Bowditch^s Remarhs on BoctoP Stewarfs Formula, 117 



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stance it happens that the numerical values of the forrautas (7), 

 (9) become nearly equal in tJiis particular case, although their 



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general analytical values are essentially different from each other. 

 I have not thought it necessary to enter into a discussion of other 

 parts of Doctor Stewart's work, since this defect in the most 

 important theorem makes the method wholly fail: but before clcs- 

 ing this paper, it may be proper to demonstrate what was remark- 

 ed above, that by following implicitly his method, the sun's com- 

 puted distance would have been infinite, if the moon had been 

 situated a little nearer to the earth. 



I shall refer to Prop 11, of Doctor Stewart's essay on the 

 sun's distance, using the figure corresponding thereto. The two 

 chief numbers mentioned in the first part of that Proposition are 

 178.725 and 357.43365, which for perspicuity I shall denote by a. 

 and h respectively. Then by going over the calculation of Prop, 

 11, precisely in the same manner as Doctor Stewart has done, It 

 will be found that AB : Bs : : 178.20^ 95 (=4a4 3&; : 0.0i633 



3 a — &), so that if w was assumed of such a value as to make 

 Sa — 6=0, the quantity B^, which represents the versed sine 

 of the arch Bv would be nothing, consequently the correspond- 

 ing sine vs would be nothing, and since by construction 

 BD = -f- • AB, the ratio BD to v s would be infinite. Bu: it is 

 said (in page 58 of that essay) that the ratio of the mean distance 



of the sun from the earth to the mean distance of the moon from 

 the earth is srreafer than that of — ; hence it would follow, that 



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the sun's distance from the earth would be greater than infinite 

 If 2 a — b= 0. Now this equation could be easily satisfied by 

 decreasing a little the value of m. For, by page 5-7 o', the same 



work, a = -j, and if we suppoae the motion of the moon from 



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