The Talmud, too, has statements where it can be implied that Pi = 3. It says (in the Eruvin tractate, as if somebody cared...) that the circumference of any circular shape is "three time" the diameter (and that the difference between a square shape and a circular one inscribed in it is "one forth" of the square -- that is, that the ratio between Pi*r^2 and (2r)^2 is 4:3, which implies Pi =3 as well).
But it cannot possibly in fairness mean that the Rabbis were making a mathematical claim. I have just, by chance, been reading that part of the Eruvin tractate. It goes more or less like this:
1). Eruvin means "mixing" -- when two meals, or areas or goods, etc., etc., count as "mixed" and as one for various ritual purposes, and when as two, such as whether one is allowed to move between them on the Sabbath, which is allowed if it counts as one courtyard, and sometimes not if it counts as two. It also discussed what one must do to separate or mix them.
2). In the place where the have Pi=3, they are discussing specifically what sort of openings or gates of what size and shape count as "connecting" two courtyards and making them count as one.
3). The relevant Talmud discussion starts with one sage asking, "what if the opening is crooked or round and not straight?"
Another sage replies, "we treat the crooked as if it were straight."
A third notes that the difference between that the difference between a crooked (circular) opening and a straight (square) one is 4:3 in area.
A fourth says the difference between the circumference and diameter of a circle is 1:3 -- like in Solomon's sea.
This naturally leads to a discussion of water for purification purposes, e.g., how much water could the sea of Solomon hold? Did it also have square baths above or below the round main one? Given the dimensions of the main bath, what is the maximum number of people could bathe in it for ritual purification? What is, equivalently, the minimum amount of water for such purposes?
It gets even more irrelevant from there, until one of the sages apparently finally remembers they're supposed to be discussing the shapes of various openings between courtyards, and they go back to that subject, for -- oh -- three sentences until the next digression.
Yes, the Talmud is in effect a huge collection of tractates (my copy fills twelve large volumes) of such meandering discussions. So much, by the way, for the attempt to find a "secret Jewish world domination plan" in that.
It is obvious they weren't thinking of exact mathematical relationships at all, but of rough, round quantities that can serve as a practical guide: they decide, for example, that the minimum depth of a ritual bath must be three cubits deep (a cubit is about half a yard), an opening must be four cubits high to count as a "gate" for various purposes, and that the minimum amount of water in a bath for ritual purification is forty se'ah (About 280 liters.)
