Whole amount at the end of 1st year = a

2nd

3d

4th

...

= a + a + ar

= a + a
a (1+r)

= a + a + a(1 + r) + ar + ar(1+r) = a + a(i+r) + a(1+r)2

= a + a + " (1 + r) +a(1 + r)3 + ar +ar(1+r)+ar (1+r)2

= a + a(1+r) + a(1+r)2 + a(1+r)s &c. = &c.

PROBLEM IX. To find the present value of an annuity a payable for t years, compound interest being allowed at the rate r.

It is manifest that the present value of this annuity must be a sum such, that if put out to interest for t years at the rate r, its amount at the end of that period will be the same as the amount of the annuity.

Hence, if we call this present value p, we shall have, by Probs. VII. and VIII.

What is the present value of an annuity of £500, to last for 40 years, com pound interest being allowed at the rate of 24 per cent. per annum.

PROBLEM X To find the present value (P) of an annuity a which is to com. mence after T years, and to continue for t years.

The present value required is manifestly the present value of a for Ti years, minus the present value of a for T' years.

By Problem IX. the present value of a for T+t years=

a for T

(1+r)

..(12)

PURCHASE OF ESTATES,

PROBLEM XI. To find the present value p of an estate or perpetuity, whose annual rental is a, compound interest being calculated at the rate r.

The present value of an annuity a, to continue for t years, by Prob. IX. is

but if the annuity last for er r, as in the case of an estate, then t∞, and

What is the value of an estate, whose rental is £1000, allowing the purcha.es

5 per cent. for his money?

PROBLEM XII. To find the present value of an estate or perpetuity, whose annual rental is a pounds, to a person to whom it will revert after T years, compound interest being allowed at the rate 1.

By Problem X., the present value of an annuity, to commence after T years and to continue for t years, is

In the present case, t = ∞, and .:. (1 + r)−(T+= 0; hence we shall

1. Find the interest of £555 for 24 years at 43 per cent. simple interest.

Ans. £65 18s. 1jd.

2. In what time will the interest of £1 amount to 15s., allowing 4 per cent. cimple interest? Ans. 16 years, 8 months.

3. What is the amount of £120 10s. for 2 years, at 43 per cent. simple inAns. £134 16s. 24d. terest?

4. The interest of £25 for 3 years, at simple interest, was found to be £3 18s. 9d.; required the rate per cent. per annum. Ans. 44.

5. Find the discount on £100 due at the end of 3 months, interest being calculated at the rate of 5 per cent. per annum. Ans. £1 4s. 8d.

6. What is the present value of the compound interest of £100 to be received five years hence, at 5 per cent. per annum. Ans. £78 7s. Od.

7. What is the amount of £721, for 21 years, at 4 per cent. per annum, compound interest? Ans. £1642 19s. 94d.

The rate of interest being 5 per cent., in what number of years, at compound interest, will £l amount to £100? Ans. 94 years, 141.4 days.

9. Find the present value of £430, due nine months hence, discount being allowed at 4 per cent. per annum. Ans. £415 198. 21d

10. Find the amount of £1000, for 1 year, at 5 per cent. per annum, compound interest, the interest being payable daily. Ans. £1051. 5s. 9d. nearly.

11. What sum ought to be given for the lease of an estate for 20 years, of the clear annual rental of £100, in order that the purchaser may make 8 per cent. of his money? Ans. £981 16s. 34.

12. Find the present value of £20, to be paid at the end of every five years, for ever, interest being calculated at 5 per cent. Ans. £72 78. 914

13. What is the present value of an annuity of £20, to continue for ever, and to commence after two years, interest being calculated at 5 per cent.?

Ans. £362 16s. 23d.

14. The present value of a freehold estate of £100 per annum, subject to the payment of a certain sum (A) at the end of every two years, is £1000, allowing 5 per cent. compound interest. Find the sum (A). Ans. A £102 10s.

15. What is the present value of an annuity of £79 4s. to commence 7 years hence and continue ter ever, interest being calculated at the rate of 41 per cont. ? Ans. 1293 5s. 111d.

GEOMETRY.

DEFINITIONS.

1. A POINT is that which has position, 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 dimensions, 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, Jies 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 its direction between its extreine 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.

11. One line is Perpendicular to another, when it inclines not more on the one side than the other, or when the angles on both sides of it are equal.

12. A line or circle is Tangential, or is a Tangent to a circle, or other curve, when it touches it, without cutting, although 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 Obiso