The Composition on Warfare of Nikêfôros is much more prescriptive about all the soldiers. Kataphraktoi were to wear helms with two layer mail hangings covering all but their eyes, klivania with upper sleeves, most probably splinted, skirts and lower sleeves of padding faced with mail rather than a full mailshirt as shown here, and grieves, again most probably splinted. Nikêfôros also recommends that a katafraktos also carry a mace holstered on each side of the saddle, and one of each sort of sword (spathion and paramêrion) in an arrangement you can see below which allowed ready access to either. Cavalry lances were the shorter kontarion in this period, and apparently still mostly used with the ancient, loose under- or overarm stabbing technique or else two handed, rather than couched in the Western manner.
Such panoply was very nearly perfect in terms of pre-gunnery warfare and clearly survived through the next two centuries, where economics permitted. Run your mouse onto the main picture and then off to see the man from other angles.
For more information see the authors Osprey Publishing, Warrior Series volume Byzantine Cavalryman: Eastern Roman Empire, c.900–1204, 2009.
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