Fidelio, Vol. IV,
No, 2. Summer 1995
|This article is reprinted from the Summer 1995 issue of FIDELIO Magazine.
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The Metaphor of Perspective
by Pierre Beaudry
The scientific idea of a nation-state, as opposed to the territorial looting of an empire, is based entirely on the willful purpose of fostering the common good of a population, and this commonwealth can only be achieved by means of improving the productive powers of labor of that population. In this fashion, the nation-state must be ruled in a dirigistic fashion, from a centralized government which commits itself to fostering mans ability to reflect this general purpose through works in art and science. In turn, the elevated individual soul will ennoble the nation-state by bringing a contribution to its advancement and progress.
This is the general outlook which became predominant in France around the 1460s, promoted chiefly through the influence of Nicolaus of Cusa and the school of the Brotherhood of the Common Life from Deventer. It was from this school that key collaborators of Leonardo da Vinci and Frances King Louis XI, such as Mathias Ringmann, Vautrin and Jean Lud, and Jean Pélerin Viator, came to establish themselves in Lorraine, the homeland of Joan of Arc and the crucial region in the creation of the nation-state of France which was then ruled by Duke René II.
From the standpoint of ennobling the individual, the nation-state should promote and defend the fundamental right of every human being to develop his mental powers of reason in imago Dei, and to perfect himself in order to get closer to the principle of composition of Divine Reason, the underlying principle of the Good that generates the changing relationships of all things in harmony with Natural Law.
In concrete terms, this means that the ruler of the nation-state must be committed to fostering mans access to scientific knowledge, i.e., the discovery of the higher principles underlying the physical processes of nature, and the mastery of how to apply these to machine-tool principles and machines more generally. From this standpoint, the nation-state cannot exist without the explicit objective of establishing the principle of what Leibniz would later call Academies or Societies:
With the help of these Academies (or Societies), which are institutions of research and development, with their own manufactures and commercial houses directly attached to them, the monopolies will be eliminated, because the Academies will always guarantee a just and low price for the goods, and very often, such goods would become even cheaper because new manufactures will be built where none exist at that time.
These are the kinds of institutions that began to flourish in France under the leadership of King Louis XI, such as the Vosges Gymnasium, a Brotherhood of the Common Life school which had its own printing and distribution house. Later, this would be followed by the school of the Oratorians, the Royal Academy of Sciences, the Ecole Polytechnique of Gaspard Monge and Lazare Carnot, and the Ecole des Arts et Métiers [Arts and Trades]. Finally, this outlook would ulitmately be exported to Germanys Göttingen University, and into the United States West Point Military Academy. In each and every case, the key to developing scientific method would be modeled on Nicolaus of Cusas teaching at the Council of Florence, and would be reflected in the rigorous approach of resolving paradoxes, expecially the paradox of the One and the Many, by means of constructive projective geometry.
This article will review the significance of projective geometry, or perspective, from the standpoint of a series of nested theorems developed over a period of three hundred years by lawful predecessors and successorsnamely, Leonardo da Vinci-Jean Pélerin Viator (1505), Gérard Desargues (1639), Blaise Pascal (1645), Gaspard Monge (1794), and Jean-Victor Poncelet (1822). This long process of maturation involved a series of theorems, all of which contributed to developing a general synthetic approach to the understanding of the Euclidean plane, and laid the foundation for a science of constructive geometry as the definite and most lawful approach to the development of the creative process itself.
Indeed, there exists perhaps no single process of geometric discovery which has contributed more to increasing relative population-density in the world for the last five-hundred years, than the invention of perspective in France from the end of the Fourteenth to the end of the Eighteenth centuries. It would not be an exaggeration in the least to say that without this discovery initiated by the secretary of Louis XI, Jean Pélerin Viator, in collaboration with Leonardo da Vinci during the 1490 period, the industrial revolution made possible by Monge and Carnot some three hundred years later would not have been possible.
It would be on the basis of these discoveries made in the field of conical and orthographic projections that industrial designing would ultimately become the sine qua non condition for developing interchangeable parts in modern tool and machine-tool construction. In point of fact, there exist no household appliances in any home today that were not planned and designed, down to the last bolt, by such methods.
To begin, however, we must first presentby means of synthetic projective geometrynot a specialized form of geometry but a geometry of principles, what Lazare Carnot called a geometry of sentiment,1 which is aimed at moving the soul beyond the mediocrity of daily routines; a perspective aimed at directing man above the banalities of everyday life toward virtue, toward more noble sentiments, such as love of God, love of mankind, and love of country. In order to achieve this, you must acquire the sentiment of elevation, the sentiment of proportion, and the sentiment of the infinite. This is the crucial dividing line in the world today, as it has been throughout human history: whether human beings are treated as animals, or they are treated as created in the image of God.
For today, when the fate of mankind hangs in the balance, the same question that was posed two hundred years ago by the founders of the Ecole Polytechnique in the France of 1794, is again posed with renewed revolutionary vigor: will humanity be subjected to the barbarity of soul-less Aristotelian formalism, or will we succeed in reviving Platonic humanism, ruled by reason and guided by what the scientists of the French Renaissance tradition identified as le sentiment?
The Sentiment of Elevation
During the opening of his class on Geometry and Mechanics Applied to the Arts at the Conservatory of Paris in 1826, Jean-Victor Poncelet spoke the following amazing words, which show how the education of Polytechnique was oriented toward teaching the most advanced conception of science to ordinary workers:
Some people began to believe that mathematical truths were by necessity unintelligible to simple workers, because they are presented in abstract and difficult forms from dogmatic schoolbooks; some believed that they could not be easily understood and palpable: they were wrong. It was just that their method was at fault. There exists no mathematical principle, applicable to the works of the arts, that one cannot, with a little bit of study, manage to render easily intelligible to any individual with an ordinary intelligence. ...
Thus, national education must be organized around this unified geometric thought, a triply-self-reflexive movement of self-consciousness:
From the standpoint of synthetic constructive geometry, what Poncelet identifies here is the process of creativity, the true sense of identity of the scientist, that is, of a true citizen of a sovereign nation-state; that is, not a specialist of some trade or art, but a universal man capable of understanding the necessity of developing the nation-state as a scientific idea. Indeed, this elevated sentiment implies for the student a mastery of the same underlying principle of change in both himself and in nature, in order for him to become the One, and his technological inventions for the nation, become the Many. In this way, Poncelet joins Lyndon LaRouche, isochronically, on the necessity of mastering the higher hypothesis2: this triply-self-reflexive principle illustrating the theorem of continuity whereby man is created in the image of God.
The Sentiment of Proportion
The most important thing to remember and to master properly is the sentiment of proportionality, that is, the theory of proportions or of the equality of relationships under consideration.
What kind of proportion exists between a bounding principle outside of the universe, and the harmonic ordering of the five Platonic solids inside of the universe? Kepler addressed this same question four-hundred years ago in attempting to understand the ratios between the celestial spheres. He wrote:
This projection is identified by Poncelet as a harmonic range, whose projective properties were well-known to the ancient Greeks, particularly Euclid. Charles Julien Brianchon, another student of the Ecole, had also arrived at the same results by establishing the following constant ratio:
AC : AD :: BC : BD = constant.
According to Poncelet, the Greeks had already defined this as the harmonic proportion in the following form:
(DA-DC) : (DC-DB) :: DA : DB.
A close examination of this last equality of relationship shows that this harmonic proportion uses only the distance of D to the other three points. This reflects the fact that the distance DC is known as the harmonic mean between the two distances DA and DB. Poncelet further noted that while the line AB is divided harmonically by points C and D, the reciprocal is also true, that is, CD is also divided harmonically by A and B.
Now, suppose that you position the three arbitrary points A,B,C in such a way that C is closer to A on the same straight line; then the fourth harmonic point D will be found on the opposite side of the same line [See Figure 2]. This amazing tilting of line A′B′ from right to left is the result of the reversing of the harmonic range, which can only occur when point D passes to infinity; and this is produced when infinite line DA is rotated into infinite line DB.
So, the reversing of the ratio is a very curious phenomenon indeed, which seems to be an exception to the rule of the theorem, and seems to cause an anomalybecause when the tilting from right to left occurs, the fourth harmonic point D is nowhere to be found on the straight line on which A,B, and C lie. Indeed, DC is conjugated to AB in both cases only because DA and DB have become two infinite lines. As we shall see, far from being the exception, this case in fact establishes the rule: that is, when D is projected at infinity, that very projection determines the harmonic ordering of the whole system.
So, to sum up. These ratios are crucial for two reasons: Firstly, because they tell us a great deal about the natural harmonic ordering of space, and most importantly that there is no such thing as arbitrariness in spatial relationships; and secondly, such ratios will tell us how far the movement of the soul must reach to access its principle, and will help us understand what we must seek in order to answer the question about our proportionality to the Infinite. But before going into that question, let us point out one thing that has to be addressed concerning the complete quadrilateral.
The complete quadrilateral is not simply what the Twentieth-century mathematician David Hilbert makes it out to be in his Geometry and the Imagination. In his Traité des Propriétés Projectives des Figures (Sec. II, Chap. I, Art. 154), Poncelet stresses that there is a difference between the simple quadrilateral (ABCD) and the complete quadrilateral (BAEDFCB) [See Figure 3], and this is, that the complete quadrilateral must have nine straight lines and as many harmonic ranges.
The nine harmonic ranges forming the complete quadrilateral are: EALB, EPGM, EDNC, EHFI, FCMB, FNGL, FDPA, BGDH, and AGCI.
Furthermore, Poncelet acknowledges that this theorem was known by the ancients, as it is reported by Pappus (Fourth century A.C.E. ) in his Collections Mathématiques, Book VII, Prop. CXLV, and that it was also reproduced by Grégoire de Saint-Vincent (Opus geometricum, Prop. X, 1647) and Laurent Lahire (Sectiones conicae, in Folio, Livre I, p. 5, 1685).
Sentiment of the Infinite
In Sec. I, Chap. I of his Traité des Propriétés Projectives des Figures, Poncelet establishes the fundamental theorem of projective geometry, which will represent a rigorous solution to Zenos paradox of the bad infinite. His theorem identifies what happens when the fourth point of a harmonic range ABCD, point D, goes to infinity, and establishes the basis for linear perspective, a perspective established by Leonardo da Vinci and Father Jean Pélerin Viator circa 1490 [See Figure 4].
The condition for Poncelets theorem to be true must flow essentially from the following two axiomatic considerations.
First, it is because the infinite ratio DA : DB corresponds to infinite unity, that CA : CB, a finite ratio, is reflected into a finite unity. And from this it must follow that all finite segments of equal partitioning of a perspective lattice shall have their receding scale formed everywhere by parallel lines.
Secondly, the partitioning of the perspective lattice into equal parts is consequent to the harmonic point D being projected at infinity; therefore the unity of the two infinite distances DA and DB, as well as their finite difference CA and CB, must be determined by the same projective property that establishes point D at infinity.
This theorem of Poncelet establishes explicitly, for the first time in history, not only that the harmonic range of the complete quadrilateral is nothing but the theorem of perspective, but also that the point at infinity, otherwise known during the Renaissance as the subject point (Jean Pélerin Viator), is a unique resolution of the Parmenides paradox of the One and the Many, and becomes the founding theorem of projective geometry.
From this, a more general theorem may be established, stating that if any number of infinite magnitudes or distances converge toward one point at infinity, they may differ from one another by some finite amount, but they cannot be affected in their cardinality by any changes in the lower finite order: that is to say, on the contrary, that it is the projective property of point D at infinity which determines the harmonic ordering of all of the finite and infinite distances of such a lattice. The point at infinity which determines an infinite number of such lines is thus a power point, which bounds every other point in the lattice from the outside, and is transfinite to them. This will become very important later for Cantors considerations in defining the transfinite numbers.
The reader should also note that this is what Lyndon LaRouche means, when he says that the higher species determines everything in the subordinated lower species, but that the lower species cannot determine anything with respect to the higher species. Indeed, this is surely the case where poetry must supersede mathematics.4
Lazare Carnot made this point very clearly in introducing the basic curriculum at the Ecole Polytechnique, where the science of linear perspective was to be superseded by the science of aerial perspective, where he says
[L]inear perspective ... is calculated mathematically, [but] aerial perspective ... can only be grasped by sentiment. By comparing these two sciences, where one is sensual, the other ideal, the methodical course of one will help penetrate the mysteries of the other. ... [Aerial perspective is] the art of generating ideas by means of the senses, of acting on the soul by the organ of vision. It is in this way that it acquires its importance, that it competes with poetry; that it can, like poetry, enlighten the mind, warm the heart, excite and nourish higher emotions. We shall emphasize the contributions that it can bring to morality and to government; and how, in the hands of the skillful legislator, it will be a powerful means of instilling horror of slavery and love of the fatherland, and will lead man to virtue.
The point is that unless you have reference to the infinite, harmonic ordering of the finite is not accessible.
Louis XI and the Institution
|Ode to Enthusiasm
by Lazare Carnot
Sublime soaring of generous souls,
|Ode a lenthousiasme
par Lazare Carnot
Sublime essor des grandes âmes,
Schiller also had a very concise description of this French Revolution, which he called A great moment which found a small people. This is how he refers to the situation in his On the Aesthetical Education of Man, especially the end of Letter IV:
Man can, however, be opposed to himself in a twofold manner: either as a savage, if his feelings rule over his principles, or as a barbarian, if his principles destroy his feelings. The savage despises art and recognizes nature as his unrestricted master; the barbarian derides and disrespects nature but, more contemptible than the savage, he frequently enough continues, to be the slave of his slaves. The educated man makes nature into his friend and honors its freedom, while he merely bridles its caprices.
When reason therefore brings her moral unity into physical society, she should not damage the multiplicity of nature. When nature strives to maintain its multiplicity in the moral structure of society, there should be no breach in the moral unity; equally far from uniformity and confusion rests the victorious form. Totality of character must therefore be found in the people, which should be capable and worthy, of exchanging the state of necessity for the state of freedom.
On the Aesthetical Education of Man, Letter IV
And since the Jacobin terror had destroyed the laboratories and guillotined the scientists (such as Lavoisier), there was no better and more necessary idea than to establish a curriculum based on geometric discoveries, as the catalyst that would lead to the discovery of the creative process of the human mind, and give France the scientists, the engineers, the metallurgists, the chemists, and so forth, that the nation-state needed so desperately. And so began the real French Revolution when, in 1794, Robespierre was defeated by Carnot, and the Committee of Public Safety passed a resolution for the creation of the Ecole Polytechnique and the Ecole des Arts et Métiers [Arts and Trades]. As Poncelet, one of the very first student brigade leaders would later express it,
We do not intend to teach you a method and a process for each art, but instead what is the principle common to all of the arts ... with the purpose of making inventors out of you, inventing new machines and new processes.Just as life gives the body its unity, projective and descriptive geometry brought to national education its vital inspiration, without which all of the arts and scientific studies would have been disparate and meaningless. In fact, national French education was organized around this unifying geometric idea, according to which students were required to master the underlying principle of a given discipline, then apply the same underlying principle to another discipline, and lastly embrace in a unique theorem the principle underlying all of the disciplines of human industry. Such a higher geometric principle is what Poncelet formulated as the basis for the development of arts and crafts throughout France, this principle of continuity which broadens the mind and embraces in a unique theorem a multitude of lesser truths.
Perspective and projective geometry, which had been the exclusive science of painters and of cathedral builders up until that time, now became the very foundation of the industrial revolution. Monge would apply the principles of projective geometry to the design and manufacture of standardized and interchangeable parts for military components. From then on, everything that had been previously built by hand had to be recast at the foundry, based on uniform and universal designs that would become the standard for each and every small part of an assembly, from a simple mechanical wheel, to a complicated piece of artillery, to a ocean-going vessel; the kinds of machinery designs that Leonardo da Vinci had developed three hundred years before, now became the standard type of models for the Ecole Polytechnique [See Figure 15].
Principles of projective geometry applied to machine design (orthographic projection). (a) Leonardo da Vinci, Machine for making bands of copper, Manuscript G, 70v. (not shown) (b) Modern machine design, as standardized by Monnge at the Ecole Polytechnique.
Correlate this with the geometry of Keplers snowflake principle of close-packing, this cubic projection which is inscribed in the dodecahedron [See Figure 16].10
Cubic projection. Close-packing, as described by Kepler in The Six-Cornered
Snowflake, derives from the properties of the dodecahedron.
There is only one way to map all of the points of a three-dimensional object onto a two-dimensional plane, and that is to discover the means of triply relating pairwise (biunivocal) relations as one. This is what the dodecahedron generates in the form of the inscribed cube when you unfold its sides onto a two-dimensional plane [See Figure 17(a)].
Orthographic projection. (a) Unfolding a dube onto a two-dimensioinal plane. (b) Three -dimensional object represented orthographically in two dimensions.
Monge would affix the different views of the orthographic projection, in the same plane. The idea was to completely represent every aspect of a three-dimensional object on a two-dimensional plane. This was done such that perpendicular lines drawn in each of the three planes of projection, are all orthographically interconnected into one single projection of (1) the frontal view; (2) the top view; and (3) the side view, as in shown in Figure 17(b).
This is done by a simple circular generative process of unfolding the different sides, whose interconnectedness is extended by parallel projective lines falling at right angles to one another. The projective angles of 120° in one domain correspond to right angle (orthographic) projections of 90° in the other. If you complete the projections of the rear, bottom, and left sides of the cube, you will have gone full circle, that is, you will have covered the six sides of the cube, or the complete sphere.
The crucial point here is that the horizontal and vertical magnitudes of a two-dimensional plane are able to contain all three dimensions of an object, provided that one can discover the unique way to relate triply, as one all of the dual relations: (1) height and width; (2) width and depth; and (3) depth and height. If these relationships taken two by two can, together, form a unity of all three variables, then you have an equivalence of relations between volumes and planes. This means that the principle which defines the ordering of the triple relation is of the same species as that which determines the duality of relationships.
Consider, however, that the ability to project such a three-dimensional object onto a two-dimensional plane is not a simple task to realize. It is not the trivial action of measuring something forward, upward, and sideways, or simply filling linear extension in all directions. It is not a simple act of adding a new dimension to a surface; you are not simply going from the square to the cube. What you are dealing with is an actual unity of reflection of the creative process, of the faculty of imagination (of which, by the way, animals are not capable), in the sense that you are projecting on a lower manifold the geometry of a higher manifold, which involves the creation of a leap caused by the generative principle of a One, a common principle, which underlies an infinity of space-connected problems, and bounds them together from the outside.
Again, that One is exemplified by the generative principle of the Viator three-point perspective. It is from this vantage point that Monge would develop his descriptive geometry, which would lead to developing the generative principle of orthographic projection for industrial design. But in his classes, Monge would make use of only two planes of projections, since the horizontal and the vertical projective planes are sufficient to convey the three dimensions.
Thus, the principle of this parallel projection does not lie in the connection between points, lines, and surfaces, as such, but more fundamentally in the ability to access and discover the region of the mind which foreshadows the higher characteristic between two distinct manifolds rather than concentrating on the variable positions of the perceived objects from one manifold or the other. This is not a question of Aristotelian reduction to sense perception, but of a Platonic approach to knowledge.
Let us take the example of a series of theorems which represent a historical sequence of predecessors and successors in isochronic collaboration with one another over a period of over three hundred years. (The term integral here is not used in the usual analytical sense of the word, but more broadly, meaning that the theorems of Viator and Desargues actually integrate, synthetically, the theorems of Pascal, Monge, and Poncelet, into one single Viator-Desargues Integral Theorem of projective geometry.) First, the Poncelet complete quadrilateral with its nine integrated harmonic ranges (Figure 3) is derived from the Monge Four-Sphere Theorem (Figure 18), which itself is derived from Desargues Theorem (Figure 9). In turn, you could easily find that Desargues Theorem is itself derived from Viators three-point perspective. Hence, a series of crucial theorems initiated from the Viator-Desargues Integral Theorem determines Euclidean geometry essentially as the science of projective synthetic geometry.11
Given four spheres (A,B,C, and D) of different positions and size in space, if you conceive of six conical surfaces which circumscribes them externally, two by two, the summits of these six cones shall be in the same plane and at the intersections of four straight lines; and if you conceive of six other conical surfaces, circumscribed internally (that is to say, which have their summits between the centers of two spheres), the summits of these six new cones will be, three by three, in the same plane with three of the first ones [See Figure 18].
Monge Four-Spere Theorem construction.
The same principle of continuity may be pursued in the case of packing of spheres in space, such that, for every three spheres of different sizes and positions in the plane which are enveloped by conic surfaces whose three apexes fall on the same straight line, there exists a fourth sphere whose center forms, with the centers of the three other spheres, and with the internal and external points of similitudes, nine harmonic ranges belonging to a complete quadrilateral in one single plane. This plane is one of the five such planes that form the Viator-Desargues Integral Theorem construction [See Figure 19].
Viator-Desargues Integral Theorem construction (shnowing only a single plane.
It can be further demonstrated that, for every three spheres of different size and position, there exists a fourth sphere which lies in the same plane and is conjugated with the other three to form a harmonic quadrilateral. The positions of the four spheres relative to one another, (or of the same sphere rotating along an elliptic path to different positions), are harmonically ordered in the plane, just as the four external points formed by the apexes of their circumscribing cones are harmonically ordered on a straight line.
Since each plane of three spheres of different size and position can have a fourth sphere (Figure 19) which is harmonically conjugated to the three others to form a harmonic cluster, the completed form of the Desargues, Pascal, Monge, and Poncelet Theorems will reflect a harmonic field.
Thus, all of the spheres of the Viator-Desargues Integral Theorem will form a harmonic field of clustered spheres which, in Poncelets terminology, will correspond to the continuous projective property of five complete quadrilaterals generated by multiply-connected circular action onto five different planes (including ten straight lines and ten harmonic ranges); or, in Monges terminology, ten conical projections tangent to eight spheres of different sizes and positions in space, oriented two by two, and forming through their internal and external centers of similarity, ten harmonic ranges. (Figures 20 (a) and (b) shows the Viator-Desargues Integral Theorem with only four spheres.)
Three-dimensional representation of the Viator-Desargues Integral Theorem construction. (a) Four-sphere model. (b) Eight-sphere model.
This transformation, this higher form of correlation between theorems, corresponds to what Carnot called natural geometryas opposed to algebraic analysis, which cannot make such non-linear correlations. This is why Baron Augustin Cauchy, the father of analysis, eliminated such basic constructions from the curriculum of the Ecole Polytechnique. Now, this historical sequence of discoveries by geometers contributed to the crucial breakthroughs which brought about the development of every major discovery of the industrial revolution. So you have here, in essence, the crux of the conflict between the Aristotelian/Venetian method and the Platonic method.
If Cauchy had been an honest analyst, he would have had to admit that one cannot make the leap from the side to the diagonal of a polygon at infinity, the place where rational and irrational numbers meet on the same line. No matter what open-endedness you may find in the two-dimensional plane, you will nonetheless find closure in the three-dimensional magnitude, by virtue of the generative principle underlying the Viator-Desargues Integral Theorem. This is why Cauchy didnt understand the Poncelet principle of continuity, and despised it.
This synthetic-constructive method of natural geometry can only be understood, wrote Carnot in his Eloge de Vauban (1783), by means of principles which are, so to speak, located in le sentiment, as opposed to algebraic analysis, which can only be acquired by memorization of formulas. Indeed, synthetic-constructive geometry aims at elevating the soul through noble emotions, such as passion for scientific discovery, or love of God and love of mankind, by means of discovering non-linear correlations between entities which otherwise have no algebraic relationship whatsoever. This is why synthetic geometry is useful for innovation, but algebraic analysis is not.
The algebraic-analysis approach, on the other hand, is (as described here by Carnot) an abstract art of building systems, the art of tracing on paper lines which are dependent in their mutual positions on quasi-arbitrary conditions to which some people have given the important name of axioms. It may be acceptable for an engineering task, but it is useless for the purpose of invention; in fact, it is detrimental to the creative process.
Ultimately, algebraic analysis will lead you easily to cultural pessimism, because it is a region of dry, passionless, deductive processes which stultifies creativity. Its main claim to fame is cold, logical proof, which its sycophants elevate to the supreme level of the elitist knowledge that they portray as science. Just to give you a taste of this pessimism, witness how Baron Cauchy himself, a Bourbon legitimist and a sworn enemy of Poncelet, conceived of the importance of human discovery:
When we take a quick look at the productions of the human mind, we are tempted to believe that human knowledge can grow and and multiply itself at infinity. ... However, if we observe that all of our intelligence and our means are enclosed within limits that can never be superseded, we will persuade ourselves that our knowledge is limited ... that if man has been unable to visit the poles, he remains in an eternal despair of ever reaching these frozen regions ... . Who will ever be able to dig a well of 1,500 leagues deep? We have managed to elevate ourselves to 1,500 fathoms in the atmosphere, but the rarity of the air ... will constantly bring back to earths surface whomever would want to reach higher ... . Exact sciences can be considered as completed sciences ... . By means of sophisms man can come to the point of doubting these truths we teach him, but he will never discover new ones!
Augustin Cauchy, Cherbourg, 1811 Such a spirit of limitation can come only from a prolonged contact with the oligarchical worldview, the view of man as an animal, and the algebraic method itself, which is defined internally from the very limitations of the axioms and postulates which generate theorem-lattices. So, by virtue of the very nature of the closed completeness of theorem-lattices, it is impossible for an algebraic-animalist mind to make the non-linear leap between sets of theorem-lattices; which is what is required for creative discoveries.
It was Louis Pasteur who continued the spirit of the Ecole into late-Nineteenth century France. He saw very clearly the acute crisis that France had been going through since 1815, and he identified precisely the problem that had crippled the nation since the Congress of Vienna. The joy of discovery had been killed in the school system, and the inner God (as he put it, recalling Carnots commitment to enthusiasm) was no longer the praised emblem and principle of the Ecole. It had been replaced by the evil of radical positivism.
By 1814, Auguste Cauchy and Auguste Comte had taken over the Ecole Polytechnique and had totally subverted its high purpose.12 They dumbed-down everything to what became known as Positivism, the new religion of manwhat was later called secular humanism. This was the context for the following beautiful statement of Pasteur:Positivism sins not only through methodological error. There is a considerable gap in its seemingly tight net of reasoning ... . The large and obvious flaw in the system consists in that the positivist conception of the world does not take into account the most important of positive notionsthat of the infinite.
What lies beyond the starry vault of the heavens? More starry heavens. So be it! And beyond? Pushed by an invisible force, the human mind will never cease asking itself: What is there beyond? Does it want to stop either in time or space? Since an endpoint would be merely a finite dimension, greater only than those that had preceded it, no longer does the mind begin to envision it than this implacable question returns, and the mind cannot quell curiositys call. ... Positivism gratuitously brushes aside this positive and fundamental notion, along with its consequences for the life of society. ...
Are not the science and passion of understanding nothing else but the effects of the spur of knowledge, put in our souls by the mystery of the universe? Where are the real sources of human dignity, of liberty and of modern democracy, if not in the notion of the infinite before which all men are equal.
The spiritual bond situated [by the positivistsPB] within a sort of lower-level religion of Man, cannot reside elsewhere than within the higher notion of the infinite, because this spiritual bond must be associated with the mystery of the world. The Religion of Man is one of those superficially obvious and suspect ideas which brought one eminent psychologist to say : I have thought for a long time that the person who has only clear and precise ideas must assuredly be a fool. For the most precious notions harbored by human intelligence are deeply behind-the-scene and in semi-daylight, and it is around these confused ideas, whose interrelations escape us, that the clear ideas gravitate, extending, developing, and germinating themselves. If we were cut off from this background, the exact sciences would lose the greatness which they draw from the secret rapport they hold with those infinite truths whose existence we can only suspect.
The Greeks understood this mysterious power below the surface of things. It is they who bequeathed to us one of the most beautiful words of our language: the word enthusiasm, [which means] inner God.
The greatness of human actions is measured by the inspiration that gives them birth. Joyous is he who carries within him an inner God, an ideal of beauty, which he obeys: an ideal of art, an ideal of science, an ideal of his nation, an ideal of the virtues of the Gospel. These are the living sources of great thoughts and great actions, and all of them are lit by the gleam of the infinite.
Speech delivered to the French Academy of Sciences, 1882
The Leonardo/Viator Method:
On the foreshortening of the square circumbscribed by a circle, Jean Pélerin Viator established in his De Artificiali Perspectiva, that perspective is based on a triply-self-reflexive rotation of intersecting visual pyramids, an approach typical of Leonardo. With the three apexes of the three cones located on a straight line representing the infinite horizon, Viators theorem reads:
The narrowing of the receding square lying in the plane A,B,B′,A′ is constructed from the inclined radial lines of the central visual pyramid P, which intersect two other visual pyramids projected from third points D and D′, which are equally removed from the subject point P at a distance twice the width AB of the tetragon, or more or less that distance depending on the closer or farther view. And the circle circumbscribing the square is generated from the sphere, and is perceived inclined as an oval or as a lens depending on the position of the frontal view.
Jean Pélerin Viator,
De Artificiali Perspectiva, Fol. 5
Although no document attests to Leonardos or Viators explicit knowledge of the harmonic range as later developed by Poncelet, both based linear perspective on a harmonic ordering of the complete quadrilateral, where AD : B′D :: AC : CB′. (Viators above-mentioned relationship PD = PD′ = 2AB = 2/1 is derivable from the generative principle of the Golden Section of the dodecahedron.)
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1. In the French tradition, sentiment is used to mean the emotion of agape that is conjoined with the activity of creative reason. This has nothing to do with feelings per se, referring instead to the higher emotions, such as freedom, love of God, love of country, and everything that relates to the common good of mankind as opposed to personal interest. For an extended discussion of this question, see Jacques Cheminade, Régard sur la France républicaine (Paris: Editions Alcuin, 1991).
2. Cf. Lyndon H. LaRouche, Jr., On the Subject of Metaphor, Fidelio, Vol. I, No. 3, Fall 1992, pp. 36-39.
3. We use the mathematical notation for expressing ratios and proportions, rather than the more familiar arithmetic notation DA / DB = CA / CB, because the former denotes geometrical relations, whereas the latter denotes algebraic ones.
5. For the life of Viator, see L. Brion-Guery, Jean Pélerin Viator (Paris: Les Belles Lettres, 1962).
6. Cf. Lyndon H. LaRouche, Metaphor, op. cit., pp. 23-36.
7. It should be noted that even the current scholarly literature, such as the Source Book in Mathematics of David Eugene Smith, continues to this day the same Venetian tradition of mistranslating and disfiguring the few remains of Pascals work, as exemplified by certain unintelligible translations of his theorems. David Eugene Smith, A Source Book in Mathematics (New York: Dover Publications, 1959), pp. 326-330.
8. Dino de Paoli, Construction of a Harmonic Golden Section, Leesurg, 1978 (unpublished).
9. Cf. Lyndon H. LaRouche, Jr., Metaphor, op. cit., Figure 1, p. 19. The author has chosen to term LaRouches diagram the aleph model, as it illustrates the first of the successive levels of the mathematical transfinite of Cantors aleph series.
10. Johannes Kepler, De Nive Sexangula (On the Six-Cornered Snowflake), trans. by Colin Hardie (Oxford: Clarendon Press, 1966; reprinted by 21st Century Science & Technology, 1991).
11. The author has shown elsewhere that the construction represented by this Integral Theorem is actually bounded by the dodecahedron, as the underlying One of the Many which Raphael Sanzio used as the architectonic idea for his School of Athens fresco. Although this discovery cannot be presented here, the reader should know that all theorems of dodecahedral Euclidean space, be they of finite or infinite magnitude, find their generative principle in the boundary conditions set from the outside by the nested projection of an icosacuboctadodecasphere, which represents a higher geometry from the standpoint of the Keplerian quantum field.
12. A Note on Polytechnique and America. As early as 1815, the Congress of Vienna forced the expatriation of the Polytechnique method into Germany and the United States, where two polytechnician students Claude Crozet and Isaac Roberdeau were sent in 1816 with a recommendation from Lafayette. Their mission was to create a corps of engineers for the industrialization of the United States. Claude Crozet developed a corps of engineers at West Point. One of the best students in the class of 1825, Alexander Dallas Bache (the great-grandson of Benjamin Franklin), was sent to Europe to study under Wilhelm Weber and Carl Friedrich Gauss at Göttingen University. Bache would later design the engine boilers for the safest locomotives in the world.
It was West Point engineers like Bache, Stephen Long, and George Washington Whistler, who developed the Baltimore Railroad in the 1830s. Whistler was also sent to Russia, to build the first railroad from St. Petersburg to Moscow in 1843. The Frenchman Isaac Roberdeau would build all of the fortifications on the East Coast, including Fort McHenry in Baltimore. Roberdeaus prize work, Fortress Monroe (1830) in Norfolk, Va., made him famous as the Vauban of the New World.
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