73 8 
A N U I T I E S. 
Y = 2$ — 25 X '019800 — 24-5050 
H = 25 — 25 X -019053 = 24-5237 
Q_— 25 — 25 X -018683 = 24-5329 
M— 25 — 25 X -018435 = 24 - 539 * 1 
Hence the difference in the value by making pe¬ 
riods of payments fmaller, for any given term of 
years, is the more as the intervals are fmaller, or the 
periods more frequent. The fame difference is alfo 
variable, both as the rate of intereft varies, and alfo 
as the whole term of years n varies; and, for any 
given rate of intereft, it is evident that the diffe¬ 
rence, for any periods m of payments, firft increafes 
from nothing as the term n increafes, when n is o, 
to fome certain finite term or value of n, when the 
difference D is the greateft or a maximum; and that 
afterwards, as n increafes more, that difference will 
continually decreafe to nothing again, and vanifh 
when n is infinite : alfo the term or value of n , for 
the maximum of the difference, will be different ac¬ 
cording to the periods of payment, or value of m. 
And the general value of n, when the difference is a 
maximum between the yearly payments and the pay¬ 
ments of m times in a year, is expreffed by this for¬ 
mula, viz. 
ml. 
m-\- r — 1 
l. r 
m-\-r — 1 , 
vil. - -- — !• • r 
where /. denotes the lo¬ 
garithm of the quantity following it. Hence, taking 
the different values of m, viz. 2 for half-years, 4 for 
quarters, 12 for monthly payments, &c. and fubfh- 
tuting, in the general formula, the term or value of 
n for each cafe, when the difference in the prefect 
worths of the annuities, will be as follows, reckon- 
intereft at 4 per cent. viz. 
2 1 . 
1 . 
!±i 
2 
/. r 
2 1 . 
'■+ 1 
-l.r 
Years 
25*3777 
for half-yearly pay 
[ments, 
4/. 
2 
2-+3 
n — 
l. r 
4/ 
r +5 
. = 25-3200 for quarterly payments, 
12I 
4 
r-]-i 1 
-l.r 
l.- 
l.r 
ill 
r-f 11 
--25-2643 for monthly pay- 
_l r [ments. 
Annuities may alfo be confidered as in arrears, or 
as forborn, for any number of years; in which cafe 
each payment is tcube confidered as a fum put out 
to intereft for the remainder of the term after the 
time it becomes due. And as il. due at the end of 
1 year amounts to r at the end of another year, and 
to r“ at the end of the third year, and to r 3 at the 
end of the 4th year, and fo on; therefore by adding 
always the laft year’s annuity, or 1, to the amounts 
of all the former years, the fum of all the annuities 
and their interefts will be the fum of the following 
geometrical feries, 1 -p .to 
r n ~~\ continued till the laft term be r n ', or till 
the number of terms be n, the number of years the 
annuity is forborn. But the fum of this geometrical 
7* /l —I 
progreflion is-, which therefore is the amount of il. an. 
r — 1 
nuity forborn for n years. And this quantity being multiplied 
by any other annuity a, inftead of i, will produce the amount 
for that other annuity. 
But the. amounts of annuities, or their prefent values, are 
eafieft found by the two following Tables of numbers for the 
annuity of il. which are ready computed from the forewoin°- 
principles. 
TABLE I. 
The Amount of an Annuity of il .at Compound Interef. 
"n 
at 3 pot 
cent. 
34 P r i 4 P e ** 
cent, j coin. 
44 P' r 
cent. 
5 per 
cent. 
6 per 
cent. 
1 
2 
3 
4 
_5 
C 
1 
8 
9 
10 
11 
12 
13 
M 
12 
10 
‘7 
18 
l 9 
10 
1 ’00000 
2- O30Oc| 
3- 09090 
4- 18363 
5 ’ 3 ° 9 i 4 
I -coooo 
2- 03500 
3- 10623 
4*21494 
5-36247 
1 *00000 
2*04000 
3*12160 
4*24646 
5-41632 
I*00000 
2*04500 
5*13703 
4- 27819 
5- 47071 
I *00000 
^2*05000 
3*15250 
4-31013 
5*52563 
1 -oocoo 
2- o6ccO 
3- 18360 
4- 37462 
5- 63709 
6- 40841 
7- 46246 
8- 89234 
10-15911 
11 -46388 
0-55015 
7*77941 
9- 05169 
10- 36850 
11*73139 
0-03298 
7-89829 
9-21423 
105828c 
12-0061 i 
6-71689 
8- 01915 
9- 38001 
10-80211 
12-28821 
6-8019; 
8- 14201 
9- 54911 
11- 02656 
12- 57789 
6-97532 
8- 39384 
9- 89747 
11-49132 
13-18079 
12-80780 
14-19Z03 
i 77 y 
17- 0^632 
18- 59891 
13- 14199 
14- 60196 
ifr 11303 
1 7-67699 
19-29568 
1 3-48635 
15- 02501 
16- 02684 
18-29191 
2 o- 3 2 * 359 
13-8411 1> 
15*46403 
17- 15991 
18- 93211 
20-78405 
14- 20679 
15- 91713 
1771298 
19-59803 
21-57850 
14-97164 
16-86994 
18-88214 
21-01507 
23-27597 
20- 150?8 
21- 76159 
2 3 - 4 , 444 
25 -n 687 
2(4 *57° ^7 
20-97103 
22-70502 
24-49969 
26-35718 
z8-2796S 
21-82453 
23-6975: 
25-04541 
27-67123 
z 9 ' 7 ; 8 o 8 
22-71934 
24-74171 
26*85508 
29-00356 
31-37142 
23-0574, 
25*84037 
28-13238 
30-53900 
33*06595 
25-67253 
28-21288 
30-90565 
33*75999 
3 6 *7 559 
2! 
22 
z 3 
24 
£5 
26 
27 
z8 
29 
50 
28-07049 
3053678 
32-45288 
34-4264? 
36-45926 
30-26947 
32-12890 
34 ' 46 ° 4 ! 
36-66653 
38-9:986 
31-96920 
34*24797 
36-61789 
39-08260 
41*64591 
3 j* 783 i 4 
36-30335 
38-93703 
u-689zo 
44 ' 5 b 52 i 
35-71925 
38-5052; 
41*43048 
44-50200 
47-72711 
39*99273 
43 - 3922 , 
46-99583 
50-81558 
54-86451 
30-55304 
40-70963 
42-93092 
45-21885 
47*57542 
41-31310 
43 * 759 ° 6 
46-29063 
48-91080 
51-62268 
54*31174 
47-08421 
49-96758 
52-9662, 
56-08494 
47-57064 
50-71132 
53*99333 
57 * 4 2 3 P 3 
61 -00707 
5 '*i '345 
^4-66914 
58-40258 
62-3227 
66-4388; 
491 5 '° 3 b 
6 3 */o 57 * 
68-52811 
73-63980 
79-05819 
3 1 
3 2 
33 
34 
35 
50-00268 
52-50276 
55-07784 
57 * 73 ° J ^ 
60-46208 
54 - 4 2 947 
57 - 3345 0 
60-34121 
63-45315 
66-67401 
59 * 3 2 834 
62-70147 
66-20955 
69-85791 
73*65222 
6475239 
68-60625 
72-75623 
77-03026 
81-49662 
70-76070 
75-29883 
80-06377 
85*06696 
90-32031 
8 -I-- 8 O 108 
90-88978 
97*34316 
104-18375 
111 -43478 
3 6 
37 
38 
39 
4 ° 
63-27594 
66-17422 
69-15945 
72-23423 
75-40126 
70-00760 
7345787 
77 -02889 
80-72491 
84* 5 5° 2 8 
77*59831 
81-70225 
85 * 97°35 
90-40915 
95'oz552 
86-16397 
91-04134 
96-13820 
101-46442 
107*03032 
95-8.6;2 
: c 1 -62814 
10770955 
114-09502 
120-7997; 
119-12087 
127-26812 
135-90421 
145-05846 
154 -16197 
4 J 
42 
+3 
44 
45 
*6 
47 
48 
19 
5 ° 
5 < 
3 2 
53 
54 
78-66330 
82-02320 
85-48389 
89-04841 
927198b 
88-50954 
92-60737 
96*84863 
101-23833 
105-78167 
99*82654 
104-81960 
110-01238 
115-41288 
! 2 I -02939 
112-84669 
118-92479 
125-27640 
131-91384 
1 38-84977 
127-83976 
135*23175 
142-99334 
151-14301 
159-70016 
165-04708 
175-95054 
187-50758 
1997580- 
2t 2 * 7435 ' 
96-50146 
100-39650 
104-40840 
108-54065 
112-79687 
110-48403 
1 1 5 ' 3 S °97 
120-38826 
125-60185 
130-99791 
iz&- 87 o 57 
' 32*94535 
139*26321 
i 45*83373 
152-66708 
146-09821 
153-67263 
101-58790 
169-85936 
i 78 -' 5 ° 3°3 
168-68516 
178-11942 
188-02539 
193-4266 
209 348CC 
-20*506 i 1 
241-09861 
256*5645.- 
272*95840 
290*33'-A 
117-1807 7 
izi-6g6zo 
126-34708 
131-13750 
136-,58284 
142-36324 
148-34595 
154-53806 
159 * 7-7377 
167-16472 
174-85131 
182-84536 
187*53566 
196-97477 
206-83863 
217-14637 
220-SI540 
232-85617 
245 *49897 
758 - 7739 ? 
.08-75606 
328-28142 
348 97831 
370-91701 
TABLE 
