MEMOIR OF LEGENDRE. 143 



Borda. M. Legendi-e calculated all the triangles situated in France, and after- 

 wards those also which extended in England as far as Greenwich. On this occa- 

 sion he went to London, where he was received with the distinction due to him, 

 and was named member of the Royal Society of London. He published at this 

 time in the Memoirs of the Academy for the year 1787 (printed in 1789) an 

 important paper entitled, Memoir on the trigonometrical operations of ivhicli the 

 results depend on the figure of the earth ;* of this he has himself explained the 

 object in terms which I take the liberty of abridging : 



The only question here is that which regards operations exacting extreme pre- 

 cision, such as the measurement of the degrees of the meridian or of a parallel, 

 and the geographical determination of the principal points of a large area from 

 the triangles which connect them. Operations of this kind may be carried 

 henceforward to a great degree of precision by means of the repeating circle. 

 In effect, the use ^vliich we have made of this instrument, in 1787, has convinced 

 us that it can give each angle of a triangle to about two seconds, or even more 

 exactly, if all circumstances are favorable. It is further necessary that the cal- 

 culations established on such data should not be inferior to the latter in exact- 

 ness ; especially is it requisite to take account of the reduction to the horizon, 

 which amounts quite often to several seconds; and thence arise triangles of infi- 

 nitely small curvature, the calculation of which demands special rules ; for, by 

 considering them as rectilinear, we should neglect the small excess of the sum 

 of the three angles over 180°, and hy considering them as spherical, the sides 

 Avould be changed into very small arcs, the calculation of which by the common 

 tables would be neither exact nor commodious. 



I have assembled in this memoir, continues M. Legendre, the necessary formu- 

 las, as well for the reduction and calculation of these sorts of triangles, as for 

 what relates to the position of the different points of a chain of triangles on the 

 surface of the spheroid. In these calculations, he adds, there are some elements 

 susceptible of a slight uncertainty, * * * * In order that 



the calculation need be made but once, and to judge by a glance of the influence 

 of errors, I have supposed the value of each principal element to be augmented 

 l)y an indetenninate quantity which denotes the correction of it. These literal 

 quantities, which are to be regarded as very small, do not prevent the calculation 

 from being proceeded with by logarithms in the usual manner. 



This was an important addition to the methods of calculation employed till 

 then, and still later he further added the method of least squares. He gives in 

 this memoir formulas for the reduction of an angle to the horizon, as also for 

 other determinations, and especially the important theorem known under the 

 name of the theorem of Legendre, through which the calculation of a spherical 

 triangle of small extent is reduced to that of a rectilinear triangle, by subtract- 

 ing from each of the three angles the third of the spherical excess of their sum, 

 tliat is to say the inconsiderable quantity by which this exceeds 1 80°. M. Legen- 

 dre has subsequently demonstrated that this fundamental theorem is applicable 

 also to spheroidal triangles, whether traced on an ellipsoid of revolution or even 

 on a spheroid slightly iiTegular. 



He also occupies himself, in the same memoir, with the value of the degrees 

 of the meridian in the elliptical spheroid, and with the detemiination of the 

 respective position of different places deduced from the nature of the shortest 

 line which can be traced on the surface of this spheroid from one extremity to 

 the other of the chain of triangles and from the intersections of that line with 

 the different sides of the triangles or with their prolongations. This line, which 

 M. Legendre, at different times and always with success, made the object of his 

 researches, bears the name of the geodesic line ; on the regular ellipsoid it is of 

 double curvature, unless it coincides with a meridian. Finally, he occupies liim- 



*Mem. dc I' Academie des Sciences, for 1787, (printed in 1789,) p. 352. 



