Note that it is a population of 10,000 Gifted for Europe, not 10,000 Gifted children.
That includes all of the magical, faerie and infernal traditions which find the Gift valuable. Given the average lifespan at the time, this means about 200 Gifted children being born every year. Or about 1,000 Gifted youths between age 5-15, which is suitable to become an Hermetic apprentice.
From these, many would be snatched by other traditions which also value the Gift. Some would have the Gentle Gift and never be found.
You also need to take into account the major investment in time that taking an apprentice represents. The master basically spends almost 4 years of his life training a single apprentice. Only in the last few years does the benefit of having an apprentice begins to outweight the investment in time -- provided the master spends a significant amount of time in the lab, where having an apprentice can speed things off.
All that being considered, I think that the average number of apprentices trained by a magus over his life would be around 2 or 3. That limits the growth rate of the Order. And as Fafnir mentioned, any big conflict (like the Schism War) can wipe out several decades of growth. Smaller conflicts (likes the Tremere Sundering, the Tytalus Infernal Purge and more recently the sack of Constatinople) can destroy severeal years' worth of growth.
That being said, if you want the Order to be much larger in your saga, go for it! Just realize that the more magi there are, the more pressure there will be for them to affect the development of Europe. If 1,000 magi can for the most part remain hidden and have subtle influence in the course of history, 10,000 represents such a large power that they would literally have the power to take control of Europe. It will make for a very different setting.
So, I found this thread four years later. I think it's very interesting and the conclusions are sensible, namely that Magi train apprentices fairly infrequently and that the Schism war set back Order growth by about a century.
I was trying to figure a heuristic about apprenticeship rate for my own purposes and I came up with the following back-of-the-envelope calculation.
Unlike the original poster I divide time in 15-year increments, not 50-year increments, because 15 years is the duration of an apprenticeship. So in the 450 years fr 767 to 1117 he gets nine periods and I get 30 "generations". But the population growth dynamics should come up to roughly the same.
We're told there were 12 founders in 767 and there are about 1200 Magi in 1220 so that's a 100-fold increase in 30 "generations" of 15 years. There were 13 founders if you count Trianoma but one house is wiped out in the Schism War. So it kinda washes out for my purposes. The calculation is not all that precise anyway.
A 100-fold increase in 30 generations is a 10-fold increase in 15 generations. This is very very close to a 1/6 increase per generation. That is, (1 + 1/6)^15 ~ 10, so let's use that.
Mage mortality is maybe 1/10 per 15-year "generation". So the rate of apprenticeship has to compensate for that. 1/5-1/6=1/30 is too little, 1/4-1/6=1/12 is about right for the mortality rate, 1/3-1/6=1/6 is too high. So pick 1/4 for the average apprenticeship rate.
This means that in every 15-year period one out of every four Magi is training an apprentice. Or that a given Magus will train an apprentice every 60 years on average. And they live about 180 years on average after apprenticeship.
I do think a malthusian analysis of the growth of the order demographics makes sense, and it is probably a factor in old tribunals. However, I think there's a few things you can't realistically take into account:
While it is convenient to look at history to evolve your game, I just don't think it makes any sense to project a demographic curve up until the 20th century, assuming the evolution of the world to be similar to our own, the ratio of gifted to remain the same throughout the centuries, and the vis sources not to vary at all, with the order of hermes still existing as it is in the 20th century. In my mind, you either have to adopt an approach that is similar to the world of darkness: e.g. technocracy with magic slowly fading from the world, and a drop in the number of gifted OR completely revise the history of mankind past a certain point to preserve the mythic feeling (e.g. no scientific revolution - make it a magical revolution, at which point magical auras would increase, not decrease);
I don't think Europe's sources are anywhere near fully tapped by the order of hermes (although the old tribunals may think otherwise). Some of those are untapped, controlled by faerie courts, controlled by hedge wizards. They will be found and annexed. How many sources are unknown simply because the closest covenant is 200 miles away? How many more covenants can be created in the wilderness to benefit from such untapped sources?
The order is in my mind on the crux of vastly increasing its vis production. It really shouldn't be too hard to start breeding magical creature lineages to the point where you're able to cull their number to get a yearly vis source. If vis gets scarce, covenants will focus more of their time on such projects. Also, if vis production becomes really tight and such projects can't keep up with demand, the likely reaction is mainstreaming mysteries like Philosophic Alchemy which would basically result in every magus in the order getting a personal vis source to fuel his projects, whose yearly production increases with age (e.g. no worries about scarcity at all - ever).
While ancient sites might get swallowed up by the Dominion and get pushed into regios if you follow RL urbanization, there is another phenomena that should play in the Order's favor - e.g. the strengthening of auras over time in most major covenant sites. This may in fact end up creating vis sources that did not exist before through slow warping or even lab story events.
We are actually doing this in my current game and I even broke it down when someone asked about our vis sources. Animals are not the only source of this however, since there is also plants and some objects you can look for.
The only animal breeding for vis we are doing is a pond full of magic carp. We do have a bunch of other animals which have vis, but they are mostly our hermetic cats (we have a mix of white lineage and black lineage).
However we have been transplanting vis producing trees to the point where we actually have several small groves of them. They are a fairly stable and easy to gather vis source. We also have grow truffles of virtue. Sacrificing one every year causes all the plants in our covenant to grow better, as well as having the effect of causing them to grow on each level of the regio.
We have been experimenting with other plants such as flowers, but have not had much success. The most luck we have had is a few of my characters rose bushes produce crystal/glass roses, with the occasional one having vis.
For vis producing objects/locations, you can get some idea from Hermetic Projects and Transforming Mystic Europe. Most of these are not possible to move to your covenant, though there are a few that can be.
Specific lineages of magical cats (of Virtue) that have been breed by the OoH since its founding. There are three lineages: the black (descendants of Jerbiton’s familiar), the white (descendants of Mercere's familiar), and the Tortoiseshell (descendants of Celtic wizards familiars, actually two lines that split during the Schism War).
If you want more details on them, they are in RoP:M, page 70. They are a nice addition to any long standing Covenant, tracing back their origins to the founding of the Order.
Our group is currently searching for a few other animals of virtue. Specifically horses, dogs, and bees. We have stables and kennels setup, though not in use since horses were not very useful until about 30 years ago in-game. That was when the whole invent portal/Mercere freakout story line happened. It has been "on the list" but very low priority since then. We do have a whole bunch of bee hives scattered throughout our Covenant, though most heavily on the first level.
A few cattle of virtue to round out our cattle herd would be nice as well. We do have a few ravens of virtue, descendants of those collected when some of our Magi were looking for familiars.
