Saturday, 2 June 2018

René Descartes By Er. Sajad Ahmad Rather


René Descartes
FRENCH MATHEMATICIAN AND PHILOSOPHER

René Descartes, (born March 31, 1596, La Haye, Touraine, France—died February 11, 1650, Stockholm, Sweden), French mathematician, scientist, and philosopher. Because he was one of the first to abandon scholastic Aristotelianism, because he formulated the first modern version of mind-body dualism, from which stems the mind-body problem, and because he promoted the development of a new science grounded in observation and experiment, he has been called the father of modern philosophy. Applying an original system of methodical doubt, he dismissed apparent knowledge derived from authority, the senses, and reason and erected new epistemic foundations on the basis of the intuition that, when he is thinking, he exists; this he expressed in the dictum “I think, therefore I am” (best known in its Latin formulation, “Cogito, ergo sum,” though originally written in French, “Je pense, donc je suis”). He developed a metaphysical dualism that distinguishes radically between mind, the essence of which is thinking, and matter, the essence of which is extension in three dimensions. Descartes’s metaphysics is rationalist, based on the postulation of innate ideas of mind, matter, and God, but his physics and physiology, based on sensory experience, are mechanistic and empiricist.
Early Life And Education

René du Perron Descartes was born in La Haye en Touraine (now Descartes, Indre-et-Loire), France, on 31 March 1596. His mother, Jeanne Brochard, died soon after giving birth to him, and so he was not expected to survive. Descartes' father, Joachim, was a member of the Parlement of Brittany at Rennes. René lived with his grandmother and with his great-uncle. Although the Descartes family was Roman Catholic, the Poitou region was controlled by the Protestant Huguenots. In 1607, late because of his fragile health, he entered the Jesuit Collège Royal Henry-Le-Grand at La Flèche, where he was introduced to mathematics and physics, including Galileo's work. After graduation in 1614, he studied for two years (1615–16) at the University of Poitiers, earning a Baccalauréat and Licence in canon and civil law in 1616, in accordance with his father's wishes that he should become a lawyer. From there he moved to Paris.

In his book Discourse on the Method, Descartes recalls,
I entirely abandoned the study of letters. Resolving to seek no knowledge other than that of which could be found in myself or else in the great book of the world, I spent the rest of my youth travelling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting upon whatever came my way so as to derive some profit from it.
Given his ambition to become a professional military officer, in 1618, Descartes joined, as a mercenary, the Protestant Dutch States Army in Breda under the command of Maurice of Nassau, and undertook a formal study of military engineering, as established by Simon Stevin. Descartes, therefore, received much encouragement in Breda to advance his knowledge of mathematics. In this way, he became acquainted with Isaac Beeckman, the principal of a Dordrecht school, for whom he wrote the Compendium of Music (written 1618, published 1650). Together they worked on free fallcatenaryconic section, and fluid statics. Both believed that it was necessary to create a method that thoroughly linked mathematics and physics.
While in the service of the Catholic Duke Maximilian of Bavaria since 1619, Descartes was present at the Battle of the White Mountainoutside Prague, in November 1620.

Philosophical Work
Initially, Descartes arrives at only a single principle: thought exists. Thought cannot be separated from me, therefore, I exist (Discourse on the Method and Principles of Philosophy). Most famously, this is known as cogito ergo sum (English: "I think, therefore I am"). Therefore, Descartes concluded, if he doubted, then something or someone must be doing the doubting, therefore the very fact that he doubted proved his existence. "The simple meaning of the phrase is that if one is skeptical of existence, that is in and of itself proof that he does exist."
Descartes concludes that he can be certain that he exists because he thinks. But in what form? He perceives his body through the use of the senses; however, these have previously been unreliable. So Descartes determines that the only indubitable knowledge is that he is a thinking thing. Thinking is what he does, and his power must come from his essence. Descartes defines "thought" (cogitatio) as "what happens in me such that I am immediately conscious of it, insofar as I am conscious of it". Thinking is thus every activity of a person of which the person is immediately conscious. He gave reasons for thinking that waking thoughts are distinguishable from dreams, and that one's mind cannot have been "hijacked" by an evil demon placing an illusory external world before one's senses.
And so something that I thought I was seeing with my eyes is in fact grasped solely by the faculty of judgment which is in my mind.
In this manner, Descartes proceeds to construct a system of knowledge, discarding perception as unreliable and, instead, admitting only deduction as a method.

Dualism

Descartes, influenced by the automatons on display throughout the city of Paris, began to investigate the connection between the mind and body, and how the two interact. His main influences for dualism were theology and physics. The theory on the dualism of mind and body is Descartes' signature doctrine and permeates other theories he advanced. Known as Cartesian dualism, his theory on the separation between the mind and the body went on to influence subsequent Western philosophies. In Meditations on First Philosophy Descartes attempted to demonstrate the existence of God and the distinction between the human soul and the body. Humans are a union of mind and body, thus Descartes' dualism embraced the idea that mind and body are distinct but closely joined. While many contemporary readers of Descartes found the distinction between mind and body difficult to grasp, he thought it was entirely straightforward. Descartes employed the concept of modes, which are the ways in which substances exist. In Principles of Philosophy Descartes explained "we can clearly perceive a substance apart from the mode which we say differs from it, whereas we cannot, conversely, understand the mode apart from the substance". To perceive a mode apart from its substance requires an intellectual abstraction, which Descartes explained as follows:
"The intellectual abstraction consists in my turning my thought away from one part of the contents of this richer idea the better to apply it to the other part with greater attention. Thus, when I consider a shape without thinking of the substance or the extension whose shape it is, I make a mental abstraction."[
According to Descartes two substances are really distinct when each of them can exist apart from the other. Thus Descartes reasoned that God is distinct from humans, and the body and mind of a human are also distinct from one another. He argued that the great differences between body and mind make the two always divisible. But that the mind was utterly indivisible, because "when I consider the mind, or myself in so far as I am merely a thinking thing, I am unable to distinguish any part within myself; I understand myself to be something quite single and complete."[
In Meditations Descartes discussed a piece of wax and exposed the single most characteristic doctrine of Cartesian dualism: that the universe contained two radically different kinds of substances - the mind or soul defined as thinking, and the body defined as matter and unthinking. The Aristotelian philosophy of Descartes' days held that the universe was inherently purposeful or theological. Everything that happened, be it the motion of the stars or the growth of a tree, was supposedly explainable by a certain purpose, goal or end that worked its way out within nature. Aristotle called this the "final cause", and these final causes were indispensable for explaining the ways nature operated. With his theory on dualism Descartes fired the opening shot for the battle between the traditional Aristotelian science and the new science of Kepler and Galileo which denied the final cause for explaining nature. Descartes' dualism provided the philosophical rationale for the latter and he expelled the final cause from the physical universe (or res extensa). For Descartes the only place left for the final cause was the mind (or res cogitans). Therefore, while Cartesian dualism paved the way for modern physics, it also held the door open for religious beliefs about the immortality of the soul.
Descartes' dualism of mind and matter implied a concept of human beings. A human was according to Descartes a composite entity of mind and body. Descartes gave priority to the mind and argued that the mind could exist without the body, but the body could not exist without the mind. In Meditations Descartes even argues that while the mind is a substance, the body is composed only of "accidents". But he did argue that mind and body are closely joined, because:
"Nature also teaches me, by the sensations of pain, hunger, thirst and so on, that I am not merely present in my body as a pilot in his ship, but that I am very closely joined and, as it were, intermingled with it, so that I and the body form a unit. If this were not so, I, who am nothing but a thinking thing, would not feel pain when the body was hurt, but would perceive the damage purely by the intellect, just as a sailor perceives by sight if anything in his ship is broken.
Descartes' discussion on embodiment raised one of the most perplexing problems of his dualism philosophy: What exactly is the relationship of union between the mind and the body of a person? Therefore, Cartesian dualism set the agenda for philosophical discussion of the mind–body problem for many years after Descartes' death. Descartes was also a rationalist and believed in the power of innate ideas. Descartes argued the theory of innate knowledge and that all humans were born with knowledge through the higher power of God. It was this theory of innate knowledge that later led philosopher John Locke (1632-1704) to combat the theory of empiricism, which held that all knowledge is acquired through experience.

Descartes on physiology and psychology

In The Passions of the Soul written between 1645 and 1646 Descartes discussed the common contemporary belief that the human body contained animal spirits. These animal spirits were believed to be light and roaming fluids circulating rapidly around the nervous system between the brain and the muscles, and served as a metaphor for feelings, like being in high or bad spirit. These animal spirits were believed to affect the human soul, or passions of the soul. Descartes distinguished six basic passions: wonder, love, hatred, desire, joy and sadness. All of these passions, he argued, represented different combinations of the original spirit, and influenced the soul to will or want certain actions. He argued, for example, that fear is a passion that moves the soul to generate a response in the body. In line with his dualist teachings on the separation between the soul and the body, he hypothesized that some part of the brain served as a connector between the soul and the body and singled out the pineal gland as connector.[  Descartes argued that signals passed from the ear and the eye to the pineal gland, through animal spirits. Thus different motions in the gland cause various animal spirits. He argued that these motions in the pineal gland are based on God's will and that humans are supposed to want and like things that are useful to them. But he also argued that the animal spirits that moved around the body could distort the commands from the pineal gland, thus humans had to learn how to control their passions.
Descartes advanced a theory on automatic bodily reactions to external events which influenced 19th century reflex theory. He argued that external motions such as touch and sound reach the endings of the nerves and affect the animal spirits. Heat from fire affects a spot on the skin and sets in motion a chain of reactions, with the animal spirits reaching the brain through the central nervous system, and in turn animal spirits are sent back to the muscles to move the hand away from the fire. Through this chain of reactions the automatic reactions of the body do not require a thought process.
Above all he was among the first scientists who believed that the soul should be subject to scientific investigation. He challenged the views of his contemporaries that the soul was divine, thus religious authorities regarded his books as dangerous. Descartes' writings went on to form the basis for theories on emotions and how cognitive evaluations were translated into affective processes. Descartes believed that the brain resembled a working machine and unlike many of his contemporaries believed that mathematics and mechanics could explain the most complicated processes of the mind. In the 20th century Alan Turing advanced computer science based on mathematical biology as inspired by Descartes. His theories on reflexes also served as the foundation for advanced physiological theories more than 200 years after his death. The Nobel Prize winning physiologist Ivan Pavlov was a great admirer of Descartes.[80]

Three types of ideas

There are three kinds of ideas, Descartes explained: Fabricated, Innate, and Adventitious. Fabricated ideas are inventions made by the mind. For example, a person has never eaten moose but assumes it tastes like cow. Adventitious ideas are ideas that cannot be manipulated or changed by the mind. For example, a person stands in a cold room, they can only think of the feeling as cold and nothing else. Innate ideas are set ideas made by God in a person’s mind. For example, the features of a shape can be examined and set aside, but its content can never be manipulated to cause it not to be a three sided object.

Descartes' moral philosophy

For Descartes, ethics was a science, the highest and most perfect of them. Like the rest of the sciences, ethics had its roots in metaphysics. In this way, he argues for the existence of God, investigates the place of man in nature, formulates the theory of mind-body dualism, and defends free will. However, as he was a convinced rationalist, Descartes clearly states that reason is sufficient in the search for the goods that we should seek, and virtue consists in the correct reasoning that should guide our actions. Nevertheless, the quality of this reasoning depends on knowledge, because a well-informed mind will be more capable of making good choices, and it also depends on mental condition. For this reason, he said that a complete moral philosophy should include the study of the body. He discussed this subject in the correspondence with Princess Elisabeth of Bohemia, and as a result wrote his work The Passions of the Soul, that contains a study of the psychosomatic processes and reactions in man, with an emphasis on emotions or passions. His works about human passion and emotion would be the basis for the philosophy of his followers, (see Cartesianism), and would have a lasting impact on ideas concerning what literature and art should be, specifically how it should invoke emotion.
Humans should seek the sovereign good that Descartes, following Zeno, identifies with virtue, as this produces a solid blessedness or pleasure. For Epicurus the sovereign good was pleasure, and Descartes says that, in fact, this is not in contradiction with Zeno's teaching, because virtue produces a spiritual pleasure, that is better than bodily pleasure. Regarding Aristotle's opinion that happiness depends on the goods of fortune, Descartes does not deny that this good contributes to happiness but remarks that they are in great proportion outside one's own control, whereas one's mind is under one's complete control. The moral writings of Descartes came at the last part of his life, but earlier, in his Discourse on the Method he adopted three maxims to be able to act while he put all his ideas into doubt. This is known as his "Provisional Morals".

Descartes and natural science

Descartes is often regarded as the first thinker to emphasize the use of reason to develop the natural sciences. For him the philosophy was a thinking system that embodied all knowledge, and expressed it in this way:
Thus, all Philosophy is like a tree, of which Metaphysics is the root, Physics the trunk, and all the other sciences the branches that grow out of this trunk, which are reduced to three principals, namely, Medicine, Mechanics, and Ethics. By the science of Morals, I understand the highest and most perfect which, presupposing an entire knowledge of the other sciences, is the last degree of wisdom.
In his Discourse on the Method, he attempts to arrive at a fundamental set of principles that one can know as true without any doubt. To achieve this, he employs a method called hyperbolical/metaphysical doubt, also sometimes referred to as methodological scepticism: he rejects any ideas that can be doubted and then re-establishes them in order to acquire a firm foundation for genuine knowledge. Descartes built his ideas from scratch. He relates this to architecture: the top soil is taken away to create a new building or structure. Descartes calls his doubt the soil and new knowledge the buildings. To Descartes, Aristotle’s foundationalism is incomplete and his method of doubt enhances foundationalism.

Descartes on animals

Descartes denied that animals had reason or intelligence. He argued that animals did not lack sensations or perceptions, but these could be explained mechanistically. Whereas humans had a soul, or mind, and were able to feel pain and anxiety, animals by virtue of not having a soul could not feel pain or anxiety. If animals showed signs of distress then this was to protect the body from damage, but the innate state needed for them to suffer was absent. Although Descartes' views were not universally accepted they became prominent in Europe and North America, allowing humans to treat animals with impunity. The view that animals were quite separate from humanity and merely machines allowed for the maltreatment of animals, and was sanctioned in law and societal norms until the middle of the 19th century. The publications of Charles Darwin would eventually erode the Cartesian view of animals. Darwin argued that the continuity between humans and other species opened the possibilities that animals did not have dissimilar properties to suffer.

Mathematical legacy

One of Descartes' most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He "invented the convention of representing unknowns in equations by xy, and z, and knowns by ab, and c". He also "pioneered the standard notation" that uses superscripts to show the powers or exponents; for example, the 2 used in x2 to indicate x squared.[103][104] He was first to assign a fundamental place for algebra in our system of knowledge, using it as a method to automate or mechanize reasoning, particularly about abstract, unknown quantities. European mathematicians had previously viewed geometry as a more fundamental form of mathematics, serving as the foundation of algebra. Algebraic rules were given geometric proofs by mathematicians such as PacioliCardanTartaglia and Ferrari. Equations of degree higher than the third were regarded as unreal, because a three-dimensional form, such as a cube, occupied the largest dimension of reality. Descartes professed that the abstract quantity a2 could represent length as well as an area. This was in opposition to the teachings of mathematicians, such as Vieta, who argued that it could represent only area. Although Descartes did not pursue the subject, he preceded Gottfried Wilhelm Leibniz in envisioning a more general science of algebra or "universal mathematics," as a precursor to symbolic logic, that could encompass logical principles and methods symbolically, and mechanize general reasoning.
Descartes' work provided the basis for the calculus developed by Newton and Leibniz, who applied infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics. His rule of signs is also a commonly used method to determine the number of positive and negative roots of a polynomial.
Descartes discovered an early form of the law of conservation of mechanical momentum (a measure of the motion of an object), and envisioned it as pertaining to motion in a straight line, as opposed to perfect circular motion, as Galileo had envisioned it. He outlined his views on the universe in his Principles of Philosophy.
Descartes also made contributions to the field of optics. He showed by using geometric construction and the law of refraction (also known as Descartes' law or more commonly Snell's law) that the angular radius of a rainbow is 42 degrees (i.e., the angle subtended at the eye by the edge of the rainbow and the ray passing from the sun through the rainbow's centre is 42°). He also independently discovered the law of reflection, and his essay on optics was the first published mention of this law.
Final Years And Heritage
In 1644, 1647, and 1648, after 16 years in the Netherlands, Descartes returned to France for brief visits on financial business and to oversee the translation into French of the Principles, the Meditations, and the Objections and Replies. (The translators were, respectively, Picot, Charles d’Albert, duke de Luynes, and Claude Clerselier.) In 1647 he also met with Gassendi and Hobbes, and he suggested to Pascal the famous experiment of taking a barometer up Mount Puy-de-Dôme to determine the influence of the weight of the air. Picot returned with Descartes to the Netherlands for the winter of 1647–48. During Descartes’s final stay in Paris in 1648, the French nobility revolted against the crown in a series of wars known as the Fronde. Descartes left precipitously on August 17, 1648, only days before the death of his old friend Mersenne.
Clerselier’s brother-in-law, Hector Pierre Chanut, who was French resident in Sweden and later ambassador, helped to procure a pension for Descartes from Louis XIV, though it was never paid. Later, Chanut engineered an invitation for Descartes to the court of Queen Christina, who by the close of the Thirty Years’ War (1618–48) had become one of the most important and powerful monarchs in Europe. Descartes went reluctantly, arriving early in October 1649. He may have gone because he needed patronage; the Fronde seemed to have destroyed his chances in Paris, and the Calvinist theologians were harassing him in the Netherlands.
In Sweden—where, Descartes said, in winter men’s thoughts freeze like the water—the 22-year-old Christina perversely made the 53-year-old Descartes rise before 5:00 AM to give her philosophy lessons, even though she knew of his habit of lying in bed until 11 o’clock in the morning. She also is said to have ordered him to write the verses of a ballet, The Birth of Peace (1649), to celebrate her role in the Peace of Westphalia, which ended the Thirty Years’ War. The verses in fact were not written by Descartes, though he did write the statutes for a Swedish Academy of Arts and Sciences. While delivering these statutes to the queen at 5:00 AM on February 1, 1650, he caught a chill, and he soon developed pneumonia. He died in Stockholm on February 11. Many pious last words have been attributed to him, but the most trustworthy report is that of his German valet, who said that Descartes was in a coma and died without saying anything at all.
Descartes’s papers came into the possession of Claude Clerselier, a pious Catholic, who began the process of turning Descartes into a saint by cutting, adding to, and selectively publishing his letters. This cosmetic work culminated in 1691 in the massive biography by Father Adrien Baillet, who was at work on a 17-volume Lives of the Saints. Even during Descartes’s lifetime there were questions about whether he was a Catholic apologist, primarily concerned with supporting Christian doctrine, or an atheist, concerned only with protecting himself with pious sentiments while establishing a deterministic, mechanistic, and materialistic physics.
These questions remain difficult to answer, not least because all the papers, letters, and manuscripts available to Clerselier and Baillet are now lost. In 1667 the Roman Catholic church made its own decision by putting Descartes’s works on the Index Librorum Prohibitorum (Latin: “Index of Prohibited Books”) on the very day his bones were ceremoniously placed in Sainte-Geneviève-du-Mont in Paris. During his lifetime, Protestant ministers in the Netherlands called Descartes a Jesuit and a papist—which is to say an atheist. He retorted that they were intolerant, ignorant bigots. Up to about 1930, a majority of scholars, many of whom were religious, believed that Descartes’s major concerns were metaphysical and religious. By the late 20th century, however, numerous commentators had come to believe that Descartes was a Catholic in the same way he was a Frenchman and a royalist—that is, by birth and by convention.
Descartes himself said that good sense is destroyed when one thinks too much of God. He once told a German protégée, Anna Maria van Schurman (1607–78), who was known as a painter and a poet, that she was wasting her intellect studying Hebrew and theology. He also was perfectly aware of—though he tried to conceal—the atheistic potential of his materialist physics and physiology. Descartes seemed indifferent to the emotional depths of religion. Whereas Pascal trembled when he looked into the infinite universe and perceived the puniness and misery of man, Descartes exulted in the power of human reason to understand the cosmos and to promote happiness, and he rejected the view that human beings are essentially miserable and sinful. He held that it is impertinent to pray to God to change things. Instead, when we cannot change the world, we must change ourselves.



Carl Friedrich Gauss By Er. Sajad Ahmad Rather


CARL FRIEDRICH GAUSS




Johann Carl Friedrich Gauss ( 30 April 1777 – 23 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including  algebraanalysisastronomydifferential geometryelectrostatics, etc.

Sometimes referred to as the Princeps mathematicorum (Latin for "the foremost of mathematicians") and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.


Early Life

Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Gauss later solved this puzzle about his birth date in the context of finding the date of Easter, deriving methods to compute the date in both past and future years. He was christened and confirmed in a church near the school he attended as a child.
Gauss was a child prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100. There are many other anecdotes about his precocity while a toddler, and he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opusDisquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Career and achievements

Algebra

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to the implicit use of the Jordan curve theorem. However, he subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), which, among other things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass. It appears that Gauss already knew the class number formula in 1801.

In addition, he proved the following conjectured theorems:

·         Fermat polygonal number theorem for n = 3

·         Fermat's last theorem for n = 5

·         Descartes's rule of signs

·         Kepler conjecture for regular arrangements

He also

·         explained the pentagramma mirificum (see University of Bielefeld website)

·         developed an algorithm for determining the date of Easter

·         invented the Cooley–Tukey FFT algorithm for calculating the discrete Fourier transforms 160 years before Cooley and Tukey


Astronomy

In the same year, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres. Piazzi could only track Ceres for somewhat more than a month, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data—three degrees represent less than 1% of the total orbit.

Gauss, who was 24 at the time, heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 31 December at Gotha, and one day later by Heinrich Olbers in Bremen.

Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work, Gauss used comprehensive approximation methods which he created for that purpose.

One such method was the fast Fourier transform. While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey,[54] Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier on the subject in 1807.

Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". Though Gauss had up to that point been financially supported by his stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support. Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical observatory in Göttingen, a post he held for the remainder of his life.

The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Theory of motion of the celestial bodies moving in conic sections around the Sun). In the process, he so streamlined the cumbersome mathematics of 18th-century orbital prediction that his work remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error.

Gauss proved the method under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares."


                                                
Geodetic survey

In 1818 Gauss, putting his calculation skills to practical use, carried out a geodetic survey of the Kingdom of Hanover, linking up with previous Danish surveys. To aid the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.


Non-Euclidean geometries

Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.

Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry.

Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."

This unproved statement put a strain on his relationship with Bolyai who thought that Gauss was "stealing" his idea.

Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, a biographer of Gauss, argues in Gauss, Titan of Sciencethat Gauss was in fact in full possession of non-Euclidean geometry long before it was published by Bolyai, but that he refused to publish any of it because of his fear of controversy.

Theorema Egregium

The geodetic survey of Hanover, which required Gauss to spend summers traveling on horseback for a decade, fueled Gauss's interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. Among other things, he came up with the notion of Gaussian curvature. This led in 1828 to an important theorem, the Theorema Egregium (remarkable theorem), establishing an important property of the notion of curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface.

That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space.


In 1821, he was made a foreign member of the Royal Swedish Academy of Sciences. Gauss was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1822.

Magnetism

In 1831, Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism (including finding a representation for the unit of magnetism in terms of mass, charge, and time) and the discovery of Kirchhoff's circuit laws in electricity. It was during this time that he formulated his namesake law. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic observatory to be built in the garden of the observatory, and with Weber founded the "Magnetischer Verein" (magnetic club in German), which supported measurements of Earth's magnetic field in many regions of the world. He developed a method of measuring the horizontal intensity of the magnetic field which was in use well into the second half of the 20th century, and worked out the mathematical theory for separating the inner and outer (magnetospheric) sources of Earth's magnetic field.


Later years and death

Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness.  For example, at the age of 62, he taught himself Russian.

In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula.

In 1845, he became an associated member of the Royal Institute of the Netherlands; when that became the Royal Netherlands Academy of Arts and Sciences in 1851, he joined as a foreign member.

In 1854, Gauss selected the topic for Bernhard Riemann's Habilitationsvortrag, "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (habilitation lecture About the hypotheses that underlie Geometry). On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.

On 23 February 1855, Gauss died of a heart attack in Göttingen (then Kingdom of Hanover and now Lower Saxony); he is interred in the Albani Cemetery there. Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer. Gauss's brain was preserved and was studied by Rudolf Wagner, who found its mass to be slightly above average, at 1,492 grams, and the cerebral area equal to 219,588 square millimetres (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius.