دانستنیهای ریاضی و علوم کامپیوتر
Important Mathematical Discoveries
Augustin Cauchy, a great French mathematician, possessed a remarkable talent from childhood, akin to Gauss, but was deeply religious. He achieved numerous discoveries in mathematics, formulating the theory of functions with a complex variable, which were significant achievements. From that year onwards, Cauchy consistently submitted his astonishing discoveries to the Academy of Sciences for publication.
The Academy's reports were overwhelmed by Cauchy's numerous articles, and Cauchy wished to publish a journal to include all his articles. Two brilliant young men, Niels Henrik Abel of Norway and Évariste Galois of France, revolutionized mathematics with their discoveries.
Abel was raised in a poor family and excelled in mathematics from childhood. In his youth, he went to Berlin and then to Paris, but he struggled to reach the heights of the scientific figures of the day, such as Gauss, Poisson, and Cauchy. However, he eventually managed to record his important discoveries, which Cauchy lost. The unfortunate Abel returned to Norway and, in despair and poverty, passed away at the age of 26.
Some time later, Cauchy found Abel's notes and brought them to the French Academy of Sciences, and a great prize was awarded for Abel's discoveries. Charles Gustav Jacob, a Dutch citizen, also made several significant discoveries. Galois was an unparalleled genius from childhood, but he managed to organize the scattered studies and discoveries of mathematicians in a systematic manner and, with his numerous and rich discoveries, increased the power of mathematical knowledge. Galois brought his discoveries to Cauchy, which, like Abel's notes, were lost, and Galois was deeply saddened until he soon fell ill and died. At the age of 20, he participated in a duel and was shot, and on the last night of his life, he wrote all his discoveries in a concise form while writhing in pain and left them as a will for the world. This scientific legacy was written on the eve of his death.
As one scientist said, "For hundreds of years, numerous generations of great mathematicians will be short of breath; Galois, the founder of group theory, died due to the absence of a doctor."
Mathematics on the Path of Astonishing Progress:
Gaspard Monge (1746-1818), a Frenchman, prepared a geographical map for his homeland in his youth, which was installed in the governor's office. Afterwards, in any school he was sent to, his superiority over the teachers there was immediately apparent. In a school, he succeeded in inventing descriptive geometry, and for the benefit of the country, it was suggested that he keep it secret so that foreigners would not discover this great invention.
As the French Revolution broke out, he joined the ranks of the revolutionaries and made great efforts in the implementation of the revolution's goals. After a few days, he succeeded in establishing the École Polytechnique and teaching there.
Jean-Victor Poncelet was one of those who was captured and imprisoned in Russia during the war between France and Russia. It was in prison that he began to study the lessons, especially geometry, until he was able to prepare his friends for the examinations of the École Polytechnique upon their return to France. Eventually, the theory of transformation by pole and polar attracted his attention before anyone else, and he created projective geometry, and when he presented it to the French Academy of Sciences, they did not pay much attention to it, so he presented it to the Brussels Academy.
Michel Chasles (1793-1880), a Frenchman, was initially a bill broker, but he went bankrupt and went to Belgium, and in his free time, he began to think until, in 1834, a book was chosen called "Machine of the École Polytechnique." Chasles achieved important discoveries every year, including inventing the theory of characteristics.
Jacob Steiner (1786-1863), a German, made numerous discoveries about curves and surfaces. Lagrange, after being chosen as the professor of the École Polytechnique, published his analytical theory and, after a while, his solution to numerical equations in 1797 and opened new ways for analysis. Lagrange was safe from any harm throughout the revolution and afterwards.
"Encyclopedia of Knowledge"
Mathematical Advances
Most mathematical discoveries are used by physicists, and many advancements have been made in the science of physics. Many physicists introduced differential equations, which express the relationship between cause and effect, into the world of physics. Then, they explored partial derivative equations and used them in mathematics. After that, they began to study and apply functions, each of which was a function of an infinite number of different causes. Integral equations have also been used in the sciences of physics. These discoveries, which were also called the calculus of mathematical functions, were followed and completed by several people, including Vito Volterra (1860-1940), but a remarkable transformation and a great honor befell mathematicians.
Because for a long time, scientists calculated whole numbers, and after a while, their calculations were decided to be fractional numbers. Then, the Pythagoreans introduced irrational numbers into mathematics, and after that, negative and imaginary numbers, and finally, the calculus of functions entered the realm of mathematics, which was considered the end point and the evolution of this science.
Introduction
Archimedes, a Greek scientist and mathematician, was born in Syracuse, Greece, in 212 BC, and went to Alexandria to study in his youth. He spent most of his life in his birthplace and had a close friendship with the ruler of this city. Here, we speak of the most famous bath that a human being has taken in the history of mankind. In the stories, it is said that more than 2000 years ago, in the city of Syracuse, the capital of the Greek state of Sicily at that time, Archimedes, a mechanic and mathematician and advisor to the court of King Hiero, made one of his most famous discoveries in the bathhouse.
Discovery in the Bath
One day, as he entered a public bathhouse and sat in it, he observed the water level rising, and suddenly an idea came to his mind. He immediately wrapped a cloth around himself and, in this form, went towards his house and kept shouting, "Eureka, Eureka!" What had he found? The king had given him the mission to discover the secret of the treacherous jeweler of the court and expose him. King Hiero had suspected the jeweler and thought that he had taken a part of the gold that had been given to him to make the royal crown for himself and mixed the rest with silver, which was much cheaper, and made the crown.
Although Archimedes knew that different metals have different specific weights, he thought until that moment that he had to melt the royal crown and cast it into a gold ingot so that he could compare its weight with a pure gold ingot of the same size. But in this method, the royal crown would be destroyed, so he had to find another way to do this. On that day, as he was sitting in the bathhouse, he saw that the water level in the bathhouse rose, and he immediately realized that his body had displaced a certain amount of water in the bathhouse.
Testing and Proving the Impurity of the Royal Crown (A Discovery of the Secrets of Nature)
He hurried back home and began to practically test this finding. He thought that objects of the same size displace the same amount of water, but if we look at the weight, a half-kilogram gold ingot is smaller than a silver ingot of the same weight (gold weighs almost twice as much as silver), so it must displace less water. This was Archimedes' hypothesis, and his experiments proved this hypothesis. For this, he needed a water container and three weights of equal weights, which were the royal crown, the same weight of pure gold, and again the same weight of pure silver.
In his experiment, he found that the royal crown displaced more water than a gold ingot of the same weight, but this amount of water was less than the amount of water that a silver ingot of the same weight displaced. In this way, it was proven that the royal crown was not made of pure gold, but the fraudulent and treacherous jeweler had made it from a mixture of gold and silver, and in this way, Archimedes discovered one of the most striking secrets of nature. That is, the weight of hard objects can be measured with the help of the amount of water they displace. This law (specific weight), which is called density today, is called Archimedes' principle. Even today, after 23 centuries, many scientists still rely on this principle in their calculations.
Activity in Other Fields
Archimedes had very high and remarkable intellectual capacities in mathematics. He invented amazing catapults to defend his lands, which proved very useful. He was able to calculate the surface and volume of objects such as a sphere, cylinder, and cone and created a new method of measurement in mathematical knowledge. Also, obtaining the number π is one of his valuable works. He wrote books about the properties and methods of measuring geometric shapes and volumes such as cones, spiral curves and spirals, parabolas, the surface of a sphere "food" and a cylinder, in addition to which he discovered laws about inclined surfaces, screws, levers, and the center of gravity.
One of Archimedes' new methods in mathematics was obtaining the number π. He devised a method to calculate the number π, that is, the ratio of the circumference of a circle to its diameter, and proved that the number is enclosed between 3.17 and 3.1071. Apart from that, he gave different methods for determining the approximate square root of numbers, and from studying them, it is known that he was familiar with connected or continuous fractions before Indian mathematicians. In the calculation, he set aside the impractical and multi-operational method of the Greeks, who used different signs to represent numbers, and invented his own counting system, with the help of which it was possible to write and read any large number.
The knowledge of fluid equilibrium was discovered by Archimedes, and he was able to apply its laws to determine the equilibrium of floating objects. He also, for the first time, clearly and accurately stated some of the principles of mechanics and discovered the laws of the lever.
Archimedes and Other Scientists of His Time
Archimedes has a saying about himself that has remained immortal despite the passage of centuries, which is: "Give me a point of support, and I will lift the earth." The same statement has been quoted in another form in the literary texts of the Greek language from Archimedes, but the meaning is the same in both cases. Archimedes was like a reclusive and isolated eagle; in his youth, he traveled to Egypt and studied for a while in the city of Alexandria, and in this city, he found two old friends, one Konon (this person was a capable mathematician whom Archimedes had great respect for, both intellectually and personally) and the other Eratosthenes, who, although a worthy mathematician, was considered a superficial man who had extraordinary respect for himself.
Archimedes had constant communication and correspondence with Konon and shared an important and beautiful part of his works with him in these letters, and later, when Konon passed away, Archimedes corresponded with a friend who was one of Konon's students. In 1906, J.L. Heiberg, a scientific historian and specialist in the history of Greek mathematics, succeeded in discovering a valuable document in the city of Constantinople.
This document is a book called "The Theorems of Mechanics and Their Method," which Archimedes had sent to his friend Eratosthenes. The subject of this book is the comparison of the volume or unknown surface of a shape with the volumes and surfaces of known shapes, by which Archimedes succeeded in determining the desired result. This method is one of Archimedes' honors, which allows us to consider him in the sense of a modern and new thinker, because he used everything and anything that was possible to use in any way to be able to attack the problems that occupied his mind.
The second point that allows us to give Archimedes the title of modern is his methods of calculation. Two thousand years before Isaac Newton and Leibniz, he succeeded in inventing integral calculus, and even in solving one of his problems, he used a point that can be considered one of the pioneers of the idea of creating differential calculus.
Farewell to the World
Archimedes' life passed with complete peace, like the life of any other mathematician who has complete security and can bring all the possibilities of his intelligence and genius to the stage of execution. When the Romans conquered the city of Syracuse in 212 BC, the Roman commander Marcellus ordered that none of his soldiers had the right to harass, insult, and injure this famous and great scientist and thinker. Nevertheless, Archimedes fell victim to the Roman victory over the city of Syracuse. He was killed by a drunken Roman soldier, and this was while he was thinking about a mathematical problem in the city market square; it is said that his last words were: "Do not disturb my circles." In this way, the life of Archimedes, the greatest scientist of all time, came to an end; this defenseless 75-year-old mathematician went to another world in 278 BC.
Source: Roshd Encyclopedia
Khayyam
Ghiyath al-Din Abu al-Fath Omar ibn Ibrahim Khayyam (Khayyami) was born in 439 AH (1048 AD) in the city of Nishapur, at a time when the Seljuk Turks had control over Khorasan, a vast area in eastern Iran. He began to learn science in his birthplace and learned the sciences of his time from prominent scholars and professors of that city, including Imam Muwafiq Nishaburi, and as it has been said, he was very young when he became proficient in philosophy and mathematics. Khayyam left Nishapur in 461 AH for Samarkand, where he wrote his outstanding work in algebra under the patronage of Abu Tahir Abd al-Rahman ibn Ahmad, the Qadi al-Qudat of Samarkand.
Khayyam then went to Isfahan and resided there for 18 years, and with the support of Malik-Shah Seljuk and his vizier Nizam al-Mulk, along with a group of scientists and famous mathematicians of his time, he conducted astronomical research in an observatory that was established by the order of Malik-Shah. The result of this research was the reform of the calendar prevalent at that time and the organization of the Jalali calendar (the title of Sultan Malik-Shah Seljuk).
In the Jalali calendar, the solar year is approximately equal to 365 days, 5 hours, 48 minutes, and 45 seconds. The year has twelve months; the first six months are 31 days each, and the next five months are 30 days each, and the last month is 29 days. Every four years, one year is called a leap year, and the last month is 30 days, and that year is 366 days. In the Jalali calendar, there is a time difference of one day every five thousand years, while in the Gregorian calendar, there is an error of three days every ten thousand years.
After the assassination of Nizam al-Mulk and then Malik-Shah, disagreements arose among Malik-Shah's children over the possession of the throne. Due to the turmoil and conflicts arising from this matter, scientific and cultural issues that were previously of special importance were forgotten. The lack of attention to scientific affairs and scientists and the observatory prompted Khayyam to leave Isfahan for Khorasan. He spent the rest of his life in the important cities of Khorasan, especially Nishapur and Marv, which was the capital of Sanjar's rule (the third son of Malik-Shah). At that time, Marv was one of the important scientific and cultural centers of the world, and many scientists were present there. Most of Khayyam's scientific works took place in this city after his return from Isfahan.
Khayyam's scientific achievements for human society have been numerous and very noteworthy. For the first time in the history of mathematics, he categorized equations of the first to third degrees in an admirable way, and then, using geometric drawings based on conic sections, he was able to provide a general solution for all of them. He used both a geometric solution and a numerical solution for second-degree equations, but for third-degree equations, he only used geometric drawings; and in this way, he was able to find a solution for most of them and, in some cases, examine the possibility of two answers. The problem was that due to the undefined negative numbers at that time, Khayyam did not pay attention to the negative answers of the equation and simply ignored the possibility of three answers for the third-degree equation. Nevertheless, almost four centuries before Descartes, he was able to achieve one of the most important achievements in the history of algebra and sciences and put forward a solution that Descartes later (in a more complete form) expressed.
Khayyam was also able to successfully define a number as a continuous quantity and, in fact, define the positive real number for the first time and finally reach the judgment that no quantity is composed of indivisible parts, and mathematically, any amount can be divided into an infinite number of parts. Also, Khayyam, while searching for a way to prove the "parallel postulate" (the fifth postulate of the first article of Euclid's Elements) in the book "Sharh Ma Ashkal min Musadarat Kitab Uqlidis" (Explanation of the Problematic Principles of Euclid's Book), became the innovator of a deep concept in geometry. In an effort to prove this principle, Khayyam stated propositions that were completely in accordance with the propositions that were stated a few centuries later by Wallis and Saccheri, European mathematicians, and paved the way for the emergence of non-Euclidean geometries in the nineteenth century. Many believe that Pascal's arithmetic triangle should be called Khayyam's arithmetic triangle, and some have gone even further and believe that Newton's binomial should be called Khayyam's binomial. Of course, it is said that both Jamshid Kashani and Nasir al-Din al-Tusi, while examining the laws related to taking roots from numbers, brought more than this Newton's rule and the law of forming the coefficient of the binomial expansion.
Khayyam's remarkable talent caused him to have achievements in other fields of human knowledge as well. Short treatises on topics such as mechanics, hydrostatics, meteorology, music theory, etc. have also been left from him. Recently, research has been done on Khayyam's activities in the field of decorative geometry, which confirms his connection with the construction of the northern dome of the Jameh Mosque of Isfahan.
Historians and scientists of Khayyam's time and those who came after him all acknowledged his mastery in philosophy, to the extent that he was sometimes considered the sage of the era and the Avicenna of the time. Khayyam's existing philosophical works are limited to a few short but deep and fruitful treatises. Khayyam's last philosophical treatise reflects his mystical tendencies.
But apart from all this, Khayyam's greatest fame in the last two centuries in the world is due to his quatrains, which were first translated into English by Fitzgerald and made available to the world, and placed his name in the ranks of the four great poets of the world, namely Homer, Shakespeare, Dante, and Goethe. Khayyam's quatrains, due to the very free (and sometimes incorrect) translation of his poetry, have caused some unacceptable misinterpretations of his personality. These quatrains have intensified the debate and disagreement among analysts of Khayyam's thought. Some rely only on the appearance of his quatrains to express his thoughts, while others believe that Khayyam's real thoughts are deeper than can be expressed simply by interpreting his poetry. Khayyam, after a fruitful life, finally passed away in 517 AH (according to most sources) in his homeland of Nishapur, and with his death, one of the most brilliant pages of the history of thought in Iran was closed
Abel
Niels Henrik Abel (1802-1829) is one of the most forward-thinking mathematicians of the 19th century and probably the greatest genius
from the Scandinavian countries. Abel, along with his contemporaries, Gauss and Cauchy, is one of the pioneers of the invention of
modern mathematics, which is characterized by an emphasis on precise proof. His life was a sharp mixture of optimism
humorous when under pressure from poverty and obscurity, and in return for his brilliant achievements
was modest in his youth, and he calmly surrendered in the face of premature death.
Abel was one of six children of a poor priest in one of the villages of Norway. He was not yet sixteen when
his great talent was revealed and encouraged by one of his teachers, and soon he began to read and
understanding the works of Newton, Euler, and Lagrange. He wrote the following point in one of his mathematical notes: "In my opinion, if someone wants to make progress in mathematics, he should
He wrote: "If someone wants to make progress in mathematics, he should study the works of the masters and not the students."
He was not more than eighteen years old when his father died and left the family in poverty. They made a living with the help of friends and neighbors, and with the financial help of a few professors,
this boy was able to enter the University of Oslo in 1821. His first research, which included solving the
the classic problem of the simultaneous curve by means of an integral equation, was published in 1823. This was the first answer
an equation of this type, and paved the way for the vast progress of integral equations in the late nineteenth and early
in the twentieth century. He also proved that the fifth degree equation ax^5+bx^4+cx^3+dx^2+ex+f=0
cannot be solved in the general case like lower degree equations, in terms of radicals, and thus solved a problem that
had plagued mathematicians for 300 years. He published his proof at his own expense in a small pamphlet.
In this scientific growth, Abel soon went beyond Norway and decided to travel to France and Germany. With the support of his friends
and his professors, he applied to the government, and after the usual formalities and delays, he received a scholarship for a scientific journey in Europe. He spent most of his first year abroad in Berlin. There
he had the great good fortune to meet a young and enthusiastic amateur mathematician named August Leopold Crelle, the famous journal of
called "Journal of Pure and Applied Mathematics" was inspired. This was the first periodical in the world that was entirely dedicated to
mathematical research. The first three volumes included 22 articles by Abel.
Abel's early studies in mathematics were limited to the old tradition of the eighteenth century, exemplified by Euler. In Berlin
he was influenced by a new school of thought led by Gauss and Cauchy, and its greatest emphasis was
on precise deduction rather than detailed calculations. At that time, apart from Gauss's great work on hypergeometric series,
there were few proofs in analysis that are still valid today. As Abel explains in a letter to one of his
friends: "If we leave aside the simplest cases, in all mathematics, not even a series can be found
that its sum is precisely determined. In other words, the most important parts of
mathematics lack a basis"
During this period, he wrote the result of his classic studies on binomial series
and in it, he established the general theory of convergence and presented the first convincing proof of the correctness of the expansion of this series.
Abel sent the pamphlet on his fifth-degree equations to Gauss, hoping that it would serve as a scientific passport, for
Gauss, for a reason that is not clear, put it aside without even looking at it, because 30 years later, after his death, it was found unopened among his papers. Unfortunately for both, Abel felt that he had been sabotaged, and decided to go to Paris without meeting Gauss.
In Paris, he met Cauchy, Legendre, Dirichlet, and others, but these meetings were superficial and he was not known as he should have been. He had published several important articles in Crelle's journal at that time, but the French were less
informed of the existence of this periodical, and Abel was ashamed to talk to new acquaintances about his work.
Shortly after his arrival, he completed his masterpiece entitled "Note on a General Property of a Wide Class of Transcendental Functions," which he considered his masterpiece. This work includes a discovery about the integral
algebraic functions, which is known today as Abel's theorem, and is the basis for his later theory about the integral
Abel, and a large part of algebraic geometry. It is said that decades later, Hermite said about Abel's note, "There is so much work left from Abel that it will keep mathematicians busy for 500 years."
Jacobi described Abel's theorem as the greatest discovery of integral calculus in the 19th century. Abel presented his manuscript to the French Academy. He hoped that this work would attract the attention of French mathematicians to him, but he waited in vain until his bag was empty and he was forced to return to Berlin. The incident was as follows: The aforementioned manuscript was given to Cauchy and Legendre for review, Cauchy took it home and put it in an irrelevant place and completely forgot about it, and it was not published until 1841, and at that time, it was lost before the printed copies were read. Finally, the original copy of the article was found in Florence in 1952. Abel completed his first revolutionary article on elliptic functions in Berlin, a subject he had been working on for years, and returned to Norway while heavily in debt.
He expected to be appointed as a professor upon his return, but his dreams were shattered again, and he made a living by teaching privately, and for a short time, he was also employed as an assistant teacher in an institution. During this period, he was completely busy working, and often worked on the theory of elliptic functions, which he had discovered as the inverse of elliptic integrals. This theory quickly established itself as one of the main branches of 19th-century analysis, with many applications in number theory, mathematical physics, and algebraic geometry. In the meantime, Abel's fame reached all the mathematical centers of Europe and he was among the great mathematicians of the world, but he was unaware of this due to his reclusiveness. In early 1829, the disease of tuberculosis, which he had contracted during his travels, progressed so much that it prevented him from working, and in the spring of the same year, Abel passed away at the age of twenty-six. Shortly after his death, Crelle wrote in a memorial that Abel's efforts had been successful, and Abel should be appointed to the chair of mathematics at the University of Berlin.
Crelle praises Abel in his journal as follows: "All his works contain signs of genius and amazing intellectual power. It can be said that he could overcome all obstacles with an irresistible power and penetrate the depths of the problem... His distinguishing feature was his purity and innate nobility, as well as an unparalleled humility that increased his value to the extent of his extraordinary genius." But mathematicians have their own methods for remembering great mathematical men, and they remember him by saying Abel's integral equation, Abel's integrals and functions, Abelian groups, Abel's series, Abel's partial sum formula, Abel's limit theorem in the theory of power series, and Abel's summability. There are few people whose name is connected to so many topics and theorems in modern mathematics, and what he could do during an ordinary life is beyond imagination.
Laplace
Pierre-Simon de Laplace (1749-1828), a French mathematician and theoretical astronomer, was so famous in his time that he was called the Newton of France. His main interests throughout his life were celestial mechanics, probability theory, and promotion.
At the age of twenty-four, he was deeply involved in the details of applying Newton's law of gravity to the entire solar system, in which the planets and their satellites are not affected by the sun, but rather they interact with each other in complex and intertwined forms. Even Newton believed that divine intervention was sometimes necessary to prevent the emergence of disorder in this complex mechanism. Laplace decided to investigate the scientific reasons for this, and he succeeded in proving that an ideal solar system in mathematics is a stable dynamical system that always remains unchanged. This achievement is only one of his many victories, which is mentioned in his great and historical work entitled "Celestial Mechanics" (which was published in five volumes from 1799 to 1825). This work has collected the works of several generations of prominent mathematicians on gravity. Unfortunately, due to his future fame, he removed all references to the discoveries of his predecessors and contemporaries, and created the possibility of the illusion that all these contents and opinions belong to him. Many stories have been mentioned in relation to this research. One of the most famous of them is about the time when Napoleon protested to Laplace to show a point of weakness, why he had written a thick book about the solar system without even mentioning his celestial mechanics. Laplace's comprehensive development of potential theory, which is widely discussed in numerous fields of physical sciences, from gravity and fluid mechanics to electromagnetism and atomic physics, is left for future generations. Although Laplace took the idea of potential from Lagrange without mentioning this matter, he used this theory so extensively that since then, the differential equation of potential theory has always been known as Laplace's basic equation.
Another masterpiece of his was the "Analytical Theory of Probabilities" (1812), which organized his forty years of discoveries about probability. He again forgot to mention the many opinions of others that were mixed with his own opinions, but nevertheless, it is unanimously agreed that his book in this field is the greatest work that has been written in this part of mathematics. In the introduction to this book, he says: "Probability theory is essentially nothing more than a common understanding that has been put into calculation." It may be so, but the 700 pages of subsequent complex analysis, in which he freely used Laplace transforms, generating functions, and many other non-elementary tools, is, according to some, even more complex than the book "Celestial Mechanics."
After the French Revolution, Laplace's political talent and his greed for power reached their peak. His compatriots mock his instability and obedience in the matter of politics. This means that whenever a change occurred in the government (which happened a lot at that time), Laplace would adapt to the environment by changing his principles - he oscillated between strong republicanism and praise of the monarchy - and each time he obtained a better job and a more important title. It is appropriate to compare him to the false representative of the Pope in the story that has come in English literature, who became Catholic twice and Protestant twice. It is said that the Pope's representative replied to the accusation of changing colors: "No, it is not so, because although I changed my religion, I was certainly committed to my principle, the principle that dictates that I must remain a bishop in the story until the end of my life."
To compensate for his mistakes, Laplace was always generous in helping and encouraging younger scientists. He occasionally helped men like the chemist Gay-Lussac, the traveler and naturalist Humboldt, the physicist Poisson, and especially the young Cauchy, who became one of the greatest creators of mathematics in the 19th century, in advancing their work.
Lagrange
Joseph-Louis Lagrange (1736-1813) hated geometry, but he has outstanding discoveries in the calculus of variations and analytical mechanics. He also contributed to the compilation of number theory and algebra, and he helped the intellectual current that was later strengthened by Gauss and Abel. His mathematical life can be considered a natural extension of the work of his older and more important contemporary, Euler, who, in many ways, advanced his work and refined it.
Lagrange was born in Turin from mixed French-Italian ancestors. During his childhood, he was more interested in art and literature than in science. But it was still in school that his interest in mathematics was ignited by reading an article by Edmund Halley on the applications of algebra in optics. He then began a period of independent studies, and he progressed so quickly that at the age of 19, he was appointed as a professor at the Royal Military School of Turin.
Lagrange's works in the calculus of variations are among his first and most important works. In 1755, he informed Euler of his method of multipliers to solve the isoperimetric problem. These problems had occupied Euler's mind for years, because they were beyond the limits of his semi-geometric methods. Euler immediately received the answer to many of the questions that were in his mind, but he responded to Lagrange with commendable kindness and forgiveness, and refrained from publishing his works, as he wrote in a letter to Lagrange: "I refrained from publishing them so as not to deprive you of any part of the honors that belong to you." He continued, and both he and Euler used it in many new types of problems, especially in mechanics.
In 1766, when Euler left Berlin for St. Petersburg, he suggested to Frederick the Great that Lagrange be invited to succeed him. Lagrange accepted this invitation and lived in Berlin for 20 years until Frederick's death in 1786. During this time, he worked extensively in algebra and number theory, and he wrote his masterpiece, the "Analytical Mechanics" (1788), in which he integrated general mechanics, and as Hamilton called Lagrange's equations of motion, generalized coordinates, and the concept of potential energy.
After the death of Frederick, the atmosphere of the Prussian court became relatively pleasant for scientists, so Lagrange accepted Louis XVI's invitation to go to Paris, and there, he was given an apartment in Auvergne. Lagrange, despite all his great genius, was humble and unbiased. And although he was a companion of the nobles and in fact one of them, he was respected and noticed by all parties during the unrest of the French Revolution. His most important work during this period was his guiding and pioneering role in establishing the metric system for weights and measures.
In mathematics, he tried to provide an acceptable basis for the basic processes of analysis, but these efforts were largely fruitless. In his later years, he felt that mathematics had reached a dead end and that physics, chemistry, biology, and other sciences would attract the most powerful minds of the future. This pessimism might have disappeared if he could have foreseen the arrival of Gauss and his successors on the scene, who turned the 19th century into the richest stage in the long history of mathematics.
Poincaré
Jules Henri Poincaré (1854-1912) was recognized worldwide as the greatest mathematician of his generation at the beginning of the twentieth century. In 1879, he began his university studies in Caen, and only two years later, he was appointed as a professor at the Sorbonne University. He spent the rest of his life there, and each year he taught a different subject. In his lectures - which were edited and published by his students - with great initiative and technical mastery, he actually discussed all the well-known fields of pure and applied mathematics, and many fields that were unknown before his discovery. In total, he wrote more than 30 technical books on mathematical physics and celestial mechanics, six books at a general level, and approximately 50
