VOL. LXXXV.] PHILOSOPHICAL TKANSACTIONS. GSQ 



Butser Hill, Dean Hill, and Wingreen on the north, and on the south by those 

 connecting the stations Nine Barrow Down, Motteston Down, Dunnose, Rook's 

 Hill, Chanctonbury Ring, and Ditchling Beacon. 



The 3d branch, is that which proceeds from the side Hanger Hill and Ban- 

 stead, to Botley Hill and Leith Hill, and thence towards Beachy Head and 

 Brightling, joining the series formerly projected at Botley Hill and Fairlight 

 Down; the branch being bounded to the westward by the sides connecting the 

 stations Hanger Hill, Banstead, Leith Hill, Ditchling Beacon, and Beachy 

 Head. 



The 4th branch, is that by which the distance between Beachy Head and 

 Dunnose is obtained, and is formed by the sides connecting the stations Beachy 

 Head, Ditchling Beacon, Chanctonbury Ring, Rook's Hill, and Dunnose. 



The account then proceeds to the selection of the angles constituting the 

 principal triangles, and the manner of reducing them for computation. The 

 angles of the several triangles, constituting the general series, are, with a very 

 few exceptions, those arising from using the means of the several observations 

 given in the foregoing part of this work ; for though the rejecting of such as 

 might apparently suit the purpose, would give the sums of the 3 angles of many 

 of the triangles, nearer to 180 degrees plus the computed excess; yet as all the 

 observations have been made with equal care, and are for the most part to be 

 considered as of equal accuracy, it has been thought proper to select those means, 

 as being the fairest mode of proceeding. 



If the observations had been made on a sphere of known magnitude, and the 

 angles accurately taken, the most natural method of computing the sides of the 

 triangles from the measured bases, would be by spherical trigonometry; but if 

 the magnitude was such, that the length of a degree of a great circle was equal 

 to a degree of the meridian in these latitudes nearly, in order to obtain the sides 

 true to a foot from such computation, with any facility, a table of the logarith- 

 mic sines of small arcs computed to every -j-i-j- of a second of a degree, would 

 be necessary, because the length of a second of a degree on the meridian is 

 about 100 feet. As the lengths of small arcs and their chords are nearly the 

 same (the difference in these between Beachy Head and Dunnose being less 

 than 4 feet) it is evident that this business might be performed sufficiently near 

 the truth in any extent of a series of triangles, by plane trigonometry, if the 

 angles formed by the chords could be determined pretty exactly. We have en- 

 deavoured to adopt this method in computing the sides of the principal triangles, 

 in order to avoid an arbitrary correction of the observed angles, as well as that 

 of reducing the whole extent of the triangles to a fiat, which evidently would 

 introduce erroneous results, and these in proportion as the series of triangles 

 extended. 



