Portrait of Simone Weil, (1909–1943),

Wikipedia Commons.

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Today’s sharing from the Blue House of Via-HYGEIA, is a text by Simone Weil, ‘Sketches of a History of Greek Science’, published posthumously in ‘Intuitions Pré-Chrétiennes‘, La Colombe, Paris 1951. From page 173 to 180. Our Via-HYGEIA English working translation. This is not just a mere article about epistemology and Greek science. It offers in a (dense) nutshell Simone Weil’s broader vision of the ‘Greek Experience’, as the essential root of her quasi-gnostic, intimate,  amorous investigation of Christianity-poles apart with the Roman Catholic Cesarian approach which she disliked deeply. ‘The influence of the Old Testament and that of the Roman Empire-whose tradition was maintained by the Catholic Papacy, are in my opinion the two causes of the corruption of Christianity.‘ (From ‘Letter to Déodat Roché‘). More to come soon🥳

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A Introductory Excerpt

From ‘On Science’,

Paris: Éditions Gallimard, 1966, 285 pp. Collection: Espoir.

‘But if Greek science is already classical science, it is also, at the same time, something entirely different. The famous formula of Plato, “Let no one ignorant of geometry enter here,” is enough to demonstrate this. What one sought when going to Plato’s school was a transformation of the soul that would enable one to see and love God; who today would think of employing mathematics for such a purpose? In Europe, since the Christian era, the period par excellence when people sought God, which we call the Middle Ages, ended with the revival of the study of mathematics. Pascal, on the verge of discovering the algebraic form of integral calculus, abandoned algebra and geometry in his longing for a connection with God. We can hardly imagine today that the same person could be both a scholar and a mystic, except at different times in their life. If a scholar has any inclination toward art or religion, these inclinations are separated within them from their main occupation by an impervious partition, and if they attempt to bring them together, it is, as many examples show, through vague clichés and significant banalities.

Similarly, over the past three centuries, those who have devoted themselves to art or religion have not thought to take an interest in science, and if Goethe seems to be an exception, he had his own unique conception of science. The strangest thing is that if we consider separately the scientific, artistic, and religious conceptions of the West since the Renaissance, Greece always appears as the source and model. But these resemblances deceive us, for science, art, and the pursuit of God, which were united among the Greeks, are separated among us. Keats hated Newton; what Greek poet would have hated Eudoxus?

If we examine Greek science closely, we find notions with multiple resonances and moving significance.’

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Original French

‘Mais si la science grecque est déjà la science classique, elle est aussi, en même temps, tout autre chose. La fameuse formule de Platon, « Nul n’entre ici s’il n’est géomètre », suffit à le montrer. Ce qu’on venait chercher quand on allait chez Platon, c’était une transformation de l’âme permettant de voir et d’aimer Dieu ; qui songerait aujourd’hui à employer la mathématique à un tel usage ? En Europe, depuis l’ère chrétienne, la période par excellence où l’on a cherché Dieu, et que nous nommons le Moyen Âge, s’est terminée quand on a rénové l’étude de la mathématique ; et Pascal, sur le point de trouver la forme algébrique du calcul intégral, a abandonné l’algèbre et la géométrie par désir d’un contact avec Dieu. Nous ne pouvons imaginer aujourd’hui qu’un même homme soit un savant et un mystique, sinon à des périodes différentes de sa vie. Si un savant a quelque inclination pour l’art ou pour la religion, ces inclinations sont séparées en lui de son occupation principale par une cloison étanche, et, s’il essaie d’opérer un rapprochement, c’est, comme le montre plus d’un exemple, par des lieux communs vagues et d’une banalité significative. De même, au cours des trois derniers siècles, les hommes qui se sont voués à l’art ou à la religion n’ont pas songé à s’intéresser à la science, et si Gœthe semble faire exception, il avait de la science une conception qui lui était propre. Le plus singulier est que, si nous considérons séparément les conceptions scientifiques, artistiques, religieuses de l’Occident depuis la Renaissance, la Grèce apparaît chaque fois comme la source et le modèle. Mais les ressemblances nous trompent, puisque la science, l’art, la recherche de Dieu, unis chez les Grecs, sont séparés chez nous. Keats haïssait Newton ; quel poète grec aurait haï Eudoxe ?

Si l’on examine de près la science grecque, on y trouve des notions à résonances multiples et à significations émouvantes.’ From: ‘Sur la Science’, Paris: Éditions Gallimard, 1966, 285 pp. Collection: Espoir.

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‘Sketches of a History of Greek Science’

starts here:

‘(The history of Greek science) It starts with the notion of similar triangles attributed to Thales (of Miletus). Thales, ‘a Phoenician by remote descent’ (Herodotus), was the teacher of Anaximander, whose fragment quoted later below, shows that the inspiration was identical with the Pythagorean inspiration. An ancient said that Thales and Pherecydes of Syros (an Ancient Greek mythographer and proto-philosopher, possibly the teacher of Pythagoras, who said: ‘When Zeus was about the start the creation, He transformed Himself into Love‘.) established water as principle of all things but Pherecydes was calling it ‘Chaos’. If the Primitive water is identical to the Chaos, it matches with the conception of the first lines of the Book of Genesis.

The similar triangles are triangles which sides are proportional,

If two similar triangles have two equal sides, without being equal themselves, we have a three-term proportion, with two extreme terms and one middle term:

If we pose the problem of constructing a triangle such that it can be divided into two similar triangles with a common side, we arrive at the construction of the right triangle, which immediately gives the theorem known as the Pythagorean theorem (the sum of the squares of the sides is equal to the square of the hypotenuse) and the theorem stating that the height of the right triangle is the mean proportional between the segments determined on the hypotenuse. The theorem of the right triangle’s inscribed circle provides the power of the circle for the construction of mean proportional. In fact, these theorems followed those concerning similar triangles.

The concept of similar triangles, it is said, allowed Thales to measure the height of the Egyptian pyramids based on their shadows and the ratio between the height and the shadow of a man at the same hour. Thus, proportion makes the forbidden dimension, the one that leads to the heavens, the height, measurable and, therefore, in a sense, comprehensible to humans. It is also similar triangles that have allowed the measurement of the distances of celestial bodies.

Moreover, these theorems allow finding a mean proportional between any two arbitrary integers.

The question arose as to whether the search for a mean proportional could be achieved either through arithmetic-through a geometric construction-or solely through geometry. It can be easily shown that the mean between one and two has a ratio with unity such that it is impossible to find two integers united by this ratio, regardless of their values. This is because the integer twice a square, in the form 2n², can never be a square. The duplication of the square can only be accomplished through the geometric construction of a mean proportional. It can also be easily demonstrated that the same applies to the mean between one and any non-square number, just as it does for the mean between one and two.

Thus, these means, although they should be counted among numbers, have only a geometric foundation. Therefore, it was necessary to establish that arithmetic operations and proportion can be rigorously defined for these quantities.

Which is perfectly well achieved in the fifth book of Euclid, whose content is attributed to Eudoxus (of Cnidus),  a friend of Plato and  student of the Pythagorean geometer Archytas. This book contains what we call nowadays the theory of the real number, at the state of perfection. After the destruction of the Greek civilisation, this theory was lost, even though we still got the works of Euclid, simply because we could not understand the state of mind it was corresponding. In the course of the last century, mathematicians having  found again the surge for a rigorous approach ‘re-invented’ this theory because they did not know it was within Euclid’s surviving works. They found it there afterwards.

The essential of this theory is a simple definition of the proportion by the mean of the notion of most big and most small. It is said that a is to b like c is to d, if ma>nb always leads mc>nd, and if ma < nb always leads mc <nd, whatever are the whole number m and n. It is easily demonstrated that this condition is made for similar triangles. Therefore, it is allowed to rigorously affirm that the height of a rectangle triangle is the proportional average between the segments of the hypothenuse.

Therefore, the ratio we can also call number, at the condition we understand by that real number, is defined only by a certain order of correspondances that mutually bind four groups of an infinity of terms. The number or the ratio (αριθμος-arithmos or λογος-logos) truly appears like a mediation between unity and what is illimited.

During Plato’s time as well, the oracle of Apollo, by ordering the doubling of the cubic temple of Delos, presented the problem of the duplication of the cube to the Greek geometers. This problem reduces to the search for two mean proportional between 2 and 1.

Menaechmus, a student of Plato, successfully carried out this research. Additionally, he is the inventor of the parabola and the equilateral hyperbola. It is through the intersection of these curves that he achieves the duplication of the cube. Now, if we consider the problem of finding two mean proportional while focusing on the construction that allows us to find such a mean using a circle, we arrive at a construction of the parabola as a section of the cone that encloses its algebraic formula. There is nothing implausible about Menaechmus having found the sections of the cone with their formulas while seeking two mean proportional between 1 and 2. In doing so, he would have invented the notion of a function. When I mention formulas here, I do not mean the letter combinations that exist in our algebra, but rather the knowledge of the varying ratios of quantities that we express using these combinations, which the Greeks did not express in this way but nevertheless clearly conceived. They possessed the notion of a function. It appears in the history of their science, linked to the search for mean proportional. The first discovered function, namely the formula of the parabola, is the function that serves as a mean proportional between a variable and a constant.

The invention of integral calculus is attributed to the same Eudoxus who formulated the theory of real numbers. He also formulated the postulate, erroneously known as Archimedes’ axiom. Here it is: “Two quantities are said to be unequal when their difference, when added to itself, can exceed any finite quantity.” This is the notion of the summation of an infinite series. By using this notion, Eudoxus found the volume of the pyramid and the cone, and later Archimedes found the quadrature of the parabola.

Thus, it is indeed about integration. The area of a parabola is measured by the sum:

It is therefore the sum of the terms of an unlimited decreasing geometric progression. It can be proven using the postulate known as Archimedes’ axiom that this sum is rigorously equal to:

It is the blending of the limit and the unlimited that appears here. The same thing appears as both unlimited and finite. This was already the case in what is mistakenly called Zeno’s paradoxes.

The same Eudoxus developed an astronomical system to address Plato’s question: “Find the set of circular and uniform motions that, concerning the stars, account for the observed appearances.” It is based on the ingenious idea of composing motions, which forms the basis of our mechanics. Just as we construct the parabolic motion of projectiles using two rectilinear motions, one uniform and the other accelerated, Eudoxus accounted for all the observed positions of the stars in his time by assuming multiple simultaneous uniform circular motions around different axes performed by a single celestial body. This conception is as bold and pure as the definition of real numbers and the summation of an infinite series. If Plato desired only uniform circular motions, it is because he considered such motion to be divine, and he stated that the stars are “images of divinity sculpted by divinity itself.” When Plato speaks of the “Other,” which rebels against unity but is harmonized with it by force, he most likely has in mind this composition of motions. In its singular motion, the sun is simultaneously driven by the circle of the equator and the circle of the ecliptic, representing the Same and the Other, yet these motions form a single unified movement.

In the following period of Greek science, Ptolemy reproduced, in a much cruder form, the system of Eudoxus; Apollonius (of Perga)  continued Menelaus’ discoveries on conic sections, and Archimedes expanded upon Eudoxus’ work on integration.

Furthermore, Archimedes laid the foundations of mechanics and physics. The branch of mechanics known as statics was nearly completed by him, specifically the theory of the balance or lever – which are essentially the same – and the theory of the center of gravity that follows from the former. Archimedes’ theory of the balance, which is rigorously geometric, is entirely based on proportion. Equilibrium is achieved when the ratio of weights is the inverse of the ratio of their distances from the pivot point. Thus, it can be rigorously said that the cross was a balance in which the body of Christ countered the weight of the world. For Christ belongs to heaven, and the distance from heaven to the point of intersection of the branches of the cross is to the distance from that point to the Earth as the weight of the world is to the weight of Christ’s body. Archimedes said, “Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” To fulfill this statement, two conditions were necessary. Firstly, the fulcrum itself should not belong to the world. Secondly, the fulcrum had to be at a finite distance from the center of the world and at an infinite distance from the acting hand. The act of lifting the world with a lever is possible only for God. The Incarnation provided the fulcrum. It can also be said that every sacrament constitutes such a fulcrum, and that every human being who perfectly obeys God constitutes such a fulcrum. For they are in the world but not of the world. They possess a force infinitely small in comparison to the universe, but through obedience, the point of application of this force is transported to heaven. It can be said that God acts in this world only in this manner, through infinitely small things that, although opposed to infinitely great things, are effective by the law of the lever.

Archimedes laid the foundation for physics by developing a branch known as hydrostatics. He constructed it purely geometrically, without any mixture of empiricism, and it is truly marvelous. Hydrostatics also relies entirely on proportion. When a body floats, the line of flotation is such that the ratio of the submerged volume to the total volume is identical to the ratio between the density of the body and that of the water. This can be proven as a theorem of geometry through symmetry, assuming that symmetry exists wherever there is equilibrium. Water thus appears as a perfect balance. This property, which is akin to justice, may not be unrelated to the symbolism of baptism in its original form. The immersed person experiences two forces, one downward and the other upward, with the latter prevailing.

We know very little about the origins of chemistry in antiquity, except that Plato proposed a theory of the four elements based on proportion. Air and water are two mean proportional between fire—which is also light and energy—and earth. In essence, there is energy, matter, and two mean proportional that connect them. Two, because space is three-dimensional.

Biology was already quite advanced during Plato’s time, as Hippocrates predates him. It is primarily based on the notions of proportion and harmony as the unity of opposites. Hippocrates defines health as a certain proportion between the pairs of opposites that concern the living body, such as cold and hot, dry and moist—a proportion that must correspond to the surrounding environment. Thus, through elimination, the living being becomes an image of the environment.’

Written in Marseille and Casablanca,

between November 1941 and May 26 1942.

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Original French

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