Constitutional question on the amendment process.?

It clearly states "a Convention". ONE. Singular. What it's there for is if two-thirds of the state legislatures think there should be an Amendment but Congress doesn't do anything about it, the states can force Congress into it - Congress has to call a convention and that can propose Amendments. What it doesn't say, as you noticed, is what majority there needs to be in this national convention for an Amendment to go forward for ratification by the states.

If an Amendment is proposed by Congress, two-thirds of each house of Congress must vote for it if it is to go forward for ratification by the states (you left out "of both Houses" from the quotation!) and three-quarters of them have to ratify it in the manner specified. So that's absolutely clear.

But the Article says absolutely nothing about how the convention will work, who should be in it or what majority it needs, so presumably a simple majority will do. It says two-thirds of each House for Congress to put forward an Amendment, but that's only what it says for Congress.

It was the fashion in the 18th century to write longer sentences with lots of punctuation so you have to think your way through this sentence. (This is why Abraham Lincoln's Gettysburg speech is so famous. It was startling at the time. The previous speaker went on and on for hours with a speech full of literary allusions, as was still the custom in the 1860s. Then Lincoln got up, said his piece in short, pithy sentences, and sat down again while the official photographer was still adjusting his camera! Now THAT'S the way to make a speech!) The way to think through this long sentence in the constitution is "The Congress,... blah blah blah... on the Application of the Legislatures of two thirds of the several States, shall call a Convention...".

I'm thinking here about how Germany elects their President - that's done by a Federal Convention, consisting of the Bundestag (their equivalent of the House of Representatives) plus an equal number of delegates from the 16 states, appointed by the state legislatures with a specified number from each state. So it's all totally clear - the Federal Convention is exactly twice the size of the Bundestag and who should be in it is all set out. This process has to be used every few years to elect a President so by now Germany knows exactly how to operate it.

Maybe the constitutional writers thought it was obvious that if they didn't specify a particular majority, a simple majority applies, same as it does in Congress for passing ordinary Bills. I can well imagine a Supreme Court case if the process were ever used, and the Supreme Court affirming that a simple majority was meant as no other kind was specified. But what occurs to me is that there is also absolutely nothing to say what the convention should consist of. Particularly, what proportions of the members each state should have. Or can Congress appoint whoever it likes? It seems it can and if anyone challenges that in the Supreme Court, the court has absolutely nothing to go on to say whether it's unconstitutional.

The process has never been used because if two-thirds of the state legislatures think there is a good reason for an Amendment, no doubt someone in Congress does too, so it will go through the process of Congress proposing it just as has always been done so far. Just as well when the problems with it are far more than you thought of!