MR WALLACE'S Formula. 149 



For example, let the sine be required, corresponding to the 

 cosine 9.9450915962. Using VLACQ'S Arithmetica Logarithmica, 

 it appears that the given number is between the cosines of 

 28 12' and 28 13'. The differences between the cosines and 

 between the sines of these angles are now to be taken, and also 

 the difference between the given cosine and the next greater in 

 the Table, as follows ; 



cos 28 12' 9.9451254712 sin 9.6744484704 cos 28 12' 9.9451254712 



cos 28 18 9.9450577094 sin 9-6746839948 given cos 9-9450915962 



Differences, 677618 2355244 338750 



and this proportion stated : 



677618 : 2355244 : : 338750 : correction. 



The fourth term, or correction, comes out 1177417, which, added 

 to the sine of 28 12', gives 9.6745662121 for the sine required. 

 This, however, is only accurate in the first seven figures, the 

 true value being 9.6745662532, and the error .0000000411. We 

 have therefore lost the advantage which may be derived from 

 the logarithms in these Tables being carried to ten figures in- 

 stead of seven, the common number. 



Having been led to this subject, by what I consider an im- 

 provement in the mode of resolving a case in Plane Trigonome- 

 try, which will form the subject of a separate paper, I have inves- 

 tigated rules for deducing the logarithms of Trigonometrical func- 

 tions from one another. The formulas are indeed only approxima- 

 tions, but they are of the second degree, and therefore sufficient- 

 ly accurate ; and from their nature, they are well adapted to lo- 

 garithmic calculation. As they appear to me to be new, and 

 to possess considerable analytic elegance by their simplicity and 

 compactness, I have ventured to lay them before the Royal So- 

 ciety. 



