On 30 November 1812, the young engineer from Cherbourg read to the Institut de France a memoir with a long title: on functions that can take only two values, equal and of opposite signs, when the variables are interchanged. Behind the title was the first general theory of determinants. Functions that change sign Take three variables $a_1, a_2, a_3$ and the product of the differences $$\Delta = (a_2 - a_1)(a_3 - a_1)(a_3 - a_2).$$If we swap $a_1$ and $a_2$, each factor either stays the same or changes sign, and altogether $\Delta$ becomes $-\Delta$. Any swap of two variables does the same. A permutation of the variables, which is a sequence of swaps, multiplies $\Delta$ by $+1$ or by $-1$: it is even or odd.
Determinants were born to solve systems of linear equations. In 1750 Gabriel Cramer published the rule that bears his name, which gives each unknown as a ratio of two expressions formed from the coefficients. But an organized way of computing those expressions when there were many unknowns was missing. In 1772, almost at the same time, Alexandre-Théophile Vandermonde and Laplace studied these expressions as objects in their own right. The word determinant, in its present sense, was fixed only by Cauchy in 1812, and the vertical bars we use today came from Arthur Cayley in 1841.