A Scientific Problem: Reclaiming the Soul of Gauss

By Michael Kirsch

Before launching into his highest achievement in Book V of the Harmony of the World, in which he demonstrated 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 demonstrated 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]

Nicolas of Cusa’s influence on Johannes Kepler in every field of his works 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, and especially the youth generation today.

The nature of the universe as demonstrated in the two webpages of the LYM on Kepler, 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 being: to have fun.  Man is a creature which cannot be bounded by any bounds, because of that which lies inside man, his soul.  It is in the nature of the human soul to have fun, but a certain kind, 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; 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. 

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, must continue so that a new generation of economic scientists are unleashed who do 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 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 did discover, and comparing them with the 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.[2]    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’ 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 Nicolas of Cusa and his student, Kepler, whose relationship released the Earth in motion from the shackles of empiricism, and with it all of modern science.  In carrying forward the scientific revolution of Cusa and Kepler, 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 Nicolas 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’ 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 provides the basis for discussing Carl Gauss’ 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’ infinitesimal in his discovery of the orbit of Ceres, and above all, his contribution to the ‘higher purpose’ of mankind.

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.[3]   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 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 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—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 only in the Maximum is found absolute Equality, 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”.

What implications did this hold for astronomy?

“…It is not the case that in any genus— even [the genus] of motion—we come to an unqualifiedly maximum and minimum. Hence, if we consider the various movements of the spheres, [we will see that] it is not possible for the world-machine to have, as a fixed and immovable center, either our perceptible Earth or air or fire or any other thing. For, with regard to motion, we do not come to an unqualifiedly minimum—i.e., to a fixed center. For the [unqualifiedly] minimum must coincide with the [unqualifiedly] maximum; therefore, the center of the world coincides with the circumference. Hence, the world does not have a [fixed] circumference. For if it had a [fixed] center, it would also have a [fixed] circumference; and hence it would have its own beginning and end within itself, and it would be bounded in relation to something else, and beyond the world there would be both something else and space (locus). But all these [consequences] are false. Therefore, since it is not possible for the world to be enclosed between a physical center and [a physical] circumference, 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.[4]

“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.”[5]  

Here in these passages Cusa, deriving 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; not even the sun is completely devoid of motion. Thus the Aristotelian Ptolemaic model system was exposed as a fraud.[6] This truth would be thoroughly demonstrated by Kepler in refuting the Equant.[7]  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.[8]  Cusa also established that since motion never occurs around a fixed point, there are no perfect Circles.[9]   This was left for Kepler to demonstrate in Chapters 40-60 of the New Astronomy. [10] 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.[11]

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!” Somewhere, a baby boomer sighs 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 “If we can fully attain unto this knowledge or 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 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”[12] 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.”[13]

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!”[14]

To proceed further toward our end, Cusa then declares, spinning Aristotle in his grave[15]:

“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.[16]

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:

“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.”[17]   

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.”[18]     

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.” [19] 

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.”[20]

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.”[21]; 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.[22]    

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

“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

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:

“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.”[23]

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. As he continued:

“There can namely be given an incidental angle that is greater than a rectilinear, and another incidental angle 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.”[emphasis added][24]

 

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.”[25]

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:

“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 + 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.”[26]

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.” [27]

He also states here in the 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 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.[28]