16. Describe an isosceles triangle having a given base, and whose vertical angle is half a right angle. 17. AB is a straight line, C and D are points on the same side of it; find a point E in AB such that the sum of CE and ED shall be a minimum. 18. Having given two sides of a triangle and an angle, construct the triangle. Examine the cases when there will be (1.) one solution; (2.) two solutions ; (3.) none. 19. Given an angle of a triangle and the sum and difference of the two sides including the angle, to construct the triangle. 20. Show that each of the angles of an equilateral triangle is twothirds of a right angle, and hence show how to trisect a right angle. 21. If two angles of a triangle be bisected by lines drawn from the angular points to a given point within, then the line bisecting the third angle will pass through the same point. 22. The difference of any two sides of a triangle is less than the third side. 23. If the angles at the base of a right-angled isosceles triangle be bisected, the bisecting line includes an angle which is three halves of a right angle. 24. The sum of the lines drawn from any point within a polygon to the angular points is greater than half the sum of the sides of the polygon. PROP. 33—48. 25. Show that the diagonals of a square bisect each other at right angles, and that the square described upon a semi-diagonal is half the given square. 26. Divide a given line into any number of equal parts, and hence show how to divide a line similarly to a given line. 27. If D and E be respectively the middle points of the sides BC and AC of the triangle ABC, and AD and BE be joined, and intersect in G, show that GD and GE are respectively one-third of AD and BE. 28. The lines drawn to the bisections of the sides of a triangle from the opposite angles meet in a point. 29. Describe a square which is five times a given square. 30. Show that a square, hexagon, and dodecagon will fill up the space round a point. * 31. Divide a square into three equal areas, by lines drawn parallel to one of the diagonals. 32. Upon a given straight line construct a regular octagon. 33. Divide a given triangle into equal triangles by lines drawn from one of the angles. 34. If any two angles of a quadrilateral are together equal to two right angles, show that the sum of the other two is two right angles. 35. The area of a trapezium having two parallel sides is equal to half the rectangle contained by the perpendicular distance between the parallel sides of the trapezium, and the sum of the parallel sides. 36. The area of any trapezium is half the rectangle contained by one of the diagonals of the trapezium, and the sum of the perpendicu. lars let fall upon it from the opposite angles. 37. If the middle points of the sides of a triangle be joined, the lines form a triangle whose area is one-fourth that of the given triangle. 38. If the sides of a triangle be such that they are respectively the sum of two given lines, the difference of the same two lines, and twice the side of a square equal to the rectangle contained by these lines, the triangle shall be right-angled, having the right angle opposite to the first-named side. * 39. If a point be taken within a triangle such that the lengths of the perpendiculars upon the sides are equal, show that the area of the rectangle contained by one of the perpendibulars and the perimeter of the triangle is double the area of the triangle. 40. In the last problem, if O be the given point, and OD, OE, OF the respective perpendiculars upon the sides BC, AC, and AB, show that the sum of the squares upon AD, OB, aná DC exceeds the sum of the squares upon AF, BD, and CD by three times the square upon either of the perpendiculars. 41. Having given the lengths of the segments AF, BD, CE, in Problem 40, construct the triangle. 42. Draw a line, the square upon which shall be seven times the square upon a given line. 43. Draw a line, the square upon which shall be equal to the sum or difference of two given squares. 44. Reduce a given polygon to an equivalent triangle. 45. Divide a triangle into equal areas by drawing a line from a given point in a side. 46. Do the same with a given parallelogram. 47. If in the fig., Euc. I. 47, the square on the hypothenuse be on the other side, show how the other two squares may be made to cover exactly the square on the hypothenuse. 48. The area of a quadrilateral whose diagonals are at right angles is half the rectangle contained by the diagonals. SECTION IT. ALGEBRA. CHAPTER I. ELEMENTARY PRINCIPLES. 1. Algebra treats of numbers, the numbers being represented by letters (symbols of quantity), affected with certain symbols of quality, and connected by symbols of operation. It is easy to see that these symbols of quantity may be dealt with very much as we deal with concrete quantities in arithmetic. Thus, allowing the letter a to stand for the number of units in any quantity, and allowing also 2 a, 3 a, 4 a, &c., to stand respectively for twice, thrice, four times, &c., as large a quantity as the letter a, it at once follows that we may perform the operations of addition, subtraction, multiplication, and division upon these symbols exactly as we do in ordinary arithmetic upon concrete quantities. For instance, 4 a and 6 a make 10 a, 9 a exceeds 5 a by 4 a, 15 a is 5 times 3 a, and 7 a is contained 8 times in 56 a. Neither is it necessary in these operations to state, or even to know the exact number of units for which any symbol of quantity stands, nor indeed the nature of these units ; it is simply sufficient that it is a symbol of quantity. Thus, in the science of chemistry, we use a weight called a crith ; and a person unacquainted with chemistry might not know whether a crith were a measure of lengtlı, weight, or capacity, or indeed whether it were a measure at all, yet he would at once allow that 6 criths and 5 criths are 11 criths, that twice 4 criths are 8 criths, &c, The Signs + and – as Symbols of Operation. 2. In purely arithmetical operations, the signs + and are respectively the signs of addition and subtraction. In this sense, too, they are used in algebra. Thus, a + b means that b is to be added to a, and a – 7 means that b is to be subtracted from a. Hence, as long as a and b represent ordinary arithmetical numbers, a + b admits of easy interpretation, as also does a - b, when b is not greater than a. But when b is greater than a, the expression a - b has no arithmetical meaning, By an extension, however, of the use of the signs + and -, we are able to give such expressions an intelligible signification, whatever may be the quantities represented by a and b. Positive and Negative Quantities.—The Signs + and as Symbols of Affection or Quality. 3. DEF.-A positive quantity is one which is affected with a + sign, and a negative quantity is one which is affected with a - sign. Let BA be a straight line, and 0 a point in the line; and suppose a person, starting from 0, to walk a miles in the direction OA. Suppose also another person, starting from the same or any other point in BA to walk a miles in the direction OB. These persons will thus walk a miles each in exactly opposite directions. Now, we call one of these directions positive (it matters not which) and the other negative. Let us take the direction OA as positive. We then have the first person walking a miles in a positive direction, and the second walking a miles in a negative direction. We represent these distances algebraically by t a and - a respectively. It will therefore be seen that the signs + and — have no effect upon the magnitudes of quantities, but that they express the quality or affection of the quantities before which they stand. Again, suppose a person in business to get a profit of £6, while another suffers a loss of £6. We may express these facts algebraically in two ways. We may consider gain as positive, and loss as negative gain, and say that the former has gained + 6 pounds, while the latter has gained – 6 pounds. Or we may consider loss as positive, and gain as negative loss, and say that the former has lost – 6 pounds, while the latter has lost + 6 pounds. We hence see that the gain of + 6 is equal to a loss of - 6, and that a gain of - 6 is equal to a loss of + 6. . The Sum of Algebraical Quantities. 4. Let a distance AB be measured to the right along the line AX. And let a further - B o distance BC be measured from B in the same direction. By the sum of these lines we mean the resulting distance of the point C from the original point A, that is to say, the distance AC. (It may be remarked that we add the line BC to the line AB by measuring BC in its own proper direction from the extremity B of AB. It is hardly necessary to remind the student that both lines are in the same straight line AX.) Let us represent the distances AB and BC by + a and + b respectively; then the algebraical sum of the lines will be represented writing these quantities side by side, each with its own proper sign of affection. Thus the sum of the distances AB and BC is expressed by + a + b, or, as it is usual to omit the + sign of a positive quantity when the quantity stands alone or at the head of an algebraical expression, the sum of AB and BC is expressed by a + b. Hence, the interpretation of a + b is that it represents the distance AC. Again, taking as above + a to represent the distance AB along the straight line AX, and _IB_X measured to the right, let a distance BC be measured from B in the same straight line AX, but a EX this time to the left. |