474 PHILOSOPHICAL TRANSACTIONS. [ANNO 1799. 



From a review of the inaccuracies to which this method, in particular cases, may 

 be liable, it appears that none of them can ever be of sufficient importance to affect 

 the mariner. If he only computes the time with each of the altitudes and the lati- 

 tude by account, and an incremental area with the greatest altitude and the former 

 latitude varied 10', the correction will generally be deduced within much less than 

 a second, and in the most unfavourable circumstances within a minute of the truth. 

 But the astronomer, in every instance, even when the latitude and declination are 

 nearly equal and of the same kind, by adopting the precautions which have been re- 

 commended, may be assured of a result perfectly exact. If however he should 

 entertain any doubts on this point, he might easily compute a second value of the 

 incremental area with the latitude already determined ; and this, it is evident, would 

 necessarily produce a conclusion not less accurate than if it were obtained from the 

 direct method. 



The most satisfactory way of proving the utility of this rule, will be to suppose 

 a particular latitude and declination ; with these to compute the altitudes, when the 

 sun is at 2 given distances from the meridian ; and thence to deduce the latitude, 

 by an application of our own principles. And for this purpose, Mr. L. here gives 

 4 several examples, the calculation of which he gives, at length; and then adds the 

 following remarks. 



1 . All the altitudes that are taken on the same side of noon, tend only to correct 

 the error which may be supposed to exist in the greatest of these altitudes, and can 

 have no effect in removing any inaccuracy to which the greatest altitude on the 

 other side of the meridian may be subject. Hence we must take more than one 

 altitude on each side of noon, if we are desirous of obtaining a very exact con- 

 clusion. 



2. When some of the observations are made in the morning, and others in the 

 afternoon, the smaller the hour- angle, in every instance, the more favourable it 

 will be for our purpose. But if we cannot procure an altitude on each side of the 

 meridian, we ought to make one observation when the hour-angle is as large as 

 possible, and with this all the rest should be separately combined. We must be 

 cautious however not to let the sun be too near the horizon, lest the apparent alti- 

 tude should be affected by the uncertainty of the refraction. 



3. If the clock were to furnish us with the true time, we might combine to- 

 gether any 2 observations made within the proper limits, without applying to the 

 first table, and deduce a very exact correction. Should there even be a small error 

 in the supposed time, we might still proceed in the same manner, without being 

 liable to any material inaccuracy, provided the difference between the hour-angles 

 was not very considerable. The error indeed occasioned by adopting this method 

 of finding the 2 areas, and taking a mean between them for the value of gb, may 

 easily be determined in any particular case. If t and t' be the respective tangents 

 of the smaller and greater hour-angles, and i their difference ; i the error of the 

 clock in minutes of time ; and i the whole error in the time computed with the 



