THE SUBSTANCE OF TENSORS:
The Ontological Matter
by Lyndon H. LaRouche, Jr.
May 21, 2009
For the relevant Classical scholar, the essential reality of human life's activity lies in that so-called "infinitesimal" which is known, otherwise, as Classical poetic irony, rather than within the medium of simply literal statements. On this account, Percy Bysshe Shelley's 1819 A Defence of Poetry must be placed adjacent to Bernhard Riemann's 1854 habilitation dissertation, Über die Hypothesen, welche der Geometrie zu Grunde liegen. So, in Johannes Kepler's uniquely original discovery of the universal principle of gravitation, as in his The Harmonies of the World, and in the assessment of Kepler's discovery by Albert Einstein, what is ontologically real, lies, for some among us, in that real universe whose mere shadows are as familiar to us as what are widely mistaken for literal sense-certainties among even what is considered a majority among the well-educated today.
To the naive person typical of academic life, but, also other persons generally, today, it is that reality which is, typically, apprehended by them as being merely the ironies, the mere overtones of Classical poetry, or, of discovered universal physical principles. Contrary to those persons, these are the ironies which are customarily viewed, mistakenly, as being the mere shadows of the relevant realities of sense-certainty; whereas, for true Classical poetry and scientific discovery, today's customary, so-called popular sense of what is substance, and which shadow, has been the reverse of what is known to the greatest scientists and poets; or to a musical genius such as J.S. Bach, Wolfgang Mozart, Ludwig Beethoven; or, for a truly great scientific discoverer, such as a Filippo Brunelleschi, a Nicholas of Cusa. a Leonardo da Vinci, a Johannes Kepler, or a Riemann, a Max Planck, an Albert Einstein, or an Academician V.I. Vernadsky.
As some recent developments in scientific studies have demonstrated, the clearest example of that same universal principle of Classical poetic irony as we encounter in the work of physical science, is presented to us in the most deeply-rooted treatments of the work of Bernhard Riemann, as in the attention to his work by the scientists Albert Einstein and Academician Vernadsky. This lies in the work of Einstein and Vernadsky, considered here, as being the subject of the ontological, rather than merely formal implications of the tensor itself.
What I have just written here, is a reflection of what had first come to me during the mid-1930s, as a hint of a future discovery which I had first made later, in 1953, in my adolescent rejection of the concept of a Euclidean geometry, and in a later time, my recognition of the more deeply underlying form of the issue of Bernhard Riemann's 1854 habilitation paper. That latter paper has defined what became my own relatively unique, and uniquely successful, present approach to long-range economic forecasting.
Any future, successful physical science, including a science of physical economy, will be an outgrowth of consideration of the deepest issues posed by treatment of the subject of the tensor, as by Albert Einstein and Academician V.I. Vernadsky respectively. For the members of our own so-called "basement team," a certain significant breakthrough in this matter occurred recently, through that team's exploration of the tensor when it is considered as a physical, rather than a formal mathematical conception. This was done by an associate's applying that notion of the tensor associated with the carefully considered conceptions of both Einstein and Vernadsky, to the related method actually used by Carl F. Gauss in his published presentation of the discovery of the principle of the orbit of the asteroid Ceres.
A subsequent, quick review of a number of the most relevant among the already familiar, crucial discoveries by Gauss, including Gauss's own reference to his suppression of public attention to his youthful discovery of the principle of an (actually) anti-Euclidean geometry, then, now calls our attention to the notoriety, among Gauss's often frustrated admirers, of Gauss's habit of generating crucial discoveries of physical principle with accompanying descriptions of his own, validated discoveries, while leaving the germ of his original generation of that discovery largely unstated.
Now, if and when we look back, here and now, to view the work of Gauss during his work of the first half of the Nineteenth Century from this present standpoint, we may recall a series of cases in which Gauss had presented an illustration of the discovery of a principle of physical-scientific work, in which certain crucial features of the process of that discovery as such, had been left in mystery for his admirers to discover later. However, now, since the treatment of the work of Bernhard Riemann by, most notably, Einstein's and Vernadsky's treatments bearing upon the subject of the Riemannian roots of the tensor, we are impelled to re-examine those discoveries by Gauss from the vantage-point of the relevant treatments of the subject of the tensor by Einstein and Vernadsky.
That had become the most immediate mission of our current "basement crew" since the preliminary, exploratory phases of the current Riemann mission had, so to speak, "settled in." Gauss left a significant number of his crucial discoveries with much about the way the actual discovery occurred unrevealed. In these cases, Gauss clearly intended that his associates and students should work through the crucial elements of the discoveries for themselves. It is time to attend to at least a significant ration of that unfinished business.
In any case, the circumstances in which a brilliant young Carl Gauss would avoid reference to the underlying principles of the method employed by him for his greatest discoveries, are not really mysterious to those among us who know the history of the conflict between Classical scientists, such as Gauss's teacher Abraham Kästner, on the one side, and, on the other, the relatively hegemonic cults of the followers of what is still politically hegemonic in physical science teaching today: the current, viciously reductionist phase of the cult of modern Liberalism in the programs of higher educational institutions.
In each case in which Gauss omitted public reference to the roots of his discovery of a principle, such as the matter of the Ceres orbit, it was that conflict between Gauss's own roots in the Classical standpoint, against the empiricists' dogma, which was the point of the conflict which Gauss was avoiding, as much as possible, in his published work. For him, mathematics was the Queen of science, but, that is the King. It was permitted, therefore, sometimes, to honor the Queen instead of the King.
The crucial issue here, is the unfortunate, misguided habit of seeking the meaning of a physical principle in the mere shadows which the principle casts upon the domain of mathematics, rather than the ontological actuality of the principle itself. When the same discoveries by Gauss are examined afresh from the standpoint in Riemannian method represented by Einstein and Vernadsky, the weight of attention is properly shifted from mathematical shadows to the substance which casts the shadows.
For such matters of the history of science as those, what might appear to some of us as a chance development in the work of what is referred to as our "basement team," prompted a round of silent moments of triumph in response to Sky Shields' programmed application of the concept of the tensor in such a way as to expose the dynamics of the Gaussian solution for the orbit of Ceres. He had treated it, in a recent application, as being a matter of a physical-experimental, rather than a formally mathematical discovery, as to the meaning of the idea of the tensor as that subject had been treated, variously by Einstein and Vernadsky.
This has much broader implications than might be suspected by some influences which were notable within the bounds of the Twentieth-century physical-science classroom. This is key to understanding, as the great English poet Shelley would have recognized, the leading reasons for the repeated failures of civilizations, a failure to be attributed as being essentially the substitution of what is termed "sense-certainty," for the reality whose nature is illustrated both by Shelley's A Defence of Poetry, and in the outcome of the work of those two great physical scientists from my own lifetime whose discoveries I have just emphasized here.
That much said in a justified spirit of optimism, the hope which I have thus expressed, while true, is also, presently, a gravely endangered expectation.
Therefore, when I hear the sententious utterance of the word "practical" in the name of policy and politics, I shudder at that chill I feel crawling up my back, as I glance at the fanaticism in the eyes of that speaker. What can I say, then, which might give honest reassurance to those children who might find a chill running, shuddering up their spines, if they sense that they might be the victims of having heard that speaker's malicious intentions? "Who," those children might ask themselves, "is that whom I sense might be soon walking on my grave?" The Obama administration, for example, so far, with its Nazi-like health-care policies and its related adoption of the British "cap-and-trade" hoax, has given much reason to fear for the early fate of all humanity, including those children, and not only their aging grandparents, or even parents, right now.
The Ontological Issue
In respect to the subjects thus placed before the reader here, in all relevant, competent sorts of known treatments of the subject of the dynamical roots of ancient, through modern physical science, the principal issue has been the dispute: whether the products of the mental-creative powers of science, are either reflections of the sense-perception of sensory experience (a view which is the standpoint of the modern academic reductionists) or, on the contrary, that the principles discovered are native to those innately creative powers, specific to the human mind, those powers which the mind employs for insight into the deepest significance of what are, on the surface of events, the mere empirical phenomena, those mere shadows of reality known to us as sense-perception.
The outlook of the actual, or prospective genius, was then the standpoint of such as the ancient Pythagoreans and Plato, and of their followers such as the great Eratosthenes, Archimedes, and the modern science of Filippo Brunelleschi, Nicholas of Cusa, Leonardo da Vinci, Johannes Kepler, Pierre de Fermat, Gottfried Leibniz, and Bernhard Riemann.
The conflict between those two opposing, categorical viewpoints, the Classical Pythagorean-Platonic, versus the empiricist, is typified in what is the most notable case for physical science today, as the standpoint of the notion of the function of the tensor in the work of such followers of Riemann as, most notably, Albert Einstein and Academician V.I. Vernadsky. The deliberations on the subject of the Riemannian tensor by those two great thinkers of modern science, mark out the territory of the investigations to be examined in my remarks here.
In response to my admonition to my younger associates, I have warned that the issue of the tensor, so situated for treatment, within the domain of a science of physical economy, must be "ontological in respect to its own physical efficiency, rather than merely formal." My associate Sky Shields applied this emphatically ontological (rather than merely mathematically formal) approach to craft a graphic form of animated generation of the reconstruction in such a restatement of what was, among us, the well known accomplishment of Carl F. Gauss's uniquely original discovery of the orbit of the Asteroid Ceres.
The immediate impact of Sky Shields' animated reconstruction was that it reminded our relevant circles of collaborators of the many cases in which Carl Gauss had made what had been fundamental discoveries, which not only proved to be essentially correct, and for which Gauss had supplied a fully competent description of the function; but, nonetheless, he had failed to supply a full account of the actual process of generation of that otherwise proven physical discovery. This frustrating experience with the practice of Gauss's restraint, had been a virtually life-long habit for him, at least since the beginning of the Nineteenth Century. The first known leading example of this, in my knowledge, is that of the matter referenced by him, to his old friend Wolfgang (Farkas) Bolyai, much later in their lives, of Gauss's youthful discovery of a general conception of an anti-Euclidean geometry. That youthful work was, clearly, a result of the influence of one among young Gauss's principal teachers, Göttingen University's Abraham Kästner.
Einstein and Vernadsky remain, today, the principal successors of the generally fundamental contributions to a Riemannian universal physical science. Einstein remains the principal initiator of a competent approach to the subject of the tensor; but, it is Vernadsky, who modified Einstein's work on the specific account of the Biosphere and Noösphere, who provides the corrected standpoint of reference in method for establishing a standard form of a science of physical economy today.
The Personality of Genius
To understand a specific, dynamic range of quality of the mentality of geniuses, of which those two scientists, Einstein and Vernadsky, are exceptionally good examples, it is necessary to understand the coincidence of certain exceptional features of their intellectual achievement with a certain tendency for exceptional aspects of their personal relationships in other respects. Compare the related cases of such geniuses as Nicholas of Cusa, Leonardo da Vinci, Johannes Kepler, Gottfried Leibniz, Johann Sebastian Bach, Moses Mendelssohn, Wolfgang Amadeus Mozart, Ludwig van Beethoven, or a Gauss, Dirichlet, Riemann, or Einstein,
Here, the concept of dynamics comes prominently into play in treating our subject here; for there is no narrowly definable, specific form of standard personality; but, rather, there is an intrinsically dynamic principle of what may be classed as types, all sharing membership in what is definitely a distinct, dynamic quality of range of variations, as Percy Bysshe Shelley considers such a set of relationships in the concluding paragraph of his A Defence of Poetry.
Many of those cases of personalities who are more readily identified as fitting a type of candidates for the designation "genius," probably fit what might seem to be, otherwise, a provisional standard of certain superficial characteristics in common. By the standards of Binet and related testing, they will often register, as I have known such cases, between 120 and 160 on the relevant scale, or higher. However, the "I.Q.," while it is not an insignificant suspect for such classification, only points toward such a range of scoring which also includes the scheming by "possible suspects" seeming to fit the models of a large ration of rather disgusting types, including some "sociopaths," who whatever their scoring, show neither actual creative-scientific characteristics, nor artistic genius, at all.
The clear distinction of the true genius is not a numerical score, but of certain recognizable, qualitative and functional characteristics. Where does the relevant person locate his, or her sense of personal identity as a person living within this world? Does he, or she locate reality as being essentially located in the physical body, and sense-perceptual experiences, as such, of that body; or, does he, or she identify with the viewpoint of the mind itself, rather than seeing themselves as a superior sort of mammal with a special added knack, a virtual gimmick, such as superior skills in mathematical formalism, or a command of sundry languages which is lacking in most other "human animals" whom they encounter, or, more simply, whom they chance to know, as on a first-name or similar basis?
In the end, those really qualified to be considered actual or potential geniuses, as manifest types, are relatively rare in our contemporary societies, much rarer in the U.S.A. today, for example, than two or three generations ago. Actual geniuses often tend to see themselves, not entirely without reason, as a "persecuted," or "potentially persecuted" minority, akin to the less noble, academic class of so-called "nerds," among both their immediate peers and society more broadly. They are, in that sense, seen, by others, and, often, by themselves, as "eccentric;" and, the usual reaction to their presence by cruder minds, evokes a view of them as "somehow alien," or "ugly ducklings" who are regarded as "somehow strange" by the set of their putative peers in their society at large.
The essential distinction of that minority which comprises the minority of which the relatively superior individual intellects are composed, when all actually relevant considerations are taken into account, is that their personal sense of identity is located, essentially, in looking outward toward the world of sense-perception from inside the domain of ideas; whereas, in the history of recent generations, the majority among secondary and university students, or graduated professionals, for example, express a directly contrary, so-called "more practical," outlook. That majority has adopted a view, contrary to that of a great Classical musical performer who is dominated by the location of a sense of innermost personal identity in that profession, as in a deeper examination of the motives of an actual genius. It is the orientation toward the experience of an act of a valid discovery of a principle, which shows the exceptional case of the creative, personal world-outlook of the truly "inner-directed," creative personality as being qualitatively different than today's majority of individuals, even most relevant cases of professionals. It is among that minority that the potential candidate for classification as "genius" is to be found.
Such definitions as that are not, however, the end of the matter.
Usually, we should speak of those who should be recognized to be more or less clear cases of geniuses. Yet, there are many others, who are potentially gifted, but whose case is not so readily a clear case of "genius" to ordinary scans. Those among us whose profession occupies them with organizing social processes of the institutions of which they tend, more or less, to be naturally leaders, are either aware of this in some degree; or, if they are not, they should have found themselves in a somewhat different occupation, one more suited to the short-comings of their insight.
My own social experience, generally, but, more emphatically in organized associations in which I have had some significant sort of relevant participation, has been that I have been mightily occupied with detecting and encouraging those whom one could recognize as partaking of some of the quality otherwise recognizable as a touch of insight akin to that of genius, a quality in them which has been often hidden within the subject of that person's more obvious outlooks and roles in day to day life and its activities.
For anyone in a role comparable to my own, these individuals inhabiting what might be termed "the twilight zone at the fringe of genius," are very important, even when they do not appear to be luminaries. We can detect them as being precisely that, because they react to important relevant clues to which the more commonplace individual usually fails to react. In their best moments, whatever they do on other occasions, such a person with significant creative impulses is operating, if only approximately, as the true genius does. In their most insightful moments, they are operating from inside the domain of human intellect. They react, in their relatively better moments, as if by an intellectually gifted, frequently insightful person, but from a place somewhat outside the category of the specifically inner-directed quality of the motives of both the actual or potential young genius.
Those in the early dawn of what borders on genius, represent persons with the potential for becoming the members of an organization, or in society around them, who serve the cause of "management by exception." They are usually more occupied with "getting the job done," than getting the position of higher rank; they enjoy being what they are as necessary eyes and ears of the relevant social process. They are not mere spies, but, rather, represent persons with certain specially tuned sensibilities which are lacking among the generalities of those from among their ostensible peers, whose reactions are of the more ordinary sort. They include what became recognizable as gifted machine-tool designers, or a comparable talent. It is my view, that they must be protected in this role, because they have a true touch of genius, which might tend to develop to a certain higher state of relative maturity under appropriate circumstances.
It was such approximate expressions of true genius, which we should recall from among Benjamin Franklin's collaborators engaged with him in introducing the "industrial revolution" to England, with ties to related talents in France who were akin to the recruits to the Monge-Carnot circles of the pre-1815 Ecole Polytechnique.
Therefore, in those cases of such potential, there is a certain tendency for short-range development of their valuable insights, on which account society, especially today's, tends to neglect, or, at least, downgrade the significance of the potential contribution by such individuals. Since the 1968-1973 interval, for example, the shift toward hatred of actually creative mental life, a shift into existentialist, sometimes even Satanic foolishness, has been the correlative of a general decline, even a correlative of a rather brutal repression of the creative potentials, and also regression in mating practices, of adolescent and young adult strata generally.
For an example of what I am referring to here, think of the many potentially gifted pupils in our educational systems, in whom we fail to recognize the potential of a talent, that, notably, at what may be a crucial point in that young person's self-development. The present policies of education in the U.S.A. tend, thus, to destroy more such minds than they enrich. In such cases, when the victims of such treatment are recognized as such, there is a tendency say, "Aw, if we had recognized and fostered their talent in a timely way, they could have developed as ? important players among us today." Unfortunately, the goals of "mass production" of program-determined social types work to the effect of leveling the pavement of stultifying conformity, and the youth with it.
In that specific sense, I am certain that society could increase the role of the maturing young potential genius in society now; but, that this means scrapping entirely what was praised by the President George W. Bush, Jr. administration as a trend in U.S. public education today. The relative, virtual mass-brain-damage evident in the late-adolescent and adult youth today, as compared with the generation of university students in the last years of the administration of President Bill Clinton, now just less than a decade ago, is a relevant illustration of this point. The evidence of MySpace, Facebook, and, now, Twitter, amounts to symptoms of an epidemic with an awful portent for international society today, a portent, expressed as the virtual grandchildren of Mark Rudd's proto-fascist circles among the "Sixty-Eighters," which is akin to that of what was in fact, the neo-Dionysian Flagellants of Europe's Fourteenth-century "New Dark Age."
That Said, Now, Back to Science As Such
The essential distinction of human from beast, is the role of that true creativity whose most characteristic expression is the increase of the potential relative population-density of successful cultures, even relative to the effects of the inevitable, relative depletion of what had been considered, in practice, as the relatively richest of relevant natural resources.
Thus, the indicative, if crudely stated, measure of the effect of scientific and related progress in development of the expressed human mental-creative powers and of the culture which is necessary to that development, requires, as if axiomatically, a rise in the level of intelligence of the knowledge and practice which guides a society's behavior in general. In other words, progress incurs the obvious depletion of what had been, previously, appropriate kinds and qualities of resources. However, the human capacity for physical-scientific and other advances in knowledge naturally tends to outrun the effects of the depletion incurred by continuation of the maintenance of the existing level of population.
There is, in short, no "law of entropy" intrinsic to human activity. Depletion occurs when man violates our inborn creative-mental nature, and chooses either a policy of regression, or even simple technological-cultural stagnation, or the frankly fascist goals of British imperialism's fraudulent doctrine of "cap and trade."
The problem represented by long periods of either stagnation, or even regression in cultural characteristics of a people, is most often traced, in known history as such, to the phenomenon of imperialism, as the playwright Aeschylus identified this sickness as the role of that evil Olympian Zeus, who prohibited man's acquisition of the knowledge of such forms of "fire" as nuclear-fission power. Since the predominant civilizations in actually known history of the internal life of cultures have been dominated by imperialist, or similarly brutish forms of systems throughout most of our planet's known cultures, in all continents, a tendency for "zero growth" has been a manifest result, a virtual habit, of cultures known from their inside so far.
However, in all cases which fit that description, the decadence occurred only because it had been effectively imposed by some dreadful, imperialist or comparable political power or similarly depraved form of culture. What the administrations of former President George W. Bush, Jr., and his successor (so far) President Barack Obama have done, has been to use repressive measures, such as President Obama's economic and health-care policies to collapse the standard of living and technological practice in the direction of Nazi-like economic and related health-care practices of types imitated, exactly from the precedent of the Nazi Hitler regime. There is no difference, on this account, between the top-down trends built into current practices under this new President (so far) and the frequently identical measures which this British-controlled Presidency and its co-thinkers in the Congress have copied directly, explicitly, and precisely from those initiatives characteristic of the Adolf Hitler regime. Obama's adoption of the exact-same policy which Adolf Hitler pre-dated to September 1, 1939 is a case of an exact copy, with a virtually identical outcome, of genocide, now already built into the system, unless a radical reversal of current Obama policies is effected very soon.
Those who would deny that fact already in evidence, are either liars or pitiable fools.
Such ugly facts taken into account, any attempts to continue the health-care and low-energy-flux-density policies (e.g., "cap-and-trade") are genocidal policies with the same characteristics as those of the Nazi Hitler regime. That said, the reversal of those implicitly pro-Nazi policies, signals a return to the kinds of policies characteristic of the impulses of the Presidency of Franklin D. Roosevelt, under which conditions, creativity as the dynamics of Einstein and Vernadsky, will be the remedy which corresponds with what will come to the surface as popular intention, as a similar pattern could be recalled from the rise of the U.S.A. under President Franklin D. Roosevelt, from out of the depths into which the nation had been drifting since the death of the assassinated U.S. President William McKinley on Sept. 14, 1901.
Any different policy than I propose, would be, in effect, tantamount to treason.
All men and women are born with the assigned intent to be geniuses in Classical modes of art and physical science. It is Classical modalities in art which supply the spark of genius on which the creation of valid discoveries in physical science depends.
It is therefore, the primary mission of constitutional government, to develop newly conceived human individuals into such geniuses, then developed to such effect. In what way they will become manifestly geniuses is not to be predetermined in any arbitrary way. Genius fostered will, like flowing water, find the pathway which chooses the course of its expression. Our essential responsibility is to nourish the abundance of the supply of flow.
In the meantime, as the summation of this matter is met in, most emphatically, the closing paragraph of Percy Bysshe Shelley's A Defence of Poetry, creativity flows from Classical artistic composition into the spark which ignites the development of the domains of the abiotic, the Biosphere, and the Noösphere. The effort to resolve the matter of the subject of the Riemannian tensor, as by Albert Einstein and Academician V.I. Vernadsky, is the currently visible approach to be taken in crafting that mission of development. It is the promotion of the development of the powers of creativity through the combined, interdependent efforts of a physical science and a Classical culture, which is crucial.
It is from poetry so defined, as Shelley said, that is the origin of human creativity to such combined effects.
Under the current trends in policy of the now incumbent President, it must be seen by those who have the courage to face the obvious truth of our situation, that we are doomed, and that soon, unless his present policies are not only dumped entirely, but reversed. Do not be so pessimistic as that. Crisis brings change; make it happen.
 Ironically, I was born exactly one-century-plus-one-month after Shelley's death. History's mere coincidences, even when slightly stretched in that manner, are sometimes like that. The alternate title for this report could be, "endangered actual and potential young geniuses situated, precariously, in a presently imperilled world."
 On the Hypotheses, Which Underlie the Principles of Geometry.
 The presentation of the notion of a modern, specifically anti-Euclidean geometry was made during the Eighteenth Century by a most celebrated scientific figure of the time, Gauss's teacher, Göttingen Professor Abraham Kästner. Gauss's known references, in his now published correspondence, including relevant correspondence with Wolfgang (aka Farkas) and Jonas Bolyai, indicate the relevant discovery by Gauss as dated from some time during the 1790s, prior to his Disquisitiones Arithmeticae. The obvious reasons for Gauss's caution, relative to the openness of the relevant Lejeune Dirichlet and Bernhard Riemann, were clearly, as Gauss's letters to the Bolyais on anti-Euclidean geometry indicate, the adverse political conditions for science imposed upon the leaders of the Ecole Polytechnique, and others, by, initially, the advent of Napoleon Bonaparte to power in France, and, with the British appointment of the Restoration monarchy, thus, creating the aversive conditions which continued within Germany under both a certain King of Prussia, as under the early Nineteenth-century British Foreign Office, then under the direction of Jeremy Bentham. Gauss' reference to his own discovery in this connection (in a letter to Farkas Bolyai of March 3, 1832), indicated his own discovery as something additional to that of his former teacher Kästner's treatment of the "parallel postulate."
 Such was the cloak used by Rudolf Clausius and Hermann Grassmann et al. in their concocting the fraud against the work of Riemann which was, shamefully, adopted by the editor of Riemann's Werke, Heinrich Weber. When the repeatedly demonstrated experiment of Wilhelm Eduard Weber, with whom Riemann collaborated for a time, is taken into account the Clausius-Grassmann note was clearly fraudulent.
 Viewed in retrospect by modern scientists, the most stunning accomplishment of Brunelleschi, is to be located in his use of the physical principle of the catenary as the principle of construction employed to craft the otherwise impossible dome of Santa Maria del Fiore. The notion of the catenary as an expression of a physical principle, rather than a mere geometrical form, was explored in a crucially important way in Leonardo da Vinci's exposition on the relationship of catenary and tractrix. The same conception turns up again, significantly to the credit of Fermat, in the development of the more advanced notion of the Leibniz calculus as expressing a principle of universal physical least action.
 Notably, the approach of Gauss's informed contemporaries relied upon a prompting by Johannes Kepler's definition of the existence of the remnant of an "exploded" planet, lying in an original orbit between those of Mars and Jupiter.
 For those readers not already familiar with this case, Kästner, born in 1719, in the Leipzig of Gottfried Leibniz and Johann Sebastian Bach, was, at this time, a leading mathematician of Germany since his adulthood, and the originator of the modern concept of an anti-Euclidean geometry. Kästner, who had early dedicated his adult life to defense of the genius of Leibniz and Bach, became also, the principal backer of the legacy of Leibniz in that time, and thus closely associated with the circle of the Gotthold Lessing and Moses Mendelssohn whose combined efforts were the chief prompters of the great, late Eighteenth-century cultural and political renaissance in trans-Atlantic civilization, including its impact on the principal authors of the conception of the American Revolution. Although a competent notion of a physical geometry existed in the work of such as the Pythagoreans and Plato, prior to Aristotle's and Euclid's hoaxes, the establishment of a true non-Euclidean geometry was first completed by Bernhard Riemann's establishment of a truly physical, rather than nominal geometry, as from the outset, in Riemann's 1854 habilitation dissertation.
 The late Norbert Brainin, of the Amadeus Quartet, is, for me, a prime example of that case for great musical performers. He exemplifies those great performers of music who performed from inside the domain of the music he performed. He qualifies as a member working from within the domain of true genius, rather than as an outsider performing a score.