Bernoulli, a significant mathematician even in that family of mathematicians, is known as an early proponent of calculus. Jacob was his elder brother, who taught and worked with him closely, and whose university position he inherited. Daniel was his son; he developed a jealously competitive relationship with both. He tutored Euler (whose father was a friend, and who corresponded for 14 years), also teaching König, Haller and Maupertuis (who lodged with him). The aristocratic de l’Hôpital paid him handsomely, and published Bernoulli’s work as if his own. Leibniz, one of the founders of calculus, corresponded extensively about it.
Playfair is known, not for original discoveries, but for his clear expounding of theories (especially Hutton’s). Hutton was a close colleague — Playfair, Black and he often spending evenings together, including at the Oyster Club, where other members (and friends of Playfair) included Smith and Ferguson, whom Playfair held a joint professorship with. Playfair encouraged Somerville, investigated Edinburgh geology with Nasmyth, and met his lifelong friend Maskelyne (who got him to publish his first mathematical paper, and introduced him to London colleagues) by visiting and helping with his experimental measuring of mass at Schiehallion.
Boscovich, not as well known as he might be, held strikingly advanced ideas about space/time relativity and the nature of atoms, among much else. His wide-ranging correspondents included Euler, Jacobi, Bernoulli, Lagrange, Priestley, Spallanzani, Méchain, Metastasio and Voltaire. Old friends in France included Nollet, Lacaille, Lalande (struck by his extreme spirituality and profundity) and especially Clairaut, who admired his vast culture and personal dynamism. He met Buffon and Franklin, impressed Burney, discussed Newton with Johnson (in French and Latin), was initiated into the mysteries of Guy Fawkes’ Night by Maskelyne, and was resented by d’Alembert.
Voltaire, who’d known her as a child, was her lover and intellectual companion for 15 years, following her relationships with Maupertuis (who’d been her mathematics tutor) and Clairaut. She and Voltaire used English to argue in. She consulted Buffon over her translation of Newton’s ‘Principia’, and corresponded with Bernoulli, a friend. König travelled to Brussels with Châtelet and Voltaire, and seems to have been one of the few mathematicians she had a major disagreement with.
Mayer is known for lunar observation, particularly in relation to navigation; he also drew superb lunar maps. Mayer wrote to Euler (who had already heard of the young man) about astronomical refraction and lunar theory; they corresponded for four years, and Euler helped him to a professorship. Lacaille and Mayer wrote to each other particularly about the 1761 Passage of Venus, each arguing that the other’s instrument was less accurate than his own (Mayer was right, though Lacaille wasn’t to know it). Lambert, interested like Mayer in colour perception, visited him in Göttingen, and extended his work.
Daniel Bernoulli was one of the founders of mathematical physics. His father Johann taught him, but (over-competitive and jealous) tried to claim the honours for Daniel’s work on hydrodynamics. Goldbach, Clairaut, Maupertuis and Euler were all close mathematical friends of Bernoulli’s, living fruitfully intertwined lives. Euler, a friend from youth, lodged and worked with him in St Petersburg (he was asked to bring tea, coffee and brandy from Switzerland). Goldbach helped him get an early paper published, and like Lagrange and Clairaut, kept up an important correspondence. König was one of his students.
Goldbach is notable for his work on number theory. On a decade-and-a-half journey around Europe, he met Moivre in London, Leibniz in Leipzig (they continued to correspond), and Nicolaus Bernoulli in Venice, who suggested Goldbach contact his brother Daniel — they corresponded for 7 years. Euler became a colleague in St Petersburg; they started a 35-year correspondence. In one of his letters, Goldbach outlined the conjecture now famously named after him; characteristically, Euler tested its validity substantially further than Goldbach, who tended to be more interested in the ideas than the hard work.
Lexell, who spent most of his career in Russia, is known for his work on cometary orbits and on spherical geometry. He arrived in St Petersburg when its eminence grise, Euler, was over 60 and practically blind. He helped Euler in his work as one of his disciples, became his friend and chosen successor, and was present when he died (Lexell himself however dying young, barely a year later). Linnaeus was one of his correspondents. Although Lexell wrote at least once to William Herschel, the discoverer of Uranus, confirming its status as a planet and calculating its orbit, the extent of any further contact is not clear.
Gibbs helped found chemical thermodynamics, and did fundamental work on vector analysis (sparked by reading Maxwell), while his work in theoretical physics paved the way for quantum mechanics. Maxwell, one of the first to recognise Gibbs’ genius, later made and sent him a plaster model. Although Gibbs, self-contained, rarely travelled, his mailing-list of 300 prominent scientists included his correspondents Peirce, Rayleigh and Ostwald (who translated his work). In Germany he sat in on lectures by Kirchhoff, Helmholtz, Bunsen and Magnus (influences all); however it’s not known whether he actually met them.
Plücker did fundamental work in two very different fields: analytical geometry and cathode-ray physics (as well as in spectroscopy). Crelle promoted the young Plücker’s work, and helped him to an academic post, thus antagonising Plücker’s bête noire, the better-known but less original Steiner. Klein was Plücker’s doctoral student and assistant; he completed his unfinished work after his death. Plücker visited Wheatstone in his laboratory and saw his wave machine, corresponded with Faraday, was joined by Hittorf in his spectroscopic investigations, and served the visiting Helmholtz “remarkably good” wine.