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4. Add two given squares.

Make a right angle, and on its sides place the sides of the squares. These lengths will be the small sides of your right angled triangle; then draw the longest side, and you have the side of a square equal to the other two.

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5. Double a square. (fig. 3.)

The triangle must be a rectangular isoceles, and the two small squares will then be equal to each other, or united they will be equal to the large square.

6. Cut off a square. (fig.2.)

If the great square and one of the small ones be given, it is easy to find the size of the other.

First draw the great side, then the semicircle, then draw from either end of the great side, (which you will notice is the diameter of the semicircle) a cord of the semicircle, which is equal to a side of the given small square. The other cord which will finish the right angled triangle, is the side of the square required.

7. Take the half of a square. (fig. 3.)

A perpendicular to the middle of the long side, will strike the semi-circle, and a cord from this point of intersection to either end of the diameter or long side, will give the side of a square, half as large as the great

square.

8. Make a graduated semicircle, usually called a Protracter. (fig. 4.)

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90 105

120

135

75

150

60

165

By general consent a circle is divided into 360 equal parts, called degrees. A semicircle of course contains 180 degrees, that is, half of 360.

After having drawn a semicircle and its diameter, draw a perpendicular radius. This radius forms a right angle with the diameter, and cutting the semicircle in two equal parts or quarters of circles, leaves 90 degrees for each of them. If 90 degrees of a circle make a right angle, 45 degrees will make half a right angle, &c.

Or by another method. A radius, if made a cord of the semicircle, will allow three cords, each of which will contain 60 degrees; halve these arcs, and you have arcs of 30 degrees; halve the arcs of 30 degrees, and you have 15 degrees; cut these into three equal parts, and you have 5 degrees; then divide the arcs of five degrees into five parts, and you have the 180 degrees of the semicircle.

Whether the circle be large or small, it is divided into the same number of degrees; for if the radii of a small circle be lengthened out, and a larger circle drawn from the same centre, the radii will form the same part of the large as of the small circle, and the angle between any two radii will be unchanged.

9. Make an angle of 30 degrees on the graduated semicircle. (fig. 4.)

A radius drawn from the centre to the number 30 on the graduated semicircle, will form an angle of 30 degrees with the diameter of the semicircle. And so for any other number of degrees. It will be seen that any number of degrees less than 90 will make an acute angle, and more than 90 degrees will form an obtuse angle thus, in fig. 4, 30 degrees form an acute angle, and the remaining 150 degrees of the half circle form an obtuse angle.

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Angles, therefore, are measured by their openings. Place the point of any angle on the centre of the semicircle, or the centre of the semicircle on the point of the angle, and then by seeing how many degrees the opening of the angle measures on the graduated edge of the semicircle, you will find the size of the angle. If the sides of the angle do not extend to the circumference, you may extend them till they do. If they extend beyond the circumference, measure the angle where the graduated circle cuts its sides.

10. After the pupil has drawn the semicircle, the monitor must require him to draw angles of various sizes, from 1 to 180 degrees. Then, laying aside the semicircle, let him draw angles of various degrees, which the monitor will test by his brass semicircle, or by angles of pasteboard previously prepared: the latter are the handiest if well cut.

11. Make a sphere and its meridians. (fig. 5.)

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Describe a circle, and draw two diameters perpendicular to each other; one for the axis, and the other for the equator, (a circle which goes round the earth at an equal distance from the ends of the axis, which ends are called poles). Then draw arcs of a circle, all passing through the poles, and whose centres are consequently on the perpendicular to the axis (that is, the equator) prolonged to the right or left hand. See Class III, problem 35. fig. 21.

These arcs have their centre as much farther off as they are nearer the axis. Their number is not important, but if five be made on each side of the axis, as in the figure, each of the spaces between them will be just 15 degrees, or one twenty-fourth part of the whole sphere. These arcs in geography are called meridians.

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12. Make a sphere, and the little circles which run parallel to the equator. (fig. 7.)

After having described a circle, and its two perpendicular diameters, as in the preceding problem, divide the circle by dots into arcs of say 15 degrees; there will then be five dots and six arcs between the equator and each pole; then divide the axis into the same number of parts. The next object is to draw an arc through the three points nearest the equator, then through the three next, and so on till all are drawn.

These arcs on a solid globe would be parallel to the equator, but do not appear so on a plane or flat surface. In geography, they are called Parallels of Latitude.

If an apple be taken and sliced from side to side, it will exactly represent these circles, which are planes cutting a sphere perpendicular to its axis.

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13. Draw a sphere which shall unite the two preceding problems. (fig. 7.)

14. Draw an ellipse. (fig. 8.)

An ellipse is an oval, which may be more or less lengthened, as in figures 9 to 13. To make an ellipse: first cross two perpendicular right lines; the upper and lower halves to be of equal length, and the right and left hand to be equal also. You thus obtain the longest and shortest diameter of the ellipse, called its great and small axis. The next thing is to draw the curved lines as in