HIPATIA e as Cônicas - Misteriosas relações entre filosofia e geometria (Subtit. English/Español)

NEW ACROPOLIS - International Organization PHILOSOPHY-CULTURE-VOLUNTEERING Hypatia and the Conics - Lúcia Helena Galvão comments on the text of Prof. José Carlos Fernández-2020 Hello, welcome to our usual conversation. Today I have something interesting to announce to you. A very special Acropolitan, who is the national director of New Acropolis in Portugal, professor José Carlos Fernández. He writes very well. He is an extremely polite person and has a series of brilliant articles on several subjects. I plan to start a series today, talking about some of those articles. Today we are going to talk about Hypatia. But, not simply

about the History of Hypatia. He produced, based on historical data, a text of his own, a literary text, where he speaks, simulating a class of Hypatia, within the Alexandria Neoplatonic School. She'll talk about what she understood the most ... that was mathematics. But, Hypatia was not just mathematics, not just the pure mathematics, but also, mathematics applied to human knowledge, to philosophy. And it wasn't just her who did that. We know that... throughout history, Isaac Newton himself, so known to us, had a whole philosophical study behind your physics. So, let's get to know a little more

about another side of Hypatia. But, of course, everyone is interested in knowing a little bit of her story. I'll be very succinct in her story, because it is very well known, the Internet is full of articles talking about it. I did a lot of research to find a reasonable version for you, because there are many contradictions, in several versions. And I brought you a very brief profile. I recommend it to everyone who wants to know a little more, but don't want to search so much in texts, to watch the movie Agora. It's not entirely faithful,

of course, some details were romanticized, as is usually done in movies. But in general, it is interesting and clearly expresses respectability of this character. At a time when it was very difficult for women to stand out, a high-level woman, who was able to silence the male voices, at the neo-Platonic school in Alexandria, bigger than your father, Theon, who was known as a great mathematician. She was a woman who knew several areas. That's what we're going to talk about a little bit. Today, I want to add to you, through this beautiful article by Professor José Carlos

Fernández, where he talks about Hypatia and the Conics. We'll talk about the hyperbola, ellipse, parabola, circumference. You don't have to worry: "Oh, I don't understand math at all." I'm not an expert either. But, we'll discuss very basic elements, that allows us to make a parallel of that with the knowledge of life that Hypatia had. I hope you enjoy this trip, of our today's adventure, and I hope it adds some reflection for you. Hypatia is born, lives, and dies in Alexandria, she dies in a very tragic way. It is not known exactly when she was born.

There is a large margin of error, from 351 to 370 AD. But, we know for sure that she died in the year 415. Her death is very well documented, including by a contemporary historian, who was Socrates, the Scholastic. Also known, as Socrates of Constantinople. She lives about 150 years after Plotinus, at the same neo-Platonic school in Alexandria. But, at a time when we have historically considered the Middle Ages, like a millennium, from the 5th to the 15th century. She lives between the 4th century and the 5th century. She is killed at the beginning of the

5th century. We cannot put up a flag and say that was here that the Middle Ages began. The process is very gradual. But we'll see that some elements which will later characterize the Middle Ages, were already appearing. It is interesting even from this point of view, to understand the dynamics of the story. So, she is born, daughter of Theon, who was already a teacher, a mathematician much noted at the Alexandria School, who boldly teaches everything to his daughter, which was very unconventional, in the case of a woman, at that time. His daughter is brilliant. Hypatia

is so brilliant, that she at the age of thirty, assumed the direction of the Neoplatonic School of Alexandria. It was a school full of brilliant thinkers. She takes over, only at the age of thirty. We have a big list of thinkers from the past, it's a huge list of everything they've done, that we sometimes get confused about, about what they were fundamentally. Hypatia was a mathematician and philosopher, astronomer, logic. Practically interested in everything, she was a universalist, interested in "lato sensu" human knowledge, more focused on these areas: Mathematics, Philosophy, Logic, and Astronomy. Within that profile,

she had several very famous disciples, including Bishop Sinesius of Cyrene, was an important bishop within the Church who learns a lot from Hypatia. Also among his disciples was the imperial mayor of Alexandria, Orestes, her personal friend. She lives her entire life in Alexandria. Although, for everyone at the time, the known world of the time Alexandria was known as a violent region. Christianity was coexisting with what they called "pagan cults". Inside Alexandria, there were Christians, Jews, and several ... various adherents of pagan religions. Furthermore, Christianity itself was not unified in Alexandria. For the world of that

time, even for Rome, Alexandria was a place that lived through a very violent moment, very complex. Today there is a lot of controversy about what happened. The safest source we have is exactly, Socrates, the Scholastic. But, there was a lot of contest, I heard the opinion of several thinkers. Everything makes us believe that, at a certain moment, Hypatia would have fallen from grace for some reason in front of this bishop, who was the patriarch of Alexandria. When Hypatia begins her studies, he had not yet gone to Alexandria. When he takes over, he comes with immediate

conflict with the imperial mayor, and this conflict had a reason, Bishop Cyril was a very radical Christian, very fundamentalist. When he arrives in Alexandria, one of the first things he does is to dissolve a Christian group that he calls: heretical. He was a great persecutor of the Christian sects that appeared at the time. He, as a fundamentalist, also had a fundamentalist group that followed him. At times, it came into conflict with the city's imperial mayor. At one point, Orestes would have ordered the death of a monk named Ammonium. So, this monk, Ammonius, was executed by

the imperial mayor of Alexandria, and the bishop wanted to take revenge on that. To get revenge, he incited hatred in the people who followed their teachings, to kill Hypatia. If we consider this story, is curious, may I have your special attention here, is the fact that this bishop, Cyril, had a certain shame to assume, that he would have sent someone to kill Hypatia. That's why there are doubts in the story, whether it was him or not. Socrates, the Scholastic clearly said that it was him. But, today there are still historians who think that he never

claimed to be the perpetrator of this crime. Let's try to understand... The mayor is on one side, on the other side, there is the bishop who was responsible for that city. In a conflict between a moderate Christianity and a fundamentalist one. The fact that the bishop, wanted to hide that he was the author of the murder of Hypatia, it’s already promising, because only a hundred years later, medieval thought had already taken place. As we're going to see is that when medieval thought is established, people were sentenced to death, without anyone being afraid of it. Let's

see if you understand my point: for Plato, "modesty" was a very interesting subject, He separated "modesty" from "shame". The shame is when you are ashamed of good things, it is bad; Modesty is when you are ashamed of mistakes, it is good, because if you display your mistakes head-on, and not be touched by it, at all, means it a shameless person. That is, it has already given up having an appearance, the person already considers that is right, truthful. Don't worry about hiding. And then, for Plato, and I agree, a person who commits a crime, orders to

do it, but it still hides, it does not justify that person in any way. But, it justifies that at that moment, in this society, there was still a moral criticism about this behavior, a century later, no more. Why am I highlighting this for you, in the middle of this story? For a simple reason. We cannot say that violence existed only in the Middle Ages, or that the violence existed only in the church, this is stupid. Nobody will deny it, if someone comes to deny that there was violence in the Middle Ages, I’ll think we’re back

in the Middle Ages. There was violence in the Middle Ages, but from there to want to attribute this to an isolated episode that the Church committed. Violence continues among us all our lives. Violence is a phenomenon of humanity. It existed in history in several other contexts. What I want to say to you, the violence that in this historic moment in which we live, it is often hidden, nobody knows how it was, who sent it, what is behind it. If we continue in this historical tone, in a century it will be explicit and will be regulated.

And that was the medieval mentality. That what in a moment caused shame; in the other, it was explicit, in just a century. Therefore, it is important to pay attention to this. The violence we are experiencing, a century from now, how it will be like? It depends on how we deal with it now. We are slowly losing our shame, and bringing all things to the surface. That's how you enter a mentality of an era of terror, a medieval era. To summarize the whole story: Hypatia would have been killed by revenge on monk Apollonius, when she was

leaving the Neoplatonic museum. She was reportedly caught in her carriage, dragged through the streets, tortured ... I am not going into details about how it was, because it is morbidity. Afterward, his body would be burned. So, the great philosopher of Alexandria dies. Many historians speak of this story that I am passing on to you in a very synthetic way. And with Hypatia, the classical thought, classical philosophy, dies. And it is very likely. It's a big impact because Hypatia's fame went very far. This marks Alexandria as a place of obscurantism, and there are many revolts after

her death; and others in name of Bishop Cyril. That made the climate in Alexandria even more radical. A city that a century and a half before, had Plotinus, had Ammonium Sacas, a little earlier, two centuries ago, had Pitolomeu. A city that had both origins: the Christian and the pagan, This region, that had been so brilliant, from one hour to the next it gets dark and lets itself be swallowed ... by a fundamentalist and enemy thought of any version other than yours. See how this is a human phenomenon. We are all subject to the same. And

this city, which was so bright, seems to have been one of the first places where this mentality was established. Then, after this initial phase, let's see another interesting thing that I saw, it is that two bishops, shortly after Hypatia, were murdered ... like her, two Christian bishops. Exactly because of these dissensions, between Cyril and the Mayor, and all this confusion ... Two monks who were Jorge de Laodicéia and Protério, they were dragged through the city, they were cut, they were burned, that is, Christians, and shortly thereafter. This is just so that we have a more

complete picture of what happened. There was a big conflict there, there was a clash between different versions of Christianity. And the different thought from the official, it was slowly becoming impossible to pronounce. I can't help drawing a parallel with the present moment. We know that our contact is mainly through the internet. How many times posted publications, simply because they say what nobody admits to hearing, receive 500 bad reviews, to the point that we have to remove them. How many times do we have simple things, that can no longer be pronounced? We are not going to

deceive ourselves by saying, "Ah, that happened at that time". how obscure it was. Let's look a little bit at the moment we are living in and we will realize that we are becoming quite a fundamentalist. Fundamentalism is ... a phenomenon that spreads throughout time, because it's not from a movement, it's from humanity. So, continuing this narration that is not the main thing in our history. I brought you a simple list of what is said ... that would have been helped, because in fact, this is also complex. What would have been written together by Hypatia and

her father, Theon. It's hard to know, because even if it was just her job, it is likely that it would have been considered co-authored, and there is nothing left of these things. Everything got lost. So she had a comment on Diophantine Arithmetic, volume 13, had a commentary on the Apollonius Conics of Pergamum, which we’ll talk about later. She edits the existing version ... of Ptolemy's Almagest. This Almagest work from where the astronomical conception comes from... that the world has followed for another thousand or twelve hundred years, it was overthrown very late in the Renaissance. She

does a recompilation, an edition of that text, on which her father works. It is interesting to say that she worked on astronomy, the Almagest, but she also worked on Tetrabiblos, which dealt with Astrology. She also had this side, she liked that knowledge, deeper, more parallel with human nature. And then he fell into so much demerit. She also worked on the work "The Elements of Euclid" and she had a text of her own, which was "The astronomical canon". And made a mapping of celestial bodies. They say that she participated in the creation of the Astrolabe, in

quotes, Astrolabe had already been created, she improves that instrument. Anyway, what's left of it? Nothing. Commentators, doxography, people who said: "it seems that she said such a thing..." Socrates himself, which we have already mentioned, and others mentioned that "perhaps..." Even some of the thinkers of the Platonic Academy, like Proclus, had mentioned a few things, which may have existed at that school in Alexandria. When the Library of Alexandria was burned, probably what was there about Hypatia disappears, So, we only have the memory of a woman. It is said that from all over the world, those who

had doubts about mathematics, sought out Hypatia, and doesn't leave without an answer, at a time when female opinion, it was very undervalued. Notice it was a woman and a young woman. She takes over the Academy at the age of 30, dies around 45 years. If we consider that your birth date is uncertain, we cannot know exactly. But she was already an authority at that historic moment. Well, let's talk then, a little bit about that particular treatise. Hypatia seems to have paid special attention, to this Treatise on the Conics The Conquest of Apollonius of Perga. This

Apollonius of Perga was studious... who made a treatise consisting of 8 books, talking especially about the conic sections. I will explain, for those who are not from the area of mathematics, a little bit of what it is about. But, this story of these conic sections, these curves like the Hyperbola, the Parabola, the Ellipse, and the Circumference, have always been the object of attention very special by several thinkers. Euclid, a century before Apolônio de Perga already talked about them. But it seems that Apollonius, in his treatise which was all reviewed and commented by Hipátia, does what

was best at the time on this subject. There is a great deal related to these conical sections. because they exist, as it is said in several schools, a parallel between these curves and some things, some conceptions about the creation of the universe, the role of the man. That is, there would be a symbolic universe that these conics were used to explain. And what are we going to do today? Let's work a little on the text of Prof. José Carlos Fernández, and let's see the conceptions he puts on these conics. A basic explanation of mathematics and

symbolism. This text that I proposed to you, talks about these symbolic relationships. It's a very short text, just two pages, and I hope this talk will help you to understand. So, imagine, is usual to talk about a double cone of revolution. What is a double cone of revolution? Take a pencil ... and makes it spin in circles. You will have an inverted cone on top, and a cone in the correct position below. This is a double cone of inversion. But, for simplicity, let's choose a simple cone. Imagine a cone in front of you ... We

can see this cone, a circumference, an ellipse a parabola and a hyperbola. Within that image presented, let's see that there is a plan that passes horizontally inside a cone. This plane that passes horizontally, forms a circumference. But, if you pass that piece of paper, or whatever you’re going through inside that cone, and it tilts a little bit, you will see that it no longer has a circumference. It is an extended circumference that, we call an ellipse. This ellipse is curious because we know that the circumference, it is a set of points equidistant from the center.

The ellipse, the elongated one, has two centers, two foci, and it is defined by a set of points, whose distance to these two foci, added, it is a constant, it is the same. So, it's like you stretch the circumference, we have an ellipse, with two foci. It is the opposite. The circumference is a special case of the ellipse. The ellipse joins the two foci until finally concentrating them in only one. This is a second geometric figure. The third geometric figure occurs when you do not pass this plane, going through all sides of the cone, you

pass it leaning. Imagine, you have the cone... you'll pass an inclined plane, parallel to one side of the cone. This shape that will result, in perfect parallel, with the angle of the side, it will be the parabola. The parabola, as we see, is always a set of curves that are formed through that plane that was passed in parallel on the side of the cone. Now, if you take this plan and do not go in parallel with the side, starts to tilt the angle, more and more until it is in parallel with the axis, all these

figures that are formed from different angles, they are called hyperbolas. These curves for a long time were considered very special, by several thinkers. Hypatia was no different. They are full of symbolism, I will try to explain a little. Because for this 5th century AD thinker, studying these curves brought so many things. Why does this bring so many relationships within? Of course, it would be necessary to be a disciple of Hypatia to understand her teachings. And, what we're going to give you are some bases, some elements to understand, that behind appearances, there may be very deep

relationship skills. Let's start! Courage, for those who are afraid of math, it will be very pleasant. Sacred Geometry, it is common in all traditions that you can imagine. We find sacred geometry in India, in Egypt, everywhere. Particularly at the Neoplatonic School in Alexandria, it seems that it was one of the favorite subjects of Hypatia. She then taught the conventional mathematical elements, and then it gave the symbolic dimension of what she was talking about. Let's see then, a little about the conics and why it was so special. The parabola in particular, has a very interesting feature.

If you take any point, make a point on a sheet of paper, and pass a straight line in it, in parallel with that if you were tracing all points that have the same distance, from that point and that line, one here; another there, successively. I would draw a parabola. Geometrically, that's how it is defined. Take a line and a point, and draws all the points that is equidistant from these two realities. Along the line, with the same distance, here and there. This is a parabola. This parabola has several interesting features. One of them, and about

which Prof. José Carlos will speak here, first, is the fact that everything you push, plots a parabolic curve, this is related in all classics as the trajectory of life. It’s as if physical bodies are launched on a trajectory, where they ascend, pass as close as possible to the world of Platonic ideas, they reach their fullness, where they flowery and bear fruit. And then, they start again to return to earth. An exit angle, identical to the entry angle, isn't it? They make this curve, arrive as close as possible to their ideals, their references, bear fruit and

bring the earth this heavenly influence, and start to decline. And according to Hypatia, in the literary text written by Prof. José Carlos, there would be the opposite too if we put the parabola upside down. You will see that there is a movement of the human soul, of that human essence that is trapped in matter. Which is the opposite, an upside-down parabola, it's coming down ... gets as close to the earth as possible, wears a physical costume, reflects in the matter, gives your message, lives your experience, and rises again ... They say that this would be

the curve of life. The appearance, the physical body, the physical world, an ascension point arrives and declines. And the essence reaches a point of descent, gets as close as possible to the matter, and ascends again. I gave a talk a long time ago about the origin of modern celebrations, that works with something with this, that point, where consciousness and the maturity of the body is at its peak and the conscience is as mature as possible. Sends its message to the world, delivers, flowery, and bears fruit. And then, each one follows its course ... the matter,

"for Caesar what is Caesar's; for God what is of God". That is, the body returns to the earth and the soul returns to its plane, whatever it may be. I will not go into theological considerations. So, growth and the decline of living organisms, and the rhythms of the soul in matter, would be parabolic, according to these traditions, according to Hypatia's Neoplatonism. The parabola represents the rhythms of life, in essence and appearance. And we'll see later that it represents even more things, such as, the true human mind, it would be parabolic too. And another element, that

we are going to show, is the hyperbola. In hyperbola, if you put, these two focuses inside them, you'll have hyperbola as follows: Put those two points, you're going to have a set of points. Their distance to the second focus, subtracted from the distance from the first, from the largest to the smallest, it is a constant. Let's tracepoints that have this configuration. Subtracts the distance from focus 2 from focus 1, and there will always be a constant that you draw on one side, and on the other side, hyperbolas. So, that would be the geometric characteristic of

hyperbolas. We realize that they are in opposition, each with its focus. hyperbola would represent, within that tradition, the attractions and repulsions of the material world. The characteristics of duality. In other words, we are attracted to one pole and turn our backs on another. But, we are always to the two. When we study the ellipse, you will understand what these two poles represent. Man has a material pole and a spiritual pole. You are always more attracted to one than the other, but it doesn't lose relation with both. But, this tendency to focus on one and not

the other, will generate the attractions and repulsions of the material world. It would be another law of the world manifested, represented by a hyperbola. And here she will explain something that I thought was beautiful! Well, I know this is complicated. Don't be prejudiced, let's go slowly! If we go back a little bit in Pythagorean mathematics, or even in Hindu mathematics. We will see that they always represent the Unit as a point. A point has no dimension, It represents something that is not in the matter because it has no dimension. Everything in the material world has

a dimension. This point then represents the spiritual plane. It is defined by something that is there, but in reality, it has no dimension at all. He represents Unity, the spiritual world. While, a straight line, as said Euclid himself, and then we will see Blaise Pascal, saying similar things, a line is defined by the union between two points. Necessarily, Pascal said: "Between two points, a single straight line passes." That is, two points define a line. One point is Unity, it is the spiritual world. Two points, it is the material world, it is duality. The material world

is defined by duality, the hot and the cold; soft and resistant; the light and the darkness; the high and low. Everything in the world is dual. So, duality is the law of the world. Whereas unity is the law of the metaphysical world. Physical and metaphysical. So, you realize that if we have a line, which is defined by two points and a dot up there, and tracing all the points that are equidistant from the two, do you have a parabola? When you draw this, you realize that the parabole, which is a curve, it is the daughter

of the dot with the line. It is born from several dots that are equidistant from the two. The parabola is thus the daughter of the unity married to duality. So, it comes with traces of the father, spirit and mother's traits, matter. The parabola would be an image of the cosmos, of creation. So, it has such interesting features, represents the great cycle of creation. Which are exactly the cycles of ascension and decline. The cycles of descent and return to "home". That is, the behavior of the parabola, represents the cycle of life. It would be the first

son of the unit married to duality, a curve. The spirit's marriage to the matter has a son, who is life. And that is represented geometrically by these traditions, by the parabola. Something very interesting is going to be talked about in the text of professor José Carlos Fernández. He says that when you are playing ball with a friend, and you throw with very strong momentum, (imagine that you had this strength and this ability, it is not a human thing, would be a rocket booster). If you could project the ball so high, could perhaps make a trajectory

that Mother Earth didn't pull her anymore. and the ball could perhaps, be lost in a region where it would no longer be attracted to Mother Earth. Although, it remains, and this is a ball, with a leather surface or something material, that would represent a wise man. He is sometimes able to cast his consciousness so high, that even though they are still inside a body, be able to not be so attracted to matter. This would be a comparison or symbolism of wisdom. Imagine then a ball a leather ball, that there is air inside it. You know

who has tried to play empty ball, that it does not rise at all. It rises exactly because it is full, with a lot of air. Every football player knows this. So, the parallel that is made, is to imagine this air inside, as a spirit. If you let go, it goes up. And the leather that surrounds it, like matter. If it deflates, it falls. These are two things that combined ... they can have very strong momentum, reach very high. But, except in rare cases, where they lose the attraction of the land, will make the parabolic curve

and return to the ground. Do you know what that means? This is life. You are your essence, trapped in a material shell, who is trying to raise your consciousness to get as close as possible of this mystery of the universe, of wisdom. But, that necessarily, at a certain moment, the ball falls, and deflates, and that essence returns to your world. This is a parallel interesting what we have here. And so, very rarely, without losing the physical body, men come to understand the mysteries of heaven. It would be such a powerful conscience, that was able to

throw that ball, beyond Earth's gravity range. Earth, through gravity, attracts its children, as if she were a great mother, she pulls her children back to her. It would be possible that the attraction by the father, spirit, was so high that you escape this attraction of Mother matter, Mater Materia. And it knows great power of conscience. We usually do this cycle: our ball, leather, full of spiritual air ... completes its cycle. Our conscience descends and returns to their world, at a certain moment, they separate. And this is what we call death, and it will say that

this is the mystery of life, represented by the parabola. That is, the parabola is a symbol born of heaven and earth, that represents the curve of life and traces between the two. What else can we talk about on this topic. Something very interesting that he keeps talking about, he, professor José Carlos Fernández, in relation to the parabola. He says that the human mind itself, when you qualify as such, the human mind becomes parabolic. Let's explain this well. You may have heard of ancient traditions, especially, Alchemy, who spoke of the four elements of the world: Earth,

Water, Air, and Fire. The material elements. Imagine that these material elements are the constituents of the ball's leather. So, we have a physical body, the energy that circulates through that body, we have emotions ... and a practical and concrete mind focused on survival. In India, they call this quaternary. When a mind arises, who is no longer just interested in how he will eat today, but begins to take an interest in the mysteries of the universe, about values, about wanting to understand the unit, the sense of your presence on Earth. Begins to want to understand and

manifest love, in an increasingly pure way ... will, love and intelligence. You start wanting to use these tools, in a purer way. This is no longer a mind for survival. Some of these values ​​are completely expendable for survival. And some can even get in the way. And then, this new mind, which is the fifth element, which is born facing up, it is a parabola. And this parabola has a very interesting feature. Then, he will talk about the parabolic mind. I put here, for you, a characteristic of a parabola, which is a receiver of light and

sound. Take, for example, the headlights of your car, the shape, it's a parabola. A light reflector and sound, to send it, or to receive it. Why is the parabola so good? Enter in this device parabolic, the satellite dish in your house is like this, Several light signals come in, sound signals, but they are very scattered and very weak. And what happens to these signs when they reach the parabola? They strike back... always in the same direction, they jump here and there, they leave in the same direction. Are output in a concentrated beam, in one direction.

Professor José Carlos, comments in this article, that the truly human mind, the fifth element, it is a parabolic mind. It can receive the impulses of the world, of all kinds, of color, light, sound, from all the senses, to receive it all. And channel it in one direction, this direction of our star, this direction ... the base of the parabola. That is, this human mind, parabolic, is able to collect everything it receives, on impulse, from raw material, and create a laser ... aiming for its ideal. Aiming for the destiny of your life, for your true identity.

That is, it channels all impulses in will, love, and intelligence. Aim for a single goal, which is your ideal. So, he says that the human mind is idealistic, and is very well represented by this idea of the parabola. So, if you use your mind, truly human, everything that comes to you, can be turned into an impulse, to walk towards your ideal. This is very interesting. Remember that phrase that has a thousand versions on its author, which appears to be a Confucian background phrase. It says that the good pilot, who knows where your port is, handling

your sails, so that all the winds are favorable. In other words, handling must be done in such a way that everything leads to its port. Isn't it the same idea? Knowing how to handle the sails in such a way that all the wind takes you to your port, previously chosen. It is to manage all the impulses that arrive in your parabolic mind, so that she focuses on a single point, take you to your ideal. The full condition of the realized human being in values, virtues, and wisdom. That's the idea, too. That parabolic element, is very

much associated with the ability to concentrate and focus of the higher human mind... always aiming upwards, for a star, for a metaphysical ideal. The ellipse, that is, a cut ... of a cone. It is an inclined cut, that goes through all sides of the cone. Only, it is an elongated circumference. It has a very interesting feature, it has two foci. The characteristic of an ellipse is any point you take around you, you take the distance from that point to a focus, the distance from that point to another focus, sum ... and always gives a constant.

Any of those points around. You add the distance from one focus to the other, always giving a constant. So you draw. People usually have a toy to make ellipses, which is interesting. You put two little pegs, attach a string that is a little bigger than the distance between the two, put the tip of your pen over there, and spinning, then you describe exactly an ellipse. You don't see it, but this string draws these points of the ellipse. The distance, adding the two pieces of string, it’s just a string, it’s always the same. Then, it generates

this flattened circumference. What we know today about the ellipse, it's a very interesting thing. The ellipse, every ray of light, or sound that enters, by one of the foci, will hit the other. This has been widely tested. Every source of light, or sound that enters through a focus of the ellipse... will hit the other. In this sense, there is the aspect of the so-called Sacred Geometry, which says that in the Universe, in general, everything is elliptical. We know that Johannes Kepler said of the Earth's elliptical orbit around the Sun. The Sun occupies one of the

focuses. It seems that several planets behave in the same way and ... also, some comets make an elliptical orbit involving several stars, within fixed periods. We know our Haley. That is, they involve several systems at fixed periods. An ellipse is a well-known form within the manifestation. It has a known focus and an unknown focus. This known focus would be the focus of our material life. You know what it is. What is the focus of your material life? Where you work, who are your family members, who are your friends, what your interests are, it's what your

material life revolves around. And it has another focus that is metaphysical or spiritual. This is the hidden focus, which you don't see. Ignores, sometimes, for a lifetime, even if you don't see it, there is a relationship between these two focuses. Everything that goes through that the material focus that you took as the center, will resonate in that subtle focus, metaphysical of your essence. And it will generate a sensitivity, about the alignment or not of your life, with your human nature. And the answers for that, return from that metaphysical focus to the physical focus. In anguish,

in depression, in stress, for a feeling that we don't know where it came from. But that is indicating: "You are not being consistent with what you are". You are consistent only with existence and not with essence. Sometimes not even with existence. So, this existential anguish that we have that comes from the metaphysical to the physical. According to this Sacred Geometry, it would be the effect of the foci of the ellipse. They are in close communication, although one is visible and the other invisible. One manifest focus, another hidden focus. And we keep thinking that in our

life, everything seems so right. I have a job, I earn well, I have a good family, I have friends, everything is fine. But, there is anguish in me that I don't know where it comes from. You have another focus on your life, and if you don't discover it, you don't integrate it, it will be in permanent communication, showing that half of you have not been realized. And that much of your physical life is not consistent with your human essence. And maybe there is a metaphysical reason, represented by the ellipse, for the fact that we have

so much imbalance. To look with a single focus, when in fact there are two. You will see that the circumference is a special case of the ellipse. What's going on with circumference? These two focuses are getting closer as the ellipse goes flattening. It is approaching perpendicular to the axis of the cone. Then, these two foci will get closer until they merge into one. When the center of your physical and metaphysical life is one, you are a wise man. That is, you turn on not only about your survival, but of your spiritual life. You revolve around

not just competition, convenience, or preference, not even that. You revolve around a life worthy of a human being. It does survive! But without sacrificing life. It does not crawl to survive, still is a human, being able to be in the world. Appearance and essence are in perfect harmony... those two things stick together. Now comes the circumference, which according to Plato, in his "Timaeus" dialogue, it is the most perfect of forms. All points of your life are equally attended by a center. This center integrates Heaven and Earth. Do you understand that? There were two. You ignored

this one. And a communication started, that produced terrible unhappiness in you. You started looking for another center in your life. He started to approach them, approach them and they merge. Now, you have integrated heaven and Earth inside you. And your whole life is harmonized around that. So, the most perfect of forms. That's why the circle is the representation of the conscience of a sage. He will say something interesting about the life of a human being. A child. The child lives a circumference, although she have no wisdom. Be still very unconscious. Has natural wisdom that nature

gives, for her purity, but she has a center, that is her parents. And gravitate towards that, it's balanced, it's happy, in that harmless world, of which many of us have many miss you. At a certain moment, you widen it, because she needs experience... puts another focus. Social life comes. The school comes, the friends come, integration with the world comes. It starts to oscillate around of these two foci. Sometimes it values one more, as the years go by... tends to value the other more. There comes a certain point in development that she doesn't want to turn

around these outbreaks. She wants to live her own experience. The ellipse is broken. She becomes a parabola. She will live this experience... her conscience will rise, which should do on her initiative, until you get as close as possible to your heaven of possibilities, the apex of the parabola. It will take your heavenly fruits, germinate on earth, carry out your work and then start your descent. And that would be interesting because when she does this process, using almost all conics. It is said that the representation of a man who comes to fulfill his trajectory successfully and

bears fruit in the world, it becomes like the sage's circumference. That's why it is said that a man's life is two ends: childhood wisdom, unconscious ... and after the parabola, a life well-lived, you become conscious wisdom. It is the circumference with that point. You realize that normally this point, this center of the wise man's life, he is no longer on the same physical plane. It is a center that is at the level of ideas it is something higher. So the sage is not a circumference, it is a cone. He goes looking for that peak of

his life, his star, that peak of the cone ... in a spiral. It’s another very mysterious way, that talks about how human consciousness evolves, going through the same point several times, but each time, with a little more light. Then, the circumference... rises to its apex and you return to the cone. And the cone will generate the spiral, which is how this consciousness rises to reach wisdom... going through the same points each time, with a little more wisdom, until you master all points of human experience. And take the summit up there. Much like your Christmas tree,

where you place a five-pointed star. It represents this success in human climbing. Do you see how everything is interconnected within this figure, which is the cone? Since the Celtic world, that date, which was closely associated with the winter solstice, is directly connected to the cone. And the winter solstice is exactly the moment to be reborn, in another plane. Reborn on the spiritual plane. Blaise Pascal lived in the 17th century. When he was 16... he wrote his first book... that was exactly an essay on the conics. For you to see how every mathematician likes these blessed

conics. 16 years old! It is a reference, until these days. In this essay, Pascal will say that a conic can be defined by five points. Just as you need two points to define a line; and he also reinforces this idea ... that came from a long time ago. It needs five points for you to define a conic. Professor José Carlos insists on this point. Look what an interesting thing. Five points! Five points are what you need to define, too, a human being. Exactly the "Pythagorean pentalpha" or "Vitruvian man" by Leonardo. It is when the four

material elements, namely the physical body, energy, emotions and the practical mind, are dominated by a fifth triangular or conical element, which is the parabolic mind, this receptive mind, that can receive all the energies of this world, and the next and channel in one direction. So, number five is the number of conics. And the number 5, even within the Pythagorean decade is the number of the man. It is the "Vitruvian Man". It is the "Pythagorean Pentalfa". For those who studied the Templars... they had very interesting teaching. They took this five-pointed star, turned it upside down and

turned into a goat or a Baphomet and they said: When in man, the superior mind is conquered by the matter, he becomes an animal, a goat, or a Baphomet. Then it was confusing because some said they worship the goat. It was a representation of the man who inverted the five-pointed star. And your superior, human mind, linked to values, linked to virtues and wisdom, was overwhelmed by survival. Many traditions speak of "five", associated with a five-pointed star or not, the five, associated with human nature. And a conical section is defined by five points according to Pascal

and confirmed by Mathematics throughout history. It can be confirmed by any of us. You all can realize some interesting things to just synthesize our talk. A women who lived at a certain point in the early 5th century, late 4th century, during conflict and divisions between Christianity and paganism, an announcement of what would become the Middle Ages. Sacrificed, no one knows exactly why. But, certainly, because she knew more than should. And that, within the Neoplatonic school of Alexandria, she taught pure abstract mathematic, but also applied to the understanding of the universe. And it is not the

only one. Sacred Geometry exists everywhere, for various ancient traditions. The times passed, and that classic thought that was dead, it was developed by some other thinkers. Let's see that Isaac Newton himself was a great researcher of things, who was more than just pure physics, and by other men throughout history. And today we have it here, not as absolute truth, but as an invitation to reflection. When we go out on the street and observe the behavior of Nature, the way the tree shakes, how the water behaves, this has some teaching, it has value. Nature is entirely

pedagogical. A woman like Hypatia took facts, Geometry, that's is, pure mathematics, and applied to understand life. She said: it is the same law! If you take the very end of Plato's Republic, they will say, "when you deeply understands a process, a law is abstracted from there, that can be applied in any other process of life." From a situation that worked perfectly well, a law is abstracted, that allows you to understand a thousand things from the manifested universe, because it’s all sewn by law, the same applied everywhere. So she takes these conics, which just seems like

a mathematical curiosity, and will explain the great mysteries of the manifested Universe. Believe it or not! Philosophy is not a belief system. I'm not asking you to believe. Nor do I believe in much. I check from my experiences, and I have what's the best I've seen so far. and I consider that as a provisional conviction. But I am always willing to grow. I am learning this. I'm a learner. And one of the things I want most is to inseminate Philosophy seeds in you. And it is a lie because they are already there. I just want

to cultivate. They are within any human being. In other words, I'm not saying to you think like me. I'm saying: think! I'm not telling you to think like Hypatia... or like Professor José Carlos. I'm just saying, think! Is all this a game of coincidences? Or is there any reflection that can be done about it? One thing is certain: for Hypatia and several other great scholars of history ... Conics are a mystery. Mystery of Nature, which was revealed only to those who were able to see and understand. I hope it was interesting for you, mathematicians or

not. Nova Acrópole is an international, independent, and non-profit philosophical movement, based on Culture, Philosophy, and Volunteering. There are about 80 Schools of Philosophy applied throughout Brazil. Look for one near you, we look forward to your visit!