How could one use the transit of Mercury to try and find the Astronomical Unit? I am trying to use the transit of Mercury to estimate the distance between the Earth and the Sun, or 1 AU. 

I know that I need to observe mercury from two antipodes, but I do not understand any of the underlying math behind this, or how to derive the astronomical unit from this.

Any help is appreciated!
Thanks to the Mercury transit, you can measure the parallax from the Earth. That happens due to TRACE , which tracks the transit of Mercury along the polar diameter of the Earth.



During that tracking, the transit of Mercury goes like that:

[

Now notice that, if TRACE remained stationary, the transit would be a straight line. So, if you calculate the maximum separation vertically of the center of Mercury and divide by two (how it's another different story), you got the parallax angle $\theta$:



You know $\theta$, you know Earth radius, so you know D, the distance from Earth to Mercury (in km) because

$$\tan\theta = R/D$$

Now, if you wanna know how many km a AU is, just realise that Mercury is 0.56 AU from us, so divide D/0.56 and you got it!

All images credited to NASA/IMAGE and/or Lockheed Martin/TRACE