Find the fourth proportion to $10$, $12$, and $15$.

Let the fourth proportion to $10$, $12$, and $15$ be $x$.
By definition of proportion:

$$\frac{10}{12}=\frac{15}{x}$$

Cross-multiplying to solve for $x$:

$$10\times x=12\times 15$$

$$10x=180$$

$$x=\frac{180}{10}=18$$

Therefore, the fourth proportion is $18$.