13. This instrument is used to lay down, or protract angles. It may also be used to measure angles included between lines already drawn upon paper. It consists of a brass semi-circle ACB divided to half degrees. The degrees are numbered from 0 to 180, both ways; that is, from A to B, and from B to A. The divisions, in the figure, are only made to degrees. There is a small notch at the middle of the diameter AB, which indicates the centre of the protractor. QUEST.-13. Describe the circular protractor. Practical Geometry.-Problems. PROBLEM VII. To lay off an angle with a Protractor. 14. Place the diameter AB on the line, so that the centre shall fall on the angular point. Then count the degrees contained in the given angle from A towards B, or from B towards A, and mark the extremity of the arc with a pin. Remove the protractor, and draw a line through the point so marked and the angular point: this line will make with the given line the required angle. GUNTERS' SCALE. 15. This is a scale of two feet in length, on the faces of which, a variety of scales are marked. The face on which the divisions of inches are made, contains, however, all the scales necessary for laying down lines and angles. These are, the scale of equal parts, the diagonal scale of equal parts, and the scale of chords, all of which have been described. QUEST.-14. Explain the manner of laying off an angle with the circular protractor. 15. What is Gunter's scale? Practical Geometry.-Problems. PROBLEM VIII. To bisect a given straight line: that is, to divide it into two equal parts. 16. Let AB be the given line. With A as a centre dius describe the arc EHI. Then join the points I and E by the line IE: the point D where it intersects AB, will be the middle of the line AB. PROBLEM IX. At a given point in a given straight line, to erect a perpendicular to the line. 17. Let A be the given point, and BC the given line. From A lay off any two distances AB and AC equal to each other. Then, from the points B and C, as centres, with a radius greater than B A QUEST.-16. Describe the manner of bisecting an angle with the dividers. 17. Explain the manner of drawing a perpendicular to a given line at a given point. Practical Geometry.-Problems. BA, describe two arcs intersecting each other in D: draw AD, and it will be the perpendicular required. SECOND METHOD. When the point A is near the end of the line. 18. Place one foot of the dividers at any point, as P, and extend the other leg to A. Then with P as a centre and radius from P to A describe the circumference of a circle. Through C, where the cir BC P D cumference cuts BA and the centre P, draw the line CPD. Then draw AD, and it will be perpendicular to CA, since CAD is an angle in a semi-circle. PROBLEM X. From a given point without a straight line, to let fall a perpendicular on the line. 19. Let A be the given point and BD the given line. From the point A as a centre, with a radius sufficiently great, describe B E point E, equally distant from the points B and D, and draw AE: and AE will be the perpendicular required. QUEST.-18. Explain the second method. 19. Explain the method of drawing a perpendicular to a line from a point without. When the given point A, is nearly opposite one end of the given line. 20. Draw AC to any point, as C of the line BD. Bisect AC at F. Then with F as a centre and FC or FA as a radius, describe the semi-circle CDA. Then draw BDA, and it will be perpendicular to BD at D. F PROBLEM XI. At a point, in a given line, to make an angle equal to a given angle. 21. Let A be the given point, AE the given line, and IKL the given angle. From the vertex K, as a cen K tre, with any radius, describe the arc IL, terminating in the two sides of the angle. From the point A as a centre, with a distance AE equal to KI, describe the arc ED; then take the chord LI, with which, from the point E as a centre, describe an arc cutting the indefinite arc DE, in D; draw AD, and the angle EAD will be equal to the given angle K. QUEST.-20. Give the second method. 21. Explain the manner of making an angle equal to a given angle. |