t 163 ] 



XXX. On Mr. WitchelPs Method of clearing a Lunar Di- 

 stance. By C. RUMKER, Esq.* 



T HAVE remarked that a very imperfect approximate fo- 

 -- reign method for clearing the lunar distance (under some cir- 

 cumstances liable to considerable errors) is now much in vogue 

 amongst British mariners, although they have better methods 

 of their own: amongst which WitchelPs appears to me one of 

 the best. I think approximate methods better calculated for 

 mariners than direct ones, since small errors are more likely to 

 vitiate the result and more easily escape discovery than in the 

 former, where the computer after a little practice can nearly 

 judge from the altitudes and distance what each correction 

 will amount to: and WitchelPs enables him morever to assign 

 to himself by a rough sketch the reasons for his proceedings. 

 But as analytical demonstrations are now more approved, I 

 offer you the following one of his formula, preceded by a simpler 

 practical rule than the one usually given. 

 Add together the logarithms of, 



Cotangent of half the sum of both apparent altitudes. 



Tangent of half their difference. 



Cotangent of half the apparent distance. 



The sum of these logarithms is the tangent of an arc A, which 

 must be added to half the apparent distance, and also sub- 

 tracted from it. Then add together the logarithms of, 



Cotangent of the sum of A and half distance. 



Cotangent of the lesser apparent altitude. 



Proportional logarithm of the corresponding correction. 



Cotangent of the difference of A and half distance. 

 Cotangent of the greater apparent altitude. 

 Proportional logarithm of the corresponding correction. 



The sums are the proportional logarithms of two corrections 

 in distance, whereof the difference must be subtracted f from 

 the apparent distance as long as A is less than half the appa- 

 rent distance ; but if A is greater, their sum must be added 

 to the apparent distance if the moon's altitude is greatest, but 

 subtracted therefrom if that altitude is least. With this cor- 

 rected distance find from Table XXXV. of Norie's Req. Tables, 

 the corrections answering to the moon's correction in altitude 

 and in distance: their difference added to the corrected distance 



* Communicated by the Author. 



f- I here suppose that the correction in distance depending on the 

 moon's altitude is greater than that from the sun. In the very rare con- 

 trary case their difference must be added ^li*n A is l^ss than half distance. 



if 



