Debeaune was born in Blois to Florimond de Beaune senior, Seigneur de Goulioux. He was educated in Paris, where he studied law, before entering military service. His second marriage to Marguerite du Lot in 1623, following the death of his first wife, brought him considerable wealth. On the basis of his legal studies, he bought himself the office of conseiller to the court of justice in Blois. An amateur mathematician and landowner, he built up an extensive library and constructed an observatory on his estate near Blois. Through both the medium of correspondence and regular visits to Paris, he participated actively in contemporary scientific discourse. He exchanged letters with men such as Mydorge, Billy, and Mersenne, while Descartes and Erasmus Bartholin visited him in Blois to discuss mathematical topics.

Little of his mathematical work survives, apart from what is published in the Schooten's Latin translation of the Geometry. Descartes believed that no one understood his work better than his wealthy friend. Debeaune's Notes brèves, translated by Schooten in 1649, served to clarify numerous difficult passages and played an important part in the reception of the work. Shortly before Debeaune died, he met with Bartholin, leading to the publication of two short papers on algebra in Schooten's second edition. Of Debeaune's other work, a treatise of mechanics is mentioned by Mersenne, and a treatise on dioptrics is mentioned by Schooten; neither has survived.

Debeaune's legacy consists primarily in his work on two problems. He was the first to point out that the properties of a curve can be deduced from the properties of its tangent, and he achieved important results on the determination of the tangent to an analytically defined curve. Furthermore, he considered for the first time the upper and lower limits of the roots of numerical equations.

He had the reputation of being the finest instrument maker of his day. In 1639 Descartes asked Debeaune to build the machine for grinding hyperbolic lenses, which he had described in Dioptrics (AT II 511–12). Having embarked on this project, at the beginning of 1640, Debeaune cut his hand badly on glass. Poor health set in. Around 1648 he retired his post of conseiller. He died following the amputation of a foot.

See also Dioptrics; Geometry; Mathematics; Mersenne, Marin