To determine all composite positive integers $n$ with the given property, we need to analyze the sequence of divisors and their differences. Let's break down the problem step by step.

1. Identify the divisors and their differences:
Let $1 = d_1 < d_2 < \cdots < d_k = n$ be all the positive divisors of $n$. The given property states that:
$$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1 : 2 : \cdots : (k-1)$$
This means:
$$d_2 - d_1 = 1, \quad d_3 - d_2 = 2, \quad \ldots, \quad d_k - d_{k-1} = k-1$$

2. **Express the divisors in terms of $k$:**
Since $d_1 = 1$ and $d_2 - d_1 = 1$, we have $d_2 = 2$. Similarly, $d_3 - d_2 = 2$ implies $d_3 = 4$, and so on. We can generalize this pattern:
$$d_i = d_{i-1} + (i-1)$$
For example:
$$d_3 = d_2 + 2 = 2 + 2 = 4$$
$$d_4 = d_3 + 3 = 4 + 3 = 7$$
However, this pattern does not hold for all $i$. Instead, we need to find a consistent pattern that fits the given ratio.

3. **Analyze the pattern for small values of $k$:**
Let's consider small values of $k$ to find a consistent pattern:
- For $k = 3$:
$$d_1 = 1, \quad d_2 = 2, \quad d_3 = 4$$
The differences are:
$$d_2 - d_1 = 1, \quad d_3 - d_2 = 2$$
This matches the ratio $1 : 2$.

4. **Verify if $n = 4$ is a solution:**
For $n = 4$, the divisors are $1, 2, 4$. The differences are:
$$2 - 1 = 1, \quad 4 - 2 = 2$$
This matches the ratio $1 : 2$.

5. **Check for larger values of $k$:**
If $k > 3$, the pattern $d_i = i$ for each integer $i$ does not hold because the differences $d_3 - d_2$ would not be equal to 1. Therefore, $k$ cannot be greater than 3.

6. Conclusion:
The only composite positive integer $n$ that satisfies the given property is $n = 4$.

The final answer is $\boxed{ n = 4 }$.