Denote by ${\mathcal D}$ the open unit disc in the complex plane and $\partial {\mathcal D}$ its boundary. The Douglas identity reads \begin{eqnarray}\label{abs} A(u)=\int\int_{\mathcal D}|\bigtriangledown U|^2dxdy&=&\frac{1}{2π}\int\int_{\partial {\mathcal D}\times \partial {\mathcal D}} \left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|,\end{eqnarray} \end{abstract} where $u\in L^2(\partial {\mathcal D}), U$ is the harmonic extension of $u$ into ${\mathcal D}$. The present work generalizes the above relation in the one complex variable case, as well as three more quantitative identical relations, to the context of the unit sphere of the $n$ dimensional Euclidean space ${\mathbb R}^n, n\ge 2.$ We also obtain the same types of relations with the quaternionic and the Clifford algebra settings. In particular, for each of the latter concerned two hyper-complex cases there exists an equivalent relation like