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Jacobi's results on the computation of the order and of the normal forms of adifferential system are translated in the formalism of differential algebra. Inthe quasi-regular case, we give complete proofs according to Jacobi'sarguments. The main result is {\it Jacobi's bound}, still conjectural in thegeneral case: the order of a differential system $P_{1}, \ldots, P_{n}$ is notgreater than the maximum $\cal O$ of the sums $\sum_{i=1}^{n} a_{i,\sigma(i)}$,for all permutations $\sigma$ of the indices, where $a_{i,j}:={\rmord}_{x_{j}}P_{i}$, \emph{viz.}\ the \emph{tropical determinant of the matrix$(a_{i,j})$}. The order is precisely equal to $\cal O$ iff Jacobi's\emph{truncated determinant} does not vanish.  Jacobi also gave a polynomial time algorithm to compute $\cal O$, similar toKuhn's "Hungarian method" and some variants of shortest path algorithms,related to the computation of integers $\ell_{i}$ such that a normal form maybe obtained, in the generic case, by differentiating $\ell_{i}$ times equation$P_{i}$.  Fundamental results about changes of orderings and the various normal forms asystem may have, including differential resolvents, are also provided.

Jacobi's Bound. Jacobi's results translated in K{\"O}nig's, Egerv{\'a}ry's and Ritt's mathematical languages