Suppose f is the cubic polynomial vanishing on the three lines through AB, CD, EF and g is the cubic vanishing on the other three lines BC, DE, FA . Pick a generic point P on the conic and choose λ so that the cubic h = f + λg vanishes on P . Then h = 0 is a cubic that has 7 points A, B, C, D, E, F, P in common with the conic. But by Bézout's theorem a cubic and a conic have at most 3 × 2 = 6 points in common, unless they have a common component. So the cubic h = 0 has a component in common with the conic which must be the conic itself, so h = 0 is the union of the conic and a line. It is now easy to check that this line is the Pascal line.
You know, it is quite easy to get discouraged about the state of the world these days as I do quite often. I listened to Pope Benedict this morning reflect on how best to engage with people of other religions in the long game of eventually bringing others to the truth of Christ. Though he was talking about religions of course it could easily be applied to our culture wars as well, atheists being no exception. Of course, reading the gospels are a reminder that the task has never been easy. Pope Francis’s reminder this week too about the favorite tactic of the prince of this world is to divide us and pit us against one another. Reflecting on their wisdom gives me hope, even if I’m unsure how to go about it myself. Evil is rampant in the world, and we’re not going to help Christ sitting in our comfortable (for now) domiciles.
By 1647, Pascal had learned of Evangelista Torricelli 's experimentation with barometers . Having replicated an experiment that involved placing a tube filled with mercury upside down in a bowl of mercury, Pascal questioned what force kept some mercury in the tube and what filled the space above the mercury in the tube. At the time, most scientists contended that, rather than a vacuum , some invisible matter was present. This was based on the Aristotelian notion that creation was a thing of substance, whether visible or invisible; and that this substance was forever in motion. Furthermore, "Everything that is in motion must be moved by something," Aristotle declared.  Therefore, to the Aristotelian trained scientists of Pascal's time, a vacuum was an impossibility. How so? As proof it was pointed out: