Measurements of different Portions of the Meridian. 347 



or, in numbers 60856 (1— -001629) = 60757 fathoms, which 

 is only one fathom less than the observed quantity. 



There is also another measurement with which we may 

 compare the elliptical elements that have been found. A por- 

 tion of the parallel to the equator in latitude 45° 43' 12", little 

 short of 13° in amplitude, has been measured, and the most 

 probable length of 1° of longitude on that parallel has been 

 fixed at 77865 m, 75 # . The large arc was measured in six 

 portions, the difference of longitude between the extremities 

 of every portion .being ascertained by means of signals made 

 by exploding gunpowder. If we compare the six degrees, 

 deduced from the several partial arcs, it must be allowed that 

 their differences are great and irregular f. There must there- 

 fore be considerable uncertainty in the arithmetical mean of 

 such irregular quantities; which mean is only three metres 

 less than the most probable value, of a degree. The length of 

 a degree deduced from the four partial arcs contained within 

 the territory of France, which together exceed 7° of longi- 

 tude, is 77885 m, 75, or 20 metres more than the length de- 

 duced from the whole arc of about 13° %. It is to be observed 

 that the errors of longitude at the intermediate stations accu- 

 mulate as the arc is extended ; so that there is not the same 

 advantage in measuring a large arc of the parallel, as in the 

 case of the meridian ; for, in one instance, the total amplitude 

 is affected by the sum of the errors at all the intermediate 

 stations ; whereas, in the other, it is affected only by the errors 

 of the two extreme latitudes. If we further add that a great 

 length of the parallel answers to a minute portion of time ; in 

 so much that an error of a single second in the amplitude of 

 the large arc, would produce a variation of no small magni- 

 tude in the mean degree of longitude ; it must be evident that 

 in point of accuracy, we cannot repose the same confidence in 

 a degree of the parallel measured in the manner described, as 

 we can in a degree of the meridian deduced from an arc of 

 nearly the same amplitude. 



The radius of the parallel to the equator at the latitude A, 

 or the perpendicular drawn to the polar axis, will be found, 

 by the properties of the ellipse, equal to 



a cos X a cos A 



And if this expression be expanded and multiplied by — , we 

 shall have the length of a degree of the parallel equal to 



a. * i i i ' <> * . •/ 3 sin 4 A. sin f *. \ 1 



A cos X < 1 -f e sin 2 A + s ( — -z 3 — ) { • 



* Conn, dcs Terns 1829, p. 291. t Ibid. p. 290. J Ibid. p. 293. 



2 Y 2 . Now, 



