It is a basic victoriana/edwardian type spindle leg that was common in one form or another with stools for a long time. I don't know of any specific name for it. I do a lot of them for someone who does renovation and reproduction on them. Not usually quite so many beads and coves as yours though.
I can see why you say Jacobean though as often as not they were with barley twists and / or a series of large beads running down the legs. That style tended to be later Jacobean perhaps? Most of the furniture tended to be heavy and often in oak. The simpler beaded spindle evolved into what we see today on so much furniture from the 19th and early 20th century and on until the 1950's. I admit to remembering the old sideboards and dressers with stunted versions of the style in solid oak. 'orrible stuff IMHO but now quite popular again in some circles.
Look closer guys, these are what I refer to as "square turned" because they are not turned at all. If you look at them they are square in section although at first glance appear to be turned. I've been asked to do similar before but have never actually figured out a way todo them on a standard lathe. I beleive the could be "thermed" if there was enough of them but they would originally have been made on a rotary cutting machine, which is a cross beteen a lathe and a spindle moulder, with spinning cutters that come in and remove wood.
Before I saw the way they had been made with rounded square section I would have said Victorian / Edwardian but I am not aware of anything like that in those eras of furniture making. Maybe there was though as the general look of them is definitely from that period.