Charles Augustin Coulomb's father was Henry Coulomb and his mother was Catherine Bajet. Both his parents came from families which were well known in their fields. His father's family were important in the legal profession and in the administration of the Languedoc region of France, and his mother's family were also quite wealthy. After being brought up in Angoulême, the capital of Angoumois in southwestern France, Coulomb's family moved to Paris. In Paris he entered the Collège Mazarin, where he received a good classical grounding in language, literature, and philosophy, and he received the best available teaching in mathematics, astronomy, chemistry and botany.
At this stage in his education there was a crisis for Coulomb. Despite his father's good standing, he had made unsuccessful financial speculations, had lost all his money and moved from Paris to Montpellier. Coulomb's mother remained in Paris but Coulomb had a disagreement with her over the direction his career should take so he left Paris and went to Montpellier to live with his father. At this stage Coulomb's interests were mainly in mathematics and astronomy and while in Montpellier he joined the Society of Sciences there in March 1757 and read several papers on these topics to the Society.
Coulomb wanted to enter the École du Génie at Mézières but realised that to succeed in passing the entrance examinations he needed to be tutored. In October 1758 he went to Paris to receive the tutoring necessary to take the examinations. Camus had been appointed as examiner for artillery schools in 1755 and it was his Cours de mathématique that Coulomb studied for several months. In 1758 Coulomb took the examinations set by Camus which he passed and he entered the École du Génie at Mézières in February 1760. He formed a number of important friendships around this time which were imporatnt in his later scientific work, one with Bossut who was his teacher at Mézières and the other with Borda.
Coulomb graduated in November 1761. He was now a trained engineer with the rank of lieutenant in the Corps du Génie. Over the next twenty years he was posted to a variety of different places where he was involved in engineering, in structural design, fortifications, soil mechanics, and many other areas. His first posting was to Brest but in February 1764 he was set to Martinique in the West Indies. Martinique fell under the sovereignty of France under Louis XIV in 1658. However Martinique was attacked by a number of foreign fleets over the following years. The Dutch attacked it in 1674 but were driven off, as were the English in 1693 and the English again in 1759. Martinique was finally captured by the English in 1762 but were returned to France under the terms of the Treaty of Paris in 1763. The French then made attempts to make the island more secure by building a new fort.
Coulomb was put in charge of the building of the new Fort Bourbon and this task occupied him until June 1772. It was a period during which he showed the practical side of his engineering skills which were needed to organise the construction, but his experiences would play a major role in the later theoretical memoirs he wrote on mechanics. As far as Coulomb's health was concerned these were difficult years and the illnesses which he suffered while on Martinique left him in poor health for the rest of his life.
On his return to France, Coulomb was sent to Bouchain. However, he now began to write important works on applied mechanics and he presented his first work to the Académie des Sciences in Paris in 1773. This work, Sur une application des règles, de maximis et minimis à quelque problèmes de statique, relatifs à l'architecture was written (in Coulomb's words, see for example):-
... to determine, as far as a combination of mathematics and physics will permit, the influence of friction and cohesion in some problems of statics.
Perhaps the most significant fact about this memoir from a mathematical point of view is Coulomb's use of the calculus of variations to solve engineering problems. As Gillmor writes in:-
In this one memoir of 1773 there is almost an embarrassment of riches, for Coulomb proceeded to discuss the theory of comprehensive rupture of masonry piers, the design of vaulted arches, and the theory of earth pressure. In the latter he developed a generalised sliding wedge theory of soil mechanics that remains in use today in basic engineering practice. A reason, perhaps, for the relative neglect of this portion of Coulomb's work was that he sought to demonstrate the use of variational calculus in formulating methods of approach to fundamental problems in structural mechanics rather than to give numerical solutions to specific problems.