So 2C, for political purposes, became the limit. Beyond that, Governments told us, we would have "dangerous warming". Nonsense - we are having dangerous warming now in the form of bigger storms, worse floods, longer droughts. And we are only at 1C.
At 2C or thereabouts, what we get is catastrophe - runaway warming that can no longer be halted just by stopping human emissions of carbon dioxide. Nature will take over, and we will be trapped on a one-way escalator that is taking us up to 3C, 4C, 5C, even 6C. Hundreds of millions or even billions of people would die as large parts of the planet ceased to be habitable.
If you don't want to risk unleashing that, then you don't want to go anywhere near 2C, so the official adoption by the world's Governments of 1.5C as the never-exceed limit is a major step forward.
But note that they have only pledged "to pursue efforts to limit the temperature increase to 1.5C" - not to succeed.
In order to avoid a collapse like the one at the last climate summit in Copenhagen six years ago, nobody even tried to put enforceable limits on national carbon dioxide emissions.
Each country was just invited to submit the emission cuts that it was willing to make.
The sum of all those promised cuts (if the promises are kept) is what we will get by way of global emission cuts in the next five years. UN experts did the maths and concluded these emission cuts fall far short of what is needed. If this is all that is done, then we are headed for at least 2.7C.
So, are we doomed to runaway warming? Not necessarily.
Most of the negotiators know that the cuts which are politically impossible now may become quite possible in five or 10 years, if the cost of renewable energy goes on dropping, if techniques like carbon capture and sequestration become economically viable, and if people are sufficiently frightened by a climate that is getting wilder and less predictable.
So there is a review built into the treaty and, every five years, starting in 2018, there will be a "stock-taking" exercise in which everybody's progress in cutting their emissions will be reviewed, and everybody will be encouraged to increase their commitments.
Whether they will do that depends on political, economic and technological factors that cannot yet be calculated, but there is no Government on the planet that is not frightened by the prospect of major climate change. In fact, most of them would have gone a lot further in Paris if they were not nervous about getting too far ahead of public opinion at home.
Public opinion will eventually change, because there is going to be a large amount of damage and suffering in the world. Will it change fast enough to allow Governments to act decisively and in time? Nobody knows.
Will new green technologies simply sweep the field, making fossil fuels uneconomic and government intervention unnecessary? Nobody knows that either.
We are not out of the woods but we are probably heading in the right direction - and it would be right to put in a good word for the much maligned UN.
It is the only arena in which global negotiations like this can be conducted, and its skills, traditions and people were indispensable in leading them to a more or less successful conclusion.