To find the Amount of any annuity forbors a certain number of years.

TAKE out the amount of 11. 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 50l. will amount to in 20 years, at 34 per cent. compound interest.

On the line of 20 years, and in the column of 32 per cent. stands 28-2797, which is the amount of an annuity of 17, for the 20 years. Then 28-2797 x 50, gives 1413·9857. = 14137. 19s. 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 50% which is to continue 20 years, at 34 per cent.-By the table, the present value of 11. for the given rate and time, is 14-2124; therefore 14.2124 x 50-71062/, or 710. 12s. 4d. is the 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 subtracting 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 81109

GEOMETRY.

1.

A

DEFINITIONS.

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, 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 its 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.

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

than

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

12. A line or circle is Tangential, or a Tangent to 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.

8

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.

A

26. A

26. A Scalene Triangle is that whose three sides are all unequal.

27. A Right-angled Triangle is that which has one right-angle.

28. Other triangles are Oblique-angled, and are either Obtuse or Acute.

29. An Obtuse-angled Triangle has one obtuse angle.

30. An Acute-angled Triangle has all its three angles acute.

31. A figure of Four sides and angles is called a Quadrangle, or a Quadrilateral.

32. AParallelogram is a quadrilateral which has both its pairs of opposite sides parallel. And it takes the following particular names, viz. Rectangle, Square, Rhombus, Rhomboid.

33. A Rectangle is a parallelogram having a right angle.

34. A Square is an equilateral rectangle; having its length and breadth equal.

35. A Rhomboid is an oblique-angled parallelogram.

86. ARhombus is an equilateral rhomboid; having all its sides equal, but its angles oblique.

37. A Trapezium is a quadrilateral which hath not its opposite sides parallel.

38. A Trapezoid has only one pair of opposite sides parallel

39. A Diagonal is a line joining any two opposite angles of a quadrilateral.

40. Plane figures that have more than four sides are, in general, called Polygons: and they receive other particular names, according to the number of their sides or angles. Thus,

41. A Pentagon is a polygon of five sides; a Hexagon, of six sides; a Heptagon, seven; an Octagon, eight; a Nonagon, nine; a Decagon, ten; an Undecagon, eleven ; and a Dodecagon, twelve sides.

42. A Regular Polygon has all its sides and all its angles equol.-If they are not both equal, the polygon is Irregular.

43. An Equilateral Triangle is also a Regular Figure of three sides, and the Square is one of four; the former being also called a Trigon, and the latter a Tetragon.

44. Any figure is equilateral, when all its sides are equal: and it is equiangular when all its angles are equal. When both these are equal, it is a regular figure.

45. A Circle is a plane figure bounded by a curve line, called the Circumference, which is every where equidistant from a certain point within, called its Centre.

The circumference itself is often called a circle, and also the Periphery.

46. The Radius of a circle is a line drawn from the centre to the circumference.

47. The Diameter of a circle is a line drawn through the centre, and terminating at the circumference on both sides.

48. An Arc of a circle is any part of the circumference.

#РОФОООООО

49. A Chord is a right line joining the extremities of an arc.

50. A Segment is any part of a circle bounded by an arc and its chord.

51. A Semicircle is half the circle, or a segment cut off by a diameter.

The half circumference is sometimes called the Semicircle.

52. A Sector is any part of a circle which is bounded by an arc, and two radii drawn to its extremities.

53. A Quadrant, or Quarter of a circle, is a sector having a quarter of the circumference for its arc, and its two radii are perpendicular to each other. A quarter of the circumference is sometimes called a Quadrant.

54. The