An Arc of Knowability

The Problem of Cubic Roots

by Merv Fansler

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 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 defend against 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 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.

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]

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 continued 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.

…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

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 - it took 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

Al-Khowarizimi’s method was picked up by people like Fibonacci, Nicolo Fontana Tartaglia, and Girolamo Cardano.  Just as al-Khowarizimi 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 ().  Tartaglia provided Cardan with his 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.

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

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  would have been regarded as absurd; for, how could two somethings, when added together, make nothing?)

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: Take , 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, . Then, following the animation below, one finds .   Combining this with the original equation, provides two new relationships: I) II) 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: