Actually, I think 8000 magi total is overestimating it a bit.
Assumptions:
There are about 1200 magi in the Order of Hermes in the year 1220
The order has existed for about 450 years
The average mage lives for about 100 years after his Gauntlet
The number of members has stayed more or less constant through the years (not really true - we know the OoH was much smaller in the early years.)
Based on those assumptions the total number of magi over the years would be around 450/100*1200 = 5400 magi.
This is of course a rough estimate, so give or take a thousand magi.
The Order grew faster than that. You have to remember a small incident that happened called the Schism War which killed off a fairly large chunk of the Order. The population fairly rapidly bloomed after the founding of the Order, then took minor hits and rebounded from later incidents. The Schism War is just the largest and most recent.
Your numbers would be accurate if the Order was stable throughout its history without any large drops and spikes in population other than the initial growth.
EDIT: As a side note, those incidents could be the reason for a reduction in the number of older books.
Sub Rosa #14 p.24ff has a good elaboration by Mark Shirley, an expert in animal population dynamics, about the number and ages of magi in the OoH over the centuries.
Personally I consider the population of 1200 mages for the entire order ridiculously low, despite being canon. The population of Europe at the time was in the millions (I looked up and posted numbers in another thread- I think it came to one mage per 10,000 or 100,000 people). It feels a bit like Harry Potter math to me (where J.K. Rowling admitted to just making up the numbers with idea of what the implications to the world at large would be)
It comes out to about 1 Magus per 100k people. The 1:10k people is the rate of The Gift. Mark Shirley's numbers in Sub Rosa #14 made the mistake of ignoring the stated cannon rate of The Gift and set it at roughly 1:50k (1:49k if I remember right, Serfs Parma).
EDIT: Ack, population was a little over 60m in 1220. That would mean 1 Magus per 50k.
The fact is the numbers are all over the place- if you go with the canon rate then there are more mages of the order in Ireland than the population should have Gifted people. Also if you are doing Gifted and the Order as percentage of population then obviously numbers did fluctuate. However the other issue is that you simply cannot maintain any sort of continent wide organization with numbers that low.
Scotland would be even worse, with roughly half the population of Ireland at the time and arguably even more Gifted. Of course since Magi have a willingness and tendency to move from areas with higher population to areas with lower population due to easier access to more abundant resources, you have to look at the population of the area occupied by the Order as a whole.
I do agree with you that the rate of the Gift (and possibly Magi) are too low. At 1:10k you are looking at only 6k total and 200~300 born every year with the gift in all of Mystic Europe. Those births most likely suffer a higher than normal death rate because of the negative effects of the Gift, though I only used the default 25%. To maintain their current growth rate (requiring roughly 24 per year), the Order would have to find roughly 10~16% of all children born with the gift who survive their first year of life. They would have to sort through and test nearly a quarter million children a year. If this duty was spread through the entire Order, each and every Magus would have to test nearly 200 children every year. Roughly 1 in every 5 Magi would have to spend a season each year testing children.
I personally think a ratio for the Gift of 1:1k would be much more workable. That would increase the total Gifted to 60k, reduce the required "Find" rate to 1~2%, requiring a testing of 24k children per year (20 per Magus) which while still a large amount can be performed by less Magi as a duty (assigned or self appointed). If you test 10 children per day (taking into account travel time and distractions), a season of dedicated testing could check 900 children. This would require roughly two duty seasons in each Tribunal every year (more in highly populated Tribunals, one or possibly none in the low population ones).
All those numbers lead me to conclude that the Order as a whole needs both a position/duty for mass testing for the Gift and a large number of enchanted items that can test for the Gift to reduce what would be a huge drain on Magi time. Possibly assign this position/duty to Redcaps (since they are already traveling all over) and the Order as a whole is responsible for providing them with the enchanted items. Providing enchanted items to assist the search would be a fair price for one of the Gifted children to serve as an apprentice. The Order would definitely not allow price gouging for something that is so critical for its survival.
As for the total number of Magi, most of the workups seem to go with an average lifespan of 100 years post gauntlet. However multiple simulations of the Order have placed the actual average lifespan in the 150+ years post gauntlet range. Of more direct relation to this thread, some of those simulations also generate books written. The results of one I most recently looked at produced nearly 18.5k books and 6.5k translated spell lab text written by the Order in its 450 year history that were still in circulation. It did not calculate lab text for enchanted items or casting tablets.
Plus look at it from the other side- if 1 in 100,000 people are mages and mages live 150 years post gauntlet, finishing the gauntlet at age 20-25 then living to age 175 means they are living roughly 3.5 times as long as other people, meaning the rate of recruitment would need to be at 1 person in every 350,000 to maintain that ratio. The problem is that with the above listed 6 million people in the whole of Europe (not sure that is right but we'll use it for the moment) and a land area of 3.93 million square miles that put an average population density of just over 2/square mile, which means that you would need to cover 175,000 square miles to find a single apprentice, and a covenant of 6 mages would only occur once in every 350,000 square miles, assuming that is 3.5 generations of magi as the math works out. that puts the average distance between covenants somewhere between 591 and 668 miles, and they would be overlapping each other looking for apprentices. The distance from Paris to Lyons is 241 miles for comparison. Modern day France is 248,573 sq miles, which means every generation the entirety of france (Normandy and Provincal) would produce less than one magus.
That's why the numbers make no sense.
60 million, not 6 million in the year 1200. You missed that 0 in my post and it threw off all of your math.
Adjusting your numbers gives a pop density of 15.26/square mile and you would have to cover 655 square miles (at the 1:10k RAW ratio) to find 1 person with the Gift (who might be anything from a baby to an old man).
My math for new born/young Gifted was based on the long term trend of a new generation born every 20 years and you are searching for someone age 0~5, along with the 25% fatality rate for the first year of life common for the time. That works out to ~5.33 the area/population to find someone in the age range to take as an apprentice (1 in every 3,500 square miles or 71 in France of age 0~5). The entirety of France (Normandy and Provincal) would have 379.5 total "native born" people with the Gift of all ages, ignoring transplants from the Order. These numbers are based on the population density of all of Europe. I could dig up the actual population of what is now "France" at the time and give more precise numbers but they should be close enough to get the gist across.
That is once again why I suggested 1:1k as a better ratio. It would reduce the square miles per person with the Gift by a factor of 10, while increasing total individuals with the Gift (both young and old) by a factor of 10.
EDIT: My math for new generations is actually fairly simplified, since there are depending on life spans roughly 3 generations alive at a given time in that time period, with only the middle one having children but they have roughly 3.33 children per pair before infant/child mortality to maintain the population and they should be growing at a rate to reach 80 million total population in roughly 100~150 years. That works out to 111 Gifted children born every year across Mystic Europe who will make it to adulthood before you work in the required increases to take into account population growth, unforseen events, and the portion of the population that does not produce children (due to Vows, sterile, etc). Why I came up with the 150~225 which does not include the death rate for the population that makes it past infancy but dies before they turn 20 and includes the required population growth.
EDIT 2: Oh, and I did dig up the population of what is now "France". It was 12.6 million which increases the population density within it to 50+/square mile, 240 Gifted children ages 0~5 (16~24% of all people born with the Gift each year in Mystic Europe), and 1282.4 "native born" with the Gift. Normandy and Provencal are both fairly population dense for their size.
And if you are wondering where I got how many apprentices are needed to sustain the current growth rate of the Order, it is based on ~30% of the actual members as settled on both by Mark Shirley in his write up for Sub Rosa #14 and the varies simulations I have looked at if you knock the population down for the key events in the Orders history by rough percentage of the order killed at that time.
EDIT: That would require Magi to train on average one apprentice every 50 years. That might sound high, but it is an average that covers the range from prolific (1 every 20 to 25 years) to minimal (1 every 100+ years or none at all). Counting in their longer lifespan and longer "fertile" period the Order is growing at a rate 20~40% faster than the mundane population.
The first generation of the Order was also training at a rate 2~4 times higher, though many of their "apprentices" required a much shorter time because they were already skilled adult magic users. 5 years to open them to the Hermetic Arts, teach them Hermetic Magic Theory, basics of the Arts, some Hermetic Spells, Parma Magica, and enough Latin/AL to function. There was not really much of any Organization Lore: Order of Hermes or Code of Hermes at the time since it was actively being constructed and those first apprentices lived it.
I also operate under the assumption that most of the littérature will be tractatii rather than summae. Simply because the market of people wanting to read the tractatii is much larger, whereas the same summae will be passed around to young mages as the older covenants see less need for them. Old mages have the means to bribe the once in a generation good teacher high comm person to crank out Q10+ tractatii in their relevant topic.
With this in mind in the game I run, the party mostly finds tractatii in hermetic topics, whereas the mundane topics are more balanced.
The once in a generation might be the one cranking out Q14, but even that is only a combination of 6 Attribute points and 3 Minor Virtues at character creation. Q10 on the other hand is a single Attribute point in Communication and 1 Minor Virtue, which is fairly common depending on which House the Magus is part of (Bonisagus).
Then of course you have magically improved Attributes. While some like to argue they are a Cult of Heroes thing, they are in the Core book. How extensively they are used is very Saga dependent (none to all over) but they are an effect that has been in AM5 since the beginning.
I agree with everything else you said, just wanted to point out that 10+ is not all that rare in the Order. 13+ is where things start getting rare.
I don’t know if my players are spoiled but they won’t “waste” a season in a tractatus with a quality of less than 9-10. They are still reading the best ones. No need to waste time with the lower quality ones... yet. So most apprentices (or magus) can’t produce competitive tractatus. The ones that can get their work copied fast.
Consider that a 9 can be produced by anyone with a com of 3 or a com 0 and the good teacher virtue, even if most magi can't produce one they consider worth their time there will still be a very large number who can. With resonant materials and glossing this becomes even easier. A quality 7 tractatus copied onto a parchment made from basic (+1) resonant materials and then glossed by someone with a higher communication will be at quality 7 after these treatments. As I read it a tractatus can be glossed while it is being read, which means having a magus with a high com can lead to an improvement of a lot of tractatus.
The options given in Covenants will result in both a higher max and average quality (Resonant materials, Clarification, Glossing, Commentaries, Florilegia). They raise the maximum quality of a Tractatus to 17 without Clarification (which is wasteful of time and vis for its additional +1).
Glossing of text by writers with average Communications and Good Teacher will be fairly common. Those writers are good at covering a subject but just need a little editing that a large portion of Magi are capable of providing.
Commentaries on Arts would actually be very common, though not in the way found in most games (written by PCs, based on only Summa on a subject they contain). Those are often of very limited circulation value. How they would commonly be written is based on the Roots and Branches, which are text both widely spread and commonly read across the Order.
You find the same thing in Ability Tractatus, where for example any written on Theology (Christianity) have a high chance of being a Commentary on the Bible. The Bible is the Authority (oh, we don't have those in 5th? so sad) on Theology (Christianity) and everyone who is interested will have studied it. The same goes for things like Physics by Aristotle on Philosophiae and Elementa by Euclid & Ars grammatica by Donatus on Artes Liberales.
It also has the potential to generate a fair amount of lower quality Tractatus through Correspondences. However I would think that only collections made from the writing of two Magi with fairly high Communication would be worth the trouble of bounding unless you are using the Research Rules.
I don't think you could get them at all for the Arts, which require one on one instruction. I just read that again while researching for a new thread I was considering.
You might, in theory, if you had a bombastic instructor prone to pontification and a scribe sitting quietly recording the one on one instruction. Theoretically, maybe...
it certainly would not be a significant contribution to the available reference material.
Lecture notes on magic theory on the other hand is definitely possible. lecture notes on parma magica work mechanically but should result in a whole bunch of people getting marched before the notes are burned...
Parma Magica is one of those weird topics for books. Magi want them to study, but they are a very big security risk. Handle them the same as all other dangerous/secret books. Not in the general library but a secure storage location. Only copied by and access given to by Magi.
Though I often wonder at many of the publicly posted libraries that have books on it in their collections. Are they just part of the general library where even the mundanes can see them? Is there several Librarians scattered across the Order who have read those books? They would already have at least a minimal skill in Magic Theory. Is the Gift required to use Parma Magica and what happens if they actually do try using it?