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Let $F=\mathbb{Q}\left(\sqrt{d}\right)$ be a real quadratic field with ring of integers $𝒪$. In this paper we analyze the number $h_d$ of ${GL}_1\left(𝒪\right)$-orbits of homothety classes of perfect unary forms over $F$ as a function of $d$. We compute $h_d$ exactly for square-free $d\le 200000$. By relating perfect forms to continued fractions, we give bounds on $h_d$ and address some questions raised by Watanabe, Yano, and Hayashi.