Page:A History of Mathematics (1893).djvu/285

harmoniously that it may fitly be called, in Sir William Rowan Hamilton's words, "a kind of scientific poem." It is a most consummate example of analytic generality. Geometrical figures are nowhere allowed. "On ne trouvera point de figures dans cet ouvrage" (Preface). The two divisions of mechanics—statics and dynamics—are in the first four sections of each carried out analogously, and each is prefaced by a historic sketch of principles. Lagrange formulated the principle of least action. In their original form, the equations of motion involve the co-ordinates $\scriptstyle {x}$, $\scriptstyle {y}$, $\scriptstyle {z}$, of the different particles $\scriptstyle {m}$ or $\scriptstyle {dm}$ of the system. But $\scriptstyle {x}$, $\scriptstyle {y}$, $\scriptstyle {z}$, are in general not independent, and Lagrange introduced in place of them any variables $\scriptstyle {\xi }$, $\scriptstyle {\psi }$, $\scriptstyle {\phi }$, whatever, determining the position of the point at the time. These may be taken to be independent. The equations of motion may now assume the form

$\scriptstyle {{\frac {d}{dt}}{\frac {dT}{d\xi \prime }}-{\frac {dT}{d\xi }}+\Xi =0}$;

or when $\scriptstyle {\Xi }$, $\scriptstyle {\psi }$, $\scriptstyle {\phi }$,… are the partial differential coefficients with respect to $\scriptstyle {\xi }$, $\scriptstyle {\psi }$, $\scriptstyle {\phi }$,… of one and the same function $\scriptstyle {V}$, then the form

$\scriptstyle {{\frac {d}{dt}}{\frac {dT}{d\xi \prime }}-{\frac {dT}{d\xi }}+{\frac {dV}{d\xi }}+{\frac {dV}{d\xi }}=0}$.

The latter is par excellence the Lagrangian form of the equations of motion. With Lagrange originated the remark that mechanics may be regarded as a geometry of four dimensions. To him falls the honour of the introduction of the potential into dynamics.^[49] Lagrange was anxious to have his Mécanique Analytique published in Paris. The work was ready for print in 1786, but not till 1788 could he find a publisher, and then only with the condition that after a few years he would purchase all the unsold copies. The work was edited by Legendre.