OF *HE EFFECtS OF FARALtA* AttD REFRACTIOtf. 45 



a b X dc, wherefore the rectangle under ab and dc being found 

 ' and added to the fquare of b c or a d, which is fuppoled to be 

 known, the Aim will be equal to the fquare of b d, the chord 

 of the true diftance, which may confequently be determined. Th eorems for 



If the half chords, or fines of half the arcs be ufed initead clearing the 

 of the chords of the whole arcs, the true diftance will, from the d,fta " ce of 



... parallax and 



foregoing principles, be obtained by the following theorems, refraaion. 



1 . Sine* / app.dift.g&Qor* \ __ ^, Aliff. app. alt. f|& or * \ ^ Q 



2 Cos - M's t rue a,t - Co-. or *'s true alt. q _ 



(Jos. ©'s app. alt. Cos. or *.f% app. alt. 



3. Sine* ( diff - t^ah.^fe 0or* \ + fine , /t niedift. ffiSr Q or* \ 



If a table of chords, or of natural fines of the 2d power, Tables necefiary. 

 were conftrucled, the fquare numbers correfponding to certain 

 given arches, and converfely, the arches correfponding to cer- 

 tain given fquare numbers, to be employed in the folution of 

 the above theorems, might be had without farther trouble, by 

 infpe&ion. A fecond table might alfo be conftrufted fo as to 

 fit the 2* theorem, and Amplify the calculation for rinding </, 

 but this happens to have been already done, as Table IX, 

 with the (hort, additional, corrective Tables X. and XL. in the 

 3 d edition of the requifite tables fully anfvver this purpofe, by 

 readily furnifhing a logarithm, which added to that of Q gives 

 the logarithm of q. The nature of thefe tables is explained 

 near the end of that work. 



The method now propofed of folving the problem of the Advantages of 

 longitude, fo far as a folution depends on finding the true dif- 

 tance between the moon and the fun, or any given fixed fiar, 

 has feveral advantages, it \s conceived, over the common me- 

 thods, as the anfwer will in every poffible cafe, be determi- 

 nate; the diverfe affections of the fides of the fpherical tri- 

 angles commonly ufed never ocean* oning in this way any era- 

 baraffment to the operator, and as there never happens any 

 change of figns from -J- to — , or from — to -f-, in any part 

 of the calculation, it is plain the operation mufi always be clear 

 of ambiguity. It will alfo be found on trial, that the nature 

 of the numbers employed in this mode of calculation tends 

 very much to preclude fmall errors, or to render their effects 

 infenfible. A table of logarithms of natural numbers, a table 

 of the fquares of chords, and Table IX, with, its corrections 



from 



