250 PHILOSOPHICAL TRANSACTIONS. [aNNO 1787. 



at the polar circle would be defective near 217 fathoms, and consequently on 

 8°4- the error would be 1 807 fathoms. 



The 7th or last ellipsoid, being that of the least flattening, has for the ratio 

 of its semi-diameters 540 to 539. The arc mp should contain 17106 fathoms. 

 The 45th degree of latitude being adhered to as the standard, the arc m Perpig- 

 nan would only exceed the truth by 46 fattioms ; but, on the other hand, the 

 degree at the equator erring in excess 1244- fathoms, and that at the polar circle 

 being defective near 303 ; therefore, in the first case, the error on 8*^4- would be 

 1037, and in the last 2524 fathoms. Hence it is obvious, that the arcs of an 

 ellipsoid, however great or small the degree of its oblateness may be, will not 

 any way correspond with the measured portions of the surface of the earth : for 

 if we retain the length of M. Bouguer's degree at the equator as the standard, 

 and make the ellipsoid extremely flat, as in N° 1, the figure will become too 

 prominent in middle latitudes, that is, the curve will rise above the real surface 

 of the earth, and in proportion to the excess of the radius, will always give de- 

 grees that exceed the measured length. On the contrary, if we give the ellip- 

 soid a small degree of flatness, as in N° 7? and adopt the measured length of 

 the 45th degree as the standard, the measured and computed arcs will nearly 

 agree in middle latitudes ; but at the equator the curve will rise very considerably 

 above the surface, and will there give degrees that are too great ; while at the 

 polar circle it will fall below it, and give degrees that are too little in the propor- 

 tion of about 24- to 1 compared with the error at the equator. From all which 

 we may conclude, that the earth is not an ellipsoid. 



The 2 columns towards the right-hand of the table, contain the arcs of 2 

 spheroids differing from the ellipsoid. The first is that adopted by M. Bouguer 

 as his first hypothesis, where the increments to the degrees of the meridian 

 above that at the equator follow the ratio of the 2d power or squares of the sines 

 of the latitudes, and to which he has suited his first table of degrees, N° 32, 

 p. 298. This spheroid differs but insensibly from the 4th ellipsoid. They have 

 both the same semi-diameters ; but the arcs of the spheroid being somewhat 

 longer than those of the ellipsoid, the former thus becomes, in a trifling degree, 

 more prominent in middle latitudes. On this hypothesis the arc mp should 

 be in length 27295 fathoms; m Perpignan exceeds the measurement 11 96 

 fathoms ; and the degree at the equator being adhered to as the standard, the 

 45th errs in excess 118, while that at the polar circle is defective only 20 

 fathoms. 



The 2d spheroid is that on which M. Bouguer founded his 2d hypothesis 

 which supposes the increments to the degrees of the meridian, above that at the 

 equator, to follow the ratio of the 4th power or squared squares of the sines of 



