tatting, is really the same 'stitch' as many fishing nets--just done in miniature.. not related to knitting or crochet really in any way!

crossing threads with you, yes, i am sure tatting (knotted mesh work) which had been around for eons, was adapted to make ever finer 'lace' as a way to skirt sumpuary laws.

"real lace' is called pillow lace. is a kind of free hand weaving. bobbins of thread are hand woven, and twisted about each other.(sometimes as many as 300, even narrow bands of lace often have over 100 bobbins! small straight pins keep the twists 'in place' until the weight and tension will. (think of braiding hair.. the braid will un'ravel' unless secured. (less so with curly hair, more so with straight hair.) the fine threads used on lace will unravel (when cut!) or if left unsecured even while working them. so the lace is made on large pillows,(giant pin-cushions) and held in place with pins till secure.--pillow lace is very much like braiding.. (weaving has warp and weft, lace only has weft!)

'lace like' things can be made by
tatting-- a form of knotted mesh (identical to mesh used by some fishermen)
crochet
knitting
sewing. (eyelet lace, 'drawn thread', and battenburg (fine ribbons loosely sewn together with open stitches.)

Its generaly held that weaving came first (knitting of any sort was unknown in new world, but there were many groups that specialized in weaving (cotton is native to both new world and old)-- so weaving was 'invented' in both new world and old--or it was so old it came with the settlers to new world.

something similar to tatting was used in New world --shuttle-made mesh for fish nets. (and basicly that is what tatting is, you use a shuttle to form a 'hitches' half hitches and double hitches (a hitch is a basic knot, consult a boy scout manual if you don't know it) but it was never miniaturized. (matter of time and desire)
extra thread between the stitches makes picot's. tatting is made of loops (rings) of hitches, and 'lenghts' of straight work.

its not that hard to learn, (i taught myself)but i never practiced enough to get fast (but i still made yards of tatted lace.. (several pillow case edges worth!)

crochet hooks were in use for almost as long as knitting, but mostly as repair tools. (think of them as darning needles for knitted items) crochet really didn't take off (no art work of anyone doing it, till late 17th century)and it seems to have been an european invention.

you can find pictures of the madonna knitting for (or with)the christ child starting in the 13th century, and much older knitting in near east (eygpt). knitting was invented somewhere near egypt, and was spread through out near east by islamic traders. (south americans (many) still knit in an islamic style, no doubt taught the skill by 'spanish' sailors who also knit in that style

there are 3 major styles of knitting (european, Eastern and combo) and with european, many sub styles. hell, the UK alone has a half dozen!-- knitting in shetland island is different than in the fair isle (what less than 50miles away?) and scot's knitting is different than 'English' and English has 2 major styles! (think of the alphabets.. most I-E languages have A (alpha, allef, etc..) and A still looks something the old cuniform character.. but just as hebrew uses an A, it also is very different than most romance languages. sure scholars can 'see' similarites..

All knitting LOOKS the same. but there are RADICALLY different ways to form the stitches. (clockwise motion vs conterclockwise, vs. 'combo' (both clockwise and counterclockwise motions!) Combo (the least common) is how i knit.