Aristotle's proof was less mathematical but to me more surprising: "you can see Earth's shadow on the Moon and it's always a circle" which point to a great deal of understanding of planets, lightning and orbital mechanics.
What he did was simple geometry, and in a way he was the right man in the right place. He heard that the sun was in zenith during the summer solstice ina certain town. He had a good runner measure the distance between Alexandria and that town and then measured the angle of the sun during the solistice in Alexandria (for simplicity, say it was 6 degrees). To get the full circle, just multiple by 60.
Is a pity that we have lost "his" library by human stupidity. There could have been a lot more to know about the state-of-art of astrography at his time.