Mediant is the mid-point between the tonic and dominant and sub-mediant is the mid-point between the tonic and sub-dominant below the tonic.
Names aren't all that important. It's more about undertanding function. Most people don't use those names anyway and almost everyone says "the three chord" or "the six chord".
Ahh I see. Yeah the next page actually handles the remaining two (Supertonic - II and Leading Tone - VII) but I'll treat those terms as just good crossword knowledge, nothing more (for now) 
It's funny how Alfred skimmed so much on theory in the first two books, and now suddenly offers page after page of it. I think it once again ties in with ranjit's original question about teaching theory to beginners... apparently this method seems to think that if you stick it out for two entire books, then obviously you must be dedicated enough to handle the tough stuff
Sophia Right now my theory lesson talks about tonic, dominant and subdominant (I - IV - V) and then proceeds to talk about the circle of 5ths. It also talks about the importance of IV being the subdominant not because it's below the dominant (V), but because it's the same distance below the tonic as the dominant is above the tonic.
This is very fascinating and I love learning new things... but my question is: what is the actual practical value of this knowledge other than just theory? I'm not sure how to ask this so it probably sounds like a clumsy question, but what I'm trying to say is how can this enhance my learning new pieces?
Some thoughts (have been discussing this with someone recently).
The most essential thing that music does is it starts at "home", goes away from home, and then you want it to come home again. If someone plays music and stops before it finishes, you're itching for it to come home - you might actually sing that last note out of frustration. "home" is the Tonic or I chord with the 1st degree notes usually being the final melody note. The Dominant - usually the V chord or even more strongly the V7 chord - is both "away from home" and also "want to go home". The V7 has a natural pull back to the Dominant. GBDF in the key of C major - B wants to slip to C, F wants to slip over to E (both a semitone apart so in a close hug), and the tritone BF creates an uneasiness we want to hear solved.
Dominant is also a function or role, as described. Later on you might hear of other chords that play the same role, this being the Dominant function.
The Subdominant "goes to the Dominant". (It is also literally below or "sub" the Dominant since F is below G). It's a nice intermediary between the Tonic and the Dominant. Much later on you might get a series of chords that hang together within these functions. [Tonic - buncha chords][Subdominant - buncha chords] [Dominant - ditto maybe] [Tonic - bunch chords hammering home that we are indeed home]. That's when i saw these three names actually used in any kind of practical way.
For all those names: Tonic, Supertonic, Mediant, Subdominant, Dominant, Submediant, Leading note -- mostly I don't see/hear people referring to these much (other than the three) but still learned them. I did recently watch someone explain things about the structure of some music, and he did use words like "mediant" because maybe it was faster to say than "3rd degree note / 3rd degree chord".
I found where I ran into the word Mediant being used.
"The local Tonic itself can most easily be transformed into a common precadential chord that leads to the upper mediant (arrow goes to A#) ...."
So it does get used, by some people, sometimes.
Here around 5:19
Thank you @keystring for taking the time to explain that 
Funny enough I didn't plan on learning those terms, but then found they managed to wedge themselves into my memory anyway. Well, there are only seven anyway, not exactly rocket science
Up to now, I found chords a mystery. I mean, of course I understand the concept... but I didn't understand how it applies to classical music as well. Just now my book explained the Alberti bass and suddenly it makes sense that the C-G-E-G sequence is the "tonic chord" (If that is the correct expression). Then B-G-F-G is the dominant septime. Something I heard a LOT throughout my life, but never understood
I'm really happy that that this book is going deeper into those concepts now. It also treats us to more and more truly gorgeous pieces like Haydn's Serenade (for string quartet), a piece I always loved. For the first time since I started learning piano I see those separate notes, and I'm beginning to recognize the chord behind it.
I still don't think that's something to burden a "first week beginner" with, but once you reach the stage where, well, one starts to learn stuff that might sound a little bit like real music, it's nice to get a little extra background knowledge. I'm really stoked for the rest of this book!
To confuse matters more 
the words tonic, dominant, and subdominant are used in two different senses. The first sense is the scale degrees and the chords built on them as explained in your book. The second (more important) sense is the harmonic function as keystring has been explaining.
There are three different harmonic functions in typical Western tonal music - tonic, dominant, and pre-dominant (or subdominant). The many different chords can act as one of these three functions. For example, the vii chord typically has a dominant function and the ii chord typically has a pre-dominant function. However, the mapping between scale degrees and harmonic function is not one to one and some chords have a certain function in one context and a different one in another context. Even the I chord doesn't always act as the tonic - for instance in second inversion it usually has a dominant function.
Confused yet? My point is that what's really important is how the harmonies are used in music (i.e. their harmonic function). The names are seconary and the reason those three particular names (tonic, dominant, subdominant) are commonly used is because they also refer to harmonic functions.
BartK Confused yet?
Heh, well, let's put it this way. It's like a skein of yarn that got all tangled. Every time you gently pull on one string, another one tightens. But with patience and perseverance, gradually you get there. The end result is a workable ball of yarn and you can finally start to create your piece of art.
I'm actually glad I asked. Even one YouTube teacher dismisses it as unimportant information ("something you just have to learn"), but if that is true, it becomes useless. And I simply refuse to believe that. So I'll continue pulling those strings and I'll probably have more questions in the future 
BartK o confuse matters more
I read the whole thing but am quoting the first line for brevity.
I once worked with a description of an internship of a dental assistant student in Europe. The very first thing she had to learn was the names of all the tools, and where they were placed on the tray. Later she could prepare different trays for the dentist, and hand him the tools he asked for, and going on to more complex tasks. This made me think of theory, and just knowing what things are called where they are.
At this point, Sophia, you are just learning some names and where those things are, so that if someone refers to it, you can find the thing. The link I gave, the gentleman referred to the "mediant". It's actually the first time I've heard anyone refer to it, but I knew what he meant because I had once learned what you're learning. I haven't found that one particularly useful tbh.
The thing about music is that it's not a static thing that came into existence one day and has stayed that way ever since. People have been tweaking, inventing, reinventing, discovering. They find things that sound good and use those patterns: then the theoreticians come along and set those patterns into stone creating rules; then creative musicians come along breaking those rules, and the original musicians say "Well no, I never meant this to be a rule." And we get to swim in this creative brouhaha.
There are patterns; they do exist; they do help us orient and find our way. There are also many angles to a same thing.
Sophia Up to now, I found chords a mystery. I mean, of course I understand the concept... but I didn't understand how it applies to classical music as well. Just now my book explained the Alberti bass and suddenly it makes sense that the C-G-E-G sequence is the "tonic chord" (If that is the correct expression). Then B-G-F-G is the dominant septime. Something I heard a LOT throughout my life, but never understood
Those who learned these things early on may be dismissive of them, but those of us who just wended our way through nameless, unidentified" "sensed and felt" things, probably don't realize the freedom and clarity this gives. I can identify with this.
keystring Those who learned these things early on may be dismissive of them, but those of us who just wended our way through nameless, unidentified" "sensed and felt" things, probably don't realize the freedom and clarity this gives.
This is absolutely true. Functional harmony just made something click in my brain. Until then, I didn't realize that you "had options" on how to harmonize a melody. Once that fell into place, everything started to make sense.
The name "dominant 7" is somewhat problematic. It might be good to look at the "dominant seven" chord and the word "dominant, even though it is still early.
Dominant is also what we call the 5th degree. The "seven chord" (four note-chord) built on the 5th degree: in the key of C major, that is GBDF. We also call that G7.
G7 very often "goes to" C, and has the "dominant function'. There are a number of reasons for this pull, as explained before: the tritone BF that wants to resolve from its unstable sound; B wanting to go to C, F wanting to go to E.
In the common chord progression I IV V7 I (C F G7 C) that's what we see.
Now let's look at GBDF (G7) by itself.
GBD is a major triad. The F is a minor 7th from the G, and a minor 3rd from D. Any chord in root position having these characteristics (major triad topped by m7) will be called a "dominant 7 chord".
However, there is a lot of music where this chord does not have the dominant function - it does not lead to the Tonic. You could have G7 doing all kinds of things. This quality of chord has ended up with the name "dominant 7" because it is used to so often as the dominant leading to the tonic, that it got stuck with the name.
It's like Mrs. Smith always brings cookies on Sunday and ended up getting the nickname "Cookie Lady" and so kids think her name is Mrs. Cookie Lady instead of Mrs. Smith. Even when she isn't bringing cookies.
keystring It's like Mrs. Smith always brings cookies on Sunday and ended up getting the nickname "Cookie Lady" and so kids think her name is Mrs. Cookie Lady instead of Mrs. Smith. Even when she isn't bringing cookies.
Hahaha that right there is my level of thinking Joking of course, I'm really grateful you taking the time to explain the concepts. It's complicated material for sure, but luckily not completely understanding it doesn't interfere with the ability to learn new pieces and enjoy music in general. It is interesting though to see how it all connects together - a marriage of mathematical precision and complicated concepts and then they create sweet poetry together... very fascinating indeed.
