How many sprays cover the space?

For all $d \geq 3$ we show that the cardinality of $\mathbb{R}$ is at most $\aleph_n$ 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 $d = 2$.