= Solution
Every proper equivalence class of a positive definite form contains a unique reduced form $[a,b,c]$, where
$$
|b|\leq a\leq c,
$$
with $b\geq0$ when equality occurs in either inequality. Its discriminant is
$$
b^2-4ac=-80.
$$
For a reduced positive definite form, $3a^2\leq80$, so $1\leq a\leq5$. Also $b$ is even and
$$
ac=20+\frac{b^2}{4}.
$$
Checking the finitely many possibilities with $|b|\leq a\leq c$ gives
$$
[1,0,20],\qquad [3,2,7],\qquad [3,-2,7],\qquad [4,0,5].
$$
Each coefficient triple has greatest common divisor one. By uniqueness of reduced representatives, these are four distinct classes. Hence the requested number is
$$
\boxed{4}.
$$
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