On finding ike Longitude ly Lunar Observations. 37 



been pleased to consider as shortened by refraction, the mistake 

 has no doubt arisen from the old established habit of always esti- 

 mating the apparent distance from the centre. But if we sup- 

 pose the disk to be elliptical*, then it may be shown, that if from 

 any point of either axis of an ellipsis but the centre, a straight 

 line be drawn perpendicular to tlie curve, — this line, sometimes 

 called the normal, is greater or less than half the other axis, ac- 

 cording as the point was taken in the conjugate or transverse axis^ 

 D E is, therefore, always greater than the greatest semidiameter; 

 so that it is obviously much safer just to account it equal the 

 augmented semidiameter than less, 



I do not mean to insinuate that I have put this subject be- 

 yond the possibility of improvement, or that there may not be 

 defects in the above way of considering it. My object is merely 

 to make a nearer approach to accuracy with as little additional 

 labour as possible. The method which I would therefore recom- 

 mend when the distance and altitude have both been measured 

 from parts of the limb which are either both above or both below 

 the centre — is, first, to find the true altitude of the limb, to which 

 the true or horizontal semidiameter being afterwards applied gives 

 the true altitude of the centre. Next to the observed altitude of 

 the limb apply the " agumented semidiameter," which will give 

 the place of D ; and then with these compute the true distance 

 as usual. But when the distance and altitude have been observed 

 from different sides of the diameter parallel to the horizon, it will 

 be necessary that the correction contained in several books for 

 the contraction of the tahole vertical diameter be subtracted from 

 the semidiameter which is to be applied to the apparent altitude 

 of the limb in order to get the place of D. 



It may be proper to anticipate an objection which might be 

 brought against this method — that when the angle at the sun or 

 moon is a right angle, the centre is the angular point of the tri- 

 angle ; and the above rule would then make the altitude erro- 

 neous by half the contraction. This cannot be denied : but it is 

 no less obvious, that a small alteration on the altitude at a right 

 angle will have no sensible effect on the distance, which is the 

 main thing to be attended to. If greater accuracy were required, 

 a careful determination of the figure of the disk would probably 

 point out the construction of a table answering more correctly to 

 D G, and which might be applicable to different angles formed 

 at the linninarics. A table could likewise be formed which would 

 give the excess of DE above the semidiameter for different angles 



• The figure of the sun or moon when near the horizon will more nearly 

 , consist of two semi-ellipHe!) having the tranx verse axis common to both ; but 

 the semi-conjugate axis of the up|)i;r half greater than that of the lower. It 

 is easy to see, that the above theorem applies on this suj)position also. 



C 3- and 



