The concept of using the length of a quarter of the meridian proved to be a great idea, but its implementation in the 18th century required a number of compromises. Measuring the huge distance between one of the poles and the equator in an almost straight north-south line would have been an impossible task at that time, since neither the North nor the South Pole had yet been reached. It was conservatively decided to measure two things. The first measurement was the exact latitudes of two points along the same meridian. The second was the exact length between these two points. The two points selected were Dunkirk, France, and Barcelona, Spain. These cities are located at approximately 41° latitude on the same meridian. The starting point in Dunkirk was approximately 51°02’55.08" latitude and 2°22’56.00" longitude, at an elevation of 4.27 m (14 ft) above sea level. In Barcelona, the measurements started at Montjuic tower, at 41°21’49.26" seconds latitude and 2°10’0.49" longitude, at 53.95 m (177 ft) above sea level.
This was a good compromise for a number of reasons. Most European countries are close to the 45° parallel. Dunkirk is 6° above the 45th parallel, while Barcelona is 3.5° below. Both locations are relatively close to sea level, so only minimal corrections for elevation were needed. Previous surveys included locations above sea level, requiring elevation corrections.
The actual measurement was an adventure in its own right, with political intrigue and war as a backdrop. Two eminent French astronomers and mathematicians received the assignment. Jean Baptiste Delambre measured the stations between Dunkirk and Rodez, France. The southern segment, from Rodez to Barcelona, was measured by Pierre Méchain. They began the project in 1792.
Delambre and Méchain decided to use triangulation to measure the length of the arc of the meridian between Dunkirk and Barcelona. Triangulation is a surveying technique that measures a distance by creating a network of triangles. The distance between two points of the triangle, called the base, is carefully measured. This forms one side of the triangle. A third point is established some distance away, and the angles from the ends of the base to this point are measured. The formula
is used to calculate the lengths of the other sides. The next triangle is established using one side of the first triangle. This continues until the necessary area is covered.
The task required the utmost care. First, the elevation of the points of the triangle had to be measured accurately. This was done by making vertical angle measurements with uncertainties as low as 1 half-second. Second, although the triangulation technique usually requires the measurement of only one base, Delambre and Méchain measured two. For example, a second base was measured between the French towns of Melun and Lieusaint (6,075.90 Peru toises) and between Salces (now Salses-le-Chateau) and Vernet (6,006.25 Peru toises), near Perpignan. The latter, computed from the chain of triangles and the base of Melun, differs from its actual measurement by only 26 cm (10.2 in), although the two bases are 700 km (435 mi) apart.
Third, angle measurements were repeated at each station. For example, the Aubassin-Puy Violent-Labastide angle was repeated 84 times, Labastide-Puy Violent-Montsalvy, 38 times, and Montsalvy-Labastide-Aubassin 58 times. The first 20 measurements of the Labastide-Monsalvy-Saint Jean de Rieupeyroux angle gave 87°43’19.24", the 2nd series of 20, 87°43’22.07", the third 20, 87°43’20.98", which means a spread of 2.83", an error of 0.68 m (2.2 ft) for 50 km (31 mi).
Triangles used to define the meter.
All together, Delambre and Méchain measured 115 triangles between Dunkirk and Barcelona. They completed the operation in 1798.
At the end of 1798, an international scientific committee scrutinized all the data available at the time. The group included representatives from the Netherlands, Denmark, Spain, Italy (Piedmont, Tuscany, Roma, Liguria, Cisalpine republic), and Switzerland. They checked the 115 triangles that were measured. They checked the latitude measurements performed at either end of the arc in Dunkirk and Barcelona, as well as in some intermediate points, including Carcassone, Evaux, and Paris.
In mid-1799 the committee announced the results. The arc of the meridian between Dunkirk and Barcelona was 9°40’25.40". It measured 551,584.72 Peru toises. Therefore, a quarter of the meridian was 5,130,740 Peru toises, and the meter was 443.296 lignes (a ligne was a unit of measure used at that time) of the Peru toise.
Delambre and Méchain approached the measurements differently. Delambre took notes of the multiple entries. He did not prejudge any of his measurements based on what he expected to find. He then provided the scientific committee with every bit of information he collected. Delambre was a true experimentalist: he accepted the fact that there might be errors in the process.
Méchain, on the contrary, was a theoretician at heart. He wanted the data to fit the model he had in mind, and he therefore wanted the measurements to be perfect. One example of this occurred near Barcelona. Méchain performed two series of latitude measurements. The first series was taken in the Montjuic fort, while the second was measured in the Fontana de Oro inn. The difference between the measurements was 3", a value too large considering that the stations were relatively close to each other. Méchain agonized for years about this inconsistency, to the point that he did not want to return to Paris for the final checks and report. He finally omitted the Fontana de Oro measurements from the report, a decision that can be considered a scientific mistake.
After Méchain’s death in 1803, Delambre finally had access to the full set of the Rodez-to-Barcelona documents and discovered the omission. Researchers now believe that the lack of consistency of the measurements was due to instrument errors and not poor measurement procedures. Modern thought on the importance of this mistake varies. Some researchers believe that the error had no consequence on the derivation of the meter. Others quantify the error at 0.2 mm (0.008 in).
Delambre and Méchain could have measured two angles per triangle and deduced the third one, since the sum of the three angles in a triangle equals 180°. Instead, they measured the value of each angle. Measurements were noted as they read, without corrections. Yet out of the 115 triangles that join Dunkirk and Barcelona, only 36 have errors larger than 1", when the three angles are added together.
Finally, the French army repeated the survey between 1870 and 1896. This second survey concluded that the 1799 results were just 0.2 mm shorter than the 1/10,000,000 part of the Earth’s quadrant on which its definition depended.
In 1980 the international reference for the Dunkirk-Barcelona arc was 551,589.3 toises, while the measurement by Delambre and Méchain was 551,584.7, a difference of only 10 m (33 ft) over more than 1,000 km (620 mi). |
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