Reading a position number
Stage 3's finger-string-or-position already told you a roman numeral over a note can mean a
string or a position, decided by context. Now that your hand actually has a 3rd
position to go to, that fact stops being trivia.
What III is asking you to do
In a real edition, III written over a passage — or, in a modern edition, the plain words "3rd
pos." — is an instruction: move your hand to 3rd position before this note, and stay there
until told otherwise. It's not naming a finger, and it's not naming a string (context tells
you which meaning is in play — a numeral sitting where fingerings usually go, next to Arabic
1-2-3-4s, is almost always a position, not a string).
The numeral doesn't repeat on every note — it appears once, at the moment the hand needs to move, and stays in force until either a new position numeral appears or the music clearly drops back to 1st (often marked, sometimes just implied by the notes themselves getting low again).
Now you can verify you're actually there
This is the part that's new: up to now, III was a fact you noted. Now it's a fact you can
check — play the passage, then confirm against the octave self-check from third-position
(D/A/E strings against their matching open string) that your hand really did land in 3rd
position, not just somewhere approximately close.
Try it
Try
Find any passage you know that uses a note now reachable in 3rd position — or use the example
above. Read the III, shift there, play the notes, then check yourself against an open
string. That loop — read the numeral, shift, verify by ear — is what "reading a position
number" actually means once you can do something with it.
The usual mistake
Reading III as "use your 3rd finger" — the exact confusion what-a-position-is warned about,
now showing up in real notation instead of in the abstract. A position numeral tells the hand
where to live, not which finger to use for any particular note.