202 PHILOSOPHICAL TRANSACTIONS. [aNNO I766. 



tan. MAN 



AM (sin. me) : : tan. man : co-tan. amn : hence — '■ X sin. me = co-tan. 



AMN ; but tan. man being the tangent of the latitude of the given place e, and 



therefore given, the quantity — '■ will also be given, and greater or less 



than unity in the proportion of the tan. of the latitude to the b. The co-tan. 

 therefore of the angle amn, that is the tan. of the complement of the angle 

 AMN to 90°, will be greater or less than the sine of the arc me, in the propor- 

 tion of the tan. of the latitude of the place, to the r. And consequently, while 

 the arc me is small (in which case the sine, arc, and tangent differ very little 

 from each other) the angular deviation of the intersection of the meridian plmn 

 with the great circle ameb, from a right angle, will contain more or fewer de- 

 grees, &c. than the arc me, nearly in the same proportion of the tan. of the 

 latitude of the place to the r. 



By this means then, the latitude of the place and the angle pme (contained 

 between the meridian pmn and the great circle amb) being given, the length of 

 the arc me will likewise be given, with great exactness. But as the angles pem 

 and PME must be taken by the observation of some star near the pole, they will 

 be less accurate, when reduced to the plane of the horizon, than at the pole, 

 in the proportion of the sine of the distance between the pole and zenith, that 

 is the cos. of the latitude to the r, which with the proportion just mentioned of 

 the tan. of the latitude to the r, makes the accuracy of this method on the 

 whole, when compared with that of the measurement of a degree of the meri- 

 dian, in the proportion of the tan. multiplied into the cos. of the latitude, to 

 the square of the r very nearly; but the tan. of any angle into its cos. is equal 

 to the sin. into the r. whence this proportion is the same as the sin. into the r. 

 to the square of the r. and dividing both by the r. simply as the sin. of the lati- 

 tude, to the R. as above. 



Having got the length of the arc me, of a great circle, in degrees, &c. to- 

 gether with the distance of the two stations m and e, it is easy to conclude from 

 these the length of a degree of the parallel of latitude, at the place of observa- 

 tion, which will be the same, without sensible error, as it would be, supposing 

 the earth was an exact sphere, to the same scale, with the degree of a great 

 circle just found. 



For, in fig. 12, let apb represent a section of the earth through its axis pch; 

 ACB an equatorial diameter; ad the radius of curvature at the point a; and ph 

 the radius of curvature at the point p ; dph the evolute of the curve aep ; ep 

 the radius of curvature at the point e ; which suppose to have the same latitude 

 with the point e in fig. 1 1 ; and let ep be produced till it cuts the axis ph in g : 

 then with the radius eg and centre g, describe the arc iek, which will be the 

 least circle that can touch the curve aep at the point b, without cutting it. 



