On reducing the Lunar Distance. 275 



effect of refraction on tlie same distance, whenever the less al- 

 titude falls on that parallel. 



The use of a linear table constructed in this form for show- 

 ing the effect of refraction, is therefore very simple : for hav- 

 ing found on what parallel the less altitude falls, trace that 

 upward, and its segment intercepted between BC and the 

 curve corresponding to the given distance, is the effect re- 

 quired, which is always additive. This table is not quite half 

 die size of that formerly described, when the corrections in 

 both are on the same scale, and there is sufficient room for 

 inserting it above M N in the table for parallax. If small al- 

 titudes are not to be used, a considerable part of the figure 

 next A C might be omitted, which would still further reduce 

 the size. 



It was already remarked that on the same distance, the 

 effect of refraction is nearly the same for any two altitudes 

 whose sines have the same ratio ; and to render the above ex- 

 planation more simple, I only introduced lines of sines. But 

 when the sines are proportionals, their like powers or roots 

 are also proportionals ; and therefore the effect of refraction 

 must be nearly the same when the like powers or roots of the 

 sines have the same ratio. Hence, if in place of lines of sines 

 we use their square roots the above construction would be 

 much improved : for by this means the curves being less 

 crowded would meet the parallels much less obliquely when 

 the effect of refraction is great ; and thus the different parts 

 of the scale from which the correction is to be read would 

 be better proportioned. But I have not yet ascertained what 

 root of the sines would answer best. It was however to effect 

 the same end in some degree, that the lines were to be drawn 

 diverging from D, and not parallel to A C. 



I siiali now j)roceed to describe the outline of a very diffe- 

 rent contrivance for clearing the distance ; if this has not every 

 property that the other possessed, it is much more easily con- 

 structed, it does not re(juire a ruler laid across, and when of 

 the same size as the foruier it shows the correction for paral- 

 lax on twice as great a scale. 



In such a spherical triangle as is formed by the distance and 

 the complements of the altitudes, it is in eflect demonstrated 

 by almost every writer on spherics, that a pcr]H'iuliciilar from 

 the zenith being let fall on the distance, divides it into two 

 segments whose cosines are as the sines of the altitudes; and 

 also, that the tangents of these segments are respectively pro- 

 porti(«ial to the cotangents of the altitudes multiplied by the 

 cosines of the adjacent angles at the base; lliat is, nearly as 

 the efll'Cts of the refractions on the <iistanct'. \\\ therefore, a 

 M m 2 raadv 



