For whenever in any three numbers, whether cube or square, there is a mean
I take this to refer to an average, and within the context of this dialogue the terms "cube" and "square" refer to geometric objects.
which is to the last term what the first term is to it
I interpret this phrase as saying a square can be made a cube, and vice versa given the similarities.
and again, when the mean is to the first term as the last term is to the mean
This section is focused on the ability to derive an average that is equivalent between the squares and cubes just mentioned.
- then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one.
This last statement resolves the argument that an average of a square can be equivalent to the average of a cube, using a function which divides until the averages are equal to one another.
I don't see how this applies to the phi ratio, and since no articles were linked to show how it can be interpreted to refer to the phi ratio I believe the example used is more generally simple geometric methods.
If I'm not mistaken, this dialog of Plato's showed Socrates demonstrate "anamnesis" by having someone "remember" mathematics they said they did not actually know?