354 PHILOSOPHICAL TRANSACTIONS. [aNNO 1704. 



in every triangle very seldom have had a difference of 15" from 180 degrees. 

 For this reason the bases, measured at several places in Seland and Jutland, have 

 very well agreed with the corresponding sides, computed through a long series 

 of triangles, begun from the observatory at Copenhagen. I believe that a dis_ 

 tance, found by those trigonometrical operations, is to be depended on to 

 ■y p^oo part of the whole. I beg leave to observe, that the Danish astronomers 

 and geographers, for 3 1 years, have been before-hand in making use of circular 

 instruments, which now begin to be of a more general use in astronomical and 

 geographical observations. The Royal Observatory at Copenhagen has, since the 

 year 1781, been adorned with a circular instrument of 4 feet radius, which, at 

 least at that time, was the only circular instrument of that size. 



By the trigonometrical operations, the meridian of Copenhagen, and of 

 several other places, and a perpendicular to the meridian of the observatory, are 

 drawn. The special position of villages, farms, and cottages, the situation of 

 the coast, woods, rivers, ponds, moors, roads, are laid down by the plain table, 

 on a scale of 2000 Danish or Rhenish feet to 1 decimal inch. After a reduction 

 to -i- part, to a scale of 1 Danish mile to 2 inches we have published Q geogra- 

 phical maps, which, as well for the geometrical exactness, as for the beauty of 

 engraving, seem not to be unworthy of the approbation of foreigners, I have 

 described the instruments, and the methods of our geometrical surveying, and 

 of the trigonometrical operations, in a treatise published in the Danish language 

 at Copenhagen, 1779» and translated into the German by Major Aster, at Dres- 

 den, 1787. In this paper I only shall lay before the r. s. a new method of com- 

 puting the longitude and the latitude of places, laid down by trigonometrical 

 (operations. 



Let eaih, fig. 11, pi. 4, be an ellipsis ; eh half the less axis ; ih half the 

 greater axis ; a the observatory at Copenhagen ; av its vertical line ; the angle v 

 the complement to the latitude of the observatory. Then by the nature of the 



ellipsis-, AV = — - — —r—: ; ; 7—7. AN is a great circle, perpendicular to 



the meridian of Copenhagen. The tangent to the same meridian af = av x 

 tang, v ; Avtnop. . . d is a series of triangles in the direction of the parallel of 

 Copenhagen; glurx.. . g a series of triangles in the direction of the meridian 

 GDE of a place g, whose longitude and latitude are to be calculated. For the 

 first series of triangles may be taken the parallel, divided into small parts ab = 

 BC = CD ; and for the 2d series may be taken the meridian gd ; because the 

 arches of those circles are known by the triangles, and computed from the 

 trigonometrical operations, fagd in fig. 1 1, is laid down in fig. 12, on a plane. 

 The angles a, b, c, d, are equal to the angles A, b, c, d. The lines of, bf, 

 cf, gdf, are equal to af, bp, cp, gdf, touching the meridians ak, be, ce, 

 GDE in A, B, c, D. The angle dfa = dpa = afb + bfc -|- cfd. For the 



