The Longitude of Greenwich and the Shape of the Earth

Jacques Dominique Cassini wrote in 1791 that the "sole aim" of the triangulation had been to ascertain the longitudinal difference between the observatories of Paris and Greenwich.^[26] On the other side, Roy had asserted in 1787 that the "chief and ultimate goal has always been considered of a still more important nature, namely, the laying the foundation of a general survey of the British Islands."^[27] But there is nothing to indicate that Roy had official support for such a scheme at this stage. Although he wished to create a national survey, the measurement he was actually conducting had a specific purpose within a scientific context.^[28] This purpose he had to defend. Consequently Roy accepted the French criticism depicting positional astronomy—or anyway British astronomy—as less accurate than geodesy. The fact that the expensive and time-consuming triangulation went forward without the cooperation of the astronomer at Greenwich suggests the strength of Roy's position within the Royal Society.

The disagreement between Roy and Maskelyne was ventilated in two papers published in 1787. Maskelyne wrote a belated reply to Cassini de Thury's mémoire ; Roy, annoyed at the delay, threatened

[25] Maskelyne, "Concerning the latitude and longitude," 151–2; Roy, "An account of the measurement of a base," 425, quote.

[26] Jacques Dominique Cassini, Pierre François Méchain, and Adrien-Marie Legendre, Exposé des opérations faites en France en 1787 pour la jonction des observatoires de Paris et de Greenwich (Paris, [1791]), 65.

[27] William Roy, "An account of the mode proposed to be followed in determining the relative situations of the royal observatories of Greenwich and Paris," PT , 77 (1787), 188–226, on 188.

[28] Seymour, History of the Ordnance Survey , 21–2. George III insisted that the financial support he gave go for "the science of astronomy"; Harley and O'Donoghue, "Introductory notes," xxi.

to publicize Maskelyne's idleness in the Philosophical transactions —thereby demonstrating that he had the official support of the Royal Society, whereas Maskelyne did not.^[29] Roy accentuated the criticism of British astronomy by adding to the uncertainty of eleven seconds of time in longitude between Greenwich and Paris, which Cassini de Thury had pointed out, an uncertainty of three seconds between Oxford and Greenwich. Geodetic methods would narrow the margin of error; Roy considered them "infallible," since they could always be controlled by verifying the bases. The planned triangulation would give a value for the longitudinal difference "sufficiently near the truth, and. . .probably considerably nearer than it will be brought for many years to come, by a mean of the best observations of the heavenly bodies."^[30]

Roy probably did not know that in 1785 Maskelyne had equipped his assistant Joseph Lindley with a number of watches and sent him on a secret "chronometer run" to Paris, to determine the time difference between the capitals. Lindley's result (9 minutes 20 seconds) verified Maskelyne's astronomically deduced value, published in 1787, which was later found to agree with the result of Roy's triangulation.^[31] Roy avoided admitting this embarrassing consistency, which he had earlier denied, by misquoting Maskelyne's data. In his final report he simply plucked from Maskelyne's paper a number about 10 seconds larger than the figure on which the astronomer finally settled, and claimed it as the astronomically deduced value.^[32] He thus demonstrated the superior accuracy of the geodetic method.

[29] William Roy to Nevil Maskelyne, 11 Dec 1786 (RS, DM.4.14). A copy of this letter is in PRO, O.S.3/4. Cf. Roy, "An account of the mode proposed," 213; Eric G. Forbes, "The geodetic link between the Greenwich and Paris observatories in 1787," Vistas in astronomy, 28 (1985), 173–81, on 174. Maskelyne complained that he had not been shown the mémoire until a year after preparations for the triangulation had begun. See Maskelyne, "Concerning the latitude and longitude," 153.

[30] Roy, "An account of the mode proposed," 213–4.

[31] Eric G. Forbes, Greenwich Observatory . Vol. 1: Origins and early history (1635–1835) (London: Taylor and Francis, 1975), 149–50, and "The geodetic link," 174; Maskelyne, "Concerning the latitude and longitude," 183–6; Roy, "An account of the trigonometrical operation," 231.

[32] Roy, ibid., 231. Roy's maneuvers to protect himself from the suspicion of deliberately misquoting Maskelyne are sufficiently transparent. Cf. Roy to Maskelyne, 11 Dec 1786 (PRO, O.S.3/4).

The question of the earth's shape came in as an important scientific side issue in the determination of longitude differences. Roy had developed a new technique for geodetic investigation, which involved astronomical observations of a kind different from those usually associated with longitude determinations. Nevertheless, as Isaac Dalby was to point out, it constituted perhaps the weakest link in Roy's geodetic work. By taking the angles between three mutually remote stations and at the same time observing the angles between the stations and the polestar, Roy calculated the longitudinal differences between pairs of stations by spherical trigonometry. The relationship between these differences and the distance on the ground gave a value for the length of a degree of longitude at a particular latitude. Polestar observations made at only a few stations furnished the basis for calculation of the longitude difference of the whole chain of triangles. Roy calculated the latitudes of the stations in relation to that of Greenwich from a spheroidal model of the earth devised by Pierre Bouguer.^[33]

To justify use of this spheroid, Roy computed the lengths of the arc between Greenwich and Perpignan (the southern extremity of the Paris meridian) on ten different hypotheses about the shape of the earth. In the model Roy favored, the lengths of degrees of latitude increased with the fourth power of the sine of the latitude. One of the models he rejected was an ellipsoid based on data from the six earlier arc measurements Roy thought most consistent. To achieve consistency, however, he had had to take a mean between the arcs of Cassini in France and Liesganig in Austria, since comparison between them gave an "absurd" result—that is, an oblong earth.^[34] Roy combined the six arcs into fifteen pairs and calculated the flattening for each. ("Flattening" is defined as the ratio of the equatorial axis to the difference between the equatorial and polar axes.) Values ranged from 100 to 850, but Roy did not present the extremes: he exhibited only the mean flattening of 190, well within

[33] Roy, "An account of the mode proposed," 216–20, and "An account of the trigonometrical operation," 200, 225–7.

[34] Roy, "An account of the mode proposed," 206–8.

the limits of what was considered reasonable.^[35] The same was true of the other six ellipsoids in Roy's table, resulting from other combinations of measurements or from hypothetical premises, and of the two spheroids as well.

Roy nonetheless proffered Bouguer's spheroid as the most probable alternative because, unlike the other hypotheses, it gave values sometimes above and sometimes below the lengths of the measured arcs: "a never failing proof" that it was "exceedingly near the truth."^[36] Roy hid the wide discrepancies that actually existed between different measurements behind the reassuring surface of averages. Mathematical analysis of error was then only in its infancy. Roy employed another, more visual technique for comparing the different solutions to the problem of the shape of the earth. He presented the results on the different hypotheses in tabular form so that the reader could judge, "by simple inspection only, which of the theories agrees best with actual measurement." He also gave the lengths of degrees of meridians, parallels, and oblique great circles according to Bouguer's spheroid—not only for the portion of the earth covered by his own triangulation, but for the whole earth, so that others could use these figures until, in the distant future, the shape of the earth would "ultimately" become known. Meanwhile, Roy thought, the spheroid would furnish data of "general utility."^[37]

Dalby disagreed with the use of Bouguer's spheroid and with Roy's method of finding differences in longitude. He criticized the method as too sensitive to observational errors. An error of one second of arc in the angles between the stations and the meridian would cause an error of six seconds in the longitude difference between Greenwich and Dunkerque.^[38] Roy had said that he

[35] The usual magnitude of the flattening ranged between 170 and 540—the values suggested by Maupertuis and Christian Huygens, respectively.

[36] Roy, "An account of the mode proposed," 210–1. In early mathematical analysis of observational error by Boscovich and Laplace, this condition usually came with another: that the sum of the absolute values of the errors should be minimized. See Stephen M. Stigler, The history of statistics: Measurement of uncertainty before 1900 (Cambridge, Mass.: Belknap Press, 1986), 11–61, esp. 47, 51.

[37] Roy, "An account of the mode proposed," 201, 222.

[38] Isaac Dalby, "The longitudes of Dunkirk and Paris from Greenwich, deduced from the triangular measurement in 1787, 1788, supposing the earth to be an ellipsoid," PT, 81 (1791), 236–45, on 237, and "Remarks on Major-General Roy's account of the trigonometrical operation," PT, 80 (1790), 593–614, on 607–8. Roy's method for obtaining longitudes is described in Roy, "An account of the trigonometrical operation," 206–25.

determined the longitude difference "by the instrument itself," meaning that the extreme accuracy of the theodolite guaranteed the precision of the result. Dalby challenged this assertion and rejected Roy's spheroid in favor of his own ellipsoid shape, with the flattening of 229 predicted by Newton.^[39]

As we know, the ellipsoid shape depended only on theories of gravity and mechanics and on the assumption that the earth had once been fluid. The theoretical implications of Bouguer's spheroid, on the other hand, are unclear, and probably they did not matter very much to Roy. He adopted it because it gave a good fit to existing measurements. Ramsden, not Newton, was the arbiter of exact geodesy; the theodolite could make errors "totally vanish." Dalby had no special interest in defending the elaborate technology adopted by Roy, and consequently he was happy to accept Maskelyne's value for the longitude difference between Paris and Greenwich as support for the result of the measurement in which he himself had participated.^[40] In modern terms, Roy's attitude was instrumentalist, whereas Dalby's might be called realist. Condillac, who advocated a strict empiricism, criticized both. Condillac's view eventually won out. When the concept later to be christened the geoid was developed in Germany in the early 19th century, irregularities in the earth's mass distribution ceased to be regarded as anomalies and became instead constitutive of the "real" shape of the earth.