Joseph louis lagrange biography

From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball.

Joseph Louis Lagrange, often hailed as the eighteenth century's greatest mathematical mind, entered the world in Turin on January 25, 1736, and his remarkable journey concluded in Paris on April 10, 1813. I find it interesting how his father, despite holding a respectable position managing the Sardinian military funds and initially possessing wealth, faced financial setbacks through speculation. This twist of fate, it seems to me, placed young Lagrange in a position where his own intellectual prowess became his sole inheritance, a testament to his inherent abilities.

His education at the college of Turin didn't immediately signal his future eminence in mathematics. It wasn't until the relatively late age of seventeen that a chance encounter with a memoir by Halley sparked his profound interest. Imagine that – a pivotal moment triggered by a seemingly random piece of writing! Driven by an innate curiosity, he embarked on a solitary and self-guided exploration of the mathematical landscape. The fact that within a single year of intense, unaided study, he had already transformed into a proficient mathematician and secured a lectureship at the artillery school speaks volumes about his extraordinary aptitude and dedication. It makes you wonder what latent talents might lie dormant within us, waiting for that singular spark.

The early fruits of his intellectual labor are truly astonishing. At just nineteen, he penned a letter to the great Euler, offering a solution to the isoperimetrical problem, a puzzle that had captivated mathematicians for over half a century. To achieve this breakthrough – essentially determining the optimal form of a function under specific constraints – he laid down the fundamental principles of the calculus of variations. I'm struck by Euler's generous recognition of Lagrange's method, acknowledging its generality and superiority to his own approach. His courteous act of withholding his own work on similar ground to allow the young Italian to claim undisputed invention is a remarkable example of scientific integrity and mentorship. Euler even coined the name for this new branch of analysis. This single memoir, I believe, catapulted Lagrange into the esteemed company of the leading mathematicians of his time.

In 1758, fueled by his burgeoning influence, Lagrange, together with his students, established a society that later evolved into the Turin Academy. The five volumes of its transactions, known as the Miscellanea Taurinensia, became the repository for many of his early groundbreaking works. These weren't mere notes; many were elaborate memoirs, showcasing the depth of his inquiries. The first volume, for instance, contained his insightful work on the theory of sound propagation. It's fascinating to learn that he identified an error in Newton's work, derived the general differential equation for motion, and even solved it for linear motion. This same volume also presented the complete solution to the problem of a transversely vibrating string. Here, he astutely pointed out the limitations in the solutions previously offered by luminaries like Taylor, D'Alembert, and Euler, culminating in his elegant equation: y = a sin mx sin nt. The article concluded with a masterful discussion of echoes, beats, and compound sounds – a testament to his comprehensive understanding. Other contributions in this volume touched upon recurring series, probabilities, and further explorations into the calculus of variations.

The second volume delved deeper into the calculus of variations, consolidating the findings of several memoirs from the first volume concerning its theory and notation. What I find particularly impressive is how he illustrated its power by not only deducing the principle of least action but also by providing solutions to a diverse range of problems in dynamics. It's as if he wasn't just developing a tool but immediately showcasing its immense utility in unraveling the complexities of the physical world.

The third volume continued this trend of applying his mathematical innovations to significant problems. It featured solutions to several dynamical problems using the calculus of variations, further explorations into the integral calculus, a solution to Fermat's intriguing problem of finding an integer x that makes (x²n + 1) a perfect square (where n is a non-square integer), and perhaps most remarkably, the derivation of the general differential equations of motion for three bodies interacting through their mutual gravitational forces. This last one, in particular, strikes me as a monumental achievement, laying the groundwork for understanding the intricate dance of celestial bodies.

By 1761, it seems Lagrange had ascended to a position of unparalleled eminence in the mathematical world. However, the relentless intellectual exertion of the preceding nine years had taken a toll on his health. The stark warning from his doctors, who expressed concerns about his mental and physical well-being unless he embraced rest and exercise, underscores the immense pressure he placed upon himself in the pursuit of knowledge. While his health was temporarily restored, it's poignant to note that his nervous system never fully recovered, leaving him susceptible to recurring bouts of profound melancholy – a reminder that even the most brilliant minds are not immune to the frailties of the human condition.

His subsequent work in 1764 focused on the libration of the moon, offering an explanation for why it consistently presents the same face to Earth. What's fascinating here is his use of the principle of virtual work to tackle this problem. I find this solution particularly significant as it contained the initial seeds of the idea of generalized equations of motion, a concept he would later formally prove in 1780. It's like witnessing the gradual unfolding of a major theoretical framework.

A planned visit to London took an unexpected turn when illness struck him in Paris. However, this unforeseen circumstance led to his warm reception in the French capital. It's easy to imagine his regret at leaving the stimulating intellectual and social circles of Paris to return to the quieter provincial life in Turin. Yet, his stay in Piedmont proved to be brief. In 1766, following Euler's departure from Berlin, Frederick the Great extended an invitation from "the greatest king in Europe" to have "the greatest mathematician in Europe" reside at his court. Lagrange accepted this prestigious offer and spent the next two decades in Prussia. This period was incredibly productive, yielding not only a substantial series of memoirs published in the Berlin and Turin transactions but also his magnum opus, the Mécanique analytique. Interestingly, his arrival in Berlin was marked by a somewhat humorous anecdote. Observing that most of his colleagues were married and hearing their wives' assurances of its necessity for happiness, he too entered matrimony. Sadly, his first marriage was not a happy one and ended with his wife's early death.

Lagrange found favor with King Frederick, who often advised him on the virtues of a perfectly regular life. This counsel seems to have resonated deeply with Lagrange. From then on, he approached the study of his own mind and body with the meticulousness of examining a machine, experimentally determining the precise amount of work he could undertake without succumbing to breakdown. He instituted a daily routine, setting himself a specific task each night for the following day. Upon completing any significant aspect of a subject, he would write a concise analysis, identifying areas for potential improvement in the demonstrations or the subject matter itself. His meticulous approach extended to his writing process; he would thoroughly conceptualize the subject of his papers before commencing composition, typically writing them straight through without a single erasure or correction – a testament to the clarity and precision of his thought.

His intellectual output during those twenty years in Berlin was nothing short of astounding. Beyond the monumental Mécanique analytique, he contributed between one and two hundred papers to the Academies of Berlin, Turin, and Paris. Many of these were substantial treatises in their own right, and all exhibited an exceptionally high level of intellectual rigor and originality. Except for a brief period of illness, he maintained an average output of about one significant memoir per month. Among these, I find several particularly noteworthy.

Firstly, his contributions to the fourth and fifth volumes (1766-1773) of the Miscellanea Taurinensia stand out, with the 1771 memoir being particularly significant. In this work, he tackled the crucial problem of how to optimally combine numerous astronomical observations to arrive at the most probable result – a foundational contribution to the field of statistical inference. Later, his contributions to the first two volumes (1784-1785) of the Turin Academy's transactions further demonstrate his continued engagement with diverse mathematical problems. The first volume featured his paper on the pressure exerted by fluids in motion, while the second included an article on integration using infinite series and the types of problems for which this technique is best suited.

A significant portion of the memoirs he sent to the Paris Academy addressed astronomical questions. Among these, I feel compelled to highlight his 1766 memoir on the Jovian system, his 1772 essay on the problem of three bodies (a notoriously complex problem in celestial mechanics), his 1773 work on the secular equation of the moon, and his 1778 treatise on cometary perturbations. What's remarkable is that these were all written in response to subjects proposed by the French Academy, and in each instance, he was awarded the prestigious prize. This speaks volumes about the high regard his work was held in by his peers.

However, the majority of his papers during this prolific period were contributed to the Berlin Academy. Several of these delved into questions of algebra. I find the following particularly noteworthy: (i) His exploration of integer solutions to indeterminate quadratic equations (1769) and more generally to indeterminate equations (1770). (ii) His treatise on the theory of elimination (1770). (iii) His memoirs on a general process for solving algebraic equations of any degree (1770 and 1771). While this method ultimately falls short for equations of an order higher than the fourth (as it introduces the need to solve an equation of even higher degree), its significance lies in its ability to unify all the solutions of his predecessors under a single underlying principle. (iv) His complete solution to binomial equations of any degree, which was included within the aforementioned memoirs. (v) And lastly, his 1773 treatment of determinants of the second and third order, along with the concept of invariants – foundational ideas in modern algebra.

Several of his early papers also explored the often-neglected yet profoundly fascinating realm of the theory of numbers. Among these, I'm particularly drawn to: (i) His proof in 1770 of the theorem that every integer not a perfect square can be expressed as the sum of two, three, or four integral squares – a beautiful and fundamental result. (ii) His proof in 1771 of Wilson's theorem, which states that if n is a prime number, then (n-1)! + 1 is always divisible by n – a cornerstone of number theory. (iii) His memoirs of 1773, 1775, and 1777, in which he provided rigorous demonstrations for several results previously stated by Fermat without proof, finally resolving long-standing mathematical mysteries. (iv) And finally, his method for determining the factors of numbers of the form x² + ay².

Beyond these areas, he also made significant contributions to various aspects of analytical geometry. Notably, in two later papers from 1792 and 1793, he achieved the reduction of the equations of quadrics (or conicoids) to their simplest, or canonical, forms, providing a more elegant and understandable way to analyze these geometric objects.

During the years spanning from 1772 to 1785, Lagrange produced a remarkable series of memoirs that essentially established the science of differential equations, at least concerning partial differential equations. I don't believe any prior work had gone beyond examining equations of specific, limited forms. A substantial portion of these groundbreaking results was later compiled in the second edition of Euler's integral calculus, published in 1794, further solidifying Lagrange's impact on this field.

Lagrange's contributions to mechanics, while substantial, don't require separate detailed discussion here, as their culmination is found in his seminal work, the Mécanique analytique, which I will describe shortly.

Lastly, his extensive work in astronomy yielded numerous important memoirs. Among these, I find the following particularly significant: (i) His work on the attraction of ellipsoids (1773), building upon the earlier work of Maclaurin. (ii) His study of the secular equation of the moon (1773), which is also notable for its early introduction of the concept of the potential. Lagrange defined the potential of a body at any point as the sum of the mass of each element of the body divided by its distance from that point. He brilliantly showed that if the potential of a body at an external point were known, the gravitational attraction in any direction could be readily determined. The theory of the potential was further elaborated in a paper he sent to Berlin in 1777. (iii) His work on the motion of the nodes of a planet's orbit (1774). (iv) His investigations into the stability of planetary orbits (1776) – a fundamental question in celestial mechanics. (v) Two comprehensive memoirs (1778 and 1783) in which he meticulously developed a method for determining the orbit of a comet from just three observations. While this method didn't prove entirely practical for direct application, his system of calculating perturbations using mechanical quadratures laid the foundation for much of the subsequent research in this area. (vi) His determination of the secular and periodic variations of the elements of the planets (1781-1784). The upper limits he established for these variations closely aligned with those later obtained by Leverrier, and Lagrange's work progressed as far as the then-current knowledge of planetary masses permitted. (vii) Three memoirs on the method of interpolation (1783, 1792, and 1793). The field of finite differences dealing with interpolation remains largely in the state that Lagrange left it, a testament to the completeness of his work.

Above and beyond these diverse and impactful papers, Lagrange composed his truly monumental treatise, the Mécanique analytique. In this work, he postulates the fundamental law of virtual work, and from this single principle, utilizing the power of the calculus of variations, he elegantly deduces the entirety of mechanics, encompassing both solid and fluid systems. The very aim of the book, as I understand it, was to demonstrate that the entire subject of mechanics is implicitly contained within a single overarching principle and to provide general formulas from which any specific result could be derived. The method of generalized coordinates, which he developed to achieve this, is perhaps the most brilliant outcome of his analytical genius. Instead of tracking the motion of each individual particle within a material system, as D'Alembert and Euler had done, Lagrange showed that by defining the system's configuration using a sufficient number of variables equal to the system's degrees of freedom, both the kinetic and potential energies could be expressed in terms of these variables. The differential equations of motion could then be derived through simple differentiation. For instance, in the dynamics of a rigid body, he replaced the consideration of specific problems with a general equation, now commonly written in a particular form. Among the other significant theorems presented in this work is the proposition that the kinetic energy imparted by given impulses to a constrained material system is a maximum, as well as the principle of least action. The elegance of his analysis was so profound that Sir William Rowan Hamilton famously described the work as a "scientific poem." It's also interesting to note Lagrange's own observation that mechanics was essentially a branch of pure mathematics, akin to a geometry of four dimensions – time and the three spatial coordinates. He reportedly took pride in the fact that his entire work contained not a single diagram. Initially, finding a publisher proved challenging, but Legendre eventually persuaded a Paris firm to undertake the project, and it was finally issued under his supervision in 1788.

In 1787, following the death of Frederick, Lagrange, who had found the Berlin climate less than ideal, readily accepted the offer from Louis XVI to relocate to Paris. He had also received similar invitations from Spain and Naples, highlighting his international renown. In France, he was welcomed with the highest honors, and special apartments were prepared for him in the Louvre. However, upon his arrival, he was afflicted by another bout of melancholy. Even the printed copy of his Mécanique, a work he had dedicated a quarter of a century to, lay unopened on his desk for more than two years. It was the unfolding events of the French Revolution that finally roused him from his lethargy, initially out of curiosity, which soon transformed into alarm as the revolution progressed. Around the same time, in 1792, the persistent sadness that had marked his life and his inherent timidity touched the heart of a young woman who insisted on marrying him. She proved to be a devoted wife, to whom he became deeply attached, bringing a newfound happiness to his life. Although the decree of October 1793, ordering all foreigners to leave France, specifically exempted him by name, he was preparing to flee when he was offered the presidency of the commission for the reform of weights and measures. The final selection of units was largely due to his insightful contributions, and it was primarily through his influence that the decimal subdivision was adopted by the commission in 1799.

Despite his initial desire to escape France, Lagrange was never actually in any danger during the tumultuous years of the revolution. In fact, the various revolutionary governments, and later Napoleon himself, showered him with honors and distinctions, recognizing the immense value of his intellect. A particularly striking testament to the profound respect he commanded occurred in 1796 when the French commissary in Italy was instructed to formally visit Lagrange's father and convey the republic's congratulations on his son's achievements, acknowledging that Lagrange "had done honour to all mankind by his genius, and whom it was the special glory of Piedmont to have produced." It's also worth noting that Napoleon, upon gaining power, actively encouraged scientific pursuits in France and generously supported them.