Dear Bingley: I searched for "falx" and found this in the Smith dictionary:
FALX, dim. FALCULA (a#rph, dre/panon, poet. drepa/nh, drepa/nion), a sickle; a scythe; a pruning-knife, or pruning-hook; a bill; a falchion; a halbert.

As CULTER denoted a knife with one straight edge, "falx" signified any similar instrument, the single edge of which was curved (Dre/panon eu)kampe/j, Hom. Od. xviii.367; curvae falces, Virg. Georg. i.508; curvamine falcis aenae, Ovid, Met. vii.227; adunca falce, xiv.628). By additional epithets the various uses of the falx were indicated, and its corresponding varieties in form and size. Thus the sickle, because it was used by reapers, was called falx messoria; the scythe, which was employed in mowing hay, was called falx foenaria; the pruning-knife and the bill, on account of their use in dressing vines, as well as in hedging and in cutting off the shoots and branches of trees, were distinguished by the appellation of falx putatoria, vinitoria, arboraria, or silvatica (Cato, De Re Rust. 10, 11; Pallad. i.43; Colum. iv.25), or by the diminutive falcula (Colum. xii.18). "

Early English foot soldiers carried a "billhook" which was like a heavy axe with forward curving tip, "J" shaped.