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Joseph Lagrange was an experienced mathematician who prospered and ascended the ladder of advancement and prosperity at his own expense. This mathematician is known to be the greatest and one of the most talented mathematicians of the 18^{th} century. The facts demonstrate that practically each mathematic branch has been enriched by this French-born Italian scientist. The mathematician is most famous for his analytical contributions and formulation of ‘calculus of variations’ and his developments in mechanics, which he considered to be a part of pure mathematics. The current paper will provide a biographical sketch of Joseph Lagrange and his contribution to mathematics and overall mathematical impact on science.

## Biography

Joseph-Louis Lagrange is typically believed to be a French mathematician; however, Italians refer to him as an Italian mathematician. They definitely have a number of justifications to support such statement as Lagrange was born in Turin in 1736 and baptized in honor of Giuseppe Lodovico Lagrangia. The mathematician’s father was known to be treasurer of the Office of Public Works and Fortifications in Turin, and his mother was a single daughter of a Cambiano’s medical doctor who lived near the city of mathematician’s birth. Lagrange was the oldest child among the 11 infants and the only infant who lived to maturity. In fact, the mathematician’s family was known to have French roots from the father's side, as his great-grandfather was a French cavalrymen commander who left France in order to help and work for the Duke of Savoy. Lagrange consistently addressed his French genealogy as since his youthfulness he demonstrated a desire to register himself as Lodovico LaGrange, thus utilizing the French mode of his family’s name. Despite the fact that mathematician’s father occupied a highly important position, serving the king of Sardinia, Lagrange’s family was not rich because the mathematician’s father lost huge amounts of money in failed and ineffective financial speculations. Mathematician’s father planned a career of lawyer for his son, and Lagrange had countenanced this plan voluntarily. The mathematician studied at the College of Turin, and classical Latin was his favorite subject. At the very beginning, Lagrange had no ardor for mathematics, regarding Greek geometry quite uninteresting.

The actual interest in mathematics arose when Lagrange looked though a copy of a work by Halley concerning the usage of algebra in optics. Lagrange became interested in physics as well. The unsurpassed teachings of Beccaria at the College of Turin helped him in this process, and Lagrange took a decision to make a career in the sphere of mathematics. The mathematician completely devoted himself to mathematics, but Lagrange was mainly self-educated and did not have the advantage of learning mathematics with principal mathematicians. His first published mathematical work appeared on 23 July 1754. This work was written in the form of a letter. It was scribed in Italian and addressed Giulio Fagnano. Probably, the most unexpected and remarkable was the name under which the mathematician wrote this letter. He named himself as Luigi De la Grange Tournier. Nevertheless, the work was not a masterpiece and demonstrated that the young mathematician was working on his own with no the counseling from any mathematical supervisor. This letter depicts a correspondence between the binomial theorem and the subsequent derivations of the functions product.

In fact, Lagrange sent the outcomes of his paper to Euler before submitting the paper to the Italian publication. Euler, who was working in Berlin at that time, received the letter in Latin. Nevertheless, Lagrange discovered that the outcomes of his work turned out to be in accordance with the outcomes between Leibniz and Johann Bernoulli one month after the paper was published. Thus, Lagrange became seriously disappointed by his discovery due to the fact that he had a fear of being marketed as a cheat who copied the outcomes of other scientists. Nevertheless, this less than prominent start did nothing more than make Lagrange double his attempts to provide outcomes of genuine praise in the sphere of mathematics. He started his work on the tautochrone (the curve of tautochrone demonstrated that a weighted particle will consistently appear at a stated post in the analogous time regardless of its primary position). Lagrange had made some crucial discoveries concerning tautochrone by the end of 1754. These discoveries seriously contributed to the innovative subject of the ‘calculus of variations’, which mathematicians only started to study. Thus, the mathematician sent his results concerning tautochrone to Euler. These outcomes contained his ‘maxima and minima’ method. Euler was amazed by Lagrange’s new concepts and ideas. Despite the fact that Lagrange was merely 19 years old, he was approved as a professor of mathematics at the Royal Artillery School in Turin in 1755. Lagrange deserved this appointment as he had already demonstrated the authenticity of his ideas and profundity of his immense talents to the world of mathematics.

Lagrange sent Euler the outcomes that the mathematician had achieved while applying the calculus of variations to mechanics in 1756. They generalized the outcomes that Euler obtained on his own, and therefore, Euler introduced Lagrange to Maupertuis (who was the Berlin Academy president) as an outstanding young mathematician. In fact, Lagrange had been not only a remarkable mathematician, but also a solid exponent of the least action principle as well. Therefore, Maupertuis had no hesitations and tried to attract Lagrange to a position in Prussia, which was more prestigious. Nevertheless, Lagrange did not search for greatness as he merely desired to be capable of devoting his time to mathematics. Thus, Lagrange timidly, but affably refused from this position. On the other hand, Euler also offered Lagrange to take part in the elections to the Berlin Academy as well. Therefore, Lagrange was elected to it in 1756. Lagrange was a founding participant of a scientific society in Turin during the whole subsequent year. This new society had one important function to perform, in particular the publishing of a scientific journal which is called ‘Mélanges de Turin’ in French or Latin. The mathematician nder the current research was the main contributor to the first volumes of the journal till 1766.

The articles written by Lagrange that appeared in the journal covered a wide variety of themes. He published his outstanding outcomes concerning the calculus of variations together with a concise work on the probabilities calculus. In addition, Lagrange published a work on the dynamic foundations, which he grounded on his evolvement of the basis of least action and on kinetic energy. This journal also allowed him publishing a serious study on the sound propagation. Lagrange made crucial contribution to the theory of vibrating strings in this publication. The mathematician utilized a prudent mass model for his vibrating string, which he demonstrated to consist of *n* masses connected by imponderable strings. In addition, Lagrange resolved the conclusive system of *n*+1 differential mathematical problem, then letting *n* become inclined to infinitude in order to receive the analogous functional decree similar to Euler’s work. The mathematician utilized the discrepant method to obtain the decree, but his method demonstrates that he was searching for less discrepant routes than those utilized by Euler, regardless the fact that Lagrange had the greatest respect to this scientist. In addition, Lagrange studied the amalgamation of differential mathematical problems and made different appliances to such topic as fluid mechanics. This work also included the methods to resolve systems of linear differential mathematical problems, which utilized the specific caliber of a linear replacement for the first time.

In 1766, Lagrange was compelled by Euler to become the Director of Mathematics at the Berlin Academy. The mathematician was welcomed cordially by the majority members of the Academy; thus, he shortly became a close friend of Lambert and Johann (III) Bernoulli. Nevertheless, not all members were contended to observe this young man having such a reputable position. Castillon, who was 32 years older than the young mathematician, was especially unpleased to see Lagrange in this position. Castillon believed that he should have been nominated as the Director of Mathematics.

In a year, the young mathematician married Vittoria Conti, who was his cousin. Nevertheless, the couple did not have children, and actually Lagrange did not demonstrate a desire to have a child. The native town of Lagrange consistently deplored losing the mathematician, and it was actually suggested that Lagrange would return. Nevertheless, Lagrange lived and worked in Berlin for 20 years. This possibility allowed him to produce a solid current of best caliber papers and win the prize from the Academia of Sciences of Paris on a regular basis. These facts show that he shared the prize for the three body problem of Euler. Then he won the prize in 1774 for the investigation of the motion of the moon. In addition, he won the 1780 prize for the study of the disturbance of the comets orbits caused by planets. In fact, his operations in Berlin embraced numerous topics, including astronomy, mechanics and fluid mechanics, the solar system stability, calculus foundations, etc. He researched and worked on number theory as well. He believed that each positive integer is the total sum of four squares. Furthermore, the mathematician approved the theorem of Wilson concerning the fact that *n* can be regarded as prime when and merely if (*n* -1) + 1 is divisible by *n.* Nevertheless, he started this work even without permission by Wilson. In addition, Lagrange conferred his crucial work known as “Réflexions sur la résolution algébrique des equations”* *in 1770. This work made an essential research demonstrating why mathematic problems of degrees after four could be resolved with a help of radicals. It was the earliest paper to conjecture the roots of a mathematical problem of theoretical numeracy on contrary to having numeral valuables. He researched transpositions of the roots and, and despite the fact that Lagrange did not create transpositions in the paper, this research is believed to be the first step in the evolvement of the group theory. His first wife died soon after the marriage because of lingering disease. Thus, in 1792, the scientist married a young daughter of an astronomer. However, he did not have children in this marriage as well. During the last years of his life he was trying to arrange a fresh edition of “Mecanique”, however, he was already weak, and his powers were failing. He died in 1813 in Paris, and there is no actual specified cause of his death. His body was buried in the Pantheon.

## The Major Lagrange’s Contributions

Despite the fact that Lagrange made a number of important contributions to mechanics, he did not produce any comprehensive work. On the other hand, Lagrange made significant contributions to numerous branches of mathematics. The most significant of them concern the variations calculus, resolution of polynomial mathematical problems, power sets and functions, etc.

## Differential Equations

The preliminary paper written by Lagrange in Turin was dedicated to the issue of the sound propagation. The mathematician considered the perturbation given over an erect line. Thus, he diminished the issue to the analogous differential equation appearing in the research of the transversal string vibrations. In accordance with the mathematician, the curve adopted by such a string can be demonstrated as “*y*=*a*s” in the formula “(*mx*s)y.” While discussing preliminary resolutions of the fractional differential equation, Lagrange supported Euler who supposed that the limiting to functions demonstrating Taylor augmentation had not been necessary. Nevertheless, the mathematician did not succeed in recognizing the general statement of the solution by Daniel Bernoulli concerning the form of a trigonometric set. Moreover, Lagrange overlooked the possibility to acknowledge the significance of Fourier's concepts, who was the first to demonstrate the ideas that were essential for the resolution of fractional differential equations with conditioned boundary circumstances. Nevertheless, it was Lagrange who converted the partial differential equations studdy into a specific branch of mathematics. In fact, formerly mathematicians demonstrated a tendency to research merely a few particular equations, having only a general method. Finally, the most important Lagrange's contribution to the subject concerned the clarification of the connections between singular resolutions and envelopes.

## Calculus of Variations

In fact, Euler named the new branch of mathematics as calculus of variations. This branch allowed the scientist to invent solutions to isoperimetric issues. Lagrange considered that the method utilized by Euler required more simplicity, which was advantageous in a subject of pure analysis. As a matter of fact, young mathematician particularly discountenanced the geometrical constituent of the Euler's method. As it was already mentioned, calculus of variations started with the letter that he sent to Euler. In 1760, Lagrange began to deal with surface locations and volumes. The mathematician outlined the surface location and the volume with the help of the following equation: “*z=f(x,y)* and *dz=Pdx+Qdy.”* In fact, the mathematician did not provide a sufficient number of explanations at that time, but he pointed out that the double-integral symbols indicated that the two integrations should be executed in sequence. However, he induced the general concept of a surface integral in 1811. He stated that if the tangent plane *dz*, which stood for the constituent of surface, made an angle with the x, y (plane), the usage of mere trigonometry allows simply to write *dxdy.* Therefore, if there is a specific function of three elements, the second integral can be taken above a specific location in the surface, while the first can be taken over the above-mentioned location in the plane. It is important to mention that maximization and minimization of integrals by variation in the form of the function amalgamated was essential for the applications of the issue. In fact, the fundamental ideas and concepts of the calculus of variations were complicated and not completely understood by coevals of Lagrange. Nevertheless, he did not try to provide a meticulous foundation of the principles, whereas the outcomes merely advocated the method.

## Algebraically Polynomial Equations

In addition, Lagrange also attempted to resolve algebraically polynomial equations of degree four and higher beginning with the methods utilized by Cardano. The mathematician attempted to colligate by regarding transpositions of the roots. Nevertheless, these attempts were futile, and Lagrange had to abandon his interrogation. However, his research formed the basis, on which all nineteenth-century works concerning the algebraic resolutions of equations were grounded. Thus, Augustin-Louis Cauchy, and Karl Gustav Jacobi further evolved higher analysis. The importance of his basic position was challenged by the destitution of a well-established theory of set. Therefore, the enrollments utilized were clumsy and thus rejected by the mathematician himself. Lagrange’s theory of functions is subdivided into three parts, where the first one clarified the general dogma of functions; the second explained the application of this dogma to geometry; and the third part dealt with its aspects concerning mechanics.

## The Research on Functions

Finally, the devotion of the mathematician to the desideratum of an algebraic basis for the calculus caused him to make the main achievement of the “Fonctions Analytiques”, in which he researched functions with the help of their power set enlargements. He considered that each function could be augmented into a power series. In fact, “Fonctions Analytiques” made it possible to create the central theorem of calculus. Lagrange demonstrated the theorem as follows: “if from *x=a* to *x=b*, for *b>a* is positive, then *f(b)-f(a)* is positive.” He proved the theorem, equipping it by the formula “*f(x+i)=f(x)+iP”*, in which *P* stands for a function of *x* and *i*. Lagrange stated that if *i=0*, it means that *P(x,i)* should be positive up to a particular caliber of *i,* which actually could be taken as small as necessary.

## Conclusion

Thus, the facts demonstrated that Lagrange was born in Turin, but he is regarded to be a French mathematician. Despite the fact that his father desired him to become a lawyer, the scientist was interested in mathematics. Being 16 years old, he started studying mathematics by his own means. During the following years, the mathematician provided numerous solutions and applied them to mechanics. The achievements of young mathematician were so tremendous that he was esteemed by the majority of his coevals as the greatest living mathematician of that time. In addition, Lagrange was selected to inherit Euler’s position as the director of the Berlin Academy. The years, which the mathematician spent in Paris, were initially dedicated to prophetic monographs totalizing all of his mathematical conceptions. Despite fame, Lagrange was known to be a timid and modest man. He made principal contributions to numerous branches in the sphere of mathematics. The most crucial contributions concern the variations calculus, resolution of polynomial equations, power sets, and functions.

The facts demonstrate that Lagrange genuinely took an outstanding part in the development of practically each branch of pure mathematics. Similarly to well-known Diophantus and Fermat, he had a specific genius and talent for the theory of numbers. He provided solutions to numerous problems, which had been put forward by other scientists, and augmented a number of theorems of his own. Most prominently, Lagrange created the calculus of variations. In addition, the theory of differential equations owns its position as a science thanks to this prominent scientist as it had been perceived merely as a collection of complicated inventions for the resolution of specific problems before Lagrange’s contribution. In case of the calculus of finite differences, the scientist created the formula of interpolation, which bears his name. Nevertheless, he majorly impressed by contributions to the sphere of mechanics (which he believed to be a branch of pure mathematics).