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III.-GEOMETRICAL EXERCISES.

The Skeleton Propositions, which form a continuation or completion of the plan pursued in the Gradations of Euclid, furnish a most useful and improving series of exercises ; and, on the ascertained fact in the art of teaching, that repetition is a most important auxiliary, they are recommended to the notice and adoption of the teachers of mathematics. At any rate, where employed, they will accustom the student to be systematic and exact, and not to advance a step without a reason; and progress in this way, though it may chance to be less rapid, will be on a solid basis, and bring into play some of the most valuable qualities of the mind.

Potts, COLENSO, COOLEY, CHAMBERS, and others, have each published Collections of Geometrical Exercises ; but as teachers may not wish to go beyond the limits of the present work, two Series of Exercises are now appended; the First Series consisting of Problems and Theorems which are inserted in the Gradations, and of which the General Enunciations may be given out to learners, for analysis, solution, or proof; and the Second Series containing Propositions which are not fully proved or not inserted in the Gradations.

SERIES I.

Problems in Book I. 1. By means of an equilateral triangle to measure an inaccessible distance. 2. To construct a scale of equal parts. 3. To ascertain an inaccessible distance when two sides and their included

angles have been measured. 4. To show, by observations on the shadows which objects cast, the perpen

dicular heights of the objects. 5. To determine without a theodolite the augle at a given point made by a

st. line from two objects meeting in that point. 6. To construct the Mariner's Compass-Card. 7. From a given point at the end of a st. line to raise a perpendicular. 8. On a given line to describe an isoceles triangle of which the perpendi

cular height is equal to the base. 9. From a given point over the end of a st. line to let fall a perpendicular

to the line. 10. Given an angle of 73°, required its complement; an angle of 96°, re

quired its supplement.

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11. By the application of the principle that vertical angles are equal, to find

the distance between two objects. 12. From a given point, A, to direct a ray of light against a mirror, so that

the ray shall be reflected to another given point. 13. To determine the number and kind of polygons which may be joined so

as to cover a given space. 14. To construct a triangle when the base, the less angle at the base, and the

difference of the sides, are given. 15. By means of a mirror placed horizontally, to construct a triangle, the

perpendicular of which shall be representative of the height of any

object. 16. Given three st. lines respectively equal to 40, 50, and 30 equal parts, to

form a triangle. 17. On a given st. line to describe an isosceles triangle having each of the

equal sides double of the base. 18. On a given st. line and with a given side to construct an isosceles triangle. 19. To construct a Line or Scale of Chords. 20. By aid of a line of Chords,-1°. to make an angle containing a certain

number of degrees, as 40° ; 2°. to measure a given angle ; 3o. from the extremity of a line to raise a perpendicular; and 4°. to construct a triangle of which the base contains 40 equal parts, one of the angles at the base 40°, and its other adjacent side 35 equal parts ; and to

measure the other side and the other angles. 21. To measure an inaccessible distance, A B, only A being approachable. 22. By the theory of Representative Values to find the distance between two

stations. 23. To measure an inaccessible distance, A B, neither A nor B being

approachable. 24. Given the vertical angle and the perpendicular height of an isosceles

triangle, to construct it. 25. On the principle that two rays of light proceeding from the centre of the

sun to two points on the earth are physically parallel, to ascertain the

earth's circumference. 26. By the method of parallel lines to ascertain the distance of an inaccessible

object. 27. To determine the Parallax of a heavenly body. 28. To construct a figure which will give the representative value of the

perpendicular height of a mountain. 29. To construct a regular Polygon,-1° when the side is given ; 2°. when

the side is not given. 30. To ascertain both the perpendicular height of a mountain, and the hori

zontal distance from the foot of the mountain to the foot of the per

pendicular. 31. To divide a finite st. line into any given number of equal parts. 32. To construct a Sliding Scale to measure the hundreth part of an inch. 33. To construct a Sliding Scale for measuring the minutes into which a

degree on a circle is divided. 34. To ascertain the continuation of a st. line when an obstacle intervenes. 35. From a given point in the side of a parallelogram to bisect the

parallelogram. 36. To convert a parallelogram into an equivalent rectangle.

37. By the method of Indivisibles to explain the equality of parallelograms

on the same base and between the same parallels. 38. To construct a Diagonal Scale. 39. To divide a triangular space into two equal parts. 40. From any point in the side of a triangle to divide the triangle into two

equal parts. 41. To find the Area of a trapezium. 42. To find the Area of a square. 43. To find the Area of a triangle, 44. To find the Area of any right-lined figure. 45. To find the Area of a regular polygon. 46. To find the Area of a circle. 47. Of the Diameter, Circumference, Area, and Ratio of the diameter and

circumference, any two being given, to ascertain the others. 48. To describe a triangle equal to a given parallelogram, and having an

angle equal to a given angle. 49. A parallelogram being given, to find another parallelogram equal to it, and

having one side equal to a given st. line. 50. Given the area of one figure, and the side of another which is to be a

parallelogram equal to the given figure, to find the other side of that

parallelogram. 51. To change any right-lined figure, first into a triangle, and then into a

rectangle of equal area. 52. To straighten a crooked boundary without changing the relative size of

two fields. 53. Given the diagonal to construct a square. 54. To ascertain the height of an inaccessible object by aid of the Geometrical

Square. 55. Given in numbers the sides of a right-angled triangle, to find the

hypotenuse. 56. Given in numbers the hypotenuse and one side, to find the other side. 57. To find a square equal to any number of squares ; or a square that is

the multiple of a given square ; or a square that equals the difference of two squares ; or a equare that is the half, the fourth, &c., of a give

square. 58. To make a rectilineal figure similar to a given rectilineal figure. 59. To make a circle the double, or the half, of another circle. 60. To construct the Chords, Natural Sines, Tangents, and Secants, of

Trigonometrical Tables. 61. To find right triangular numbers. 62. To compute Heights and Distances from the curvature of the earth.

Theorems in Book I.

1. By the bisection of the vertical angle of an isosceles triangle to show that

the angles of the base are equal ; and also that the bisecting line bisects

the base at right angles. %. Only one perpendicular can be drawn from a given point to a given st.

line.

3. The perpendicular is the shortest line from a given point to a given st. line. 4. From the same point only two equal st. lines can be drawn to a given st.

line. 5. All heavy bodies free to move continually descend, or seek the point

which is nearest to the earth's centre. 6. Of all st. lines that can be drawn from one point to a plane surface, and

reflected to a third point, those are the shortest which make the angle

of incidence equal to the angle of reflection. 7. The chord of 60° is equal to the radius of the circle. 8. A st. line which is perpendicular to one parallel, is also perpendicular to

the other. 9. If a st. line falling on two other st. lines make the interior angles on the

same side less than two rt. angles, those two lines on being produced

shall intersect. 10. A parallel to the base of a triangle through the point of bisection of one

side, will bisect the other side. 11. The st. lines which join the middle points of the three sides of a triangle,

divide it into four triangles which are equal in every respect. 12. The st. line joining the points of bisection of each pair of sides of a triangle,

is equal to half the third side. 13. A trapezium is equal in area to a parallelogram of the same altitude, and

of which the base is half the sum of the parallel sides. 14. The squares of equal st. lines are equal ; and if the squares are equal, the

lines are equal. 15. Every parallelogram having one rt. angle, has all its angles rt. angles. 16. If a perpendicular be drawn from the vertex of a triangle to the base, the

difference of the squares of the sides is equal to the difference between

the squares of the segments. 17. If a perpendicular be drawn from the vertex of a triangle to the base, or

to the base produced, the sums of the squares of the sides and of the alternate angles are equal.

Problems in Book II.

1. From Propositions 1, 2, and 3, deduce various methods for the Multiplicar

tion of Numbers, and demonstrate the rule. 2. From Prop. 4, point out a practical way of extracting the Square root of a

number, and prove the correctness of the formula. 3. To find the difference between the squares of two unequal numbers without

squaring them. 4. To find Quantities in Arithmetical Progression. 5. To find the value of an Adfected Quadratic Equation in Algebra. 6. By aid of Prop. 6, to ascertain the diameter of the earth. 7. Given the sum and the difference of two magnitudes, to find the magni

tudes themselves. 8. From the Area of a rectangle and one side given, to obtain the other side. 9. To divide a given Line a, so that its parts & and a -a may make a (a – x)

= x. Let the solution be given both algebraically and arithmetically. 10. To ascertain the Area of a triangle when the three sides are known. 11. From the three sides of a triangle given, to obtain the perpendicular :

1°. when the perp. falls within the base ; and 2° when it falls without

the base. 12. To find a mean proportional to two given línes. 13. To approximate to the square of any curve-lined figure. 14. To calculate the Area of any right-lined figure.

Theorems in Book II.

1. The difference of the squares of two quantities, equals the rectangle o

their sum and difference. 2. The difference of the squares of two quantities is greater than the square

of their difference, by twice the rectangle of the less and their difference. 3. The square of the sum of two lines is equal to four times the rectangle

under them, together with the square of their difference. 4. Four times the square of half the sum is equal to four times the rectangle

under the lines, together with four times the square of half the

difference. 5. The sum of the squares of any two st. lines is equal to twice the square of half their sum, together with twice the square of half their difference. 6. The sum of the squares is equal to half the square of the sun, together

with half the square of the difference.

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1. To find a point which is equidistant from the three vertical points of a

triangle. 2. To bisect a triangle by a st. line drawn from a given point in one of its

sides. 3. Describe a circle which shall pass through two given points, and have its

centre in a giyen line. 4. Through a given point to draw a st. line that shall be equally inclined to

two given st. lines. 5. Given a triangle A BC, and a point D in AB ; to construct another triangle

ADE equal to the former, and having the common angle A. 6. To change a triangle into another equal triangle of a given altitude. 7. To draw a st. line which, if produced, would bisect the angle between two

given st. lines, without producing them to meet. 8. To trisect a right angle.

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