(2) "Find the amount of £P. at compound interest for n years, the interest being payable yearly." (1) Let x the number of days in which B can complete the work by himself, 1 Then = the part B would do in 1 day. Now A does it in 10 days 1 10 = the part A does in 1 day. The part done in 1 day by A and B working to 1 gether would be, + 1 10 XC But they can together complete it in 7 days; and therefore they can do 1 1 per day. 1 7 Hence, + = 10 (2) Let r the interest of £1 for a year; the amount of £1 for a year will then be 1 + r. To find the amount of succeeding years, we have the following proportions: as the principal of the first year is to the principal of any succeeding year, so is the amount of the first year to the amount of that succeeding year, Thus 1:1+r::1+r: (1+r)2: the amount of 2nd year 1:(1+r)2:1+r: (1+r)3= 1:(1+r)::(1+r): (1+r)n = (1+r)" being the amount of £1 for n years, the amount of £P will be P. (1+r)". See Appendix, Algebra. 30 GEOMETRY. SECTION I. 1. Define a plane superficies, and a circle. 2. Draw a straight line perpendicular to a given straight line from a given point in it. 3. The angles at the base of an isosceles triangle are equal to each other, and if the equal sides be produced the angles on the other side of the base shall be equal. SECTION II. 1. Prove that the sum of the three angles of a triangle equals two right angles. Given the value of two angles of a triangle, how is the value of the third ascertained? 2. Prove that parallelograms on the same base and between the same parallels are equal to one another. 3. In any right angled-triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the sides containing the right angle. SECTION III. 1. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts together with the square of the line between the points of section is equal to the square of half the line. 2. In any triangle the square of the side subtending any acute angle is less than the squares of the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle ;-prove only the first case of this proposition. SECTION IV. 1. Construct a triangle whose area shall be equal to that of a given trapezium. 2. Show how to make a square double a given square. 3. Show that the diagonals of a parallelogram bisect each other. N.B.-As it may fairly be assumed that every schoolmaster is possessed of a copy of Euclid's Elements ; and as the following questions may be answered in the very words of Euclid; it will scarcely be necessary to give other solutions than direct reference to the particular proposition, the knowledge of which is presupposed in the questions. Schoolmasters will find the recently published "Geometry and Mensuration," by Mr. Tate, a most valuable auxiliary in acquiring a rapid knowledge of the essentials of the science. It may be briefly characterised as Euclid divested of undue technicalities and redundancy. A regard to economy in publication requires the use of as few diagrams as possible; hence the reason of aiming to dispense with them, as in this paper. 1. 66 SECTION I. Define a plane (1) superficies, and (2) a circle." (1) A plane superficies is a surface, any two points of which being joined the straight line between them will be wholly within the superficies. (2) A circle is a plane figure bounded by one line, called the circumference, which is such that all points of it are equally distant from a point within the circle called the centre. 2. "Draw a straight line perpendicular to a given straight line from a given point in it." Euclid i. 11. Tate, Art. 23. 3. "The angles at the base of an isosceles triangle are equal to each other; and if the equal sides be produced the angles on the other side of the base shall be equal." Euclid i. 5. Tate, Art. 18. SECTION II. 1. "Prove that the sum of the three angles of a triangle equals two right angles. Given the value of two angles of a triangle, how is the value of the third ascertained?" Euclid i. 32. Tate, Art. 34. The sum of the three angles being two right angles, or 180°; if two be known, they must be deducted from 180°, to leave the third. 2. "Prove that parallelograms on the same base and between the same parallels are equal to one another." Euclid i. 35. Tate, Art. 48. 3. "In any right-angled triangle the square which is described upon the side subtending the right angle is equal to the square described upon the sides containing the right angle." Euclid i. 47. Tate, Art. 45. SECTION III. 1. "If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts together with the square of the line between the points of section is equal to the square of half the line." Euclid ii. 5. 2. "In any triangle the square of the side subtending any acute angle is less than the squares of the sides containing the angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle ;-prove only the first case of this proposition." Euclid ii. 13. Tate, Art. 50. SECTION IV. Construct a triangle whose area shall be equal to that of a given trapezium." ༤. From Euclid ii. 14, describe a square equal to the given trapezium. A triangle with a base equal to that of the square so found, and with twice the altitude of the square; or with an altitude equal to that of the square, but a base twice as large will be equal to the square (Euclid i. 42), and consequently to the trapezium. 2. "Show how to make a square double a given square." By constructing a square on the diagonal of the given square. 3. "Show that the diagonals of a parallelogram bisect each other." Playfair's Euclid ii. Prop. B. |