Practical Geometry.-Instruments.

REMARK II. When the length of a line is given on the paper, 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 an 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.

1. Suppose the length of a line drawn on the paper, to be 3,55 inches, the scale being 40 feet to the inch: then,

3,55 × 40=142 feet, the length of the line.

2. If the length of a line on the paper is 6,25 inches, and the scale be one of 30 feet to the inch, what is the

Take ab for the

10. This scale is thus constructed.

unit of the scale, which may be one inch, 2, or 3 of an

QUEST.-10. Explain the construction of the scale of equal parts?

Practical Geometry.-Instruments.

inch, in length. On ab describe the square abcd. Divide the sides ab and dc each into ten equal parts. Draw af and the other nine parallels as in the figure.

Produce ba to the left, and lay off the unit of the scale any convenient number of times, and mark the points 1, 2, 3, &c. Then, divide the line ad into ten equal parts, and through the points of division draw parallels to ab as in the figure.

Now, the small divisions of the line ab are each one tenth (1) of ab; they are therefore,1 of ad, or,1 of ag or gh.

If we consider the triangle adf, the base df is one tenth of ad the unit of the scale. Since the distance from a to the first horizontal line above ab, is one-tenth of the distance ad, it follows that the distance measured on that line between ad and af is one-tenth of df: but since one-tenth of a tenth is a hundredth, it follows that this distance is one hundredth (,01) of the unit of the scale. A like distance measured on the second line will be two hundredths (,02) of the unit of the scale; on the third, ,03; on the fourth, ,04, &c.

If it were required to take, in the dividers, the unit of the scale and any number of tenths, place one foot of the dividers at 1, and extend the other to that figure between a and b which designates the tenths. If two or

Practical Geometry.—Instruments.

more units are required, the dividers must be placed on a point of division farther to the left.

When units, tenths, and hundredths, are required, place one foot of the dividers where the vertical line through the point which designates the units, intersects the line which designates the hundredths: then, extend the dividers to that line between ad and be which designates the tenths: the distance so determined will be the one required.

For example, to take off the distance 2,34, we place one foot of the dividers at 7, and extend the other to e: and to take off the distance 2,58, we place one foot of the dividers at p and extend the other to q.

REMARK I.—If a line is so long that the whole of it cannot be taken from the scale, it must be divided, and the parts of it taken from the scale in succession.

REMARK II.—If a line be given upon the paper, its length can be found by taking it in the dividers and applying it to the scale.

QUEST.-Show how to take off 1,35, also 2,47, also 1,78. If the line is so long that the whole of it cannot be taken at once, what do you do? If the line be given on paper, what do you do?

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

Through A, 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 CнO.

QUEST.-11. Explain the construction of the scale of chords. What chord is equal to the radius of the circle?

Practical Geometry.-Problems.

PROBLEM VI.

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

12. Let AB be the line, and A the given point.

Take from the scale the chord of

A

B

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 degrees, and with this line as a radius, and B as a centre, describe an arc cutting BC in C. Through A and C draw the line AC, and BAC will be the required angle.

QUEST.-12. Explain the manner of laying off an angle with the scale of chords.