Find the third proportional to a - b and a² - b².

Let the third proportion to a - b and a² - b² be x,

$\Rightarrow a - b : a^2 - b^2 = a^2 - b^2 : x \\[1em] \Rightarrow \dfrac{a - b}{a^2 - b^2} = \dfrac{a^2 - b^2}{x} \\[1em] \Rightarrow x = \dfrac{(a^2 - b^2) \times (a^2 - b^2)}{(a - b)} \\[1em] \Rightarrow x = \dfrac{(a + b)(a - b) \times (a^2 - b^2)}{(a - b)} \\[1em] \Rightarrow x = (a + b)(a^2 - b^2)$

Hence, third proportional to a - b and a² - b² is (a + b)(a² - b²).