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26. This scale is thus constructed. Take ab for the unit of the scale, which may be one inch,,, or of an 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, we see, that 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, is 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 more units are required,
the dividers must be placed on a point of division further 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 embraced 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.
NOTE 1.—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.
NOTE 2. 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.
27. 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.
If through C, and each point of division, a chord be drawn,
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.
III. To lay off, at a given point of a line, with the scale of chords, an angle equal to a given angle.
28. 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 degrees, and with this distance 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.
29. 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 semicircle, ABC, 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.
IV. To lay off an angle with a Protractor.
30. 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 toward B, or from B toward 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.
31. The sector is an instrument generally made of ivory or trass. 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
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 égales,” 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:
Let it be 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 two sides of a triangle, of which the line 40, measured across, is the base, it is plain, that if any other line be likewise measured across the angle of the sector, the bases of the triangles, 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 and similarly for other lines.
V. 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