Have the Roots of the Arts and Branches of the Arts (Covenants P.93) ever been defined in a level / quality / author format?
Covenants states that the Roots can be purchased via Redcap for "a very small consideration" - what would such a small consideration be, especially in a backwater Tribunal like Hibernia?
No, they have not been defined.
How much is a small consideration depends on how you have set up your game world.
De Lapii from the box on page 23 the Normandy book is the only one I can think of.
Currently, I'm trying to rate the worth of anything in terms of equivalent seasons of study / lab work.
IME, even a single season is a decent amount, while "a very small consideration" sounds like it would be considerably less so.
However, I have trouble coming up with things worth the equivalent of less than a season...
"Make it up yourself" is all well and good as an answer, but I'm looking for how others have adressed the same question, as a guideline on how I can tackle / quantify the question myself.
I appreciate your frustration. The 'We Hate Definitions' group has made deep inroads into ArM.
For my own game, a Root should be available for half or less than half the price of a Sound summa (see p. 95 Covenants), possibly much less if you have been of good service to that Redcap or House Mercere or there's some other reason the covenant might be rewarded.
The Roots would probably be of a high category, possibly the 30 minus level ranking, as they are supposed to be the best possible primers. They may also be of a fairly low level, say Level 5, as they're supposed to be foundational works. Such a level and resultant high Quality (Q25) should take a mage from 0 to 5-and-change in one season. Not very useful for a mage at basic mastery level, but they are supposed to be primers, after all.
Someone will be along shortly to have different definitions or opinions.
The roots have been defined as being able to quickly bring a student up to level 5, to enable them to take an apprentice, so that puts them at a minimum level/quality of 5/15
A small consideration is a pawn of vis, even in Hibernia. Roots seem to be the equivalent of open-source literature (likely all the original authors are dead regardless), freely copied and traded to the point of being ubiquitous. The pawn is just pay for the cost of copying it, and even then it's more than it's actually worth, but trading silver for magical knowledge would probably feel wrong to most magi.
A Root should be at least L5/Q15. Which, in the economics of my saga, has a retail value of 5 pawns of Vim.
I believe there is a common agreement that a root is at least 5/15, allowing a magus to achieve within a season of study the base knowledge to train an apprentice without penalty (eg: to have a level 5 in every Art).
Depending on your Saga and average power level, you might also consider 6/21. There are sufficient discussions on this forum on the power level and Order knowledge growth curve to support the existence of magi able to write roots of this quality, even without requiring resonant materials and high level craftsmen, by using the possiblity to increase the quality level by one and reducing the level by 2 (or is it 3 ? - serf's Parma).
Higher level Roots are conceivable (7/28), but require optimization of the author (Good Teacher, High Com, high level Art) and very likely use of resonant material and competent craftsmen. They are definitely rarer, or even non-existent.
In the rulebook, it is also written that such Roots only exist for 10 Arts, none being defined on purpose, to leave freedom for each Saga for PCs to have the opportunity to become the author of a Root or possibly to emphasize a shortage of (written) knowledge in one Art.
I believe that it is also written that you can borrow one of these Roots for one pawn, through the RedCap network, kind of implying that several copies exist in the Order, possibly one per Tribunal. It is open to interpretation.
Covenants pg 93 states that there are Roots for 9 arts and Branches for 8 arts.
I should have hold my Serf's parma higher 
Thanks for correcting me.
If such books are in such abundence. Why does every new gauntlet mage not start with a 5 in almost every art? A covenant would pick up one of every form available. And on a rotating basis each Magi would pay to have their apprentice read them. If you can get to a 5 in 1 season, why wouldn't you have your apprentices use one of his non-lab seasons reading the book? Heck, You're training your apprentice, without having to waste time teaching the pain in the ass. This seems like an awesome plan. Starting Magi would took incredibly different if these books are in such high availability.
Not to mention that for /most/ Magi. SQ 15 is /really/ hard to get. That requires on average 26 in the appropriate art to get a 5/15 summae written. At least it can be on average written in 1 season. But that Magi with a 26 in an art, is much better off writing a Tractatus, that he can trade with other magi for tractatus he cares about.
Now if you argue that the roots are 5/8? That's much more reasonable. You only really need a 12 in an art or so to do that And it would take 2 seasons per form to get to 5, which means it's no longer cost effective to have your apprentice collect them all. It makes much more sense to narrow is field of work. You get him to read 2-3 of those to get him the basics he needs, so you can then turn around and teach him Ignem directly.
Because some masters prefer their apprentices in the lab rather than reading books.
This. Keep in mind that the apprentice is basically slave labor for 3 seasons of the year to recompense his master's 1 season of instruction. The Order's apprenticeship system is not about turning out the best magi possible, but limiting the exploitation of the Gifted students (enforcing 1 season of training per year, setting a limit to the length of apprenticeship).
And L5 SQ15 is only hard for the 'average' magus, but there are enough Good Teachers and/or high Communication writers in the history of the order to have produced a multitude of these things. Something being defined as a 'root' or 'branch' has more to do with politics/reputation than anything else; the Hermetic community has to 'agree' that a book is a 'definitive' starter or comprehensive summae for it to be labelled a 'root' or 'branch'; that's why not all 15 arts are represented. Probably there are 3-4 potential candidates for all of the 'other' roots but magi are so busy arguing over which one is best that it will take a century before the anyone gathers a consensus by force of the other loudmouths dying out. This is academia, after all. 
Also, if newly Gauntleted magi had 5+ in all/most of the Arts, then they'd be able to manage an apprentice (almost) right off the bat and that's probably not a good idea.
Apprenticeship is 12 years give or take.
You have them taught in Latin by a normal teacher early on.
One season a year for the first 3, you train him in magic theory. (because it's free, and it adds to your total and you still get to have him assist)
One season a year you sit him down with a book. He picks up a 5 in an art and leaves you alone to do real work.
The other 2 he's in your lab as slave labor, picking up exposure exp to what ever.
The Magi still gets 4 seasons of lab work.
The Apprentice gets his 1 season of study.
And in 12 years, he's got a 5 in every art that's available, a MT of 3-6 and enough exposure exp for Parma of his choice. It's not like his arts help you in the lab. And you only waste maybe 5 seasons teaching him personally.
Canonically, a Hermetic Apprenticeship lasts 15 years, except in Greece where it lasts 14 years (to coincide with Tribunals).
This is often done, sure. But you still have a requirement to teach them one season a year. And a season where they are learning Latin is a season where they aren't assisting you in the lab.
How does it add to your lab total, when he's spending the time learning Latin? I know it will later, but it doesn't solve the issue entirely...
Whenever I have an apprentice, I always have real work for him to do. If, for some reason, my character isn't using his lab, then the apprentice can distill vis, or with house rules, refine and improve the lab, fix arcane connections, or even learn a spell from a lab text, to name just a few things.
This is a great deal for the apprentice. It, however, does not provide any advantage to the master. These kinds of discussions should focus on how it improves the master's quality of life. Apprentices are property in the Hermetic sense, never forget that and use that as the basis for deciding how they will be treated by magi of the Order.
I have a Mercere (illegitimate lineage) who was recently forced to take an apprentice. You can be darn sure that he's making sure he gets the full benefit of 3 seasons of work from her for every season he teaches. There may be a season or two where she does her own things as directed by my character. I recently had her learn a spell from a lab text, because I was doing something else (reading to improve an Art, IIRC), didn't need my lab and felt she could stand to learn a spell. Her Arts and our aura made distilling vis unappealing in a time use sense. Learning the spell meant that it was something she could do instead of her master, to free him up to focus on other things.
Any number of masters would be against a scheme where their apprentices start out with a minimum score of 5 in the Arts and an MT score of 6...
But you /are/ getting the most out of your Apprentice.
You get 3 full seasons of lab assisting, and you get to never have to actually teach them in person, wasting your valueable time.
Year 1:
Season 1: Open the damn gift.
Season 2: Have someone teach him some latin because well, he can't help you if he doesn't know latin. (You get to work in your lab just like normal)
Season 3: train him in magic theory while working in your lab.
Season 4: work in your lab with your now.. exciting +2 or +3 to all lab activities.
Year 2:
Season 1: Have someone teach him latin so he gets to a 3. You work in your lab
Season 2: Lab work for all
Season 3: Lab work for all
Season 4: Lab work for all
Year 3:
Season 1: Apprentice, sit down read this book. Lab work for Magi.
Season 2: Lab work for all train him in Magic theory
Season 3: Lab work for all
Season 4: Lab work for all
He's got a 3 latin, roughly a 3 MT and he's getting his 1 season of teaching a year and you don't have to bother with him.
Now you can inteject that with letting him learn some spells, and stuff. But that almost certainly doesn't happen till year 10+ and by then he's read all of the accessible books anyways.
How is that plan not better for the Magi, then actually having to teach the rug rat something for 1 season a year?
How does this fulfill the Apprentice (more or less only) right, that of personal, 1-on-1 teaching, for no less than 1 season each year?
Not 15 seasons, spread over 15 years, but 1 or more seasons, each year.
Because that's mandatory? Because if you don't, your apprentice has the right to leave you, usually for another master?
Does the code /actually/ say it must be 1 on 1 teaching? All it says is 1 season of training.