Biography of joesph louis lagrange

His papers on the general process for solving an algebraic equation of any degree, and ; this method fails for equations of an order above the fourth, because it then involves the solution of an equation of higher dimensions than the one proposed, but it gives all the solutions of his predecessors as modifications of a single principle. The complete solution of a binomial equation of any degree; this is contained in the papers last mentioned.

Lastly, in , his treatment of determinants of the second and third order, and of invariants. Several of his early papers also deal with questions connected with the neglected but singularly fascinating subject of the theory of numbers. Among these are the following:. His proof of the theorem that every positive integer which is not a square can be expressed as the sum of two, three or four integral squares, His papers of , , and , which give the demonstrations of several results enunciated by Fermat, and not previously proved.

There are also numerous articles on various points of analytical geometry. In two of them, written rather later, in and , he reduced the equations of the quadrics or conicoids to their canonical forms. During the years from to he contributed a long series of papers which created the science of partial differential equations. A large part of these results were collected in the second edition of Euler's integral calculus which was published in Lastly, there are numerous papers on problems in astronomy.

Of these the most important are the following:. Attempting to solve the three-body problem resulting in the discovery of Lagrangian points, On the secular equation of the Moon, ; also noticeable for the earliest introduction of the idea of the potential. The potential of a body at any point is the sum of the mass of every element of the body when divided by its distance from the point.

Lagrange showed that if the potential of a body at an external point were known, the attraction in any direction could be at once found. The theory of the potential was elaborated in a paper sent to Berlin in Two papers in which the method of determining the orbit of a comet from three observations is completely worked out, and this has not indeed proved practically available, but his system of calculating the perturbations by means of mechanical quadratures has formed the basis of most subsequent researches on the subject.

His determination of the secular and periodic variations of the elements of the planets, the upper limits assigned for these agree closely with those obtained later by Le Verrier, and Lagrange proceeded as far as the knowledge then possessed of the masses of the planets permitted. Three papers on the method of interpolation, , and the part of finite differences dealing therewith is now in the same stage as that in which Lagrange he left.

In this he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics, both of solids and fluids. The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained.

The method of generalized co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D'Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation.

For example, in dynamics of a rigid system he replaces the consideration of the particular problem by the general equation, which is now usually written in the form. T for the Kinetic energy and V for the Potential energy. Amongst other minor theorems here given it may mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action.

All the analysis is so elegant that Sir William Rowan Hamilton said the work could only be described as a scientific poem. It may be interesting to note that Lagrange remarked that mechanics was really a branch of pure mathematics analogous to a geometry of four dimensions, namely, the time and the three coordinates of the point in space; and it is said that he prided himself that from the beginning to the end of the work there was not a single diagram.

At first no printer could be found who would publish the book; but Legendre at last persuaded a Paris firm to undertake it, and it was issued under his supervision in He received similar invitations from Spain and Naples. In France he was received with every mark of distinction, and special apartments in the Louvre were prepared for his reception.

Curiosity as to the results of the French revolution first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed. It was about the same time, , that the unaccountable sadness of his life and his timidity moved the compassion of a young girl who insisted on marrying him, and proved a devoted wife to whom he became warmly attached.

Although the decree of October, , which ordered all foreigners to leave France, specially exempted him by name, he was preparing to escape when he was offered the presidency of the commission for the reform of weights and measures. The choice of the units finally selected was largely due to him, and it was mainly owing to his influence that the decimal subdivision was accepted by the commission of Though Lagrange had determined to escape from France while there was yet time, he was never in any danger; and the different revolutionary governments and at a later time, Napoleon loaded him with honours and distinctions.

The next work he produced was in on the libration of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work. His solution is especially interesting as containing the germ of the idea of generalised equations of motion, equations which he first formally proved in Already by , Euler and Maupertuis , seeing Lagrange's mathematical talent, tried to persuade Lagrange to come to Berlin, but he shyly refused the offer.

In , d'Alembert interceded on Lagrange's behalf with Frederick of Prussia and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin. In , after Euler left Berlin for Saint Petersburg , Frederick himself wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court.

Lagrange was finally persuaded. In , he married his cousin Vittoria Conti. Lagrange was a favourite of the king, who frequently lectured him on the advantages of perfect regularity of life. The lesson was accepted, and Lagrange studied his mind and body as though they were machines, and experimented to find the exact amount of work which he could do before exhaustion.

Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or the subject-matter were capable of improvement. He carefully planned his papers before writing them, usually without a single erasure or correction. Nonetheless, during his years in Berlin, Lagrange's health was rather poor, and that of his wife Vittoria was even worse.

She died in after years of illness and Lagrange was very depressed. In , following Frederick's death, Lagrange received similar invitations from states including Spain and Naples , and he accepted the offer of Louis XVI to move to Paris. In France he was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the French Academy of Sciences , which later became part of the Institut de France Curiosity as to the results of the French Revolution first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed.

She insisted on marrying him and proved a devoted wife to whom he became warmly attached. In September , the Reign of Terror began. Under the intervention of Antoine Lavoisier , who himself was by then already thrown out of the academy along with many other scholars, Lagrange was specifically exempted by name in the decree of October that ordered all foreigners to leave France.

On 4 May , Lavoisier and 27 other tax farmers were arrested and sentenced to death and guillotined on the afternoon after the trial. Lagrange said on the death of Lavoisier:. Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments and at a later time, Napoleon gave him honours and distinctions.

This luckiness or safety may to some extent be due to his life attitude he expressed many years before: " I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable ". It may be added that Napoleon, when he attained power, warmly encouraged scientific studies in France, and was a liberal benefactor of them.

Lagrange was involved in the development of the metric system of measurement in the s. He was offered the presidency of the Commission for the reform of weights and measures la Commission des Poids et Mesures when he was preparing to escape. After Lavoisier's death in , it was largely Lagrange who influenced the choice of the metre and kilogram units with decimal subdivision, by the commission of His lectures there were elementary; they contain nothing of any mathematical importance, though they do provide a brief historical insight into his reason for proposing undecimal or Base 11 as the base number for the reformed system of weights and measures.

The discourses were ordered and taken down in shorthand to enable the deputies to see how the professors acquitted themselves. However, Lagrange does not seem to have been a successful teacher. Fourier , who attended his lectures in , wrote:. The inscription on his tomb reads in translation:. Count of the Empire. Grand Officer of the Legion of Honour.

Grand Cross of the Imperial Order of the Reunion. Member of the Institute and the Bureau of Longitude. Born in Turin on 25 January Died in Paris on 10 April Lagrange was extremely active scientifically during the twenty years he spent in Berlin. Some of these are really treatises, and all without exception are of a high order of excellence.

Except for a short time when he was ill he produced on average about one paper a month. Of these, note the following as amongst the most important. First, his contributions to the fourth and fifth volumes, —, of the Miscellanea Taurinensia ; of which the most important was the one in , in which he discussed how numerous astronomical observations should be combined so as to give the most probable result.

And later, his contributions to the first two volumes, —, of the transactions of the Turin Academy; to the first of which he contributed a paper on the pressure exerted by fluids in motion, and to the second an article on integration by infinite series , and the kind of problems for which it is suitable. Most of the papers sent to Paris were on astronomical questions, and among these, including his paper on the Jovian system in , his essay on the problem of three bodies in , his work on the secular equation of the Moon in , and his treatise on cometary perturbations in These mechanics are called Lagrangian mechanics.

The greater number of his papers during this time were, however, contributed to the Prussian Academy of Sciences. Several of them deal with questions in algebra. There are also numerous articles on various points of analytical geometry. In two of them, written rather later, in and , he reduced the equations of the quadrics or conicoids to their canonical forms.

During the years from to , he contributed a long series of papers which created the science of partial differential equations. A large part of these results was collected in the second edition of Euler's integral calculus which was published in Lastly, there are numerous papers on problems in astronomy. Of these the most important are the following:.

In this book, he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics , both of solids and fluids. The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained.

However, he became involved in quite a number of speculative ventures and lost a great deal of his wealth as a result. The formal education of Lagrange really was not one of any note. He did attend the college of Turin and never really became interested in mathematics until he reached the age of Once his interest was piqued, he traveled down the path of being a very accomplished mathematician.

Over time, his expertise led him to becoming the director of mathematics at the Prussian Academy of Sciences. The discovery of this new interest in mathematics was one that emerged by accident. Lagrange had read a paper by Edmund Halley on the subject and then became so interested in it that he launched himself into a great deal of study.

In a very short time, he would go on and become a very skilled and visionary mathematician. The accomplishments of Joseph-Louis Lagrange were not reflected by his ability to simply perform high-level mathematics work. Not only was Lagrange an outstanding mathematician but he was also a strong advocate for the principle of least action so Maupertuis had no hesitation but to try to entice Lagrange to a position in Prussia.

He arranged with Euler that he would let Lagrange know that the new position would be considerably more prestigious than the one he held in Turin. However, Lagrange did not seek greatness, he only wanted to be able to devote his time to mathematics, and so he shyly but politely refused the position. Euler also proposed Lagrange for election to the Berlin Academy and he was duly elected on 2 September The following year Lagrange was a founding member of a scientific society in Turin, which was to become the Royal Academy of Sciences of Turin.

The papers by Lagrange which appear in these transactions cover a variety of topics. He published his beautiful results on the calculus of variations, and a short work on the calculus of probabilities. In a work on the foundations of dynamics, Lagrange based his development on the principle of least action and on kinetic energy. He had read extensively on this topic and he clearly had thought deeply on the works of Newton , Daniel Bernoulli , Taylor , Euler and d'Alembert.

Lagrange used a discrete mass model for his vibrating string, which he took to consist of n n n masses joined by weightless strings. His different route to the solution, however, shows that he was looking for different methods than those of Euler , for whom Lagrange had the greatest respect. In papers which were published in the third volume, Lagrange studied the integration of differential equations and made various applications to topics such as fluid mechanics where he introduced the Lagrangian function.

Also contained are methods to solve systems of linear differential equations which used the characteristic value of a linear substitution for the first time. Another problem to which he applied his methods was the study the orbits of Jupiter and Saturn. The topic was on the libration of the Moon, that is the motion of the Moon which causes the face that it presents to the Earth to oscillate causing small changes in the position of the lunar features.

Lagrange entered the competition, sending his entry to Paris in which arrived there not long before Lagrange himself. In November of that year he left Turin to make his first long journey, accompanying the Marquis Caraccioli, an ambassador from Naples who was moving from a post in Turin to one in London. Lagrange arrived in Paris shortly after his entry had been received but took ill while there and did not proceed to London with the ambassador.

D'Alembert was upset that a mathematician as fine as Lagrange did not receive more honour. He wrote on his behalf [ 1 ] :- Monsieur de la Grange, a young geometer from Turin, has been here for six weeks. He has become quite seriously ill and he needs, not financial aid, for the Marquis de Caraccioli directed upon leaving for England that he should not lack for anything, but rather some signs of interest on the part of his native country In him Turin possesses a treasure whose worth it perhaps does not know.

Despite no improvement in Lagrange's position in Turin, he again turned the offer down writing:- It seems to me that Berlin would not be at all suitable for me while M Euler is there. By March d'Alembert knew that Euler was returning to St Petersburg and wrote again to Lagrange to encourage him to accept a post in Berlin.

Biography of joesph louis lagrange

Full details of the generous offer were sent to him by Frederick II in April, and Lagrange finally accepted. However, not everyone was pleased to see this young man in such a prestigious position, particularly Castillon who was 32 years older than Lagrange and considered that he should have been appointed as Director of Mathematics.

Just under a year from the time he arrived in Berlin, Lagrange married his cousin Vittoria Conti. He wrote to d'Alembert :- My wife, who is one of my cousins and who even lived for a long time with my family, is a very good housewife and has no pretensions at all. They had no children, in fact Lagrange had told d'Alembert in this letter that he did not wish to have children.

Turin always regretted losing Lagrange and from time to time his return there was suggested, for example in