I know that in towns and cities there might be libraries built, but is it possible or likely that peasants can have access to it as well? I've heard that every village has an least a monastery or church, but I don't think they keep books there.
No, there would not have been a public library.
The majority of the world was illiterate before 1950. Even in Europe, most people were illiterate prior to 1800 (the protestant nations seemed to have a higher literacy rate at that time - probably because they had translated their bibles and prayer books from Latin, and held their services in English/Dutch/Swedish/etc).
In the middle ages, almost all Western literature was written in Latin, not the native tongues that the general peasantry would have spoken (and they likely still wouldn't have been literate in their own language, anyway - they didn't "need" to be).
Most churches probably wouldn't have had more than a bible and service book which the local priest would have used for his sermons. Scholarly and historical works would have been held at universities and other centres of learning, such as monasteries and cathedrals - but they would only have been available to teachers, students, and visiting scholars. Books would have typically been chained so as to prevent removal from the library.
"Free" public libraries - ones without (or at least, minimal) restrictions started to appear about the 17th century - these were typically associated with cathedrals or universities. Also, you wouldn't have been able to remove books from the library.
The kind of public lending library we're all familiar with didn't start appearing until the 19th century.
Medieval times span ten centuries and a continent. An English village in 1400 would be far from a Norwegian village in 500. That makes generalizations difficult. Here I'm thinking of the 11th or 12th century, England, France or the HRE.
- Many villages had a church, but that did not mean there was a full-time priest. (That would be a chapel of ease, unlike a filial church with a priest.) Monastries were kind of villages in their own right, not part of the average village.
- Illiterate priests were common enough to be a concern. (A bit of googling got me lots of tertiary sources, but no neat secondary or primary ones.)
- The rights and duties of a village might be described in the manorial roll of the manor. Business records were kept e.g. on tally sticks.
Literacy rates in the 15h century were on the order of 10%, and that would have been concentrated in professions (clergy, law, government) - so let's halve that for the village population. I wish I could dig into the numbers to distinguish between literacy and functional literacy.
the 10th largest town in England had approximately 5000 people - we can assume that the average village had less than 50% of that number. A second source lists 50-300 as a more reasonable number. Although I suspect the distribution was a power curve rather than a bell curve, let's be generous and assume that there are 175 people in the average village.
So there are less than 17 literate people in this village, and probably less than 9. Again, given the incredible cost of learning to read (hours of non-productive labor), they are probably related to one another and already share a household.
Prior to the invention of a printing press books were fantastically expensive. This source refers to print runs of less than 20 books - so there are very few books in the world. This source suggests that all of Germany printed less than 100 books/year. - and most of those are Bibles.
So what is the point of a library? Of the 17 people in the village who can use the book for more than fire kindling, they all know each other and can share books. Most of them own the same book - a Bible.
Books are an incredibly expensive luxury- elsewhere I've seen records of monastic libraries that were admired for their extensive collection of less than a dozen books. (The annual book production of a major country).
The notion of a medieval village library is absurd.
Before the printing press was invented in 1439 the only books were hand copied ones. Building a "large" library first required that you trained a team of scribes to write, and then found places to borrow books that they could copy.
Fun fact: actually, you didn't need to train most of your scribes to read as well as write, because book copying was often done by dictation, as a means of low-volume mass production: if you could borrow one copy of a book, a single reader and say 10 scribes taking dictation could make 10 simultaneous copies, which you might then hope to sell. Of course, this "chinese whispers" copying method explains why it was rare to find two copies of any book with identical texts!
Only the very rich could afford to do that, and the cost of books was still prohibitive after the invention of the printing press. The original selling price of Gutenberg's bible was about 3 years' average wages for a clerk - who would at least have been literate enough to read it, if he had found a way to live without money for three years in order to buy it.
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I would say for the majority of inhabitants the answers presented here are correct, however there are a few notable exceptions, especially in renaissance italy (and other cities with a strong merchant class). For example, the Biblioteca Marciana in Venice is an early public manuscript depository available since 1362. The English Wikipedia lacks a bit in detail and only starts with the finished building 200yrs later, but the collection was open to the public from the start, since the initial collection was endowed "ad communem hominum utilitatem".
No. Not for peasants. Wealthy families might have a primer, a book with the alaphbet, prayers, and stories that a child might learn from. Nicholas Orme's book Medieval Schools: From Roman Britain to Renaissance England describes many examples of schooling but it was still for the wealthy or religious. By the late 14th century there were free schools established by wealthy patrons but the students would not have been of the peasant class.
Now, peasants and/or the illiterate were occasionally in the presence of books (if that is what you mean by access) they would see books being used during mass and sometime attribute magical religious significance to the object of the books or the ink and paint on the pages.
There may have been "libraries" in villages, but they were private libraries. Typically maintained by the local church or monastery, or by a handful of wealthy citizens.
They would have had maybe a few dozen books, the Bible, religious books, and maybe the "classics." These were rare and expensive. Most people couldn't read, and apart from the Bible, most had no access to books. Only a handful of favored people, basically the "best friends" of owners could use these private libraries.
The idea of mass production (including that of books and paper), mass education, and mass libraries with "many" books got started with the industrial revolution of the 19th century with paperbacks appearing about mid-century, even though this trend did not fully develop until the 20th century.
There were Gurukuls (Vedic Schools) in south west Asia. They taught Vedas, Upnishads and other Sanskrit literature. After the invasion of Mughals, the system break down as the mughals started destroying Indian Vedic literature and Gurukuls. The Gurukul system of education still exist in India. (The Ved contains complete knowledge regarding medicine, vehicles, war, peace, law and spiritual science.)
Most Medieval Libraries were located either in major cities or Academic Libraries-(which were either based in cities or medium sized towns).
In the Byzantine Empire, the University of Constantinople had an Adjacent Library, though to my knowledge, there were no villages within the Eastern Roman Empire which had Libraries.
In Northern and Western Europe, most Libraries were associated with the various Universities, such as Padua and Bologna in Italy or Oxford and Cambridge in England, though it is difficult to find evidence of major Libraries in Medieval rural England, Italy or elsewhere within continental Europe.