Find two numbers such that their mean proportion is $12$ and third proportion is $768$.

Let the two numbers be $a$ and $b$.The mean proportion of $a$ and $b$ is $12$:
$$\sqrt{ab}=12⟹ab=144\text{— (Equation 1)}$$
The third proportion of $a$ and $b$ is $768$:
$$\frac{b^2}{a}=768⟹b^2=768a\text{— (Equation 2)}$$
From Equation 1, we can substitute $a=\frac{144}{b}$ into Equation 2:
$$b^2=768\times \left(\frac{144}{b}\right)$$
$$b^3=768\times 144$$
$$b^3=110592$$
$$b=\sqrt[3]{110592}=48$$
Now, substitute $b=48$ back into Equation 1 to find $a$:
$$a=\frac{144}{48}=3$$
Hence, the two numbers are $3$ and $48$.