144 MEMOIR OF LEGENDKE. 



self w'tli the operations wLicli liave for their object the measurement of the 

 degrees of the meridian, and cunchides with some theoretical and practical reflec- 

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

 relate to that oVy'ect. 



These reflections were judicious ; hut at the moment of recording them, ]\I. 

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

 recently made, did not foresee those improvements w'hich it was even then on 

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

 tions of 1787 were found to be inferior in the measurement of angles and bases, 

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

 Hence it resulted that the geodesic connection of Dunkirk and Greenwich 

 required to be recommenced in 1817. This new undertaking was confided to 

 MM. 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 M. Legendre the occasion of estab- 

 lishing, and which in the sequel he still further developed 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 

 measured, in 1739 and 1740, in the great and admirable geodesic operation which 

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

 eflect, 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 MM. Borda, Lagrange, Laplace, JMonge, and Condorcet, on the 

 choice of a unit of measure. The report, 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 Duukkk 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 seiies of triangles 

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

 to Barcelona. 



It was afterwsxds agreed that MM. Cassini, Mechain, 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 (Ajiril 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. IMechain and Delambre to exe- 

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

 urement of the dependent bases of the meridian, and it was 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, w-e 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 verifj'ing the whole work W'as 

 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 satisfactor}' 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 the ulterior calcula- 

 tions and the difl'erent 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 established the basis in his memoir of 1787. In applying it on so 

 extensiv* a scale, he improved and developed it, and gave a large number of new 

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



