To find two numbers based on the given conditions of mean proportion and third proportional, follow these steps:

Understanding Mean Proportion
- The mean proportion between two numbers \( a \) and \( b \) is given by the formula:
\[
\sqrt{a \cdot b} = 18
\]
- Squaring both sides gives:
\[
a \cdot b = 18^2 = 324
\]

Understanding Third Proportional
- The third proportional to two numbers \( a \) and \( b \) is the number \( c \) such that:
\[
\frac{a}{b} = \frac{b}{c}
\]
- This can be rearranged to:
\[
c = \frac{b^2}{a}
\]
- According to the problem, \( c = 144 \).

Setting Up the Equations
- From the third proportional, substituting \( c \):
\[
144 = \frac{b^2}{a}
\]
- Rearranging gives:
\[
b^2 = 144a
\]

Combining the Equations
- Now we have two equations:
1. \( a \cdot b = 324 \)
2. \( b^2 = 144a \)
- From the first equation, express \( b \) in terms of \( a \):
\[
b = \frac{324}{a}
\]
- Substitute into the second equation:
\[
\left(\frac{324}{a}\right)^2 = 144a
\]
- Simplifying this leads to:
\[
\frac{104976}{a^2} = 144a
\]
- Multiplying through by \( a^2 \):
\[
104976 = 144a^3
\]
- Thus:
\[
a^3 = \frac{104976}{144}
\]
- This simplifies to:
\[
a^3 = 729 \implies a = 9
\]

Finding the Second Number
- Substitute \( a \) back into the equation for \( b \):
\[
b = \frac{324}{9} = 36
\]

Final Result
- The two numbers are:
- \( a = 9 \)
- \( b = 36 \)
- They satisfy both the mean proportion and third proportional conditions.