The difference of the squares of two positive integers is 180.

Let the larger number be x.

Then according to the question,

Square of the smaller number = 8x, then

$x^{2}-8 x=180$

$\Rightarrow x^{2}-8 x-180=0$

$\Rightarrow x^{2}-18 x+10 x-180=0$

$\Rightarrow x(x-18)+10(x-18)=0$

$\Rightarrow(x+10)(x-18)=0$

$\Rightarrow x+10=0$ or $x-18=0$

$\Rightarrow x=-10$ or $x=18$

Since, x being a positive integer so, x cannot be negative,

Therefore, larger number = 18.

then the smaller number $=\sqrt{8 \times 18}=12$

Thus, the two positive numbers are 12 and 18 .