The Day the World Discovered the Sun by Mark Anderson Page A

seemed to be a quick and easy route to the solar parallax, such a crucial number in science.
    However, one of Halley’s protégés, the French astronomer Joseph-Nicolas Delisle, examined the English method in closer detail and found it wanting. In 1761, for instance, the difference between the shortest and longest transit times would be just 13 minutes, making crucial the accuracy of each Venus transit duration measurement down to the second. It would also require the weather’s cooperation for five or more hours. And the location of the longest transit time would be in the Indian ocean, notorious for its changeable weather conditions.
    Delisle thus developed a supplementary method of discovering solar parallax from the Venus transit. Delisle’s technique required an observer to mark the exact local time for just one of the four points of contact between Venus and the sun. (Those four points are the planet’s outer and inner limbs touching the sun’s edge on entry and exit—external and internal points of ingress and egress, respectively.) Delisle’s method did only require observation of a single moment in the transit, thus making stations of multiple observers more likely to have at least one overlapping data pointeven in very uncooperative weather. Crucially, however, Delisle’s method also required knowing both latitude and longitude of the observing station.
    The story in the present book describes the compromise Halley-Delisle method that astronomers used in the 1761 and 1769 Venus transit voyages: measure latitude, longitude and as much of the Venus transit as possible. Astronomers after the fact would then use both Halley’s and Delisle’s methods to discover solar parallax, ideally enabling them to cross-check their results as well. 3
    For the purposes of the present appendix, we’ll also consider two approaches to deriving solar parallax or distance. Neither is strictly the Halley or Delisle approach. Rather the first is much simpler—although far less accurate—than the other.
    In both approaches, the basic idea is the same. Venus’s silhouette follows different paths across the solar disk depending on where on the earth one observes the Venus transit. The greater the path length across the sun, the longer Venus will take to cross that path length. Observe Venus crossing the sun from two widely separated locations on earth, and the difference between the transit times will ultimately depend on three factors: the exact locations on earth where the astronomers observed the transit, the physical distance to Venus, and the distance to the sun.
    Because of Kepler’s third law, the proportional distance between Venus and the sun was well established. And if the observers independently determine their latitude and longitude, then the only remaining free variable is the distance to the sun.
    The simpler approach, then, only requires some high school mathematics: 4

    (relative sizes and distances not to scale)
    Define the physical distance between two sets of Venus transit observers as b , and the distance between the earth and Venus as p . So by trigonometric definition—opposite over adjacent—the tangent of the apparent angular difference between the transit of Venus as seen by one observer compared to the other observer is:

    But the angle θ here is small, so tan( θ ) ≈ θ . Therefore,

    Multiply both sides of the equation by the ratio, where a is the distance between the earth and the sun, and equation (3) becomes:

    Andis already a known quantity,, courtesy of Kepler’s third law.
    Ideally, one could take a time-lapse photograph of Venus as it traced a chord across the sun’s face in, say, Vardø and compare that to the same time-lapse photograph of the Venus transit as seen by Chappe in Baja, Mexico, or by Cook and Green in Tahiti. The angular separation of the two Venus transits would be θ , and some smart cartography