How can we establish the distance of the Earth from the Sun?

The knowledge of the solar parallax is crucial for this calculation. Generally, the parallax is an angle of rotation of the ray that is heading to the observed object when the place of the observation changes. If the Sun is that given object, we call it the solar parallax. In order to better explain this issue, a picture is used. When two observers P1 and P2, placed on the Earth, are measuring the position of an object T respectively to a static (i.e. not moving) background, the difference between their measured values will equal to an angle p. The triangle TSP1 has an right angle (in P1). We can say following statement about the distance a:   a=r/sinp. If we use very little values of the angle p, we can replace the sinus function by its argument, then we get following dependance a=r/p. So, the distance is proportional to size of the parallax.

Can the transit observation be used for establishing the distance of the Earth from the Sun?

The principle of the measurement was interpreted by a Scotish mathematician James Gregory in a work Optica promota in 1663. While observing from two locations on the Earth with different latitude, two observers will measure different time intervals of endurance of the transit of Venus over the Sun. We can exemplify this fact schematicaly by a simple picture (scale and distance ratio are not proper in order to achive better illustration). The observers whose post on the northern hemisphere is placed southward will see the route of Venus’s disk nearer to the center of the Sun than an observer whose latitude is bigger will see. The whole phenomenon lasts a longer time for the southern observer. It is most essential to determine the moments of both inner and outter contacts. Using the knowledge of the difference of latitudes and length of both time periods, we can determine the difference of the Venus’s and the Sun’s parallaxes. By dint of the 3rd Kepler’s principle, which says that the relationship between the third power of the main half-axle of the route and the square of the orbital time period is constant for all the planets, we can get a relationship between both parallaxes and then, finally, a resulting solar parallax.