How many sprays cover the space?

For all $d\ge 3$ we show that the cardinality of $\mathbb{R}$ is at most ${\mathrm{\aleph }}_n$ if and only if ${\mathbb{R}}^d$ can be covered with $(n+1)(d-1)+1$ sprays whose centers are in general position in a hyperplane. This extends previous results by Schmerl when $d=2$.