4 MATHEMATICAL PROBLEM. 



1.1. 2—^*6 , .5— 4 fir* ft . 14— 155^*6+3^*6*— :j*c , 



— ^ — --4 1 _ \ : 1- 



g g* g* g^ g^ 



42-.56^*6 + 28^H~6.^*o— 7g*6 « u . ■ .V. 

 2 : 2 — , &c. ; but V IS the com- 

 fit 

 plement of z; therefore zzzQ — y — 1'570796 — v nearly. 



Example to (H) For example, let a rz 9 X Q ; then g zi 10 X Q — 



the last para- i -i 2 "^^ b 



graph. 15-70796; whence —r: '063662; -,—-000266; — -f— — 



g g^ 5 



'000089. Stopping here we get w z: — + -^ -i 2 — ~ 



- ^^ " ~ g g y 



•063839; but arc C Pr:Q — u =: 1-506957 : dividing this 

 by '017453 the length of a degree to the radius unity, we 

 get 86° 20' 37" for the angle subtended by the arc C P, 

 or c. 



»inthelastar- (K) Since ^zr^f— « = 15*644121 ; and q=v + -—+— &c. 

 tide IS too J 15 



great. z:'063992; we have tyi^qzzl-OOlOQS, &c. ; but t qzzl uni- 



versally, which shows, that g — v exceeds the true value of 



/, or — — 15*62386, &c. ; therefore v is a httle too great, 



which makes z too little : but it is to be remembered, that 

 the true place of P has been nearly found in an even qua- 

 drant. - : 

 a found when (L) It appears from (F), that we may put ^—w s ; more- 

 t = nz. over, we have (by trigohometVy) ^f — r^zrl, henCe qzzz 



— ; put 2r— Q — y, Q being a quadrant or 1*570796; also, 



put 9 — u-f 6i;^ + ct)' + da;', &c., as in (G), and we have 



Qv — v^-\-Qbv^ — hv^-^-Qcv^ — cv^ hc.~. — ; and by re- 



n 



II 2-Q^6 

 5 — 4Q*6 



versmg the series, we get v zz. 1 -z — :; 4- — h 



^ ' ^^*» v'here ?i is always greater than unity; but 

 ^ ft 



when r is known, z is given =:Q>— v. 



Example to (M,) Put nrz 100, then — - — '000636, —7—5. =: *000028 ; 



the last para- Qn , • Q^ tr 



graph. stopping here, we get rr:-000664, and 5;= 1-570132, or 89* 



57' 4b". 



(N) When 



