be
integrated,
which
is
not
feasible by the preax
−
x
√
√
ax2 − x2 2
methods,
but
which
can
be
done
in
the
following manner.
x
ax −vious
xax −
2
a2 x2
√
ax − x
2
2
ax − x2 is an irrational quantity, then to make it rational,ax − x = m2
ax − xSince
2 − x2 must become a square. Let therefore
ax − xax
a2 x2
ax − x2 =
ax − x2
a2 x2
m2
2 2
ax − x2 =
2 x a=xam2 : (a2 + m2 )
m2
ax − x = a2 x22 a2 x2
2 .
m
2
− x 2=
2 2
ax − xax=√
x
2 2 a
m
ax−−xx2m
==a2 m2 : (a2 + m2 )
ax
x = am2 : (a2 + m2 )
m
x = am2 : (a2 + m2 ) and consequently
From this
assumption
2
x = am√2ax
: (a−2 x+2 m
)
2
dx = 2a32a
m3dm
dm: (m2 + a2 )2 ,
= a m : (a2 + m2 )a, 3 dx
2 2 : (a22 + m2 ) √
x2=: (a
am
√
x
=
am
+
m
)
23 =
2 2a3 2dx
2
3
2a3 dm3
: (a m+ m )
− x = ax2m
√
aax
dx
dm
therefore
whole
given
22
2
2
3 quantity
3
2=the
2 am
2 =x
√
=
2a3 dm
a dx
am
:+
(am
+
)x2 ax=−2a
√
2a
dx
dm
a
m
:
(a
)
ax
−
x
3
2
2
2√
√
2
√
2
dx2 = 2a2 2m dm22 : (mx √
+
a
)
m
=
2
x
ax
−
x
3
m
=
2ax − 2x
2 2a3 dm
a m+:m
(a2dx
)32m dm3m
2 2
axm=: (a
)ax
ax − xax=−√
2 2
x ax −√xa2 dx m=
3+=m
x
−
x
2a
:
(m
+
a
)
2a
a
dx
dm
2
2
2
3
2m : (a + m )
ax
− x2 2+=a2a)√
m2
=
dx = 2a
m3dm
: (m
x ax − x2
2
−2a
:m
3
22 ax −
2 2x2
m
3 = 2a
2 : (m
2
x
,
dx
m
dm
+
a
)
dx = 2a m dm : (m + −2a
a )3:m
−2a33 : m 
3
m dm : (m2 + a2 )2 −2a3 : m
−2a :m
m= dxa2=x 2a
: (a − x) 
whose integral
is easy to find,
and indeed is −2a3 : m .
m = a2 x : (a − x)  2
x) now
m =3 
a22x : (aIf−one
−2a
:m
inserts the value m = a x 
: (a − x) , the
m = a x : (a − x)

 a2 x : (a − x)
result is
m
=
a−x
m = a2 x : (a − x)  − 4a5 − 4a4 x =

2
−2a
5 − 4a4 x


4a5 − 4a4 x x
a−x
4a

a
−
x
2
x
= −2a4a5 
a −
x
− 4a4 x
− 4a5 − 4a4 x = −2a22− a − x x
2
x 5 = −2a
−
x
4x
= −2a  x
−
a−x
−
4a
4a
= −2ax2
− x
a −x x
4a5 −x4a4 x
2
√
x
x
= −2a
a3√dx : x ax−
− x2
x
x
√
3
3
x+a=y
the
a33 dx :as
x√
axintegral
− x22 of a dx : x ax − x2a.3 dx : x√ax − x2
a dx : x ax −Likewise
x
√
to integrate √
3
2
√
dx
−ax=
3
x+
y3
2 + 2ax a
2 : x ax
x
dx
+
a
√
a3 dx : x ax − x2
√
3
2
2 √
3
dx
3
dx √
x2 + 2ax +xa2
ax3 + 2ax + a dx 3 x2 + 2ax + a2
dx 3 x2 + 2ax + a2x, x=+ya −
=y x
√
x
√
dxx3 x2 + 2ax + a2
3
x
3
2
2
2
dx
x =xy +−2ax
a + a dx = 3y dy
x
x2 + 2ax + a2 must become axcube. Let therefore x + a = y 3 .
√
2
2
3
x a+
2ax
2
2
x22 + 2ax
+ will
a2 x = y 3 −
=+3ya2 dy and
then
, dx
x2 +
x2 + 2ax
+ 2ax
a2 + a = y
x + 2ax + a2
Therefore the whole quantity
√
x22 + 2ax + a2
3
x
x2 + 2ax + a2 dx = 3y 2√
3 2 + 2ax + a2 = y
dy
√
x=y −a
3y 4 dy
dx 3 x2 + 2ax + a2
3y 4 dy
dx 3 x2 + 2ax + a2
√
.
= 3
=
3
2
2
2
2 y
x + 2ax
y −s
x
x=
y3 − a √
dx +
=a3y
dy
3y 4 dy
dx 3 x2 + 2ax + a2
If one can obtain the√integral of this, then one also has the integral = 3
3
y −s
x
a2 √
= y2 2
3
of the given quantity. x2 + 2ax +dx
3y 4 dy
x + 2ax + a2
= 3
y −s
x
√
1
3y 4 dy
dx 3 x2 + 2ax + a2
= 3
y −s
x
1
1
1
1
1
1
December 2008
48
What’s the Matter with Descartes
Vance
Exclusive Interview: René Descartes
What’s the Matter with Descartes?
Timothy Vance
T
here’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 coursTim
es 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 ...
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-
December 2008
49
Vance
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?
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 – MYSELF! You know: Cogito ...
D – Yes, all corporeal substance.
V – Ergo sum?
V – Substance made of what?
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
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.”
René Descartes Meditations – No. 1
V – Uh–huh ... Right.
2
René Descartes Principles of Philosophy Part II – ‘On Motion’
Translated by M.S. Mahoney, 1977
4
Mahoney, ibid.
3
5
Mahoney, ibid.
Mahoney, ibid.
December 2008
50
Vance
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
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.
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
D – Mais oui, for what you say is to me both “clear and
distinct.”
6
Mahoney, ibid.
7
Mahoney, ibid.
8
Meditations – “I. Of the Things of Which We May Doubt”
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.
December 2008
51
Vance
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?
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 Leibniz11 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–] <POOF!>
[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.
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.
11
i.e. Dynamics
December 2008

december 2008 Vol 3 No. 3

Transcription

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