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

could yield b .
    However, this approximation is too simplistic to do justice to the precision of the transit measurements and the needed precision of the calculations. For starters, it doesn’t take into account the fact that b is measured over the curved surface of the earth. And, of course, there was no such thing as photography in 1769.
    The approximation needs to be better.
    Like many complex problems, there is no single correct way of arriving at an answer. Perhaps the most straightforward conceptually is the “educated guess” approach.
    Namely, take the best result of the solar parallax from the 1761 Venus transits, and then blindly calculate the transit time one would expect to observe at any given latitude and longitude in the 1769 transit. Then factor out the estimated solar parallax and factor in the 1769 observations of transit times and known latitudes and longitudes. 5
    Mathematically, if solar parallax being calculated is π Sun and the estimated solar parallax is π est , then the ultimate expression would take the form:

    Where dt O is the observed difference in Venus transit times, discussed below, and dt C is the calculated difference in transit times.
    Before getting into theoretical calculations, it’s important to spell out precisely what data the three teams at the core of this book—Hell and Sajnovics at Vardø, Chappe at San José del Cabo, Cook and Green at Tahiti—brought home.

    Table A.1. Results of 1769 Venus transit expeditions as measured by teams led by Maximilian Hell, Jean-Baptiste Chappe d’Auteroche, and James Cook
    So one of the two possible dt o s here would be t 3 -t 2 for Chappe subtracted from t 3 -t 2 for Hell: 5 h 53 m 14 s - 5 h 37 m 23 s = 15 m 51 s . The other would be t 3 -t 2 for Cook subtracted from t 3 -t 2 for Hell: 5 h 53 m 14 s - 5 h 30 m 04 s = 23 m 10 s .
    What remains, then, is the calculation of dt C , whose full derivation exceeds the scope of this appendix. Nevertheless, some essential equations and numerical results in these calculations can still yield a numerical answer.
    It begins with so-called direction cosines of the three observers’ locations on earth. Consider them x, y , and z components of unit vectors pointing to each observer’s station (Vardø, San José del Cabo, Tahiti)—where this particular Cartesian coordinate system has its origin at the center of the earth and its x-y plane as the earth’s equator and its x-z plane the Greenwich meridian.
    Here, for example, are the direction cosines for Hell’s observatory at Vardø:
α Vardø = cos φ Vardø cos λ Vardø = (0.33599) · (0.85699) = 0.28794
    Î² Vardø = cos φ Vardø sin λ Vardø = (0.33599) · (0.51534) = 0.17315
    Î» Vardø = sin φ Vardø = 0.94186
(6)
    Since the direction cosines describe unit-length arrows locating the various Venus transit observing stations, then similarly splitting up the dt C calculation into its three Cartesian components might help to simplify this difficult problem. For instance, dt C between Vardø and Tahiti would then take the form:
dt C = A ( α Vadrø – α Tahiti ) + B ( β Vadrø – β Tahiti ) + C ( γ Vadrø – γ Tahiti )
(7)
    Where the coefficients A, B , and C (whose units are in seconds) represent components of the differential transit time weighted to expected values of its x, y , and z contributions. The coefficients are not particular to any location on earth but rather to the geometry of the sun’s position in the sky and Venus’s typical path across the sun’s face during the June 3, 1769, transit.
    The (present-day) French astronomer François Mignard derived the first approximations of A, B , and C as follows—in this case, particular to each contact. (So for the following formulas, one would calculate separate coefficients for the 1769 transit’s external ingress, internal ingress, internal