June 2007
Vol. 1 No. 4
www.seattlelym.com/dynamis
EDITORS
Peter Martinson
Jason Ross
Riana St. Classis
ART DIRECTOR
Chris Jadatz
LAROUCHE YOUTH
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On the Cover
The Virgin and Child with
St. Anne and St. John the Baptist
(Burlington House cartoon).
Leonardo da Vinci, ca. 1500.
Cusa's conception of the
Maximum, coinciding with the
Minimum, in the Lamb of God
2
From the Editors
3
The Tragedy of Leonhard Euler
– Lyndon H. LaRouche, Jr.
4
A Scientific Problem: Reclaiming the Soul of Gauss
– Michael Kirsch
23
Translation: Some Geometrical Writings of Nicholas of Cusa
– Abraham Kästner
29
Translation: Cardinal Cusa’s Dialogue on Static Experiments
– Abraham Kästner
“…God, like one of our own architects, approached the task of constructing
the universe with order and pattern, and laid out the individual parts
accordingly, as if it were not art which imitated Nature, but God himself
had looked to the mode of building of Man who was to be.”
Johannes Kepler
Mysterium Cosmographicum
1
From the Editors
At the beginning of 2007, a website was put up by five
LaRouche Youth Movement members, dedicated to taking future statesmen thoroughly through a view of the Harmonically
composed universe, from the perspective of Johannes Kepler.1
This exhaustive website was only the tip of the iceberg, though,
as the “graduates” of that program returned home equipped with
a new scientific capability, which is today lacking within established scientific institutions. Along with “graduates” of the first,
New Astronomy phase of the educational program, they are now
providing intense seminars on the method of Johannes Kepler to
the ranks of the LYM, who will become capable of mastering
the discoveries of the world’s leading physical economist, Lyndon LaRouche. When the “Harmony” group finished, they
opened the door to the third phase of study – Carl Gauss’s application of Kepler’s Harmonies to the determination of the first
asteroid orbit.
However, the challenge confronting a student of Gauss,
differs from that facing a student
of Kepler. Kepler tells the reader
everything
about
how
he
developed the principles of
universal gravitation, and the
underlying principles of harmony.
In contrast, Gauss never told
anybody how he really produced
his discoveries. In fact, the predominant belief in the halls of
academia today is that nobody can
know just how Gauss discovered
anything. All one can do is study
the math formulas he wrote down,
and memorize how he gets from
the beginning of a derivation to
the end, even though it is obvious
that these chains of equations
were produced after Gauss
generated his insight. According
to this Ivory Tower doctrine, the
man dies with the body, and all
we can do is helplessly speculate
on what he was thinking.
Of course, this belief is bunk. If we couldn’t recreate the
ideas of other human souls, then there couldn’t have been any
economic progress! But, how does one study something invisible, like Gauss’s mind? LaRouche gave two bits of advice on
performing this intellectual archeology: 1) The key to Gauss’s
discoveries were Kepler’s harmonies, and 2) the legacy of Kepler and his followers, Bach and Leibniz, was defended and
carried through the hell of the Enlightenment, to Gauss by his
teacher at Göttingen University, Abraham Gotthelf Kästner.
LaRouche further elaborated on harmonies in his paper, Man
and the Skies Above:2
As Kepler's original discoveries of astrophysical and
related principles show us, we must turn to the faculty of hearing to provide us a method for correcting
the inherent errors embedded in naive readings of the
sense of sight. To be specific, we require harmonics.
We must do as the Pythagoreans and Kepler have
done, force the suggestions provided by merely seeing to be corrected by warnings heard from the domain of harmonics. In a more adequate regard for
experience, we must treat all of our other senses as
relevant modification of a world-view premised on
the integrated faculties of sight and sound alone.
This insight has formed a necessary
“magnetic North pole” for the
preliminary investigation of Gauss’s
discovery by the current group.
Thus armed, they dove into the
work
of
Kästner
and
his
collaborators, and especially their
work on Kepler.
The issue of ∆υναµις you now
hold in your hands is the first of a
series of issues that will document
the “Gauss” group’s preliminary
investigation of the space between
Kepler and Gauss. The next issues
will also provide a body of
reference material, in the form of
first-ever English translations of
papers crucial for rediscovering
Gauss’s mind. This issue contains a
paper written by Michael Kirsch, A
Scientific Problem: Reclaiming the
Soul of Gauss, which makes
comprehensible the conceptions of
Nicholas of Cusa, and demonstrates
how they were experimentally
elaborated by Johannes Kepler. Also included, are two sections
of Kästner’s Geschichte der Mathematik, which contain his
investigations of some of Cusa’s geometrical writings. These
were also translated by Kirsch.3
We begin with a short comment by LaRouche, on the celebrations occurring globally in honor of the 300th birthday of
Leonhard Euler.
Peter Martinson
Jason Ross
Riana St. Classis
2
Lyndon H. LaRouche, Jr. Man and the Skies Above, May 11, 2007.
http://www.larouchepac.com/pages/writings_files/2007/0522_skies.shtml.
1
3
http://www.wlym.com/~animations/harmonies.
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For online versions, see: http://www.wlym.com/~animations/ceres.
June 2007
2
The Tragedy of Leonhard Euler
LaRouche
The Tragedy of Leonhard Euler
Lyndon H. LaRouche, Jr.
June 3, 2007
Today the Fachschaft Physik der Uni-Dortmund 2007
is opening a festival commemorating the birth of the celebrated
Leonhard Euler three hundred years ago.
Euler, who rose to justly acclaimed fame under the influence of Gottfried Leibniz and the guidance of Jean Bernoulli,
had been celebrated as an accomplished representative of the
work of Leibniz and Bernoulli, until he changed his allegiances
in science rather radically, to the anti-Leibniz camp, as this is
typified in the clearest and most flagrant fashion, by his
wretched 1763 Letters To a German Princess.
In the matter of this about-face, it is neither useful nor
necessary to debate each Euler work one at a time. There is an
absolutely crucial and fundamental issue of science at stake.
Every other topic which might be dragged in as a kind of academic foliage, is essentially irrelevant to both the fact and implications of Euler’s apostasy, most notably that which put him
in embittered opposition to Gottingen’s Abraham Kaestner, and
Gotthold Lessing and Moses Mendelssohn at Berlin.
The issue is that treated with bold precision by Carl F.
Gauss in his own 1799 doctoral dissertation, the same issue for
which the famous student of both Gauss and Lejeune Dirichlet,
Bernhard Riemann, was celebrated by such as Albert Einstein
later: the most important of the issues of scientific method in all
known science to the present day, the issue of the ontological
actuality of the infinitesimal which remains the principal issue
of modern European science, from Nicholas of Cusa’s discovery
of the systemic error in Archimedes’ mistaken effort to treat the
circle as an expression of quadrature, and with Kepler’s celebrated treatment of higher order of the methodological fallacy of
the quadrature of the circle, in his definition of the principle of
motivated action in the generation of the planetary elliptical
orbit.
In fact, the entirety of the mainstream of actual progress from the work in Sphaerics by the Pythagoreans, and the
combined work of the Pythagoreans and the circles of Socrates
and Plato, is the conception of the infinitesimal as an ontologically efficient actuality, rather than, as Euler attempts, as do de
Moivre, D’Alembert, Lagrange, et al. to treat the concept of the
infinitesimal as merely a fantastic formality, rather than the ontological actuality recognized by Cusa, Kepler, Leibniz, Bernoulli, et al., the actuality of the catenary principle of the Leibniz-Bernouilli universal principle of physical least action, that of
the Leibnizian complex domain and the actually physical hypergeometries of Bernhard Riemann.
pretext for denying society the right to access to practical
knowledge of the use of various ordinary, and also higher forms
of “fire.”
In physical science, as opposed to mere mathematical
formalism, the central issue of these discoveries, of the use of
“fire,” or related kinds of scientific principles and technologies,
is the nature of knowledge of an efficient form of universal
physical principle. The crucial issue in the teaching and application of physical science for the promotion of the general welfare
of society, is the issue of whether or not a physical principle of
mathematical work is merely an enticing formality, or, as Kepler
defines the universal principle, of motivation of the planetary
orbital pathway as a physical motive, as Gauss saw the motive
of planetary action expressed in such forms as the asteroid orbits
of Ceres and Pallas.
The central achievement of Bernhard Riemann has
been that of the revolutionary advancement in methods of scientific practice which came boldly to the surface with Riemann’s
1854 habilitation dissertation, and the development of the physical conceptions in hypergeometry which came along the same
pathway cleared by that dissertation. That is the pathway opened
by Cusa’s exposure of the error of Archimedes’ quadrature of
the circle, by Kepler’s discovery of the physical science of elliptical functions and the calculus, by Fermat’s opening the gates
on the physical concept of least action, by Leibniz’s and Jean
Bernoulli’s development of the concept of a universal principle
of physical least action, by Gauss’s insights into the nature of
physical motivation, and the discoveries of Riemann.
It should be recalled by anyone claiming competence
in physical science, that the Kepler-Leibniz infinitesimal is not
the mere formality which de Moivre, D’Alembert, Euler, Lagrange, et al., proposed. It is expressed as a constant rate of
change of the direction of the motivated orbital pathway. It was
this conception of the infinitesimal, which was already implicit
in Archytas’ construction of the doubling of the cube, already
clear in Nicholas of Cusa’s rejection of the use of mere quadrature for the circle, and Kepler’s taking the attack on the fallacy
of the “equant.”
The world of today, is gripped by the onrushing force
of what threatens to become, soon, the gravest, planet-wide crisis in all modern history. The remedies for this are available,
provided we abandon the ivory-tower mathematical fantasies of
information theory which had mostly replaced emphasis in employment, on a return to physical-scientific progress in agriculture, industry, and basic economic infrastructure. The efforts of
Euler’s turn into awful ideologies such as those expressed by his
1763 Letter to a German Princess, is not the sort of thing we
should promote under the specific kinds of breakdown of the
production process which Europe and North America are suffering today.
The Issue Is Humanity
The essential issue implicit in Euler’s descent into
mere mathematician’s formalism, instead of physics, is not a
mere issue of formalities. The issue, as since Aeschylus’ Prometheus Bound, is whether or not the high priesthood reigning over
the opinions which society is permitted to believe, shall be a
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A Scientific Problem: Reclaiming the Soul of Gauss
Kirsch
A Scientific Problem: Reclaiming the Soul of Gauss
Michael Kirsch
Before launching into his highest achievement in Book V of
The Harmony of the World, in which he demonstrates that it is
through harmonics that the physics of the solar system are
known, thus redefining the nature of humanity as a whole, Johannes Kepler demonstrates that the causes of those harmonic
proportions with which we measure the universe, have their
origin from within the rational soul, as “abstract quantities.” At
the height of his argument he declares:
Nicholas of Cusa’s influence on Johannes Kepler in
every field of his work had its origin in Cusa’s establishing the
nature of the human soul’s relationship with the universe and
the Creator of that universe.
This relationship addresses the greatest challenge facing mankind, particularly today’s youth generation.
The nature of the universe as demonstrated in the two
web pages of the LYM on Kepler,2 has pointed to the reality,
that the principles which man discovers, never begin with necessity, or mere practical use. Science is, in fact, not a means to
an end, but an end itself: to address the higher purpose of mankind. What is this higher purpose? In all the aims of science,
mankind has been driven by an inner desire to accomplish the
greatest function of the human animal: to have fun. Man is a
creature which cannot be bounded by any bounds, because of
that which lies within him, his soul. It is in the nature of the
human soul to have fun, but a certain kind of fun, which can
only be called, real fun.
Today the “Boomer” generation filling the institutions
of government and science have lost an understanding of how to
have real fun, and in doing so, they have misplaced a thorough
conception of their own souls. Since they lack this freedom,
they also fail to understand the deeper implications of science,
and its relation to humanity. The effect of an entire generation
having lost the conception of the immortality of the human soul,
has been a dynamic and multilayered collapse of the U.S. and
world economy, the U.S. institutions of Government, and a
rabid empiricism which dominates science. Therefore, given
the need and possibility of such events as the recent Russian
proposal for joint U.S.-Russia cooperation on the Bering Straits
project, what is required today is a clear conception of the human soul.
Three months ago, and none too soon, a sea change
occurred in modern science; the elaboration by the LYM of Kepler’s achievement in actually redefining the potential of the
human species, the human soul, and the nature of all human
knowledge, put modern empiricism on notice and has shaken
the rotting foundations of current thinking. This revolution in
science sparked by the Kepler Two project3 must continue, so
that a new generation of economic scientists will be unleashed,
which will not fail to bring the essence of the human soul as
defined by Kepler in The Harmony of the World fully into the
domain of modern science.
In a fantastic irony, the needed challenge for such a
change in science intersects the specific task of this report: the
third phase of “Animating Creativity,” on Gauss, begs the question: by what means, might we discover the thought process that
allowed Carl Gauss to discover the orbit of Ceres? Understanding the principles he discovered, and comparing them with the
Finally there is a chief and supreme argument, that
quantities possess a certain wonderful and obviously
divine organization, and there is a shared metaphoric
representation of divine and human things in them. Of
the semblance of the Holy Trinity in the spherical I
have written in many places… We come, therefore, to
the straight line, which by its extension from a point at
the center to a single point at the surface sketches out
the first rudiments of creation, and imitates the eternal
begetting of the Son(represented and depicted by the
departure from the center towards the infinite points of
the whole surface, by infinite lines, subject, to the most
perfect equality in all respects); and this straight line is
of course an element of a corporeal form.
If this is spread out sideways, it now suggests a
corporeal form, creating a plane; but a spherical shape
cut by a plane gives the shape of a circle at its section,
a true image of created mind, which is in charge of ruling the body. It is in the same proportion to the spherical as the human mind is to the divine, that is to say as a
line to a surface, though each is circular, but to the
plane, in which it is also placed, it is as the curved to
the straight, which are incompatible and incommensurable. Also the circle exists splendidly both in the plane
which cuts, circumscribing the spherical shape, and in
the spherical shape which is cut, by the mutual concurrence of the two, just as the mind exist in the body, giving form to it and to its connections with the corporeal
form, like a kind of irradiation shed from the divine
face onto the body and drawing thence its more noble
nature.
Just as this is a confirmation from the harmonic
proportions of the circle as the subject and the source of
their terms, equally it is the strongest possible argument
for abstraction, as the suggestion of the divinity of the
mind exists… in a circle abstracted from corporeal and
sensible things to the same extent as concepts of the
curved, the symbol of the mind, are separated and, so to
speak, abstracted from the straight, the shadow of bodies.1
2
1
3
Johannes Kepler, The Harmony of the World, Book IV, Chapter 1
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See http://wlym.com/~animations.
See http://wlym.com/~animations/harmonies.
June 2007
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A Scientific Problem: Reclaiming the Soul of Gauss
Kirsch
method employed in his 1799 Fundamental Theorem of Algebra, it is furthermore clear that Gauss greatly obscured the nature of his thoughts throughout almost all his work. The
Napoleonic tyranny that swept Europe, and later the cultural
collapse of Romanticism following the Congress of Vienna,
were the conditions in which Gauss decided to take such a
course.4 However, since the nature of “harmonics” as discovered uniquely by Kepler must be carried forward and applied to
the domain of modern science, the implications of Carl Gauss’s
discoveries and the thinking he had concerning them, must be
fully comprehended.
To this end, there are no means more suitable for such
an immortal task—in reviving the nature of mankind in science
today, and the consequences which that implies—than to study
the mind of Nicholas of Cusa and his student, Kepler, whose
relationship of motion released the Earth from the shackles of
empiricism, and with it all of modern science. In carrying forward the scientific revolution of Cusa and Kepler, and without
losing the freedom of thinking involved in the completely integrated epistemology contained therein, the hidden genius of
Gauss will become accessible. In other words, how did Cusa
and Kepler think, as reflected in what is explicit in their work—
which can be a guide to reflect back onto Gauss’s work—
thereby drawing out the substance of what was implicit in his
unspoken thoughts?
Abraham Kästner, the architect of the German renaissance and the teacher of Carl Gauss, considered Nicholas of
Cusa to be a founder of many fields of science, which preceded
the work of many, including Kepler and Leibniz. This is cause
for celebration, and also indicates the great likelihood of
Gauss’s acquaintance with Cusa’s ideas.
Therefore, what we now show is how the discoveries
of Cusa and his conception of the human soul, took root in Johannes Kepler, and today provide the basis for discussing Carl
Gauss’s elaboration of: an anti-Euclidean harmonic solar system, his comprehension of the transcendental nature of the Kepler Problem, the applications of the method of Leibniz’s
infinitesimal in his discovery of the orbit of Ceres, and above
all, his contribution to the “higher purpose” of mankind.
Nicholas of Cusa sought to demonstrate that the Creator of the Universe was not something able to be reduced to a
particular metaphor or described in any way, but only known
inconceivably by the mind of man, and that all knowledge
sought and captured by man came from seeking after this
knowledge of the Creator. Cusa investigated the nature of such
a universe, that which he calls a “contracted maximum,” as the
medium between the absolute infinite and the plurality of finite
things. Here he returns the conception of the universe to the
Pythagorean conception of forms, which make up the “world
soul” in a universe which is not a duality, as defined by Aristotle, of, on the one side, unknowable principles and, on the
other, the world of the changeable sense, but rather a universe
with an infinite Creator whose perfection reaches through the
universe to all matter. Although there are many paradoxes he
sets forward concerning how the idea of a maximum existing in
plurality is known, we go here to the heart of the issue.
In the course of investigating the Absolute Maximum—a subject to which we will return—he makes the following observation: of things admitting of more or less, we never
come to an unqualifiedly maximum or minimum. Therefore, he
states, since only the cause of all causes is the Maximum, and is
the only absolute infinite not subject to being greater or lesser
by any degree, we never come therefore to Absolute Equality,
except in the Maximum. That is, only the Maximum which
contains all things in it, including the minimum, is equal to itself. Since absolute Equality is found only in the Maximum, all
things differ. From this comes an immortal statement by Cusa:
“Therefore, one motion cannot be equal to another; nor
can one motion be the measure of another, since, necessarily,
the measure and the thing measured differ,” and, “with regard to
motion, we do not come to an unqualifiedly minimum.”6
What implications did this hold for astronomy?
It is not the case that in any genus— even [the genus] of motion—we come to an unqualifiedly maximum and minimum. Hence, if we consider the various
movements of the spheres, [we will see that] it is not
possible for the world-machine to have, as a fixed and
immovable center, either our perceptible Earth or air
or fire or any other thing. For, with regard to motion,
we do not come to an unqualifiedly minimum—i.e., to a
fixed center. For the [unqualifiedly] minimum must coincide with the [unqualifiedly] maximum; therefore, the
center of the world coincides with the circumference.
Hence, the world does not have a [fixed] circumference.
For if it had a [fixed] center, it would also have a
[fixed] circumference; and hence it would have its own
beginning and end within itself, and it would be
bounded in relation to something else, and beyond the
world there would be both something else and space
(locus). But all these [consequences] are false. Therefore, since it is not possible for the world to be enclosed
between a physical center and [a physical] circumfer-
Part I: The Edifice of the World
Abraham Kästner, in 1757, in his Praise of Astronomy,
declared Nicholas of Cusa to be one of two “revivers of the edifice of the world” along with Copernicus.5 Even though Cusa
had written specifically on astronomy, as with his collaborator,
the famous astronomer Toscanelli, Kästner is most probably
making reference to Cusa’s De Docta Ignorantia. In that work,
there lies a principle so vast, that its implications will guide us
through the entirety of this investigation.
4
Tarranja Dorsey, First Thoughts on the Determination of the Orbit of
Gauss: http://tinyurl.com/2tzdnl/OrbitOfGauss.pdf.
5
See http://tinyurl.com/2tzdnl/KaestLobderSternk.pdf.
6
Nicholas of Cusa, De Docta Ignorantia, Jasper Hopkins translation.
Added words in square brackets are translator’s.
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ence, the world—of which God is the center and the
circumference— is not understood. And although the
world is not infinite, it cannot be conceived as finite,
because it lacks boundaries within which it is enclosed.7
Therefore, the Earth, which cannot be the center,
cannot be devoid of all motion… Therefore, just as the
Earth is not the center of the world, so the sphere of
fixed stars is not its circumference…
And since we can discern motion only in relation to
something fixed, viz., either poles or centers, and since
we presuppose these [poles or centers] when we measure motions, we find that as we go about conjecturing,
we err with regard to all [measurements]. And we are
surprised when we do not find that the stars are in the
right position according to the rules of measurement of
the ancients, for we suppose that the ancients rightly
conceived of centers and poles and measures…
Neither the sun nor the moon nor the Earth nor any
sphere can by its motion describe a true circle, since
none of these are moved about a fixed [point]. Moreover, it is not the case that there can be posited a circle
so true that a still truer one cannot be posited. And it is
never the case that at two different times [a star or a
sphere] is moved in precisely equal ways or that [on
these two occasions its motion] describes equal approximate-circles—even if the matter does not seem
this way to us.8
Cusa moved the Earth out of a fixed center, and set it into motion, an idea which would later be taken up by Copernicus.
Cusa sets up the paradox that since all motion is derived from
the comparison with something fixed, all astronomical knowledge of his time is thrown into error, since the platform of observations is itself moving. This would later be taken up by
Kepler in calculating the orbit of the Earth in Chapters 22-30 of
The New Astronomy.11 Cusa also established that since motion
never occurs around a fixed point, there are no perfect circles.12
This was left for Kepler to demonstrate in Chapters 41-60 of
The New Astronomy.13 Likewise the non-circular orbits are
constantly adjusting themselves to a different center, and thus
cause the orbits of the bodies to take a different course. Lastly,
Cusa did away with the idea that the there is a limit to the universe, at the “eighth sphere” of the fixed stars.
Thus a constantly changing universe was established,
with no fixed center. Within such an “imprecise” universe with
no place devoid of motion, how could the cause of motion be
determined, if motion is not determined by simply comparing
two objects, assuming one to be at rest? This higher concept of
motion was left untouched until Kepler established the true
physical causes in the New Astronomy in chapters 32-40.14
Part II: What is Science?
What therefore is man that he exists within such a universe? How must mankind approach the challenge of a uni-
In these passages, Cusa, considering the universe as a
product of a Maximum Creator with a certain paradoxical relation to the universe, derived principles which are seen today,
after the work of Johannes Kepler, to be entirely true. The universe which is infinite with respect to all things is such that it
even coincides with the minimum. And if we are talking about
the boundary of the universe, it is such that the center coincides
with the circumference. Since motion never comes to a minimum, there is no fixed center; neither the Earth nor the Sun is
completely devoid of motion. Thus the Aristotelian Ptolemaic
model system was exposed as a fraud.9 This truth would be
thoroughly demonstrated by Kepler in refuting the equant.10
11
See http://wlym.com/~animations/part3/index.html.
In Cusa’s Theological Complement he proves again why there can
be no perfect circles, referencing back to his De Docta Ignorantia.
Kepler is reported to have most certainly read this work. See Commentary Notes on Chapter II in The Mysterium Cosmagraphicum, and
Eric Aiton, “Infinitesimals and the Area Law” in F.Kraft, K.Meyer, and
B.Sticker, eds., Internationales Kepler Symposium Weill der Stadt,
1971 (Hildesheim, 1973), p. 286. Given Kepler’s knowledge of this
fact he most likely already knew what to look for when arriving at
Tycho Brahe’s house in 1600.
13
http://.wlym.com/~animations/part4/index.html
14
http://.wlym.com/~animations/part3/index.html. This higher understanding of motion was also the central question in Leibniz’s determination of dynamics, in opposition to the fraud of Descartes, as the
following quote from Leibniz’s 1692 Critical Thoughts on the General
Part of the Principles of Descartes shows: “If motion is nothing but the
change of contact or of immediate vicinity, it follows that we can never
define which thing is moved. For just as the same phenomena may be
interpreted by different hypotheses in astronomy, so it will always be
possible to attribute the real motion to either one or the other of the two
bodies which change their mutual vicinity or position. Hence, since
one of them is arbitrarily chosen to be at rest or moving at a given rate
in a given line, we may define geometrically what motion or rest is to
be ascribed to the other, so as to produce the given phenomena. Hence
if there is nothing more in motion than this reciprocal change, it follows
that there is no reason in nature to ascribe motion to one thing rather
than to others. The consequence of this will be that there is no real
motion. Thus, in order to say that something is moving, we will require
not only that it change its position with respect to other things but also
that there be within itself a cause of change, a force, an action.”[emphasis added]
12
7
Since it is not the maximum, the universe could have been greater,
but since in the possibility of being, matter cannot be extended unto
infinity, the universe could not be greater. Thus it is unbounded and
with respect to all that can be in actuality, nothing is greater than it.
8
In De Ludo Globi, Cusa, discussing the motion of the irregularly
shaped ball used for the game, and the conditions of the ground, and the
way in which each different player sets the ball on the ground, says “It
is not possible to do something the same way twice, for it implies a
contradiction that there be two things that are equal in all respects
without any difference at all. How can many things be many without a
difference? And even if the more experienced player always tries to
conduct himself in the same way, this is nevertheless not precisely
possible, although the difference is not always perceived.” Abraham
Kästner in his review of Cusa says that this is Leibniz’s Principle of
Indiscernibility. http://tinyurl.com/yv8kca/makKaestnerCusareview.pdf
9
For Kepler’s discussion of the Aristotelian and Ptolemaic models, see
Part I of his New Astronomy.
10
March 2007 Vol. 1 No. 3 http://wlym.com/~seattle/dynamis
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A Scientific Problem: Reclaiming the Soul of Gauss
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verse, which, as Cusa says, is a “contracted” image of the Absolute Maximum, in which imprecision enters into all considerations of measurement? Therefore, how does the human mind
then proceed to investigate the causes in such a universe?
In Nicholas of Cusa’s De Docta Ignorantia, he begins
by stating that all things desire to exist in the best possible manner, and that they use their judgment that this desire be not in
vain, allowing each being to attain rest in what they seek. With
the power of number, mankind judges the uncertain, proportionally, by comparing it with the certain. Cusa states an apparent
paradox that arises:
mally small, or the minimum, thus the maximum is such that it
coincides with the minimum. Since the maximum is not greater
or lesser, it does not allow opposition; there are no opposites in
the maximum, and therefore, he states what appears to be logically inconsistent: “Thus the Maximum is beyond all affirmation and negation: it is not, as well as is, all things conceived to
be, and is as well as is not, all things conceived not to be. It is
one thing such that it is all things, and all things such that it is
no thing, maximum such that it is minimum.”16
But how can such contradictions be combined? If we
are created to seek maximum ignorance, but such a maximum
only creates inconsistencies in our understanding, how can the
human intellect not have been created in vain? Cusa, throwing
Aristotle’s maxim “each thing either is or is not” out the window, stated that infinite truth must therefore be comprehended
not directly, as by means comparisons of things greater or
lesser, but, rather, “incomprehensibly comprehended!”17
To proceed further toward our end, Cusa spins Aristotle
in his grave by declaring:18
Both the precise combinations in corporeal things and
the congruent relating of known to unknown surpass
human reason to such an extent that Socrates seemed
himself to know nothing except that he did not know.
If we were created with a desire to seek knowledge and
given only these means of comparative relation, then a paradox
seems to arise. If all we come to know in our seeking is that we
don’t know, weren’t we created in vain?
Rather, we must desire to know that we do not know!
“No! It’s a trap,” an Aristotelian shouts, “don’t you
see? This proves that you can’t know anything about the invisible universe. All you can do is make assumptions a priori and
set up set of definitions and axioms that follow. Forget about
whether the initial axiom is true, just see if you can make it
work!” Somewhere, a Baby Boomer sighs with relief, “Thank
goodness you alerted me! I thought I was going to have to think
to get past this one. I like beliefs so much better. They just feel
right, you know?”
Instead, Cusa concludes:
16
De Docta Ignorantia Book I, Chapter 4. Cusa continues to elaborate
the characteristics of the Maximum in the following chapters.
He goes on to say that everything is limited and bounded
with a beginning and an end, and so all finite things never proceed to
infinity because then infinity would be reduced to the nature of finite
things, and thus the Maximum is the beginning and end of all finite
things. Every finite thing is originated: it could not come from itself,
because it would then exist when it did not.
In De Ludo Globi, he similarly demonstrates the necessity for
the maximum, stating that since all things must be something, and all
things exist, and in all existent things there is being, without which they
couldn’t exist, so, therefore, the being of all things is present in all existing things, and all existing things exist in being. Thus the most simple being is the exemplar of all existing things, and this exemplar, the
being of all things, or Absolute Being, is the Creator of all existing
things, for the exemplar of something generates that something as an
image of itself. Therefore, nothing exists without Absolute Being.
17
John Wenck accused Cusa of asserting that absolutely nothing could
be known. Cusa replied in his Apologia Doctae Ignorantiae: “For in
an image the truth cannot at all be seen as it is [in itself]. For every
image, in that it is an image, falls short of the truth of its exemplar.
Hence, it seemed to our critic that what is incomprehensible is not
grasped incomprehensibly by means of any transcending. But if anyone realizes that an image is an image of the exemplar, then leaping
beyond the image he turns himself incomprehensibly to the incomprehensible truth. For he who conceives of each creature as an image of
the one Creator sees hereby that just as the being of an image does not
at all have any perfection from itself, so its every perfection is from that
of which it is an image; for the exemplar is the measure and the form
(ratio) of the image.”
Cusa had been sent to Constantinople as part of his attempts
to reunite the Greek and Roman Churches. He returned in February
1438. At the end of De Docta Ignorantia, Cusa states, “while I was at
sea en route back from Greece, I was led (by, as I believe, a heavenly
gift from the Father of lights, from whom comes every excellent gift) to
embrace—in learned ignorance and through a transcending of the incorruptible truths which are humanly knowable—incomprehensible
things incomprehensibly.”
18
Aristotle in his metaphysics, after a lengthy attack on the Pythagorean conception of number states in his final conclusion:“the objects of
mathematics are not separable from sensible things, as some say, and
they are not the first principles."
If we can fully attain unto this knowledge of our
ignorance, we will attain unto learned ignorance... The
more he knows that he is unknowing… the more
learned he will be.
Now, after wrestling with this, ask the question: if we
seek to become learned in our ignorance, what must we study, to
attain the maximum learning of our ignorance?
Cusa proceeds, bringing us with him to measure the
Maximum, to that very end. But how can you measure the absolute Maximum? If measuring is done by means of comparative
relations, what can be compared to the absolute Maximum?
There is no comparative relation of the finite to the infinite.
Things greater or lesser partake in finite things, and the maximum does not. The “rule of learned ignorance”15 is that in
things greater something can always be greater, in things lesser,
always lesser, and thus in comparing two things we never find
them to be so equal that they could not be more equal indefinitely.
Cusa elaborates the paradox which the intellect faces
with such an incomprehensible maximum. Since the maximum
is not greater or lesser, it is both maximally large, and maxi15
http://cla.umn.edu/sites/jhopkins/DeLudo12-2000.pdf, Book II, section 96
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the mind a light-of-truth is present; through this light the mind
exists, and in it the mind views itself and all other things.”21
But, since truth is the form of the mind, it is not something greater or lesser, and thus as it is a Maximum to the mind,
it is not seen directly. Cusa likens the truth to an invisible mirror in the mind. And as is the rule of learned ignorance, that
which is not the maximum can always be a greater or lesser; that
which is not truth can never measure truth so precisely that it
couldn’t surpass the former measure: “Now, the mind’s power is
increased by the mind’s viewing; it is kindled as is a spark when
glowing. And because the mind’s power increases when from
potentiality it is more and more brought to actuality by the
light-of-truth, it will never be depleted, because it will never
arrive at that degree at which the light-of-truth cannot elevate it
more highly.”22
Astonishingly, this unsurpassable tension of the mind
in its search for Maximum truth is described by Cusa as, “the
most delectable and inexhaustible nourishing of the mind,
through which it continuously enters more into its most joyful
life!” 23
But wait, since our desire to know everything about the
universe clashes with the Maximum truth being infinitely distant, then, logically, wouldn’t the Creator be evil?
In truth, there is nothing more fun, as Cusa perfectly
describes:
“Moreover, that movement is a supremely delightful
movement, because it is a movement toward the mind’s life and,
hence, contains within itself rest. For, in moving, the mind is
not made tired but, rather, is greatly inflamed. And the more
swiftly the mind is moved, the more delightfully it is conveyed
by the light-of-life unto the Mind’s own life.”24
Therefore, although the view of the likes of Norbert
Wiener and his information theorist followers claim that mankind is in a race against entropy, and will never be able to discover everything fast enough, making them “[S]hip-wrecked
passengers on a doomed planet,”25 in truth, this paradox of the
mind’s inability to comprehend the entire universe, is not part of
an evil design, but is in fact what drives the universe forward.
The speculation of mankind is not a sign of an entropy of the
mind, but is nourishment itself, and in the process of mankind’s
discoveries, the universe develops.26
Since this is the purpose of mankind’s nature–to ascend
with the intellect–Nicholas of Cusa demonstrated that the universe itself is a reflection of this relationship of the mind of man
and the universe as a whole. The comparison for how the mind
seeks the truth in measuring the “Maximum Number” was demonstrated in Cusa’s extensive treatment of the relationship of the
curved and straight, which formed the basis for all of modern
science, and the ascent of which we will no longer delay.
We must leave behind the things which, together
with their material associations, are attained through the
senses, through the imagination, or through reason[leave them behind] so that we may arrive at the most
simple and most abstract understanding, where all
things are one, where a line is a triangle, a circle, and a
sphere, where oneness is threeness (and conversely),
where accident is substance, where body is mind (spiritus), where motion is rest, and other such things.
In conducting an inquiry into unseen truths, visible
images must be used to reflect the unseen as a mirror or metaphor. However, for the visible image to truly reflect the invisible, there must be no doubt about the image.19
As Cusa said before, the mind invokes comparative
relations of the known to the unknown to come to knowledge.
But all perceptible things are in a state of continual instability
because of the material possibility abounding in them. For example, when a geometer uses mathematical figures for measuring things he seeks not the lines in material, as he cannot draw
the same figure twice, but seeks the line in the mind. For perceptible figures are always capable of greater precision, being
variable and imperfect. Cusa says that the eye sees color as the
mind sees its concepts, but the mind sees more clearly, as insensible things are unchangeable.
As Plato’s Socrates said:
And do you not also know that [geometers] further
make use of the visible forms and talk about them,
though they are not thinking of them but of those things
of which they are a likeness, pursuing their inquiry for
the sake of the square as such and the diagonal as such,
and not for the sake of the image of it which they
draw?... The very things which they mold and draw,
which have shadows and images of themselves in water, these things they treat in their turn as only images,
but what they really seek is to get sight of those realities
which can be seen only by the mind.20
The triangle in the mind, which is free of perceptible
otherness, is therefore the triangle which is the truest. Cusa says
the Mind is to the mathematical figures it contains, as forms are
to their images. Then, since mathematical things in the mind are
the forms, and thus do not admit of otherness, the mind could be
said to be the form of forms.
The mind views the figures in its own unchangeability:
“But its unchangeability is its truth. Therefore, where the mind
views whatever [figures] it views: there the truth of it itself and
of all the things that it views is present. Therefore, the truth
wherein the mind views all things is the mind’s form. Hence, in
21
Nicholas of Cusa, Theological Complement
Ibid.
23
Ibid.
24
Ibid.
25
Norbert Weiner, The Human Use of Human Beings, Chapter II:
“Progress and Entropy”
26
Norbert Wiener, The Human Use of Human Beings
22
19
Abraham Kästner remarks on the importance of this concept in his
review of Cusa’s De Venatione Sapiente. See Translations from the
Geschichte http://wlym.com/~animations/ceres/index.html
20
Plato’s Republic, Book VI
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smaller than the rectilinear, and nevertheless never one
equal to the rectilinear. Therefore with incommensurable magnitudes this conclusion does not hold. That is
to say, if one could give one incidental angle that is larger than this rectilinear angle by a rational fraction of
the rectilinear, and another that is smaller than this rectilinear by a rational fraction of the rectilinear, then one
could also give one equal to the perimeter. But since
the incidental angle is not proportional to the rectilinear, it cannot be larger or smaller by a rational fraction of the rectilinear, thus also never equal. And since
between the area of a circle and a rectilinear enclosed
area there can exist no rational proportion…. Therefore
the conclusion is also here not permissible.28
Part III: On the Curved and the Straight
As Cusa’s criticism of the error of Archimedes on the
subject of the isoperimetric principle expressed by the
circle, echoes the relevant conception, the cognitive
power of the specifically human individual mind is not
a secretion of the living body, but a principle which
subsumes the living body dynamically. This dynamical
principle of human reason, reflects the idea of the image of the Creator.
– Lyndon LaRouche, Cusa and Kepler
Nicholas of Cusa demonstrated a fundamental truth about the
nature of the curved and straight. The mind’s attempt to relate
the curved and the straight represents its capability to measure
the universe as a bounding array of Maximum numbers, which
once identified—and distinguished in the same way as the human mind is distinguished from the Maximum—could be incomprehensibly comprehended.
Cusa begins his On the Quadrature of the Circle:
There are scholars, who allow for the quadrature of the
circle. They must necessarily admit, that circumferences can be equal to the perimeters of polygons, since
the circle is set equal to the rectangle with the radius of
the circle as its smaller and the semi-circumference as
its larger side. If the square equal to a circle could thus
be transformed into a rectangle, then one would have
the straight line equal to the circular line. Thus, one
would come to the equality of the perimeters of the circle and the polygon, as is self-evident.27
Cusa states that the central premise of Archimedes is: since one
can have a greater or a lesser polygonal perimeter, then one
can have also an equal perimeter.
Those who followed Archimedes thought therefore,
says Cusa:
Cusa had challenged this already in his De Docta Ignorantia:
[T]here can never in any respect be something equal to
another, even if at one time one thing is less than another and at another [time] is greater than this other, it
makes this transition with a certain singularity, so that it
never attains precise equality [with the other]… And
an angle of incidence increases from being lesser than a
right [angle] to being greater [than a right angle] without the medium of equality.29
If the square that can be given is also not larger or
smaller than the circle by the smallest specifiable fraction of the square or of the circle, they call it equal.
That is to say, they apprehend the concept of equality
such that what exceeds the other or is exceeded by it by
no rational—not even the very smallest—fraction is
equal to another.
See animation:
http://tinyurl.com/yv8kca/Moving%20Inciden
tal%20Angle.swf
But, Cusa says, there were those who disagreed that where one
can give a larger and a smaller, one can also give an equal. This
applies to the angles which arise in the relations of the circle and
polygon. He continues:
The nature of the incidental angle compared to the rectilinear angle drives the point home, that if the circle could be
converted into the polygon, then each of the parts of the circle
and each of the parts of the rectilinear polygon could be a part of
There can namely be given an incidental angle that is
greater than a rectilinear, and another incidental angle
28
Emphasis added. This question of incidental angles was a great epistemological debate with grand implications. See Will Wertz: Nicholas
of Cusa’s “On the Quadrature of the Circle” at
http://www.schillerinstitute.org/fid_97-01/012_Cusa_quad_circ.html
29
De Docta Ignorantia Book III Chapter I
27
All quotes in this section, unless otherwise indicated, are taken from
Nicholas of Cusa’s On the Quadrature of the Circle, translated by Will
Wertz. See: http://www.schillerinstitute.org/fid_9196/941_quad_circle.html
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the other, but a segment of the circle cannot be transformed into
a rectilinear area because of the nature of the incidental angles.
After showing this incommensurability of the curved
and straight angles, Cusa concludes:
minimum which is maximum and from a maximum to which a
minimum is not opposed.”31
He also states in his On the Quadrature of the Circle:
“In respect to things which admit of a larger and smaller, one
does not come to an absolute maximum…” and since “polygonal figures are not magnitudes of the same species…” a polygon
never becomes small enough or large enough to equal a circle.
“Namely, in comparison to the polygons, which admit of a larger and smaller, and thereby do not attain to the circle’s area,
the area of a circle is the absolute maximum, just as numerals do
not attain the power of comprehension of unity and multiplicities do not attain the power of the simple.
“The more angles the inscribed polygon has, the more
similar it is to the circle. However, even if the number of its
angles is increased ad infinitum, the polygon never becomes
equal to the circle unless it is resolved into an identity with the
circle.”
If a circle can be transformed into a square, then it
necessarily follows, that its segments can be transformed into rectilinearly enclosed figures. And since
the latter is impossible, the former, from which it was
deduced, must also be impossible.
Thus, the following property of the circle arises:
Just as the incidental angle cannot be transformed
into a rectilinear, so the circle cannot be converted into
a rectilinearly enclosed figure.
But how close could you get? Cusa says there is a incommensurability between the two kinds of angles, but what
exactly is it?
Just how close can one get to precision, and why is
absolute precision impossible with the curved and straight? To
demonstrate this Cusa says that it if one uses the contingent angle – a very small angle – it is possible to give: 1) an incidental
angle smaller than a rectilinear angle by the contingent angle,
which is not any rational fraction of the incidental angle and 2) a
rectilinear angle larger than the incidental angle by a contingent
angle which is also not any rational fraction of the rectilinear.
That is, an incidental angle + contingent angle = rectilinear angle, and a rectilinear angle – contingent angle = incidental angle.
But wait a second – Cusa says the contingent angle “is
not a rational fraction of the incidental or contingent angle.”
One cannot add and subtract incommensurable magnitudes to
attain equality.
The Characteristic of Learned Ignorance
All of the above was the gist of Cusa’s overview as to
what the nature of the problem is. Afterwards, Cusa identifies
the degree of incommensurability that exists when seeking for
the isoperimetric circle. Although he identified the incommensurability between the different angles, he had yet to identify the degree of imprecision that exists. What follows
therefore, is Cusa’s elaborate process of setting up incommensurable proportionals to box-in the nature of the species difference.
Isoperimetric means: equal perimeter. In the Mathematical Complement, the idea of isoperimetric takes a broader
meaning, in looking at triangles and squares and other polygons
that all have equal perimeters, and what the relationship of the
radii would be that circumscribe those figures.
Here, in On the Quadrature of the Circle, Cusa is
looking for the radius of the circle whose perimeter would be
equal to the perimeter of a given triangle which is inscribed in a
circle. Where would such a radius be? What would be its characteristics?
See animation:
http://tinyurl.com/yv8kca/Moving%20Contigent.swf
In the same way he says, one can give a square that is
larger in a perimeter by the circle, yet not by a rational proportion of the square, and one can give a smaller circle than a
square, yet not by a rational proportion of the circle. Therefore
a smaller and larger square can be given to the circle but never
come so close which is smaller or larger by a rational fraction.
As he said in De Docta Ignorantia, “Similarly, a
square inscribed in a circle passes—with respect to the size of
the circumscribing circle—from being a square which is smaller
than the circle to being a square larger than the circle, without
ever arriving at being equal to the circle.”30
He then remarks on what necessarily follows. In his
On Conjectures, Cusa had identified the nature of numbers such
as the circle: “Hence, species are as numbers that come together
from two opposite directions—[numbers] that proceed from a
30
See animation: http://tinyurl.com/yv8kca/QofC2nd.swf
First, he shows that the simple idea of an equality between the triangular perimeter and the circular perimeter creates
a paradox which yields the defining characteristic of the isoperimetric radius. This provides the pathway to box in where it
must dwell.
To demonstrate the equality of the circular to the triangular perimeter, he had to show that the “radius must be to the
sum of the sides of the triangle, as the radius of the [isoperimetric] circle is to the circumference.” But – and here is the crux –
since the radius has no rational proportion to the circumference
of the circle, such a radius would not be proportional to the sides
of the triangle, because if the radius is not proportional to the
circumference, and if the triangular circumference were equal to
31
De Docta Ignorantia, Book III, Chapter I
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the circle, then it would share in the lack of proportionality with
the radius.
tional to the one we are looking for, the extension must also be
proportional. But, the line drawn to the side of the triangle from
d can never be exactly proportional to the one sought since the
sought length is not proportional to the side of the triangle. It
cuts the side larger or smaller. So if the line cutting the side of
the triangle is extended by the proportion of the side of the triangle, its extension can never be exact either. So which extension is least non-proportional to the one sought?
The fact that we can find a length that is smaller than
the one sought, and one larger than the one sought, means there
should be a length where we can cut the line such that it is neither larger nor smaller, right? The closest we can come, Cusa
says, is when both extensions are equal to each other and thus
the amount by which the created length is larger or smaller than
the sought length is the smallest it can be, even though it is not
the sought length by the amount smaller or larger but not by a
rational fraction; again, because of the incommensurability between the isoperimetric radius and the perimeter of the triangle.32
See animation: http://tinyurl.com/yv8kca/QofCIncPer.swf
The sought-for line – the radius of the isoperimetric
polygon – cuts the side of the triangle. But what follows from
the above statement is, that since it is not proportional to the
circumference of the polygon, it would not be proportional to
any part of it, or proportional in square to any part of it. Therefore, in this diagram, since the radius of the isoperimetric circle
we are looking for, dl, is not proportional to the perimeter of the
triangle, then also the line dk – which is proportional to dl by
construction – would not be proportional to eb, de, or db. Nor
would the line ek, created by dk, be proportional to eb, de, or db.
See animations:
http://tinyurl.com/yv8kca/inscribed%20triangle.swf
http://tinyurl.com/yv8kca/Pi.swf
After finding the closest value for the isoperimetric radius, he
makes his point:
True, that is not the precise value, but it is neither larger
nor smaller by a minute, or a specifiable fraction of a
minute. And so one cannot know by how much it diverges from ultimate precision, since it is not reachable
with a usual number. And therefore this error can also
not be removed, since it is only comprehensible through
a higher insight and by no means through a visible attempt. From that alone you can now know, that only in
the domain inaccessible to our knowledge, will a more
precise value be reached. I have not found that this realization has been passed along until now. [emphasis
added]
And what this points to, is an extremely important affirmation by Cusa. Since, as was shown, no line can be drawn
that stands in rational proportion with the sides of the triangle,
no point on eb could be given precisely that the “sought length”
would be drawn through.
Thus, any length along eb, which is in proportion to eb,
would not be in proportion to the length sought. And also, any
length which is drawn from d such that it would be in proportion
to a length along eb, would not be the “sought length.”
So this gives us the method of approach to boxing in
our isoperimetric radius, right? Since the sought line is not proportional to eb and db, what we are looking for then, must be to
find the line which is the most non-proportional to them, and
then we will have the line which is the least non-proportional to
the “sought length.” The length we are looking for, when compared to the known lengths of the triangle, is the minimum with
respect to its degree of knowability. Therefore, we are looking
for the radius which brings us the most ignorance relative to the
known triangle!
Where must the cut be? One extends the length, dk,
which cuts the side of the triangle, proportionally as the line on
the side of the triangle – eb, created by the cutting line – is to
the whole side of the triangle ab [see animation below] and also
the line on the other side of the cut to the whole side. However,
since the line cutting the side of the triangle has to be propor-
At the conclusion, having thus demonstrated what he called a
“species” difference, which even Archimedes failed to see, Cusa
remarks on the “higher purpose” of seeking truth.
32
As an example of a non-proportionality between magnitudes, he says
that the lines bounding the incidental, rectilinear, and coincidental angles share in the non-proportionality that their angles share. They are
magnitudes which are larger or smaller than each other by a magnitude
larger or smaller than a rational fraction. This line he says is “before all
divisibility of the line… by which a straight line can cut a straight line
in two… It is like an unattainable endpoint [of a line]… nonetheless…
in its way, divisible by a curve.” The point he makes is that the normal
divisibility of a line which lies between two endpoints is different than
the divisibility of the line bounding the contingent angle, and yet it is
still divisible in its way. This contingent angle length is the difference
between proportionality and non-proportionality. This magnitude is the
type which describes how close one can approach the sought length.
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The measure with which man strives for the inquiry of
truth has no rational proportion to Truth itself, and consequently, the person who 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.
paring the curved and straight, the straight line participates more
in the infinite line than a curved line participates in it.36
Mathematics of the Infinite
See animation: http://tinyurl.com/yv8kca/infinitecircle.swf
Later, in his Theological Complement, Cusa introduces the needed conceptions that the ancients were missing. It
was not that they presupposed the coincidence in equality of the
circle and square, which Cusa says all seekers do,33 but that they
endeavored to manifest what they presupposed by means of
reason. “But they failed because reason does not admit that
there are coincidences of opposites.”34
“But the coincidence of those features which are found
to be diverse in every polygon… ought to have been sought
intellectually, in terms of a circle; and [then those inquirers]
would have arrived at their goal.”
Having demonstrated the species difference of the circle, Cusa introduced the exact method of approach to the “incomprehensible maximum” in his De Docta Ignorantia, again,
here, in the case of this maximum “number” indicated by the
species difference.
He writes in De Docta Ignorantia: “But since from the
preceding [points] it is evident that the unqualifiedly Maximum
cannot be any of the things which we either know or conceive:
when we set out to investigate the Maximum metaphorically, we
must leap beyond simple likeness.”35 In other words, to represent the infinite, which bounds all things, we must move from
mathematical relations in the finite, to mathematical relations in
the infinite, and only then compare these infinite mathematical
figures to the absolute infinite.
For it is the nature of the intellect to conceive of such
infinite relations, as the mind itself conceives everything in such
a way. When a mathematician draws a triangle or circle, he
looks to the infinite exemplar. The triangle drawn is actually
infinite in the mind, and not subject to size. The triangle that is
imagined in the mind, it is not thought of as large or small, it is
not imagined as 4 feet, 10 feet, or 1000 feet, but as the potential
of all triangles.
Applying the rule of learned ignorance from the De
Docta Ignorantia: any curve which admits of more or less cannot be a maximum or minimum curve. And measuring a curve
with the rule of learned ignorance, we see that the maximum
curved line is straight, and the minimally curved line is straight,
therefore, a curve is in reality nothing but partaking in a certain
amount of straightness to a greater or lesser degree. Now com-
Then Cusa says: “At this point our ignorance will be
taught incomprehensibly how we are to think more correctly and
truly about the Most High as we grope by means of a metaphor.” In the Theological Complement, with this “Most High”
number, Cusa applied this method of the infinite to a true solution of the quadrature of the circle. Cusa shows that the relations between the circle and polygons is only comprehended in
the infinite, that in the infinite all polygons coincide with the
infinite circle.
His point is best expressed in the 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 and he failed to grasp the higher concept.
Cusa, however, answered the question as follows: “We
come to the truth of the equality of curved and straight only
through considering the isoperimetric circle as triune through
the coincidence of opposites in polygons… The triune isoperimetric circle is the coincidence of three circles in which the perimeter of the circle is found whose measure is a straight line.
In such a circle, the inscribed circle and circumscribed coincide… and the polygon in the middle too.”
36
Cusa says on this topic “the most congruent measure of Substance
and accident is the Maximum.” Leibniz later demonstrated this issue of
substance, that if the predicates were in the substance, then a clear concept was had of the substance. (As Cusa says, the Creator creates, and
Man forms conceptions of the created. The clearest concept of the substance is when nothing interferes with predicate’s expression of the
substance, as is the case of the catenary curve, as the clearest expression of the principle of least action, as shown in the Leibniz construction of the catenary which most clearly expresses the irony of the
paradox of physical action: that is, the complex domain. Afterwards,
the implications of Cusa’s principle of Maximum-Minimum were developed in the infinitesimal calculus.
33
Cusa said that the knowledge is presupposed, to which the mind is
guided by a light of truth in the mind. And all who seek knowledge are
instigated by that infinite art or science.
34
All quotes in this section are taken from Nicholas of Cusa’s Theological Complement
35
De Docta Ignorantia, Book I, Chapter 12
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What is Cusa talking about? His point is, that real
isoperimetric circle is in the infinite. The solution exists in the
intellect, where the relations between different species becomes
clear. The infinite brings the boundaries of a species into the
understanding, thereby illuminating the concept of a generating
principle.
Cusa had made this point in De Docta Ignorantia as
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, similarly in the maximum, 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.”
Cusa concludes the discussion of his solution as such:
cannot be a smaller were as small as its arc. Cusa says, as
learned ignorance teaches, since neither the chord nor the arc
could become so small that they could not become smaller, both
are capable of being smaller, “since the continuum is infinitely
divisible.”38
Cusa: the arc is to the sine, as triple the radius
is to the sum of the cosine plus twice the radius.
r × a : r sin a = 3r : r cos a + 2r
See animation: http://tinyurl.com/yv8kca/kastneranimation.swf
The ancients sought after the squaring of a circle…
If they had sought after the circularizing of a square,
they might have succeeded… a circle is not measured
but measures… [I]f you propose to measure the maximal truth… as if it were a circular line—you will be
able to do so only if you establish that some circular
line is the measure of a given straight line.
Given a finite straight-line, a finite circular-line
will be its measure. Thus, given an infinite circularline, an infinite straight-line will be the measure of the
infinite circular-line… Because the infinite circularline is straight, the infinite straight-line is the true
measure that measures the infinite circular-line…
Therefore, the coincidence of opposites is as the circumference of an infinite circle; and the difference between opposites is as the circumference of a finite
polygon.37
At the end of Cusa’s Mathematical Perfection, after
investigating the minimal arc of a circle to determine the relation between the half arc and sine,39 he states:
In a similar manner, you yourself may derive the relationship with regard to the minimum in other curved
surfaces. What can be known in mathematics in a human manner, from my point of view, can be found in
this manner.40
In what is historically of great importance, Abraham
Kästner, in his review of Cusa’s works, remarked about this
statement:
That sounds like bringing in the infinitesimal calculus
(analysis of the infinite). Thus one could say something
to the cardinal which he had not considered. In fact, he
contemplated evanescent magnitudes, only he did not
know how this conception would be used.41
Infinitesimals?
In Cusa’s Mathematical Perfection, the aim of which was “to
hunt for mathematical perfection from the coincidence of opposites,” he investigates whether the smallest chord of which there
The Infinitesimal: An Imprecise Measure for the
Transcendental
Lyndon LaRouche, in his Paper For Today’s Youth:
Cusa and Kepler, wrote:
37
Nicholas of Cusa’s Mathematical Complement is not available in
English, thereby making many of the mathematical theorems in the
Theological Complement very vague. Among them is the following:
“There cannot be found a straight line equal to a circular line, unless
first the opposite is found, i.e. a circular line equal to a straight line.
Once this is found, then, from a proportion between circular lines, the
unknown straight line is found, through both the known line and known
proportion of circular lines… There can be exhibited a circular line that
is equal to a given straight line, but not conversely. For only if the former equality is known can the latter equality be known—and then
[only] as proportionally [equal], as is explained in my oft-mentioned
book Complementum.”
38
Kästner’s Review of Cusa’s Geometrical Writings, translated by Michael Kirsch. See elsewhere in this issue.
39
Cusa had also stated in On Conjectures, Part II, Chapter II: “For if
every chord is smaller than the arc that it subtends, and if the chord of a
smaller arc is more like its own arc than the chord of a larger arc [is
like its arc], then if we were to admit that the two chords of the halfarcs were equal to the chord of the whole arc, it would be evident that a
coincidence of chord and arc would be implied.”
40
Ibid.
41
Kästner, Review of Cusa’s Geometrical Writings, this issue.
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Cusa’s treatment of the circle, in correcting the error of
Archimedes, is… of crucial clinical significance, in our
search for insight, for our reaching out in our zeal to
touch the substance of the human soul within ourselves,
or in others.
they have for Gauss’s discovery of the orbit of Ceres? Kepler’s
conception of the entire universe was shaped most prominently
by Cusa, particularly on the question of “quantity.” In the second chapter of his Mysterium Cosmographicum, before putting
forward his conception of the nested Platonic solids as the organization of planets, it is Cusa’s curved and straight which
guides the way:
Cusa’s investigation of the curved and straight is a
model for the identification of the nature of the human soul. It
is more than a simple likeness. There is no other way to ascend
to the identification of species differences in magnitude. It is
the capability of the human mind, to conceive and discover the
relations between transcendental magnitudes through ascending
to the intellect and in viewing as if through a mirror, the image
of a higher principle reflected in the intellect as a species difference, and comprehended incomprehensibly. The transcendental
magnitude delivers mankind to an understanding of power, an
understanding of universal principles which express themselves
to the visible domain as an image of creativity.
Cusa concluded his On the Quadrature of the Circle
with this discussion: “And they are entities that have a circular,
interminable movement around the being of the infinite circle.
They encompass within themselves the power of all other species on the path of assimilation, and, beholding everything in
themselves, and viewing themselves as the image of the infinite
circle and through just this image—that is, themselves—they
elevate themselves to the eternal Truth or to the Original itself.
These are creatures bestowed with cognition, who embrace all
with the power of their mind.”
Indeed, for Nicholas of Cusa, the relation of the curved
and straight is no mere comparison, as such; that is, it is not a
case of “this is like that.” Nicholas of Cusa saw every human as
conceiving in their mind an infinite circle, which is the measure
of all things, as an image of the absolute maximum. All finite
things, all expressions of number, every polygon, and every
other shape is measured by this eternal conception of the infinite
circle. The intellect being continually guided forward by this
exemplar in the mind toward ever higher understanding of how
this measurement reveals the truth in all things.
Cusa saw the form of circular movement precedes all
circular movement and is altogether free of time. The form of
the circle is seen in reason, which exists in the rational soul. But
where is reason except in the rational soul? Therefore, if the
soul sees within itself the form of the circle, which is beyond
time, then it must be beyond time. Thus it cannot cease or perish.42
It was matter which God created in the beginning… I say what God intended was quantity. To
achieve it he needed everything which pertains to the
essence of matter; and quantity is a form of matter, in
virtue of its being matter, and the source of its definition. Now God decided that quantity should exist before all other things so that there should be a means of
comparing a curved with a straight line. For in this one
respect Nicholas of Cusa and others seem to me divine,
that they attached so much importance to the relationship between a straight and a curved line and dared to
liken a curve to God, a straight line to his creatures; and
those who tried to compare the Creator to his creatures,
God to Man, and divine judgments to human judgments
did not perform much more valuable a service than
those who tried to compare a curve with a straight line,
a circle with a square…
To this was also added something else which is far
greater: the image of God the Three in One in a spherical surface, that is of the Father in the center, the Son in
the surface, and the Spirit in the regularity of the relationship between the point and the circumference…
Nor can I be persuaded that any kind of curve is more
noble than a spherical surface, or more perfect. For a
globe is more than a spherical surface, and mingled
with straightness, by which alone its interior is filled.
But after all why were the distinctions between
curved and straight, and the nobility of a curve, among
God’s intentions when he displayed the universe? Why
indeed? Unless because by a most perfect Creator it
was absolutely necessary that a most beautiful work
should be produced.
This pattern, this Idea, he wished to imprint on the
universe, so that it should become as good and as fine
as possible; and so that it might become capable of accepting this Idea, he created quantity; and the wisest of
Creators devised quantities so that their whole essence,
so to speak, depended on these two characteristics,
straightness and curvedness, of which curvedness was
to represent God for us in the two aspects which have
just been stated… For it must not be supposed that
these characteristics which are so appropriate for the
portrayal of God come into existence randomly, or that
God did not have precisely that in mind but created
quantity in matter for different reasons and with a different intention, and that the contrast between straight
and curved, and the resemblance to God, came into existence subsequently of their own accord, as if by accident.
Part IV: Unfolded Implications
Cusa’s higher understanding of the purpose of mathematics was fully alive in the mind of Kepler. Kepler also found
that these conceptions and demonstrations of Cusa were necessary to continue forward to a higher understanding of the universe. Many of his discoveries were influenced by Cusa’s
thinking. Here we take a look at the broad range of such discoveries, keeping in mind the question: what implications do
42
For more on Cusa’s conception of the human soul, see Appendix.
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It is more probable that at the beginning of all
things it was with a definite intention that the straight
and the curved were chosen by God to delineate the divinity of the Creator of the universe; and that it was in
order that those should come into being that quantities
existed, and that it was in order that quantity should
have its place that first of all matter was created.43
muted proportion of density with extension, are
equals.45 Hence, between the point and the surface
there is everywhere an utterly absolute equality, a most
compact union, a most beautiful conspiring, connection,
relation, proportion, and commensurateness. And since
these are clearly three—the center, the surface, and the
interval—they are nonetheless one, inasmuch as none
of them, even in thought, can be absent without destroying the whole… The sun is accordingly a particular body, in it is this faculty of communicating itself to
all things, which we call light…46
In various of Kepler’s letters, he expressed the same sentiment concerning Cusa’s view of man:
“Geometry is one and eternal, a reflection out of
the mind of God. That mankind shares in it is one of
the reasons to call man an image of God.”
Infinitesimal Considerations
However, although Cusa discovered the method to investigate the Maximum, i.e. universal principles, he did not indicate how these principles express themselves at every moment
of change. But, as Kästner remarked, Cusa's investigation in his
Mathematical Perfection47 appeared to be introducing infinitesimals into the construction. One wonders, therefore, what
influence did this have on Kepler's discovery of such magnitudes?
Kepler, moving beyond geometry, into the domain of
physics, discovered the form in which the motion along the orbit
expresses the unseen physical principle at every moment. Kepler had found out he was wrong in the small, by 8' of arc. But
in order to correct his error, he had to know the whole orbit.
Working on calculating the motion of the Earth, Kepler,
in Chapter 32 of the New Astronomy, derives the principle that
the time needed to traverse an arc of the orbit is proportional to
the distance from the sun, stating: “But since[the daily arc of the
eccentric at aphelion] and [the daily arc of the eccentric at perihelion] are taken as minimal arcs, they do not differ appreciably
from straight lines.” Why did he do this? Kepler was the first
to discover the principles of planetary motion. They were not
self-evident! In order to know the whole orbit, he had to discover the relationship expressed at each moment. Thus, in
thinking how to represent a path that reflects the power of the
Sun, he conceived of the idea of using “minimal arcs” that represent moments of a process of continual change along the orbit.48 Kepler was able to determine the whole orbit by
understanding the relationship expressed in the smallest possible
“Man’s intellect is created for understanding, not
of just anything whatsoever but of quantities. It grasps
a matter so much the more correctly the closer it approaches pure quantities as its source. But the further
something diverges from them, that much more do
darkness and error appear. It is the nature of our intellect… the study of divine matters concepts which are
built upon the category of quantity; if it is deprived of
these concepts, then it can define only by pure negations.”
“No eerie hunch is wrong. For man is an image of
God, and it is quite possible that he thinks the same way
as God in matters which concern the adornment of the
world. For the world partakes of quantity and the mind
of man grasps nothing better than quantities for the recognition of which he was obviously created.”44
Later, in Kepler’s investigation of light in his Optics in
1604, this influence of Cusa concerning the curved and straight
and his conception of the infinite sphere, would again present
themselves as the opening conception concerning the internal
relations of space:
For when the most wise founder strove to make
everything as good, as well adorned and as excellent as
possible… [there] arose the entire category of quantities, and within it, the distinctions between the curved
and the straight, and the most excellent figure of all, the
spherical surface. For in forming it, the most wise
founder played out the image of his reverend trinity.
Hence the point of the center is in a way the origin of
the spherical solid, the surface the image of the inmost
point, and the road to discovering it. The surface is understood as coming to be through an infinite outward
movement of the point out of its own self, until it arrives at a certain equality of all outward movements.
The point communicates itself into this extension, in
such a way that the point and the surface, in a com-
45
In Cusa’s De Docta Ignorantia, Book I, Chapter 23, he said: “The
center of a maximum sphere is equal to the diameter and to the circumference… for in an infinite sphere the center, the diameter, and the
circumference are the same thing.”
46
Kepler, Optics, Chapter I
47
Kepler is also said to have certainly read this work. See Eric Aiton,
“Infinitesimals and the Area Law” in F.Kraft, K.Meyer, and B.Sticker,
eds., Internationales Kepler Symposium Weill der Stadt, 1971 (Hildesheim, 1973), p. 286.
48
Gauss in his Summary Overview very often finds himself dealing
with higher order magnitudes. Like Kepler, he swapped curved areas
with straight areas in the small. In the Summary Overview, g represents
the sector of an orbit between to positions of a heavenly body and the
sun, and f represented the triangle formed between those two observations and the sun. In one calculation, Gauss stated, “We can set f’ : g’
= 1, since the difference is only of the second order.”
43
Johannes Kepler, Mysterium Cosmographicum, Chapter II
These are taken from three different letters. All are found in the
book Kepler, written by Max Caspar.
44
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part of the orbit. In what is similar to the later physical differential outlined in Bernoulli's lectures on the Catenary, Kepler
found that there is a physical relationship which the motion
along an orbit must maintain at every moment: the motion expresses a continuity of area in relation to the time that the planet
expends in moving along the orbit.
Leibniz later generalized the method for the actual
physical actions of the universe so that the infinite may be accessible to the human mind. Leibniz showed with the calculus,
that the many physical curves which he and the Bernoullis investigated were the reflection of an unseen physical principle, a
dynamic, which represented itself as knowable to the human
mind in the form of an infinitesimal relationship, as a metaphor
for that dynamic. However, Leibniz moved even further than
the recognition of these infinitesimal relationships and discovered the ability to identify the principles that draw out the action
of motions. At his highest point, after exposing the fraud of
Cartesian physics by posing the challenge of the curve of
isochronous descent, he then discovered the complex domain,
(involving the “integral” of the catenary), a higher geometry in
which the action of physical principles could be represented.49
As we work forward through Gauss’s discovery, the
reader should keep in mind that the higher-order magnitudes
that Gauss uses, found their basis in Cusa’s ideas, were first
applied by Kepler, and were later generalized by Leibniz. The
mind measures the infinite, not directly, but, as Cusa showed,
metaphorically, in the form of the idea of an infinitesimal as a
reflection of the infinite.
Kepler writes in his Optics:
Speaking analogically rather than geometrically,
there exists among these lines the following order, by
reason of their properties: it passes from the straight
line through an infinity of hyperbolas to the parabola,
and thence through an infinity of ellipses to the circle.
For the most obtuse of all hyperbolas is a straight line;
the most acute, a parabola. Likewise, the most acute of
all ellipses is a parabola, the most obtuse, a circle. Thus
the parabola has on one side two things infinite in nature–the hyperbola and the straight line–and on the
other side two things that are finite and return to themselves–the ellipse and the circle. It itself holds itself in
the middle place, with a middle nature. For it is also infinite, but assumes a limitation from the other side, for
the more it is extended, the more it becomes parallel to
itself, and does not expand the arms (so to speak) like
the hyperbola, but draws back from the embrace of the
infinite, always seeking less although it always embraces more.
With the hyperbola, the more it actually embraces
between the arms, the more it also seeks. Therefore,
the opposite limits are the circle and the straight line:
The former is pure curvedness, the latter pure straightness. The hyperbola, parabola and ellipse are placed in
between, and participate in the straight and the curved,
the parabola equally, the hyperbola in more of the
straightness, and the ellipse in more of the curvedness.
For that reason, as the hyperbola is extended farther, it
becomes more similar to a straight line, i.e. to its asymptote.50 The farther the ellipse is continued beyond
the center, the more it emulates circularity, and finally
it again comes together with itself… the lines drawn
from these points touching the section, to their points of
tangency, form angles equal to those that are made
when the opposite points are joined with these same
points of tangency. For the sake of light, and with an
eye turned towards mechanics, we shall call these
points “foci.”51
“Maximum” Conic Sections
In a letter to his friend J.G. Brenegger on April 5th
1608, Kepler wrote, among other matters: “Cusa said the infinite
circle is a straight line.” Cusa’s idea led to a breakthrough in
conics by Kepler in his Optics, achieving a continuity of conic
sections.
While investigating the hyperbola and the relation between the chord and the sagitta, as the focus moves closer to the
base, he writes, “The sagitta52… is ever less and less until it
vanishes and the chord at the same time is made infinite since it
coincides with its own arc (speaking improperly since the arc is
a straight line).” 53
50
What implications did this have for Gauss’s later use of this continuity of conic sections in the Theoria Motus? In an interesting echo of
this sentiment Gauss also treats the parabola as an infinite ellipse. “If
the parabola is regarded as an ellipse, of which the major axis is infinitely great…”
51
Kepler, Optics, Chapter 4.
52
In the diagram the sagitta it is the length A, the focus of the hyperbola, to S on the axis of the hyperbola.
53
Kepler, Optics, Chapter 4.
See animation: http://tinyurl.com/yv8kca/radius%20equals.swf
49
For more on the Leibniz Calculus, see the October 2006 issue of this
journal, Vol. 1, No. 1, at http://wlym.com/~seattle/dynamis. More on
the Leibniz-Bernoulli breakthrough of integration and its implications
for Gauss’s work will be forthcoming at a later time on this Orbit of
Ceres webpage.
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Echoing the infinite metaphors of Cusa, he continues:
“For geometrical terms ought to be at our service for analogy. I
love analogies most of all: they are my most faithful teachers,
aware of all the hidden secrets of nature. In geometry in particular they are to be taken up, since they restrict the infinity of
cases between their respective extremes and the mean with
however many absurd phrases, and place the whole essence of
any subject vividly before the eyes.”54
Later, Leibniz applies what he called the “law of continuity”55 as the measurement for the error in Descartes’ rules of
motion:
This principle has its origin in the infinite and is absolutely necessary in geometry, but it is effective in physics as well, because the sovereign wisdom, the source
of all things, acts as a perfect geometrician, observing a
harmony to which nothing can be added. This is why
the principle serves me as a test or criterion by which to
reveal the error of an ill-conceived opinion at once and
from the outside, even before a penetrating internal examination is begun. It can be formulated as follows.
When the difference between two instances in a given
series or that which is presupposed can be diminished
until it becomes smaller than any given quantity whatever, the corresponding difference in what is sought or
in their results must of necessity also be diminished or
become less than any given quantity whatever… A
given ellipse approaches a parabola as much as is
wished, so that the difference between ellipse and parabola becomes less than any given difference, when
the second focus of the ellipse is withdrawn far enough
from the first focus, for then the radii from that distant
focus difference from parallel lines by an amount as
small as can be desired…56
“Given the mean anomaly, there is no geometrical method of
proceeding to the equated, that is, to the eccentric anomaly. For
the mean anomaly is composed of two areas: a sector and a triangle. And while the former is numbered by the arc of the eccentric, the latter is numbered by the sine of that area multiplied
by the value of the maximum triangle, omitting the last digits.
And the ratios between the arcs and their sines are infinite in
number. So, when we begin with the sum of the two, we cannot
say how great the arc is, and how great its sine, corresponding to
this sum, unless we were previously to investigate the area resulting from a given arc; that is, unless you were to have constructed tables and to have worked from them subsequently.”
– Johannes Kepler, New Astronomy, Chapter 60
The question arises, what was Kepler’s Problem?
What did he do that caused such ferment after his death? Why
was there a political operation to get rid of his Problem?59 Reflect on Cusa’s discussion of the nature of the human mind’s
relationship to infinite truth as the true relation of curved and
straight.
Above all, this was Kepler’s “problem.” It was the
“problem” which led him to seek the relationship between the
physical causes and the true motions of the planets.
After Kepler succeeded in demonstrating the physical
cause of the motions of the planets, he then ventured forth to
correlate that cause with the motions. This required not merely
associating a known principle with observations; the power of
the species from the sun caused the motions of the planets to
express themselves in the form of the countless paradoxes of
Chapters 41-60 and led Kepler into an unexplored domain of the
mind. And only by the passion with which he chased after it,
with a presupposition of the existence of the truth, willing to
The Transcendental
Lastly, and perhaps of greatest importance, is the foundation of the transcendental magnitude discovered by Cusa and
its contribution to the “higher purpose” of mankind. 57,58
54
Ibid.
G.W. Leibniz, “Critical Thoughts on the General Part of the Principles of Descartes,” 1692, in Leroy Loemker, ed, Leibniz: Philosophical
Papers and Letters, Vol II, No. 42
56
A letter of Leibniz, 1687, in Loemker, Vol II, No. 37
57
Cusa identified the nature of the species difference in the Quadrature
of the Circle. His solution to “rectify” the curved line, was to apply his
method of coincidence of opposites with the maximum circle. How
would Cusa’s method be applied to solve the Kepler problem, which
expresses the inability to relate the arcs and sines? Further, how does
Cusa’s method therefore lead into the higher functions of Gauss and
Riemann which address the Kepler Problem?
58
Transcendental equations and magnitudes are employed and encountered by Gauss throughout the Theoria Motus. Gauss discusses the
Kepler Problem, and makes advancements toward solving the problem.
In one location there, Gauss remarks that it is possible to determine the
whole orbit by two radii vectors if their magnitude and position are
given together with the time taken to move from one radii vector to the
55
next(between the two positions). But, “This problem”, he says, “considered among the most important in the theory of the motions of the
heavenly bodies, is not so easily solved, since the expression of the
time in terms of the elements is transcendental…”
59
Peter Martinson Neither Venetians nor Empiricists Can Handle Discoveries,
http://www.wlym.com/~animations/ceres/PDF/Peter/Astronomy.pdf
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man as in the image of the Creator.65 As he said: “Finally there
is a chief and supreme argument, that quantities possess a certain wonderful and obviously divine organization, and there is a
shared metaphoric representation of divine and human things in
them…”66
With these harmonies established as proportions from
the soul, Kepler then took up his edifice of the world from his
Mysterium and bringing in his New Astronomy, sought to demonstrate the causes of the motions. Kepler determined that the
extreme motions of the planets at perihelion and aphelion were
the area to seek for harmony in the heavens, and he proceeded to
calculate every possible proportion between each of the planets’
diverging, converging, and extreme motion in pairs. Once he
then fit the planets’ harmonies to the musical scale, he went on
to determine the origin of the eccentricities of the planets and
also, to look at the Solar system as a harmonic whole.
As soon as Kepler began to organize the Solar System
as a whole as one harmonic system in the second part of chapter
nine of Book V of The Harmony of the World, the echo of
Cusa’s principle of “imprecision” in the universe—with which
we began this investigation—could be heard.
become sufficiently knowledgeable of his ignorance, did Kepler
succeed in relating the unseen principle to the sense perceptions
– the observations, the distances, and equations – and brought
the understanding of his intellect into actuality. And while the
unseen principle was finally brought into visible distance with
the mind’s eye, and seen to take the form of an ellipse, even this
was still a shadow of a paradoxical motion of a higher power, a
“maximum” truth, which was unknowably knowable in the form
of the same species identified by Cusa: the transcendental nature
of the arc and sine.
The Newtonians, in their attempt to reduce transcendental magnitudes to lower algebraic magnitudes with their infinite series, in their attempt to bury Kepler’s “Problem” had
already been proved wrong by Cusa.60
“Number is always greater or lesser and never one, for
then it would be the maximum or minimum number and then,
number, being all things, would necessarily no longer be multiple but absolute oneness, therefore, the Maximum must be that
minimum and maximum number, One.”61
In other words, one never can come to the Maximum
number through an infinite succession of numbers, because then
number would cease to exist, and “all finite things never proceed to infinity because then infinity would be reduced to the
nature of finite things.”62
However, the true intention in banning the “Kepler
problem” was to outlaw such thinking as Kepler’s, for this
higher paradox served as a mirror of our own likeness to the
image of the Creator, driving mankind toward the infinite truth.
Conformably to the rule, there is no precision in
music. Therefore, it is not the case that one thing [perfectly] harmonizes with another in weight or length or
thickness. Nor is it possible to find between the different sounds of flutes, bells, human voices, and other instruments comparative relations which are precisely
harmonic— so [precisely] that a more precise one
could not be exhibited. Nor is there, in different instruments [of the same kind]—just as also not in different men—the same degree of true comparative
relations; rather, in all things difference according to
place, time, complexity, and other [considerations] is
necessary. And so, precise comparative relation is seen
only formally; and we cannot experience in perceptible
objects a most agreeable, undefective harmony, because
it is not present there. Ascend now to [the recognition]
that the maximum, most precise harmony is an equality-of-comparative-relation which a living and bodily
man cannot hear. For since [this harmony] is every
proportion (ratio), it would attract to itself our soul's
reason [ratio] — just as infinite Light [attracts] all
light—so that the soul, freed from perceptible objects,
would not without rapture hear with the intellect’s ear
this supremely concordant harmony. A certain immensely pleasant contemplation could here be engaged
in—not only regarding the immortality of our intellectual, rational spirit (which harbors in its nature incorruptible reason, through which the mind attains, of
itself, to the concordant and the discordant likeness in
musical things). But also regarding the eternal joy into
Part V: An Imprecise Harmony
In Book I of The Harmony of the World, Kepler discovered the causes of the harmonic proportions mathematically,
as no one had ever done before, and developed how these quantities are intellectual, knowable, and derived from the mind.
Before Kepler, they were studied as something outside the
mind.63 The only divisions of a circle which are “knowable” to
the human mind, turn out later in Book III to also be the only
divisions of a string which are harmonic to the human ear.64
Thus, with such a relationship to Nicholas of Cusa, through all
of his work, it should be no surprise that before launching into
Book V of his Harmony of the World, he looked to Cusa’s conception of the curved and straight to demonstrate that the proportions of the harmonies had their foundation in the nature of
60
John Keil claimed to have “solved” the Kepler Problem with an infinite series. For more on John Keil, see Peter Martinson’s report, Neither Venetians nor Empiricists Can Handle Discoveries, at
http://www.wlym.com/~animations/ceres/PDF/Peter/Astronomy.pdf.
61
De Docta Ignorantia, Book I Chapter VI
62
Ibid.
63
Johannes Kepler, The Harmony of the World, Introduction to Book
I, Book I, and Book IV
64
What does it mean, that the reason why proportions are harmonic,
and why they sound “musical” to the human ear, is because they are
knowable to the human mind? What does this say about the human
mind? Is it looking as from outside the universe, analyzing sense perceptions from the outside, or rather, from within?
65
See the coming article on Book IV, to be posted at:
http://wlym.com/~animations/harmonies.
66
The reader is encouraged to return to the beginning of this article,
where the entire quote is placed.
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which the blessed are conducted, once they are freed
from the things of this world.67
musical comma! Cusa identified the universe as one of “imprecision,” in which the physics of orbits of planets were in a state
of continual change, but Kepler has identified the method to
make this “imprecision” knowable. The continuous change
expressed itself in the form of a comma. The comma is not a
“thing” but occurs—as in other places in Chapter 9 of the Harmony of the World—as a consequence of the musicality of the
system as a whole. Here the musicality of the system, in the
region containing the key to both kinds of harmony, soft and
hard, demanded the dissonance be spread out equally, which
took the form of a comma.70
And in the face of those who would demand a fixed
universe, those who would argue, “Well aren’t you just fudging
this? Aren’t you accepting this small change just to impose
your hypothesis onto the universe?” Kepler, understanding the
nature of imprecision of a universe based on change said:
For, in proposition XXVI of chapter nine, while constructing the intervals between Venus and Earth, Kepler ran into
such “imprecision.” In propositions XXIII-XXV he developed
the fact that the characteristics necessary to have a solar system
with both hard and soft melody depended on the hard sixth, 3/5,
between their aphelial motions (that is, the aphelia of Venus and
Earth), and a soft sixth, 5/8, between their perihelial motions.
This created the necessity for very small changes to each
planet’s own individual motions. He said that “harmonic
beauty” urged that these planets’ own motions—that is, the proportion between one planet’s perihelion and aphelion—since
they were very small and cannot be any of the harmonic intervals, should at least be of the melodic intervals, that is the diesis
24/25, or the semitone 15/16.68 In this case, Kepler had shown
that the two intervals of Earth’s and Venus’s own motions
would have to differ by a diesis in themselves, but these two
melodic intervals, the 24:25 and 15:16, differ by only 125:128,
which is smaller than a diesis. Therefore, Kepler showed that
only one of the planets could have the melodic interval. Either
the Earth would have the semitone, 15:16, and Venus the
125:128, a non-melodic interval, or Venus would have the diesis
24:25, and Earth would have 12:13, a non-melodic double
diesis.
Do you ask whether the highest creative wisdom would
have been taken up with searching out these thin little
arguments? I answer that it is possible for many arguments to escape me. But if the nature of harmony has
not supplied weightier arguments… it is not absurd for
God to have followed even these, however thin they
may appear, since he has ordered nothing without reason.
For it would be far more absurd to declare that God
has snatched these quantities, which are in fact below
the limit of a minor tone prescribed for them, accidentally. Nor is it sufficient to say that He adopted that
size because that size pleased Him. For in matters of
geometry which are subject to freedom of choice it has
not pleased God to do anything without some geometrical reason or other, as is apparent in the borders of
leaves, in the scales of fishes, in the hides of wild
beasts, and in their spots and the ordering of their spots,
and the like.71
But since the two planets have equal rights, therefore if the nature of melody had to be violated in their
own proportions, it had to be violated equally in both
cases, so that the difference between their own intervals
could remain exactly a diesis, to differentiate the necessary kinds of harmonies… Now the nature of melody
was equally violated in both cases if the factor by which
the superior planet’s own proportion fell short of a double diesis, or exceeded a semitone, was the factor by
which the inferior’s own proportion fell short of a simple diesis, or exceeded the interval 125:128.69
Kepler’s method of hypothesis cures the mental diseases of entropy found so frequently in modern science today.
70
Although more is needed to demonstrate it, this also points to question: is the relationship between the orbits of the planets transcendental? Riana St. Classis discussed this question in the LYM Harmony of
the World website: “The harmonic nature of the relationship of the
individual planets and the sun is reflected in the total orbital period of
each planet, the total area of the orbit swept out as equal areas in equal
times, or better, as Kepler views it, the area swept out by the planet is
the time it has traveled. This is echoed in the fact that within an individual orbit, at two moments, the proportion of the apparent (from the
sun) speeds has an inverse relationship to the proportion of the squares
of the distances of the planet from the sun at those moments. But this
relationship does not hold between planets. If the area a planet sweeps
out is the time it has traveled, this time is unique to this individual
planet. 100 units of Mars’s orbit are not equal to 100 units of Jupiter’s
orbit. If we were to evaluate these two portions from the standpoint of
how we think of time on the earth, according to the earth’s rotation
about its axis, the number of days Mars took to travel 100 units would
be different than the number of days Jupiter took to travel 100 units.”
71
Kepler, The Harmony of The World, Part V, Chapter 9.
So instead of the Earth’s motion having either the melodic semitone of 15:16 or the unmelodic interval of 12:13, it
has 14:15, and instead of Venus having the melodic diesis of
24:25 or the unmelodic interval of 125:128, it had 35:36. And
14:15 and 35:36, both differ from 15:16 and 24:25 by 80:81, a
67
De Docta Ignorantia, Book II, Chapter I
On harmonic versus melodic intevals, see LYM Harmonies website
http://tinyurl.com/yq26fx/melodic.html.
69
The Harmony of the World, Book V, Chapter 9, Proposition XXVI
68
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That the human soul’s own proportions are found throughout the
universe, creates the conviction that we are inside the universe,
and that we understand it as a reflection of ourselves. This
thinking is exactly opposite to the empiricism that struck Europe
after the death of Leibniz.
The underlying axiom of science today is immediate
skepticism at one’s mind’s ability to know the reason for the
creation of the universe. And so, when a human discovers such
intricacies as the comma, which create a harmonic organization,
the immediate reaction is to say, “Well, this universe may be
harmonic, but, it sure is held together pretty thinly. You’re telling me it hangs on the difference of 15/16 to 12/13 to 14/15?
And 9/10 to 24/25, to 35/36? You must be imposing your assumptions on to this.”
Rather than looking at such matters, and remarking at
the absolute perfection that exists, and celebrating in the mind’s
capability, there is the fear of the popular ideal that there is no
God in science, and thus, we are imposing our thoughts onto the
universe.72 Such thinking is entropic, because in that thinking
one must force the universe into harmony, one has to put it together piece by piece, and it is delicately holding together,
rather than the idea that one is on the inside of it, and have detected in the small the reason for its perfection. Such imprecisions as commas and infinitesimals are not seen as a fragile
argument that needs to be held together with great convincing,
but are the reflection of the relationships indicating a new unseen dynamic.
Inquire further. How did Kepler determine the causes
for the eccentricities? Did the physics of the orbital elements
randomly create harmony, or did the necessity for harmony generate each orbit as it is? Further, if each orbit necessitated creating harmony, how did the Solar System become one whole
harmonic system? Take a few examples for the relations of the
Solar System as a whole.
Why do Earth and Venus have the smallest eccentricities of all the planets; that is, why are the physical orbits of the
planets the way they are? Kepler shows that it is upon these two
planets that the hard and soft sixths depend, and thus the crux of
the whole musical system rests on them. After working out this
question of how hard and soft harmonies are distributed
throughout to form one harmonic system, Kepler writes:
Therefore, you have here the reasons, for the disagreements over very small intervals, smaller in fact
than all the melodic intervals.73
The region of most importance for the harmony of the
whole Solar System,74 that between Earth’s aphelion and Venus’s perihelion, forms harmony in octaves with the outermost
parts of the Solar System. Saturn, the highest planet, is in harmony at aphelion with the Earth at aphelion forming a ratio of
1
/32 (which is continuous repetition of the octave 1/2), and Mercury, the innermost planet, is in harmony at perihelion with Venus’s perihelion forming 1/4 (one doubling of an octave 1/2).
Here the whole system is seen to sing in grand counterpoint,
echoing octaves within itself.
Also, in these outer planets, perfect harmonies were
found among the converging motions in the pairs of planets, but
not in each individual planet’s motions, while in the inferior
planets, the opposite was the case.
And as was said above, Earth and Venus had two perfect harmonies, 5/8 and 3/5, between their extreme motions, so
that they make the harmony either soft or hard, whereas between
Mercury and Venus there are two perfect harmonies in the motions, but the motions do not change their kind of harmony.
And as Venus is the most imperfect in its own proportions and
has the smallest eccentricity, so Mercury is the most perfect,
forming a perfect 5/12, and has the largest eccentricity.
In conclusion, Kepler showed that the physics of the
system, that is the orbital elements of each planet, occur as a
secondary product to the musicality of their motions, which in
turn itself, is secondary to the idea of the Great Composer.
Physics is an afterthought to the principle of perfection and reason. An intention to create a harmonic organization of the system as a whole generated each particular harmonic proportion,
and as a consequence, each particular physical characteristic.
Kepler then went on to derive all the orbital elements as shadows of the harmonies.75
In demonstrating that the physics of the entire Solar
System could only be known through harmony, how does that
transform the definition of humanity as a whole?
Wrestle with this question: how can it be that the solar
organization of the heavens is based on the same harmonic ratios that human beings created music with before we even knew
the ratios of the motions of the planets?
72
The Case of Leibniz’ discovery of the catenary principle is an example of the folly of modern thinking concerning science, and an example
which irreparably dooms the credence of its modern ways. Leibniz and
Bernoulli demonstrated that the change in direction at every possible
moment of a curve, is guided by a constant physical relationship between vertical and horizontal tension, i.e., the physical differential relationship. However, Leibniz, who had launched a scientific political
movement against the Cartesians, had turned physics into a problem of
finding the dynamic, i.e. the individual substance, determining the effects. Therefore, he sought more than the physical relationship guiding
the chain. And although Bernoulli found his own construction for the
catenary: Leibniz’ was unique. Because of his passion to demonstrate
the perfection with which the Creator created the universe, only he
discovered the true concept of the substance, a construction which expressed such perfection, both in its beauty, and in its power; his construction captured the irony of the paradox of the physical action of the
curve. The relationship between the substance and the sense perceptible
physical curve, is only knowable to the mind in form of a higher transcendental, the geometry of the complex domain. Therefore, modern
critics who shriek, “but why must we talk of a Creator in relation to the
universe? Science has nothing to do with it!”, should well pay heed to
these historical truths. For, like Cusa’s transcendental, the existence of
the physical complex domain, upon which modern science depends,
would never have been discovered without Leibniz’ knowledge and
demonstration of “the best of all possible constructions”, in the image
of the best all possible Creators.
73
The Harmony of the World, Book V, Chapter 9, Proposition XLIV
The Harmony of the World, Book V, Chapter 9, Proposition XIV
75
http://tinyurl.com/yq26fx/proposition48.html
74
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Look at the harmonics in human music. In the human
organism, we can use our reason, our intellectual inquiry, to
detect the relations of the sounds we make with our vocal chords
to create pleasing tones. Those are instinctual if the ear and
mind are trained to focus on certain properties of the voice. The
harmonies are then organized to express even more. And as
Kepler showed, when we turn our ears, our inner ears, to the
heavens, we detect an ordered development which is the same as
the way human beings communicate ideas in music. Thus, not
only are we tuning ourselves to the universe when we sing, we
then tune to and compose with the principles that the Composer
used.
And if music is nothing other than harmony detected
by the human ear, then the same harmonic organization, the
same geometrical proportion exists in the small and in the large,
in fact, in all physical principles. Therefore, as Kepler “listened” to the Solar System to determine its characteristics, all
these ratios can be examined with the “inner ears” first to see if
they are the correct ones. If they are harmonic, then the organization is true, if they are not, then it is not true. What area of
physical science is not affected by this discovery?
Such was Kepler’s revolution. He demonstrated all of
the indicated paradoxes of an “imprecise,” continuously changing universe that Cusa had indicated, and applied Cusa’s investigations into the infinitely small and large. But Kepler, having
demonstrated all of the implications of Cusa’s physics, went
further, to change the universe as a whole, in redefining its “imprecision” as only knowable, through measurements with the
same proportions—the ones Kepler most prominently derived
from Cusa’s conceptions—found within the human soul.
Therefore, the human soul is shown in the organization
of the entire solar system, as a universal principle.
APPENDIX: Cusa on the Human Soul
There are four elements of the soul, the intellect, the
rationality, the imagination, and the senses. The rationality is
aroused by the senses, and it in turn arouses the intellect.
Cusa relates the capacity of each part of the soul
through a metaphor of a sphere.
When the senses perceive a sphere, only the part of the
sphere seen by the eyes, or touched by the hands, is real, therefore, no sphere actually exists for the senses. But for the imagination, a round sphere is conceived, even though the eyes see
only a part of it. The imagination has the power to conceive all
parts of the sphere, thus making it whole. Further, the rational
soul understands the sphere in its rational form, as equal radii
from the center in all directions. But the intellect conceives of a
sphere, which is infinite, with the center coinciding with the
circumference. Cusa says, that the true sphere is the one the
intellect perceives. Likewise with the circle, the rational concept of it is not the true one, if it is merely that all lines to the
center are equal. The true circle in absolute unity is without
lines and circumference.
And that is “real fun.”
Marvelous is this work of God, in which the discriminative power ascends stepwise from the center of
the senses up to the supreme intellectual nature… in
which the ligaments of the most subtle corporeal spirit
are constantly illuminated and simplified, on account of
the victory of the power of the soul, until one reaches
the inner cell of rational power, as if by way of the
brook to the unbounded sea, where we conjecture there
are choirs of knowledge, intelligence, and the simplest
intellectuality.
Since the unity of humanity is contracted in a human way, it seems to enfold everything according to the
nature of this contraction. For the power of its unity
embraces the universe and encloses it inside the
boundaries of its region, such that nothing of all of its
potentiality escapes… Man is indeed god, but not absolutely, since he is man; he is therefore a human god.
Man is also the world, but not everything contractedly,
since he is man. Man is therefore a microcosm or a
human world.
—Nicholas of Cusa, On Conjectures
See sphere animations:
Sensible:
http://tinyurl.com/yv8kca/sensiblesphere.swf
Imaginative:
http://tinyurl.com/yv8kca/imaginativesphere.swf
Rational:
http://tinyurl.com/yv8kca/rationalsphere.swf
Intellectual:
http://tinyurl.com/yv8kca/Michael/Intellectual%20Sphere.swf
The intellect depicts the sense perceptible in the imagination. The imaginative representation is then enfolded by the
rationality into a unity of knowledge. It unites the otherness of
the senses in the imagination, and then unites the otherness of
the imagination in the rationality, and, lastly, the intellect enfolds the varied otherness of the rationality into the unity of itself. Likewise, the intellect becomes actual through the descent
to the senses. The unity of the intellect descends to the otherness of rationality, and the unity of the rationality descends to
the otherness of the imagination and so on.
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in that way. Thus, the soul is not corruptible in motion, nor is it
subject to time. Thus it is eternal, and immortal.
The intention of the intellect is to become actual. In
that way, Man submits himself to the senses in order to attain
understanding. He says our intelligence is like a spark of fire
concealed under green wood, which needs the senses to draw
forth the heat in the wood. The more powerful is the actuality
of fire, the more rapidly it causes the ignitable to become actual.
And as the imagination needs the rationality to be intelligible, so
colors need light to be seen, as one’s vision cannot move directly to color without light.
Ascend higher therefore: the rationality is conveyed
into the intellect through itself, as light is into vision, and the
intellect descends through itself into rationality, as the vision
proceeds to light. Now all things are defined by that which
measures it, and so the rationality is defined by, and is the intellect descending into it.
Although the rationality partakes in the otherness of the
senses, the intellect is the unity of the rationality, and thus precedes otherness. Cusa says, that the rational higher nature,
which also absorbs the unity of imagination, and which is concealed in the light of the immortal intellect, is also immortal,
like light that cannot be obscured.
Therefore, the difference between men and the beasts is
that human rationality is absorbed in the immortality of the intellect. It is always intelligible through itself light as light is
visible through itself. Animals have an otherness of rationality,
like the otherness of colors which are not visible through themselves.
The absolute intellect embraces truths that have been
unified by the rationality. Taking the origin of truth from sensible things is not absolute knowledge. But, if the otherness of
the senses enfold into a unity in the rationality of the soul, and
all of the different rational operations enfold into a unity in the
intellect, what is the intellect an otherness of, in which it is enfolded as a unity?
Cusa says that the intellect is the otherness of the infinite Unity.
And so, although the intellect can never attain infinite unity, it
moves as far from otherness as possible to attain the highest
unity. The perfection of the intellect is its continual ascension
toward the infinite cause of all causes.
Without the rational soul, then time, the measure of
motion could neither be, nor be known, since the rational soul is
the measuring scale of motion, or the numerical scale of motion.
And conceptual things are created by Man, as things existent by
god. Soul creates instruments to discern and know. They unfolded their conceptions in perceptible material. And man creates instruments like temporal measures. Since time is the
measure of motion, it is the instrument of the measuring soul.
Therefore, the soul’s measuring does not depend on time, rather
the scale for measuring motion, time, depends on the soul. As
for the eye and sight, the eye is the instrument of sight, likewise
the rational soul does not measure motion without time, but the
soul is not subjected to time. We are not the slaves of our instruments. Thus, the soul’s movement of distinguishing cannot
be measured by time, its movement cannot come to end at some
time, and thus its movement is perpetual. And its nature is not
corruptible as all things subject to motion dissolve, but rather,
the soul measures motion with time; therefore, that which measures motion, is the form of motion and is not subject to motion
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Some Geometrical Writings of Nicholas of Cusa
Kästner
Some Geometrical Writings of Nicholas of Cusa
Abraham Kästner
title, but rather begins with: In the Holy and Indivisible name of
the Trinity. Amen. On an empty page is written: Pietro di Crescenzi | Diverse treatises of Nicholas of Cusa | Note on the Treatise on the Koran of Mohammed. Thus this treatise is noteworthy for the old owner, which admittedly, it is not for me.
Also with Petri de Crescentiis book, the date of the
printing is not denoted. The general time period can be determined from the name of the printer. It certainly does not need
proof that the printing of both books falls in the 15th Century
year, only on account of Cusa’s works do I include the verification, that therein numeral 7 throughout is expressed as was customary toward the end of this century. But 4 is written as it is
presently.
The following translation, by Michael Kirsch, is from Volume IV
of Kästner’s Geschichte der Mathematik and the first section of
the book which contains separate investigations of elementary
geometry. The text was translated from German. Italicized text
was translated from Latin by William F. Wertz, Jr.
Some Geometrical Writings of Nicholas of Cusa
1. I own a compilation, on the cover of which is written: Diverse
treatises by Nicholas of Cusa, which extend over pages. On the
other side of this page is a Prohemium.
The beginning of this introduction reads: In this volume
certain treatises and books of the highest contemplation and
knowledge are contained: in the clear memory of most excellent
and learned individual Nicholas of Cusa, most Holy Roman
Church, Cardinal-Presbyter of St. Peter in Chains: published
among many others ....
The first letter I is missing, in its place is so much
space, that it would have reached until the row beneath, where
the section which I transcribed ends.
Similarly all the initial letters are missing throughout.
That it is all in Gothic script, it is unnecessary for me to remind
any expert that this is a known sign that this print belongs
among the oldest.
At the end of the mentioned side is an index of the
works contained in the compilation. I place it here in its entirety, even though it does not all pertain to mathematics. Each
title has its own line but I separate them with |:
3. Allow me to cite something from the first part of the compilation of the Cardinal’s works, that will be able to be drawn out
for mathematics, in so far has the art of geometrical perspective
and optics as a basis. The book, The Vision of God, p. 402 is
addressed to ad abbatem et fraters in Tegernsee. The preamble
gives, for an allegory, a picture that views every face wherever
one stands. The Cardinal recounts a few examples [of these
images], where they are located, and sends: a painting: containing the figure of an omnivoyant individual, which I call the
“icon of God.” Which if they hang it on a wall, and stand in
front of it, then the face would look at to everyone, regardless of
where they were standing, and if someone walks around in front
of it he will experience that the immobile face is moved toward
the east such that it is moved simultaneously toward the west...
and that it observes one motion in such a way that it observes
all motions simultaneously. And while he considers in what
manner this sight deserts no one, he sees how diligently it is
concerned for each one as if it is concerned only with him who
experiences being seen by it and not for anyone else.
I find the Cardinal’s prayerful meditation of the likeness, theoretically truer, and practically more heart lifting, than
what has become stated in the philosophy of our time: God
reigns over the whole, without troubling over the individual
parts. The rest of the Cardinals thoughts, in which indeed there
is much rightness and goodness, don’t belong to the present
purpose.
De visione dei | De pace fidei | Reparatio kalendarii |
De mathematicis complementis | Cribratio alchoran
libri tres | De venatione sapientiae | De ludo globi libri
duo | Compendium | Trialogus de posesst | Contra bohemos | De mathematica perectione | De berillo | De
dato patris luminum | De querendo deum | Dyalogus de
apice theorie
2. The format is a short folio, the pages are not numbered at the
bottom as is usual, but they are marked with letters, one letter
for every six pages, the first the letter is a, b, ... then A; my copy
goes until the 4th page of letter C; upon whose first side begins
with: Treatise On Beryllus Expounded.
It is thus not complete; however, it is in a very fine
binding, and following tradition is decorated with engraved figures, well preserved. Bound by: the Book of Rural Arts (Ruralium Commodorum) by Pietro di Crescenzi at whose bottom
reads: this industrious style characterizes the present Book of
Rural Profits by Pietro di Crescenzi as a whole printed for the
service of the Omnipotent God in the house of John of Westphalia. Nourishing and flourishing at the University of Louaniensi. No dates, Gothic script. Here the initial letters are
inscribed with thick red ink. Also this book has no particular
4. I can now arrive at my actual intention:
On Mathematical Complements1 To the Most Blessed
Father, Nicholas V. Nicholas, Cardinal of St. Peter in Chains.
Great is the power of the pontifical office which you
hold, most blessed Father Nicholas V: all who consider his
powers with attention, equate it to a certain extent to the
strongest power, that is there, to transfer the circle into the
square and the square into the circle....
1
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Some Geometrical Writings of Nicholas of Cusa
Kästner
diameter, draw then through
the circles center, parallel with
the line π until at the
hypotenuse. This parallel is
the
semi-circle’s
half
circumference.
When the radius of
the circle is longer than the
line named I, then the parallel
hits the extended hypotenuse.
As the proportion
which I call π : I, the Cardinal
uses, as is easy to consider:
the half of the straight line,
which he had assumed, and
the radius of the circle, which
he had located.
Recently you have transmitted to me the geometrical
writings of the great Archimedes, after they were translated
from the Greek, as you received them, through your efforts into
Latin. They have appeared so admirable to me, that I had to
devote myself to them with all my commitment, and thus has it
occurred that as the result of my own research and work I have
attached a complement, which I permit myself to dedicate to
your Holiness...
5. Archimedes had measured the circumference through a
straight line, attempted by means of the spiral, but the velocity
of [one the one side] the point, which on the radius moves away
from the center, and [on the other] the point, which moves at the
end of the radius of the circle, are in proportion as the radius and
circle, and this very proportion was sought.
6. The Cardinal begins with an examination of regular polygons.
The perpendicular from the center of such a polygon to its side,
he calls prima linea, and the straight line from the center to the
vertex of the angle of the polgon, secunda linea. This [latter] is
the radius of the circle, which can be described around the polygon.
2nd Line
2nd Line
1st Line
1st Line
Then he pictures a series of such polygons, all having
the same perimeter as the sides grow in number. The first and
second lines differ less, the greater number of sides the polygon
has. Thus as the number of sides grows larger, so much closer
does the polygon become to a circle, which would have the
same circumference. About this polygon, polygonias issoperimetras, he undertakes an investigation, [and] gives theorems, for the relationship of the area of such a polygon to the
circle, and presents the following problem:
Given a straight line, discover the radius of a circle,
whose circumference is as long as this straight line. In his proof
he uses nothing more than the first and second lines of the
isoperimetrical triangle and square.
9. The transformation of a square into a circle, among other
things. To find the sine and chord, for 1, 2, 3, 4…. degrees
which no one yet knew.
Between the half of the straight line, which he assumed
for the length of the circumference, and to the radius which he
found for it, he takes the
geometric mean proportional line
which is the side of a square
having equal area to the circle.
Correct, except that his
construction does not correctly
give him the ratio of the radius to
the circumference.
Then he produces from
the apex of a right angle on both
sides, the radius and half the side
of the square, and draws the hypotenuse, so again he gets an
angle to which he shapes from
copper or wood, and by means of
it finds the square equal to every circle and the circle equal to
every square.
7. If I have correctly understood his discourse and accurately
calculated, then he gives for the circumference = c, the radius =
c x 0.102384. Thereby the proportion of the diameter to the
circumference would come to 1: 4.8835.
8. He then also inversely transforms the circumference of a given circle into a straight line. The
method is theoretically correct and ingenious: from
the vertex of a right angle one applies straight lines
to both sides, which are in the ratio as I : π, and the
hypotenuse is drawn, which is however made longer
than between the end points of the sides. This figure becomes
constructed from brass or wood (in ere aut lingo). If a circle is
now given, then the acute angle [which is opposite the line π] is
laid in circumference of the circle and the line = 1, along the
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Here a circle had been drawn with the square that
should be equal to it, whose side clearly will cut the circle, but
intersect outside of it. This square’s lower side is extended, and
the extension is tangent to an equal circle. Both circles’ centers
lie above the extended line. From the point of contact a curved
line goes upwards, then again downwards, through the center of
the square, until about the middle of the square’s side which was
extended.
I don’t find this curved line mentioned in the text. It
could occur to someone [that] the circle, [of] which the extended
side is tangent [to] at the bottom, should roll along the straight
line, and its point which is initially the lowest describe a cycloid: therefore, the straight line, over which the circle turns,
must also be tangent with the end of the cycloid, like at the beginning, and the straight line of the figure is tangent to only one
of the two circles.
Also I find the rotation of the circle nowhere mentioned here by Cusa, which could so easily occur to one who
seeks the quadrature of the circle: perhaps he did not think of it,
because here he did not intend to square a given circle, but
rather the inverse, to transform a straight line into the circumference of a circle.
7 3
= 147
165 6/16
14
= 196
9/16 x 147
7
= 49
= 82 11/16
If the square root of 49 is now subtracted from the square root
of 82 11/16, then this difference denotes the excess of the radius
of the circle inscribed in the square over that in the triangle and
amounts to something more than 2; if one subtracts the square
root of 165 6/16 from the square root of 196, then this difference
amounts to something more than 1. Thus, you have the
differences between the prime on the one side and the second on
the other, and everything further can be pursued from the
relationship of these differences.
196 - 165 6/16
= a little more than 1
49 - 82 11/16
= a little more than 2
10. Then the cardinal said: after that which I have previously
treated, one can now also attempt what was until today
unknown in geometry, namely a final theory of curves and
chords (de sinibus et chordis). No one could ever indicate the
chord of a curve of one, two, four degrees and so forth; now one
can find it. It is certain: in order to produce the radius of an
isoperimetric circle, each regular polygon adds a fixed fraction
of the difference between its second and first line to its first line.
Moreover: in all polygons, the same relationship is always
preserved between the excess by which the first line of any
arbitrary polygon exceeds the triangular first line, and the
excess by which the triangular second exceeds the second of the
other polygon. From this, the general theory of curves and
chords is elicited; without this theory geometry remained
incomplete up to now. But you will find how one can arrive at
the practical implementation in approximation numbers in the
following. It is impossible in whole numbers, because the
square root of 2 (medietas duplae-literally, the mean of two)
cannot be expressed in numbers, for this relationship has a
quantity which is neither even nor odd.
The radius of the circle circumscribed by the triangle is
therefore 14; then the radius of the associated inscribed circle is
7 [I have mentioned how this number is expressed (2)], the
square thereof is 49 and the square of half of the side of the
triangle is three times as much, namely 147, the square of the
radius of the circle is four times as much, namely 296 [this is
what it says in the text, but it should be 196]. Half of the side of
the tetragon is now the root of nine-sixteenths of the square over
half of the side of the triangle, that is, the square root of 82
11/16 [He means 9/16. 147 = the square root of 82 11/16].
That is also the radius of the circle inscribed in the square. The
radius of the circle circumscribed in the square is the root of the
doubled number, that is, the square root of 165 6/16 [2. (the
square root of 82 11/16) = the square root of 165 6/16].
Namely, if you subtract this difference from the sagitta of the
side of the triangle, that is from 7, the sagitta of the square
remains; if you now divide 7 according to the relationship of the
difference given above and add the larger section to the radius
of the circle inscribed in the triangle, you have the radius of the
isoperimetric circle.
49 - 2 + 1 = 4
Sagitta A - Differences of the 1st and 2nd lines = Sagitta B
A
B
"Divide [A] according to the relationship of the difference given above and
add the larger section to the radius of the circle inscribed in the
triangle, you have the radius of the isoperimetric circle"- Cusa
In this way you can also provide the square of any
arbitrary polygonal side from the square of the side of the
triangle and of the side of the square; from this and from the
relationship of the differences one comes to the sagitta and to
the radius of the inscribed circle, and thus one knows the
curvature of the chord, and this is the final completion of the
geometrical theory, to which the ancients, as far as I have read,
had not advanced. Now the theory of the geometrical
transformations is also completed, which earlier I have
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line equal to the circumference, and sought the radius which
belongs to it. He was led to it by the isoperimetric polygons.
adequately described more briefly, as far as it concerns the
quadrature of the circle.2
I have placed this passage here, because in it sinus and
sagitta became named.3 At the beginning of it I could anticipate, that it would be proven how its chord or sine of a degree
etc, would be given, but at the end that was far from the case.
Nevertheless, such chords had already been given by Ptolemy,
and accordingly it also was given by the Arab in the Almagest,
which could not have been unknown to the Cardinal. I thus do
not see, how he promised to accomplish something that the ancients did not accomplish, for in any case he only wanted to
yield approximately that which was desired.
Admittedly the accomplishment would have been
rather difficult due to the very incomplete state of his arithmetic,
and thus did not achieve complete accuracy.
From 82 + 11/16 = 8.6875 gives the logarithm =
1.9173874, which halved = 0.95869378 which belongs to
9.0927. Subtracting 7 from this, leaves 2.0927, which the
Cardinal called a little more than 2. With such an entirely
superficial estimation of figures he could not advance further,
even if the theory were accurate, from which he derived it. He
thus flattered himself too much, to the Pope, when he said about
its construction with angles: to whomever wants to exert his
genius, it becomes clearly accessible. Hence this invention
rightfully obtains the name Complement, and deserves to
become generally well-known through your wonderful power,
Most Blessed Father, which astonishes all Catholics, so much
that they name you after the name of admiration, father of
fathers.
About lines and figures, which arise, if a straight line
moves or rotates, while a point moves along it. At the conclusion: to find the sides of polygons which are equal to the circle.
12. The content of the book De venatione sapientiae, is shown
by its title.5 Among the means which the Intellect employs to
hunt wisdom, Chapter V calls to notice also: Quomodo exemplo
geometric perficit. The content is, that the geometrical ideas in
the mind are never perfectly represented through their sensual
images; one seeks only that the images of the ideas are as precise as possible and as precise as required by the image.
13. Perhaps Mathematics could also be expected in the book De
ludo globi.6 It is a dialogue, in which are presented: Nicholas,
Cardinal of St. Peter in Chains, and John, Duke of Bavaria.
The Duke begins: Since I have seen that you have withdrawn to
your seat, perhaps tired by the game of spheres, I would like to
confer with you about this game, if it is agreeable to you. The
duke observes that there must indeed be something more to be
considered about this game because it so pleasing to men, and
the Cardinal acknowledged this, for some sciences also have
their own game: Arithmetic has its number games, music its
monochord, nor does the game of chess lack a moral mystery.
The Cardinal observes further: no brute beast moves a
ball to its goal. Therefore you see that the works of man originate from a power which surpasses that of other animals of this
world.
The ball used in this game must have had a certain
metaphor. I do not think you are ignorant of why the ball,
through the art of the turner, assumes a hemispherical shape
that is somewhat concave. For it if did not have such a shape,
the ball would not make the motion that you see: helical/vertiginous, that is spiral or involuted. For part of the ball,
which is a perfect circle, would be moved in a straight line,
unless its heavier and corpulent part retarded that motion and
drew the ball centrally back to itself. Based on this diversity the
shape is capable of a motion, which is neither entirely straight
nor entirely curved, as it is in the circumference of the circle,
which is equidistant from its center. From this you will first
observe the reason for the shape of the ball, in which you will
see the convex surface of the larger half sphere and the concave
surface of the smaller half sphere. And the body of the sphere is
contained between them. You will then see that the ball can be
varied in infinite ways according to the various conditions of the
described surfaces and can always be adapted to one or the
other motion.
11. For the times in which the Cardinal lived, it indicates an
extraordinary spirit and passion to perceive what was to be discovered, and to attempt the discovery, even if that attempt was
not sufficient.
The comparison between his first and second lines and
sides of the isoperimetrical polygon can presently be given
through the formulas of analytical trigonometry; he could hardly
represent it exactly for every individual polygon solely through
common arithmetic. I surmise that he had even determined the
first and second lines for the triangle and square solely through
diagrams, because he conveyed everything onto diagrams; and
when he wants to illustrate its composition with numbers, he is
absolutely not concerned to be accurate or to come close to being exact, but only to use it as a example.
Among those, who have occupied themselves with cyclometry,4 I don’t know any one else, who took a given straight
14. The cardinal gives the following report, not far from the end
of this book: However, it was my intention to apply this recently
invented game, which everyone easily understands and gladly
plays, because of the changing and never certain course of the
ball, in a manner useful to our purpose. I have made a mark
where we stand when throwing the ball, and a circle in the center of the level ground. In its center, enclosed in the circle, is
the seat of the king, whose kingdom is the kingdom of life. And
in this circle are nine others. However, the law of the game is
2
For more on the quadrature of the circle, the reader is referred to Part
III of “A Scientific Problem: Reclaiming the Soul of Gauss,” this issue.
3
In case the reader did not follow Cusa, the sagitta is the distance between where the radius intersects the midpoint of a side of a polygon
and where it intersects the circle. The more sides a polygon has, the
smaller the sagitta.
4
Cyclometry is the study of circles.
5
6
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there be two things that are equal in all respects without any
difference at all. How can many things be many without a difference? And even if the more experienced player always tries
to conduct himself in the same way, this is nevertheless not precisely possible, although the difference is not always perceived.
Here one has Leibniz’s principium indiscernibilium
(Principle of the Indiscernible).
to make the ball stop moving inside the circle. And the closer it
comes to the center, the more it acquires, corresponding to the
number of the circle, in which it comes to rest. And whoever is
the first to attain 34 points, the number of the years of Christ, is
the winner.
This game, I say, signifies the motion of the soul from
its kingdom to the kingdom of life, in which is peace and eternal
happiness. Jesus Christ, our King and the giver of life, governs
in its center. Since he was similar to us, he moved the sphere of
his person, so that it came to rest in the middle of life, leaving us
the example, so that we would act just as he acted. And our
sphere would follow him, even though it would be impossible for
another sphere to attain peace in the same center of life where
the sphere of Christ rests. Inside the circle there is an infinity of
locations and mansions. For each person’s locus rests on its
own point and atom, which no one else could ever attain. Nor
can two spheres be equidistant from the center, but one is always more, the other less so. Therefore it is necessary that all
Christians contemplate how some do not have the hope of another life and that they move their sphere in the earthly domain.
Others have the hope of happiness, but they attempt to achieve
that life by their own powers and laws without Christ. And they
make their sphere run to higher things by following the powers
of their own genius and precepts of their own prophets and
teachers. And their sphere does not reach the kingdom of life.
There is a third group, which embraces the life that Christ, the
only begotten Son of God preached and walked. They turn to
the center where the seat of the king of virtue and of the mediator of God and man is. And following the vestige of Christ, they
bring their sphere onto a moderate course. These alone acquire
a mansion in the kingdom of life. For only the Son of God, descending from heaven, knew the way of life, which he revealed
in word and deed to the believers.
I thought the long passage deserved to be distinguished, because in addition to demonstrating the composition
of the game, it also demonstrates a remarkable theological intention. Maybe a game with a ball, which must be left to rest inside a certain boundary was common, and the Cardinal adjusted
it for his purpose. In any event he gives himself as the inventor
somewhat before the quoted passage. Freedom, he says, is
man’s superiority over the beasts, as beasts of one species all act
the same concerning prospecting their food, building nests etc.,
always one as the other; while each man acts according to his
own wisdom: When I invented this game, I thought, considered,
and determined, that which no one else thought, considered, or
determined.
Indeed, the structure of what he calls a ball is also peculiar, of which could well be desired a more exact description… but an intelligible description, a useful illustration, was
not required in that time. If such a thing did exist, the shape and
path that it would take by a given impulse, could keep an Euler
busy.
16. The visible rounding could not be perfect: the outermost
edge of the roundness is terminated in an indivisible point that
remains entirely invisible to our eyes. For nothing can be seen
by us unless it is divisible and has size. The significance is well
only this: whether the spherical curvature were geometrically
perfect, or depart insensibly from it, cannot be perceived with
the senses. Then the dialogue passes on to the roundness of the
universe, motion, philosophy, morality, and even theological
teachings. Even if there was place for it here, it would be too
much effort to clearly represent it, as even the Verses at the
close of this book say in praise of the same. They begins thus:
What genius you desire at present in our little book
First repeat the holy reason three times, four times,
And more than once: understanding as soon as you
survey the heights]
At the top: and titles are reduced to empty reason.
17. There follows yet a second book De ludo globi, where the
people in discussion are: the young man Albert, The Duke of
Bavaria and Nicholas of Cusa. Albert has seen that his relative
Johann read the book De ludo globi, and comes to the Cardinal
in request of further explanation. It didn’t seem to me, he says,
that you explained the mystical meaning of the circles of the
region of life. Theorems appear here as before in the first book,
which are sometimes explained with geometrical likenesses, as,
for example, through circles and rotation of the circle.
18. The book De mathematica perfectione7 is dedicated: to the
Most Reverend Father in Christ, the Lord Antonius, of the Holy
Roman Church, Cardinal-Presbyter of St. Chrysongonus, by
Nicholas, Cardinal of St. Peter in Chains. Then he says: However, that mathematical insights lead us to the entirely absolutely divine and eternal, your paternal Grace knows better than
I, according to the extent of your high erudition, You who are
the summit of theologians.
19. He begins with: would the smallest chord of which there
cannot be a smaller have no sagitta8 and be as small as its arc?
Reason conceives this, although it knows that neither the chord
nor the arc could become so small, that it cannot become
smaller, “since the continuum is infinitely divisible.”
20. He now imagines a right angled triangle, whose hypotenuse
linea prima, is the radius of a circle, whose arc measures the
angle opposite the smallest side (its linea secuda)… Thus, this
angle can be no larger than 45 degrees… He calls the third side
linea tertia, the arc simiarcus, the second line semicorda… That
15. At the time all that was known, was that with every shot of
the ball it would take a different path, because each time it
would, in a different manner, be held, be let go from the hand,
be laid onto the ground, and collide: It is not possible to do
something the same way twice, for it implies a contradiction that
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Mathematical Perfection. This work has also never been translated
into English.
8
See footnote 2.
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Some Geometrical Writings of Nicholas of Cusa
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is to say the half of the chord, of the arc, of which the given
would be half…
Then he says the following: the named half arc is to the
half chord, as the triple of the first line is to the sum of the third
line plus the twice the first.
That sounds like an introduction of the analysis of the
infinitesimal calculus. Thus one could say something to the
Cardinal which he had not considered. In fact, he had contemplated evanescent magnitudes, only he did not know how this
conception would be used.
21. I call Cusa’s first line or the hypotenuse r; the triangle’s
angle a; thus the length of arc described with r = r a; the second
line = r sin a, the third line = r cos a; and the Cardinal says:
r a / r sin a = 3r / (2r + r cos a); thus a = 3 sin a/2 + cos a =
3 tang a/2 sec a + 1.
The composition is only true, when the Arc and Sine
vanish together. Thus for small arcs truth is close, for the
greater is always removed more from it, and furthest, when the
angle = 45 degrees; since I find by means of the logarithm 3/2
Sec 45 +1 = .78361; therefore the arc for 45 degrees is =
0.78539; thus the maximum defect known in those times, when
only the Archimedean proportion of the diameter to the circumference was known, which was limited to hundredths of the
diameter.9
24. In the book De berillo there are frequently straight lines and
angles which are meant to explain philosophical, theological
teachings. Beryllus is a lucid, white, and transparent stone. It
is given at the same time a concave and convex form, and looking through it, one attains to things with intellectual eyes which
were previously invisible. This book meant to accomplish the
same for the intellect.
25. More efforts of the Cardinal about the quadrature of the circle of which Regiomantus spoke, find themselves drawn out in
the book De triangulos, where I also discussed it.
Betrachtung bei Gelegenheit des Kometen
A.G. Kästner, 1742
22. The cardinal could not have proved the theorem. His justification of it is fairly obscure, and to explain it would only be
worth the trouble if it could contain the truth. Only so much of
it deserves to be brought forward as would give an idea as to
how he might have come upon the theorem. He assumes one
and the same straight line, added to the first and third side of
every triangle, and gives a sum, which is proportionate as the
arc to the second side. Then it can be solved from his discourse,
that this line would be the double of the first side, in the biggest
triangle, whose angle opposite the second side is = 45 degrees.
Where he knows that, he does not mention; maybe he has discovered it through trials, and thereby assumed this magnitude of
the quadrant as well as he knew; his operation could not have
been very precise, otherwise he would have perceived that it did
not concur with his assumption.
Then he said, what occurs in this maximum triangle,
occurs also in the minimum, if the same thing could happen, as
when the third link would not surpass the second; thus it occurs
also with all the triangles in between. And that is the root of this
teaching. From it follows: if I find the line, which is to be added
to the right-angled triangle with bc as half-chord of the quadrant and in the hexagon with bc as half-chord, then the sums
found are in the same ratio as the arcs, i.e. they are as 3 to 2. It
is clear that I have therewith found the line, which is to be
added in all cases and there is no doubt about it.
At any rate, it is unquestioned that the Cardinal expressed himself very incomprehensibly.
Durch Glas, das unsre schwachen Blicke
Zur Kenntniss ferner Welten stärkt,
Ward gestern, mit verschiednem Glücke,
Der Erdball, der jetzt brennt, bemerkt.
Des heitern Himmels blaues Leere
Stellt sich des Einen Auge dar;
Der findet in dem Sternen heere,
Statt des kometen, den Polar.
Wohl! endlich hab ich ihn gefunden,
So ruft der Dritte halb entzückt;
Er ruft, und sieht sein Glück verschwunden,
In dem die Hand das Rohr verrückt.
Reflection Upon the Occasion of a Comet
A. G. Kästner, 1742
By lens, through which our feeble gazes
The ken of realms afar gains might,
Was yesterday, the globe that blazes,
With certain luck, revealed to sight.
The jovial Heaven’s azure reaches
Present themselves before the eye;
Amongst the starry host is seated
No comet, but a Pole on high.
23. A number of applications of this theorem to the measure of
the circle and the sphere. The close of the book is: In a similar
manner, you yourself may derive the relationship with regard to
the minimum in other curved surfaces. What can be known in
mathematics in a human manner, from my point of view, can be
found in this manner.
“Aha! I now at last have found it,”
Cries out a watcher, filled with hope;
He cries, and sees his luck diminish:
His hand has bumped the telescope.
– Translated by Tarrajna Dorsey
9
Sections 20 and 21 are clarified with the following animation:
http://wlym.com/~animations/ceres/PDF/Michael/kastneranimation.swf
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Cardinal Cusa’s Dialogue on Static Experiments
Kästner
Cardinal Cusa’s Dialogue on Static Experiments
Abraham Kästner
The following translation, by Michael Kirsch, is from Kästner’s
Geschichte der Mathematik. It is the first of a section of
writings on the mechanical sciences.
lignis, in maiori est plus leuitatis. Philo. Video et
placet multum.2
The philosopher receives continuous pure lessons from
the mechanic, and expresses gratitude. Therefore he had
conformed his take [Nahmen gemass] to reflect the mechanic’s experience, adjusting to the same concepts, and
testing the conclusions. But sure enough terms and conclusions of the philosopher were at that time still quite imperfect, as were the mechanic’s.
The lightness of wood in water had to be caused by the
water forcing the wood up, since wood in air is not light.
Sure enough, the larger piece of wood is forced upwards by
a greater body of water, but it is larger by the same ratio,
whereas the force that forces it up is equal to the force that
lifts up the smaller piece. However the surface of the bigger wood conducts [verhält sich] itself to the surface of the
smaller, if both parts have similar figures = root 2 : 1 since
the masses themselves, and conducts the upwards propelling moving power = 2 : 1. Since then the resistance of the
water, if all the rest is the same, still adjusts [richtet] the
surface, thus might consider in conjunction with the moving
power upwards to be the bigger part of wood not as much
as by the smaller, and those might climb faster, since the
mentioned masses are somewhat a hindrance to the resistance.
Cardinal Cusa’s Dialogue on Static Experiments1
1.
Nicolai Cusani, De staticis experimentis dialogus, finds itself with M. Vitruvii Pollionis de Architectura Libri X.
Strassburg 1550:
A philosopher entertains himself with a mechanic; the
mechanic observes that these scales serve the purpose of
recognizing the nature of bodies. Water of equal mass does
not have equal weight. Sure enough its weight changes, as
the water at its source, is different from the same water at a
distance from it, though these barely appreciable differences can be set aside. The weight of blood and urine are
not equal for the sick and healthy, the young and old, German and African. So a physician would do well to make
note of these distinctions. Also, recording the division of
weight and juices of plants, with their origin, would teach
more about their nature than the deception of their taste.
Comparing these weights with the weights of blood and
urine determined the doses and taught the diagnosis. Thus
easily through weight experiments, one can ascertain such
knowledge. Water is allowed to flow from the narrow hole
of a water clock, thus lasting as long as a hundred heartbeats of a healthy young man, and in turn a hundred of a
sick man. One will not find the same weight of emanating
waters. Thus, with the differences of pulses, weight renders
knowledge of diseases. In the same way compare a hundred respirations of a sick person and a healthy person.
People could be weighed in air and water; even animals.
Accordingly, make modifications and write down that
which is measured.
3.
The mechanic also discoursed concerning the resistance of
water, but not in the manner which we currently use the
word: he meant, it resists, as the more resists the less (ut
maior gravitas minori). If a round piece of wood is pressed
in wax, and fills the cavity with water, weighing more than
the wood, it will float, and a part of it remains above the
water, of which the excess corresponds to the weight of water. If the piece of wood is not round but flat, it makes a
bigger space, and floats more, thus, ships in shallow water
have level bottoms.
2.
4.
These modifications which the philosopher does not understand, the mechanic explains thus: he takes a piece of
wood, whose weight, compared to the weight of water filling up an equally big space, is in a 3:5 proportion; he divides it into two unequal parts, one twice as big as the
other; he puts them in a large tank, presses them down to
the bottom with a stick, pours water in, filling the tank, and
then pulling the stick away, both parts ascend, the bigger
one faster than the smaller. Ecce tu vides diuersitatem motus in identitate proportions ex eo euenire quia in leuibus
Also a proposal to research the attractive power of magnets,
and the like power of diamonds, which, as he would say, resist the attraction of magnets, and the power of other stones,
in conjunction with their magnitude…
5.
If one hundred pounds of earth which had been weighed
with its plants and seeds growing from it, is put in a pot
with the plants and seeds removed, and weighed again, then
2
1
Behold, you see diversity of motion in identical proportions, hence it
results that in smooth wood, there is to a greater degree more smoothness. Philo. I see and it is very pleasing.
Des Cardinal Cusasus Gespräch von statischen Versuche.
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Cardinal Cusa’s Dialogue on Static Experiments
Kästner
one would find that it had lost a little weight, and accordingly therefore, plants receive their weight mostly from water. If the ashes of herbs were weighed, the amount of
weight the water has contributed would be revealed. The
elements transform themselves partly to one another, as water becomes stone… with the balance, earth, oil, salt… will
lead to much research.3
weight of the day of the month and hour of the day can also
be conjectured; but on a short day these changes are uncertain. Thus the water could also be weighed, that flows in
between two transits of a fixed star through the meridian,
and that in between two risings of the sun, and further, concluding from the motion of the sun in the zodiac, the inequality of the motion of the sun itself. When the sun is
rising on the equator, the water that flows in between the
rise of its upper part to lower part provides the relation between the solar body to its sphere. Thus the mechanic will
also need a water clock with the moon, with a lunar eclipse,
to determine the relationship of the moon to the earth’s
shadow.
6.
Also the entire globe’s weight can be conjectured from the
weight of a cubic inch because its circumference and diameter are known. The philosopher thereby recollects, In
maximo ista vix conscriberentur. More understandings
from these matters might have given other recollections.
7.
12.
Perhaps, said the philosopher, would one also come,
through such subtle conjectures, to the weight of air? The
mechanic responded thus: Much dry, pressed together wool
is put into a bowl of a large balance, and in the other bowl
stones of equal weight. That, in temperate air. The weight
of the wool would be found to increase, or decrease, accordingly as the moisture or dryness of the air. That would
lead to conjectures of the weather.
If in March the certain weights of water, of wood, of air,
were found, and compared with the weight of other years
and the seasons, one would thereby deduce the bigger or
smaller fertility, as from astronomical laws. If at the beginning of winter fish and creeping animals are found to be fat,
a long and harsh winter is conjectured, because nature protects creatures against it. The weight of a bell, pipes, and
the water that fills the pipes, gives the measure of the notes.
Measure of circles and of squares, and all as regards spatial
figures, also provides the truth, more easily through weight
than other methods…4 So, one can weigh how much space
lines, planes, and bodies contain and from such a measure,
like measures can be inferred.5
The philosopher recognized: a book in which such
measurements were collected, would be very instructive, to
be conveyed everywhere. And the mechanic concluded:
yes, if you care for me, be diligent in the task.
8.
If one would weigh a thousand grains of wheat or barley,
from fruitful fields and varying climates, then he would
learn from this something about the force of the sun in these
declinations. Also thus from mountains and in valleys, of
the same geographic parallels, (in eadem linea ortus et occalus).
9.
If a rock falls from a high tower, and water flowing from a
pierced hole is collected during the time of the fall; then,
doing the same with a piece of wood of like magnitude, the
philosopher believed that the differences of the weight of
these three things would yield the weight of the air. The
mechanic judged: repetitions from various equal sized towers, and of various times, would confer endless speculations. The air can be weighed yet easier, if one fills the
same bellows equivalently for various times and at various
places, the same motion observed through equal heights,
and water which had flowed in this time from a waterclock
is weighed.
4
Cusa elaborates that the ratios of polygons areas could be found by
weighing the water that would fill up cylinders cut in the shape of those
polygons.
5
Kästner passes over Cusa’s discussion of harmony which has bigger
meaning with regard to Kepler’s work, and the current investigation by
Larry Hecht on the relation between harmonics and the moon model. I
excerpt it here:
Layman: Experiments done with weight-scales are very useful with
regard to music. For example, from the difference of the weights of
two bells of consonant tone, it is known of which harmonic proportion
the tone consists. Likewise, from the weight of music-pipes and of the
water filling the pipes there is known the proportion of the octave, of
the fifth, and of the fourth, and of all harmonies howsoever formable.
Similarly, the [harmonic] proportion—from the weight of mallets from
whose striking on an anvil there arises a certain harmony, and from the
weight of drops dripping from a rock into a pond and making various
musical notes, and from the weight of flutes and of all musical instruments—is arrived at more precisely by means of a weight-scale.
Orator: So too, [as regards the harmonic proportion] of voices and
of songs.
Layman: Yes, all concordant harmonies are, in general, very accurately investigated by means of weights. Indeed, the weight of a thing
is, properly speaking, a harmonic proportion that has arisen from various combinations of different things.
– from the Jasper Hopkins translation of The Layman on Weights and
Measures by Nicholas of Cusa.
10.
To find the depth of the ocean, an approximate a procedure,
which I have written in Puehler Geometrie Book I page
674. The power of men to weigh green wood, and its varying weight depending on its degree of warmness and coldness, and its dryness and wetness.
11.
If the entire year through each day from the rising of the
sun until it sets, water flowed from the water clock, and
would be weighed, thus, from these recorded weights, the
3
The ellipses (…) are present in Kästner.
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