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from ed. Harari, Josue V. And David F. Bell, Hermes; Literature, Science, Philosophy. the Johns Hopkins University Press, Baltimore, 1982
Hieronymus informs us that he [Thales] measured the height
of the pyramids by the shadow they cast, taking the observation
at the hour when the length of our shadow equals our height.
- Diogenes Laertius
The height of a pyramid is related to the length of its
shadow just as the height of any vertical, measurable object is
related to the length of its shadow at the same time of day.
- Plutarch
The text attributed to Hieronymus by Diogenes is supposedly one which tells of the Greek miracle, of the emergence of an abstract form and line of reasoning against the ground of an earlier practice or perception. How should we read this tale of an origin which eludes our attempts to classify it as reality or myth? Here are a few of its legends.
The tale dramatizes the theorem of Thales. Two triangles: the first is constituted by the pyramid, its shadow and the first or last ray of light; the second by any object whose height is accessible, its shadow and a ray of light. The triangles are similar since their angles, one of which is 900, are equal. Hieronymus relates a particular case where the triangles are isoceles, Plutarch the general case where they are not. (1) This depends on the moment of the day: the particular form is observable at a single instant. Both texts are diagrams of Thales’s theorem and tell less the story of its origin than the possibility of its application.
Let us assume, then, the existence of the pyramid and its shadow. In this schema the following elements are accessible: the black region which I can measure directly, the peg planted in the earth and its desolate shadow. Inaccessible, however, are the height of the tomb and that of the sun. As Auguste Comte says: "In light of previous experience we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know. It is this general fact which makes the science of mathematics necessary. For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics." (2) Geometry is a ruse; it takes a detour, an indirect route, to reach that which lies outside immediate experience. In this case the ruse is the model: the construction of the summary, the skeleton of a pyramid in reduced form but of equivalent proportions. In fact, Thales has discovered nothing but the possibility of reduction, the idea of a module, the notion of model. The pyramid itself is inaccessible; he invents a scale, a type of ladder.
Hence, again, Auguste Comte: "It is thus that, for example, Aristarchus of Samos estimated the relative distance of the sun and the moon from the earth by measuring the sides of a triangle constructed to be as similar as possible to the right triangle formed by the three heavenly bodies when the moon is at a right angle to the sun and when, consequently, in order to define the triangle one had only to observe its angle with the earth." (3) Like Thales, Aristarchus constructs the reduced model of an astronomical situation. To measure the inaccessible consists in mimicking it within the realm of the accessible. This is true in many instances, one of which is the example of ships at sea. Glossing the twenty-sixth proposition of the first book of Euclid’s Elements, Proclus writes: "Eudemus in his history of geometry attributes the theorem itself to Thales, saying that the method by which he is reported to have determined the distance of ships at sea shows that he must have used it." (4) Tannery, in his Geometrie grecque (p. 90), reconstructs the measuring technique of the famous fluminis variatio used by the Roman agricultural surveyor Marcus Junius Nipsus. In any case, the point is to transpose some unreachable figure into a more immediate realm in the form of a miniaturized schema.
Accessible, inaccessible, what does this mean? Near, distant; tangible, untouchable; possible or impossible transporting. Measurement, surveying, direct or immediate, are operations of application, in the sense that a metrics can be used in an applied science; in the sense that, most often, measurement is the essential element of application; but primarily in the sense of touch. Such and such a unit or such and such a ruler is applied to the object to be measured; it is placed on top of the object, it touches it. And this is done as often as is necessary. Immediate or direct measurement is possible or impossible as long as this placing is possible or is not. Hence, the inaccessible is that which I cannot touch, that toward which I cannot carry the ruler, that of which the unit cannot be applied. Some say that one must use a ruse of reason to go from practice to theory, to imagine a substitute for those lengths my body cannot reach: the pyramids, the sun, the ship on the horizon, the far side of the river. In this sense, mathematics would be the path these ruses take.
This amounts to underestimating the importance of practical activities. For in the final analysis the path in question consists in forsaking the sense of touch for that of sight, measurement by "placing" for measurement by sighting. Here, to theorize is to see, a fact which the Greek language makes clear. Vision is tactile without contact. Descartes knew this, just as he understood better than most what measurement is. The inaccessible is at times accessible to vision. Can one measure visually the distance to the sun, to the moon, to a ship, to the apex of a pyramid? This is the whole story of Thales, who discovered nothing but the precise virtues of the human gaze, just as, somewhat later, Berkeley organized in an erudite manner a spectacle of light beneath his microscope, a rigorous organon of optical representation. Since he cannot use his ruler, he sets up lines of sight or, rather, he lets light project them for him. As far as I know, even for accessible objects, vision alone is my guarantee that the ruler has been placed accurately on the thing. To measure is to align; the eye is the best witness of an accurate covering-over. Thales invents the notion of model, of module, but he also brings the visible to the tangible. To measure is, supposedly, to relate. True, but the relation implies a transporting: of the ruler, of the point of view, of the things lined up, and so on. In the realm of the accessible, the transporting is always possible: in the realm of the inaccessible, vision must take care of displacements: hence the angle of sight, hence the cast shadow. Measurement, the problem of relation; sight, the cast shadow; in any case, the essential element is the transporting.
Let us return to the schema, that of Diogenes or Plutarch. It deals with things in motion or at rest. The constant factor is the pyramid, immobile for ten centuries beneath the Egyptian sky. The apparent movement of the sun, the length and the position of the shadow are all variable. Everyday experience tells us that the latter depend on the former. Hence the initial idea: the clock. The pyramid is a gnomon and the line of its shadow tells time. Measurement and the gauging of the shadow’s variations mark the rhythm of the sun’s course. The gnomon, stable and arbitrary, is only an intermediary object; the variances echo one another. The goal is either civil or astronomical. With a sundial, the measurement of space only measures time. The sundial, whose origin is lost somewhere at the dawn of time, (5) will disappear during the quarrel of the Ancients and the Moderns, a quarrel particularly acerbic in regard to clocks. Hence, in Diogenes and Plutarch, the remains of what was once the problem of time: to wait for the moment of equality between an object’s shadow and its height, or to observe the two shadows at the same time of day; to keep the sun and its daily course in mind. This is why I quoted Aristarchus: we begin with astronomy.
Thales’s idea (for we must give it a name) consists simply in turning the process around, that is, in considering and then resolving the reverse problem of the gnomon. Instead of letting the pyramid speak of the sun, or the constant determine the scale of the variable, he asks the sun to speak of the pyramid; that is, he asks the object in motion to provide a constant flow of information about the object at rest. This ruse is much more clever than the one we described earlier: the constant is no longer what gauges the regular intervals of the variable; on the contrary, Thales gauges, within the variable realm, the stable unknown of the constant. Or rather, with the gnomon, whoever measured space also measured time. By inverting the terms, Thales stops time in order to measure space. He stops the course of the sun at the precise instant of isoceles triangles; he homogenizes the day to obtain the general case. And so do Joshua and Copernicus. Hence it becomes necessary to freeze time in order to conceive of geometry. Once the gnomon has disappeared, Thales enters into the eternity of the mathematical figure. Plato will follow him. This is the old Bergsonian conclusion.(6)
An initial summary: the proliferation of geneses. How did geometry come to the Greeks? A practical genesis: build a reduced model, have a notion of the module, bring the distant to the immediate. A sensorial genesis: organize a visual representation of that which defies physical contact. A civil or epistemological genesis: take astronomy as a starting point, reverse the question of the gnomon. A genesis that is either conceptual or esthetic: erase time in order to measure and master space. Exchange the functions of the variable and the invariable. The origin of geometry is a confluence of geneses. We must follow the other effluents.
Thales’s schema presents an optical diagram which is stable with regard to the apparent movement of the sun, at least in its second version. Vision and its spectacle presuppose the following: a site or a point of view, a source of light, and finally the object, either luminous or in shadow. We have said that the essential element is the transporting. For even if measurement can be exact or precise, only the relation is rigorous: the reference of a giant schema to a reduced model. These initial geneses are acts of transporting: reduction, the transition from touching to seeing, and vice versa, the reversal of the gnomonic function, the exchange of the stable and the variant, the substitution of space for time. Hence a new series of questions.
1. Where is the point of view? Anywhere. At the source of light. Application, relation, measurement are made possible by aligning landmarks. One can line up the sun and the top of the tomb, or the apex of the pyramid and the tip of its shadow. This means that the site need not be fixed at one location.
2. Where is the object? It too must be transportable. In fact, it is, either by the shadow that it casts or the model that it imitates.
3. Where is the source of light? It varies, as with the gnomon. It transports the object in the form of a shadow. It is in the object; this is what we will call the miracle.
We are dealing less with the story of how something came about than with the dramatization of a preexisting form, Thales’s theorem. The first legend, made up of several geneses, is a mathematical decipherment. One must extract the implicit schema from an anecdote whose "local color" has been used by traditional scholars to show that the Greek sage learned everything from Egyptian priests. The relationship of the circumstantial form to the schema leads one to think less about the invention of the second in the action related by the first than about the covering up of the latter by the former. Or, in case I wish to recall Thales’s theorem, the story of the pyramid can serve as a mnemonic device. In a culture with an oral tradition, story takes the place of schema, and theater equals intuition. The diagram of the theorem can only be transmitted in written form, but, in an oral culture, drama is the vehicular form of knowledge. Myth then, the mythical tale, is less a legend of origin than the very form of transmission; it does not bear witness to the emergence of science so much as it communicates an element of science. Here mathematics is the key to history, not the contrary. The schema is the invariant of the tale instead of the tale being the origin of the schema. To know, then, and, in this case, to know Thales’s theorem is to remember the Egyptian tale. To teach the theorem is to tell the pseudo-myth of origin. We know that all mythical tales are merely the dramatization of a given content. Only the mathematical decipherment of the text can demonstrate the relationship of the implied schema to the mobilization which turns it into a transmissible tale.
Thales’s theorem is itself anecdotal in relation to the invariable concept that it expresses in its own genre: that of similarity. Curiously enough, when the schema is analyzed thoroughly one rediscovers the lived variety of the tale. On the one hand, the theorem is only possible because of the space of similarities; it may only be inscribed on or in that space which is the space of transport. On the other hand, it is perfectly natural to take a trip to the Nile delta; at the foot of the pyramids, what Thales or anybody else perceives cannot be anything other than objects of the same form but of different dimensions. The perception of three pyramids is developed within the space of similarities, and this space is constituted by choice in this place: each pyramid is different and yet the same, like the triangles in Thales’s theorem. Hence the story is perfectly faithful to the concept, and similar to the idea of similarity. It is more a question here of technology than of perception: similarity is the secret to the triple edifice, the secret to its construction. The pure knowledge implicit within the design of the pyramids is certainly homothetic. In order to build them one must have (but not necessarily know) Thales and homothesis. The size and the position of the stones are an application of homothesis. Whether this is an application ignorant of this knowledge or an operation executed according to a clearly explicit concept is hard to decide. In any case, the passage in Diogenes is twice deciphered mathematically; further, in each instance the articulation of the various concepts and of the tale is clearly visible. The circle has been completed: Khephren and Mykirionos are reduced models of Cheops.
What is the status of the knowledge implied by a certain technique? A technique is always an application that envelops a theory. The entire question-in this case the question of origin-boils down to an interrogation of the mode or the modality of that enveloping process. If mathematics arose one day from certain techniques it was surely by making explicit this implicit knowledge. That there is a theme of secrecy in the artisans’ tradition probably signifies that this secret is a secret for everybody, including the master. There is an instance of clear knowledge that is hidden in the workers’ hands and in their relation to the blocks of stone. This knowledge is hidden there, it is locked in, and the key has been thrown away. It is in the shadow of the pyramid. Here is the scene of knowledge, the dramatization of the possible origin, dreamed about, conceptualized. The secret that the builder and the rock-cutter share, secret for him, for Thales, and for us, is the shadow-scene. In the shadow of the pyramids, Thales is in the domain of implicit knowledge; on the other side of the pyramid, the sun must make that knowledge explicit in our absence. Henceforth the entire question of the relationship between the schema and history, of the relationship between implicit knowledge and the artisans’ practice, will be posed in terms of shadow and sun, a dramatization in the Platonic mode, in terms of implicit and explicit, of knowledge and practical operations: on the one hand, the sun of knowledge and of sameness; on the other, the shadow of opinion, of empiricism, of objects.
These first two readings reveal convincingly the implicit knowledge that a fabricated object hides within itself. In general, it is easy to determine the nature of theoretical knowledge mixed with actual practice. It is easy as the usual path of science is easy, that is, it is difficult but not impossible, complicated but eventually solvable. The thing which is difficult and ultimately inextricable, which we attempt indefinitely to render explicit without being able to explain it completely, and which is thus forever clouded over, is the modality, the "how" of this implication, which, in an actual application, is clearer. The articulatory mode of luminous knowledge and blind practice is blinder in implication, more luminous in application. The origin of knowledge acquired through everyday practice is on the side of shadow; the origin of a practice acquired through knowledge is on the side of light. One could learn a great deal about the emergence of a theory by diligently asking oneself about its various realizations a posteriori and by reversing the analysis. This theoretical-practical relationship, that of sun and shadow, is also what interests Diogenes.
A shadow adequately designates the folds of hidden knowledge. In the initial technical activity, knowledge is in shadow, and we are also in the dark as acting beings, trying to situate theory in light. We will soon discover that Thales failed in this last attempt. The pyramid has its shadow, and beneath the sun of Egypt everyone has a shadow. What else can I know and do, except measure the relationship between the two shadows (a relationship identical to that between object and subject), except measure the relationship between the secret which is entombed in the stones and the one which lies in the worker? The relationship between these two secrets says, designates, describes the secret of the relationship between man and his wrought object. In the legend, Thales’s geometry expresses the relationship between two blindings, that which practice engenders and that which the subject of practice engenders. His geometry says this and measures the problem, but does not resolve it; dramatizes his concept but does not explain it; designates the question admirably without answering it; tells of the relationship between two numbers, the mason’s and the edifice’s, without deciphering either one. And perhaps one can never do anything but that if one confines oneself to the problem of the logos. The relationship between the two shadows is the problem of designation, the pure naming of the enveloping mode of a piece of knowledge by its technique. The technique of measurement which is still a ruse of application, or, as Auguste Comte says, an indirect path, repeats the implication but does not explain it. Thales extracts a technique from a technique, and from a practice he gets another practice. Of course, architecture and mensuration both envelop the same knowledge, homothesis and the celebrated theorem; still, the application is repeated. The homology of repetition eventually designates the homothesis, but in each instance within the gangue of the applied. The theory expressed by shadows remains in shadow. It was not born in its pristine form that day. There is no longer any original miracle: different techniques give rise to other ones and perpetuate themselves in repetition; measurement and architecture see the theorem differently, that is all. And we remain in the immense shadow of the secret. For, again, one cannot conceive of the origin of technique except as the origin of man himself, faber as soon as he emerges, or rather, emerging because he is faber. Technique is the origin of man, his perpetuation and his repetition. Hence Thales repeats his very origin, and our own: his mathematics, his metrics of geometry, repeats in another way (and as simply as possible) and designates in another way the modality of our technical relationship to objects, the homology of the fabricator to the fabricated. His mathematics takes its place in the open chain of those utterances and designations, but it does not provide the key to the cipher; it does not excavate the secret articulation of knowledge and practice in which the essential element of a possible origin is located. His mathematics is the relation between two shadows, two secrets, two forms and two ciphers, relation or logos, relationship and utterance to be transmitted, utterance which transmits a relationship. As is commonly said, it measures the problem, takes its dimensions, poses it, weighs it, demonstrates it, relates it, but never resolves it. The logos of shadows is still the shadow of the logos.
Still, what Thales’s mathematics recounts, at its very inception, is the de-centering of the subject of clear thought with regard to the body that casts its shadow: the subject is the sun, placed beyond the object, on the other side of the shadow. This was also Copernicus’ lesson. What this mathematics articulates is the Platonic decision that a geometry of measurement is but a propaedeutic. What it announces, for the first time, is a philosophy of representation, dominating both the pure diagram and its dramatization beneath the torches of the solstice. From whence one returns to the size of the stones and to the pyramid. The edifice is a volume of volumes, a polyhedron composed of cut-out blocks of stone. Now how is one to study and learn about a volume if not by means of a planar projection? And how can one lay hold of it if not by attacking its surfaces? Thales’s geometry says this, and so do architectural technique and the mason’s daily practice. In each of the three cases it is a matter of studying a solid in terms of all the bits of information that have been gathered at the relevant levels: the secrets of an object’s shaded surfaces and its cast shadow. I know nothing about a volume except what its planar projections tell me. But a projection assumes a point of view and a drawing on a smooth surface, a surface without any shaded area and without any hidden fold. I can know a stone, a solid, even the pyramid, only by its contour described by the sun on the plane of the desert sand. The sun-subject writes a form in the sand, a form that is changing and infinite like the profiles of a Ptolemaic perception, a form that describes a cycle of representation. Each moment of the representation, arrested, fixed in the sand, is nonetheless equipped with a constant: a stable relationship with the same shadow, at the same moment, of another object-with me, for example. Here the geometry of perspectival measurement articulates the invariant in the variations of representation.(7) The cast shadows vary, the secrets are transformed, but they share among them a secret which remains constant and which is the unknown, the pyramid’s secret: its inaccessible height. As variable as representation may be, it still designates, suddenly, a portion of the real, a stability proper to the object, its measurement. Which is why, from this position, I can only know about the volume that which is said, written, or described by cast shadows-the bits of information transported onto the sand by a ray of sunlight after its interception by the angles and summit of an opaque prism. This geometry is a perspective (an architecture), it is a physics, an optics: the shadow is a black specter.
The theater of measurement demonstrates the decoding of a secret, the decipherment of a writing, the reading of a drawing. The sand on which the sun leaves its trace is the screen, the wall at the back of the cave. Here is the scene of representation established for Western thought for the next millennium, the historically stable form of contemplation from the summit of the pyramids. Thales’s story is perhaps the instauration of the moment of representation, taken up ad infinitum by philosophers, but also and above all by geometers, from Descartes and his representational plane to Desargues and his point of view, from Monge and his descriptive diagram to Gergonne and his legislative transfers:(8) the first word of a perspectival geometry, of an architectural optics of volumes, of an intuitive mathematics immersed in a global organon of representation, the first instance of the Ptolemaic model of knowledge. But from Thales’s time to the present day we have forgotten that the shadow was cast, transported by some supporting device, that it itself transported certain information. We have read that first spectral analysis without analyzing its condition. The most important question-which messenger transports (and how?) which message?-was covered over for centuries by the blinding scenography of the shadow-light opposition.
Thales’s story is not unanalogous to that of Desargues: the size of the stones, a perspectival geometry, the theory of shadows.(9) Nor, after all, to Plato’s stories: the sun of the same, the other and empirical object, the cast shadow of the shaded surface, similarity, the cave of representation. Is it a tale of origin? Yes, and in several ways: the origin of a technology, of an optics, of a philosophy of representation. Of a geometry? Perhaps-if geo-metry is that triangulation which Plato scorned for being pre-mathematical. It is a mnemonic recipe, friend of the cultural memory because of its forceful dramatization and mythification under the sun of Ra, easy to transmit within a homogeneous cultural setting,(10) the ruse of applied mathematics, of an architect and of an expert builder. Even Descartes, followed by Desargues and Monge, remains in the domain of applied geometry as well as that of representation; they perpetuate an engineering geometry that is metric and descriptive. They exhibit the archaic forms of pre-mathematics that run through history. Like Thales, they impede the formation of pure mathematics. And the latter will emerge as soon as this geometry dies-very recently. And Husserl will write The Origin of Geometry as the bell tolls its disappearance, as if an immense historical cycle had finally come to an end. Thales’s story tells something like the story of the birth of a geometry, the measured division of the earth and the differences in shadow and light written on the earth by solid figures and the sun; it does not tell of the birth of mathematics. As proof, let us cite Plato, who, in order to bring about this miracle, requires something else: the essential reality of idealities. Question: how can the pyramid be born as an ideal form?
To answer this question, let us return to our spectral analysis. Everything transpires as if Plato had relegated Thales’s story to the depths of his cave. The flat, even wall is always bright: on it the volume casts a shadow; light creates a shaded area. My knowledge is limited to these two shadows; it is only a shadow of knowledge. But there is a third shadow of which the two others provide only an image, or a projection, and which is the secret buried deep within the volume. Now it is probable that true knowledge of the things of this world lies in the solid’s essential shadow, in its opaque and black density, locked forever behind the multiple doors of its edges, besieged only by practice and theory. A wedge can sunder the stones, geometry can divide or duplicate cubes, and the story, indefinitely, will begin again; the solid, whose surfaces cannot be exhausted by analysis, always conserves a kernel of shadow hidden in the shade of its edges. Thales, while reading and noting the volume’s traces, deciphers no secret except that of the impossibility of penetrating the volume’s arcana, in which knowledge has been entombed forever, and from which the infinite history of analytical progress bursts forth as if from a spring. In this case, his history tells the conclusion of a story, that of the confrontation with solid objects, that of the attack on compact volumes, comprehended like theoretical, objective, unconscious elements, like theoretical, objective, indefinite unknowns. In this case the thing exists qua thing, like an unknown and a correlate, like a secret involuted into thousands and thousands of replicas. Two decisions: either I recognize the object by its shadow, which gives rise to geometry or, better yet, to the idealism of representation, or I allow for a kernel of shadow within the object. In the latter case, theory and practice develop this secret infinitely in a perpetually open history, the history of science, which admits that the solid always envelops something that can be rendered explicit. In Plato, the idealism of representation appears repressed in the depths of his cave, and realism is assumed. However, the story begun in the Nile delta will soon be completed by a sudden and incredibly audacious coup d’etat: the radical negation of interior shadows. The Sun of Thales and Ra, the sun whose rays are shut out for an impeccable definition,(11) is reduced to the meager fire of the prisoners of representation. Thales’s theorem, schema of this story, is in the cave’s shadow. Outside, the new sun gives off a transcendent light which pierces things and transmits an all-seeing vision. This is how the marvelous miracle is accomplished: the transparency of volumes, the metaphorical naming of the realism of idealities. From the cave to the world outside, the scenography turns into an ichnography: the shadow of solids played on the plane of representation and defined them by boundaries and partitions now light goes through them and banishes the interior shadow.(12) In place of a planar triangulation of geometry there is now a stereometry of empty forms in the epiphany of diaphanousness. The archaic Thales of mensuration gives way to pure geometry, pure because it is cut through by the intuition of transparency and emptiness. Then and only then can the pyramid be born, the pure tetrahedron, first of the five Platonic bodies. By this miracle the sun is in the pyramid: the site, the source of light, the object, all in the same place.
Beneath this new sun, solids no longer have a shadow or a secret; light passes through them without being interrupted, just as it glides along a straight line or a plane; the world they constitute is thoroughly knowable. One can understand the importance that Plato and his school constantly attribute to the stereometry of volumes.(13) The open history of infinite explicitations is closed by this power move, by this stroke of lightning that rips away the veils of shadow; this history is reoriented toward the transcendency of forms. There is no more specter, or analysis; the three shadows (the one on the shaded area of the surface, the one cast, and the one buried within) are snatched away by the sun of the Good. And, as if to close the circle in all rigor and for the coherence of global history, the Timaeus will constitute the world by means of these five bodies: tile first, the simplest, the tetrahedron in fact, will be fire. Plato has the pure pyramid come into existence beneath the fires of the sun, and from this tetrahedron he has fire born again: a double miracle that fulfills the scriptures, the Egyptian legend, and the initiation of intuition by positioning the source of light within the polyhedron. When the pyramid is itself fire (did its name influence its legend?), the sun passes through it. The entire myth of origin, even that of The Republic, is thus immersed in a vision of fire and dramatizes a solar rite. The new Thales can no longer see any shadow beneath the furnace that pure form and the solar hearth constitute: original conjunction of mathematical stereometry and the mythical element, blinding atmosphere of the first philosophies of intuition. The kernel of knowledge is continually enveloped by myth, and the myth is ceaselessly generated within the theater of representation. Theory, vision; light, fire.(14) We have here a new genesis with four branches where two tributaries are mixed: science and the history of religions. From astronomy to solar mythology.
Nevertheless, this power move is not exactly a revolution. Plato kills the hen that laid the golden eggs: by cutting through the solids he nullifies history; the eternity of transcendency freezes the diachrony and the genealogy of forms. The future of the square and the diagonal is decided as much on the sand where we describe them through the language that names them as it is decided in the sky of ideas. The realism of transparent idealities is still immersed in a philosophy of representation. Of course, ichnography is substituted for scenography, but the former is a trans-representation from a divine point of view. To go beyond Thales’s scene, the shadowless theater is still a theater. The inevitable realism is still an idealism: the geometric form clearly expresses this difficulty. This form is pre-judged to be without shadow or secret, it exists itself and in itself, but it never hides anything that could exceed the definition one has fixed for it. It exists as an ideality, transparent to vision, transparent to noesis. It is a theoretical element known thoroughly, something seen and known without residue. Intuition is blinded by its existence, but intuition passes through it. Its identity guarantees that it is ubiquitously identical, and hence its perception is not interrupted. Vision and knowledge are white specters. Now, precisely when this pure geometry, inherited from Plato, dies, when it is no longer possible to assume intuitive principles, when the theater of representation is closed, the secret, the shadow, and the implication will explode again among these abstract forms before the eyes of dumbfounded mathematicians - explosions that had been announced before all these deaths throughout history. The right angle, the plane, the volume, their intervals and their areas, will be recognized as chaotic, dense, compact-again teeming with folds and dark hiding places. Pure and simple forms are neither that simple nor that pure; they are no longer complete, theoretical knowns, things seen and known without residue, but rather theoretical, objective unknowns infinitely folded into one another, enormous virtualities of noemes, like the stones and the objects of the world, like our stone constructions and our wrought objects. Form hides beneath its form transfinite kernels of knowledge which, one might fear, history will never exhaust; these highly inaccessible instances become our new tasks. Mathematical realism is weighed down and takes on the old density that Plato’s sun had dissolved. Pure and abstract idealities create shaded areas; they are full of shadows; they become again as black as the pyramid. Present-day mathematics, although maximally abstract and pure, is developing in a lexicon that derives in part from technology. It is a new way of listening once more to Thales’s old Egyptian legend.
The solar myth envelops an implicit knowledge. Oral legend dramatizes an implicit schema and concept. The philosophy of vision, of intuition, and of representation includes and acts out an implicit theory. The technology of construction is the kernel of an implicit science. A triple, quadruple tunic whose surroundings present a new problem: what are the relationships of a technique, of a myth, of a communication, and of a philosophy? Again, the idealities implicit in technology, mobilized in representation, dramatized by myth, and transported by a particular language are filled to the brim with an implicit knowledge. The birth of beauty never stops; Harlequin has never donned his last costume. The myth is perpetuated; representation is spread further and further; archaisms resound through the centuries and are ferried to our feet like alluvia. What Thales saw at the base of the pyramids (the sun, the homothetic edifice, the shaded surface and the cast shadow), what Thales did alongside the pyramids (the partitioning off and the measurement of similar triangles in the parallelism of two gnomons, one of which is our body), are the thousands and thousands of implications that the history of science is slowly developing and that the eternal geometers will see, without always seeing them, and will create, without always knowing it. These implications express nothing less than the obscure articulations of rigorous knowledge and the totality of other human activities, indefinitely abandoned to their obscure fate. If by the birth of geometry one means the appearance of an absolute purity on an ocean filled with these shadows, then let us say, a few years after its death, that it was never born.
The history of mathematical sciences, in its global continuity or its sudden fits and starts, slowly resolves the question of origin without ever exhausting it. It is constantly providing an answer to and freeing itself from this question. The tale of inauguration is that interminable discourse that we have untiringly repeated since our own dawn. What is, in fact, an interminable discourse? That which speaks of an absent object, of an object that absents itself, inaccessibly.
(1) See Tannery’s discussion of the tradition that opposes Diogenes and Plutarch, Geometrie grecque (1887; reprint ed., New York: Arno Press, 1976), pp. 88-94; see also the texts that Kirk and Raven have gathered for The Presocratic Philosophers (Cambridge: At the University press, 1962). Cf. P. 253, Speusippus: "For 1 is the point, 2 the line, 3 the triangle and 4 the pyramid. All these are primary, the first principles of individual things of the same class."
(2) Auguste Comte, Philosophie premiere: Cours de philosophic positive, lecons 1 a 45, ed. Michel Serres, Francois Dagognet, and Alial Sinaceur (Paris: Hermann, 1975), Troisieme lecon, pp. 67-68.
(3) Ibid., Onzieme lecon, p. 176.
(4) Proclus, A Commentary on the First Book of Euclid’ Elements, trans. Glenn R. Morrow (Princeton: Princeton University Press, 1970), p. 275.
(5) Herodotus (Histories, 2:109) believes that the Greeks learned the rise of the gnomon and the division of the day into twelve parts from the Babylonians.
(6) See Henri Bergson, Oeuvres, ed. André Robinet (Paris: Presses Universitaires de France, 1959), pp. 51-92. -Ed.
(7) The moments of equality (isomegethes) is a special case. Hieronymus’s lesson, though old, delves further into the application.
(8) Rene Descartes (1596-1650), Gerard Desargues (1591-1661), Gaspar Monge (1746-1818), and Joseph Gergonne (1771-1859) were all instrumental in the development of descriptive and perspective geometry. -Ed.
(9) One of Desargues’s geometrical treatises dealt with methods for cutting stone, and another (which has been lost), sometimes referred to as Lecons de tenebres (Lessons on Shadows), dealt with conic sections. -Ed.
(10) The terms currently used, "formation" and "production," are concepts borrowed in part from biology. Have we really progressed in philosophy since the Hellenistic age, when the same problems had different names? The Greeks called it "Thales," whose proper name is close to words meaning "sprout," "young shoot," "to bloom," "to become green," "to grow," and so on; in other words, to take form or to be produced. The mythical age said mythically -by means of a symbolic system all its own- what the metaphysical age says conceptually, by means of its own symbolic system. Perhaps one should decipher the names of the Seven Wise Men, symbolically equal in number to the planets and to the Roman kings. Perhaps one should write about the beginnings of Greek knowledge using Dumezil’s method.
(11) One can dream, but only dream, that geometry could only have been born on a soil and in a climate where things finally appear exact. Beneath a blinding sun that diffuses its metallic light in a transparent and pure atmosphere, the world is outlined like a definition. The sailor’s horizon is a circumference without undulating lines; the edges of things are precise; their shadow is rigorously delineated; the blue-black sky is homogenous space; and so forth. These conditions, proper to the Greek climate, are necessary, as are so many others - but they are far from being sufficient.
(12) Ichnography is defined as "a horizontal section, as of a building, showing its true dimensions according to a geometric scale; ground plan, map, also, the art of making such plans" (Webster’s) -Ed.
(13) The message of Book VII of the Republic is a message of origin; that the republic comes into light is not surprising (526d). Stereometry is due, in large part, to Theaetetus.
(14) Liddell and Scott favor the etymology pyr ("fire") for the term "pyramid". The Timaeus (56b) associates the two terms, and Plato never says "tetrahedron," a term which came into use starting with Pappus. The Pauly-Wissow encyclopedia (5.V. "Pyramids") has nothing definite to say about the origin of this term.
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