The tenth century saw a revival of Byzantine military manuals, which gives us an insight into how the army commanders thought about warfare. For example, there are a succession of these army manuals starting from about the 950s to the 1000s AD that cover the use of large hollow square formations. In order these are the Sylloge Tactictorum (c. 950 AD) , the Praecepta Militaria (c. 965 AD), the Tactica of Nikephoros Ouranos (c. 1000 AD). [Infantry versus Cavalry: The Byzantine Response by Eric McGeer].

Some passages are verbatim quotes of the earlier manuals, however, what I find fascinating is that other passages have been edited with newer tactics. For example, the later Praecepta Miltiaria describes keeping the pikemen (menuvlatoi) close to the front line of spearmen (hoplites - no tactical relation to the ancient Greek hoplites aside from both having shields) where the Sylloge describes keeping them separate. These slight changes lend credence to the idea that these manuals were actually used and refined through periods of loss, such as in the early 950s soon after the Sylloge was written.

It is possibly worth providing the context that the Praecepta Militaria was a military manual likely edited together from the notes of the mid-tenth century general, Nikephoros Phokas, in 965 AD. The intervening 15 years (or so) had seen a very positive reversal in Byzantine fortunes, which had swept Nikephoros to the throne in 963 AD and his sponsorship as emperor led to the wide publication of the Praecepta Militaria. This has been a very wordy way of saying that Nikephoros was a very competent general who cared deeply about military literature and that we should pay attention to his writings and writings that he's sponsored.

Near the beginning of the Praecepta Militaria, the following quote is provided that relates to the usage of ancient formations in current times:

"The heavy infantrymen must be deployed two deep in a double faced formation, and keep two infantrymen in front and two in the back. Between them are three light archers, so that the depth of the formation is seven men. We do find the ancient Macedonians making their phalanx sixteen men deep, occasionally twelve or ten. But because their adversaries were borne by elephants with wild beasts set loose among their formations, as we find the Ethiopians did against Alexander the Great, they also employed other methods in their wars in addition to these and for these reasons made use of such formations.

In our own day, however, such formations are no longer employed and this type of phalanx is impractical. When compared with the wars of the ancients, even the offspring of Hagar have greatly reduced the depth of their formations. " [Chapter 1, lines 63-74, Page 17 of Sowing the Dragon's Teeth by Prof. Eric McGeer]

What is interesting here is that Alexander the Great did encounter elephants, they weren't the primary threat he faced when using his deep pike formations against Achaemenid Persia.

It's difficult to highlight an absence of evidence, however, another manual published under (or soon after the death of) Nikephoros II Phokas, called On Skirmishing, makes no reference to earlier tactics and strategies, instead keeping to very contemporary approaches of tracking the enemy, working out their size and force composition, shadowing them, and either attacking smaller raiding parties or forcing the invading force to not send out raiding parties at all. [Three Byzantine Treaties, George T. Dennis]

In conclusion, the generals of tenth century Byzantium broadly knew of the different approaches that had been used and intentionally dismissed them in favour of novel tactics that earlier Roman and Greek armies hadn't used. This reflects the much more constant state of border warfare and raiding in Anatolia and so the focus on cavalry as the offensive arm of the military with the infantry being used in a much more defensive role.

​

Sources:

Infantry versus Cavalry: The Byzantine Response by Eric McGeer

Chapter 1, lines 63-74, Page 17 of Sowing the Dragon's Teeth by Prof. Eric McGeer

Three Byzantine Treaties by George T. Dennis

Unfortunately I won't be able to go into the Byzantine Empire's rich literary, architectural or scientific history and I hope there are some additional answers that explore this in more depth then I am able. That being said, I am familiar with Byzantine military traditions of the early and middle Medieval period and hope to illustrate how Byzantine generals thought about the past.

I would like to start by clarifying that those who lived in the Medieval period did not see themselves as inferior to their ancient predecessors. Their view of the past wasn't that progress was lost and had to be restored, but that they were the beneficiaries of a grand history that allowed them to move further and see farther than any who came before them. The following quote attributed to Bernard of Chartres by John of Salisbury is a great example of this:

We are like dwarves seated on the shoulders of giants, and thus we are able to see more and farther than the latter. And this is not at all because of the acuteness of our sight or the stature of our body, but because we are carried aloft and elevated by the magnitude of the giants.

Two point are fascinating about this quote, first, the acknowledgment of the incredible contributions to their world that the Roman giants made, and second, that they can see farther and stand taller than their predecessors. The modern notion that the people of the Medieval period were backwards or inferior is just that: a modern notion. The individuals actually living in the Medieval period did not see the world this way. While they understood the impact of the ancient Roman civilization on their own world, they saw themselves as continuing the legacy forward.

We can see a similar sentiment among the Byzantine military sources I am familiar with. The authors are aware of the massive empire and powerful military of their ancient ancestors, both Greeks and Romans, but are focused on present problems, available resources, enemies, and circumstances. What worked for the ancients was not always relevant to their current situation. Here's a passage by Nikephoros Phokas reflecting on infantry formations in the 10th century CE:

The heavy infantrymen must be deployed two deep in a double-faced formation, and keep two infantrymen in from and two in the back. Between them are three light archers, so that the depth of the formation is seven men. We do find the ancient Macedonians making their phalanx sixteen men deep, occasionally twelve or ten. But because their adversaries were borne by elephants with wild beasts set loose among their formations, as we find the Ethiopians did against Alexander the Great, they also employed other methods in their wars in addition to these and for these reasons made use of such formations. In our own day, however, such formations are no longer employed and this type of phalanx is impractical. When compared with the wars of the ancients, even the offspring of Hagar have greatly reduced the depth of their formations.

An anonymous treatise on strategy from the same general time period echoes this position: some learnings from the past are valuable, others are obsolete:

There are four principal ways in which men have fought one another in the past. They have waged war on foot, on horseback, in chariots, and on elephants. In these pages we shall not bother to discuss fighting with elephants and chariots. Why talk about them when even the terminology for their tactics has become obsolete? It is on the infantry and the cavalry forces, as they are each called, that we shall concentrate.

When we look at an earlier 7th century source, the Strategikon of Maurice, where the military situation of the empire is in a much worse state, Maurice begins by admitting that "the state of the armed forces has been neglected for a long time and has fallen so completely into oblivion, so to speak, that those who assume the command of troops do not understand even the most obvious matters." He goes on to say they make no attempt to improve upon the system of the Ancients, as evident by their grand conquests, however, the text describes tactics, equipment, formations, and training relevant to his current period. He admires his ancestors for their success, but the text is focused on the task at hand, not attempting to recreate an ancient phalanx or legion.

That being said, there is one point that is consistent from the Roman Republic all the way through the 10th century military manuals I've been quoting: the fortified camp. Having a specific, organized and consistent camp layout is something that absolutely shows up time and time again. Of course having a fortified camp is by no means unique to the Romans or Byzantines, but this is one facet of Roman military dogma that stood the test of time and remained relatively unchanged for a millennia or more.