A translation (February, 2019) by Randolph Burks of Origine de la géométrie, 3, from Michel Serres, Hermès V: Le passage du nord-ouest (Paris: Éditions de Minuit, 1980), 165-174.
Origin of Geometry, 3 By Michel Serres The first book of the Elements doesn't open with the classic five postulates and five axioms. It begins with twenty-three definitions: of the point, the line, the angle, and so on. As though it were a question of an ordinary grammar: first morphology, then syntax. Let syntax particularly be retained and we have a system whose rigor and formal purity have caused the admiration of its officiating ministers for almost two millennia. Thus has Euclid been read and reread, with the straight grain and with good reason. I want to consider here rather the definitions in their meanings. Has it been observed that the very first word of the text was σεμεῖου [semeion], sign? Under the said metric, under the unsaid topology, do the Elements imply something concerning meaning? When at the beginning of the century Hilbert reconstructed geometry by means of ideal objects he proposed, having a bit of fun, to call table, glass or bottle, indifferently, he was in fact criticizing what has a meaning or meaning in Euclid. And, in eliminating it, he attained Geometry, the one we are now considering as such. This also said, at least by paralipsis, that it wasn't at all a question in the Elements, not yet or not quite, of geometry. What was it a question of then? By working through the meaning of the definitions, I don't think it's impossible to answer this question. Hilbert's ironic sentence is the end of a long history and the beginning of a new science.1 It brings meaning to zero. If Euclid's geometry is not completely or is not yet pure, abstract or formalized, it's because it drags kernels of unanalyzed meaning in its vocabulary and morphology. We have known this for at least a century, during which our immediate predecessors picked out, for example, facts of topology that were drowned in the metric. They proceeded to a, to some filterings, which produced results become classic today. It is less known that the history of Greek mathematics, before Euclid, has itself functioned as a succession of filters. It didn't content itself with accumulating inventions. The Elements forms a deductive system; it is a historical balance sheet of the results that were known at the date it was written, but it lastly constitutes, in part, the remainder of choice and previous analyses. The Platonic school for example purified the ancient lexicon of geometry, seeking, as Mugler has said, to desensitize it. The Pythagoreans called surface: color; the Meno prefers the term: limit. This is a case of analysis of meaning and rectification of vocabulary. A variety of space is being defined differently than by perception, even if we do think they are passing from sight to touch. The express aim nonetheless remains forming an ideality. Thus Plato didn't like the very term geometry, no doubt because it recalled practices such as surveying. These discussions and analyses aren't confined to the Academy, many are found in Aristotle; they don't stop at Euclid's Elements since Proclus perpetuates them during the final days of the school of Athens. Hilbert marks the end of a history of meaning; Euclid writes at a given moment of its course. Thus I can give myself the right to analyze the meaning of the Euclidean terms while leaving aside deduction, system and syntax, imitating the Greek geometers and philosophers. I will take up again, once more, a forgotten thread of history left by the pure and abstract Geometry in the trash cans into which Hilbert threw his glasses and bottles. 1
It might be helpful to be specific. Hilbert's famous quip [my translation]: “'Tables, chairs, and beer mugs' can be said at any time instead of 'points, straight lines, and planes'.” [All footnotes are the translator's.]
Serres: Origin of Geometry, 3
Let there be, first of all, in the Definitions, two idealities, two objects or two geometric beings, the plane and the trapezium.2 I won't form any hypothesis regarding their reality or mode of existence or, as is said without thinking, status. Here are words. The plane, ἐπίπεδος [epipedos], and the quadrilateral defined as being neither a square nor a rectangle nor a diamond nor a rhomboid, τραπέζιου [trapezion]. In the first case, it is literally a question of what is positioned on the ground, at foot-level on a flat, noninclined terrain. In the second one, of what is supported on four feet, a tetrapod, for example a table. For all the relatively high points of the table, for all the lowest points of what lies on the ground, there is rest; and there is rest all the more so because the plane, or the flat, is introduced before the angle or inclination. What is thus supported or positioned remains stable in any case. The two words thus brought into connection, the two “beings” thus designated are figures of statics. I still know nothing about their mode of existence, but I know their status, which is tautological with statics. Here are stable states. This is confirmed by the use of the verb κεῖται [keitai] in Definitions 4 and 7, the first verb used after the verb ἐστιν [estin] and used before the angle and inclination. In the geometric system of reference it designates situation, like its English translation “to lie” for example. But, like this translation, it is used to say that a thing is lying down, horizontally positioned, stretched out. In any case, it's always a question of rest, of motionlessness, of stable state. A stative verb, in a first system of reference of statics. Suddenly I'm no longer talking about geometry. But you have never talked about it, you will say. For your analysis remains oblique. It has from the outset considered plane and trapezium outside of geometry. So it doesn't talk about Euclid, but the dog, a barking animal, when it's a question of the Dog, Canis Major, a celestial constellation. I will begin again: the Platonic school and the set of Greek filters didn't proceed any differently than the way I am, and didn't open any other way than the one I'm following and which ends at Hilbert. If color diffuses in space, the surface or the plane, if color, an invariant of space, since it never appears without space nor space without it, hampers Eudoxus, Plato and Theaetetus, it's surely due to a tail of meaning which the geometry practiced by them had long forgotten. They eliminate a remainder that's outside the system; they insist on erasing the smear of meaning. Hence my business of ground and table, even if the plane has left the originary earth, even if the process of geometry has turned its back to this meaning from its own dawn. Sometimes, often, words remain fossils. And translation masks the memory of this fossilized state. So I recommence the Platonic operation. It is clear, for example, that the term ἐπιφάνεια [epiphaneia], for “surface,” the sudden appearance in the light, epiphany, is a very recognizable fossil from the Pythagorean times of color. The word “surface” translates both this appearance and this memory poorly. Thus the term “plane” very poorly translates ἐπίπεδος, what's on the ground, and “trapezium” is only a translation: it has forgotten the four feet of its childhood. Let's continue. The term κλίσις [klisis], inclination, used for the definition of the angle, is, as we know, new in Euclid, who takes it up again in Book 11, where stereometry begins. Archimedes, of course, Pappus and Proclus as well, constantly made use of it, but it was unknown to the Greek geometrical tradition from Thales all the way up to and including Aristotle's lexicon. An angle would evoke for this tradition rather a broken line: here, the word would instead be κλάσις [klasis], and the verb κλᾶυ [klan], often used by the vocabulary of optics. Definition 8, in which κλίσις appears, already contains ἐπίπεδος and κειμένων [keimenon]. The proximity of these three words is instructive. 2
To understand what follows, it is very useful to have a Greek-English edition of Euclid's definitions from Book 1.
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Serres: Origin of Geometry, 3 Something tilts or is positioned in divergence from an equilibrium. The balance inclines, lowers and rises at the same time. Proclus certainly read a schema of this type here since he criticized the definition as productive not of one angle, but of two. Statics reappears, but it is accompanied by a beginning of kinematics. For κλίνω [klino], again, designates a support, but also a fall, a stretched-out situation, on a bed or a table. Better, on a triclinium when the Greeks were feasting.3 But, by inclination, it designates a detour, an arrow and already almost a movement. Έπίπεδος, κλίσις, τραπέζιου, successive equilibriums, constructed at increasing levels. The resemblance between κλίσις and κλάσις is of the same order as that which exists between ἐπίπεδος and ἐπιφάνεια. Their difference marks the distance between statics and optics. Plato refuses to adopt the term ἐπιφάνεια, too luminous, too visible, too much of appearance. Euclid refuses κλάσις, for analogous reasons no doubt, since he writes άπτομέυωυ [aptomenon], which belongs to the zone of touch, but in introducing κλίσις for the first time, he acknowledges, without saying as much and perhaps without knowing it, completely different reasons from the order of mechanics. Inclination is not first and foremost an event in space but the rupture of an equilibrium that's already there and the search for a new stability. I lean over and lie down. Κεῖται disappears; σταθεῖσα [statheisa] now appears, the word I was in need of, and its epistemological corollary, ἐφεστηκυῖα [ephesteknia] or ἐφέστηκεν [ephesteken]. Here is the right angle, the metric norm of course but also the schema of equilibrium. Episteme is first equilibrium. Thus the straight can become inclined. The straight, that is to say, εὐθὐς [euthus], εὐθεῖα [eutheia]. Now εὐθὐς, which is the right course, is opposed to πλάγιος [plagios], oblique, to στρογγύλος [stroggulos], the round or rounded, to καμπύλος [kampulos], the curve or curved, to περιφερής [peripheres], that which turns, which rolls, which moves circularly. Not here, in the Euclidean text or words, but in language in general. In other words, here are three forms and three movements: what is straight and goes straight; what tilts and inclines; what is round, which turns in a circle. This order is precisely the order of the set of Definitions. First the straight, straight line and flat plane; then the angle and its inclination in divergence from equilibrium, an angle that can be right, in its turn, but also obtuse or acute according to said divergence; and immediately after, the circle. Let's note in passing that the acute, ὀξεῖα [okseia], signifies very quick and rapid as well and that ἀμβλεῖα [ambleia], the obtuse, is quite close to the verb ἀμβλύνω [ambluno], which sometimes designates the slowing of a given movement. We're simply moving from statics to phoronomy. The movement of rotation appears with the angle or inclination, themselves appearing on the straight course. This result isn't merely obtained by the lateral meanings in the diverse semantic zones, but also with the order and through the very construction of the text. Let's lastly note that, from the introduction of the circle, in the preceding definition and in its own, the word σχῆμα [schema] appears, whose link to ῥυθμόσ [ruthmos] is known in the Democritean lineage as well as the Aristotelian. Of course, περιφερεία [periphereia], the circumference, from which I started, is used here. In Euclid, the circle, that is to say, rhythm, is in some way the first schema. We are returning to equilibrium, or rather we are reaching a new equilibrium, beyond inclination and circular or angular movement. The diameter represents this stability just as much as the center. A new inclination appears with the second plane figure, the triangle or better, the trilateral. Euclid, as we know, defines three of them: the equilateral, the isosceles and the scalene in general. This classification is commonly read by genus and differentia. But what's the situation with meaning again? Ίσοσκελές [isoskeles] literally designates two equal legs. Plato uses this word in the Euthyphro (12d) to 3
Triclinium=lit de table, which literally reads as table bed.
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Serres: Origin of Geometry, 3 say an even number; rhetoric repeats it for a discourse with equal or equilibrated parts, this is the periodic sentence. Rhythm again. But σκέλος [skelos] shows the leg. In the same Platonic place, σκαλενός [skalenos] says the odd, but in general it designates something or someone who limps. Proclus links it to σκολιός [skolios], oblique or winding, and σκάζω [skazo], limping, being unequal. Consequently, statics returns, the scalene tilts, the isosceles recovers lost equilibrium, and lost in a movement, that of walking. We should note in this connection that γωνία [genia], angle, whether acute, right or obtuse, thanks to which we can class triangles into right triangles and other ones, designates a corner, the pillar of a bridge, but is especially related to γόνυ [genu], the knee. I'll finish, in part, with where I began, with one of the quadrilaterals. The most interesting thing here, I think, is not the trapezium or the tetrapod table, balanced in every instance, but the rhombus and the rhomboid. For the term ρόμβος [rombos] derives from ρὲμβω [rembo], turning or rather spinning round, like a whirlpool. And ρόμβος is a top, or any object with a circular form that can turn round an axis. Archimedes of course gives a stereometric follow-up to this figure and calls two cones with a common circular base and vertices opposed along the same axis a solid rhombus. Here is the top, and here is, in Euclid, a planar top. We know, from a well-known passage in the Republic (IV, 436d), that the top's spinning had posed the difficult problem of motion and rest to Plato. For it is stable in turning, a contradiction. He escapes from this by affirming that it is at rest in respect to the straight and in movement in respect to the round, something true only on condition of ignoring that the axis becomes all the more stable the more quickly it turns. The theorem was of course not known to the Greek engineers, but the fact has never been unknown, I suppose, to children themselves. They play with the contradiction that delays the philosopher. They enjoy rest in and through circular motion. Hence one can enjoy what causes fear, to return to Plato's text. The top is not a bad pharmakon, poison and cure. The top constructs a contradiction. And Euclid's Definitions construct it in turn, more childlike than the Republic. You Greeks, you'll always be children, says the old Egyptian priest of the Timaeus. Children of science and geometry, freeing themselves from repetitive traditions. In short, the Definitions end, or almost, with two cases of figures, trapezium and rhombus, in which equilibrium is at stake: either on a high place, or without base, on a single point and through a motion – refined, complex, difficult and sophisticated cases. In a way, everything moves toward the rhombus: the point on which it's positioned, the acute point (the ancient στιγμή [stigma], the ὀξεῖα of the angle, the needle or spur of the κέντρον [kentron]) on which it is supported, the angle formed by this point, the circle described by the top in rotation, the double triangle visible as stable in motion, and the quadrilateral plane called diamond. Nothing is straight, everything is straight; nothing is stable, everything is stable. The text constructs the whirling rhombus piece by piece; in brief, it assembles the whirlpool, before the major drawing that is the bundle of parallel lines that never meet however long they continue in both directions. It looks, once again, like the model I brought to light in Lucretius' physics, the model mathematized in the Archimedean system: turbulence and cataract. The Democritean tradition is legible here just as much as the one that goes back to Plato. Everything happens, consequently, as though the Definitions constructed, term by term, case by case, and parts by parts, equilibriums, stabilities that are increasingly complex starting from the simplest ones. From the lowest low point, from what's positioned right on the ground to either the high point or the most refined, the most difficult case, exactly the contradictory case, through successive disruptions of previous equilibriums and through access to new stabilities: inclination, movement, rotation, the unequal gait of the lame (two feet, four feet, a single foot...), lastly all these ruptures at the same time. This is no longer the announcement of a Geometry. These are prolegomena for a Mechanics. Lagrange, it seems, appears in Euclid. Analytical Mechanics seems to emerge from the Elements, the idea that 4
Serres: Origin of Geometry, 3 statics dominates phoronomy, and almost the principle of virtual velocities. We at least understand why Euclidean space has always seemed to be the familiar space of our ordinary technologies more than an abstract, formal and pure space. It's because it is already or still a Lagrangian or Archimedean space, in short, a space of statics. The space of the ground, right-angled walls, tables, supports and doors. No, Euclidean geometry was neither pure nor abstract, and Hilbert is right, and Klein before him even more so: it was still an applied mathematics. The group of movements is still tied to practical adhesions. It couldn't be any other way. Here is the major monument of Greek science, its exemplary achievement. Yet this science, ὲπιστήμη [episteme], in its meaning and its project, remains a knowledge of equilibrium; this comprehensive word tells us so. Euclid repeats it with ἐφεστηκυῖα or ἐφέστηκεν. Science as such, in its definition, is inscribed in the Definitions. The monument, on its facade, bears its inscription. This knowledge of stabilities endures from the Greeks to Lagrange and no doubt beyond, through positivism, and after it this homonymous science of equilibrium comes all the way to us, all the way to recent times when knowledge is becoming rather one of divergences from this equilibrium. Western science is the science of stabilities; the Euclidean system doesn't hide this. The latter is a syntactical system and a system for semantics. Let's consider a ball whose contour is poorly defined, whose border or periphery is not precise. A ball whose ὄρος [horos] or πέρας [peras] is not well cut out at the outset. The general question of definition can be depicted by this form, which can be drawn in a space as the semantic zone of a word. In common language, this zone has fluctuating edges. Let's mark a little closed ball, for example a point, in the ball; it suffices if this point is inside. Let's thus consider two, three, etc., several balls, and respectively as many points marked in their interiors. From points to points, let's trace as many lines as it's possible to trace. Here in all is a connected network. The relations between the points determine the points inside the semantic zones, and reciprocally the points inside the zones determine the relations. This double determination in practice stabilizes and resolves the problem of definition. The Definitions form a well-connected network that can be constructed and drawn. Let's lastly observe that, in order to construct this network, we only had need of three words present in the text itself: ὄρος or πέρας, boundary, σημεῖον, point, and γραμμή [gramme], line. We will return to these three words. The method used up to now consists in choosing a ball and moving in its zone starting from the point marked by the text. The essential thing of course is that we never near the fluctuating border, much less go beyond it. Assume then this movement, which can crudely be called a change of sense.4 It adopts, in the zone, a certain direction, a certain sense. Question: in how many balls can this movement be carried out, on condition that it's the same, in the same direction and in the same sense? Answer: in only a subset. For it is impossible in the zone of ἐπιφάνεια, of ἐτερόμηκες [eteromekes] or of παράλληλοι [paralleloi], for example, to locate a point that can be referred to movement or rest. And if it were possible to do so for the whole of the network of the Definitions, these latter would be ambiguous. Statics would have always been read in Geometry. And so, if one links the new points of the subset in the same way as before, a subnetwork is obtained. The subnetwork highlighted up to now has been that of mechanics – of Lagrange or Archimedes immersed in Euclid in order to clarify ideas. Yet this subnetwork is constructed on common language. And it is reconstructed in such a way that we are sure, from the successive filterings carried out by the Platonic school for example on the vocabulary of 4
Sense=sens, which can mean meaning or direction. I will translate this word as “sense” in this passage, appealing to the less common directional meaning of the term as well as to the semantic one.
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Serres: Origin of Geometry, 3 geometry, that it would have been reduced or eliminated by local change of the lexicon if it had run counter to the practice of the Greek geometers. Thus Euclid as well substitutes κλίσις for κλάσις in the interest of erasing all reference to either optics or the visible, in such a way that ἐπιφάνεια is a fossil or remainder of this evolution. Yet the mechanical subnetwork has remained present, has not been subject to a filtering, has been preserved. Why? The reason was given in part when word ὲπιστήμη was invoked. Inscribed in its own term, the global idea of science is the idea of equilibrium. The lexicon recreates this idea. But it doesn't let it be seen directly. In a certain way, this lexicon conceals it. Beneath the definition of abstract identities, this lexicon conceals a schema, one which is perhaps Democritean, since it could be read again in Lucretius's physics, which is tied to the Greek idea of science. Consequently, as for a painting, the original can be read beneath the repaints. Consequently, we are perhaps holding here something having to do with the origin of geometry, the considerable residue of a very old filtering. The origin question is easier to resolve in language itself than by history or worse, metaphysics. Yet, the subnetwork of mechanics, from equilibrium to the whirling top, covers a large extent of the global network. Can this operation be iterated and subnetworks discovered having less extent than the above and hence perhaps more buried? Are there repaints that conceal from us more than mechanics? We would have to translate “trapezium” by banking table or money-changing table, after having translated it merely as table, and to construct the associated subnetwork. Could an anthropology be read at the second level of this palimpsest? I think so. Other paths lead to a similar result.
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