Despite the strong military connections of the family, Borda's father had wanted him to become a magistrate but he allowed his son to follow his wishes and, at this stage, Borda began a career as a mathematician in the army.
Sa kabila ng malakas na militar na koneksyon ng pamilya, Borda's ama had to sa kanya upang maging isang hukom ngunit siya pinapayagan ng kanyang anak na sundin ang kanyang mga hangarin at, sa stage na ito, Borda nagsimula ng karera bilang isang dalub-agbilang sa hukbo.