MAT II KM. \TICS. TRIGONOMETRY. 



[LOGARITHMIC TABLES. 



li. :..',. 



taking the logarithms on both rides. 



4n - m' 

 :-<f + l* 



Log. sin.^. r- kg. - 



- log. m log. n + log. ir - log. 2 + log. (4m 2 TO*) 

 - log. 4 J 



/I 1 1 .' 1. 1 m \ 



" [& + -2 4*' n* "" 3 4' n ~" " j 



M p mi + Li- t + 1 - 1 m " + ^ 



I n> """ ' "j 



Mf 1 -'.I i*j.L J \ 



M ^8"n' + 2-8'u* + 3'"- " "* ' J 



A'O. 



Where M is the modulus of the common logarithms. 



m IT 

 .'.Log. sin. -.y- 



log. m -f log. (2 m) + log. (2n -f- 3) 3 log. n 

 -f log. log. 8. 



, 



6* "*" 8 1 



10* 





- + g 6 - + go + 10 + ' 



Now log. TT log. 8 9-594059885702190 10 

 And M = -434294481903:>.YJ 



Hence reducing the coefficients of 5, ^ 



to num- 



bers (in the same manner as we calculated the value of 

 v in article 53), and remembering that L. sin. 90 



= log. sin. 90 + 10 we obtain 



L. sin. -- 90 = log. m + log. (2n m) + log. ( 2 + ) 

 3 log. n. 



-^f X 070022820605902- 

 * 



+ 9-594059885702190. 



n* 



X 0-001117206441662 



~ X -000039229146454 - * X 0-000001729270798 



- m ] a X 000000084362986 -^-!x 0-000000004348716 

 n lu n 1 * 



_!?_ x 0-000000000231931- ,.X 0-000000000012659 



fl ''' ' 



il B rt O 



_" x 000000000000703-'"-- XO 000000000000040 



11 H 



- Ac. 



It can be proved in the same manner, by means of the 

 expression 



Cos. x-=(l- 4x *\ fl- J) that 



L cos. !?90-10 +log.(n m)+log.(n + m) 21og n 



terms involving ^ , <tc. , similar to those in L. sin. -90 



n* n* n 



By means of the former of these expressions we may 

 calculate logarithmic linos of angles from 45 to 90, or 



by means of the latter logarithmic cosines of angles 

 from to 45, which clearly comes to the same thing. 

 \\> c in then calculate remaining logarithms by means 

 of the formula 



L. sin. A-L. sin. 2A L. cos. A + 9 69897000433C019 

 To prove this formula 



Since sin. 2A = 2 sin. A cos. A 



log. sin. 2A log. 2 -f log. sin. A + log. cos. A 

 .'. L. sin. 2A = L. sin. A + L. cos. A + log. 2 10. 

 An.l 10 log. 2 = 9-698970004336019. 



Hence the formula. 



AVo may also employ the formula for L sin. 90 to 



calculate the L. Fines fur angles of lariro intervals, as for 

 1 2 3 .... and OK n airily the method of interpolations 

 referred to in artii-lu iiT. 



(69). Hfms. 



If wo take a table of logarithms of numbers we shall 

 observe that the numbers run quite regularly, viz., 

 10000, 10001, 1000L', A'- , and the corresponding loga- 

 rithms are entered in the table. But if we take a table 

 of logarithms of gines, Arc., we shall find that the loga- 

 rithms of the sines and tangents of small angles are given 

 for much smaller divisions of the angle than of the larger 

 angles. Thus, in Hulsst-'s edition of Vega's '/'.<'<''., we 

 find the logarithmic sines, cosines, tangents, and cotan- 

 gents for every minute from 6 0' up to 45 0', while 

 they are given for every 10" from ty 0", up to 6 0' 0*; 

 and to this general table are prefixed two others, the 

 former giving the sines and tangents for every tenth 

 part of a second from O'O" up to 0' 59". 9, the latter 

 giving the sines and tangents for intervals of 1" from 

 0' 0" up to 1 2y 59"; the need for these minute 

 calculations appears from the following considerations: 



To show that 



where n is a small number, 

 we have 



Log. sin. (fl+ f)-log. sin. 0=log. 



= log. (cos. + cotan. sin. ). 

 Now if 6 be small, so that we may omit e 3 .... * 



* In this aud the following articles are several in- 

 stanees of approximation which must be carefully at- 

 tended to. \Ve say if S be so small that we may 



omit f, 3 and all the higher powers of 5, then cos. S = 



M 

 1 and sin. 8 = S. For this, see cor. to article (50). 



We then obtain 



log. sin. (0 + )-log.sin. 0=M [ (cotan. 0. ^-^' 



which is of course =M (cotan. 0.3- -le 2 cotan. 1 

 \ 22 



-L - ootan. 0. 3- ^ + <tc. ) 

 2 8 / 



which =M | cotan. ft^-| & (1 + cotan.' ff) j 



if we omit f 3 , *, <bc., and hence the result of the text. 



So again in another article. If we omit 0*, and all 

 higher powers of 0. we have 



0> . sin. 0* 



w 



Now 



/ 02 



( 1 c 



0* 



by the Binomial Theorem 



.'.(l | 2N \i= 1 ^ if we omit 0* &o. 



