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X. Investigation of Formula, for finding the Logarithms of Tri- 

 gonometrical Quantities from one another. By WILLIAM 

 WALLACE, F. R. S. Edin. and Professor of Mathematics in 

 the University of Edinburgh. 



(Read November 3. 



-I.HE most simple and obvious use of the Trigonometrical 

 Tables, is to find the logarithmic sine, tangent, &c. correspond- 

 ing to a given angle ; and, reversely, the angle corresponding to 

 a given sine or tangent. However, in their more general appli- 

 cation, we have often to find a logarithmic cosine, having given 

 the corresponding sine, or the contrary ; also the sine or cosine 

 from a given tangent, and, in these cases, it may be of no conse- 

 quence to know the exact angle. 



Supposing a logarithmic cosine to be given, to find the sine, 

 it is usual, first, to find the angle from the cosine, and then the 

 sine from the angle *. In this way, as the given cosine may not 

 be found exactly in the Table, the differences must be taken, and 

 two proportions made to find a correction, to be added to the ta- 

 bular sine next less to that which is required. Indeed the two 

 proportions may be brought into one, by leaving out the ratio of 

 the angles, which is common to both, and introduced unnecessa- 

 rily ; but, in either way, the result is an approximation, only of 

 the first degree, near enough, indeed, for most purposes, but 

 which, in certain cases, may not be sufficiently correct. 



* CAGNOLI Trigonometric, Art. 428. 2de Edition. 



