Vandermonde did not take up math until he was already 35 years old. At 35, he wrote a paper that created enough of a stir in the scientific community of France to get him selected to the Academie des Sciences in 1771. In total, he wrote only four mathematical papers.
In 1778, he chose to work on paper regarding musical theory. Many might have expected a mathematical analysis that extended the work of the Ancient Greeks. Instead, Vandermonde argued that there should be no theory of music as mathematical theory, but, instead, musicians should judge music solely based on their trained ear.
Throughout his life, Vandermonde collaborated with the greatest talents of his time. In 1777, he collaborated with Etienne Bezout and Antoine-Laurent Lavoisier to study the effects of the severe frost of 1776. In 1787, he worked with Gaspard Monge and Claude Louis Berthollet to research the manufacturing of steel with the aim of improving bayonets. In this effort, he investigated different combinations of iron and carbon.
In Vandermonde's first paper on mathematics, he presented a solution to the eleventh root of unity using ideas of symmetrical polynomials and permutations. These ideas would predate both the work of Carl Friedrich Gauss and Evariste Galois.
Henri Lebesgue wrote (quoted from Jean Tignol, see below):
Surely, any man who discovers something truly important is left behind by his own discovery; he himself hardly understands it, and only by pondering it for a long time. Vandermonde never came back to his algebraic investigations because he did not realize their importance in the first place, and if he did not understand them afterwards, it is precisely because he did not reflect deeply on them: he was interested in everything, he was busy with everything; he was not able to go slowly to the bottom of anything...Today, Vandermonde is best known for something that may have been accidentally named after him: the Vandermonde determinant. The idea is not found in any of his papers. Still, the last of his four math papers did focus on determinants in which for the first time, a determinant is presented as a mathematical function and Vandermonde proceeds to investigate its mathematical properties. One mathematician, Thomas Muir, has gone so far to say that Vandermonde is "the only one fit to be viewed the founder of the theory of determinants."