144 MEMOIR OF LEGENDRE. 



self W'tli tlie operations wliicli have for their object the measurement of the 

 deg-rees of the nieriilian, and concludes with some theoretical and practical reflec- 

 tions on the use of the repeating circle of Borda in the delicate operations which 

 relate to that object. 



These reflections were judicious ; ])ut at the moment of recording them, M. 

 Legendre, struck with the progress which the construction of instruments had 

 recently made, did not foresee those improvements which it was even then on 

 tlie point of receiving. They were such that at the end of 30 years the opera- 

 tions of 1787 were fcnmd to he inferior in the measurement of angles and bases, 

 the oV)servation of night-signals, &c., to those generally executed in this way. 

 Hence it resulted that the geodesic connection of Dunkirk and Greenwich 

 required to l>e recommenced in 1817. This new undertaking was confided to 

 M5I. Arago and Mathieu, associated with Captain Kater and other English 

 savants. What remained and will always remain of the operations of 1787 are 

 the formulas and theorems which it furnished JI. Legendre the occasion of estab- 

 iisliing, and which in the sequel he still further dev'eloped and improved. 



His memoir was written in the anticipation of new and more extended appli- 

 cations; for the project already existed of resuming the measurement of the 

 meridian which traverses France from north to south, and which had been once 

 measm-ed, in 1739 and 1740, in the great and admirable geodesic operation which 

 liad supplied the basis of the chart of Cassini. The National Assembly, in 

 efl"ect, having adopted the plan of establishing a new system of weights and 

 measures for all France, a report was made to the Academy of Sciences, March 

 19, 1791, by JOI. Borda, Lagrange, Laplace, Monge, and Condorcet, on the 

 choice of a unit of measure. Tlie re})ort, after a profound discussion of the sub- 

 ject, proposed to take as the unit of measure the metre, representing the ten- 

 millionth part of a quarter of the meridian, calculated from the measured length 

 of the arc comprised between Dunkirk and Barcelona. It proposed at the same 

 time the execution of different preliminary operations, one of the most important 

 of which was the verification, by new observations, of the series of triangles 

 employed for the measurement of the meridian of Cassini and its prolongation 

 to Barcelona. 



It was afterwards agreed that ]\IjM. Cassini, ]\Iechain, and Legendre, the same 

 who had connected the meridian of Paris with that of Greenwich, should be 

 charged with this new operation. Yet M. Legendre is not comprised in the num- 

 ber of the 12 commissioners nominated (April 17, 1795) to conduct all the 

 labors necessary for fixing the bases of the metrical system. These commis- 

 sioners designated from their own number MM. Mechain and Delambre to exe- 

 cute the measurement of the angles, the astronomical observations, and the meas- 

 lu'ement of the dependent bases of the meridian, and it w-as they in effect who, 

 in very difficult times, had the merit of executing this vast operation with means 

 often greatly restricted ; yet, a few years afterwards, we find M. Legendre among 

 the members of the mixed commission, formed of a union of French and foreign 

 savants, to which the duty of examining and verifying the whole work was 

 entrusted. All the triangles were separately calculated by four persons, MM. 

 Tralles, Van Swinden, Legendre, and Delambre, each employing the method he 

 preferred, and the results were only admitted when there was a satisfactory agree- 

 ment between the four calculations. M. Legendre signed with the other com- 

 missioners the report made to the National Institute, June 17, 1799, on the basis 

 of the metrical system, and he continued to take part in all tlie ulterior calcula- 

 tions and the different verifications rendered necessary by certain discordances 

 which had been remarked, and by some doubts which had arisen on the exact- 

 ness of several parts of the operation. The method he followed was that of 

 which he had estalilished the basis in his memoir of 1787. In applying it on so 

 exteusiv a locale, he improved and developed it, and gave a large number of new 

 theorems leading to more rapid reductions, to more convenient formulas. He read 



