From A as a centre, with a

radius greater than the shortest

distance from A to BC, describe

the indefinite arc ED: from the

B

point E as a centre, with the same radius, describe the arc AF; make ED AF, and draw AD: then will AD

=

be the parallel required.

VI. Two angles of a triangle being given, to find the third

53. Draw the indefinite line

DEF. At the point E, make the angle DEC equal to one of the given angles, and the angle CEH equal to the other: the remaining angle HEF will be the third angle required.

D

E

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VII. To represent, on paper, a line of a given length, so that any number of its parts shall correspond to the unit of the scale.

54. Suppose that the given line were 75 feet in length, and it were required to draw it on paper, on a scale of 25 feet to the inch.

The length of the line 75 feet, being divided by 25, will give 3, the number of inches which will represent the line on paper.

Therefore, draw the indefinite line AB, on which lay をB

L

A

off a distance AC equal to 3 inches: AC will represent the given line of 75 feet, drawn on the required scale.

REMARK I. This problem explains the manner of repre enting a line upon paper, so that a given number of its parts shall correspond to the unit of the scale, whether that unit be an inch or any part of an inch.

When the length of the line to be laid down is given,

be represented on the paper by a distance equal to the unit of the scale, we find the length which is to be taken from the scale by the following

RULE.

Divide the length of the line by the number of parts which is to be represented by the unit of the scale: the quotient will show the number of units which is to be taken from the scale.

EXAMPLES.

1. If a line of 640 feet is to be laid down on paper, on a scale of 40 feet to the inch; what length must be taken from the scale?.

40)640(16 inches.

2. If a line of 357 feet is to be laid down on a scale of 68 feet to the unit of the scale, (which we will suppose half an inch), how many parts are to be taken?

3. A line of 384 feet is drawn on paper, on a scale of 45 feet to the inch; what is its length on the paper? Ans. 8.53 inches.

REMARK II. When the length of a line on the paper is given, and it is required to find the true length of the line which it represents, take the line in the dividers and apply it to the scale, and note the number of units, and parts of a unit to which it is equal. Then multiply this number by the number of parts which the unit of the scale represents, and the product will be the length of the line.

For example, suppose the length of a line drawn on the paper was found to be 3.55 inches, the scale being 40 feet to the inch: then,

VIII. Having given two sides and the included angle of a tri angle, to describe the triangle.

=

55. Let the line B= 150 feet, and 120 feet, be the given sides; and A= 30 degrees, the given angle: to describe the triangle on a scale of 200 feet to the inch.

Draw the indefinite line DG, and

at the point D, make the angle GDH

equal to 30 degrees: then lay off DG equal to 150, equal to three quarters of an inch, and DH equal to 120, equal to six tenths of an

B

120

inch, and draw GH: DHG will be the required triangle.

IX. The three sides of a triangle being given, to describe the triangle.

56. Let A, B and C, be the sides. Draw DE equal to the side A. From the point D as a centre, with a radius equal to the second side B, describe an arc: from E as a centre, with a radius equal to the third side O, describe another arc inter

D

AH

B

CH

secting the former in F; draw DF and EF, and DFE will be the triangle required.

X. Having given two sides of a triangle and an angle opposite one of them, to describe the triangle.

57. Let A and B be the given sides, and the given angle, which we will suppose is opposite the side B.

Draw the indefinite line DF and make the angle FDH equal to the angle C: take DHA, from the point H, as a centre, with a radius equal to the other given side, B, describe an arc cutting

AH

BH

H

D

DF in F; draw HF: then will DHF be the required tri

If the angle C is acute, and the side B less than A, then the arc described from the centre E with the radius EF = B will cut the side DF in two points, F and G, lying on

A

BH

D

the same side of D: hence, there will be two triangles, DEF, and DEG, either of which will satisfy all the condi tions of the problem.

XI. The adjacent sides of a parallelogram, with the angle which they contain, being given, to describe the paral lelogram.

58. Let A and B be the given sides, and O the given angle.

Draw the line DH, and lay off DE equal to A; at the point D, make the angle EDF= 0; take DF=B: describe two arcs, the one from

=

D

A

BH

F

EH

F, as a centre, with a radius FG DE, the other from E, as a centre, with a radius EG = DF; through the point G, where these arcs intersect each other, draw FG, EG; DEGF will be the parallelogram required.

XII. To find the centre of a given circle or arc.

59. Take three points, A, B, C, any where in the cir cumference, or in the arc:

draw AB, BC; bisect these two lines by the perpendiculars, DE, FG: the point 0, where these perpendiculars meet, will be the centre sought.

The same construction serves for making a circumference pass through three given points A, B,

C, and also for describing a circumference, about a given

PLANE TRIGONOMETRY.

SECTION III.

DEFINITIONS.-APPLICATION TO HEIGHTS AND DISTANCES.

1. In every plane triangle there are six parts: three sides and three angles. These parts are so related to each other, that when one side and any two other parts are given, the remaining ones can be obtained, either by geometrical construction or by trigonometrical computation.

2. Plane Trigonometry explains the methods of com puting the unknown parts of a plane triangle, when a sufficient number of the six parts is given.

3. For the purpose of trigonometrical calculation, the circumference of the circle is supposed to be divided into 360 equal parts, called degrees; each degree is supposed to be divided into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds.

Degrees, minutes, and seconds, are designated respectively, by the characters ". For example, ten degrees, eighteen minutes, and fourteen seconds, would be written 10° 18' 14".

4. If two lines be drawn through the centre of the circle, at right angles to each other, they will divide the circumference into four equal parts, of 90° each. Every right angle then, as EOA, is measured by an arc of 90°; every acute angle, as BOA, by an arc less than 90°; and every obtuse angle, as FOA, by an arc greater than 90°.

5. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc EB is the complement of AB. The sum of an arc and its complement is equal to 90°.

6. The supplement of an arc is what remains after subtracting the

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B

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