Mathematical and Computer Science Facts
Important Mathematical Discoveries
Augustin Cauchy, a great French mathematician, possessed immense talent from childhood, much like Gauss. However, he was deeply religious and achieved numerous discoveries in mathematics. He formulated the theory of functions with an imaginary variable. These were very significant discoveries. From [Year Missing], Cauchy consistently made astonishing discoveries, which he sent to the Academy of Sciences for publication. The Academy's reports were in a state of panic because Cauchy's papers were excessively numerous, and Cauchy wished to publish a journal containing all his articles. Two teenage geniuses, one named Niels Henrik Abel from Norway, and the other Évariste Galois from France, brought about a profound transformation in mathematics with their discoveries.
Abel was raised in a poor family. His mathematical genius shone from childhood. In his early youth, he came to Berlin and then to Paris. Despite his efforts to reach the lofty peaks of contemporary science like Gauss, Poisson, and Cauchy, he was unsuccessful. Ultimately, he managed to compile his notes containing his important discoveries, which Cauchy lost. Poor Abel returned to Norway and passed away at the age of 26 in despair and poverty.
A few days later, Cauchy found Abel's notes and took them to the French Academy of Sciences, where Abel's discoveries earned a great prize. Charles Gustave Jacob Jacobi of the Netherlands had also achieved several significant discoveries. Galois, from childhood, was an unparalleled prodigy. He managed to systematically organize the scattered studies and discoveries of mathematicians and enriched the field of mathematics with his numerous and profound discoveries. Galois brought his discoveries to Cauchy, which, like Abel's notes, were lost. Galois was also deeply distressed and soon fell ill, as he participated in a duel at the age of 20, sustaining a bullet wound. On his last night, he wrote down all his discoveries in summary form, twisting in pain, leaving them as a testament for the world. This scientific legacy was inscribed on the brink of his final night.
According to one scientist, "For hundreds of years, successive generations of great mathematicians will suffer from shortness of breath. Galois, the founder of group theory, died due to the absence of a physician."
Mathematics on the Path to Astonishing Advancements:
Gaspard Monge (1746-1818), a Frenchman, created a geographical map of his homeland in his early youth, which was displayed at the governor's office. Subsequently, his superiority over his teachers became evident in every school he was sent to. At a school, he succeeded in inventing descriptive geometry and was offered to keep it secret for national interests, so that foreigners would not discover this great invention.
When the French Revolution broke out, he joined the revolutionaries and made great efforts to implement 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 France's war with Russia. While in prison, he engaged in the study of the aforementioned subjects, particularly geometry, in his mind, until he was able to prepare his friends for the entrance examinations of the École Polytechnique upon their return to France. Ultimately, the theory of poles and polars captivated him the most, and he created projective geometry. When he presented it to the French Academy of Sciences, they did not pay much attention, so he offered it to the Brussels Academy.
Michel Chasles (1793-1880), a Frenchman, was initially a stockbroker but went bankrupt and moved to Belgium. He contemplated during his free time, and in 1834, a book named [Book Title Missing] was selected by the École Polytechnique. Chasles made important discoveries every year, including the invention of the theory of characteristics.
Jakob Steiner (1786-1863), a German, made numerous discoveries concerning curves and surfaces. Lagrange, after being selected as a professor at the École Polytechnique, published his theory of variations and, a few years later, [Year Missing] his solution of numerical equations, opening new paths for analysis. Lagrange remained unharmed throughout the revolution and its aftermath.
"Encyclopedia Britannica"
Mathematical Advancements
Most mathematical discoveries were utilized by physicists, leading to significant advancements in physics. Physicists had long incorporated differential equations, which express the relationship between cause and effect, into the realm of physics. They then explored partial differential equations and applied them in mathematics. Following this, they focused on studying and applying models where each was a function of infinitely many diverse causes. Integral equations were also applied to the sciences of physics. These discoveries, known as the calculus of mathematical functions, were pioneered by Vito Volterra (1860-1940). This important discovery was pursued and completed by several individuals, bringing great transformation and immense honor to mathematicians.
For a long time, scientists calculated with integers. After a while, their calculations extended to fractional numbers. Then, the Pythagoreans introduced irrational numbers into mathematics, followed by negative and imaginary numbers, and finally, the calculus of functions entered the domain of mathematics, considered the ultimate point and culmination of this science.
Introduction
Archimedes, a Greek scientist and mathematician, was born in Syracuse, Greece, in 212 BC. In his youth, he traveled to Alexandria to pursue knowledge. He spent most of his life in his homeland and had a close friendship with the ruler of the city. Here, we speak of the most famous bath taken by a human in the history of mankind. According to legends, over 2000 years ago in Syracuse, the capital of the then Greek province of Sicily, Archimedes, a mechanician, mathematician, and advisor to King Hiero, made one of his most famous discoveries in the royal bath.
A Discovery in the Bath
One day, as he stepped into a public bath and sat down, he observed the water level rising. Suddenly, an idea struck him. He immediately wrapped a towel around himself and rushed home, shouting repeatedly, "I found it, I found it!" What had he found? The king had tasked him with uncovering the secret of a treacherous court jeweler and exposing him. King Hiero suspected the jeweler, believing he had kept a portion of the gold given to him for crafting the royal crown, mixed the rest with silver, which was much cheaper, and then made the crown.
Although Archimedes knew that different metals have different specific weights, until that moment, he thought he would have to melt the crown, cast it into a gold ingot, to compare its weight with a pure gold ingot of the same size. However, this method would destroy the crown, so he had to find another way. That day, as he sat in the bath, he saw the water level rise and immediately realized that his body had displaced a certain amount of water in the bath.
Experiment and Proof of the Crown's Impurity (A Discovery of Nature's Secrets)
He hurried home and began to experiment with this finding. He reasoned that objects of the same size displace the same amount of water, but if we consider weight, a half-kilogram gold ingot is smaller than a silver ingot of the same weight (gold is about twice as heavy as silver). Therefore, it should displace less water. This was Archimedes' hypothesis, and his experiments proved it. For this, he needed a container of water and three weights of equal mass: the royal crown, a pure gold ingot of the same weight, and a pure silver ingot of the same weight.
In his experiment, he determined that the crown displaced more water than the gold ingot of the same weight, but less water than the silver ingot of the same weight. Thus, it was proven that the crown was not made of pure gold but that the deceitful and treacherous jeweler had made it from a mixture of gold and silver. In this way, Archimedes discovered one of nature's most remarkable secrets: that the weight of solid objects can be measured by the amount of water they displace. This law (specific weight), which we now call density, is known as Archimedes' principle. Even today, after 23 centuries, many scientists rely on this principle in their calculations.
Activities in Other Fields
Archimedes possessed exceptionally high intellectual capacities in mathematics. He invented remarkable catapults for the defense of his lands, which proved very useful. He was able to calculate the surface area and volume of objects such as spheres, cylinders, and cones, and introduced a novel method of measurement in mathematics. The determination of pi was also among his valuable contributions. He wrote books on the properties and measurement methods of geometric shapes and volumes, including cones, the spiral, parabolas, spherical surfaces, and cylinders. Additionally, he discovered laws concerning inclined planes, screws, levers, and centers of gravity.
One of Archimedes' novel methods in mathematics was the determination of pi. He devised a method to calculate pi, the ratio of a circle's circumference to its diameter, and proved that the number is between 3.14 and 3.23. Furthermore, he developed various methods for approximating square roots, and studying them reveals that he was familiar with continued fractions before Indian mathematicians. In arithmetic, he abandoned the impractical and multi-operation system of the Greeks, who used different symbols to represent numbers, and invented a counting system with which any large number could be written and read.
The science of fluid equilibrium was discovered by Archimedes, and he was able to apply its laws to determine the equilibrium of submerged objects. He also clearly and precisely stated some principles of mechanics for the first time and discovered the laws of the lever.
Archimedes and Other Scientists of His Time
Archimedes made a statement about himself that has endured for centuries: "Give me a place to stand, and I will move the Earth." The same sentiment, in a different form, is attributed to Archimedes in Greek literary texts, but the meaning is the same in both cases. Archimedes, like a reclusive eagle, was solitary. In his youth, he traveled to Egypt and studied in Alexandria for a period, where he found two old friends: Conon (a capable mathematician whom Archimedes greatly respected intellectually and personally) and Eratosthenes, who, although a competent mathematician, was considered superficial and held an extraordinary opinion of himself.
Archimedes maintained constant correspondence with Conon, sharing a significant and beautiful portion of his works in these letters. After Conon's death, Archimedes corresponded with a friend who was one of Conon's students. In 1906, J.L. Heiberg, a Danish historian and specialist in Greek mathematics, discovered a valuable document in Constantinople.
This document is a book titled "The Method of Mechanical Theorems," which Archimedes had sent to his friend Eratosthenes. The subject of this book is the comparison of the volume or surface area of an unknown shape with the volumes and surface areas of other known shapes, by which Archimedes succeeded in determining the desired result. This method is one of Archimedes' claims to fame, allowing us to consider him a thinker with a modern and innovative approach, as he utilized everything and anything that could be used to tackle the problems that occupied his mind.
The second point that allows us to call Archimedes innovative is his calculation methods. Two thousand years before Isaac Newton and Leibniz, he invented integral calculus and even employed a concept in solving one of his problems that could position him as a precursor to the idea of differential calculus.
Farewell to the World
Archimedes' life passed in complete tranquility, like that of any mathematician who has complete security and can bring all the possibilities of their intellect and genius to fruition. When the Romans conquered Syracuse in 212 BC, the Roman general Marcellus ordered that none of his soldiers were to harm, insult, or injure this famous and great scientist and thinker. Nevertheless, Archimedes became a victim of the Roman conquest of Syracuse. He was murdered by a drunken Roman soldier while contemplating a mathematical problem in the city market. It is said his last words were: "Do not disturb my circles." Thus ended the life of Archimedes, the greatest scientist of all time. This defenseless 75-year-old mathematician departed for the afterlife in 278 BC.
Source: Encyclopedia of Growth
Khayyam
Ghiyath al-Din Abu'l-Fath, Omar ibn Ibrahim Khayyam (Khayyami), was born in 439 AH (1048 AD) in the city of Nishapur, during the Seljuk Turks' dominion over Khorasan, a vast region in eastern Iran. He pursued his education in his birthplace, learning the sciences of his time from prominent scholars and teachers, including Imam Muwaffaq Nishapuri. It is said that he mastered philosophy and mathematics at a very young age. In 461 AH, Khayyam left Nishapur for Samarkand, where, under the patronage of Abu Tahir Abd al-Rahman ibn Ahmad, the chief judge of Samarkand, he authored his distinguished work on algebra.
Khayyam then moved to Isfahan and resided there for 18 years. With the support of Malik-Shah Seljuk and his vizier Nizam al-Mulk, he, along with a group of renowned scientists and mathematicians of his time, conducted astronomical research at an observatory established by order of Malik-Shah. The outcome of this research was the correction of the prevailing calendar and the establishment of the Jalali calendar (named after Sultan Malik-Shah Seljuk).
In the Jalali calendar, a solar year is approximately equal to 365 days, 5 hours, 48 minutes, and 45 seconds. The year has twelve months; the first six months have 31 days each, the next five months have 30 days each, and the last month has 29 days. Every four years, a leap year occurs, with the last month having 30 days, making the year 366 days. In the Jalali calendar, there is a one-day difference in time every five thousand years, whereas the Gregorian calendar has an error of three days every ten thousand years.
After the assassination of Nizam al-Mulk and then Malik-Shah, disputes arose among Malik-Shah's sons over succession. Due to the ensuing chaos and conflicts, scientific and cultural matters, which had previously held great importance, were neglected. The lack of attention to scientific affairs, scholars, and the observatory prompted Khayyam to leave Isfahan for Khorasan. He spent the remainder of his life in important cities of Khorasan, particularly Nishapur and Merv, which was the capital of Sanjar's (Malik-Shah's third son) rule. At that time, Merv was considered one of the world's major scientific and cultural centers, attracting many scholars. Most of Khayyam's scientific work was realized in this city after his return from Isfahan.
Khayyam's scientific achievements for humanity have been numerous and highly significant. For the first time in the history of mathematics, he impressively classified equations from the first to the third degree and then, using geometric constructions based on conic sections, provided a general solution for all of them. He used both geometric and numerical solutions for quadratic equations, but for cubic equations, he only employed geometric constructions, thus finding a solution for most of them and examining the possibility of two solutions in some cases. The difficulty was that, due to the undefined nature of negative numbers at the time, Khayyam did not pay attention to negative solutions of the equation and easily overlooked the possibility of three solutions for a cubic equation. Nevertheless, approximately four centuries before Descartes, he achieved one of humanity's most important breakthroughs in the history of algebra and sciences, proposing a solution that Descartes later (more completely) articulated.
Khayyam also successfully defined continuity of quantity and, in fact, for the first time, defined a positive real number, ultimately concluding that no quantity is composed of indivisible parts and that, mathematically, any magnitude can be divided infinitely. Furthermore, while seeking a way to prove the "parallel postulate" (the fifth postulate of Euclid's first book) in his book "Explanation of the Difficulties in the Postulates of Euclid," Khayyam introduced a profound concept in geometry. In his attempt to prove this postulate, Khayyam stated propositions that were entirely consistent with those later presented by European mathematicians Wallis and Saccheri, paving the way for the emergence of non-Euclidean geometries in the 19th century. Many believe that Pascal's triangle should be called Khayyam's triangle, and some go further, suggesting that Newton's binomial theorem should be named Khayyam's binomial theorem. It is said, however, that Jamshid al-Kashi and Nasir al-Din al-Tusi, while examining the laws related to extracting roots from numbers, presented more than Newton's formula and the law for forming binomial expansion coefficients.
Khayyam's extraordinary talent led him to make contributions in other fields of human knowledge as well. He left behind short treatises on subjects such as mechanics, hydrostatics, meteorology, and music theory. Recently, research has also been conducted on Khayyam's activities in decorative geometry, confirming his connection to the construction of the northern dome of the Jameh Mosque in Isfahan.
Historians and scholars contemporary to Khayyam, and those who came after him, all acknowledged his mastery in philosophy, to the extent that he was sometimes referred to as the "Sage of the Age" and the "Avicenna of his time." Khayyam's existing philosophical works are limited to a few short but profound and rich treatises. His last philosophical treatise reflects his mystical inclinations.
However, beyond all these, Khayyam's greatest fame in the last two centuries worldwide is due to his quatrains, which were first translated into English by FitzGerald and made accessible to the world, placing him among the four great poets of the world: Homer, Shakespeare, Dante, and Goethe. The quatrains of Khayyam, due to their highly free (and sometimes erroneous) translation, have led to some unacceptable misinterpretations of his character. These quatrains have intensified the debate and disagreement among analysts of Khayyam's thought. Some rely solely on the superficial appearance of his quatrains to express his ideas, while others believe that Khayyam's true thoughts are deeper than can be expressed merely by a superficial interpretation of his poetry. After a fruitful life, Khayyam passed away in 517 AH (according to most sources) in his hometown of Nishapur, and with his death, one of the brightest pages of intellectual history in Iran was closed.
Abel
Niels Henrik Abel (1802-1829) was one of the most pioneering mathematicians of the 19th century and arguably the greatest genius to emerge from Scandinavia. Abel, along with his contemporaries Gauss and Cauchy, was a pioneer in the development of modern mathematics, characterized by its emphasis on rigorous proof. His life was a sharp blend of cheerful optimism when under the pressure of poverty and obscurity, and humble in the face of his brilliant and abundant achievements in his youth, and calmly resigned in the face of an early death.
Abel was one of six children of a poor pastor in a Norwegian village. Before he was sixteen, his immense talent became apparent, and he was encouraged by one of his teachers. Before long, he was reading and understanding the works of Newton, Euler, and Lagrange. He later wrote in one of his mathematical notes, as a commentary on this experience: "If one wishes to progress in mathematics, one must study the works of the masters, not the disciples." He was not yet eighteen when his father died, leaving the family in destitution.
They subsisted with the help of friends and neighbors, and with financial assistance from several professors, this young man managed to enter the University of Oslo in 1821. His first research, which included solving the classic problem of the tautochrone curve using an integral equation, was published in 1823. This was the first solution of such an equation and paved the way for the extensive development of integral equations in the late 19th and early 20th centuries. He also proved that the quintic equation ax^5+bx^4+cx^3+dx^2+ex+f=0 cannot, in general, be solved in terms of radicals, unlike lower-degree equations, thus solving a problem that had troubled mathematicians for 300 years. He published his proof at his own expense in a small pamphlet.
In this scientific growth, Abel soon surpassed Norway and decided to travel to France and Germany. With the support of his friends and professors, he applied to the government, and after the usual formalities and delays, he received a scholarship for a long scientific journey across Europe. He spent the first year of his travels mostly in Berlin. There, he had the great fortune to encounter young and passionate amateur mathematicians named August Leopold Crelle, whose famous journal, the Journal für reine und angewandte Mathematik, was inspired. This was the world's first periodic journal entirely dedicated to mathematical research. The first three volumes contained 22 articles by Abel.
Abel's early studies in mathematics were confined to the old tradition of the 18th century, exemplified by Euler. In Berlin, he was influenced by a new school of thought led by Gauss and Cauchy, which placed the greatest emphasis on rigorous deduction rather than detailed calculations. At that time, apart from Gauss's monumental work on hypergeometric series, there was little proof in analysis that would be considered valid even today. As Abel explained in a letter to one of his friends: "Except for the simplest cases, there is not a single infinite series in all of mathematics whose sum is precisely determined. In other words, the most important parts of mathematics lack a foundation."
During this period, he wrote the result of his classic studies on binomial series, in which he established the general theory of convergence and provided the first convincing proof of the validity of the expansion of these series.
Abel had sent his pamphlet on quintic equations to Gauss in Göttingen, hoping it would serve as a scientific passport. However, for reasons that are unclear, Gauss set it aside without even glancing at it, as it was found unopened among his papers 30 years later, after his death. To the regret of both, Abel felt he had been obstructed 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 recognized as he should have been. At that time, he had published several important articles in Crelle's journal, but the French were scarcely aware of this periodical, and Abel was too shy to discuss his work with new acquaintances. Shortly after his arrival, he completed his distinguished work titled "Memoir on a General Property of a Large Class of Transcendental Functions," which he considered his masterpiece. This work includes a discovery concerning the integral of algebraic functions, now known as Abel's theorem, and forms the basis for his later theory of Abel integrals and a significant part of algebraic geometry. It is said that decades later, Hermite commented on this memoir: "Abel left behind so much work that it will occupy mathematicians for 500 years." Jacobi described Abel's theorem as the greatest discovery in integral calculus in the 19th century. Abel submitted his manuscript to the French Academy. He hoped this work would attract the attention of French mathematicians, but he waited in vain until his purse was empty and he had to return to Berlin. What happened was that the manuscript was given to Cauchy and Legendre for review. Cauchy took it home, misplaced it, and completely forgot about it. This work was not published until 1841, and even then, the printed copies were lost before they could be read. Finally, the original copy of the article surfaced in Florence in 1952. In Berlin, Abel completed his first revolutionary paper on elliptic functions, a subject he had worked on for years, and, heavily indebted, returned to Norway.
He expected to be appointed a professor upon his return, but his hopes were dashed again. He made a living through private tutoring and briefly worked as an assistant teacher at an institution. During this period, he worked incessantly, often 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 numerous applications in number theory, mathematical physics, and algebraic geometry. Meanwhile, Abel's fame spread to all mathematical centers in Europe, and he was ranked among the world's great mathematicians, though he remained unaware of this due to his reclusive nature. By early 1829, the tuberculosis he had contracted during his travels had progressed so far that it prevented him from working, and in the spring of that year, Abel died at the age of twenty-six. Shortly after his death, Crelle ironically wrote in an obituary that Abel's efforts had been successful, and Abel should be appointed to the chair of mathematics at the University of Berlin.
Crelle praised Abel in his journal: "All his works bear the marks of genius and astonishing intellectual power. It can be said that he could overcome all obstacles with irresistible force and penetrate to the depth of the problem... His distinguishing features were his inherent purity and nobility, as well as an unparalleled modesty that elevated his worth to the level of his extraordinary genius." However, mathematicians have their own ways of remembering great mathematicians, and they remember him by mentioning Abel's integral equation, Abel's integrals and functions, Abelian groups, Abel's series, Abel's partial summation formula, Abel's limit theorem in power series theory, and Abelian summability. Few people have their names associated with so many topics and theorems in modern mathematics, and what he could have accomplished in a normal lifetime is beyond imagination.
Laplace
Pierre-Simon de Laplace (1749-1828) was a French mathematician and theoretical astronomer so renowned in his time that he was called the "Newton of France." His primary interests throughout his life were celestial mechanics, probability theory, and career advancement.
At the age of twenty-four, he was deeply involved in the details of applying Newton's law of gravitation to the entire solar system, where planets and their moons are not only influenced by the sun but also interact with each other in complex ways. Even Newton believed that divine intervention was sometimes necessary to prevent disorder from arising in this intricate mechanism. Laplace decided to seek scientific explanations for this, and he succeeded in proving that an ideal solar system in mathematics is a stable dynamical system that remains unchanged. This achievement is but one of his many victories, detailed in his great and historic work titled "Celestial Mechanics" (published in five volumes from 1799 to 1825). This work compiles the efforts of several generations of distinguished mathematicians concerning gravity. Unfortunately, due to his future fame, he omitted all references to the discoveries of his predecessors and contemporaries, creating the illusion that all these ideas and opinions belonged to him. Many anecdotes are related to this endeavor. One of the most famous recounts a time when Napoleon, to highlight Laplace's shortcomings, objected that he had written a voluminous book about the universe without even mentioning his celestial mechanics. His lasting contribution to future generations is the comprehensive development of potential theory, which is widely discussed in numerous branches of physical sciences, from gravity and fluid mechanics to electromagnetism and atomic physics. Although Laplace took the idea of potential from Lagrange without mentioning it, he utilized this theory so extensively that since then, the differential equation of potential theory has always been known as Laplace's fundamental equation.
His other masterpiece is the treatise "Analytical Theory of Probabilities" (1812), in which he organized his forty years of discoveries in probability. Again, he forgot to mention the numerous ideas of others that he had integrated with his own, but it is unanimously agreed that his book is the greatest work written in this area of mathematics. In the introduction to this book, he states: "The theory of probability is essentially nothing more than common sense reduced to calculation." This may be so, but the subsequent 700 pages of complex analysis, in which he freely uses Laplace transforms, generating functions, and many other non-elementary tools, are, according to some, even more complex than the book "Celestial Mechanics."
After the French Revolution, Laplace's political talent and ambition for position reached their peak. His compatriots mocked his instability and compliance in politics. This means that whenever there was a change in government (which happened frequently at that time), Laplace easily adapted to the environment by changing his principles—he oscillated between staunch republicanism and flattery 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 Bri, mentioned in English literature, who became Catholic twice and Protestant twice. It is said that the Pope's representative, in response to the accusation of changing his colors, said: "No, that is not the case, for although I changed my religion, I certainly adhered to my principle, the principle that dictates I should remain Bishop of Bri for the rest of my life."
To compensate for his shortcomings, Laplace was always generous in helping and encouraging younger scientists. He occasionally assisted men like Gay-Lussac the chemist, Humboldt the traveler and naturalist, Poisson the physicist, and especially, the young Cauchy, who became one of the greatest builders of 19th-century mathematics, in advancing their work.
Lagrange
Joseph-Louis Lagrange (1736-1813) disliked geometry but made outstanding discoveries in the calculus of variations and analytical mechanics. He also contributed to the formulation of number theory and algebra, and to the intellectual current that was later strengthened by Gauss and Abel. His mathematical life can be seen as a natural extension of the work of his older and more important contemporary, Euler, whom he advanced and refined in many respects.
Lagrange was born in Turin to a mixed French-Italian ancestry. In his childhood, he was more interested in arts and literature than in sciences. However, while still in school, his interest in mathematics was ignited by reading an article by Edmond Halley on the applications of algebra in optics. He then began a period of independent study and progressed so rapidly that at the age of 19, he was appointed professor at the Royal Military School of Turin.
Lagrange's works on the calculus of variations are among his earliest and most important. In 1755, he communicated his method of multipliers to Euler for solving the isoperimetric problem. These problems had occupied Euler's mind for years, as they went beyond his semi-geometric methods. Euler immediately received answers to many of the questions that were on his mind, but he responded to Lagrange with admirable kindness and generosity, refraining from publishing his own work, as he wrote in a letter to Lagrange: "I have 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 applied it to 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 remained in Berlin for 20 years until Frederick's death in 1786. During this period, he worked extensively in algebra and number theory and authored his masterpiece, the treatise "Analytical Mechanics" (1788), in which he unified general mechanics, and as Hamilton named Lagrange's equations of motion, generalized coordinates, and the concept of potential energy.
After Frederick's death, the atmosphere at the Prussian court became relatively pleasant for scientists, so Lagrange accepted Louis XVI's invitation to move to Paris, where he was given an apartment in the Louvre. Despite his immense genius, Lagrange was humble and unbiased. And although he was an associate of aristocrats and, in fact, one of them himself, he was respected and regarded by all parties during the turmoil 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 attempted to provide a sound foundation for the fundamental 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 been dispelled if he could have foreseen the advent of Gauss and his successors, who transformed the 19th century into the richest period in the long history of mathematics.
Poincaré
Jules Henri Poincaré (1854-1912) was recognized globally at the beginning of the 20th century as the greatest mathematician of his generation. In 1879, he began his university studies at Caen, and only two years later, he was appointed professor at the University of Paris. He spent the rest of his life there, teaching a different subject each year. In his lectures—edited and published by his students—he discussed, with great originality and technical mastery, virtually all known fields of pure and applied mathematics, as well as many fields unknown before his discovery. In total, he wrote over 30 technical books on mathematical physics and celestial mechanics, six books for a general audience, and nearly 500 research articles in mathematics. He was a quick-witted, powerful, and tireless thinker who did not dwell on details and, as one of his contemporaries said, "was a conqueror, not a colonizer." He also possessed a remarkable memory and, by habit, would do mathematics in his mind while walking in his study, only putting it on paper after completing it mentally. He was elected to the Academy of Sciences before he was 32. An academician who nominated him for membership said that "his work is beyond ordinary praise, and consequently, it reminds us of what Jacobi wrote about Abel: he solved problems that had not been conceived before him."
Poincaré's first major mathematical achievement was in analysis. He invented the theory of automorphic functions, generalizing the concept of periodicity of a function. Elementary trigonometric and exponential functions are singly periodic, and elliptic functions are doubly periodic. Poincaré's automorphic functions form a broad generalization of these functions, as they are invariant under an infinite group of fractional linear transformations and encompass the rich theory of elliptic functions as a part. He used them to solve linear differential equations with algebraic coefficients and also showed how they could be used to uniformize algebraic curves, i.e., to express the coordinates of any point on such a curve in terms of single-valued functions x(t), y(t) of a single parameter t. In the 1880s and 1890s, automorphic functions became a broad branch of mathematics that, in addition to analysis, extended into the realms of group theory, number theory, algebraic geometry, and non-Euclidean geometry.
Another fundamental point of Poincaré's thought can be found in his research on celestial mechanics ("New Methods of Celestial Mechanics" - in three volumes, 1892-1899). During this work, he presented his theory of asymptotic expansions (which drew attention to divergent series), studied the stability of orbits, and laid the foundation for the qualitative theory of nonlinear differential equations. His famous investigations into the evolution of celestial bodies led him to study the equilibrium shapes of rotating fluid masses whose particles are held together by gravitational attraction, and he discovered pear-shaped forms that later played an important role in the work of Sir G. H. Darwin (son of Charles Darwin).
Poincaré summarizes these discoveries: "Consider a rotating fluid body that contracts due to cooling, but assume that this contraction occurs so slowly that the body remains homogeneous and the rotation of all parts of the body is the same. The shape of the body, initially approximately spherical, transforms into a rotating ellipsoid that becomes wider and wider, then, at a certain moment, transforms into an ellipsoid with three unequal axes. Then, the body ceases to be elliptical and becomes pear-shaped, until finally, the mass of the body, becoming increasingly slender at the waist, splits into two separate and unequal bodies." These ideas have gained more attention in our time, as geophysicists have recently discovered with the help of artificial satellites that the
