Why didn't the crossbow replace the bow as the primary ranged weapon in medieval europe? Why did they co-exist? Did the crossbow fill a different niche than the bow?

As u/voyeur324 has mentioned, it is very difficult to prove a negative. On top of that, nobody in medieval Europe sat down and wrote an essay on Why I Love The Crossbow or anything like that. We have only very limited access to what people thought at the time, so we have to speculate based on what evidence we do have.

I’ll address the last part of your question first because there is a popular conception in medieval military history that the two weapons were best suited to different kinds of warfare. The theory goes that the crossbow’s ability to stay loaded and ready to shoot made it more advantageous during a siege while the longbow’s faster rate of fire made it far superior in an open battle. I think this looks like a tempting explanation because at its core it is true: crossbows could stay loaded but were slow to reload, longbows were faster to shoot but holding them at full draw was exhausting. However, the evidence of how these weapons were actually used provides only minimal support for this theory, so let’s dig into it a bit.

It’s not an argument entirely without merit. It makes logical sense and there is good evidence of crossbowmen showing up in city and castle garrisons throughout the central and later Middle Ages. The Calendar of Documents Relating to Scotland includes a bunch of references to moving crossbows between various garrisons, the Tower of London (effectively the royal armoury for most of the Middle Ages) had quite a few, and Calais seems to have maintained a highly paid force of crossbowmen for its defence, especially after the rest of France was lost in the mid-1400s.

Here’s the problem: those crossbows did not exist in isolation. There are tons of records of bows being moved between castles, bows were also kept in the Tower of London, and Calais always had way more archers than it did crossbowmen. While crossbowmen in garrisons seem to have been paid more, possibly indicating a greater value for their contribution, in general crossbowmen seem to have received higher wages than archers and this is arguably because their equipment cost more – wages were often based on how expensive it was to kit yourself out for the role, military positions that required a horse paid more, for example.

This same pattern is evident in the composition of medieval armies, crossbowmen and archers were not mutually exclusive. Situations like the famous duel between English longbowmen and Genoese crossbowmen at Crécy were a rarity rather than the norm. Crusader armies usually had a mix of archers and crossbowmen (interestingly, so did Muslim armies at the time). Except for England during the 14th and 15th centuries, there doesn’t seem to be any evidence of a medieval army showing a significant preference for bows over crossbows or vice versa. And even then, it's not like England never used crossbows they were just vastly outnumbered by the archers with bows. Henry V took crossbowmen with him to Harfluer (and very possibly on to Agincourt), but he only brought ~30 of them in an army of several thousand.

So, if we’re fairly confident that these two weapons did coexist, then why? Again, there’s no definitive answer to this, historians are left with only informed speculation. The best explanation actually goes back to the very start: the differences between the two weapons. Crossbows could stay loaded, which was nice, but they were also generally more powerful and, importantly, their power was divorced from the strength of the archer. To get the most out of a longbow you need some really fit archers, to get the most out of a crossbow you just need someone who can handle the spanning equipment – that requires some physical fitness but not the same degree. The cost of that power and ease of use was that crossbows were slower to reload than longbows so if you wanted to lay down a constant barrage of missiles (which you might well want to do) you either needed multiple crossbowmen working together with one person reloading and the other shooting or you needed a bunch of bows. The English certainly made a convincing case that massed longbows was a pretty effective battle strategy, there is a lot to recommend it. At the same time, you could just have both – and a lot of medieval armies did.

Another factor to consider is cost. Crossbows were way more expensive than longbows and crossbowmen tended to come equipped with more gear – they needed spanning equipment, but they also often wore pretty heavy armour. Not everybody wanted to pay top dollar for a bunch of what were effectively medium infantry equipped with crossbows when archers were cheaper. The daily wage of a crossbowmen at 8d.-10d. a day vs. an archer at 6d. (8d. if mounted) might not seem like much, but if you’re hiring thousands of them that adds up. Anne Curry has certainly argued that part of the reason why Henry V had so many damn archers with him at Agincourt was cost, although she was comparing it to the cost of a man-at-arms which tended to clock in at 12d. a day. Why hire one man-at-arms when you can get two archers for the same cost? The math for crossbowmen isn’t as extreme, but the same principle applies.

This is a bit of a messy answer, but the tl;dr is that bows and crossbows often served complementary roles within medieval warfare rather than directly competing with each other. There is also room for personal preference, some rulers seem to have really liked crossbows – Richard I shot one in battle at Acre in 1191 and Jaffa in 1192 – and that preference may be reflected in the composition of their armies. There is also a caveat to add that medieval terminology sometimes gets messy, and soldiers classified as one type might only have been for financial records rather than actually serving that role. As an example, for a while in the Calais garrison ‘archers’ was the catch all term for anyone receiving 6d. a day and a lot of supposed ‘archers’ were armed with spears or bills rather than bows. A more extreme version came in the form of the francs-archers of Renaissance France which started out as archers but over a little less than a century evolved into a form of light cavalry armed with lances but still called the same thing.

There’s not a lot of good books on the history of crossbows to recommend (and I probably can’t get away with providing a book I’m still writing as a source), but on medieval archery in general absolutely check out Strickland and Hardy’s The Great Warbow or Jim Bradbury’s The Medieval Archer (the former is the best book on the subject but the latter is more widely available).

I also pulled from (among many others):

Grummitt, David, The Calais Garrison: War and Military Service in England, 1436-1558

Potter, David, Renaissance France at War: Armies, Culture and Society, c.1480-1560

Prestwich, Michael, Armies and Warfare in the Middle Ages: The English Experience

Tibble, Steve, The Crusader Armies

Alm, Josef, European Crossbows: A Survey