Here, a = £1000 r = £.05 REVERSION OF PERPETUITIES. 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 r. By Problem X., the present value of an annuity, to commence after T years and to continue for t years, is = p 1000 .05 = 20000, or 20 years' purchase. = = {(1 + 1) −TM − (1 + ») − (T+0} a 7' In the present case, t = ∞, and .:. (1 + r) −(T +1) = 0; hence we shall have ..........(14). EXAMPLES FOR PRACTICE. 1. Find the interest of £555 for 2 years at 43 per cent. simple interest. Ans. £65 18s. ltd. 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 23 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. 4. 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. 84d. 6. What is the present value of the compound interest of £100 to be received years hence, at 5 per cent. per annum. Ans. £78 78. Ožd. five 7. What is the amount of £721, for 21 years, at 4 per cent. per annum, compound interest? Ans. £1642 19s. 94d. 8. The rate of interest being 5 per cent., in what number of years, at compound interest, will £1 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 19s. 24d. 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. 3. 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 7s. 9žd. 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 £19 4s. to commence 7 years hence and continue for ever, interest being calculated at the rate of 44 per cent. ? Aus. £1293 5s. 11 d. 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, 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. tance; 9. Parallel Lines are always at the same perpendicular disand 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 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 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 ob tuse 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. A Parallelogram 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, or all its sides equal, and all its angles equal. 35. A Rhomboid is an oblique-angled parallelogram. 36. A Rhombus is an equilateral rhomboid; having all its sides equal, but its angles oblique. 37. A Trapezium is a quadrilateral which has 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; 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 equal.—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 everywhere 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. 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 ar and its chord. |