Arctinus of Miletus, The Aethiopis Fragment 1 (from Proclus Chrestomathia 2) :

"The Iliad of Homer, is followed in turn by the five books of the Aithiopis, the work of Arktinos of Miletos. Their contents are as follows. The Amazon Penthesileia, the daughter of Ares and of Thrakian race, comes to aid the Trojans, and after showing great prowess, is killed by Akhilleus and buried by the Trojans. Akhilleus then slays Thersites for abusing and reviling him for his supposed love for Penthesileia. As a result a dispute arises amongst the Akhaians over the killing of Thersites, and Akhilleus sails to Lesbos and after sacrificing to Apollon, Artemis, and Leto, is purified by Odysseus from bloodshed."

Thersites was ugly inside and out, and the son of Agrius.

Ctesibius Alexandrinus was the first who found out the properties of the wind, and of pneumatic power, the origin of which inventions is worthy of being known. Ctesibius, whose father was a barber, was born at Alexandria. Endowed with extraordinary talent and industry, he acquired great reputation by his taste for his mechanical contrivances. Marcus Vitruvius Pollio, de Architectura

Ctesibius or Ktesibios of Alexandria, Egypt, was a Greek physicist and inventor around (285 - 222) BC. His lost work On pneumatics still earns him the title of father of pneumatics, for the first treatises on the science of compressed air and its uses in pumps and even a cannon, are his. Like all his other works, however, this work has not survived. His Memorabilia, a compilation of his research, cited by Athenaeus, is also lost.

Ctesibius was probably the first head of the Museum of Alexandria. Unfortunately, very little is known of his life and work. His work is chronicled in Vitruvius, Athenaeus, Philo of Byzantium who repeatedly mentions him, adding, with an almost audible sigh, that the first mechanicians had the advantage of being under kings who loved fame and supported the arts. Proclus, the commentator on Euclid and Hero of Byzantium the last of the engineers of antiquity also mention his, though nothing of his has survived.

Ctesibius was the son of a barber from Aspondia in Alexandria. He also started his life as a barber and his first invention was a counter-weighted mirror. The mirror was on one end of a pole and a lead weight, weighing the same as the mirror, was on the other. This acted as a counterbalance and enabled the mirror to adjust to the height of different customers. Ctesibius had the lead counter-balance weight running inside a tube and noticed that the weight sometimes made a whistling noise as the air escaped when the weight moved. This effect provoked his curiosity about both the powers of air, and musical instruments. When he was small, he dropped a lead ball in a tube and the air escaped with a loud sound because the ball compressed the air. Because of this finding, Ctesibius realized that air was also a substance, so his inventions were based on this fact.

Ctesibius is known for his improvement of the waterclock. This invention is similar to the mechanism used in the 20th-century flush toilet. The floating valve is the predecessor of the floating ball in the upper chamber of the toilet. After a flush, the floating ball sinks with the declining water level, pulling open the water valve with its metal arm. The incoming water fills the vessel again, raising the ball so that its arm closes the flow of water at the precise level of "full..

Mankind was a long time in developing adequate measures for fighting fire. The early Greeks and Romans used a device called the Ctesibius pump to create a stream of water. The idea was lost, ironically, in the burning of Alexandria, Ctesibius home city. This technology was forgotten, however, during the Middle Ages. By the time the colonists arrived in America, townfolk still depended on the bucket brigade; two lines of people connecting the water source and the fire, one passing full buckets, the other returning the empties. Wet blankets protected threatened structures. By the eighteenth century the principle behind the Ctesibius pump had been rediscovered and hand-operated fire engines began making their appearance. As better water systems were developed, higher water pressure became possible. Heavy riveted hoses were developed to handle the increase. The improvement in hoses allowed the firemen to take the water directly from the water plug to the source of the fire. A standard fixture of the early volunteer department was the hand-drawn hose cart consisting of a large reel mounted on two wheels.

The hydraulis ( Ύδραυλις )

About B.C. 284 to 246 there lived at Alexandria, under Ptolemy Euergetes, a man named Ctesibius, who followed his father's trade of barber. Being of a mechanical turn of mind, he observed that the counterweight of a movable mirror, used for the purposes of his trade, produced a musical sound by the force with which it drove the air out of the tube in which it moved. Experimenting with the principle thus noticed, he succeeded in making a machine consisting of a hollow vase inverted, with an opening on the top, to which was attached a trumpet, and, on water being pumped into the vase the air was driven forcibly through the trumpet, producing a very powerful sound; and the machine caused so much admiration that it was consecrated in the temple of Venus. Charles Francis Abdy Williams The Story of the Organ, 1903

Ctesibius invented a water organ, which was an air pump with valves on the bottom, a tank of water in between them and a row of pipes on top.

The hydraulis was the first keyboard musical instrument and the ancestor of the modern church organ. It utilized a large chamber partly filled with water. Ctesibius evidently took the idea of his organ from the Syrinx or Pandean pipes, a musical instrument of the highest antiquity among the Greeks. His object being to employ a row of pipes of great size, and capable of emitting the most powerful as well as the softest sounds, he contrived the means of adapting keys with levers (agkoniskoi), and with perforated sliders (pomata) to open and shut the mouths of the pipes (glossokoma), a supply of wind being obtained, without intermission, by bellows, in which the pressure of water performed the same part which is fulfilled in the modern organ by a weight. The instrument was called the water-organ (hydraulis). The wide mouth of a funnel-like extension from the wind chest was set in the top of the water; as air pressure in the wind chest fell, water rose in the funnel and compressed the air, thus keeping the air pressure constant. The hydraulis was used for public entertainments in ancient Rome and Byzantium. Nero the mad Roman Emperor is said to used a hydraulis. Bellows-type organs were also known to the ancient world. This was the organ that reappeared in Europe in the 8th and 9th centuries, imported from Byzantium and from Arabs who had discovered ancient Greek treatises.

A Reconstructed Hydraulis

The Hydraulis of Dion,

The Hydraulis of Dion, Dion Archaeological Museum

In 1992 Greek archaeologists recovered a fragmentary hydraulis with 19 bronze tubes dating from the 1st Century B.C. at the Greek city of Dion, at the foot of Mt. Olympus. Based on this example and documentary evidence, the European Cultural Centre of Delphi finished reconstructing the instrument in 1999.

...The writings speak of water-blown pipes, which were used to simulate the singing of birds and the sound of a trumpet blown by a statue of Memnon in Thebes. This statue has been called one of the world's wonders. When the sun shone upon the statue from a certain angle (particular time of day), an awe-inspiring sound was emitted, according to Tacitus, Pausanius and other writers. They compared the sound, produced by two organ pipes, to the sound of the bursting strings of a lyre or harp. The sun's rays fell onto a sealed tank, which was partially filled with water. When the water was heated and had expanded sufficiently, it was forced through a siphon into a second tank. The air, which was displaced from the second tank, blew the two pipes. During the cooler night, a vacuum was created in the first tank causing water to be drawn in from a reservoir thus making the instrument ready for the next day. The tank was shielded from the sun so that the sun's energy would warm it only at a very specific time of day. In another Greek example, a pipe or whistle was blown to imitate the chirping of a bird. An artificial bird was placed on top of an artificial tree on top of a mechanism similar to the one described above. The warbling of the bird was imitated by the inversion of the sounding pipe into a tank filled with water. This instrument was not solar-powered, and had to be activated by turning on a tap. It could only play until all of the water had flowed from the first tank into the second one. Aristokles (second-century BC) speaks of an instrument he calls the organon referring to a water organ, which made figures play wind, string and percussion instruments. Joseph R. Curtis, The Water Organ and Other Related Sound-Producing Automata

The Story of the Organ

Image: Salpinx and Hydraulis Player (1st century BC)

THE ORGAN IN CLASSICAL LITERATURE

Pump of Ctesibius

By Ctesibius around 250 BC.

It is now necessary to explain the machine of Ctesibius, which raises water to a height. It is made of brass, and at the bottom are two buckets near each other, having pipes annexed in the shape of a fork, which meet at a basin in the middle. In the basin are valves nicely fitted to the apertures of the pipes, which, closing the holes, prevent the return of the liquid which has been forced into the basin by the pressure of the air. Above the basin is a cover like an inverted funnel, fitted and fastened to it with a rivet, that the force of the water may not blow it off. On this a pipe, called a trumpet, is fixed upright. Below the lower orifices of the pipes the buckets are furnished with valves over the holes in their bottoms. Pistons made round and smooth, and well oiled, are now fastened to the buckets, and worked from above with bars and levers, which, by their alternate action, frequently repeated, press the air in the pipes, and the water being prevented from returning by the closing of the valves, is forced and conducted into the basin through the mouths of the pipes; whence the force of the air, which presses it against the cover, drives it upwards through the pipe: thus water on a lower level may be raised to a reservoir, for the supply of fountains.

Book 10, Vitruvius

Renan had the best reasons in the world for calling the advent of mathematics in Greece a miracle. The construction of geometric idealities or the establishment of the first proofs were, after all, very improbable events. If we could form some idea of what took place around Thales and Pythagoras, we would advance a bit in philosophy. The beginnings of modern science in the Renaissance are much less difficult to understand; this was, all things considered, only a reprise. Bearing witness to this Greek miracle, we have at our disposal two groups of texts. First, the mathematical corpus itself, as it exists in the Elements of Euclid, or elsewhere, treatises made up of fragments. On the other hand, doxography, the scattered histories in the manner of Diogenes Laertius, Plutarch, or Athenaeus, several remarks of Aristotle, or the notes of commentators such as Proclus or Simplicius. It is an understatement to say that we are dealing here with two groups of texts; we are in fact dealing with two languages. Now, to ask the question of the Greek beginning of geometry is precisely to ask how one passed from one language to another, from one type of writing to another, from the language reputed to be natural and its alphabetic notation to the rigorous and systematic language of numbers, measures, axioms, and formal arguments. What we have left of all this history presents nothing but two languages as such, narratives or legends and proofs or figures, words and formulas. Thus it is as if we were confronted by two parallel lines which, as is well known, never meet. The origin constantly recedes, inaccessible, irretrievable. The problem is open.

I have tried to resolve this question three times. First, by immersing it in the technology of communications. When two speakers have a dialogue or a dispute, the channel that connects them must be drawn by a diagram with four poles, a complete square equipped with its two diagonals. However loud or irreconcilable their quarrel, however calm or tranquil their agreement, they are linked, in fact, twice: they need, first of all, a certain intersection of their repertoires, without which they would remain strangers; they then band together against the noise which blocks the communication channel. These two conditions are necessary to the diaIogue, though not sufficient. Consequently, the two speakers have a common interest in excluding a third man and including a fourth, both of whom are prosopopoeias of the,powers of noise or of the instance of intersection.(1)Now this schema functions in exactly this manner in Plato's Dialogues, as can easily be shown, through the play of people and their naming, their resemblances and differences, their mimetic preoccupations and the dynamics of their violence. Now then, and above all, the mathematical sites, from the Meno through the Timaeus, by way of the Statesman and others, are all reducible geometrically to this diagram. Whence the origin appears, we pass from one language to another, the language said to be natural presupposes a dialectical schema, and this schema, drawn or written in the sand, as such, is the first of the geometric idealities. Mathematics presents itself as a successful dialogue or a communication which rigorously dominates its repertoire and is maximally purged of noise. Of course, it is not that simple. The irrational and the unspeakable lie in the details; listening always requires collating; there is always a leftover or a residue, indefinitely. But then, the schema remains open, and history possible. The philosophy of Plato, in its presentation and its models, is therefore inaugural, or better yet, it seizes the inaugural moment.

To be retained from this first attempt at an explanation are the expulsions and the purge. Why the parricide of old father Parmenides, who had to formulate, for the first time, the principle of contradiction. To be noted here again is how two speakers, irreconcilable adversaries, find themselves forced to turn together against the same third man for the dialogue to remain possible, for the elementary link of human relationships to be possible, for geometry to become possible. Be quiet, don't make any noise, put your head back in the sand, go away or die. Strange diagonal which was thought to be so pure, and which is agonal and which remains an agony.

The second attempt contemplates Thales at the foot of the Pyramids, in the light of the sun. It involves several geneses, one of which is ritual. But I had not taken into account the fact that the Pyramids are also tombs, that beneath the theorem of Thales, a corpse was buried, hidden. The space in which the geometer intervenes is the space of similarities: he is there, evident, next to three tombs of the same form and of anotherdimension -the tombs are imitating one another. And it is the pure space of geometry, that of the group of similarities which appeared with Thales. The result is that the theorem and its immersion in Egyptian legend says, without saying it, that there lies beneath the mimetic operator, constructed concretely and represented theoretically, a hidden royal corpse. I had seen the sacred above, in the sun of Ra and in the Platonic epiphany, where the sun that had come in the ideality of stereometric volume finally assured its diaphaneity; I had not seen it below, hidden beneath the tombstone, in the incestuous cadaver. But let us stay in Egypt for a while.

The third attempt consists in noting the double writing of geometry. Using figures, schemas, and diagrams. Using letters, words, and sentences of the system, organized by their own semantics and syntax. Leibniz had already observed this double system of writing, consecrated by Descartes and by the Pythagoreans, a double system which represents itself and expresses itself one by the other. He sometimes liked, as did many others, to privilege the intuition, clairvoyant or blind, required by the first [diagrams] over the deductions produced by the second [words]. There are, as is well known, or as usual, two schools of thought on the subject. It happens that they trade their power throughout the course of history. It also happens that the schema contains more information than several lines of writing, that these lines of writing lay out indefinitely what we draw from the schema, as from a well or a cornucopia. Ancient algebra writes, drawing out line by line what the figure of ancient geometry dictates to it, what that figure contains in one stroke. The process never stopped; we are still talking about the square or about the diagonal. We cannot even be certain that history is not precisely that.

Now, many histories report that the Greeks crossed the sea to educate themselves in Egypt. Democritus says it; it is said of Thales; Plato writes it in theTimaeus. There were even, as usual, two schools at odds over the question. One held the Greeks to be the teachers of geometry; the other, the Egyptian priests. This dispute caused them to lose sight of the essential: that the Egyptians wrote in ideograms and the Greeks used an alphabet. Communication between the two cultures can be thought of in terms of the relation between these two scriptive systems (signaletiques). Now, this relation is precisely the same as the one in geometry which separates and unites figures and diagrams on the one hand, algebraic writing on the other. Are the square, the triangle, the circle, and the other figures all that remains of hieroglyphics in Greece? As far as I know, they are ideograms. Whence the solution: the historical relation of Greece to Egypt is thinkable in terms of the relation of an alphabet to a set of ideograms, and since geometry could not exist without writing, mathematics being written rather than spoken, this relation is brought back into geometry as an operation using a double system of writing. There we have an easy passage between the natural language and the new language, a passage which can be carried out on the multiple condition that we take into consideration two different languages, two different writing systems and their common ties. And this resolves in tum the historical question: the brutal stoppage of geometry in Egypt, its freezing, its crystallization into fixed ideograms, and the irrepressible development, in Greece as well as in our culture, of the new language, that inexhaustible discourse of mathematics and rigor which is the very history of that culture. The inaugural relation of the geometric ideogram to the alphabet, words, and sentences opens onto a limitless path.

This third solution blots out a portion of the texts. The old Egyptian priest, in the Timaeus, compares the knowledge of the Greeks when they were children to the time-wom science of his own culture. He evokes, in order to compare them, floods, fires, celestial fire, catastrophes. Absent from the solution are the priest, history, either mythical or real, in space and time, the violence of the elements which hides the origin and which, as the Timaeus clearly says, always hides that origin. Except, precisely, from the priest, who knows the secret of this violence. The sun of Ra is replaced by Phaethon, and mystical contemplation by the catastrophe of deviation.

We must start over -go back to those parallel lines that never meet. On the one hand, histories, legends, and doxographies, composed in natural language. On the other, a whole corpus, written in mathematical signs and symbols by geometers, by arithmeticians. We are therefore not concerned with merely linking two sets of texts; we must try to glue, two languages back together again. The question always arose in the space of the relation between experience and the abstract, the senses and purity. Try to figure out the status of the pure, which is impure when history changes. No. Can you imagine (that there exists) a Rosetta Stone with some legends written on one side, with a theorem written on the other side? Here no language is unknown or undecipherable, no side of the stone causes problems; what is in question is the edge common to the two sides, their common border; what is in question is the stone itself.

Legends. Somebody or other who conceived some new solution sacrificed an ox, a bull. The famous problem of the duplication of the cube arises regarding the stone of an altar at Delos. Thales, at the Pyramids, is on the threshold of the sacred. We are not yet, perhaps, at the origins. But, surely, what separates the Greeks from their possible predecessors, Egyptians or Babylonians, is the establishment of a proof. Now, the first proof we know of is the apagogic proof on the irrationality of .

And so, legends, once again. Euclid's Elements, Book X, first scholium. It was a Pythagorean who proved, for the first time, the so-called irrationality [of numbers]. Perhaps his name was Hippasus of Metapontum. Perhaps the sect had sworn an oath to divulge nothing. Well, Hippasus of Metapontum spoke. Perhaps he was expelled. In any case, it seems certain that he died in a shipwreck. The anonymous scholiast continues: "The authors of this legend wanted to speak through allegory. Everything that is irrational and deprived of form must remain hidden, that is what they were trying to say. That if any soul wishes to penetrate this secret region and leave it open, then it will be engulfed in the sea of becoming, it will drown in its restless currents."

Legends and allegories and, now, history. For we read a significant event on three levels. We read it in the scholia, commentaries, narratives. We read it in philosophical texts. We read it in the theorems of geometry. The event is the crisis, the famous crisis of irrational numbers. Owing to this crisis, mathematics, at a point exceedingly close to its origin, came very close to dying. In the aftermath of this crisis, Platonism had to be recast. The crisis touched the logos. If logos means proportion, measured relation, the irrational or alogon is the impossibility of measuring. If logos means discourse, the alogon prohibits speaking. Thus exactitude crumbles, reason is mute.

Hippasus of Metapontum, or another, dies of this crisis, that is the legend and its allegorical cover in the scholium of the Elements. Parmenides, the father, dies of this crisis-this is the philosophical sacrifice perpetrated by Plato. But, once again, history: Plato portrays Theaetetus dying upon returning from the the battle of Corinth (369), Theaetetus, the founder, precisely, of the theory of irrational numbers as it is recapitulated in Book X of Euclid. The crisis read three times renders the reading of a triple death: the legendary death of Hippasus, the philosophical parricide of Parmenides, the historical death of Theaetetus. One crisis, three texts, one victim, three narratives. Now, on the other side of the stone, on the other face and in another language, we have the crisis and the possible death of mathematics in itself.

Given then a proof to explicate as one would a text. And, first of all, the proof, doubtless the oldest in history, the one which Aristotle will call reduction to the absurd. Given a square whose side AB = b, whose diagonal AC = a:

We wish to measure AC in terms of AB. If this is possible, it is because the two lengths are mutually commensurable. We can then write AC/AB = a/b. It is assumed that a/bis reduced to its simplest form, so that the integers a and b are mutually prime. Now, by the Pythagorean theorem: a² = 2b². Therefore a² is even, therefore a is even. And if a and b are mutually prime, b is an odd number. If a is even, we may posit: a = 2c. Consequently, a² = 4c². Consequently 2b² = 4c², that is, b² = 2c². Thus, b is an even number.

The situation is intolerable, the number b is at the same time even and odd, which, of course, is impossible. Therefore it is impossible to measure the diagonal in terms of the side. They are mutually incommensurable. I repeat, if logos is the proportional, here a/b or 1/, the alogon is the incommensurable. If logos is discourse or speech, you can no longer say anything about the diagonal and is irrational. It is impossible to decide whether b is even or odd. Let us draw up the list of the notions used here. What does it mean for two lengths to be mutually commensurable? It means that they have common aliquot parts. There exists, or one could make, a ruler, divided into units, in relation to which these two lengths may, in turn, be divided into parts. In other words, they are other when they are alone together, face to face, but they are same, or just about, in relation to a third term, the unit of measurement taken as reference. The situation is interesting, and it is well known: two irreducibly different entities are reduced to similarity through an exterior point of view. It is fortunate (or necessary) here that the term measure has, traditionally, at least two meanings, the geometric or metrological one and the meaning of non-disproportion, of serenity, of nonviolence, of peace. These two meanings derive from a similar situation, an identical operation. Socrates objects to the violent crisis of Callicles with the famous remark: you are ignorant of geometry. The Royal Weaver of the Statesman is the bearer of a supreme science: superior metrology, of which we will have occasion to speak again. What does it mean for two numbers to be mutually prime? It means that they are radically different, that they have no common factor besides one. We thereby ascertain the first situation, their total otherness, unless we take the unit of measurement into account. What is the Pythagorean theorem? It is the fundamental theorem of measurement in the space of similarities. For it is invariant by variation of the coefficients of the squares, by variation of the forms constructed on the hypotenuse and the two sides of the triangle. And the space of similarities is that space where things can be of the same form and ofanother size. It is the space of models and of imitations. The theorem of Pythagoras founds measurement on the representative space of imitation. Pythagoras sacrifices an ox there, repeats once again the legendary text. What, now, is evenness? And what is oddness? The English terms reduce to a word the long Greek discourses: even means equal, united, flat, same; odd means bizarre, unmatched, extra, left over, unequal, in short, other. To characterize a number by the absurdity that it is at the same time even and odd is to say that it is at the same time same and other.

Conceptually, the apagogic theorem or proof does nothing but play variations on the notion of same and other, using measurement and commensurability, using the fact of two numbers being- mutually prime, using the Pythagorean theorem, using evenness and oddness. It is a rigorous proof, and the first in history, based onmimesis. It says something very simple: supposing mimesis, it is reducible to the absurd. Thus the crisis of irrational numbers overturns Pythagorean arithmetic and early Platonism.

Hippasus revealed this, he dies of it -end of the first act.

It must be said today that this was said more than two millennia ago. Why go on playing a game that has been decided? For it is as plain as a thousand suns that if the diagonal or are incommensurable or irrational, they can still be constructed on the square, that the mode of their geometric existence is not different from that of the side. Even the young slave of the Meno, who is ignorant, will know how, will be able, to construct it. In the same way, children know how to spin tops which the Republicanalyzes as being stable and mobile at the same time. How is it then that reason can take facts that the most ignorant children know how to establish and construct, and can demonstate them to be irrational? There must be a reason for this irrationality itself.

In other words, we are demonstrating the absurdity of the irrational. We reduce it to the contradictory or to the undecidable. Yet, it exists; we cannot do anything about it. The top spins, even if we demonstrate that, for impregnable reasons, it is, undecidably, both mobile and fixed. That's the way it is. Therefore, all of the theory which precedes and founds the proof must be reviewed, transformed. It is not reason that governs, it is the obstacle. What becomes absurd is not what we have proven to be absurd, it is the theory on which the proof depends. Here we have the very ordinary movement of science: once it reaches a dead-end of this kind, it immediately transforms its presuppositions.

Translation: mimesis is reducible to contradiction or to the undecidable. Yet it exists; we cannot do anything about it. It spins. It works, as they say. That's the way it is. It can always be shown that we can neither speak nor walk, or that Achilles will never catch up with the tortoise. Yet, we do speak, we do walk, the fleet-footed Achilles does pass the tortoise. That's the way it is. Therefore, all of the theory which precedes must be transformed. What becomes absurd is not what we have proven to be absurd, it is the theory as a whole on which the proof depends.

Whence the (hi)story which follows. Theodorus continues along the legendary path of Hippasus. He multiplies the proofs of irrationality. He goes up to . There are a lot of these absurdities, there are as many of them as you want. We even know that there are many more of them than there are of rational relations. Whereupon Theaetetus takes up the archaic Pythagoreanism again and gives a general theory which grounds, in a new reason, the facts of irrationality. Book X of the Elements can now be written. The crisis ends, mathematics recovers an order, Theaetetus dies, here ends this story, a technical one in the language of the system, a historical one in the everyday language that relates the battle of Corinth. Plato recasts his philosophy, father Parmenides is sacrificed during the parricide on the altar of the principle of contradiction; for surely the Same must be Other, after a fashion. Thus, Royalty is founded. The Royal Weaver combines in an ordered web rational proportions and the irrationals; gone is the crisis of the reversal, gone is the technology of the dichotomy, founded on the square, on the iteration of the diagonal. Society, finally, is in order. This dialogue is fatally entitled, not Geometry, but the Statesman.

The Rosetta Stone is constructed. Suppose it is to be read on all of its sides. In the language of legend, in that of history, that of mathematics, that of philosophy. The message that it delivers passes from language to language. The crisis is at stake. This crisis is sacrificial. A series of deaths accompanies its translations into the languages considered. Following these sacrifices, order reappears: in mathematics, in philosophy, in history, in political society. The schema of Rene Girard allows us not only to show the isomorphism of these languages, but also, and especially, their link, how they fit together. For it is not enough to narrate, the operators of this movement must be made to appear. Now these operators, all constructed on the pair Same-Other, are seen, deployed in their rigor, throughout the very first geometric proof. just as the square equipped with its diagonal appeared, in my first solution, as the thematized object of the complete intersubjective relation, formation of the ideality as such, so the rigorous proof appears as such, manipulating all the operators of mimesis, namely, the internal dynamics of the schema proposed by Girard. The origin of geometry is immersed in sacrifical history and the two parallel lines are henceforth in connection. Legend, myth, history, philosophy, and pure science have common borders over which a unitary schema builds bridges.

Metapontum and geometer, he was the Pontifex, the Royal Weaver. His violent death in the storm, the death of Theaetetus in the violence of combat, the death of father Parmenides, all these deaths are murders. The irrational is mimetic. The stone which we have read was the stone of the altar at Delos. And geometry begins in violence and in the sacred.

Geometry was documented by the Sumerian civilization around 5,000 years ago; it was used by them in setting out temples (look up Gilgamesh). The Egyptians also studied and applied a sophisticated knowledge of geometry. In building Stonehenge, around 2,500 BC Neolithic surveyors also used survey methods based on geometric principals.
Geometry started with the first masons. They studied a camp fire while discussing the question of how they had managed to master fire and why where they here(existance)?They decide it was to impersonal to ask what so they decided on whom was the creator. and the natural order would logically be 1 the creator 2 the woman or vessel to make life and 3 the male to impregnate. (note 2+3 =5 the numbers used to make the metric system) They saw the flame and could see the shape (a pyramid). one constructed a model of this shape and experimented with it and found that when the legs where even and the joining lash hung in the centre it would always find the same centre when struck. this was the first ever level. The next thing he became aware of was it could make a circle when tapped clockwise or anti clockwise and it would evenly cross its path on the way to the centre from this he devised with the help of two sticks joined in the centre he could scribe a circle and divide it into equal parts of varying numbers/sections. This was the first set of dividers. This group were the first scientists and mathematicians and being smarter then their peers they decided to keep the secrets to themselves and formed a hierarchy and devised to build a monument to their discoveries a pyramid. The hierarchy built a system and devised through architecture a way to build it. the first ever pyramid. This group still exist to day and still keeps profiting at the top of the chain of command. Most early science break throughs where by masons. Emerson was a mason , he could only have discovered how to make a light bulb work when he understood the world, the element could only live when it was in a controlled atmosphere like us on the planet. This group has perpetuated greed through the centuries and has forgotten the balance of the elements that formed the world. i have recently discovered by mathematics that moving iron and gold from one point/location (Western Australia) in the world around the surface of the globe it will have a slowing effect causing seasons to change, earth quakes etc. the planetary system is what we engineers have seek ed to perfect, a perpetual movement. due to friction It is only possible in a perfect vacuum which is what space is. The elements gold and iron act as a counter balance ,more to the point is this effect going to be felt through out the universe. Like the butterfly effect.