Since -8, \(m\), \(n\), and 19 are in arithmetic progression, the common difference \(d\) between consecutive terms is the same:

\(d = m - (-8) = n - m = 19 - n\).

From the first equation, we have \(m + 8 = d\).

From the second equation, we have \(n = m + d\).

From the third equation, we have \(n = 19 - d\).

Now, we can solve for \(m\) and \(d\) using the first equation:

\(m = d - 8\).

Substitute this into the second equation:

\(n = (d - 8) + d = 2d - 8\).

Now, substitute this into the third equation:

\(2d - 8 = 19 - d\).

Solve for \(d\):

\(3d = 27\) => \(d = 9\).

Substitute \(d\) back into \(m = d - 8\):

\(m = 9 - 8 = 1\).

And substitute \(d\) into \(n = 2d - 8\):

\(n = 2(9) - 8 = 18 - 8 = 10\).

So, the values of \(m\) and \(n\) are 1 and 10, respectively.