Componendo-Dividendo: if $a:b = c:d$, then—

Componendo-dividendo divides the componendo result by the dividendo result: $\dfrac{(a+b)/b}{(a-b)/b}=\dfrac{(c+d)/d}{(c-d)/d}$, so $\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}$.