SECT. VI. ARCS OF THE MERIDIAN. 47 



exactly in proportion as the curvature is less. A comparison of 

 the length of a degree in different parts of the earth's surface will 

 therefore determine its size and form. 



An arc of the meridian may be measured by determining 

 the latitude of its extreme points by astronomical observations 

 (N. 125), and then measuring the distance between them in 

 feet or fathoms. The distance thus determined on the surface 

 of the earth, divided by the degrees and parts of a degree 

 contained in the difference of the latitudes, will give the exact 

 length of one degree, the difference of the latitudes being 

 the angle contained between the verticals at the extremities 

 of the arc. This would be easily accomplished were the distance 

 unobstructed and on a level with the sea. But, on account 

 of the innumerable obstacles on the surface of the earth, it is 

 necessary to connect the extreme points of the arc by a series 

 of triangles (N. 126), the sides and angles of which are either 

 measured or computed, so that the length of the arc is ascertained 

 with much laborious calculation. In consequence of the irre- 

 gularities of the surface each triangle is in a different plane. 

 They must therefore be reduced by computation to what they 

 would have been had they been measured on the surface of the 

 sea. And, as the earth may in this case be esteemed spherical, 

 they require a correction to reduce them to spherical triangles. 

 The officers who conducted the trigonometrical survey, in mea- 

 suring 500 feet of a base in Ireland twice over, found that the 

 difference in the two measurements did not amount to the 800th 

 part of an inch ; and in the General Survey of Great Britain, five 

 bases were measured from 5 to 7 miles long, and some of them 

 400 miles apart, yet, when connected by series of triangles, the 

 measured and computed lengths did not^ differ by more than 

 3 inches, an unparalleled degree of accuracy ; but such is the 

 accuracy with which these operations are conducted. 



Arcs of the meridian have been measured in a variety of lati- 

 tudes in both hemispheres, as well as arcs perpendicular to the 

 meridian. From these measurements it appears that the length 

 of the degrees increases from the equator to the poles, nearly in 

 proportion to the square of the sine of the latitude (N. 127). 

 Consequently, the convexity of the earth diminishes from the 

 equator to the poles. 



Were the earth an ellipsoid of revolution, the meridians would 

 be ellipses whose lesser axes would coincide with the axis of 



