The Day the World Discovered the Sun

by Mark Anderson

  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:

  And is 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 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 πSun and the estimated solar parallax is πest, then the ultimate expression would take the form:

  Where dtO is the observed difference in Venus transit times, discussed below, and dtC 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 dtos here would be t3-t2 for Chappe subtracted from t3-t2 for Hell: 5h53m14s - 5h37m23s = 15m51s. The other would be t3-t2 for Cook subtracted from t3-t2 for Hell: 5h53m14s - 5h30m04s = 23m10s.

  What remains, then, is the calculation of dtC, 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:

  αVardø = cosφVardø cosλVardø = (0.33599) · (0.85699) = 0.28794

  βVardø = cosφVardø sinλVardø = (0.33599) · (0.51534) = 0.17315

  λVardø = sinφVardø = 0.94186

  (6)

  Since the direction cosines describe unit-length arrows locating the various Venus transit observing stations, then similarly splitting up the dtC calculation into its three Cartesian components might help to simplify this difficult problem. For instance, dtC between Vardø and Tahiti would then take the form:

  dtC = A(αVadrø – αTahiti) + B(βVadrø – βTahiti) + C(γVadrø – γTahiti) (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 egress, and external egress.)

  The variables X and Y (and their respective time derivatives) concern the position (and components of its speed) of Venus on the sun’s disk, as seen through a telescope. The origin of the X-Y coordinate system is the sun’s center, with X directed toward increasing right ascension6 and Y toward celestial north. The values rVenus and rSun represent the distances in AU from the earth to the respective bodies; thus 0.28 and 1. And HSun and δSun are the right ascension and declination7 of the sun as measured at Greenwich.

  Despite the intimidating formulas above, there are just two key points to equations 8–10. First is the significance of any observer’s additional Venus transit measurements—beyond timing, latitude, and longitude of the observatory. For example, Chappe’s almost obsessive chronicling of every moment of Venus’s transit—its angular speed across the sun’s disk, its angular distance from the edge of the sun—would translate easily into Ẋ, Ẏ, X and Y in the above formulas and thus help any solar parallax calculation enormously.

  The second point is to demonstrate that A, B, and C are each directly proportional to the initial estimate of solar parallax from equation 5. So πest merely cancels out in this first approximation, leaving πSun independent of one’s initial guess for the solar parallax.

  The above represents only a first approximation of the coefficients A, B, and C, however. Mignard performed more detailed calculations of the three values and found—again, specific to the June 3, 1769, transit—A = 476.5 seconds, B = 376.5 seconds, and C = 516.1 seconds.8

  At last we may be able to derive our own value of the solar parallax, πSun, from equations 5–7 and the data in Table A.1.

  Sticking with the Vardø and Tahiti measurements, then, equations 6 and 7 yield:

  dtC = 476.5(1.1096) + 376.5(0.65753) + 516.1(1.24227) seconds

  = 1417.4 seconds = 23m37.4s

  (11)

  Equation 5 along with the dtO enumerated on p. 237 produces:

  Mignard used the modern-day value (8.794 arc seconds) for deriving his A, B, and C coefficients, which, as shown above, to first approximation is irrelevant to the final solar parallax calculation anyway. So, using just Hell’s and Cook’s 1769 Venus transit measurements and a reasonable facsimile of the information available to an eighteenth-century astronomer, we calculate a solar parallax value of:

  Note that this value is very close to related eighteenth-century calculations. In 1771, for instance, the Oxford University astronomy professor Thomas Hornsby used differ
ent equations to find that comparing Vardø-Tahiti data yields = 8.639 arc seconds.9

  Finding the corresponding distance to the sun is just one more step, involving RE, the radius of the earth:

  NOTES

  CHAPTER 1: A STAR IN THE SUN

  1. Benjamin Martin, Venus in the Sun . . . (London: W. Owen, 1761), xi.

  2. Council Minutes of the Royal Society, 4:254, cited in Harry Woolf, The Transits of Venus (Princeton: Princeton University Press, 1959), 85. The smartest mathematicians and philosophes of the day knew that vast improvements in the accuracy of measurement of the sun’s distance might yield only marginal improvements in predicting the moon’s and planets’ positions in the skies months and years in advance (e.g., Tobias Mayer, discussed in S. A. Wepster, Between Theory and Observations [London: Springer, 2010]). Discovering the physical size scales of the solar system was the most pressing scientific problem in astronomy at the time, and astronomers—because of their ability to generate crucial nautical charts for mariners and admirals the world over—enjoyed no small degree of access to royal funding because of their solutions to the problems of longitude.

  3. Johann Pezzl, “Sketch of Vienna,” in H. C. Robbins Landon, ed., Mozart and Vienna (New York: Schirmer, 1991).

  4. Don Michael Randel, “Joseph Haydn,” in The Harvard Biographical Dictionary of Music (Cambridge: Harvard University Press, 1996), 367.

  5. In the 1760s, outer space was considered the realm of giant, exalted objects like planets and stars. No one dared suggest that mundane things like rocks could be floating out there too. F. Brandstätter, “History of the Meteorite Collection of the Natural History Museum of Vienna,” in The History of Meteoritics and Key Meteorite Collections (London: Geological Society, 2006), 123.

  6. Van Swieten never convinced Chappe that electroshock therapy had any curative powers.

  7. Science, Technology, and Warfare: Proceedings of the Third Military History Symposium, ed. Monte D. Wright and Lawrence J. Paszek (Honolulu: University Press of the Pacific, 2001), 78–79; Derek Edward Dawson Beales, Joseph II: In the Shadow of Maria Theresa, 1741–1780 (Cambridge: Cambridge University Press, 1987), 338; Bruce McConachy, “The Roots of Artillery Doctrine: Napoleonic Artillery Tactics Reconsidered,” Journal of Military History, July 2001, 619–620.

  8. Jean-Baptiste Chappe d’Auteroche, A Journey into Siberia Made by Order of the King of France (London: T. Jefferys, 1770), 25.

  9. Chappe, Journey, 23.

  10. Carol Jones Neuman, “The Historical Background,” in Drawings by Jean-Baptiste Le Prince for the Voyage en Sibérie (Philadelphia: Rosenbach Museum and Library, 1986), 21–26.

  11. Michel Mervaud, introduction to Chappe d’Auteroche: Voyage en Sibérie fait par ordre du roi en 1761 (Oxford, U.K.: Voltaire Foundation, 2004), 6–15; Per Pippin Aspaas, private communication to author, January 3, 2012.

  12. Chappe, Journey, 46.

  13. Chappe, Journey, 47. Chappe says “the eldest [of the girls] was not above seventeen.” Elsewhere he notes that girls in the Russian provinces were married at thirteen.

  14. Chappe, Journey, 48.

  15. Chappe, Journey, 49; Olga Yu. Elina, “Private Botanical Gardens in Russia: Between Noble Culture and Scientific Professionalization (1760s–1917),” in The Global and the Local: The History of Science and the Cultural Integration of Europe: Proceedings of the Second ICESHS (Cracow, 2006), www.2iceshs.cyfronet.pl.

  16. Chappe, Journey, 52–53.

  17. Chappe, Journey, 301.

  18. Solar noon also coincides with the sun crossing an imaginary “meridian” line in the sky extending from directly north to directly south.

  19. The present description is a facsimile of Chappe’s actual latitude-calculating technique, chosen for its comparative simplicity. Technically, stellar charts give a star’s “declination,” the altitude of a star as measured from the “celestial equator,” the projection of the earth’s equator onto the sky. Calculating latitude via a star’s altitude, then, involves first calculating the altitude of the celestial equator—which is the observed stellar altitude minus its declination, corrected for additional effects like atmospheric refraction. One’s latitude, then, is 90 degrees minus the altitude of the celestial equator. This is the process Chappe actually followed. “Mr. L’Abbé Chappe d’Auteroche”: Memoire du passage de Venus sur le soleil (St. Petersburg: Imperial Academy of the Sciences, 1762), http://tinyurl.com/262daq8. Translated by Mark Anderson.

  20. Ibid. Astronomer Eric W. Elst discovered the present-day location of Chappe’s observatory, at latitude 58 degrees, 11 arc minutes, 43 arc seconds; longitude 68 degrees, 15 arc minutes, 30 arc seconds. Chappe settled on 58 degrees, 12 arc minutes, and 22 arc seconds as his final answer (Ibid.). This represents a 39-arc-second difference between Chappe’s calculation and modern determinations. Each arc second of latitude is 0.02 miles on the earth, which puts Chappe’s error at 0.78 miles. www.holbachfoundation.org/astro/Details_obs.Chappe.htm.

  21. Le Prince likely did not journey with Chappe to Tobolsk but instead traveled around Russia and Siberia separately. Upon returning to Paris, though, the artist did work in close collaboration with Chappe when preparing the drawings (on which the book’s engravings would be based) for Voyage en Sibérie. Neuman, “Historical Background,” 11.

  22. Neuman, “Historical Background,” 79–80.

  CHAPTER 2: THE CHOICEST WONDERS

  1. On the Ramillies wreck: http://sn.im/116pyp; www.submerged.co.uk/boltheadtobolttail.php.

  2. “To the Author of London Magazine,” London Magazine, or, Gentleman’s Monthly Intelligence 30 (1761): 199, http://sn.im/11gqm0.

  3. RS Misc. MSS 10/114. The East India Company was not a charitable organization. But every improvement in the art of navigation also improved its bottom line. The company’s eagerness to ensure England had the best Venus transit measurements in the world bespeaks a recognition of some commercial potential in the science it was supporting.

  4. The position of the planets and the moon was calculated using Starry Night Pro.

  5. S. P. Rigaud, Miscellaneous Works and Correspondence of the Rev. James Bradley (Oxford, 1832), 388–390.

  6. See http://sn.im/12xzim on the Byzantine role of white flags in British naval signaling. http://sn.im/12y07n; Encyclopedia Britannica (1797 ed.) on the white flag signaling “no hostile intention.”

  7. A Universal Dictionary of the Marine, “Rates,” http://sn.im/126987.

  8. The New Bath Guide (Bath: R. Cruttwell, 1789), 75–77, http://sn.im/198256.

  9. Henry Francis Whitfield, Plymouth and Devonport: In Times of War and Peace (Plymouth: E. Chapple, 1900), 157.

  10. RS Misc. MSS 10/128.

  11. RS Misc. MSS 10/130.

  12. RS Misc. MSS 10/129–131.

  13. RS Misc. MSS 10/129.

  14. James Pritchard, Louis XV’s Navy (Montreal: McGill-Queen’s University Press, 1987), 126.

  15. RS Misc. MSS 10/134.

  16. Table Bay descriptions: http://sn.im/19p306; http://sn.im/19p37h, 28ff.

  17. Re Ryk (Rijk) Tulbagh, http://sn.im/19igqj; http://sn.im/19igs2.

  18. Linnaeus, letter to Ryk Tulbagh, n.d., in A Selection of the Correspondence of Linnaeus and Other Naturalists, ed. James Edward Smith (London: Longman, Hurst, Rees, Orme & Brown, 1821), 2:570, http://sn.im/19qd40.

  19. Abraham Bogaert (1702), quoted in Leonard Thompson, A History of South Africa (New Haven: Yale University Press, 1990), 39.

  20. Kerry Ward, Networks of Empire; Forced Migration in the Dutch East India Company (Cambridge: Cambridge University Press, 2008).

  21. Thompson, History of South Africa, 36–37; Hymen W.J. Picard, Gentleman’s Walk: The Romantic Story of Cape Town’s Oldest Streets, Lanes, and Squares (Cape Town: C. Struik, 1968), 255–128, http://sn.im/1ao3p0.

  22. Theodore MacKenzie, in “Mason and Dixon at the Cape,” Monthly Notes of the Astronomical Society of South Africa 10, no. 100 (1951), finds the observatory’s location between present-day
St. Johns and Hope streets near St. Mary’s Cathedral. Using period maps (e.g. Picard, Gentleman’s Walk, frontispiece and p. 16), the 1761 equivalent of this location can readily be found.

  23. RS Misc. MSS 10/135.

  24. RS Misc. MSS 10/143, 144.

  25. Technically, the Venus transit observations ultimately yielded a number known as the solar parallax—one-half of the angular size of the earth, as seen from the sun. Because there are no stars visible in the daytime sky to measure the sun’s position against, no one had ever been able to find out the solar parallax before Halley discovered the clever triangulation trick that enabled the Venus transit to yield the answer. The distance to the sun D can then be calculated from the equation D = R/π, where R is the earth’s radius and π is the solar parallax. See the Technical Appendix in this book for more details.

  26. Using Starry Night Pro set to Cape Town on June 5, 1761, one sees Antares rising nearly directly to the east at 8:40 PM. (Given the observatory’s location, above, siting the mountains and landmarks above which it would rise can readily be done in Google Earth.) Altair rises later, toward the northeast, at 11:40 PM.

  27. Maskelyne, autobiographical notes, in Derek Howse, Nevil Maskelyne: The Seaman’s Astronomer (Cambridge: Cambridge University Press, 1989), 216.

  28. RS Misc. MSS 10/151.

  29. “The Description and Use of Hadley’s Octant, Commonly Called Hadley’s Quadrant,” in John Hamilton Moore, The Practical Navigator (London: B. Law & Son, 1791), http://sn.im/1dfkn3.

  CHAPTER 3: FLYING BRIDGES

  1. Guillaume-Thomas-François Raynal, A Philosophical and Political History of the Settlements and Trade of the Europeans in the East and West Indies (London: W. Strahan & T. Cadell, 1783), 8:190.

  2. The archbishop, a notorious reactionary, had a few previous run-ins with Chappe. The archbishop hated the pope, whom he chastised for taking Communion sitting down. And he refused to believe that the earth orbited the sun. Chappe didn’t sway the Russian prelate on the Copernican question. But as for the papal sacrilege, Chappe recalled, “I assured him the Pope was a cripple.” Chappe, Journey to Siberia, 290.