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Besides, the point being made by Tri-ring (though overstated) is that the edge of a battlesword of that type generally isn't razor sharp -- otherwise it'd be a mess after a battle. Such swords were generally "sharpened" to wider angles so they wouldn't be nicked into oblivion.
The velocity of the swing drives the cut rather than a razor thin edge in that type of blade. The japanese swords (katana) do run a lot *sharper* but their blades are made with a stiffer spine and a razor-edged softer iron.
Cutting force worked just fine until advanced chain and plate became common -- remember the whole sword/mace/spear vs leather/scale/chain/plate was a thousand+ year arms race, not a static picture. Besides even in partial plate or advanced chain, many vulnerable points remained.
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