At eighteen, Poisson caught Legendre’s attention with a paper on finite differences (he later gave Legendre’s funeral oration, saying he had only wanted to be spoken about in terms of his work). Poisson was a student, and later close friend, of both Laplace and Lagrange, and a member of the influential Société d’Arcueil that Laplace and Berthollet founded. Galois, barely twenty, sent Poisson a paper on equation theory, which Poisson thought unclear. Arago, another société d’Arcueil member, wrote Poisson’s biography, reporting his words that he was good for only two things — doing mathematics, and teaching it.
König met his fellow-student Clairaut, as well as Maupertuis, through his studies with Johann Bernoulli (Daniel Bernoulli also taught him, as did Wolff); Clairaut became a correspondent and noted collaborator of König’s. Maupertuis introduced Voltaire and Châtelet to him, then had a famous mathematical quarrel with him, Euler putting the case against König, and Voltaire defending him in print. He travelled with his friends Voltaire and Châtelet to meet Réaumur, though Châtelet (to whom he taught algebra) later had a serious disagreement with him. Réaumur, also a friend, passed König a problem concerning the geometry of honeycombs.
A letter to d’Alembert on mathematical principles got Laplace a professorship at the École Militaire. He taught Fourier (who however thought Lagrange and Monge better teachers), and encouraged Cauchy. Lagrange was a professional rival, but both gained from the mutual flow of ideas. Laplace and Berthollet founded the influential Société d’Arcueil, whose members included Arago, Poisson, Biot, Gay-Lussac, Malus and Humboldt. With Lavoisier, he showed that respiration was a form of combustion. Biot helped prepare his work for publication, but said that Laplace often forgot his original reasoning, substituting the line “it is easy to see.”
Maupertuis was taught by Bernoulli, who encouraged his development of Newton’s theories; König was a fellow-student, though a famous and bitter mathematical quarrel marked their last years. Marivaux was a friend as a young man in Paris. Maupertuis was Châtelet’s geometry tutor and lover, and also taught Buffon. Clairaut accompanied him on a year-long expedition to Lapland (they met Celsius en route), though the friendship eventually deteriorated. His great friend Euler wrote to him for two decades, and deputised for him in Berlin. Voltaire had been a good friend, but mocked his ideas and his relations with Lapp women.
D’Alembert praised the 16-year-old’s gifts, taught him and became a close friend. The youngest of Diderot’s Encyclopaedists, he was a regular at d’Holbach’s salon, and wrote a biography of another good friend, Voltaire. He encouraged Monge to submit his research to the Académie des Sciences, and Legendre to write what became a classic geometry textbook. He was one of the first mathematicians Lagrange met when he came to Paris, and helped liberalise Franklin’s views on slavery and racial equality. He posthumously edited out pious references to God in his correspondent Euler’s published letters.
Johann Bernoulli taught Euler unofficially; his friend and fellow-student Daniel Bernoulli helped persuade Euler’s father that his great gift was for mathematics, collaborated with him, and invited him to settle in St. Petersburg, where Goldbach was among his colleagues. Condorcet and Euler had an extensive working correspondence: other correspondents included Lomosonov, Clairaut, d’Alembert, Legendre, and the unreliable Stirling. Euler deputised for Maupertuis in Berlin. When Lagrange wrote to him with a new kind of calculus, he withheld his own work to let the 19-year-old get the credit. Lexell helped him when he was virtually blind.
At 19, Lagrange wrote to Euler, proposing a new form of calculus. Poisson and he spurred each other on to refine planetary mathematics. D’Alembert supported him, Laplace was a mathematical correspondent and rival, and Lavoisier intervened to ensure that Lagrange, Italian-born, would not meet hostility in post-revolutionary Paris (Lavoisier was himself beheaded a few months later). Lagrange taught Fourier and encouraged Cauchy in his studies. A paper Germain sent led her to become (as a woman and outsider) his protégée. Lambert was a close friend, and Monge visited him when he was dying.
Lagrange, Laplace and Monge all taught him, and may have helped his release from imprisonment on political charges. He submitted a paper on algebra to Montucla while still undecided on his career. Berthollet, Monge and Malus were fellow members of the Institut d’Egypte, with Fourier elected secretary. He taught Malus at the Ecole Polytechnique, then run by Carnot and Monge, and stimulated his protégé Champollion’s determination to decode Egyptian hieroglyphs. Delambre arranged for publication of his work on heat-conduction, and Cuvier was his fellow ‘Perpetual Secretary’ of the Académie des Sciences.
Unknown to Adams, Le Verrier had reached the same mathematical conclusions as him about the existence of a new planet (Neptune). Adams had already left a paper with Airy, the Astronomer Royal, asking for observations to be made to test for the predicted planet’s existence, but Airy fatally delayed his response, and when a row erupted over the truth of the discovery of Neptune, cold-shouldered Adams. Le Verrier became a good friend to Adams, despite getting all of the credit. Stokes was a professional collaborator and frequent correspondent; Babbage also corresponded with him.
Arago and he worked together early in their careers, Arago coming to feel that Biot had sabotaged his results (he then revenged himself by working with Biot’s protégé Fresnel on the polarisation of light). Assisting Laplace in preparing his findings for publication, Biot observed that Laplace used the stock phrase “it is easy to see” when he’d forgotten his original reasoning. Biot and Gay-Lussac were the first to ascend in a balloon for scientific purposes. He did pioneering electro-magnetic work with Savart, and asked the young Pasteur to come and demonstrate his findings about the handedness of some molecules.