36 On Jinding the Longitude hij Lunar Observations. 



when sensibly affected by the distortion of the disk. Now the 

 position of this point is evidently of more importance than that 

 of the centre itself; because the centre is not necessarily the ap- 

 parent place of the angular point of the triangle. 



Suppose the verticarsemidiameter oF the moon to be shortened 

 by the small space AB. Then if ,, 

 we take the distance of the moon's '^ 

 limb at E, from an object which 

 lies in the direction E F, this arc 

 E F will be perpendicular to the 

 limb at E, and if produced, will 

 cut the vertical diameter at some 

 point D, v/hich is below the cen- 

 tre C, bv a space nearly equ;d 

 A B. For tlie upper part of the 

 disk will not sensibly differ from 

 a circle whose radius is equal the 

 semidiameter parallel to the horizon, unless the altitude be very 

 small. If, therefore, we have observed the altitude of the upper 

 limb, the place of D (not the centre) is found by subtracting the 

 augmented semidiameter. But when the distance has been taken 

 from a part of the disk on a different side of the diameter which 

 is parallel to the horizon from the limb whose altitude has been 

 observed, then this method fails; for in the present case, had the 

 altitude of the lower limb been observed, and the semidiameter 

 added, it would have given the altitude of a point G as much 

 above the centre as it ought to be below. When this takes place, 

 recourse must be had to a table answering to the space DG. 

 From this it is evident, that when the apparent altitude of an 

 angular point of the spherical triangle is about 7°, it may some- 

 times, in the common way of workintr, be erroneous by 36"; 

 and even in the method hitherto used for remedying this evil, the 

 error will rarely be less than IS" at the altitude of 7°- But 

 both are much more considerable when the altitude is smaller. 



The method referred to, for correcting the altitudes for the 

 distortion of the disk, by a table answering to the diminution of 

 the vertical semidiameter, is thus productive of an evil little in- 

 ferior to the one i. was intended to remove ; for although by it 

 the true altitude of the centre were obtained, and also its appa- 

 rent altitude; yet, as has been remarked, the centre is seldom 

 the point from which the apparent distance should be reckoned; 

 so that what is gained in getting the true altitude of the centre 

 is just lost in departing from the point from which the distance 

 should be taken ; and to which, as we have already seen, the com- 

 mon way of working often makes a much nearer approach. 



With regard to the line D E, which some learned authors have 



been 



