Forgive me, but why all this talk about Arius? It is rather inconvenient to have to look up sources, etc. from 2000 years ago when we have modern day non-trinitarians among us today to use for examples! (In fact, if you go over to the un-orthodox theology section, i'm sure you will be able to find many of them, and perhaps some on here as well.)
Ok, so let's take a scenario then. Let's say we put you in a room with a Oneness Pentecostal (who denies the Trinity) and we give you each a bible. Let's say that all the silly traditions, like the councils, etc. never existed, and we don't have any writings of the church fathers to worry about (i'm sure some of you are fantasizing right now). Both of you are to use only scripture alone to defend your points. You defend the Trinity, and the pentecostal defends the non-trinity.
After some time has elapsed, you both gather your verses which support your side. You may pull verses up like John 10:30:
And the pentecostal may pull up verses like John 14:28:
You both have your verses which 'seemingly' support your side, and both of you are insistent upon your view. So who wins? Neither of you have authority over another, both of you insist that the only authority in the room is scripture. Both of you insist that the Holy Spirit taught you the correct doctrine from scripture. How would you convince him without recourse to tradition, injecting your own tradition (i.e. personal interpretation) or deferring to some type of outside authority (e.g. the Church)?
This is a practical, real world example of why sola scriptura simply fails to establish doctrine with any consistent or persuasive ability. There must be (and in fact always is) other factors at play which ultimately win out, and it usually comes down to the person's ability to persuade another with their own interpretation of scripture (the fallible interpretation of man, mind you). Arguments about doctrine which use only sola scriptura will always result in a stalemate with both sides ardently defending their opposing viewpoints from scripture, simply because you are pitting the ultimate authority against the ultimate authority; i.e. an unstoppable force against an immovable object. Surely this strange paradox isn't the plan that God had in mind to leave us with??