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Contributo di enrica Maragliano

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Renè descartes Renè descartes Presentation Transcript

  • BRIDGES THROUGH HISTORY WITH MATHS RENÈ DESCARTES 1596 - 1650 FABIO BARABINO & BIANCA FANTINO Liceo Classico Statale “Cristoforo Colombo” GENOVA (ITALY) YEAR 2009-2010
  • Descartes’ life and the Scientific Revolution
    • René Descartes , also known as Renatus Cartesius , was a French philosopher , mathematician , physicist , and writer of the
    • XVII century.
    • In this part we will deal with his life and with the “Scientific Revolution” which characterizes the
    • period in which Descartes lived.
  • Descartes’ work (1)
    • Descartes’ mathematical works are very important: he invented the “Cartesian coordinate system”, from which we now have the “ Analytical Geometry” , the bridge between Algebra and Geometry, crucial to the discovery of the infinitesimal calculus and analysis.
  • Descartes’ work (2)
    • He is also important as a philosopher: his thoughts are still closely studied in this days.
    • In particular, his work “ Meditations on First Philosophy ” is a standard text at most university philosophy departments and his philosophical statement " Cogito ergo sum " (in Latin “I think, therefore I am” ) is very famous still today and for this he is considered the “Father of the Modern Philosophy”.
  • Biography (1596-1597)
    • Descartes was born on 31 st March in 1596 in France at La Haye en Touraine , Indre-et-Loire .
    His birth and his family
    • His mother Jeanne Brochard died of tuberculosis one year later and his father Joachim was a member in the provincial parliament .
  • 1607 – 1616: his studies
    • At the age of eleven , he entered the Jesuit Collège Royal Henry-Le-Grand at La Flèche where he was educated.
    • After graduation, he studied at the University of Poitiers , taking a degree in law in 1616 , in accordance with his father's wishes that he should become a lawyer.
  • 1618: the meeting with Beeckman
    • In the summer of 1618 he joined the army of Maurice of Nassau in the Dutch Republic and on 10 th November, while he was walking through Breda , he met Isaac Beeckman , who sparked his interest in mathematics and the new Physics, particularly the problem of the fall of heavy bodies
  • 1620-1628: his travels
    • While he was in the service of the Duke Maximilian of Bavaria , Descartes took part at the Battle of the White Mountain outside Prague , in November 1620 .
    • In 1622 he returned to France , and during the next few years spent time in Paris and other parts of Europe.
    • He arrived in La Haye in 1623 , selling all of his property, investing this remuneration in bonds which provided him with a comfortable income for the rest of his life.
    • He returned to the Dutch Republic in 1628 , where he lived until September 1649.
  • 1629: Descartes returns to study
    • In April 1629 he joined the University of Franeker and the next year, under the name "Poitevin " , he enrolled at the Leiden University to study Mathematics with Jacob Golius and Astronomy with Martin Hortensius .
  • 1630 – 1640: Descartes quarrels with Beeckman and the Utrecht period
    • In October 1630 he had a falling out with Beeckman , whom he accused of plagiarizing some of his ideas.
    • In Amsterdam , he had a relationship with a servant girl, Helène Jans , with whom he had a daughter, Francine , who was born in 1635 , but she died five year later ; in that period Descartes was teaching at the Utrecht University .
  • His works
    • In the next twenty years, while he was living in the Netherlands , he wrote his mathematical and philosophical works, among which there are the two most important:
    the “ Treatise on the World ”, which he abandoned to publish because the astronomer Galileo Galilei was condemned by the Roman Catholic Church and he was scared for himself because he shared Galileo’s ideas, and the " Discourse on the Method ", which was published in 1637 , and there were also other books concerning both mathematics and philosophical.
  • 1643–1650: The last years of his life
    • In 1643 , Cartesian philosophy was condemned at the University of Utrecht , and Descartes began his long correspondence with Princess Elisabeth of Bohemia .
    • In 1647 , he was awarded a pension by the King of France .
    • Descartes died for pneumonia on 11 th   February 1650 in Stockholm ( Sweden ) where he was working as a teacher for Queen Christina of Sweden . Actually he is buried in Paris, in Saint-Germain-des-Près Church.
  • The Scientific Revolution Descartes lived in a period in which in Europe it was started what the historians call “The Scientific Revolution ”. It was called “revolution ” because it entirely changed the way of studying the nature: in fact it was passed from the “ Aristotelian way ” to the “ mechanistic way”.
  • “ The Aristotelian Way”
    • The “Aristotelian way ” was invented by Aristotle , a Greek philosopher of the fourth century b. Ch., and it was used from the ancient times to the Middle Ages: it consisted in finding the last aim of every natural phenomenon.
    Aristotle
  • “ The mechanistic way”
    • Then the “ mechanistic way ” was the new view of studying the nature: it was called “ mechanistic ” because the nature was considered like a big “ machine ” and each natural phenomenon like an its “ mechanism ”, of whom it was not searched the aim anymore, but how it happens .
  • Two important years: 1543 and 1687 (1)
    • The year 1543 is considered the becoming of the Scientific Revolution because is the year when the Polish astronomer Copernicus published his book titled “ The revolutions of the heavenly bodies ”, a book about Copernicus’ theory of the heliocentric system , in which the Sun is in the centre of the universe and all the planets turn around it.
  • Two important years: 1543 and 1687 (2)
    • The year 1687 is considered the end of the Scientific Revolution because the English scientist Newton published his book titled “ The mathematical principles of natural philosophy ”, in which he explained the theory of the bodily nature of the light and the law of the universal gravitation, which tells that all the bodies are attracted by the Sun and by planets .
    These two years were chosen because these two books represent the big turning points of the history of the science.
  • Descartes and the Rationalism
    • Descartes is one of the most important figure of the science of the XVII century: he founded the “ R ationalism ”, a new method to study the nature which consists in formulating mathematical hypothesis and demonstrations . Descartes considered Maths like the basis to study all the sciences and it was thank to this that he succeeded to formulate his physical, geometrical and mathematical laws.
  • Francis Bacon and the Empiricism
    • An other important figure was the
    • English philosopher Francis Bacon who invented the method of the “ Empiricism ”, the opposite of the Rationalism, based on
    • the repetition of an experiment in a
    • laboratory.
  • Galileo Galilei and the “experimental method ”
    • Surely the most important scientist was the Italian Galileo Galilei who founded the “ experimental method ”, which is used still today.
    This method is composed by an empirical part , the experiments, and by a rationalist part , the mathematical demonstrations, and for this it is also called “ double method ”, and with it Galileo formulated some of the most important laws of the modern Physics.
    • Descartes is one of the most important mathematician of the history.
    • He made lots of very important studies in Maths, both in Algebra and in Geometry .
    • In this part we will deal with his principal discoveries.
    Descartes’ mathematics studies
  • The Algebraic studies
    • Descartes did not make
    • lots of very important
    • discoveries in Algebra, but
    • surely the most important one
    • is the rule of the signs , called
    • “ Cartesian” in his honour.
  • The Cartesian rule of the signs (1)
    • This rule allows to know if the solutions of
    • an equation are positive or negative only
    • watching the signs of the coefficients of the
    • same equation: in particular it is based on the
    • permanence and on the variation of the
    • signs among the coefficients.
    + -
    • In every equation (solvable in the Real Numbers set) for each permanence of the signs between its coefficients corresponds a negative solution and for each variation a positive one.
    The Cartesian rule of the signs (2)
  • Example This is a third degree equation and it has three solutions X = ? X = ? X = ? 1 2 3
  • - 6X 3 + 5X 2 +2X -1 = 0
    • The first coefficient is negative and the second is positive so between them there is a variation of the signs and so the first solution must be positive
    +X 1
  • -6X 3 + 5X 2 + 2X -1 = 0
    • The second and the third coefficients are both positive and so between them there is a permanence and so the second solution must be negative
    -X 2
  • -6X 3 +5X 2 + 2X - 1 = 0
    • The third coefficient is positive and the fourth is negative so between them there is a variation and so the last solution must be positive .
    +X 3
  • To summarize:
    • The equation presents two variations and one permanence of the signs of its coefficients and so it must has two positive and one negative solutions.
    +X 1 -X 2 +X 3
  • Something more…
    • If a second degree equation presents one permanence and one variation it is also possible to know which of the two solutions has the bigger absolute value :
    • if the permanence precedes the variation the negative solution must has the bigger absolute value
    • if the variation precedes the permanence the positive solution must has the bigger one.
  • Some more examples (1)
    • This equation presents one variation between the first and the second coefficient and one permanence between the second and the third so the positive solution must has the bigger absolute value : in fact if we solve the equation we will find that the solutions are:
    Q.E.D.
  • Some more examples (2)
    • This equation presents one permanence between the first and the second coefficient and one variation between the second and the third so the negative solution must has the bigger absolute value : in fact if we solve the equation we will find that the solutions are
    Q.E.D.
  • Be careful!
    • The “ Cartesian rule of the signs” is valid only if we put in order the equation from the coefficient with the biggest exponent of the power to the coefficient with the smallest one.
    X V
  • Other basic rules
    • Apart the rule of the signs in Algebra Descartes did not formulate other mathematical laws, but he established same basic rules :
    • he introduced:
      • the use of the first letters of the alphabet (like “a, b, c”) to indicate the known terms
      • the last letter (like “x, y, z”) to indicate the unknown ones
      • the symbols “+” and “–“ to indicate the additions and the subtractions
    • he invented:
      • the method of the indexes ( like x 2 , x 3 , x 4 …) to represent the exponent of the power
      • the sign of the square root
  • The Geometrical studies
    • Descartes is one of the first people who studied the “Analytical” Geometry also called “ Cartesian ” in Descartes’ honour.
    • It was studied for the first time in the Middle Ages by Nicole Oresme , but surely Descartes is considered its best scholar of the history.
    • The master artists of the Renaissance used a grid as a tool for breaking up the component parts of their subjects they painted.
    • In the same period the French mathematician Pierre de Fermat (1601 - 1665) came up to the same idea, but in 3 dimensions. He didn’t publish his works.
  • Anecdote
    • The fame says that Descartes came up with the idea for his coordinate system while he was lying in bed and he watched a fly crawl on the ceiling of his room.
    • The man who invented Analytical Geometry, never got out of bed before 11 in the morning!!
    !?
  • The Cartesian Coordinate System
    • This system is formed by two axes, the x -axis and the y -axis , posed on a plane called “Cartesian plane”.
    • At every point P in the plane corresponds a pair of coordinates :
    • the first corresponding to the point where the parallel to the y -axis passing through P intersects the x -axis
    • the second at the point where the parallel to the x -axis passing through P intersects the y -axis
  • Example
    • In this figure it is represented a point, called P , on the Cartesian plane and i t corresponds to the intersection between the parallel line to the y-axis passing through P and the x-axis and the parallel line to the x-axis passing through P and the y-axis : the point P has coordinates +2 and -1,5.
    NB : To write correctly a coordinate of a point it must be put in bracket the x-point and then the y-point, separated with the semicolon, for example P(+2; -1,5).
  • Something more…
    • Now we use for axes two straight lines
    • orthogonal between each other, but they
    • were introduced by Fermat : in fact Descartes
    • used only oblique coordinates . And then both
    • of the them used only positive coordinates ,
    • instead of us that use both positive and
    • negative ones.
    • Analytical Geometry consists in studying the Geometry through the “Cartesian coordinate system” . With it is possible to draw every geometrical shape on the Cartesian plane and so it is also possible to associate them to their coordinates , that are numbers , and for this the Analytical Geometry is considered the bridge between Algebra and Geometry : we can associate to the geometrical figures (like a polygon, a line or a curve) an equation or an inequality or a system of them and so we can also translate geometrical problems into algebraic ones.
    WHY IS IT IMPORTANT?
  • “ La Gèomètrie”
    • Descartes dealt with the Analytical Geometry in his work “La Gèomètrie “ , which is a part of his most important work “Discourse on the Method” (published in the 1637 ), and it is divided in three books
  • The First Book
    • In the first book he posed the basic rules of the coordinate system and then he wrote the instructions to solve a second degree equation in a geometrical way and he represented geometrically the solutions of the equation and then he explained the connections between Algebra and Geometry.
  • The Second Book (1)
    • The second book is surely the most important: in it he studied the curves .
    • He divided the curves in two kinds: the “ geometrical curves ” and the “ mechanical curves ”:
    • the geometrical curves are can be associated to an algebraic equation , like the conics (sections);
    • the mechanical curves can’t be associated to an equation, like the spiral.
  • The Second book (2)
    • After divided the curves and established the general
    • equation of a conic, he divided again the conics, in base on their equation and on their form , distinguishing among:
    And then he determined the normal (a perpendicular line) to a generic plane algebraic curve in a generic point and its tangent. 1) a parabola 2) an ellipse 3) an hyperbola 1) 2) 3)
  • The Third Book
    • In the last book he dealt with how to solve
    • an equation having degree bigger than the
    • second one using intersections of curves
    • and he formulated the “Cartesian rule of
    • the signs” .
  • The last geometrical discovery
    • Finally, apart of all of these studies, Descartes discovered an other curve called with the Latin name Folium of Descartes .
    It is an algebraic curve defined by the equation x 3 +y 3 -3axy=0. It forms a loop in the first quadrant with a double point at the origin and asymptote x+ y +a=0 . It is symmetrical about y = x . The fame says that it was discovered when Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do.
  • Descartes’ Physics studies
    • Descartes was also an important physicist : in fact he made some important discoveries in Physics and he formulated some interesting theories.
    • In this part we will deal with them.
  • His Physical Works
    • We can read these discoveries in
    • his physical works :
    • the “Dioptre” and the “Meteors” , included in his “Discourse on the Method”
    • the “Treatise on the World” .
  • Descartes and Optics
    • His most important physical studies are in Optics , the branch of Physics that studies the propriety of the light allowing to invent glasses and cameras.
  • The Law of the Refraction of the light:
    • Among Descartes’ studies in Optics the most important one is the formulation of the law of the refraction of the light . This law studies a particular phenomenon that happens when the water reflects the light : for example, if we put a stick in the water at half it will seem to be snapped on the surface of the water . This happens because the rays of the light are deflected when they pass from the air to the water .
    • Descartes succeeded to explain this phenomenon thank to this law, that we can state in this way:
    The incidental ray, the refracted ray and the normal to the surface in the point of incidence all lie in the same plane ; the ratio among the sine of the angle of incidence (formed by the normal and by the incidental ray) and the sine of angle of refraction (formed by the normal one and by the refracted ray), independently from the angle of incidence, is constant .
  • The hypothesis of the wavy nature of the light
    • Another important Descartes’ study in Optics was his hypothesis about the light: he considered the light as a pressure on the solid bodies and so he introduced the theory about the wavy nature of the light.
  • Descartes and Astronomy
    • Descartes was also interested in Astronomy : in fact in his “Treatise on the World” he partially agreed with Copernicus’ theory about the heliocentric system and he explains a new theory, called “theory of the vortexes”.
    • With this theory Descartes hypothesized that the Sun was surrounded by a whirling vortex of matter that made all the planets turn around it. Descartes gave up to publish this book because the Roman Catholic Church condemned the heliocentric theory and he was scared for himself.
  • Descartes and Physiology
    • Descartes formulated a theory also in Physiology : he believed that our blood was partially composed by “animal spirits” that, in contact with the heavy matter in the brain, flowed into the nerves and so they made the muscles move.
  • Descartes and Kinematics
    • Descartes’ name appears also in the books of Kinematics , the branch of Physics that studies the body in movement : it is possible to describe the movement of a body only considering the “Cartesian reference system” .
  • The Cartesian reference system
    • The Cartesian reference system is the Cartesian coordinate system applied to all of the three dimensions and it is used to describe the movement of a body in the different points of view of all the bodies near it.
  • Descartes Philosophy
    • René Descartes is considered the father of modern philosophy and the founder of the Rationalism as applied to philosophical research. In fact, he is the first philosopher to begin with the innate ideas which are in our intellect, and lay down the laws which reason must follow in order to arrive at reasonably certain philosophical data. Descartes reaches metaphysical conclusions which aren’t so different from those of Scholastic philosophy. In fact he maintains the transcendent nature of God, upholds human liberty and Christian morality .
  • Philosophical works
    • The philosophical works published by the author were four:
    • Discourse on Method ;
    • Meditations on First Philosophy , in which he proves the existence of God and the immortality of the soul;
    • Principles of Philosophy , in four books, a systematic work reviewing the entire thought of the author;
    • The Passions of the Soul , treating of the problem of morality.
  • The Cartesian Method
    • Descartes, in his work Discourse on Method , after criticising the education which he had received (an indirect attack to the Scholastic method), goes on to set up the new method that, according to him, must be the basis of all scientific and philosophical research and which consists in four laws.
  • These laws are:
    • 1) To accept nothing as true that is not recognized by the reason as clear and distinct ;
    • 2) To analyze complex ideas by breaking them down into their simple constitutive elements , which reason can intuitively apprehend;
    • 3) To reconstruct , beginning with simple ideas and working synthetically to the complex;
    • 4) To make an accurate and complete enumeration of the data of the problem, using in this step both the methods of induction and deduction .
    • Having arrived at this starting point ( clear and distinct ideas ), the intellect begins its discursive and deductive operation (represented by the second and third rules). The second law (called analysis ) directs that the elementary notions be reunited with the clear and distinct ideas (the minor of the Scholastic syllogism). The third law ( synthesis ) presents them as the conclusion flowing from the premises. The final law ( complete enumeration ) stresses that no link in the deductive chain should be omitted and that every step should be logically deduced from the starting point.
    • Working
    • from one step
    • to the next , there will
    • be achieved a system of truths
    • all of which are clear and distinct, because
    • all participate in the same degree of truth enjoyed
    • by the first idea , which was clear and distinct. This ,
    • as we know, is the method adopted in mathematics.
    • Descartes transferred it to Philosophy with the intention of finding clear and distinct concrete ideas , and of deducing from these, through reason alone, an entire system of truths which would also be real or objective . In Descartes ideas do not come from experience , but the intellect finds them by itself . Descartes declares that only these ideas are valid in reality. So, the evidence of an idea is dependent upon its own clearness and distinction.
  • Methodical Doubt: “Cogito ergo sum”
    • Descartes, had first of all to seek out a solid
    • starting point ( evident idea ), and from this opens his deductive process. To arrive at this solid starting point,
    • he begins with methodical doubt , that is, a doubt
    • which will be the means of arriving at certitude .
    • You can doubt all the ideas that exist within your
    • knowing faculties, whether they come through the senses or through the intellect , you can doubt even mathematical truths, in so far as it could be that the human intelligence is under the influence of a malignant genius which takes sport in making what is objectively irrational appears to me as rational.
    • So doubt is brought to its extreme form . But, in spite of this, the hyperbolic doubt causes to rise the most l uminous and indisputable certainty . Even presupposing that the entire content of thoughts is false, the incontestable truth is that “I think” : one cannot doubt without thinking; and if I think , I exist : "Cogito ergo sum."
  • For Descartes the validity of " Cogito ergo sum " rests in this, that the doubt presents intuitively to the mind the subject who doubts, that is, the thinking substance .
    • " Cogito ergo sum" is assumed as a foundation for the primary reality (the existence of the res cogitans ), from which the way to further research is to be taken. With Descartes, philosophy from the science of being becomes the science of thought (epistemology): at first, being conditioned thought, now it is thought that conditions being.
  • The Proof of the Existence of God
    • Once established the certainty of Cogito , Descartes still has to handle the problems raised by Scepticism: the existence of the world (out of Cogito ),
    and the existence of other minds (the other Cogito ). So he analyzes the contents of Cogito ( the ideas), separating them depending on their origins : adventitious , innate and fictitious . He rejects the adventitious ones because they come from the senses (unreliable); he rejects the fictitious because they are invented . Only the innate ones seem to be reliable to find if a reality out of Cogito exists. Between these Descartes privileges the idea of God because, as a perfect idea, it cannot be created by the Cogito itself.
  • At this point Descartes introduces the proofs of the existence of God :
    • " Cogito " has given me, limited and imperfect being, a consciousness . This proves that I have not given existence to myself , for in such a case I would have given myself a perfect nature, not subject to doubt .
    • I have the idea of the perfect : If I did not possess it, I could never know that I am imperfect. Now, where comes this idea from ? Not from myself, for I am imperfect, and the perfect cannot come from the imperfect. So it comes from a Perfect Being , God .
    • The analysis of the idea of the perfect includes the existence of the perfect being, so also existence is included in the idea of the perfect.
  • The nature of God
    • Regarding the nature of God ,
    • Descartes gives to it more or
    • less the same attributes as
    • traditional Christian thought .
    • In Descartes, however, these
    • attributes assume a different significance and value . God, above all, is absolute substance : the only substance, properly so-called. An attribute which has great value for Descartes is the veracity of God. God, the most perfect being, cannot be deceived and cannot deceive.
  • The Malignant Genius
    • So, the veracity of God serves as a guarantee for the entire series of clear and distinct ideas. They are true because if they are not true, proved the existence of God, it would seem that He is deceiving by creating a rational creature who is deceived even in the apprehension of clear and distinct ideas.
    So, with the proof of the existence of God, the hypothesis of a malignant genius falls of its own weight. Ah ah ah!! I’m a malignant genius!!!
  • The origin of ideas
    • Regarding the origin of ideas, Descartes says that the idea of God, primitive notions , logical, mathematical, moral principles , and so forth, are all innate .
    • God is the guarantee of the truth of these innate ideas. The adventitious are derived from the senses , and the fictitious are fashioned by the thinking subject out of the first ones.
    • These two last ones are considered of little worth by Descartes because they do not enjoy the guarantee of the divine veracity, and are fonts of error .
    • Only innate ideas and the rational deduction made from them have the value of truth .
  • The “ Res Extensa ” and the Cartesian World
    • Descartes proves the existence of the world , not from the testimony of our experience but from the innate idea of the " res extensa ." We have a certain idea that is clear, and it is distinct from the " res cogitans ." This idea, granted the veracity of God, cannot be false; so the world exists , and its principal attribute is extension . Concerning the nature of world, Descartes distinguishes what is presented to us through the senses and what through the intellect . The first, not guaranteed by the veracity of God, do not have objective value; they are secondary qualities.
    1
    • The second which must be innate ideas, are primary qualities and are guaranteed by the veracity of God: they are real and objective. The Cartesian World is characterized by the essential attribute of extension ( res extensa ), which is infinite . In this extension God has placed force and movement, which are determined by absolute causality: not purpose, but mechanical determination governs the physical world.
    2
  • THE DUALISM OF SUBSTANCES
    • The entire Cartesian system rests upon a metaphysical dualism : " res cogitans " (God and the human soul) and " res extensa " (the corporeal world). These two realities are irreducible, in so far as thought, liberty and activity are essential to the world of the thinking being, and extension, mechanical determinism and passivity are essential to the world of the " res extensa ." All reciprocal action between the two substances is excluded because it is impossible.
    • So there is opened up the problem which was later to be taken up by rationalism:
    • the determination of the relationship between
    • spirit and matter ;
    • between God
    • (the infinite spirit)
    • and the world
    • (finite matter).
    • This problem presented even graver difficulties in connection with the Cartesian concept of substance, which exists without need of the concursus of any other to exist . Descartes considers thought not as an act , but as the thinking substance ( res cogitans ), that is, as a soul, whose essence is thought. Now such an identification belongs only to God; it is easy to see in this teaching of Descartes the
    • danger of unifying the concepts
    • of man and God and
    • the latent danger of pantheism .
  • ETHICS
    • For Descartes, ethics is the science
    • of the end of man , and this end
    • must be determined by reason .
    • Before reason can arrive at the knowledge of
    • such an end and of the means of
    • reaching it, only the philosopher must construct a provisory morality , a model of life capable of assuring him tranquillity. Granted the present order of creation, Descartes recognizes that the end of man is virtue and happiness .
    • The actuation of this end is brought about through reason , through the knowledge of God, of the soul, and of the world. It is attained through knowledge of God because God is the creator and unifier of the universe; of the soul, because the soul makes clear to us our superiority over material nature; of the physical world, because, governed by causal necessity , it teaches man the virtue of resignation and indifference in the face of the evils of life.
  • THE END