To the Four Corners of the Earth

During a stay on the island of St.Helena in 1677, Edmond Halley (1656-1742) observed a transit of Mercury over the Sun. He realised as a result of making careful timings of the contacts of Mercury with the Sun's limb at ingress and egress, that a method based on these timings could be used to determine the Solar Parallax. If the transit were to be observed from different latitudes, then the ordinary parallax effect would cause the observers to see Mercury trace a different length of chord across the Sun's disk. The length of the respective chords could be deduced from the timings. What is more, the parallax effect would be much greater for Venus than for Mercury, given the greater distance of the former from the Sun. Halley knew that he himself would not live to see the next transit of Venus: no more would occur until 1761 and 1769. In a paper which he gave to the Royal Society in 1716 he appealed directly to future generations of astronomers:

"We therefore recommend again and again, to the curious investigators of the stars to whom, when our lives are over, these observations are entrusted, that they, mindful of our advice, apply themselves to the undertaking of these observations vigorously. And for them we desire and pray for all good luck, especially that they be not deprived of this coveted spectacle by the unfortunate obscuration of cloudy heavens, and that the immensities of the celestial spheres, compelled to more precise boundaries, may at last yield to their glory and eternal fame."

Halley prophesied that his paper would be "immortal", and indeed it was. His exhortations and detailed plans, projected beyond the grave, caused well-equipped expeditions to set out to far flung corners of the Earth to observe the 1761 and 1769 transits.

In 1761, Charles Mason and Jeremiah Dixon observed from South Africa, William Wales from Canada, Jean-Baptise Chappe d'Auteroche from Siberia and Alexandre-Gui Pingré‚ from Madagascar. The French astronomer, Guillaume le Gentil set out to observe the transit from India, but unfortunately arrived too late.

Having got there eventually, he decided to stay 8 years to make sure of seeing the 1769 transit. Tragically, he missed this one too, because in an otherwise cloud-free month a cloudy morning managed to totally obscure the transit. To add insult to injury, when he returned to France, he discovered that his relatives had given him up for dead and had divided up his estate!

An even greater number of expeditions set out to monitor the 1769 transit. Amongst these was the famous voyage of Captain James Cook, which set up an observing station in Tahiti. An interesting point is that, even though France and England had only recently been at war, the French Government instructed that its navy should leave Captain Cook's ships unmolested, as they were out on enterprises that were of service to all mankind! [1]

A summary of the results of some of these expeditions is shown in Figure 10. As can be seen, the timings of ingress and egress permitted different chords to be plotted.

From the perpendicular distance between these chords, relative to the known angle subtended at the Earth by the diameter of the Sun, it is possible to compute the solar parallax [2].

In Figure 11, D = d . Lv/(LE - Lv ) (from similar triangles)

the ratio Lv / LE however was known from Kepler's Third Law. In cruder terms, it was equal to sin θ (where θ was the angle of greatest eastern elongation of Venus - see Figure 4)

Therefore, D = d . sin θ. LE/(LE( 1 - sin θ)) = d . sin θ/( 1 - sin θ)

Consequently, from the ratio D / H (based on Figure 10), H (the diameter of the Sun) can be calculated and hence the solar parallax and the distance to the Sun.

Figure 10: The Observed Tracks of Venus across the Face of the Sun during the Transits of 1761 and 1769

Halley had pointed out that the duration of the transit of Venus would be of the order of 7 hours. Venus would make approximately 1 arcsecond of progress per 14 seconds of time (for a central transit, given that the angle subtended by the Sun was about 31' 30" ). He reckoned that an error of 3 seconds in the measurement of time would produce only a 1% error in the parallax.

The results were not as good as expected. It proved difficult to discern exactly when the moments of contact took place (due to the 'black-drop' effect) and also the longitude of some of the observing stations was not known with sufficient accuracy. Nevertheless, the accuracy of the measurements represented a great achievement.

Figure 11: Halley's Method for computing the Solar Parallax

When all the results were fed back, the calculated solar parallax varied between 8.55" and 8.88". The modern accepted value is 8.794148".

It can be truly said, that the real distance from the Earth to the Sun - the 'Astronomical Unit' - was at last discovered. Kepler's laws had already enabled the relative distances of the planets from the Sun (in Astronomical Units) to be determined. Now - as a result of the selfless efforts and dedication of numerous astronomers, explorers, surveyors and sailors - the absolute dimensions of the Solar System could be fathomed.

Note [1]: Or, at least, so say some books! This story is given by A.Pannekoek 'A History of Astronomy', Dover 1989,p.287, and is repeated in the French novel 'Le rendez-vous de Venus' by Jean-Pierre Luminet,1999,p.265. It seems unlikely, given the state of peace between France and England at the time, but it is a nice thought.

Note [2]: Please note that this simple explanation is only intended to show that geometrical relationships exist which allow the solar parallax to be determined by measuring durations of the Transit of Venus at different locations. In practice, Halley's method of durations requires the use of quite complex trigonometry. To see how exactly this is done, old textbooks on spherical astronomy need to be consulted - e.g "A Treatise on Spherical Astronomy" by R.S.Ball (1908, Cambridge University Press)

7. Useful References to the Transit of Venus