and the Value of increasing Annuities, &c.&c. VSI 



05037 

 ^5 



•00037 



17-0745 17-0745---860184 ( = -05037 which ex- 



37 



ceeds the true rate or 5 per cent, by the ■■ part of itself. 



The present value of an annuity of 1/. for fifty years is 

 18,25592^ — what is the rate of interest? 



In this case also we see that the rate is exactly -05 ; but let it 

 be supposed, as before, either -0475 or -052 : then putting Z' = 

 •0403, and making use of the second formula, we find 



In these two instances, selected promiscuously, it will be seen 

 that the formula is rather erroneous in point of excess j and this 

 in general will be the case when there is much difference be- 

 tween the assumed rates : hence, under such circumstances, if 

 the rate appears to consist of an integer and some even frac- 

 tional quantity, we may properly take such rate, and fractional 

 part or that nearest it, for the exact rate per cent., rejecting all 

 the succeeding decimal places. 



But this is not necessary; since from the tables we can always 

 deduce approximate rates nearer than those above, and the value 

 of r thereby will in every case be rendered extremely correct. 



The rate of interest may also be pretty correctly obtained by 

 the aid of trigonometrical lines, sines, and tangents. 



To do this let m = - — : v — -, and assume ^— ' = 



''Z' 



tang^O - — = sine* O. 



Then log tang' O = -^(20 + log 2 + log n + 1 +IogZ-log 6) 



log sine 0= -i-(20-f-log2-|-logn-l-|-logZ'— log6) 

 And in case of an amount 



log r= (log m + log tang' + log tang' - — 20) ; 

 And in the present worth 



log rs: (log 2i;-f- 2 log sine — — 20) . 



