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found by extending the dividers from the sixty-seventh division on one arm, to the sixty-seventh division on the other.
32. This is a scale of two feet in length, on the faces of which several 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.
SOLUTION OF PROBLEMS.
L. At a given point, in a given straight line, to erect a perpendicular to the line.
33. 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 at D: draw AD, and it will be the perpendicular required.
II. From a given point, without a straight line, to let fall a perpendicular on the line.
34. Let A be the given point, and BD the given 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, and draw
III. At a point, in a given line, to make an angle equal to a
35. 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, termi
nating 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.
IV. To divíde a given angle, or a given arc, into two equal parts.
36. Let ACB 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.
V. Through a given point, to draw a parallel to a given line.
37. Let A be the given point, and BC the given line. From A as a centre, with a radius greater than the shortest distance from A to BC, describe the indefinite arc ED. Then, from the point E as a centre, with the same radius, describe the arc AF; make ED
= AF, and draw AD: thon will AD be the parallel required.
VI. Two angles of a triangle being given, to find the third.
38. 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.
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.
39. 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 off
from C, a distance AC equal to 3 inches: AC will represent the given line of 75 feet, drawn to the required scale.
NOTE. This problem explains the manner of representing 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, and it has been determined how many parts of it are to 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
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?
5.25 parts, or
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.
NOTE. 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,
3.55 X 40 142 feet, the length of the line.
VIII. Having given two sides and the included angle of a triangle, to describe the triangle.
40. Let the line B 150 feet, and C 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 feet, equal to three-quarters of an inch, and DH equal to 120 feet, equal to six-tenths of an inch, and draw GH: then, DHG will be the required triangle.
IX. The three sides of a triangle being given, to describe the triangle.
41. Let A, B and C be the sides. Make 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
C, describe another arc, intersecting 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.
42. Let A and B be the given sides, and C the given angle, which we will suppose to be opposite the side B.
Draw the indefinite line DF: then, A
at any point of it, as D, make the Br angle FDE equal to the angle C: take
DEA, and from the point E, as a
centre, with a radius equal to the other
given side, B, describe an arc, cutting DF in F; draw EF: then will DEF be the required triangle.
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