An Arc of Knowability The Problem of Cubic Roots

Preface

As the group constituting this current project took up our first task, that of working through Kepler’s Harmonices Mundi as well as the previous group’s product on that subject, we couldn’t help but read with a constant anticipation of arriving at the work of Gauss, which was our nominal domain of investigation. With that preconception in mind, many of us were awestruck when we found in Kepler’s work much of the germ ideas – even then, quite developed - that Gauss would further advance upon in his own work. In short, we saw it not only as a hypothetical that Gauss’ work was an extension of Kepler’s, but it was an undeniable, apparent fact.

It is the intention of those constituting this group to elaborate that relationship over the course of this project. This report is a first step – in one of many possible directions – to begin to draw out that relationship.

In the following pages one will find a presentation that attempts to pull together the historical background to Gauss’ entrance onto the battleground of ideas. The particular thread chosen to unfurl that skein of history of which Gauss is a crucial element, was that of the Delian Problem as it manifested in various ways in modern history through Cusa, Cardano, Kepler, Leibniz, Euler, Kästner, and eventually arriving at Gauss.

The report before you now should be considered the first in a series.

Introduction

Athenian: Let us then first consider what single science there is, of all those we have, such that were it removed from mankind, or had it never made its appearance, man would become the most thoughtless and foolish of creatures. Now the answer to this question at least is not overhard to find.

- Epinomis 1

It was the Golden Age of Athens. Pericles was beloved by the population – he had a power beyond all to sway the demos with his gripping orations. Athens under his direction was confident in its pursuit of a war in Peloponnese. The preparations were underway: Athens, the strongest naval power in the Mediterranean, had decided to take progressive action to reinforce its major vulnerability – a land attack by the Spartans. Pericles had proposed the enclosing of the entire city of Athens in a wall whose only access point to the outside world, would be through the ‘Long Walls’ stretching to Piraeus, a sea port miles away; the populace envisaged it as the foundation of an unstoppable success. It was a plan that made Athens invincible; all built upon the great edifice of Rhetoric.

Socrates: Come then, let us see now what we ought to say of rhetoric. For I, indeed, am not yet able to understand what I should say. When an assembly is held in a city, for the choice of physicians, or shipwrights, or any other kind of artificer, is it not the case that the rhetorician will refrain from giving his advice? for it is evident that, in each election, the most skilful artist ought to be chosen. Nor will he be consulted when the question is respecting the building of walls, or the construction of ports or docks, but architects only….What would be the consequence to us, Gorgias, if we should put ourselves under your instructions? …

Gorgias: I will endeavour, Socrates, to develop clearly the whole power of rhetoric: for you have admirably led the way. You doubtless know that these docks and walls of the Athenians, and the structure

1 Hamilton, Edith and Huntington Cairns,ed. The Collected Dialogues of Plato. Princeton University Press: Princeton, 1989.

of the ports, were made partly on the advice of Themistocles, and partly on that of Pericles, but not of artificers.

Socrates: This is told of Themistocles, Gorgias: and I myself heard Pericles when he gave us his advice respecting the middle wall.

Gorgias: And when there is an election of any such persons as you mentioned, Socrates, you see that the rhetoricians are the persons who give advice, and whose opinion prevails in such matters.

Socrates: It is because I wonder at this, Gorgias, that I have been for some time asking you, what is the power of rhetoric. For when I consider it in this manner, it appears to me almost divine in its magnitude.

Gorgias: If you knew all, Socrates, that it comprehends under itself almost all powers! -Plato’s Gorgias 2

Upon the assent of the population, construction was begun. The rural dwellers were drawn in from the out- stretching farmlands, which lay out beyond the city limits. Everyone snugly made their new place inside, behind the protective barrier; all were safe from the potential invaders whom Athens planned to provoke. The Peloponnesian War had begun.

With all the defensive preparation, no one could have imagined the carnage which would ensue within the city walls. Yet it was not the onslaught of an invading army, which struck its fatal blow upon the city – that would become the least of Athens’ worries. No one even suspected that the barrier had been penetrated when this dreadful lurker made his stealthy entrance, but once he had unleashed himself upon the population, all knew the brutal killer’s name: the Plague.

Once it broke out, there was no stopping it. The population was so densely packed in that there was little hope. One cannot begin to imagine the terror that must have struck the inhabitants of Athens. It is said that the population was beyond desperation: there was a breakdown in any semblance of order in the society. People abandoned their families, fleeing into wild spending sprees. Whether or not one would survive from one day to the next was left to Fate; all lived as though there were no future.

The delusion of Athenian imperial might was coming to an end, and in the face of such apposite catastrophes the true powers of Rhetoric were duly comprehended. Epitomizing the collapse of Athens was an anecdote, which Theon of Smyrna related thus:

“Eratosthenes, in his work entitled Platonicus relates that, when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an altar double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.” 3

Over 30,000 Athenians died of the Plague, a quarter of the population.

Socrates: We were never conquered by others, and to this day we are still unconquered by them, but we were our own conquerors, and received defeat at our own hands.

- Plato’s Menexenus

It would not be for over 30 years, and then, in the shadow of the collapse of Athens and the judicial murder of Socrates, that Plato’s collaborator, Archytas, would finally provide a solution to what, today, is known as the Delian Problem.

2 Cary, Henry, trans. The Works of Plato, Vol. I. London, 1881. 3 Heath, Thomas L. A History of Greek Mathematics. Vol. I. Dover: New York, 1981.

Unfortunately, with the deaths of Archimedes and Eratosthenes in 212 BC and 194 BC, respectively, the implications of Archytas’ solution, commonly understood amongst the Pythagoreans and Plato’s circles, would be lost for centuries.

Athenian: For, if we, so to say, take one science with another, ‘tis that which has given our kind the knowledge of number that would affect us thus, and I believe I may say that ‘tis not so much our luck as a god who preserves us by his gift of it.

- Epinomis

The subject of the present report is a meager attempt to render visible to the layman’s eyes but one facet of those divine machinations, which, thus far, have preserved us so.

The Cave

Socrates: Behold men, as it were, in an underground cave-like dwelling, having its entrance open towards the light and extending through the whole cave, - and within it persons, who from childhood upwards have had chains on their legs and necks, so as, while abiding there, to have the power of looking forward only, but not to turn round their heads by reason of their chains, their light coming from a fire that burns above and afar off, and behind them; and between the fire and those in chains is a road above, along which one may see a little wall built along, just as the stages of conjurers are built before the people in whose presence they show their tricks.

Glaucon: I see. Socrates: Behold then by the side of this little wall men carrying all sorts of machines rising above

the wall, and statues of men and other animals wrought in stone, wood, and other materials, some of the bearers probably speaking, others proceeding in silence.

Glaucon: You are proposing a most absurd comparison and absurd captives also. Socrates: Such as resemble ourselves, - for think you that such as these would have seen anything

else of themselves or one another except the shadows that fall from the fire on the opposite side of the cave?

Glaucon: How can they, if indeed they be through life compelled to keep their heads unmoved? Socrates: But what respecting the things carried by them: - is not this the same? Glaucon: Of course. Socrates: If then they had been able to talk with each other, do you not suppose they would think it

right to give names to what they saw before them? Glaucon: Of course they would. Socrates: But if the prison had an echo on its opposite side, when any person present were to speak,

think you they would imagine anything else addressed to them, except the shadow before them? Glaucon: No, by Zeus, not I. Socrates: At all events then, such persons would deem truth to be nothing else but the shadows of

exhibitions. Glaucon: Of course they would.

- Plato’s Republic, Book VII 4

After the collapse of Greek civilization, science in Europe not only fell stagnant, but it retrogressed. The knowledge of the Pythagoreans was lost in almost all respects. The accumulated discoveries of the Pythagorean School, who so dearly valued the reenactment of the discovery, were codified by Euclid in his Elements, thus severing the mind from the discovery. 5

4 Adaptation based on Henry Davis translation in The Works of Plato. Vol. II. William Clowes and Sons: London, 1871. 5 Euclid took up the task of arranging the discoveries of the Greeks prior to him into an axiomatic system in which that accumulation of ideas could be logically deduced from a few simple axioms, postulates, and definitions – setting

Perhaps the decay of Astronomy was the most severe loss, for it was, of course, the first science, the origin of Man’s concept of Number.

Athenian: How did we learn to count? How, I ask you, have we come to have the notions of one and two, the scheme of the universe endowing us with a native capacity for these notions? There are many other creatures whose native equipment does not so much as extend to the capacity to learn from our Father above how to count. But in our own case, God, in the first place, constructed us with this faculty of understanding what is shown us, and then showed us the scene he still continues to show. And in all this scene, if we take one thing with another, what fairer spectacle is there for a man than the face of day, from which he can then pass, still retaining his power of vision, to the view of night, where all will appear so different? Now as Uranus never ceases rolling all these objects round, day after day, and night after night, neither does he ever cease teaching men the lore of one and two until even the dullest scholar has sufficiently learned the lesson of counting. For any of us who sees this show will form the notion of three, four, and many.

- Plato’s Epinomis

This astrophysical knowledge, which extended from the Egyptian tradition by Thales up through Eratosthenes and Aristarchus, was replaced with the Aristotelian philosophy of Ptolemy, which propagated the belief that Man in no way possessed concepts commensurate with the modes of actions in the celestial sphere.

For it is not right for our human things to be compared on a basis of equality with the immortal gods, and for us to seek the evidence for very lofty things from examples of very unlike things.

- Ptolemy, Almagest Book XIII, Chapter 2

With such casualties suffered, European civilization would fester in what today is known as the Dark Age, which would last centuries.

The Liberation

Socrates: Let us inquire then, as to their liberation from captivity, and their cure from insanity, such as it may be, and whether such will naturally fall to their lot; - were a person let loose and obliged immediately to rise up, and turn round his neck and walk, and look upwards to the light, and doing all this still feel pained, and be disabled by the dazzling from seeing those things of which he formerly saw the shadows; - what would he say, think you, if any one saw more correctly, as being nearer to the real thing, and turned towards what was more real and then specially pointing out to him every individual passing thing, should question him, and oblige him, to answer respecting its nature: think you not he would be embarrassed, and consider that what he before saw was truer than what was just exhibited?

- Plato’s Republic, Book VII

Humanity would not stay bound forever, and soon enough there was a revival of the teachings of the Pythagoreans and Plato. Central to this rediscovery of the Pythagorean knowledge was the leading Renaissance figure Cardinal Nicholas of Cusa. 6

Cusa’s major work De Docta Ignorantia would define the epistemological basis for all subsequent advances in what became modern science. It was the first major stride in freeing the mind from centuries of pedantic Aristotelian philosophy.

the trend for more modern imitators such as Isaac Newton, Augustin Louis Cauchy, and Bertrand Russell. The dispassionate display of corpses found in his Elements goes to show that not only did he not acquire any real scientific understanding from the Egyptians, but he also didn’t gain any skill in their developed mortician’s art either. 6 See Reclaiming the Soul of Gauss.

“…all those who make an investigation judge the uncertain proportionally, by means of a comparison with what is taken to be certain..”

“Therefore, every inquiry is comparative and uses the means of comparative relation. Now, when, the things investigated are able to be compared by means of a close proportional tracing back to what is taken to be [certain], our judgement apprehends easily; but when we need many intermediate steps, difficulty arises and hard work is required…Therefore, every inquiry proceeds by means of a comparative relation, whether an easy or a difficult one. Hence, the infinite, qua infinite, is unknown; for it escapes all comparative relation. But since comparative relation indicates an agreement in some one respect and, at the same time, indicates an otherness, it cannot be understood independently of number. Accordingly, number encompasses all things related comparatively. Therefore, number, which is a necessary condition for comparative relation, is present not only in quantity but also in all things which in any manner whatsoever can agree or differ either substantially or accidentally. Perhaps for this reason Pythagoras deemed all things to be constituted and understood through the power of numbers.

- Cusa, De Docta Ignorantia, Book I, Chapter I 7

In reviving Platonism, Cusa faced a mammoth task - there was still much ground to be gained, for much had been lost.

Socrates: Therefore, even if a person should compel him to look to the light itself, would he not have pain in his eyes and shun it, and then, turning to what he really could behold, reckon these as really more clear than what had been previously pointed out?

- Plato’s Republic, Book VII

The reaction against Cusa was viciously outspoken, as the case of John Wenck typifies. 8 Cusa, however, was never lacking a pointed response.

“This sect regards as heresy the coincidence of opposites. Hence, this method, which is completely tasteless to those nourished in this sect is pushed far from them, as being contrary to their undertaking. Hence, it would be comparable to a miracle – just as it would be the transformation of the sect – for them to reject Aristotle and to leap higher.”

- Cusa’s Apologia Doctae Ignorantiae 9

Socrates: But if a person should forcibly drag him thence through a rugged and steep ascent without stopping, till he dragged him to the light of the sun, would he not while thus drawn be in pain and indignation, and when he came to the light, having his eyes dazzled with the splendour, be unable to behold even any one thing of what he had just alleged as true?

- Plato’s Republic, Book VII

Through Cusa the path back to the Greeks was lain open, and although the habituated modes of the Aristotelian School would still persist and even take new form, a way was given such that Mankind might recover his first science.

Delian Problem Revived

7 Cusa translations by Jasper Hopkins are available on his website: http://www.cla.umn.edu/sites/jhopkins/. 8 For a more detailed discussion on the historical significance of the attacks on Cusa see, Lyndon H. Larouche, Jr., For Today’s Young Adults:Kepler & Cusa. EIR: March 2, 2007, pg 10-33. 9 Cf. 7. See also, Nicholas of Cusa’s Debate with John Wenck. Jasper Hopkins, trans. Banning Press: Minneapolis, 1981.

Socrates: He would require, at least then, to get some degree of practice, if he would see things above him: - and first, indeed, he would most easily perceive the shadows, and then the images of men and other animals in the water, and after that the things themselves.

- Plato’s Republic, Book VII

Needless to say, the revival of science was not an instantaneous success, taking some time to readjust. Amongst the various pursuits taken up by thinkers in the Renaissance, a revisiting of the Delian Problem was in place, but from a different vantage point.

Science had not been altogether abandoned since the Age of the Greeks, in fact, though most of Europe had been in a dark age, there had been significant technological advances made, particularly in and around the Islamic Renaissance, which eventually became a conduit feeding into the resuscitation of Europe.

One of the inventions which found its birth in the Islamic Renaissance, and was adopted by thinkers in the Italian Renaissance, was that of Al-Jabr, or Algebra. This new art made its way there through the writings of al- Khowarizimi.

“In the name of God, tender and compassionate, begins the book of Restoration and Opposition of number put forth by Mohammed Al- Khowarizmi, the son of Moses. Mohammed said, Praise God the creator who has bestowed upon man the power to discover the significance of numbers. Indeed, reflecting that all things which men need require computation, I discovered that all things involve number and I discovered that number is nothing other than that which is composed of units.” - Al-Khowarizmi, Book of Algebra and Almucabola

See Box I.

Al-Khwarizimi’s method was picked up by people like Fibonacci, Nicolo Fontana Tartaglia, and Girolamo Cardano. Just as al-Khwarizimi sought to apply his method to achieve a generalized treatment of relations between surfaces and lengths, Cardano and others would seek to extend this in full to comprehend the relations between volumes, surfaces, and lengths.

As Kästner accounts it, Scipio Ferreus and Tartaglia had discovered a method for solving the problem of a cube equal to the sum of some roots and a number ( b ax x + = 3 ). Tartaglia provided Cardan with his

Box I

Al-Khowarizimi’s work represented an initial investigation into the problems that arise pertaining to the relations between areas. Below are some of the typical problems that his work dealt with. Of note is that he only took into consideration problems that had a physical representation, while neglecting ones that perhaps could be stated symbolically, but lacked any meaning (e.g., the equation

0 10 2 = + x would have been regarded as absurd; for, how could two somethings, when added together, make nothing?)

solution, but did not permit him to see the proof. Cardano, being quite capable, derived the proof of it himself and published it, rightfully accrediting Tartaglia with the discovery.

However, there was more than meets the eye with their solution. See Box II.

Box II

In Cardan’s Ars Magnae, he restates all the problems that the Arabs had addressed in their treatment of Algebra and went further to extend the method to the problem of volumetric relationships. One of the more significant problems Cardan “solved” was the cubic equation of this form: b ax x + = 3

Take 12 13 3 + = x x , for example. Now, before going through Cardan’s method, one should, as always, try to find solutions to this equation oneself and compare results with those of Cardan.

The solution that Cardan proposes, consists of the following:

Suppose that the solution will have the form that x can be stated in terms of the sum of the edges of two other cubes. That is,

3 3 B A x + = .

Then, following the animation below, one finds

) ( ) * 3 ( 3 3 B A x B A x + + = .

Combining this with the original equation, provides two new relationships:

I) 3 * 3 13 B A = II) B A+ = 12

Thus far everything seems have a physical meaning. All that remains is to figure out A and B from the two equations.

As shown in the next animation, if one sets 13 as the edge of a new cube, the volume, 2197 = 27*A*B. So A*B=2197/27. But, from (II) it was found that B=12 – A. Thus, A(12 - A) = 2197/27, which yields the equation: A A 12 27 / 2197 2 = +

[Animation 2]

Solving for this equation (which can be done using Al- Khowarizimi’s method, right?) one finds:

27 / 2197 36 6 − − = A ; and, from (II),

27 / 2197 36 6 − + = B ;

[Animation 3]

So, the problem has been solved: 3 3 27 / 2197 36 6 27 / 2197 36 6 − + + − − = x

But what does that mean?

If one looks closer there’s something that doesn’t quite fit.

27 1225

27 2197

27 972

27 2197 36 − = − = −

But that means 27 1225 6 − − = A and

27 1225 6 − + = B .

That is, it’s expression contains square roots of negative numbers!

Thus, Cardan’s final solution gives:

3 3

27 1225 6

27 1225 6 − + + − − = x

How does this compare to the solutions that the reader found? Is this a new solution, or perhaps it is equivalent to one, which the reader found? Why did Cardan’s seemingly physical derivation give something without physical meaning?

The paradox arising from their solution would occupy the minds of future geometers over the coming centuries.

A Powerful Sense of Wonder…

Socrates: - and after this he would more easily behold the things in heaven itself, by night, looking to the light of the stars and the moon than after daylight to the sun and the light of the sun.

- Plato’s Republic, Book VII

Athenian: …and what god is it…of whom I speak in this solemn fashion? Why, Uranus, to be sure, whom it is our bounded duty to honor, as it is to honor all divinities and gods, and to whom we are specially bound to pray. All of us will confess that he is the source of all the other good things we enjoy, and we in particular assent that ‘tis he who has in very deed given us number, and will renew the gift if men will only follow his leading. If a man will but come to the right contemplation of him, he may call him by the name of Cosmos, Olympus, or Uranus as he pleases; only let him follow him in his course as he bespangles himself and wheels his stars through all their courses in the act of providing us all with seasons and daily food.

- Plato’s Epinomis

Johannes Kepler, thoroughly steeped in the teachings of Cusa, would lead the way to restoring astronomy, and scientific method in general, back to its Pythagorean heritage. His “right contemplation” of the heavens, brought him to rediscover that original science of number. His three major works, Mysterium Cosmographicum, New Astronomy, and Harmonices Mundi, thoroughly Platonic in their origin, refined the modern notion of epistemology.

In his Harmonices Mundi 10 , before he could bring the reader to share in his discoveries, he had first to answer the question, ‘What does it mean to know something?’ 11

VII Definition In geometrical matters, to know is to measure by a known measure 12 , which known measure in our present concern, the inscription of Figures in a circle, is the diameter of the circle.

- Harmonices Mundi, Book I

Knowing that, he could then proceed to answer another elementary question: If that is what it means to know, what then can be known?

VIII Definition A quantity is said to be knowable if it is either itself immediately measurable by the diameter, if it is a line; or by its square if a surface: or the quantity in question is at least formed from quantities such that by some definite geometrical connection, in some series [of operations] however long, they at last depend upon the diameter or its square. The Greek for this is γνωριµον, ‘intelligible.’

- Harmonices Mundi, Book I

He thenceforth proceeded to unfold an ordering of “Degrees of Knowability” by which magnitudes might be compared as to a definite quality of magnitude as opposed to merely a quantity of magnitude.

For example, compare the side of a square of 2 and the side of a square of 4. Although in quantitative value, the side of a square of 2 is more akin to unity; in terms of the qualitative value, the side of a square of 4 is more akin to unity, because, with respect to the mind’s ability to grasp it, it resolves itself more readily.

10 All quotes referenced to the Harmonices Mundi refer to Kepler, Johannes. The Harmony of the World. Aiton, Duncan, and Field, trans. American Philosophical Society: Philadelphia, 1997. 11 See pedagogical on Knowability. 12 If this definition seems a little circular, don’t worry: that’s exactly what it is! For more on the self-evident nature of circular action see Sufficient Harmony.

However esoteric this initial beginning might appear, it accomplished something quite profound: the establishment of a concept of a species of magnitude.

On the basis of this concept, he delved into the task of ordering the various constructions of divisions of the circle and tessellations of the plane and sphere with respect to their degrees of knowability. From there he would go further to demonstrate that the reason why musical intervals manifest as either consonant or dissonant was wholly dependent upon the degrees of knowability to which they corresponded. In fact there is an intimate connection between harmony and knowledge.

Underlying Kepler’s conception is an important epistemological point which brought a renewed insight into how to investigate physical causes. What Kepler defined did not axiomatize physical science, but it did state explicitly the Pythagorean understanding that all causality was relegated to presenting itself to the mind in a knowable form. Furthermore, the way in which the Mind represents knowledge to itself is in the form of harmonics. Thus, in all of Kepler’s pursuits of principle in the universe, he knew that the way in which those principles must manifest themselves is always as an harmonic expression.

Hence, Kepler’s treatise on the nature of knowledge exhibits its title: Harmonices Mundi.

Yet, Kepler’s restoration of Astronomy through this rigorous grounding in epistemology did not occur with him resting on the shore of a peaceful lake. No, he, a sole navigator, guided this ship through a most torrential storm: the brewing and outbreak of the Thirty Years’ War.

…on the Austrian side, timorous and unwarlike astronomy is warned by the conditions, dangers, terrors, disasters, and troubles to look round for assistance. She crossed in the year 1600 from Styria into Bohemia, so that just as she had put her first roots under the shelter of the Austrian house she might also grow to maturity under it... After being tossed to and fro there by the tempests of both civil and foreign wars, in the end after the death of the Emperor Rudolph in the year 1612, with unceasing zeal for the Austrian house, she returned to Austria. Would that she could have been honored there with the devoted attention of eminent minds (no less than by myself, who restored her) as much as she was accepted and favored with goodwill. Yet, alas, of what great goods do miserable mortals despoil one another, by their shameful itching for quarrels. How profound an ignorance of their fate overwhelms them, as they have deserved. With what deplorable perverseness do we rush into the midst of the flames, in fleeing from the fire.

Would that even now indeed there may still, after the reversal of Austrian affairs which followed, be a place for Plato’s oracular saying. For when Greece was on fire on all sides with a long civil war, and was troubled with all the evils which usually accompany civil war, he was consulted about a Delian Riddle, and was seeking a pretext for suggesting salutary advice to the peoples. At length he replied that, according to Apollo’s opinion Greece would be peaceful if the Greeks turned to geometry and other philosophical studies, as these studies would lead their spirits from ambition and other forms of greed, out of which wars and other evils arise, to the love of peace and to moderation in all things.

- Kepler’s Mysterium Cosmographicum 13

With Kepler’s familiarity with the Delian Problem, especially in its cultural aspects, his comments upon the techniques of his predecessors’ approach to the problem become especially useful. His thoughts are most explicitly

13 Kepler, Johannes. Mysterium Cosmographicum. A. M. Duncan, trans. Abaris Books: New York, 1981. It should be noted that, although the Mysterium Cosomographicum was one of Kepler’s earliest works, first published in 1596, the quote given here appeared only in his second edition, which he published in 1621, three years after the beginning of the Thirty Years’ War.

given in his criticism of the various proposed methods to solve the homeomorphic problem of trisecting the angle, 14

which emerges in the attempt to construct a regular Heptagon. 15

Here it might be suggested that I should use the Analytic art called Algebra after the Arab Geber, its Italian name being Cossa: for in this art the sides of all kinds of Polygon seem to be determinable….

- Harmonices Mundi, Book I Kepler, took this suggestion up, following the procedure afforded him by his collaborator, Bürgi, which yielded the following algebraic equation, whose roots correspond to the three different sides constituting the elementary triangles of the regular Heptagon: 16

6 4 2 7 14 7 x x x − + − = 0

…you will note, first, that one may ask what this algebraic chord of Bürgi’s signifies? It certainly signifies that if seven lines are constructed in continuous proportion, the proportion being that between the side of the heptagon and the semidiameter of the circle, and the first proportional is made equal to the side of the heptagon: then seven lines equal to the first proportional plus seven equal to the fifth will add up to the same as fourteen lines equal to the third proportional plus one line equal to the seventh.

…this does not tell us how to construct the continuous proportion for which this relationship will hold, nor does it express the lengths of the proportionals in terms of things already known, but it tells us, once the [system of continuous] proportion is set out, what relationship will follow. So I am instructed to represent the relationship (affection), for it will then come about that I obtained the proportion also. But how am I to represent the relationship, by what Geometrical procedure? No other means of doing it are afforded me save using the proportion I seek; there is a circular argument: and the unhappy Calculator, robbed of all Geometrical defenses, held fast in the thorny thicket of Numbers, looks in vain to his algebra (cossa). This is one distinction between Algebraic (Cossicas) and Geometrical determinations.

- Harmonices Mundi, Book I

The question thus remains, ‘What is the difference in nature of an Algebraic determination of magnitude as compared to a Geometric one?’

…all this reasoning of Bürgi’s depends upon the nature (essential) of a discrete quantity, namely that of numbers.

But Geometry does not deal with figures in this way…though it does designate sides Expressible in length by Numbers; but inexpressible ones it in no way attempts to capture with numbers, but states their magnitudes according to their particular kinds, so that it is clear that we are dealing not with discrete quantities but with continuous ones, that is with lines and surfaces. 17

I am not told how to bring the matter to a conclusion but only how to stalk the quarry, from a distance. For since the kinds of line, according to their [degree of] knowledge, are found among the Inexpressibles (that is,

14 For a discussion of the relationship between trisecting the angle and cubic roots, see Jonathan Tennenbaum’s From Cardan’s Paradox to the Complex Domain. 15 It is expected of the reader to work through the elaboration of this aspect of Kepler’s Harmonices Mundi on the Heptagon, provided in the second part of this project. 16 If one sets 2 x y = , then the problem reduces to the cubic equation, 0 7 14 7 3 2 = − + − y y y , or, expressed in the form of Cardano, 7 7 14 2 3 + = + y y y . Can you find the solution to this using Cardan’s Formula? 17 Here we also have a foreshadowing of the grounds on which Kepler will reject the notion of a minimal musical interval as a unit. The point is made in Book III of the Harmonies that the consonant intervals are primary, and are not composed of the sum of smaller intervals, although they may be commensurate with the sum of those magnitudes. Similarly, although the terms of an infinite series may each be contained within that which it approaches, it does not follow that those discrete terms comprise the whole.

they are not numerable but reject numbers), there will accordingly be no multiplicity of numbers that can exhaust the ratio without leaving some uncertainty in it: on the other hand, this ratio [from the Algebra - MF]…takes no refuge except in numbers,…but this never gives a completely exact value; and, in short: this is not to know the thing itself but only something close to it, either greater or less than it; and some later calculator (computator) can always get closer to it [still]; but to none is it ever given to arrive at it exactly. Such indeed are all quantities which are only to be found in the properties of matter of a definite amount; and they do not have a knowable construction by which in practice they might be accessible to human knowledge.

- Harmonices Mundi, Book I

Kepler was thus justified in banishing the Heptagon from the architectonics of the Cosmos on account of its lack of Knowability. However, this generates a much deeper paradox, echoing out of his earlier work, the New Astronomy. For there, he confronted the obstacle that the manifestation of the principle, which governs the motion of a planet in its orbit, is not expressible through any of the geometrical methods existing in his time. Were it not capable of being subjected to a geometrical construction, would not the conclusion follow that it itself were unknowable?

But given the mean anomaly, there is no geometrical method of proceeding to the equated, that is to the eccentric anomaly.

It is enough for me to believe that I could not solve this a priori, owing to the heterogeneity of the arc and the sine. Anyone who shows me my error and points the way will be for me the great Apollonius.

- Kepler’s New Astronomy, Chapter 60 Such was Kepler’s oracular challenge.

…a desolating war of thirty years, which, from the interior of Bohemia to the mouth of the Scheldt, and from the banks of the Po to the coasts of the Baltic, devastated whole countries, destroyed harvests, and reduced towns and villages to ashes; which opened a grave for many thousand combatants, and for half a century smothered the glimmering sparks of civilization in Germany, and threw back the improving manners of the country into their pristine barbarity and wildness. Yet out of this fearful war Europe came forth free and independent.

- Schiller’s History of the Thirty Years’ War

Though Kepler would never see the war brought to a peace, one thing would survive steadfast: the force of the paradox he uncovered would persist as long as there was a mind to grasp it.

Socrates: …some reports of our perceptions do not provoke thought to reconsideration because the judgement of them by sensation seems adequate, while others always invite the intellect to reflection because the sensation yields nothing that can be trusted.

Glaucon: You obviously mean distant appearances and shadow painting. Socrates: You have quite missed my meaning. Glaucon: What do you mean? Socrates: The experiences that do not provoke thought are those that do not at the same time issue

in a contradictory perception. Those that do have that effect I set down as provocatives, when the perception no more manifests one thing than its contrary, alike whether its impact comes from nearby or afar. … Glaucon: Yes, indeed, these communications to the soul are strange and invite reconsideration.

Socrates: Naturally, then, it is in such cases as these that the soul first summons to its aid the calculating reason and tries to consider whether each of the things reported to it is one or two….For, if unity is adequately seen by itself or apprehended by some other sensation, it would not tend to draw the mind to the apprehension of essence…But if some contradiction is always coincidentally with it, so that it no more appears to be one than the opposite, there would forthwith be need of something to judge between them, and it would compel the soul to be at a loss and to inquire, by arousing thought in itself, and to ask,

whatever then is the one as such, and thus the study of unity will be one of the studies that guide and convert the soul to the contemplation of true being.

- Plato’s Republic, Book VII

Empiricism

Kepler’s Problem to future geometers may have still carried with it the same force with which it was originally conceived, but that did not mean that the scientific method capable of arriving at that paradox would survive intact.

Like an unfolding fugue, the theme and countersubject would continually develop contrapunctally with each introduction of a new voice; a resolution of that tension was still far from surfacing.

Kepler’s Pythagorean method of seeking out harmonies in the universe was applied to other problems.

Now we find that the ancients, among others Ptolemy, already used this hypothesis of the easiest path of a ray which falls on a plane, to account for the equality of the angles of incidence and reflection, the principle at the basis of catoptrics. It is by this same hypothesis that Mr. Fermat provided a reason for the law of refraction according to the sines, or to formulate it in other terms as Snell did, according to the secants. But what is more, I have no doubt whatever that this law was first discovered by this method. It is known that Willebrord Snell, one of the greatest geometricians of his time and well versed in the methods of the ancients, invented it, having even written a work which was not published because of the author’s death.

- Leibniz’s Tentamen Anagogicum 18

These applications, however, were not without their imposters.

But since he [Snell] had taught it to his disciples, all appearances point to the conclusion that Descartes, who had come to Holland a little later and who was most interested in this problem, learned it there. For the way Descartes has tried to explain the law of refraction by efficient causes or by the composition of directions in imitation of the reflection of bullets is extremely forced and not intelligible enough. To say no more about it here, it shows clearly that it is an afterthought adjusted somehow to the conclusion and was not discovered by the method he gives.

- Leibniz’s Tentamen Anagogicum

Such was a typical course of events in those decades following the death of Kepler.

With respect to Kepler’s direct work, some astronomer’s argued that Kepler’s New Astronomy was ungeometrical. Others, acknowledging the superiority of Kepler’s results, went so far as to have claimed to have resolved Kepler’s Problem using the very algebraic techniques which Kepler had so explicitly admonished. 19

Implicit in the work of this latter faction, was a willful attempt to extract the practical results of Kepler’s discoveries from out of the method which made those discoveries possible. This latter group would come to be known as the Empiricist School.

Leibniz

The fiercest opponent to the Empiricist School, and, what goes hand in hand, the greatest advocate of Kepler’s method, in the second half of the 17 th century was Wilhelm Gottfried Leibniz. The battles couldn’t have been any more explicit, and Leibniz led with full frontal attacks. Of the numerous antagonists he engaged, one of the most significant was that of John Locke on the subject of the nature of human understanding. In John Locke’s Essays on

18 Leibniz, Tentamen Anagogicum: An Analogical Essay in the Investigation of Causes. Loemker, trans. pg. 477-485. 19 See Venetians and Empiricists Can’t Handle Discoveries.

Human Understanding, he attempted to eliminate many of the crucial epistemological insights that had been established through the revival of Platonic studies.

…although the author [Locke] of the Essay says a thousand beautiful things which I commend, our systems are very different. His has more relation to Aristotle, mine to Plato…Our differences are upon subjects of some importance. The question is to know whether the soul in itself is entirely empty as the tablets upon which as yet nothing has been written (tabula rasa) according to Aristotle, and the author of the Essay, and whether all that is traced thereon comes solely from the senses and from experience; or whether the soul contains originally the principles of many ideas and doctrines which external objects merely call up on occasion, as I believe with Plato, and even with the schoolmen, and with all those who interpret in this way the passage of St. Paul (Rom. 2:15) where he states that the law of God is written in the heart.

- Leibniz’s New Essays on Human Understanding, Preface

The conception of number promoted by the Empiricist School was that of a mere linearized quantity (e.g., the number 2, the doubled line, the doubled square, and the doubled cube are all considered equal) – exactly contrary to Kepler’s sharp distinction between the continuous and the discrete.

Philalethes. [Locke] The different modes of numbers are capable of no other difference than that of more or less; this is why there are simple modes like those of extension.

Theophilus. [Leibniz] That may be said of time and of the straight line, but not of figures, and still less of numbers, which are not only different in size but further unlike. An even number may be divided into two equal numbers, but not an uneven. Three and six are triangular numbers, four and nine are squares, eight is a cube, etc. And this principle has place in numbers still more than in figures, for two unequal figures may be perfectly similar to each other, but never two numbers. But I am not astonished that you are deceived thereupon, because one does not commonly have a distinct idea of what is similar or dissimilar. You see then, sir, that your idea or your application of simple or mixed modes is greatly in need of correction.

- Leibniz’s, New Essays on Human Understanding, Book II, Chapter XVI

Much of Leibniz’s considerations were shaped by his struggle in dealing with Kepler’s Problem. He clearly understood that the approximations that the Empiricist School settled with were useless in advancing a science focused upon discovery of principle, as opposed to merely promoting a statistically accurate model making. It was with the intent of dealing with the paradox Kepler unearthed that Leibniz would develop his infinitesimal Calculus. 20 Leibniz’s method culminated with his discovery of the principle of the catenary, which would be his gateway into the complex domain lurking behind Cardan’s Formula. 21 A new insight gained, led to conclusions that were beyond the belief of some.

…when I told the late Mr. Huyghens that 6 3 1 3 1 = − − + − + , he found this so remarkable that he replied that there is something incomprehensible to us in the matter.

- Leibniz, Letter to Varignon, Feb. 2, 1702

How could it be that this seemingly impossible magnitude could actually be knowable? How must have Leibniz been thinking about this magnitude in order to make it comprehensible?

20 Leibniz was quite conscious that his method of the Calculus was superior to the use of infinite series in matters of principle. With that knowledge, he would often release challenges which he knew could not be solved from any other method than his own. For more on Leibniz’s defense of Kepler and his method, see Venetians and Empiricists Can’t Handle Discoveries. 21 For more on the relationship of the catenary to the complex domain see What do Cones Have to do with Gravity?.

Amongst his calculations in one of his manuscripts in which he described his derivation of this equation, he put in writing a very telling remark: “Let this formula be divided into two parts, the binomial…and the apotome…” Of course! Kepler’s degrees of knowability! 22

See Box III.

Leibniz didn’t stop there, but continued on to take up Cardan’s Formula.

How can it be that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others….This difficulty has been too much for all writers on algebra up to the present, and they have all said that in this case Cardan’s rules fail…I do not remember to have noted a more singular and paradoxical fact in all analysis; for I think I am the first one to have reduced irrational roots, imaginary in form, to real values without extracting them.

By applying his method of binomial and apotome to Cardan’s general formula, he then took up the specific case.

Take the equation, which also Albert Girard used:

22 The referenced manuscript is in Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern, ed. C. J. Gerhardt, vol. 1, Berlin 1899. pg. 547-8. The english translations provided here are taken from McClenon, R. B. A Contribution of Leibniz to the History of Complex Numbers. The American Mathematical Monthly, Vol. 30, No. 7. (Nov., 1923), pg. 369-374.

Box III

Leibniz begins his recollection of how he found 6 3 1 3 1 = − − + − + by writing:

“It will be useful to mention how my mind was led to the solution of this problem. I once came upon two equations of this kind: b y x = + 2 2 ; c xy = ”

One might conclude, “So, he was examining the circle and the hyperbola?”...Perhaps.

If one goes back to reexamine Kepler’s Degrees of Knowability, the observations might be made that: 1) obviously, there are various degrees of knowability; but also, 2) amongst the degrees treated by Kepler there are two different types of degrees.

a) the first type, is a form of knowability that results from the relationship of a single magnitude’s comparative relationship to something known b) the second type, is a form of knowability which originates in a relationship between two magnitudes that each have no knowability of the first type, but acting upon one another, generate a synthetic comparative relationship to the known

This second type of relationship is what Kepler develops for his 5 th , 6 th , and 7 th degrees of knowability. He elaborates this characteristic of the 5 th degree like so:

“Note therefore that in this degree we shall measure not the lines themselves, nor their individual squares, but instead we shall measure both the Rectangle formed from them and the sum of their squares; so what is lacking in one square, making it less expressible, is exactly compensated by the other square that is associated with it.”

Thus, in the 5 th degree, each of the magnitudes are themselves inexpressible, but that which makes each inexpressible with respect to the known finds origin in a common subsuming relationship. What is this subsuming relationship?

“…let the fifth degree of knowledge be when we have two lines which are not both Expressible, nor both Medial, and further are completely incommensurable with one another, and they make both the sum of their squares and their common rectangle an expressible quantity.”

How would one express that relationship algebraically? How about this?

“…two equations of this kind: b y x = + 2 2 ; c xy = ”

That is, given the two lines, x and y, the sum of their squares is equal to b, and their common rectangle is equal to c, where b, c would both be Expressibles. If one were to generalize the domain of all 5 th degree magnitudes itself, this would be the way to express it, so to say.

Kepler had been examining magnitudes which arise in the construction of polygons, to determine their degrees of knowability; Leibniz was beginning with the general domain of a specific degree of knowability, and arriving at the paradoxes in the particular.

Although, Leibniz details more of his discovery, anyone who takes the time to dialogue between Kepler and Leibniz on this matter, should be able to discover the rest on their own, with perhaps one last question.

Leibniz considered x and y to be a ‘binomial’ and ‘apotome’. In many cases this relationship is best grasped as being generated through the partitioning of a line. Does that yield the relationships found here? What about trying the partitioning of an area instead? What, then, are the implications of the nature of the domain of investigation?

0 12 13 3 = − − x x , whose true root is 4. From the formulas of Scipio Ferro or Cardan,

3 3

27 1225 6

27 1225 6

− − +

− + = x .

I will prove that this expression is correct and real, and must be admitted. Put 3 1 2

3 1 2 − − + − + = x ,

and certainly x will be equal to 4, as the equation postulated. Now let us see if the Cardan formula can be derived from this. 23

After taking into account Leibniz’s solution to the Cardan Formula, one must still ask, ‘Has the problem of imaginary numbers been resolved?’ Leibniz may have vindicated Cardan’s Formula as a tool to find real roots of cubic equations, but does it follow that all roots of the cubic equation are now knowable? What degree of knowability is an imaginary number? It seems that the result that algebra yields still carries no meaning in itself.

…the multitude of considerations…causes some very great difficulties in the science of numbers themselves; for short methods are sought and sometimes we do not know whether nature has them within its folds for the case in question. This it is which also makes algebra as yet so imperfect, although nothing is better known than the ideas which it makes use, since they signify only numbers in general; for the public has not yet the means of extracting the irrational roots of any equation beyond the fourth degree (excepting in a very limited case), and the methods which Diophantus, Scipio Ferreus, and Lewis Ferrari used respectively for the second, third, and fourth degrees in order to reduce them to the first, or in order to reduce an affected equation to a pure, are wholly different from each other, i.e. that which is used for one degree differs from that used for another…And this makes us also judge that algebra is very far from being the art of invention, since it needs a more general art;…Descartes has extended the application of this calculus [Algebra] to geometry, indicating lines by equations. Nevertheless, even after the discovery of our modern algebra, Bouillaud…regarded only with wonder the demonstrations of Archimedes upon the spiral, and could not understand how this great man had thought…[T]he new calculus of infinitesimals which proceeds by the method of the differences which I have thought of and successfully shared with the public, gives a general [method], wherein this discovery concerning the spiral is mere play and a sample of the easiest…The reason of the advantage of this new calculus is, moreover, that it relieves the imagination in the problems, which Descartes excluded from his geometry under the pretext that they most frequently lead to mechanics, but at bottom because they did not agree with his calculus.

- Leibniz’s New Essays on Human Understanding, Book IV, Chapter XVII

Yet, it would be that very Calculus of Leibniz, his fruits produce from taking up Kepler’s Problem, which would be so brutally attacked in his later years, and most especially after his death in 1716. With Leibniz gone, empiricism could begin to make its way into Continental Europe – and it did.

Living History

On Kepler

No mortal yet as climbed so high, As Kepler climbed and died in need, unfed:

He only knew to please the Minds And so, the bodies left him without bread.

A. G. Kästner 24

23 Surely the reader would rather prove this on their own than see it continued. 24 Translation from Bruce Director’s Riemann for Anti-Dummies, No. 3.

Alas! That I must name Kepler’s name, to our disgrace. Germany, the fruitful but negligent mother of great souls, permitted Kepler to struggle with poverty and misery when he was occupied with assigning laws to the heavenly bodies, and died on a journey which he made to collect his promised and long owed pay. Ungrateful Fatherland! have you been worthy of a Newton?…Passion leads me too far. Yes, Germany, you have not been unworthy of Newton, for you had produced Leibniz.

- Kästner’s Praise of Astronomy

It was quite early in the career of Abraham Gotthelf Kästner when those words were published in the first issue of the Hamburgisches Magazin, yet despite his age the thoughts of this impassioned youth could not but resonate throughout the circles of Leipzig and beyond. Those who would heed that call then would emerge to become the leading figures of their generation in Germany.

The Leibnizian tradition was under assault – the Royal Academy of the Sciences at Berlin had been taken over by a new faction of empiricists, headed up by Euler and Maupertuis, whose leading objective was to wipe clean any memory of Leibniz’s (and implicitly Kepler’s) role in developing continental science. 25 Kästner, born and raised in Leipzig, came to maturity in a region surrounded with a culture immersed in Leibniz and Bach. It would only be appropriate that the fight which Kästner waged were an expression of that deeply instilled tradition.

At the University of Leipzig, Kästner recruited two of his leading students, Gotthold Lessing and Christlob Mylius. It was under Kästner’s tutelage, that Lessing, who would later be credited as the founder of the German classical literary tradition, wrote his first play: a satire on the infamous prize essay contests underway at the Berlin academy. Simultaneously, Mylius and Kästner were using the Hamburgisches Magazin as a powerful tool to develop German as a scientific language by translating treatises from all over the world, as well as writing criticisms of the ongoing operations at the Berlin Academy.

After Lessing and Mylius left Leipzig, for Berlin, closer to enemy territory, the other lifelong member of this close- knit group, Moses Mendelssohn would join the battle. Lessing and Mendelssohn’s first major collaboration was a defense of Leibniz aimed against the Berlin Academy, entitled, Pope A Metaphysician!.

Although others would move in and out of their orbits, these few formed the principle persons in Europe who unleashed a cultural renaissance in 18 th Century Europe and America.

In the meantime, D’Alembert in 1746 and Euler in 1749, both at the Berlin Academy released their supposed proofs of the fundamental theorem of algebra. Both of their approaches attempted the same presumptive tactic: to limit the concept of number to that of simple extension. By having done so, they could circumvent the problem of the so-called impossible solutions, by treating them as mere symbols, possessing in no way any topologically significant difference. In essence, their route was to deny the complex domain anything beyond a mere symbolic existence.

§ 3… We call a quantity imaginary when it is neither greater than zero, nor less than zero, nor equal to zero. This will be then something impossible, as for example 1 − , or in general a + bi, since such a quantity is neither positive, nor negative, nor zero. …§ 5. Although it seems that knowledge of imaginary roots might not have any use, considering that they do not provide any solution to whatever problem there may be, nevertheless it is very important in all of analysis to become familiar with the calculation of imaginary quantities. For not only will we acquire from it a more perfect knowledge of the nature of equations, but analysis of the infinite will derive very considerable benefit from it. For each time it comes up that we must integrate a fraction, it is necessary to resolve the denominator into all its simple factors, be they real or imaginary, and from there we finally derive the integral, which although it contains imaginary logarithms, we have the means to reduce them to arcs of real circles. Besides that, it often happens that an expression which contains imaginary quantities is nevertheless real, and in these cases calculation with imaginary quantities is absolutely necessary.

25 See David Shavin’s The Courage of Gauss.

- Euler’s Investigations on the Imaginary Roots of Equations 26

A key driver to counter the Berlin maneuvers was the project of Göttingen University – a direct effort to elevate the population of Germany. 27 Although Leibniz had been dead for 20 years when it was founded in 1737, it would be his student Caroline, from the position of Queen of England, who would pull the project together. Baron von Münchhausen, appointed to oversee the project, would, together with the Rector, Gesner, of Bach’s Thomas Schule in Leipzig, seek to bring together Germany’s brightest minds.

At Göttingen the chief stress was laid on the culture of the essentially modern sciences. In the foremost rank stood the administrative and historico-political branches where Pütter, Achenbach, Schlözer, Gatterer, Heeren, gave to the university her worldwide fame; the mathematical and scientific branches are marked by the brilliant names of Haller, Lichtenberg, Blumenbach, Kästner; the philological branches by Gesner, Heyne, Michaelis…Münchhausen arranged in 1756 that a member of each faculty should deliver a public course on the whole field of the sciences taught there; in the philosophical faculty Gesner treated philologico-historical, Kästner physico-mathematical subjects. 28

For Kästner’s first public lecture in this series, held on July 30, 1756, he choose as his subject, the generalized problem of equal divisions of an arc, citing Kepler as having treated some special cases of the problem. His second lecture, held on the occasion of Easter 1757, would take up the Cardan Problem in a more thorough treatment than anyone before him had ever done. 29

Besides these annual lectures and the classes he regularly taught, Kästner was also preparing another weapon to be released in 1758: his voluminous Anfangsgründe – a series of textbooks, which would become the center of all mathematical physics education at Göttingen for the entire second half of the Eighteenth Century. These, however, were not just mere dry axiomatic Mathematics textbooks like those of Christian Wolff (which Kästner begrudgingly had to use when he first arrived at Göttingen), rather, Kästner’s compositions were rigorous pedagogicals which had a significant emphasis on the history of all the ideas contained within. Kästner was preparing these such that, those attending Göttingen not only graduated instilled with the traditions of Cusa, Kepler, and Leibniz, but also would be armed with a capacity to recognize what the crucial epistemological battles were, as well as knowing who the actors were that had participated in refining the ideas which the student would be recreating.

Thus, it should come as no surprise to find contained within the Anfangsgründe an extended treatment of the problem of Divisions of a Circle as well as that of the Cardan Problem.

In the Anfangsgründe, after he developed for his students the various takes on Cardan’s Formula, he ended the entire piece with the comment:

I would have omitted Cardan’s Formula, which according to the Credentials of much Mathematical Understanding…is of very little Use, were the numerous and peculiar Efforts which the Algebraicists have exerted upon a Difficulty which is nonexistent, not remarkable. Perhaps, after we have convenient Methods

26 Euler’s paper from Memoires de l'academie des sciences de Berlin 5, 1751, pp. 222-288, is available both in its original form and in translation (by Todd Doucet – used here) at The Euler Archive, http://math.dartmouth.edu/~euler/pages/E170.html. 27 See David Shavin, From Leibniz to Franklin on ‘Happiness’, for a fuller treatment of the historical arc which established the University. 28 Merz, John Theodore. A History of European Thought in the Nineteenth Century. Vol. I. Blackwood and Sons: Edinburgh, 1907. p 165. 29 The report on the lecture is given in the GGA of April 14, 1757. It should also be noted here that Lessing had also taken up a research project on Cardano. It is known that during the years 1750-54, Lessing was involved in examining Reformation History, which included a study of Cardan. One of the products of this work was a writing entitled Vindication of Hieronymus Cardanus, which appeared in 1754.

to find the rational and irrational Roots, their Theory, which they themselves would not employ for actual Application, just as I have reported it here, also will not be disagreeable.

This statement seems as bold as that with which Kepler condemned the Cossa of Bürgi. Perhaps something was overlooked in the original investigation of the Cardan Formula. Might there be a more fundamental problem with the form of the supposed solution? See Box IV.

Kästner, thus, was on the search for a more physical geometry, which did not make such presuppositions as the arbitrarily defined symbolism which the Empiricist School had adopted for their treatment of imaginary quantities. Hence, in developing the curriculum for the future generations in his Anfangsgründe, he would provide a new epistemological basis for what would become his Anti-Euclidean Geometry.

§3…no one is justified in subjecting the metaphysical concepts to the geometrical ones, or to attribute divisibility without end to physical extension just because it applies to geometrical extension. It applies to the latter because the general concept of division, sets no limit. Where each piece is distinguished from another through nothing other than the magnitude and position, one can make out of every part, things that do not assume exactly the same position, but have the same magnitude (Grösse), making new parts. In nature however, we find that parts distinguish themselves through more than magnitude and position alone, and thus nature somewhat hinders divisibility without end which is allowed in geometry. …§4. …the extension of the surface contains nothing from the physical, which terminates in it. Therefore, surfaces cannot be piled one upon another to make up a body, just as many nothings never make up something. Similarly, no surface can consist of lines laid against one another.

§5. The point is the limit of the line, and thus of all extension. Consequently, it itself has neither extension nor part, and an aggregate of points joined together does not constitute a line.

- Kästner’s Anfangsgründe, Volume I, Geometry

Despite Kästner’s, Mendelssohn’s, and Lessing’s efforts to put a halt to the anti-Leibnizian propaganda being injected into Germany’s intellectual circles, Euler continued his attack on Leibniz, and now Kästner, reaching its ultimate low in his 1761-63 Letters to a German Princess.

You know that extension is the proper object of geometry, which considers bodies only in so far as they are extended, abstractly from impenetrability and inertia; the object of geometry, therefore, is a notion much more general than that of body, as it comprehends, not only bodies, but all things simply extended, without

Box IV

In Kästner’s Anfangsgründe, after he derives the equations for Cardan’s Formula, and treats the matter of all the extensive commentary upon it which had accumulated over the years, he then attempts to use it to find a practicable value (i.e., a decimal approximation) for the roots of the equation. Here, he finds himself somewhat astonished as to the amount of work necessary to yield any results.

In one instance, after describing the method provided by Christian Wolff, he finishes the section remarking, “Could he not have directly found the Value of x in this Equation just as easily?”

What seems to be the problem? What does it really mean to express x as the sum of the edges of two other squares?

In illustration, if, being asked for the edge of a cube, one provided the edges of two other cubes, wouldn’t it still be necessary to have a means of knowing what the edges of those new cubes were? If then, it was inquired as to the edges of those cubes and one responded with the edges of still 4 more cubes, perhaps the absurdity would come more to the fore. It seems that every time one gets lobbed off, two more grow back in its stead.

In the end, many had taken up the battle against this monster, but it took Kästner’s keen insight to slay this fictitious mathematical Hydra that Cardan had concocted.

Underlying Kästner’s critiques of the practical value of the Cardan Formula is a more important matter: the manner in which the problem is approached completely avoids the fundamentally paradoxes that one confronts in examining volumetric relationships. One cannot merely invent symbols to solve physical problems – physical problems must be solved physically.

Perhaps, then, the presentation also had more than just a mere historical documentation of the problem as its target. Kästner would have been well aware of the problematic features of Euler’s treatment of ‘imaginary numbers’. Maybe it was no coincidence that Kästner chose Euler’s form of the Cardan Formula to be the one he would present in his textbook.

impenetrability…Hence it follows that all the properties deduced in geometry from the notion of extension must likewise take place in bodies, inasmuch as they are extended; …There are however philosophers, particularly among our contemporaries, who boldly deny that the properties applicable to extension in general, that is, according as we consider them in geometry, take place in bodies really existing. They allege that geometrical extension is an abstract being, from the properties of which it is impossible to draw any conclusion with respect to real objects; thus, when I have demonstrated that the three angles of a triangle are together equal to two right angles, this is a property belonging only to an abstract triangle, and not at all to one really existing.

- Euler’s Letters to a German Princess, VII: The True Notion of Extension

It is then a completely established truth, that extension is divisible to infinity, and that it is impossible to conceive parts so small as to be unsusceptible of further division…If then bodies, which infallibly are extended beings, or endowed with extension, were not divisible to infinity, it would be likewise false that divisibility in infinitum is a property of extension. Now those philosophers readily admit that this property belongs to extension, but they insist that it cannot take place in extended beings. This is the same thing with affirming that the understanding and will are indeed attributes of the notion of man in general, but that they can have no place in individual men actually existing.

- Euler’s Letters to a German Princess, IX: Whether this Divisibility in infinitum takes place in existing Bodies.

Kästner’s immediate reaction was to begin to lecture against Euler’s doctrines publicly. One of the main flanks he took on was Leibniz’s concept of dynamics. Kästner again attacked Euler on the issue a priori conceptions.

If it were possible to imagine a man that had never felt his own or another body, upon seeing bodies move one another, he would not think the same as we would in seeing the same. They [the bodies] would probably occur to him to follow each other like none other than silhouettes of a projector. However, if we attribute that which taught our senses [i.e. sensation] to what we must conclude from bodies which we have only seen, then we assume generally, matter to be an extended being which resists the change of its state, and believe inertia to be derived a priori from the concept of matter, even though we derived this concept out of experience alone. Inertia is thus, as [with] our other sensations, an appearance.

- GGA March 12, 1763

Kästner and his circles were preparing to escalate; in fact their counteroffensive in this period would be a sudden surge of culturally optimistic masterpieces. In 1765, Kästner, together with Erich Raspe, who had control of the Leibniz archives, would publish Leibniz’s New Essays on Human Understanding, to which Kästner would write a preface. 30 The year 1766 was a landmark: Lessing would publish his Laocoön: or On the Limits of Painting and Poetry, a treatise on how the arts are used to communicate conceptions of space and time; Mendelssohn would release his Phaedon, a reworking of Plato’s masterpiece, the Phaedo, in which the concept of the immortality of the soul was elaborated from the standpoint of Leibniz; Kästner, earlier in the year, before Mendelssohn, would write his own treatise on the immortality of the soul, and later in the year, he would see to completion the fourth volume of his Anfangsgründe, treating the subject of applied mathematics, complete with a thorough defense of Leibniz’s conceptions of a physics based upon dynamics. 31

A significant battle had been won, but the war was still far from complete. Within the next decade the American Revolution were to break out. It would be a tremendous victory not only for Kästner’s circles, who were direct collaborators of Benjamin Franklin, but, further, for the entirety of mankind.

Out of the optimism predominating this period, a new generation of great minds, such as Mozart and Friedrich Schiller, would emerge to take up the challenges of communicating such profound conceptions to an entire population. Unfortunately, Lessing would not survive to see the victory of the American Revolution. At this loss

30 See David Shavin, From Leibniz to Franklin on ‘Happiness’. 31 See Kästner’s Anfangsgründe, Volume IV, Part I – Mechanics, Section on The Measurement of Power.

of Lessing (who, one might colloquially say, had the biggest bark of them all), the European faction of republicans became immediately vulnerable, and that opening was not to be let to pass unexploited.

Criticism Is Not Creativity

It was with the opening, which the death of Lessing in 1781 provided, that Immanuel Kant, a quite obscure academic, whose career up to that point had been quite unsuccessful, proclaimed, with the air of a self-aggrandized gatekeeper, that all metaphysics must stop in its path and undergo the interrogation of Kant’s Critiques, before any philosophers would be allowed to proceed through the pass. The subject of Kant’s Critiques encompassed an outright slam on Leibniz’s philosophy by name, an assertion of the a priori nature of conceptions of space and time, and a refutation of Mendelssohn’s argument for the immortality of the soul. In one premeditated attack, he attempted to eradicate all the crucial ideas which Lessing, Mendelssohn, and Kästner had sought to inspire their generation with.

With Mendelssohn preoccupied with defending the legacy of his deceased friend, Lessing, against Jacobi’s slander that Lessing was really an anti-Leibnizian Spinozist, Kant’s novel ‘critical philosophy’ had the effect of a sensation among many circles. Kant’s ‘bread-fed scholasticism’ would take its grip most predominately in two universities: Königsberg, where Kant was a professor, and Jena.

Yet, some might say that a successor to Lessing and Mendelssohn was to be found in the spirit of Friedrich Schiller. 32 It was a turn of fate, that the much beloved dramatist and poet would find an appointment as a history professor at Jena University.

When Schiller went to Jena, May, 1789, to assume his duties as professor “extraordinary” of history, to which he had been appointed mainly through Goethe’s influence, he found himself in a very nest of Kantians, and heard the Kantian principles extolled until, as he says, he was nearly surfeited with them. Jena was known at this time as the second home of Kantian philosophy, and its fame was attracting students from all parts of Germany and the northern countries, many of whom were either already enthusiastic adherents of the Kantian philosophy, or were filled with curiosity to know more about the destructive and novel theories which, as they were told, must produce a revolution in philosophy. That Schiller was on very intimate terms with the teaching force of the University does not appear from his correspondence; indeed, the enthusiasm which the students displayed on Schiller’s arrival was by no means shared by his colleagues, some of whom were only glad enough of an opportunity to humiliate him. 33

Schiller would launch his arrival at the university with a frontal assault and would continue to battle what he perceived as Kant’s enslavement of creativity, for the rest of his life. He wrote the following, in a letter to Körner, after having studied Kant in preparation for his monumental work, Aesthetical Letters.

I feel that ever since I have acted according to laid-down rules, I have lost that boldness and living fire I formerly possessed. I now see what I create and form. I watch the progress of the fruits of inspiration; and my imagination is less free, since it is aware that it is watched....For practice sake, I like a philosophical discussion on theorics, and criticism must now remedy what it has spoiled, - for it has spoiled me. ...When I have succeeded in making the laws laid down by art second nature, in like manner as education makes the polished man, imagination will then reassert its former freedom, and will prescribe its own limits.

While Schiller was battling Kant in order to set free ‘that boldness and living fire’ which his Critiques crushed, Kästner would also continue to wage cultural warfare in his own, now taking up, more boldly and more directly than ever, a scathing attack on Euclidean geometry, and in particular, Kant’s notion of the a priori nature of the knowledge thereof. In 1790 he would publish his essay, On the Conceptions that Underlie Space.

32 See David Shavin, Philosophical Vignettes from the Political Life of Moses Mendelssohn. 33 Wilm, Emil Carl. The Philosophy of Schiller in Its Historical Relations. J. W. Luce : Boston, 1912. See also, Schiller’s Universal History Lecture: What Is, and to What End do We Study, Universal History.

§30. The reason why one does not find the same degree of evidence for this axiom [Euclid’s 11 th Axiom – the Parallel Postulate] does not lie in the concept of infinite space in the newer meaning of the word, but rather: that we have merely a clear concept of straight line, and not a definite concept…what it means in Euclid’s geometry to be possible. …§34…the difficulty concerns the distinction between curved and straight lines. A curved line means, a line in which no part is straight. This concept of a curved line is distinct, because the concept of straight line is clear; but it is also incomplete, because the concept of straight line is merely clear. §35. So we see, that the difficulty of the axiom does not concern infinite space, but rather the indistinctness and incompleteness of the concepts.

- Kästner’s On the Conceptions that Underlie Space

These battles themselves, however, did not take place in an empty space. As Lessing and Mendelssohn were taking their leave, the American Revolution was being fought and won, with all the eyes of Europe closely attentive of the outcome. It was hoped for that in Europe a similar process would ensue – instead, they would suffer the horrors of the French Revolution. Despite the valiant efforts to continue the ennoblement of the population, the people were becoming demoralized. 34

By this time, Kästner was now an old man. His generation, which had accomplished so much, was in decline and, although he had collaborators and students who understood his work, his dedication, he had, as of yet, no successor. In his old age, he must have realized that he would not be around much longer to transmit the history which he had committed himself to being the bridge for. It was probably with such in mind that he began the composition of his Geschichte der Mathematik, a history of the development of ideas from the time of Cusa to Kästner’s own day. In this capacity, he would revisit his first inspiration, Kepler, who would occupy not only a significant part of his Geschichte, but would also become a critical aspect of his lectures in his later years. Perhaps, he still had hope that some mind, would again be kindled with a passion for that great genius who had been so neglected by his fatherland.

To Mr. Christlob Mylius: Along with sending over Kepler's Harmonice Mundi

Friend, your tender ear perceives the graceful art of tones, The world-form's harmonies, your deeper thoughts explore,

In here, Newton's teacher writes of both them, Deutschland let him starve, and remains unworthy of him.

A. G. Kästner 35

Socrates: Last of all, then, methinks, he might be able to perceive and contemplate the nature of the sun, not as respects its images in water or any other place, but itself by itself in its own proper station? Glaucon: Necessarily so. Socrates: And after this, he might reason with himself concerning the sun, that it is the body which gives us the seasons and years and administers everything in its stated place, being the cause also in a certain manner of all natural events.

- Plato’s Republic, Book VII To be continued…

34 For more on the collapse of the German Classical Period, see Determining the Orbit of Gauss. 35 Cf. 24.