JUNE 2007 ISSUE. - World - LaRouchePAC

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 

3 

4 

23 

29 

From the Editors 

The Tragedy of Leonhard Euler 

– Lyndon H. LaRouche, Jr. 

A Scientific Problem: Reclaiming the Soul of Gauss 

– Michael Kirsch 

Translation: Some Geometrical Writings of Nicholas of Cusa 

– Abraham Kästner 

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 300 th 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. 

For online versions, see: http://www.wlym.com/~animations/ceres. 

∆υν ∆υναµι ∆υν ∆υναµι 

αµις αµι Vol. 1, No. 4 June 2007 

2


LaRouche 


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. 

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 

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. 



3


Kirsch 


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: 

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 

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 project 3 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 

1 

Johannes Kepler, The Harmony of the World, Book IV, Chapter 1 

2 

See http://wlym.com/~animations. 

3 

See http://wlym.com/~animations/harmonies. 



4

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. 


Kirsch 

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? 

� � � � 

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 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. 

Even though Cusa Hence, the world does not have a [fixed] circumference. 

had written specifically on astronomy, as with his collaborator, For if it had a [fixed] center, it would also have a 

the famous astronomer Toscanelli, Kästner is most probably [fixed] circumference; and hence it would have its own 

making reference to Cusa’s De Docta Ignorantia. In that work, beginning and end within itself, and it would be 

there lies a principle so vast, that its implications will guide us bounded in relation to something else, and beyond the 

through the entirety of this investigation. 

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 

4 between a physical center and [a physical] circumfer- 

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. 

5

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 


Kirsch 

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. 

12 

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 

7 

Since it is not the maximum, the universe could have been greater, following quote from Leibniz’s 1692 Critical Thoughts on the General 

but since in the possibility of being, matter cannot be extended unto Part of the Principles of Descartes shows: “If motion is nothing but the 

infinity, the universe could not be greater. Thus it is unbounded and change of contact or of immediate vicinity, it follows that we can never 

with respect to all that can be in actuality, nothing is greater than it. define which thing is moved. For just as the same phenomena may be 

8 

In De Ludo Globi, Cusa, discussing the motion of the irregularly interpreted by different hypotheses in astronomy, so it will always be 

shaped ball used for the game, and the conditions of the ground, and the possible to attribute the real motion to either one or the other of the two 

way in which each different player sets the ball on the ground, says “It bodies which change their mutual vicinity or position. Hence, since 

is not possible to do something the same way twice, for it implies a one of them is arbitrarily chosen to be at rest or moving at a given rate 

contradiction that there be two things that are equal in all respects in a given line, we may define geometrically what motion or rest is to 

without any difference at all. How can many things be many without a be ascribed to the other, so as to produce the given phenomena. Hence 

difference? And even if the more experienced player always tries to if there is nothing more in motion than this reciprocal change, it follows 

conduct himself in the same way, this is nevertheless not precisely that there is no reason in nature to ascribe motion to one thing rather 

possible, although the difference is not always perceived.” Abraham than to others. The consequence of this will be that there is no real 

Kästner in his review of Cusa says that this is Leibniz’s Principle of motion. Thus, in order to say that something is moving, we will require 

Indiscernibility. http://tinyurl.com/yv8kca/makKaestnerCusareview.pdf not only that it change its position with respect to other things but also 

9 

For Kepler’s discussion of the Aristotelian and Ptolemaic models, see that there be within itself a cause of change, a force, an ac- 

Part I of his New Astronomy. 

10 

March 2007 Vol. 1 No. 3 http://wlym.com/~seattle/dynamis 

tion.”[emphasis added] 

6

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: 

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: 


Kirsch 

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 

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 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.” 

is that in 

Cusa had been sent to Constantinople as part of his attempts 

things greater something can always be greater, in things lesser, to reunite the Greek and Roman Churches. He returned in February 

always lesser, and thus in comparing two things we never find 1438. At the end of De Docta Ignorantia, Cusa states, “while I was at 

them to be so equal that they could not be more equal indefi- sea en route back from Greece, I was led (by, as I believe, a heavenly 

nitely. 

gift from the Father of lights, from whom comes every excellent gift) to 

Cusa elaborates the paradox which the intellect faces embrace—in learned ignorance and through a transcending of the in- 

with such an incomprehensible maximum. Since the maximum corruptible truths which are humanly knowable—incomprehensible 

things incomprehensibly.” 

is not greater or lesser, it is both maximally large, and maxi- 18 

Aristotle in his metaphysics, after a lengthy attack on the Pythagorean 

conception of number states in his final conclusion:“the objects of 

15 

http://cla.umn.edu/sites/jhopkins/DeLudo12-2000.pdf, Book II, sec- mathematics are not separable from sensible things, as some say, and 

tion 96 

they are not the first principles." 

7

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 


Kirsch 

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. 

19 

Abraham Kästner remarks on the importance of this concept in his 

21 

Nicholas of Cusa, Theological Complement 

22 

Ibid. 

23 

Ibid. 

24 

Ibid. 

review of Cusa’s De Venatione Sapiente. See Translations from the 

25 

Norbert Weiner, The Human Use of Human Beings, Chapter II: 

Geschichte http://wlym.com/~animations/ceres/index.html 

“Progress and Entropy” 

20 

Plato’s Republic, Book VI 

26 

Norbert Wiener, The Human Use of Human Beings 



8

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: 

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. 

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: 

There can namely be given an incidental angle that is 

greater than a rectilinear, and another incidental angle 


Kirsch 

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 

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 

See animation: 

http://tinyurl.com/yv8kca/Moving%20Inciden 

tal%20Angle.swf 

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 

27 

All quotes in this section, unless otherwise indicated, are taken from 

28 

Emphasis added. This question of incidental angles was a great epistemological 

debate with grand implications. See Will Wertz: Nicholas 

Nicholas of Cusa’s On the Quadrature of the Circle, translated by Will of Cusa’s “On the Quadrature of the Circle” at 

Wertz. See: http://www.schillerinstitute.org/fid_91- 

http://www.schillerinstitute.org/fid_97-01/012_Cusa_quad_circ.html 

96/941_quad_circle.html 

29 

De Docta Ignorantia Book III Chapter I 



9

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: 

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. 

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 


Kirsch 

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.” 

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/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 

30 

De Docta Ignorantia, Book III, Chapter I 

31 

Nicholas of Cusa, On Conjectures 



10

the circle, then it would share in the lack of proportionality with 

the radius. 

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. 


Kirsch 

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 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 

And what this points to, is an extremely important af- a higher insight and by no means through a visible atfirmation 

by Cusa. Since, as was shown, no line can be drawn tempt. From that alone you can now know, that only in 

that stands in rational proportion with the sides of the triangle, the domain inaccessible to our knowledge, will a more 

no point on eb could be given precisely that the “sought length” precise value be reached. I have not found that this re- 

would be drawn through. 

alization has been passed along until now. [emphasis 

Thus, any length along eb, which is in proportion to eb, added] 

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 At the conclusion, having thus demonstrated what he called a 

to a length along eb, would not be the “sought length.” 

“species” difference, which even Archimedes failed to see, Cusa 

So this gives us the method of approach to boxing in remarks on the “higher purpose” of seeking truth. 

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 32 

As an example of a non-proportionality between magnitudes, he says 

the “sought length.” The length we are looking for, when com- 

that the lines bounding the incidental, rectilinear, and coincidental anpared 

to the known lengths of the triangle, is the minimum with 

gles share in the non-proportionality that their angles share. They are 

respect to its degree of knowability. Therefore, we are looking magnitudes which are larger or smaller than each other by a magnitude 

for the radius which brings us the most ignorance relative to the larger or smaller than a rational fraction. This line he says is “before all 

known triangle! 

divisibility of the line… by which a straight line can cut a straight line 

Where must the cut be? One extends the length, dk, in two… It is like an unattainable endpoint [of a line]… nonetheless… 

which cuts the side of the triangle, proportionally as the line on in its way, divisible by a curve.” The point he makes is that the normal 

the side of the triangle – eb, created by the cutting line – is to divisibility of a line which lies between two endpoints is different than 

the whole side of the triangle ab [see animation below] and also 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 

the line on the other side of the cut to the whole side. However, 

between proportionality and non-proportionality. This magnitude is the 

since the line cutting the side of the triangle has to be propor- 

type which describes how close one can approach the sought length. 



11

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. 

Mathematics of the Infinite 

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- 


Kirsch 

paring the curved and straight, the straight line participates more 

in the infinite line than a curved line participates in it. 36 

See animation: http://tinyurl.com/yv8kca/infinitecircle.swf 

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 

33 

Cusa said that the knowledge is presupposed, to which the mind is substance, as is the case of the catenary curve, as the clearest expres- 

guided by a light of truth in the mind. And all who seek knowledge are sion of the principle of least action, as shown in the Leibniz construc- 

instigated by that infinite art or science. 

tion of the catenary which most clearly expresses the irony of the 

34 

All quotes in this section are taken from Nicholas of Cusa’s Theo- paradox of physical action: that is, the complex domain. Afterwards, 

logical Complement 

the implications of Cusa’s principle of Maximum-Minimum were de- 

35 

De Docta Ignorantia, Book I, Chapter 12 

veloped in the infinitesimal calculus. 



12

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: 

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 

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 


Kirsch 

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 

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 

The Infinitesimal: An Imprecise Measure for the 

Transcendental 

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: 

Lyndon LaRouche, in his Paper For Today’s Youth: 

Cusa and Kepler, wrote: 

“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. 

38 

Kästner’s Review of Cusa’s Geometrical Writings, translated by Michael 

Kirsch. See elsewhere in this issue. 

Once this is found, then, from a proportion between circular lines, the 39 

Cusa had also stated in On Conjectures, Part II, Chapter II: “For if 

unknown straight line is found, through both the known line and known every chord is smaller than the arc that it subtends, and if the chord of a 

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.” 

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. 



13

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. 

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 

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 


Kirsch 

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: 

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. 

42 

For more on Cusa’s conception of the human soul, see Appendix. 



14

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 

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.” 

“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- 


Kirsch 

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 

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 Perfection 47 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 

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 

43 Johannes Kepler, Mysterium Cosmographicum, Chapter II sun, and f represented the triangle formed between those two observa- 

44 

These are taken from three different letters. All are found in the tions and the sun. In one calculation, Gauss stated, “We can set f’ : g’ 

book Kepler, written by Max Caspar. 

= 1, since the difference is only of the second order.” 



15

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. 

“Maximum” Conic Sections 

In a letter to his friend J.G. Brenegger on April 5 th 

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. 


Kirsch 

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 

While investigating the hyperbola and the relation between 

the chord and the sagitta, as the focus moves closer to the 

base, he writes, “The sagitta 52 … 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 

See animation: http://tinyurl.com/yv8kca/radius%20equals.swf 

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 infi- 

49 

For more on the Leibniz Calculus, see the October 2006 issue of this nitely great…” 

journal, Vol. 1, No. 1, at http://wlym.com/~seattle/dynamis. More on 

51 

Kepler, Optics, Chapter 4. 

the Leibniz-Bernoulli breakthrough of integration and its implications 

52 

In the diagram the sagitta it is the length A, the focus of the hyper- 

for Gauss’s work will be forthcoming at a later time on this Orbit of bola, to S on the axis of the hyperbola. 

Ceres webpage. 

53 

Kepler, Optics, Chapter 4. 



16

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 


Kirsch 

“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 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 

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? 

54 

Ibid. 

55 

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 

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 

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 

17

ecome 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. 

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 


Kirsch 

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. 

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 

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 

65 

See the coming article on Book IV, to be posted at: 

knowable to the human mind? What does this say about the human http://wlym.com/~animations/harmonies. 

mind? Is it looking as from outside the universe, analyzing sense per- 

66 

The reader is encouraged to return to the beginning of this article, 

ceptions from the outside, or rather, from within? 

where the entire quote is placed. 



18

which the blessed are conducted, once they are freed 

from the things of this world. 67 

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. 

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 


Kirsch 

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: 

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 

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 indi- 

So instead of the Earth’s motion having either the mevidual orbit, at two moments, the proportion of the apparent (from the 

lodic semitone of 15:16 or the unmelodic interval of 12:13, it sun) speeds has an inverse relationship to the proportion of the squares 

has 14:15, and instead of Venus having the melodic diesis of of the distances of the planet from the sun at those moments. But this 

24:25 or the unmelodic interval of 125:128, it had 35:36. And relationship does not hold between planets. If the area a planet sweeps 

14:15 and 35:36, both differ from 15:16 and 24:25 by 80:81, a 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 

67 

De Docta Ignorantia, Book II, Chapter I 

how we think of time on the earth, according to the earth’s rotation 

68 

On harmonic versus melodic intevals, see LYM Harmonies website about its axis, the number of days Mars took to travel 100 units would 

http://tinyurl.com/yq26fx/melodic.html. 

be different than the number of days Jupiter took to travel 100 units.” 

69 71 

The Harmony of the World, Book V, Chapter 9, Proposition XXVI Kepler, The Harmony of The World, Part V, Chapter 9. 



19

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 


Kirsch 

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, 

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. 



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? 

73 

The Harmony of the World, Book V, Chapter 9, Proposition XLIV 

74 

The Harmony of the World, Book V, Chapter 9, Proposition XIV 

75 

http://tinyurl.com/yq26fx/proposition48.html 

20

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. 

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 


Kirsch 

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. 

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. 



21


Kirsch 

The intention of the intellect is to become actual. In in that way. Thus, the soul is not corruptible in motion, nor is it 

that way, Man submits himself to the senses in order to attain subject to time. Thus it is eternal, and immortal. 

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 



22

Some Geometrical Writings of Nicholas of Cusa 

Kästner 


Abraham Kästner 

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. 


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 |: 

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 15 th 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. 

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 

De visione dei | De pace fidei | Reparatio kalendarii | all motions simultaneously. And while he considers in what 

De mathematicis complementis | Cribratio alchoran manner this sight deserts no one, he sees how diligently it is 

libri tres | De venatione sapientiae | De ludo globi libri concerned for each one as if it is concerned only with him who 

duo | Compendium | Trialogus de posesst | Contra bo- experiences being seen by it and not for anyone else. 

hemos | De mathematica perectione | De berillo | De 

I find the Cardinal’s prayerful meditation of the like- 

dato patris luminum | De querendo deum | Dyalogus de ness, theoretically truer, and practically more heart lifting, than 

apice theorie 

what has become stated in the philosophy of our time: God 

reigns over the whole, without troubling over the individual 

2. The format is a short folio, the pages are not numbered at the parts. The rest of the Cardinals thoughts, in which indeed there 

bottom as is usual, but they are marked with letters, one letter is much rightness and goodness, don’t belong to the present 

for every six pages, the first the letter is a, b, ... then A; my copy purpose. 

goes until the 4th page of letter C; upon whose first side begins 

with: Treatise On Beryllus Expounded. 

4. I can now arrive at my actual intention: 

It is thus not complete; however, it is in a very fine 

On Mathematical Complements 

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 



1 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 

This work has not yet been translated into English. 

23

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 

1st Line 

2nd Line 


Kästner 

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. 

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 

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. 

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. 



24

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. 

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]. 


Kästner 



25 

14 

= 196 

7 3 

= 147 

7 

= 49 

165 6/16 

9/16 x 147 

= 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 

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 

"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 

B

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. 

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 


Kästner 

line equal to the circumference, and sought the radius which 

belongs to it. He was led to it by the isoperimetric polygons. 

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. 

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 cen- 

2 

For more on the quadrature of the circle, the reader is referred to Part ter of the level ground. In its center, enclosed in the circle, is 

III of “A Scientific Problem: Reclaiming the Soul of Gauss,” this issue. the seat of the king, whose kingdom is the kingdom of life. And 

3 

In case the reader did not follow Cusa, the sagitta is the distance be- 

in this circle are nine others. However, the law of the game is 

tween 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 

5 

smaller the sagitta. 

The Hunt for Wisdom 

4 6 

Cyclometry is the study of circles. 

On the Game of Spheres 



26

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. 

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 


Kästner 

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). 

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 perfectione 7 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 sagitta 8 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 

7 Mathematical Perfection. This work has also never been translated 

into English. 

8 

See footnote 2. 



27

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. 

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 

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. 

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. 


Kästner 

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. 

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 

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. 

“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 



28

Cardinal Cusa’s Dialogue on Static Experiments 

Kästner 


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. 

1. 

2. 

Cardinal Cusa’s Dialogue on Static Experiments 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. 

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 

3. 

4. 

5. 

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. 

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. 

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… 

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 

Behold, you see diversity of motion in identical proportions, hence it 

results that in smooth wood, there is to a greater degree more smooth- 

1 

Des Cardinal Cusasus Gespräch von statischen Versuche. 

ness. Philo. I see and it is very pleasing. 



29

6. 

7. 

8. 

9. 

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 

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. 

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 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). 


Kästner 

12. 

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. 

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. 

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 4 

Cusa elaborates that the ratios of polygons areas could be found by 

philosopher believed that the differences of the weight of weighing the water that would fill up cylinders cut in the shape of those 

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 

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: 

same bellows equivalently for various times and at various Layman: Experiments done with weight-scales are very useful with 

places, the same motion observed through equal heights, regard to music. For example, from the difference of the weights of 

and water which had flowed in this time from a waterclock 

is weighed. 

10. 

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. 

To find the depth of the ocean, an approximate a procedure, Similarly, the [harmonic] proportion—from the weight of mallets from 

which I have written in Puehler Geometrie Book I page whose striking on an anvil there arises a certain harmony, and from the 

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. 

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 

11. 

of songs. 

If the entire year through each day from the rising of the Layman: Yes, all concordant harmonies are, in general, very accu- 

sun until it sets, water flowed from the water clock, and 

would be weighed, thus, from these recorded weights, the 

rately 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 

3 

The ellipses (…) are present in Kästner. 

Measures by Nicholas of Cusa. 



30

JUNE 2007 ISSUE. - World - LaRouchePAC

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