# When was the Laguerre's method first used to approximate roots?

Is there a specific date when Laguerre published his root finding method? I found his 1880 note Résolution des équations numériques, but I am not sure if this is the source because I can not read French.

Under this condition the usual "trick" in the derivation also makes sense, as when $$\alpha$$ is the largest root, and $$x$$ is a close approximation, then $$\frac1{x-\alpha}$$ is large in $$G=\frac{f'(x)}{f(x)}=\frac1{x-\alpha}+\frac1{x-\beta}+...$$ and $$H=\frac{f'(x)^2-f(x)f''(x)}{f(x)^2}=\frac1{(x-\alpha)^2}+\frac1{(x-\beta)^2}+...$$ and the remaining terms are small and of the same sign and can thus be considered as approximately equal. Setting them equal to zero, as Laguerre discusses first, gives in the first equation Newton's method and in the second $$0<\frac1{(x-\alpha)^2}\le H \\ \implies \alpha\le x-\frac{|f(x)|}{\sqrt{f'(x)^2-f(x)f''(x)}} ~\text{ or }~ x+\frac{|f(x)|}{\sqrt{f'(x)^2-f(x)f''(x)}}\le \alpha$$ giving a root exclusion region around $$x$$.