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becomes barely v= ; that is, any annuity divided by the

R-1 interest of 11. for 1 year, gives the value of the perpetuity. So, if the rate of interest be 5 per cent.

Then 1000 + 5 = 20a is the value of the perpetuity at 5 per cent : Also 100a • 4 = 25a is the value of the perpetuily at 4 per cent: And 100a 3 = 33ļa is the value of the perpetuity at 3 per cent: and so on.

Again, because the amount of ll. in n years, is Rn, its increase in that time will be R -1; but its interest for one single year, or the annuity answering to that increase, is R-1; therefore as R - 1 is to R" – 1, so is a to m; that

RD-1 is, m = -X a. Hence, the several cases relating to

R-1 Annuities in Arrear, will be resolved by the following equations :

R - 1

Xa= VR" ;

1
R" - 1
х
R"

1
x m =

X VR";
RO-1

RO-1

MR-mta

R

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R-1

R*

R

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R

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n

1.

1
g=(--

-) X

RP RN R-1 In this last theorem, q denotes the present value of an annuity in reversion, after n years, or noi con mencing till. after the first p years, being found by taking. the difference

RP-1 between the two values х and

Х

for 12 R-1 RO R1 RP years and years.

But the amount and present value of any annuity for any number of years, up to 21, will be most readil.

a

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11.2961 10.9205 10-5631 10-2228
119379 11:5174 11:1184 10.7396) 10-3797
12-56111 12-0941| 11.6523 11-2340 10-8378 | 101059
13-1661), 12-6513/ 12-1657 11.7072 112741 10 4773

13 7535 13-1897 12 6593 | 12 1600 | 11.6896| 108276 2014-87751 14.2124 13.5903) 13:0079 12:4622 11 4699 1541501 146980' 14 0292| 134047| 12.82121 11.7641}

TABLE I.
The Amount of an Annuity of 1.. at Compound Interest.
Yrs. at 3 perc. 34 nerc. 4 per c. 43 per C., 5 per c. 6 per c.

100001 1 0000 10000 1.0000 1.0000 10000
2
2-0300

2:0350 2 0400 2:0450 2:0500 2:0600 3:0909 3 1062 3-1216 3.1370 3:1525 3:1836 4 1836

4:2149 4.2465 4 2782 4.3101 4 37 46 5 5 3091

5 3625 5'4163 5.4707 5 5256 5.6371 6 6.4684

65502

6 6330 6.7169 6.8019 6.9753 76625

77794 708983 8:0192 8.1420 8 3938 8.8923

9:0517 9.2142 9.3800 9:5191 9 8975 10-1591 10°3685) 10-5828 10'8021 11.0266) 11:4913 10 11.4639 117314 12-0061 12•2882 12:5779 13-1808 11

12 8078 13:1420 13 4864 13.8412 14 2068) 14.9716 12

14. 1920 14:6020 15.0258) 15-4640 15-9171) 168699 13

15.6178 16:1130 16.6268 17 1599 17.7130 18-8821 14

17 0863 17.6770 18.2919 18.9321| 19:5986 21.0151 15

18.5989 19.2957 20-3236 20 7841 21.5786 23:2700 16

20-1569) 20 9710 21.8245 22.7193) 23.6575 25 6725 17

21:7616 22:7050| 23 6975 24 7417| 25.8404 28 2129 18

23.4141 24 4997 25.6454 26-8551 28.1324 30 2057 19

25.1169 26.3572 27.6712 29 0636) 305390 33.7600 20

26-870428.2797 29.7781) 31.3714 33 0660 36 7856 21

28.67651 30-269531.96921 33 78311 35 71931 39.9927
TABLE II. The present Value of an Annuity of 1..
erc (3) per c, 4 per c. 14, per c. 5 er c.

0.9662 09615 0 9569 09524 0.9434 2

1.8997 1.8861 1.8727 18594 1.8334 3 2.8016

2:7490 2.7233 2-6730

3.5875 3.5460 3:4651 4.5151 4:4518 4 3900

4 3295 4.2124 7

5 3286 5.2421 5 1579 5.0757 49173 8

6.1145 6 0020 5.8927 57864 5:5824 9

6-8740 6 7327 6:5959 6.4632 6'2098 10

7.6077 7 4353 7.2688 7.1078 68017 11

8.3166 8.1109 7 9127 7.7217 73601 12

9.0116 8 7605 8.5289 8.3054 7.8869 13

9.6633 9-3851 9.1186 8.8633 83838 14

9-9857 9.6829 9.3936 8 8527 15

9.8986 9•2950 16

9.7123 17 18 19

8) 13 1339| 12 5933 12 0853 11 1581

Yrs.jat 3

6

per c.

3673136299

5

0-9709
1.9135
2-8286
3•7171
4-5797
564172
62303
7.0197
7.7861
8.5302
9.2526
9-9540
10-6350 10:3027

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14-3238 137098

21

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To find the Amount of any annuity forborn a certain number

of years.

Take out the amount of 12. from the first table, for the proposed rate and time ; then multiply it by the given annuity; and the product will be the amount, for the same number of years, and rate of interest. And the converse to find the rate or time.

Exam. To find how much an annuity of 501. will amount to in 20 years, at 34 per cent. compound interest.

On the line of 20 years, and in the column of 34 per cent. stands 28.2797, which is the amount of an annuity of il for the 20 years.

Then 28.2797 x 50 gives 1413-985). = 14131. 198. 8d. for the answer required.

To find the present Value of any annuity for any number of years.-Proceed here by the 2d table, in the same manner as above for the 1st table, and the present worth required will be found.

Exam. 1. To find the present value of an annuity of 501. which is to continue 20 years, at 35 per cent. By the table, the present value of 11. for the given rate and time, is 14 2124 ; therefore 14 2124 X 50 = 710.621. or 7101. 128. 4d. is ihe present value required.

Exam. 2. To find the present value of an annuity of 201. to commence 10 years hence, and then to continue for 11 years longer, or to terminate 21 years hence, at 4 per cent. interest. In such cases as this, we have to find the difference between the present values of two equal annuities, for the two given times; which therefore will be done by subtract. ing the tabular value of the one period from that of the other, and then multiplying by the given annuity. Thus,

tabular value for 21 years 14:0292
ditto for · - 10 years 8.1 109

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GEOMETRY.

DEFINITIONS.

1. A POINT is that which has position,

1. but no magnitude, nor dimensions ; neither length, breadth, nor thickness.

2 A Line is length, without breadth or thickness.

3. A Surface or Superficies, is an extension or a figure, of two dimensions, length and breadth ; but without thickness.

4. A Body or Solid, is a figure of three diinensions, namely, length, breadth, and depth, or thickness.

5. Lines are either Right, or Curved, or Mixed of these two.

6. A Right Line, or Straight Line, lies all in the same direction, between its extremities; and is the shortest distance between two points.

When a line is mentioned simply, it means a Right Line.

7 A Curve continually changes it direction between its extreme points.

8. Lines are either Parallel, Oblique, Perpendicular, or Tangential.

9. Parallel Lines are always at the same perpendicular distance; and they never meet though ever so far produced.

10. Oblique lines change their distance, and would meet, if produced on the side of the least distance.

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11 One line is Perpendicular to another, when it inclines not more on the one side

than

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than the other, or when the angles on both sides of it are equal.

12. A line or circle is Tangential, or a Tangent 10 a circle, or other curve, when it touches it, without cutting, when both are produced.

13. An Angle is the inclination or opening of two lines, having different directions, and meeting in a point.

14. Angles are Right or Oblique, Acute or Obtuse.

15. A Right Angle is that which is made by one line perpendicular to another. Or when the angles on each side are equal to one another, they are right angles.

16. An Oblique Angle is that which is made by two oblique lines ; and is either less or greater than a right angle.

17. An Acute Angle is less than a right angle.

18. An Obtuse Angle is greater than a right angle.

19. Superficies are either Plane or Curved.

20. A Plane Superficies, or a Plane, is that with which a right line may, every way coincide. Or, if the line touch the plane in two points, it will touch it in every point. But, if not, it is curved.

21. Plane figures are bounded either by right lines or curves.

22. Plane figures that are bounded by right lines have names according to the number of their sides, or of their angles ; for they have as many sides as angles; the least number being three.

23. A figure of three sides and angles is called a Triangle. And it receives particular denominations from the relations of its sides and angles.

24 An Equilateral Triangle is that whose three sides are all equal.

25. An Isosceles Triangle is that which has two sides equal.

26. A

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