This topic is one I've thought about and discussed several times before. It doesn't have a single answer, since it depends on so many factors.
If we're talking about teleporting a Roman legion from between 107 BCE and about 300 CE somewhere in Europe sometime between 751 CE and about 1500 CE, I think they'd trounce any army they could march to, and legions are good for nothing if not marching. The reasons for their victory here have already been discussed enough in this thread, though I think that the legion's iron discipline and ability to maneuver on the battlefield without breaking formation are its biggest advantages over any army of this period, as opposed to any technological or numerical superiority. Another thing to take into account is that, unlike hereditary magnates commanding feudal levies, a legion's officers would all be reliable and well-trained and -experienced, never shirking their jobs or growing lax at the first taste of success. The only things I think would slow the legions would be big fortresses, which were not ubiquitous anywhere except maybe northern France and England, and even a royal castle could only slow a legion which had control of the land around it. I don't think anyone who knows well about Masada and its fate will disagree here.
If we look at the years between the two periods I defined above, though, things are messier. An old legion teleported into this era might well end up facing an army created in its own image, and things would be more of a toss-up. Later legions tended to be a lot more lax as time went on, they were allowed to spend most of their time sitting around in comfortable barracks or patrolling around cities as lighter, cheaper troops took their places guarding borders and fighting bandits and highwaymen, which became a more serious issue as time went on. Part of it was that legionaries could expect pay increases whenever Rome went into a crisis, as every would-be Augustus tried to curry their favor. Later legions carried less equipment, after the fourth century often used personal arms and armor of inferior quality, and would much more heavily rely on their auxiliaries. They still had tactics and training, though, and could probably best an old legion after an initial shock if given superior numbers, advantageous terrain, particularly fierce auxilia, or some other advantage(s) along those lines. The odds would be close in any case.
I'm not going to talk about how a legion would do after the fall of Constantinople. I'm no expert on the armies of the renaissance, and don't want to make false claims. It seems to me, however, that the tercio brought back something that had been missing for nearly a millennium: large scale, well organized formations able to independently move across a battlefield, using fire-and-advance strategies not dissimilar to a cohort with javelins and swords, and with enough defensive capability to neutralize most threats. Of course, tercios had their weaknesses, but with the reintroduction of unit tactics and diverse weaponry the legion could be countered again, even if it might have initial success. And by the 19th century, warfare had evolved to the point where I don't think anyone would deny an ancient legion was obsolete (except maybe in terms of logistics, discipline, and movement
)
No European state had quite the manpower of 1st c. Rome before 1600.
I don't think after that date for a while yet, either. I'm just speaking from memory right now, but I think the population of Europe as a whole (not centers of ancient population) had only just recently reached the numbers it had before the dark age population crash. Keep in mind that Europe was still very much divided into petty states with often inconsistent policies as to military service, while Rome had a relatively centralized government spanning three continents, including all the parts of Europe worth the cost of guarding and policing. Going from memory again, between Augustus and Diocletianus there were 25-30 standing legions, each with about 10,000 soldiers serving in it. Legions likely to see action could also be reinforced to 15,000 men. This means 250,000-400,000 fighting men available at all times, plus the ability to raise more soldiers on demand: Conscription was never implemented after Marius started recruiting volunteers. The problem the Romans had in raising additional armies came from a lack of money rather than a lack of manpower. As far as I know, no European army except the Ottoman sultan's (which was a seasonal, conscripted army anyway!) could match this number until the late 18th century, except
maybe the Russian army.
But numbers is the most important factor in tactics. Removing it just doesn't make military sense.
I think people were trying to compare equal-sized forces in order to assess the armies by how they fight in a battle. Of course, battles are only the culmination of a long process of training, supplying, maneuvering, and planning for every situation....
And after all life is cheap. Wouldn't it be a better measurement that the sum of the two armies' equipment should cost the same?
Factoring in prices skews everything even more. Very different things were available for very different prices in 200 and 1200.
Chainmail is an artisan product. Shields and pila aren't cheap either. Training the legionnaires to the presumed standard is probably vastly more expensive than raising new pikemen.
The expensive items are gendarmes and cannon, both of which are going to be disproportionally effective against the Romans. Everything else, with 16th c, inftastructure, is much cheaper than chainmail is.
On the other hand, transporting a 16th c. army into 1st c. Europe makes all 16th. c gear unreasonably expensive.
With 1st century infrastructure, 1st century gear was also cheaper, better, and present in larger numbers. Roman legionary equipment came from armories. These armories were characterized by the unsurpassed quality of their work, their large output, their efficient use of materials, and a degree of standardization that only machines could in the 19th century, not unlike the Roman pottery industry. The armories, like the big pottery workshops, have been characterized as early factories. Even the ballistae and onagers legions dragged with them were standardized, down to livestock being raised for the express purpose of providing sinew to make their strong ropes. They continued to produce equipment and supply the western legions until after 409, when barbarian marauders and bandits would loot and burn them frequently enough to make them more a liability than an asset. I would argue it would be cheaper to outfit a Roman army than anything in the sixteenth century, and that the army would end up better equipped (in quality rather than usefulness of equipment.)
The crusaders experimented with that because, even if they faced a totally inferior cavalry, worse than the ancient one, they were losing because of inferior tactics and flexibility.
Inferior cavalry? Horse archers were the one thing that could reliably beat legions. They made the premodern world's most flexible infantry look like a Philippian phalanx. They totally slaughtered the heavy cavalry a lot of legions were still pulling around. Whenever you see the horse archers losing to non-horse archers in a pre-gunpowder battle, it's because they've been isolated from the rest of their army and somehow immobilized, or because they've been rushed at by cavalry riding lighter horses than they are (rare in these east vs. west fights). There's a reason that the later legions recruited so many Sarmatian auxiliaries, all lancers and horse archers. You might be interested to know that horse archers actually formed a core part of the medieval Eastern Roman army, recruited right from Anatolia! Manzikert was a time of structural weakness in the army, an exception rather than a rule.
I don't think I overestimate the skill of leaders, since the medieval commanders didn't actually pre-plan anything, usually the only tactic of commanders like Richard I was to charge with the whole army hoping for the best. By contrast, some ancient commanders either were able to develop a brilliant pre-plan (Cannae) or adapt to the conditions (Ad Decimum, Pharsalus, and many others). Medieval generals didn't even had a reserve usually, all control was lost once the 1st charge was unleashed on the enemy, and often the fate of the battle totally depended on the success of that charge.
Very true. The study of tactics and strategy was kept alive, though stagnated and reduced, but armies were not well trained enough to use most strategies and tactics, low discipline meant no plans would stay in place for long, and a commander couldn't rely on his lieutenants to not suddenly rush the enemy's strong point for glory, or rush away for their lives....
Err, in what way was turkish cavalry inferior to byzantine one? One of them managed to successively maul the other, and we don't call the place "Byzantium" any more.
There wasn't too much of a difference, except that the Roman cavalry tended to be more specialized, with one unit carrying javelins and light weapons, one carrying big shields and spears, one carrying compound bows, one carrying extra-long lances, etc. Turkish tribal cavalry was a bit more versatile because almost every man would carry at least a bow and long sword. After Turkish tribesmen had overrun eastern Anatolia following the battle of Manzikert, most of the Eastern Roman cavalry
were Turkish tribesmen anyway.
I think it's pretty clear that ancient cavalry flat-out sucked compared to their medieval counterparts. (while the reverse is pretty much true when comparing infantry)
Yes, though western heavy cavalry remained inferior to eastern light cavalry, which was more versatile, could be fielded in larger numbers, and could use the same weapons. Not to mention horse archers. In fact, I could argue that horse archers and light cavalry are ancient in their tactics and equipment, barring their use of stirrups and modern breeds.
Edit: Sorry about the extremely long reply. I started writing and forgot to stop myself for an hour and a half. I need to find a job.