B Observational data needed
1) Distance between observers at points A and B
The distance AB can be deduced from the latitude of the two points of observation. In the diagram, j1 and j2 are the latitudes of A and B, and R is the radius of the Earth.
In the right angled triangle that divides the isosceles triangle RAB
sin ((j1 + j2)/2) = AB/2)/R.
Then the distance AB is
AB = 2 R sin ((j1 + j2)/2)
Be careful. If both cities are in the same hemisphere, the angle is (j1 - j2)/2 and also the geometrical situation changes if both cities are on different longitudes.
2) Distance Db between two observed paths of Venus
In order to calculate Db we need the data obtained by two observers at the points A and B on the same longitude (meridian). In any case it is necessary to have a "photograph" of the paths of Venus visible from each location or the times that Venus crossed the Sun's disk. We offer three possible methods:
a) Calculation of Db by direct measurement
Measure the diameter of the Sun D and the distance between the two paths Db, that is to say A'B', on a photograph. The angular diameter of the Sun, seen from the Earth is 30' (minutes of arc or 30 / 60°). By means of simple proportion, the distance between the observations of Venus is linked to the Sun's diameter by
Db / 30' = A'B' / D
therefore
Db = (30') ( A'B' / D )
but the formula requires the Sun's angular diameter to be expressed in radians. Therefore
Db = (30 p / 10800) ( A'B' / D )
Db = (p / 360 ) ( A'B' / D )
b) Calculation of Db by measurement of the chords
The distance Db between the chords A and B, is difficult to measure because the two lines are always very close to each other in comparison to the diameter of the Sun. We can replace the measurement of A'B' by the measurement of the chords A1A2 and B1B2, the apparent distances covered by Venus on the face of the Sun for our two observers at A and B.
Using the theorem of Pythagoras, we obtain the relation
B'S = ½
(D2 -B1B22)
A'S' = ½(D2 - A1A22)
Dividing by the diameter D gives
A'B' / D = ½[(1 - (B1B2/ D )2) - (1 - A1A2 / D ) 2)]
c) Calculation of Db by measurement of the times of transit
It is very difficult to take measurements from a photograph, and this reduces the precision of the final result. We assume that the apparent motion takes place at a constant velocity which is the same for all the observers. This is a good approximation as the velocity depends only on the relative motion of Venus and the Earth round the Sun and on the rotation of the Earth on its own axis.
We can therefore replace the measurement of distance by the measurement of the length of time of the transit.
If tA and tB are the durations of the transits A1A2 and B1B2, and we introduce t0 as the hypothetical duration of a transit along a diameter, we can write:
A1A2/tA = B1B2/tB = D/t0
Then the time relation t'/t0 corresponding to A'B'/D in the previous formula is,
t'/t0 = ½[(1 -(tB /t0 ) 2) - (1 - (tA /t0) 2)]
Be careful which times of observation you use in your calculations. Note that there are times of external and internal contact. Only the internal ones are well measured. Using the angular diameter of the Sun as seen from Earth as 30', the corresponding time from the entry internal contact to the exit internal contact along the diameter of the Sun is to = 26755 s.
