Nearly every undergraduate student of mathematics learns how to solve linear systems with the help of determinants, so it may come as a surprise that the history of the development of the determinant is not better known than it is. In fact, there may be a good reason for this: befitting the complexity of the idea, its history is also quite complicated. The story of its genesis and evolution involves the interplay of a number of different problems, perspectives and approaches, and contributions were made by dozens of people over centuries. We plan to survey a key period of this history, from the time of Leibniz at the end of the 17th century, up to the watershed day of November 30, 1812, when Binet and Cauchy both presented papers on the determinant at the same meeting in Paris.