On June 3, 1769, Lieutenant James Cook (1728-1779) of the Royal Navy and astronomer Charles Green (1735-1771) of the Royal Society measured the transit of Venus from an observatory set up at the north-east end of Matavai Bay, Tahiti. Several decades earlier, British astronomer Edmond Halley (1656-1742) had suggested that measuring the passage of Venus across the face of the Sun from several locations on Earth would provide the data needed to determine, through the principles of triangulation, the distance to the Sun and, from that measurement, the distances to all the other bodies in the solar system. Halley knew that he would not see a transit in his lifetime, as the celestial phenomenon occurs in pairs every 105 to 122 years. The next transits would occur on June 6, 1761 and June 3, 1769, the pairs always occurring eight years apart. Although the scientific community was ready in 1761, the measurements obtained were essentially a failure.
(At right, Chart of the Island Otaheite, by Lieut. J. Cook 1769) 
Having obtained disappointing results from observations of the first of the pair of Venusian transits, the Royal Society made plans for a voyage to the Pacific to observe the 1769 transit. The Royal Society would take responsibility for the scientific part of the expedition and the Royal Navy would provide the vessel and nautical expertise. The Navy chose thirty-nine year old James Cook to lead the expedition. Commanding His Majesty's Bark Endeavour, Lieutenant Cook sailed from Plymouth, England on August 26, 1768. After brief stops at Madeira, Rio de Janeiro, and Tierra del Fuego, the Endeavour rounded Cape Horn and reached Tahiti on April 13, 1769. Anchoring in Matavai Bay, Cook selected a spot on the north-east point and marked out the ground for a small fort (named Fort Venus) within which would be the portable observatory.
On May 1, 1769, Cook and the scientific members of the expedition set up the observatory. The calculation of the distance depended on noting the exact time at which the silhouette of Venus first entered, and then exited, the Sun's disk. Among the instruments the group brought ashore were a clock, reflecting telescopes, and an astronomical quadrant. When Cook and Green were ready to set up the quadrant the next morning, they discovered to their consternation that the packing case containing it had disappeared. They learned that one of the island's inhabitants had taken it. Fortunately, relations between the members of the expedition and the Tahitians had been friendly. Several of the party set out to reclaim the instrument and brought it back to the fort early in the evening. 
(At right, detail of engraving of Matavai Bay showing Endeavour and Fort Venus)
Saturday, June 3 "proved as favourable to our purpose as we could wish, not a cloud was to be seen the whole day & the Air was perfectly clear, so that we had every advantage we could desire in observing the whole of the passage of the Planet Venus over the Suns Disk." To ensure that the observation of the transit was a success, Cook had sent a small party to the nearby island of Moorea on June 1 and another group to the eastward of Fort Venus on June 2. They, too, met with success in their observations.
In December 1771, five months after the Endeavour returned to England, the fellows of the Royal Society met to hear the results of the calculations based on the Tahitian observations and those of several other astronomers taken in different parts of the world. British astronomer Thomas Hornsby (1733-1810) determined that the distance between the Earth and the Sun was 93,726,900 miles. That figure is less than 800,000 miles off from the current calculation of 92,960,000 miles.
(Below, Green's and Cook's observations of Venus)
Quote from:
Cook, James. The Journal of H.M.S. Endeavour 1768-1771. Guilford, Surrey, England: Genesis Publications Limited in association with Rigby Limited, 1977. [facsimile of the original manuscript] Call Number: G420.C66 C632 1977
Illustrations from:
Volume 2 of Hawkesworth, John. An Account Of The Voyages Undertaken By The Order Of His Present Majesty For Making Discoveries In The Southern Hemisphere, And successively performed by Commodore Byron, Captain Wallis, Captain Carteret, And Captain Cook, In the Dolphin, the Swallow, and the Endeavour. 3 vols. London: W. Strahan and T. Cadell, 1773. Call Number: G419.H38 1773
Volume 11 of Hutton, Charles, et. al. The Philosophical Transactions Of The Royal Society Of London, From Their Commencement, in 1665, To The Year 1800; Abridged, With Notes And Biographic Illustrations. 18 vols. London: C. and R. Baldwin, 1809.
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