The Volozhin Yeshiva is known as the "first yeshiva".
What was the system of Torah learning in Europe before Volozhin and in what way did Volozhin change the system?
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First of all, it's worth noting that, with only certain key exceptions, there weren't really any yeshivas in Eastern Europe before Volozhin, their having been replaced by the beis midrash, and by a system known as kest. This was a phenomenon by which young, married men were supported by their fathers-in-law to learn Torah; they would typically live with him, and would learn in the local beis midrash or shul. Their ability to do so effectively was resultant of their success in the heder system - itself largely dependent upon how much money their own parents had. (Teaching in heder was not a well-paid position, and most heders had ridiculously large class sizes). You can read more about this in this YIVO article, and in Shaul Stampfer's essay on early marriage in his Families, Rabbis and Education.
As for Volozhin, Stampfer (Lithuanian Yeshivas of the Nineteenth Century) suggests that R' Hayim introduced several innovations, on the basis of which it has come to be viewed as the first yeshiva in a modern sense. The most important of these is that of fund-raising.
Before Volozhin, a yeshiva (or beis midrash) was established by the community in which it existed. They built it, they hired the rosh yeshiva, they paid any teachers and they, more or less, supplied the students. If students travelled from a distance, they slept in the beis midrash itself, and were reliant upon the hospitality of the townspeople. As such, the local community was intimately involved, and if there was anything they didn't like they could see to it that it was changed.
R' Hayim promoted Volozhin as an institute for advanced Talmudic study, and due to his reputation as the great disciple of the Gra, funding came from far and wide. So too, incidentally, did students. As a result, the local community had nothing to do with the establishment of the yeshiva, and the students (who were given a stipend by R' Hayim) could pay them for their hospitality. As such, the yeshiva of Volozhin was completely distinct from the town of Volozhin, and this became something of a precedent. In time, it would happen that the Rosh Yeshiva of a major institution would be the authority within the local community, even hiring the town's other rabbonim and effectively replacing the beis din. For the replacement of the beis din by the Rosh Yeshiva in Lithuania - see the YIVO article on Orthodoxy (the paragraph on Hatam Sofer).
Other major innovations included the fact that R' Hayim appointed a deputy - something hitherto unheard of - and hired maggidei shiur to supplement his own teaching. This provided the yeshiva with an opportunity to attract more students, and to have guest lecturers who could increase their own reputation in the process.
I don't know the extent to which it was an innovation of Volozhin, but in terms of comparing it to other yeshivas it is worth noting that the chavruta system was a distinctly Polish/Lithuanian phenomenon and that yeshivas in Hungary were quite different. They had more student/teacher interaction, formal class times and exams. Study at Volozhin was largely private, and although shiurim were given by the Rosh Yeshiva, his deputy and by other maggidei shiur, students could learn whatever they want. By the late 19th century, and perhaps as a result, Volozhin had become a beacon for young maskilim, and many of the students left it to pursue a university education.
My understanding is that until that point, if you wanted to study you'd have to pay your own way and find someplace to study. The Volozhin yeshiva raised funds, and didn't charge tuition. So it decoupled the education from paying for it. It was also more officially organized and centralized; usually you'd have to wander from town to town and unofficially study with the town rabbi.
Rabbi Berel Wein has good lectures on this. here's an article based on his lectures
First, was to have a central place of learning. Until then, the system of learning in Eastern Europe was that boys went to cheder (literally “a room,” but more broadly an elementary school) from about the time they turned three until about bar mitzvah. Some stopped at 11 or 12. Most young Jewish men were working by the time they were bar mitzvah.
Achsanios Shel Torah - published in (5650) 1890 - by Rabbi Moshe Reinas says that in the year (5563) 1803 Rabbi Chaim Volozhin laid the cornerstone for Yeshivas Volozhin. This was opened due to Rabbi Chaim Volozhin perceiving the general state of learning Torah disintegrating. Originally it was supported by Rabbi Chaim Volozhin's personal wealth, however as the Yeshiva expanded he was unable to support it on his own. The Yeshiva was the first Yeshiva that did away with the up until then "Essen Teg" system, where the Bochurim ate at different people's houses every day, and a stipend was provided for the Bochurim instead.
I found that The Haskalah Movement in Russia also says that Rabbi Chaim Volozhin's Yeshiva was the one that did away with Essen Teg.
To me this is very interesting as I was always under the impression that Rabbi Meir Shapiro was the first one to do away with Essen Teg.