no, they didn't, because there weren't any (not in the ones I study: I am a member at
www.kronoskaf.com). Lancers in the Napoleonic wars were a reintroduction inspired by Polish cavalry (which itself was a bit of a throwback, and rarely took the field during this time--the 18th century, not the Napoleonic wars, calm down
). Certain units were equipped with lances prior to that, but they were few, largely irregulars, and in small numbers (and as mentioned, no known manuals). As to carbines: these were not used in a standard charge, but often on patrol or dismounted. in fact by 1750, hardly anyone used pistols on the charge, instead preferring to directly have at it with the sword (which was held forward in order to act as a super-short lance). the only exception I know of after 1750 was in the Austrian army, where horsemen formed super tight ranks and fired their pistols at the trot (if even moving), but that was a measure against Ottoman cavalry, not regular European varieties (the same enemy also meant that Austrian cuirassiers only wore the full cuirass and a lobster-tail helmet against the Ottomans--as late as 1789 in fact: against other Europeans they wore only the breastplate and a
secret, which was a latticework of iron fitted into the tricorne to provide protection from downwards cuts by the pallasch or sabres of European cavalry..
and again, even ignoring all the other stuff: the formations really are much tighter than what I have gathered on ancient cavalry formations thus far: the men are riding knee to knee in most armies. this would make cavalry using lances of the Hellenistic period way overpowered.
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