I see a lot of math-ed connections with next month's transit of venus.
It occurs to me that one can today probably achieve impressive accuracy on measuring the distance to Vanus, and hence, of the length of 1 AU, by having two widely separated observers each make a single, simultaneous, hydrogen-alpha image of the sun and the dot of venus on its face. One doesn't have to wait to time first, second, third, and fourth contacts.
My reason for saying this is that an Ha filter allows anyone to see all sorts of sunmarks; some careful scruitiny of where venus is in relation to those sunmarks will allow us to get the angular parallax fairly easily. Then a little bit of geometry will allow us to figure out how far the observers are from each other with respect to Venus (i.e., what's the length of the baseline). And then the rest of the calculations are pretty straightforward, at least in principle.
Oviously repeated observations will help. And obviously we can now directly measure the distance to Venus or Mars by using radar or even our own spacecraft. But I still think this is a neat idea.
After all, this is why Captain Cook went to the Pacific Ocean!