差す appears to have a lost of uses, most of which seem to involve making something pass through a point in space (and then, for lack of a better word, 'shoot out' from there) or the motions related to doing so. None of them seem to have anything to do with measurement though.
On the other hand it seems like the Kanji 差 is related to difference and subtraction which strikes me as much closer to the function of a ruler.
So why the 連用形 of 差す? And for that matter why after 物 unlike similar words such as 着物 or 買い物 (or even 指し物 :p)?