The Science of Nicholas of Cusa and Its Impact on Johannes Kepler
By Michael Kirsch
Nicolas of Cusa influenced the thinking of Johannes Kepler in nearly every field of the latter's work. This article investigates that relationship in great detail. The reader is walked through all of the major works of Cusa and provided the experience of rediscoverying his method of thought. The direct and indirect expressions of Cusa's method and discoveries in Johannes Kepler's major astronomical works are then drawn forth.
Part I: The Edifice of the World
Abraham Kästner, in 1757, in his Praise of Astronomy declared Nicolas of Cusa to be, one of two “revivers of the edifice of the world” along with Copernicus. Although Cusa was a close collaborator of Toscanelli, who was a famous astronomer at the time, the most probable reference is to Cusa’s De Docta Ignorantia. In that work, there lies a principle so vast, that its implications will guide us through the entirety of this investigation.
Nicolas of Cusa sought to demonstrate that the Creator of the Universe was not something able to be reduced to a particular metaphor or described, 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 Aristotle's duality, of, on the one side, unknowable principles and, on the other, the world of changeable sense, but instead 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 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 only in the Maximum is found absolute Equality, all things differ. From this comes the historic statement by Cusa, full of implications:
“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”.
What implications did this hold for astronomy?
By establishing the universe as a product of a Maximum Creator with paradoxical relation to that universe, Cusa 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; not even the sun is completely devoid of motion. Thus the Aristotelian Ptolemaic model system was exposed as a fraud. This truth would be thoroughly demonstrated by Kepler in refuting the Equant. 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. Cusa also established that since motion never occurs around a fixed point, there are no perfect Circles. This was left for Kepler to demonstrate in Chapters 40-60 of the New Astronomy.  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, as motion was derived from more than simply comparing two objects, with one 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.
Part II: What is Science?
What therefore is man that he exists within such a universe? How must mankind approach the challenge of a universe, 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 Nicolas of Cusa’s De Docta Ignorantia, he begins by stating that all things desire to exist in the best manner possible, and use their judgment so that this desire is 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 assume a priori and set up set of definitions and axioms that follow. Forget about whether the initial axiom is knowable, it will work.”
Now, after wrestling with this, ask the question: if we seek to become learned in our ignorance, what must humans 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” is that in things greater something can always be greater, in things lesser, always lesser, thus in comparing two things we never find them to be so equal that they could not be more equal indefinitely.
Cusa elaborates the paradox which the intellect faces with such an incomprehensible maximum. Since the maximum is not greater or lesser, it is both maximally large, and maximally 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.”
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, by means comparisons of things greater or lesser, but, rather, “incomprehensibly comprehended.”
To proceed further toward our end, Cusa then declares, spinning Aristotle in his grave:
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.
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 said:
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 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.”
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.” 
But if man's desire to know everything about the universe clashes with the Maximum truth being infinitely distant, then logically wouldn’t the Creator be evil?
Cusa resolves this matter:
“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.”
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.”; in truth, this paradox of the mind’s inability to comprehend the entire universe, is not part of an evil design, it is in fact what drives the universe forward. The speculation of mankind is not a sign of an entropy of the mind, but is the nourishment itself, and in the process of mankind’s discoveries, the universe develops.
Since this is the purpose of mankind’s nature, to ascend with the intellect, Nicolas 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 prolong.
Part III: On the Curved and Straight
-Lyndon LaRouche, Cusa and Kepler
Nicolas 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.
In Cusa’s on the Quadrature of the Circle he begins:
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,
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. As he continued:
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.”
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 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 the point:
Thus, the following property of the circle arises:
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 + contigent angle = rectilinear angle
a rectilinear – contigent 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.
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.”
He then remarks on what necessarily follows.
In ‘On conjectures’ Cusa had identified what the nature of a numbers such as the circle were: “Hence, species are as numbers that come together from two opposite directions—[ numbers] that proceed from a minimum which is maximum and from a maximum to which a minimum is not opposed.” 
He also states here in the On the Quadrature of the Circle:
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 in this section 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. It is as though: 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 Compliment, the idea of isoperimetric takes a broader meaning, in looking at triangles and squares and other polygons that all have equal perimeter, and what the relationship of the radius’ 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 give triangle which is inscribed in a circle. Where would such a radius be? What would be its characteristics?
First, he shows that the simple idea of an equality between the triangle 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, such a radius would not be proportional to the sides of the triangle, because if the radius is to the circumference, and if the triangular circumference were equal to the circle, then it would share in the lack of proportionality with the radius.
The sought 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, so 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 would be proportional to dl— would not be proportional to eb, de, or db. Nor would the line ek, created by dk, be proportional to eb, de, or db
And what this points to, is an extremely important affirmation by Cusa. Since, as was shown, no line can be drawn that stands in rational proportion with the sides of the triangle, no point on eb could be given precisely that the ‘sought length’ would be drawn to. Thus, any length along eb, which is in proportion to eb, would not be in proportion to the length sought. And also, any length which is drawn from d such that it would be in proportion to a length along eb, would not be the ‘sought length’.
So this gives us the method of approach to boxing in our isoperimetric radius right? Since the sought line is not proportional to eb and db, what we are looking for then, must be to find the line which is the most non-proportional to them, and then, we will have the line which is the least non-proportional to the ‘sought length”. The length we are looking for compared to the lengths that are known, those of the triangle, is the minimum with respect to its degree of knowability. Therefore, we are looking for the radius which brings us the most ignorance relative to the known triangle.
Where must the cut be? One extends the length that cuts the line, by the proportion of the line on the side of the triangle—created by the cutting line—to the whole side of the triangle[see animation] and also the line on the other side of the cut to the whole side. However, since the line cutting the line has to be proportional 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 sides of the triangle. It cuts it larger or smaller. So if it extends it 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.
After finding the closest value for the isoperimetric radius , he makes his point:
At the conclusion, having thus demonstrated what he called a ‘species’ difference, which even Archimedes failed to see, Cusa remarks on the ‘higher purpose’ of seeking truth.
Mathematics of the Infinite
Later, in the 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, 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.”
“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 the De Docta Ignorantia, again, here, in the case of this maximum ‘number’ indicated by the species difference.
From the De Docta Ignorantia:
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 comparing the curved and straight, the straight line participates more in the infinite line than a curved line participates in it.
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 Compliment, 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:
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 the 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 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:
In Cusa’s Mathematical Perfection whose aim was “to hunt for mathematical perfection from the coincidence of opposites,” he investigates whether the smallest chord of which there cannot be smaller were not 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.” 
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, he states:
In what is historically of great importance, Abraham Kästner, in his review of Cusa’s works, remarked about this statement:
Infinitesmal: Imprecise Measure for the Transcendental
Cusa’s investigation of the curved and straight is a model for the identification of the nature of the human soul. It is the capability of the human mind to conceive and discover the relations between transcendental magnitudes by ascending to the intellect, and 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 in his 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 Nicolas of Cusa, the relation of the curved and straight was no mere comparison, as such, that is, not a case of “this is like that.” Nicolas 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.
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. Kepler’s conception of the entire universe was shaped most prominently by Cusa; particularly on the question of ‘quantity’. In his Mysterium Cosmographicum, in the Second Chapter, 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:
In various letters of Kepler he expressed the same sentiment concerning Cusa’s view of man:
Later in Kepler’s investigation of Light in his Optics in 1604, again this influence of Cusa concerning the curved and straight and his conception of the infinite sphere, would present itself as the opening conception concerning the relationship 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 commuted proportion of density with extension, are equals. 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….
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 that Cusa’s investigation in his Mathematical Perfection appeared to be introducing infinitesimals into the construction, one wonders, 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 an arc. But in order to correct it, he had to know the whole orbit.
Working on calculating the motion of the Earth, Kepler, in Chapter 32 of the New Astronomy, as he is deriving the principle that the time needed to traverse an arc of the orbit is inversely proportional to the distance from the sun, states: “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. Kepler was able to determine the whole orbit by understanding the relationship expressed in the smallest possible part of 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 a 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. 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, a higher geometry in which the action of physical principles could be represented.
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 a friend J.G. Brenegger on April 5th 1608, among other matters, Kepler wrote: “Cusa said the infinite circle is a straight line.” This idea of Cusa led to a breakthrough in conics by Kepler in his Optics, achieving a continuity of conic sections.
While investigating the hyperbola and the relation between the chord and the sagitta, as the focus moves closer to the base, he says “ The sagitta … 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.)” 
Echoing the infinite metaphors of Cusa, he continues:
Lastly, and perhaps of greatest importance is the foundation of the transcendental magnitude discovered by Cusa.
For the question arises, what was Kepler’s Problem? What did he do which caused such ferment after his death? Why was there a political operation to get rid of his Problem? 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 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 truth, willing to become sufficiently knowledgeable of his ignorance, did Kepler succeed in relating the unseen principle to the sense perceptions –of 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, who attempted to reduce transcendental magnitudes to lower algebraic magnitudes with their infinite series, had already been proved wrong by Cusa.
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.”
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 the Harmony of the World, Kepler took these questions of the human soul and geometry as we discussed earlier, and the issue of imprecision was approached again, this time with the harmonies. In this work,he furthered the relation of man and the universe in a more profound way than ever, moving beyond simply the curved and straight as expressed in his solids, to the moving form of the soul itself in the heavens. In this work Johannes Kepler demonstrated that it is through harmonics that the physics of the solar system are known. Once again, the work of Nicholas of Cusa is seen as an inspiration.
In Book I of 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.  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. Thus, with such a relationship to Nicolas 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 the nature of the proportions of the harmonies. The influence of Cusa appears in a crucial step in his Kepler's discovery, where he shows that the causes of the harmonic proportions with which we measure physics have their origin from within the rational soul, as “abstract quantities”.
With these harmonies established as proportions from the soul, Kepler then took up his edifice of the world from his Mysterium and bringing together his New Astronomy, sought to demonstrate the causes of the motions. Kepler determined the extreme motions of the planets at Perihelion and Aphelion as the area to seek harmony in the heavens, and proceeded to calculate every possible proportion between each of the planets’ diverging, converging, and extreme motion in pairs. Once he then fit the planet’s 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 in 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.
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 Venus’ aphelion and Earth aphelion, and a soft sixth, 5/8, between their perihelial motions. This created the necessity for very small changes to each planets 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. But in this case, Kepler had shown that the two intervals of Earth and Venus’ own motion would have to differ by a diesis in themselves, these two melodic intervals the 24:25 and 15:16, differ by 125:128, which is smaller. 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.
So instead of the Earth’s motion having either the melodic semitone of 15:16 or the unmelodic interval of 12:13, it has 14:15, and instead of Venus having the melodic diesis of 24:25 or the unmelodic interval of 125:128, it had 35:36. And 14:15 and 35:36, both differ from 15:16 and 24:25 by 80:81, a 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.
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:
Kepler’s method of hypothesis cures the mental diseases of entropy found so common in modern science today. The human soul’s own proportions found throughout create the circumstances that we are inside the universe, and that we understand it as a reflection of ourselves.
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 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 minds 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. Such thinking is entropic, because in that thinking one must force the universe into harmony, one has to put it together piece by piece, and it is delicately holding together, rather than the idea that one is on the inside of it, and have detected in the small the reason for its perfection. Such imprecisions as commas and infinitesimals are not seen as a fragile argument that needs to be held together with great convincing, but are the reflection of the relationships indicating a new unseen dynamic.
Inquire further. How did Kepler determine the causes for the eccentricities? Did the physics of the orbital elements randomly create harmony, or did the necessity for harmony generate each orbit as it is? Further, if each orbit necessitated creating harmony, how did the Solar System become one whole harmonic system? Take a few examples for the relations of the Solar System as a whole.
Kepler investigates why Earth and Venus have the smallest eccentricities of all the planets, that is, why the physical orbits of the planets are the way they are. So, why the small eccentricities? Because it is on our planets that the hard and soft sixths depend, and thus upon which the crux of the whole musical system hangs. After working out how hard and soft harmony is distributed throughout to form one harmonic system Kepler said :
The region of most importance for the harmony of the whole Solar System, that between Earth’s Aphelion and Venus’ 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 1/32 (which is continuous doubling of the octave ½), and Mercury, the innermost planet, is in harmony at Perihelion with Venus’ Perihelion forming ¼(one doubling of an octave ½). Here the whole system is seen to make a giant counterpoint, echoing each other in an octave.
Also, in these outer planets, perfect harmonies were found among the converging motions in the pairs of planets, but not in each individual planets motion’s, 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 change the kind of harmony either soft or hard, whereas between Mercury and Venus there are two perfect harmonies, but which do not change their kind of harmony. And as Venus is the most imperfect in its own proportions and the smallest eccentricity, so Mercury is the most perfect forming a perfect 5/12 and the largest.
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.
In demonstrating that the physics of the entire Solar System could only be known through the harmonies, how does that transform the definition of humanity as a whole?
How can it be that the solar organizationis based on the same harmonic ratios with which human beings created music before we even discovered this?
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 not, then 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, 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 it’s ‘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.
—Nicolas of Cusa, On Conjectures
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, which 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 only see 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. I intellect perceives the potentiality of all spheres of all sizes, and even a sphere beyond all sphere’s. As if in a flash, the mind perceives a point becoming an infinite sphere, the sphere of all spheres. Likewise with the circle, the rational concept of it, is not the true one, if it is merely that which all lines to the center are equal. The true circle in absolute unity is without lines and circumference. The true circle is infinite, which is infinite straightness, and serves as a measure for all things.
See Sphere Animations:
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.
The intention of the intellect is to become actual. In that way, Man submits himself to the senses in order to attain understanding. He says our intelligence is like a spark of fire concealed under green wood, which needs the senses to draw forth the heat in the wood. The more powerful is the actuality of fire, the more rapidly it causes the ignitable to become actual. And as the imagination needs the rationality to be intelligible, so colors need light to be seen, as one’s vision cannot move directly to color without light.
Ascend higher therefore: the rationality is conveyed into the intellect through itself, as light is into vision, and the intellect descends through itself into rationality, as the vision proceeds to light. Now all things are defined by that which measures it, and so the rationality is defined by, and is the intellect descending into it.
Although the rationality partakes in the otherness of the senses, the intellect is the unity of the rationality, and thus precedes otherness. Cusa says, that the rational higher nature, which also absorbs the unity of imagination, and which is concealed in the light of the immortal intellect, is also immortal, like light that cannot be obscured.
Therefore, the difference between men and the beasts is that human rationality is absorbed in the immortality of the intellect. It is always intelligible through itself 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, the question is: what is the intellect an otherness of, in which it is enfolded as a unity? 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 as Man’s conceptions are to the Creator’s creations so the soul creates instruments to discern and know. Man unfolds conceptions in perceptible material. And man creates instruments like temporal measures. Therefore, since time is the measure of motion, it is the instrument of the measuring soul; hence, the soul’s measuring does not depend on time, rather the scale for measuring motion, time, depends on the soul. And as the eye is the instrument of sight, so 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, soul’s 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 in that way. Thus, the soul is not corruptible in motion, nor is it subject to time. Thus it is eternal, and immortal.