Abstract: The teaching relevent issues are treated for Lagrange’s equations of the first kind in theoretical mechanics. Firstly, the history of Lagrange’s equations of the first kind are clarified including Lagrange’s original idea of the equations and the multipliers, Bertrand’s formalization in general terms, Jacobi’s nomenclature of the first and the second kinds, and Rorth’s extension to nonholonomic systems. Then the Lagrange’s equations in terms of redundant coordinates are revisited by suggesting the redundant coordinates named as pseudo-generalized coordinates, reformulating the derivation of Lagrange’s equations of the first kind with the forcus on the introduction of the multipliers, and highlighting the difference between generalized forces and pseudo-generalized forces corresponding to the pseudo-generalized coordinates. Finally the relations are constructed among the different forms of Lagrange’s equations of the first kind.