Mathematical Insights
Important Discoveries in Mathematics
Augustin Cauchy, a great French mathematician, showed great talent from childhood, like Gauss, but was strictly religious. He achieved numerous discoveries in mathematics and expressed the theory of functions that have an imaginary variable, which are very great discoveries. Cauchy, from that year onwards, constantly succeeded with amazing discoveries that he sent to the Academy of Sciences to be printed.
The Academy's reports were terrifying because Cauchy's articles were infinitely numerous, and Cauchy wanted to publish a magazine in which he would include all of his articles. Two brilliant young men, one named Menrik Abel of Norway and the other Evariste Galois of France, brought about a deep transformation in mathematics with their discoveries.
Abel was raised in a poor family and from childhood, his genius in mathematics shone. In his early youth, he came to Berlin and then to Paris, and no matter how much he tried to reach the lofty peaks of science of that day, such as Gauss, Poisson, and Cauchy, it was not easy for him, but finally, he was able to find notes that contained his important discoveries, which he lost. Poor Abel returned to Norway and, in despair and poverty, closed his eyes to the world at the age of 26.
A few years later, Cauchy found Abel's notes and took them to the French Academy of Sciences, and a great prize was awarded to Abel's discoveries, and Charles Gustav Jacobi of the Netherlands also achieved several great discoveries. Galois was an unparalleled genius from childhood, but he was able to organize the scattered studies and discoveries of mathematicians in a systematic way and increase the power of mathematical knowledge with his numerous and rich discoveries. Galois brought his discoveries to Cauchy, which were lost like Abel's notes, and Galois was also very upset until he soon fell ill, because at the age of 20, he participated in a duel and a bullet entered his body, 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 testament for the world. This scientific legacy was written on the night of his death.
As one of the scientists said, "Hundreds of years will cause multiple generations of great mathematicians to suffer from shortness of breath. Galois, the creator of the group theory, died due to the absence of a doctor."
Mathematics on the path of amazing progress:
Gaspard Monge (1746-1818), a Frenchman, prepared a geographical map for his homeland in his early youth, which was installed in the governor's office. After that, 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 many efforts to implement the goals of the revolution. After a few days, he succeeded in establishing the Polytechnic School 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 in question, especially geometry, until he was able to prepare his friends for the Polytechnic exams upon their return to France. Eventually, the theory of conversion into the method of pole and polar attracted his attention before anyone else and created descriptive 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 spent his free time thinking until he chose a book called "Machine of the Polytechnic School" in 1834. Chasles achieved important discoveries every year, including inventing the theory of characteristics.
Jakob Steiner (1786-1863), a German, made numerous discoveries about curves and surfaces. Lagrange, after being chosen as the head of the Polytechnic School, published the analytical theory and, after a while, the solution of his numerical equations in 1797 and opened new ways for analysis. Lagrange was safe from any harm during the entire revolution and after it.
"Encyclopedia Book"
Mathematical Advances
Most mathematical discoveries are used by physicists, and much progress has 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 went beyond partial derivative equations and used them in mathematics. After that, they began to study and apply the 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 calculation of mathematical functions, were followed and completed by several people from Vito Volterra (1860 and 1940), but a great transformation and a great honor came to 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 calculation of functions entered the realm of mathematics, which was considered the end point and the evolution of this science.
Introduction
Archimedes, the Greek scientist and mathematician, was born in 212 BC in the city of Syracuse, Greece, and went to Alexandria to learn knowledge 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 are talking about the most famous bath that a human has taken in the history of mankind. In the stories, it is said that more than 2,000 years ago, in the city of Syracuse, the capital of the Greek state of Sicily at that time, Archimedes, the mechanic and mathematician and advisor to the court of King Hieron, made one of his most famous discoveries in the bathhouse.
Discovery in the bath
One day, when he entered a public bath and sat in it, and while doing so, he observed the water in the bathhouse rising, a thought suddenly occurred to his mind. He immediately wrapped a cloth around himself and, in this form, went towards his house and kept shouting, "I found it, I found it." What had he found? The king had given him the mission to discover the secret of the traitorous jeweler of the court and expose him. King Hieron had suspected the jeweler and thought that he had taken 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, when he was sitting in the bathhouse, he saw the water in the bathhouse rise, and he immediately realized that his body had pushed back and 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 subject from a weight point of view, 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 should displace less water. This was Archimedes' hypothesis, and his experiments proved this hypothesis. For this, he needed a water container and three weights with 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 traitorous jeweler 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 the field of mathematics. He invented amazing catapults to defend his lands, which proved to be very useful. He was able to calculate the surface and volume of objects such as spheres, cylinders, and cones and created a new method for measurement in mathematical knowledge. Also, obtaining the number 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 spiral lines, parabolas, the surface of a sphere "food" and a cylinder, in addition to that, he discovered laws about inclined surfaces, screws, levers, and the center of gravity.
One of Archimedes' new methods in mathematics was obtaining the number. He gave a method to calculate the number pi, 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 root of numbers, and from studying them, it is known that he was familiar with connected or continuous fractions before the Indian mathematicians. In the account, he put aside the impractical and multi-operational method of the Greeks, who used different signs to represent numbers, and invented a counting device for himself, with the help of which it was possible to write and read any large number.
The knowledge of the equilibrium of liquids was discovered by Archimedes, and he was able to use its laws to determine the equilibrium of floating objects. Also, for the first time, he clearly and accurately expressed 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, and that 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 secluded and isolated eagle. In his youth, he traveled to Egypt and studied for a while in the city of Alexandria, and he found two old friends in this city, one Konon (this person was a capable mathematician whom Archimedes respected very much, both intellectually and personally) and the other Eratosthenes, who, although a worthy mathematician, was a superficial man who held extraordinary respect for himself.
Archimedes had a permanent 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 "Mechanics and Their Methods," which Archimedes had sent to his friend Eratosthenes. The subject of this book is the comparison of the unknown volume or surface of a shape with the known volumes and surfaces of other shapes, by which Archimedes succeeded in determining the desired result. This method is one of Archimedes' honors, which allows us to consider him as the concept of a new and modern thinker, because he used everything and anything that was possible to use in a way to be able to attack the problems that occupied his mind.
The second point that allows us to give the title of modern to Archimedes is his methods of calculation. He succeeded in inventing integral calculus two thousand years before Isaac Newton and Leibniz, and even in solving one of his problems, he used a point that can be considered as 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. 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 was a victim of 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 destroy my circles." In this way, the life of Archimedes, the greatest scientist of all time, came to an end. This defenseless mathematician passed away at the age of 75 in 278 BC.
Source: Rosh Encyclopedia
Khayyam
Ghiyas al-Din Abul-Fath, Omar ibn Ibrahim Khayyam (Khayyam) was born in 439 AH (1048 AD) in the city of Nishapur and at a time when the Seljuk Turks ruled Khorasan, a vast area in eastern Iran. He began to learn science in his birthplace and learned the sciences of his time from the scholars and prominent professors of that city, including Imam Movafegh Nishaburi, and as it is said, he was very young when he became proficient in philosophy and mathematics. Khayyam left Nishapur for Samarkand in 461 AH and there, under the support of Abu Taher Abd al-Rahman ibn Ahmad, the Qazi al-Qudat of Samarkand, he wrote his outstanding work in algebra.
Khayyam then went to Isfahan and stayed there for 18 years and, with the support of Malik Shah Seljuk and his minister Nizam al-Mulk, along with a group of scientists and famous mathematicians of his time, conducted astronomical research in an observatory that was established by the order of Malikshah. The result of this research was the reform of the calendar prevalent at that time and the arrangement of the Jalali calendar (the title of Sultan Malikshah 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 of each month have 31 days, and the next five months have 30 days each, and the last month has 29 days. Every four years, a year is called a leap year, and the last month has 30 days, and that year has 366 days. In the Jalali calendar, there is a one-day time difference every five thousand years, while in the Gregorian calendar, there is a three-day error every ten thousand years.
After the assassination of Nizam al-Mulk and then Malikshah, a dispute arose among the children of Malikshah over the possession of the throne. Due to the turmoil and conflicts resulting 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 (the third son of Malikshah) rule. 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 after his return from Isfahan in this city.
Khayyam's scientific achievements for human society have been numerous and very noteworthy. For the first time in the history of mathematics, he classified 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 he only used geometric drawings for third-degree equations; 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. However, almost four centuries before Descartes, he was able to achieve one of the most important human achievements in the history of algebra, or rather, the sciences, and put forward a solution that Descartes later (in a more complete form) expressed.
Khayyam was also able to successfully define the 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 infinity parts. Also, Khayyam, while searching for a way to prove the "principle of parallelism" (the fifth principle of the first article of Euclid's principles) in the book "Sharh Ma Ashkal Min Musadarat Kitab Uqlidus" (Explanation of the problematic principles of Euclid's book), became the inventor of a deep concept in geometry. In an effort to prove this principle, Khayyam expressed statements that were completely in accordance with the statements that were expressed a few centuries later by Wallis and Saccheri, the 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 Tusi have mentioned Newton's instruction and the law of forming the expansion coefficient of the binomial while examining the laws related to taking roots from numbers.
Khayyam's extraordinary talent caused him to have achievements in other fields of human knowledge as well. Short treatises in fields such as mechanics, hydrostatics, meteorology, music theory, etc. have also been left from him. Recently, research has also 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.
The historians and scientists of Khayyam's time and those who came after him all acknowledged his mastery in philosophy, to the point 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 expresses 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 the analysts of Khayyam's thoughts. 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 only by the superficial interpretation of 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 the pressure of poverty and anonymity, and in return for his brilliant achievements
He was modest in his youth and was calmly submissive 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 more than sixteen years old 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 as an interpretation of this experience:
"In my opinion, 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 classic problem of the simultaneous curve by the integral equation, was published in 1823. This was the first answer
an equation of this type, and it paved the way for the wide progress of integral equations in the late nineteenth and early
It became 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 applied to the government, which, after the usual formalities and delays, received a scholarship for a scientific long trip 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 Crell, the magazine
famous for his name, Journal of Pure and Applied Mathematics, was inspired. This was the first periodical magazine in the world that was completely dedicated to
was 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, the example of which is 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, except for 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 put aside the simplest cases, in all of mathematics, not even an infinite series can be found whose sum is exactly 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 established the general theory of convergence in it 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 in Göttingen, hoping that it would serve as a scientific passport.
But, for a reason that is not clear, Gauss put it aside without even looking at it, because 30 years later, after his death, it was found sealed 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 Crell's magazine at that time, but the French were less aware of the existence of this periodical, and Abel was ashamed to talk about his work with new acquaintances. Shortly after his arrival, he completed his masterpiece under the title "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 of algebraic functions, which is known today as Abel's theorem, and is the basis for his later theory about Abel's integral, and a large part of algebraic geometry. It is said that decades later, every Hermite said, "From Abel, so much work has been left that 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 that happened was as follows: The mentioned manuscript was given to Cauchy and Legendre for review, Cauchy took it home and put it in an irrelevant place and completely forgot it, and it was not published until 1841, and at that time, it was lost before its printed copies were read. Finally, the original version 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 wishes were again shattered, and he made a living by private teaching, and for a short time, he was also assigned as an assistant teacher in an institution. During this period, he was completely busy and often worked on the theory of elliptic functions, which he had discovered as the inverse of elliptic integrals. This theory quickly found its place 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 event due to his seclusion. 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, Crell 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.
Crell praises Abel in his magazine as follows: "All his works contain signs of genius and amazing intellectual power. It can be said that he could pass through 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 modesty that increased his value to the extent of his extraordinary genius." But, mathematicians have their own methods to remember the great men of mathematics, 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 theoretical mathematician and 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, where the planets and their satellites are not affected by the sun, but rather, they affect 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 dynamic 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). In this work, the works of several generations of prominent mathematicians on gravity have been collected. 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 exploration. One of the most famous of them indicates the time when Napoleon protested to Laplace to show a weak point, why he wrote a thick book about the system of the world without even his celestial mechanics. For future generations, the development of the all-encompassing theory of potential is what remains, which is widely discussed in numerous fields of physical sciences, from gravity and fluid mechanics to electromagnetism and atomic physics. Although Laplace took the idea of potential from Lagrange without mentioning this, he used this theory so extensively that since then, the differential equation of the potential theory has always been known as Laplace's basic equation.
His other masterpiece was the Treatise on the Analytical Theory of Probabilities (1812), which organized his discoveries of forty years on 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 is the greatest work in this field that has been written in this part of mathematics. In the introduction to this book, he says: "The theory of probability is essentially nothing more than a common understanding that has been turned into a 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 gaining position reached their peak. His compatriots ridicule 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 calmly 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 Pope's fake representative in the story that has come in English literature, who became a Catholic twice and a Protestant twice. It is said that the Pope's representative said in response 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 the bishop of 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 had 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. In 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 coefficients 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 that I would not deprive you of any part of the honors that belong to you." He continued, and both he and Euler used it for many new types of problems, especially in mechanics.
In 1766, when Euler was leaving Berlin for St. Petersburg, he suggested to Frederick the Great that Lagrange be invited to succeed him. Lagrange accepted this invitation and stayed 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 treatise (1788), in which he unified general mechanics, and as Hamilton mentioned, 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 the invitation of Louis XVI to go to Paris, and there, he was given an apartment in Auvergne. Despite all his great genius, Lagrange was modest 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 in this period was his guiding and pioneering role in creating the metric system for weights and quantities.
In mathematics, he tried to offer 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 physics, chemistry, biology, and other sciences would attract the most capable minds of the future. This pessimism might have disappeared if he could have predicted the arrival of Gauss and his successors on the scene, who turned the nineteenth 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 career 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 he taught a different subject every year. In his lectures - which were edited and published by his
