154 PHILOSOPHICAL TRANSACTIONS. [aNNO 17^7' 



berg, be observed by means of the pole star corrected for its distance for the 

 day; and suppose the first to be 75° 10', and the last 125° 5'; thus we shall 

 have 2 spherical triangles to compute, in each of which, 2 sides and the con- 

 tained angle are known, and one side, viz. the co-latitude, is common to both. 

 Now from these data, making use of the half sum and half difference of the 

 sides, we shall have the angles in these 2 triangles, and the angle of longitude 

 between Bottle-hill and the Brunemberg, equal to that at the pole, will be found 

 to be l°55'56''.l. If from this angle we deduct about 30'' or 35", for the 

 space that the Bottle-hill seems to be to the westward of the meridian of Green- 

 wich, there will then remain 1° 55' 21" for the east longitude of the Bru- 

 nemberg. 



As far as we are enabled to judge at present, from the examination of the 

 divisions of Mr. Ramsden's instrument, there is every reason to believe, that in 

 taking angles around the horizon, the mean of several repetitions of the same 

 angle, as referred to different parts of the circumference of the circle, will 

 differ very little from the truth, so little indeed, that in many cases the error 

 will totally vanish. But in elevating the telescope towards the pole, let us 

 suppose that an error of 5" on each of the contained angles at Tatterlees has 

 been committed; and further, that even an error of 5" of latitude, equal to 

 about 84-i- fathoms on the meridian, may have been fallen into, in estimating 

 the co-latitude (which never can happen, but is only here admitted, to place the 

 example in the most disadvantageous circumstances possible;) then whoever will 

 recompute the 2 triangles with these new data, will find the result in longitude 

 not to be varied, in the first case above J-th part of a second, or ^th part of a 

 second in time; and in the last not quite l", or -rV^h part of a second in time. 

 Hence I conclude, that the best mode of determining the differences of longi- 

 tude will be by the instrument itself, applied in this way^, in taking the angles 

 between the pole star and very remote stations, distinguishable at night by the 

 help of the Indian lights, and whose distance is accurately known. Thia 

 method will, it is true, be liable, as well as astronomical observations, to the 

 imperfections of the instrument, particularly those of the telescope, and the 

 unavoidable error in its application; but, on the other hand, it will be entirely 

 free from the irregularities of clocks, and the imperfections of vision in marking 

 the instantaneous explosion of light. When both methods have been repeated 

 a sufficient number of times, with all imaginable care, we shall then, and not 

 till then, be able to iudge to which the preference may be due. Thus 5 or 6 

 long stations, in or nearly in the parallel of Greenwich, such, for instance, as 

 that of Shooter'S'hill Tower, would reach from the east quite to the west of 

 the island: and as a very considerable degree of consistency might be expected 

 among the results for equal portions of the parallel, this method seems to be as 



