Gurunet has, for the origin of peculate:
[Latin pecūlārī, pecūlāt-, from pecūlium, private property.]

For speculate, it has:
[Latin speculārī, speculāt-, to observe, from specula, watchtower, from specere, to look at.]

Note: where there are carets over the vowels, there should actually be lines. And, while I'm on a roll, G-net shows the origin of caret as:
[Latin, there is lacking, third person sing. present tense of carēre, to lack.] Which, I suppose, is apropos, since the capability for displaying lines over letters seems to be lacking...

EDIT: !!!!!!!! What the heck??? NOW it finally shows as an e with a line over it! [throwing hands in air e]