On reduciag the Lunar Distance. 277 



stances is to be taken ; because only one of them (the sum or 

 difference) can be equal the distance, unless when one of the 

 remainmg zenith distances is nothing ; and the sum or differ- 

 ence of the refi-actions is to be taken according as the sum or 

 difference of the zenith distances was used. The case that 

 recjuires their difference is easily known in following up the 

 altitudes, since the greater altitude will then seem to go above 

 the top before their sum could be so small as the distance. 



Next, to find the effect of parallax : Obsen-e where the lio- 

 rizontal line on which the effect of the moon's refraction was 

 found cuts the circle A D ; from this point trace down an 

 oblique line till it cut the parallel of the moon's altitude, and 

 the segment of this parallel reckoned from C D, is the princi- 

 pal effect of parallax in terms of a horizontal parallax of 60'. 

 For the tangent of segment of distance adjacent to the moon, 

 is equal the cotangent of the moon's altitude multiplied by co- 

 sine of angle at the moon ; and the quantity we have obtained 

 is obviously equal to tangent of the above segment multiplied 

 by sine moon's altitude ; that is, to cosine of angle at the 

 moon multiplied by cosine of her altitude. The above effect 

 is easily reduced to suit any horizontal parallax by the com- 

 mon rule of practice, as was directed in the other method ; and 

 the final effect of parallax may be allowed for, when it is 

 deemed necessary, by half correcting the altitudes and distance, 

 and then repeating the operation to find the true effect of pa- 

 rallax ; though this cannot be done quite so conveniently here 

 as in the other method. The correction for parallax is addi- 

 tive, when the difference of the refractions is used and the 

 moon at tlie same time the higher body. In all other cases it 

 is subtractive. 



It may be observed, that in place of a square, A B D C 

 might have been an oblong, and AD the quadrant of an ellipse. 

 Nor is it absolutely necessary that the vertical lines should be 

 lines of sines, or lines of their })owers ; they miglit, on the 

 contrary, be divided in any projjortion that may be thought 

 to suit better; but then tlie lines diverging from C could not 

 always be straight lines, neither could A D be a circular el- 

 lipse. If, however, the rectilineal part of such a diagram as 

 that already described were first constructed, it would be easy 

 from it to draw another in which the vertical lines were di- 

 vided in any other proportion. For having joined tlie like 

 points of a different division in another set of vertical lines, 

 transfer the divisions ol'the horizontal lines in the f()rnier figure 

 tothecorixsp()ndingc(|ual lines in the new figure, and through 

 lh(."-.e pohUs (haw cm ve 'lines which, in theory at least, will 



answer 



