Ex. 1. What is the present value of a pension of L.100, payable yearly, for 20 years, at 5 per cent compound interest? Ans. L. 1246, 4s. 5 d. 2. What is the present value of a perpetual annuity of L.100, payable yearly, interest at 5 per cent ? Ans. L.2000. 3. What is the present value of an annuity of L. 100, payable half-yearly for 20 years, interest at 5 per cent. per annum, also payable half-yearly? Ans. L.1255, 2s. 10 . 4. What is the present value of a perpetuity of L.100 per annum, payable half-yearly, interest at 5 per cent. per annum, being also payable half-yearly? Ans. L. 2000. 5. What is the present value of an annuity of L.100 to commence 10 years hence, and then continue for 30 years, interest at 4 per Ans. L.1168, 3s. 7 d. 6. What is the present value of an annuity of L.50, to commence 8 years hence, and then to continue for 42 years, interest at 5 per cent. ? Ans. L.589, 12s. 8d. 7. In what time will a pension of L.50 amount to L.1000, interest at 5 per cent. ? Ans. 14.2 years. 8. To what sum will an annuity of L.24 amount in 20 years, when improved at 5 per cent. ? Ans. L.793, lls. 8d. PROMISCUOUS EXERCISES. 1. It is required to divide each of the numbers 11 and 17 into two parts, so that the product of the first parts of each may be 45, and of the second 48. Ans. 5, 6, and 9, 8. 2. Divide each of the numbers 21 and 30 into two parts, so that the first part of 21 may be three times as great as the first part of 30, and that the sum of the squares of the remaining parts may be 585. Ans. 18, 3, and 6, 24. 3. A gentleman left L.210 to three servants, to be divided in continued proportion, so that the first shall have L.90 more than the last : find their legacies. Ans. L.120, L.60, and L.30. 4. There are two numbers, whose product is 45, and the difference of their squares is to the square of their differencé as 7 is to 2: what are the numbers? Ans. 9 and 5. 5. A and B engage in partnership with a capital of L.100: A leaves his money in the partnership for 3 months, and B for 2 months, and each takes out L.99 of capital and profit: determine the original contribution of each. Ans. A L.45, and B L.55. PLANE GEOMETRY. GEOMETRY is that branch of Mathematics which treats of the properties of measurable magnitudes. Magnitudes are of three kinds, viz. lines having length only, surfaces having length and breadth, and solids having length, breadth, and thickness. That branch of Geometry which treats of lines and surfaces is called Plane Geometry, and that which treats of the properties of solid bodies is called Solid Geometry. DEFINITIONS. is that which has position but no magnitude. 2. A line is length without breadth. 3. The extremities of a line are points. 4. A straight line is that which lies evenly between its extreme points. 5. A superficies is that which hath only length and breadth. 6. The extremities of a superficies are lines. 7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 8. A plane rectilineal angle is the inclination of two straight lines which meet in a point, but are not in the same straight line. NOTE. When there A are several angles at one point, as at B, each of the angles must be named by three letters, and the letter at the angular point must be placed between other two ; thus, the angle formed by the lines AB and BD meeting in the point B, is called the angle ABD or DBA; also the angle formed by the straight lines DB and BC, is called the angle DBC or CBD; but when there is only one angle at the point, as at E, it may be called simply the angle at E. D the B E с 9. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. 10. An obtuse angle is that which is greater than a right angle. 11. An acute angle is that which is less than a right angle. 12. A term or boundary is the extremity of any thing. 13. A figure is that which is enclosed by one or more boundaries. 14. A circle is a plain figure bounded by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within it to the circumference are equal to one another; 15. And this point is called the centre. 16. The diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 17. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. 18. A straight line drawn from the centre to the circumference of a circle is called a radius. 19. A straight line which is terminated both ways by the circumference, but does not pass through the centre, is called a chord. 20. The part of the circumference cut off by the chord is called an arc. 21. The figure bounded by the chord and arc is called a segment. 22. Rectilineal figures are those which are contained by straight lines. 23. Trilateral figures, or triangles, are contained by three straight lines. 24. Quadrilateral figures are contained by four straight lines. 25. Multilateral figures, or polygons, are contained by more than four straight lines. 26. An equilateral triangle has all its sides equal. 27. An isosceles triangle has two equal sides. 28. A scalene triangle has three unequal sides. 29. A right angled triangle is that which has one right angle. 30. An obtuse angled triangle is that which has one obtuse angle. 31. An acute angled triangle has all its angles acute. 32. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles. 33. A rectangle is that which has all its angles right angles, but all its sides are not equal. 34. A rhombus is that which has all its sides equal, but its angles are not right angles. 35. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor are its angles right angles. 36. All other four-sided figures besides these are called trapeziums. 37. Parallel straight lines are such as, being in the same plane, and being produced ever so far both ways, do not meet. 38. A parallelogram is a four-sided figure whose opposite sides are parallel. 39. A postulate requires us to admit the possibility of doing something, without being shown how to do it. 40. A proposition is a distinct portion of science, and is either a problem or a theorem. 41. A problem is an operation proposed to be performed. 42. A theorem is a truth which it is proposed to prove. 43. A lemma is a preparatory proposition to render what follows more easy. 44. A corollary is an obvious consequence resulting from a preceding proposition. 45. A scholium is an observation, or remark upon something preceding it. 46. An axiom is a self-evident truth. 47. The side opposite to the right angle of a right-angled triangle is called the hypotenuse; one of the sides about the right angle is called the base; and the remaining side is called the perpendicular. 48. In a triangle which is not right-angled, any side may be called the base; the intersection of the other two sides is called the vertex ; and the angle at that point the vertical angle. 49. The space contained within a figure is called its surface; and in reference to that of another figure with which it is compared, is called its area. 50. A polygon is a figure contained by more than four straight lines; when its sides are all equal, and also its angles, it is called a regular polygon. 51. A polygon of five sides is called a pentagon; that of six sides, a hexagon; that of seven sides, a heptagon; that of eight sides, an octagon; that of nine sides, a nonagon; that of ten sides, a decagon; that of eleven sides, an undecagon; that of twelve sides, a dodecagon; that of fifteen sides, a quindecagon. POSTULATES. 1. Let it be granted that a straight line may be drawn from any point to any other point. 2. Let it be granted that a terminated straight line may be produced to any length in a straight line. |