42. If, with any radius, as AC, we describe the quadrant CD, and then divide it into 90 equal parts, each part is called a degree.

Through C, and each point of division, let a chord be drawn, and let the lengths of these chords be accurately laid off on a scale: such a scale is called a scale of chords. In the figure, the chords are drawn for every ten degrees.

The scale of chords being once constructed, the radius of the circle from which the chords were obtained, is known; for, the chord marked 60 is always equal to the radius of the circle. A scale of chords is generally laid down on the scales which belong to cases of mathematical instruments, and is marked CHO.

To lay off, at a given point of a line, with the scale of chords, an angle equal to a given angle.

43. Let AB be the line, and A the given point. Take from the scale the chord of 60 degrees, and with this radius and the point A as a centre, describe the arc BC. Then take from the scale the chord of the given angle, say 30 de

grees, and with this line as a radius, and B as a centre, describe an arc cutting BC in C. Through A and C

44. This instrument is used to lay down, or prot angles. It may also be used to measure angles included between lines already drawn upon paper.

It consists of a brass semicircle, ABO, 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 made only to degrees. There is a small notch at the middle of the diameter AB, which indicates the centre of the protractor.

To lay off an angle with a Protractor.

45. 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.

30 40 50 60 70

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46. The sector is an instrument generally made of ivory or brass. It consists of two arms, or sides, which open by turning round a joint at their common extremity.

There are several scales laid down on the sector: those, however, which are chiefly used in drawing lines and angles, are, the scale of chords already described, and the scale of equal parts now to be explained.

On each arm of the sector, there is a diagonal line that passes through the point about which the arms turn: these diagonal lines are divided into equal parts.

On the sectors which belong to the cases of English instruments, the diagonal lines are designated by the letter L, and numbered from the centre of the sector, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, to the two extremities. On the sectors which belong to cases of French instruments, they are designated, "Les parties egales," and numbered 10, 20, 30, 40, &c., to 200. On the English sectors there are 20 equal divisions between any two of the lines numbered 1, 2, 3, &c., so that there are 200 equal parts on the scale.

The advantage of the sectoral scale of equal parts, is this

When it is proposed to draw a line upon paper, on such a scale that any number of parts of the line, 40 for example, shall be represented by one inch on the paper, or by any part of an inch, take the inch, or part of the inch, from the scale of inches on the sector: then, placing one foot of the dividers at 40 on one arm of the sector, open the sector until the other foot reaches to the corresponding number on the other arm: then lay the sector on the table without varying the angle.

Now, if we regard the lines on the sector as the sides of a triangle, of which the line 40, measured across, is the base, it is plain, that if any other line be likewise meas ured across the angle of the sector, the bases of the tri angles, so formed, will be proportional to their sides Therefore, if we extend the dividers from 50 to 50, this distance will represent a line of 50, to the given scale.

Let it now be required to lay down a line of sixty. seven feet, to a scale of twenty feet to the inch.

Take one inch from the scale of inches: then place one foot of the dividers at the twentieth division, and open the sector until the dividers will just reach the twentieth division on the other arm: the sector is then set to the proper angle; after which the required distance to be laid down on the paper is found by extending the dividers from the sixty-seventh division on one arm, to the sixty-seventh division on the other.

GUNTER'S SCALE.

47. This is a scale of two feet in length, on the faces of which a variety of scales is 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.

SOLUTION OF PROBLEMS REQUIRING THE USE OF THE INSTRUMENTS THAT HAVE BEEN DESCRIBED.

I. At a given point in a given straight line, to erect a perpen

dicular to the line.

48. 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 BA, describe two arcs intersecting each other in D: draw AD, and it will be the perpendicular required.

B

A

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II. From a given point without a straight line, to let fall a perpendicular on the line.

From the point A as a centre, with a radius sufficiently great, describe an arc cutting the line BD in the two points B and D: then mark a point E, equally distant from the points B and D,

B

A

and draw AE: AE will be the perpendicular required.

III. At a point, in a given line, to make an angle equal to a given angle.

50. Let A be the given point, AE the given line, and IKL the given angle.

From the vertex K, as a centre, with any radius, describe the arc IL, terminating in the two sides of the angle. From

E

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.

IV. To divide a given angle, or a given arc, into two equal

parts.

51. Let C be the given angle, and AEB the arc which measures it.

From the points A and B as centres, describe with the same radius two arcs cutting each other in D: through D and the centre C draw CD: the angle ACE will be equal to the angle ECB, and the arc AE to the arc EB.

/B

E

V. Through a given point to draw a parallel to a given line.

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