If (X,\(\perp)\) is an orthogonality space and \((Y,+)\) an Abelian group, then a mapping \(f: X\to Y\) is said to be orthogonally additive if \((1)\quad f(x_ 1+x_ 2)=f(x_ 1)+f(x_ 2)\) for all \(x_ 1,x_ 2\in X\) with \(x_ 1\perp x_ 2\). Two of sixteen results obtained in this paper are as follows: Theorem 6. If (X,\(\perp)\) is an orthogonality space, \((Y,+)\) an Abelian group and \(g: X\to Y\) an even solution of (1), then g is a quadratic mapping, i.e., \(g(x_ 1+x_ 2)+g(x_ 1-x_ 2)=2g(x_ 1)+2g(x_ 2)\) for all \(x_ 1,x_ 2\in X.\) Theorem 9. If (X,\(\perp)\) is an inner product space and \((Y,+)\) an Abelian group, then \(g: X\to Y\) is an even solution of (1) if and only if there exists an additive mapping \(\ell: R\to Y\) such that \(g(x)=\ell (\| x\|^ 2)\) for every \(x\in X\).