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of rest : the direction which it assumes, is called the magnetic meridian.
In the interior of the compass box, there is a graduated circle divided to degrees, and sometimes to half degrees; the degrees are usually numbered from the two extremities of the diameter NS, which is in the same plane with the hairs of the vertical sights, both ways, to 90°.
The length of the magnetic needle is a little less than the diameter of the graduated circle, so that the needle can move freely around its centre, within the circle, and its positions be noted on the graduated arc.
In using the compass, it is important to ascertain the exact angle, included between the magnetic meridian and the direction which may be given to the sights AF and BG, by turning the compass around on the stand.
For this end, a small arc, HI, is described on the bar AB, having its centre at the centre of the compass box. This are
. is divided to degrees, and sometimes even to the parts of a degree, and a vernier is permanently attached to the compass box. When the point of the vernier coincides with the 0 point of the graduated arc HI, the line of the compass box, marked NS, or the diameter from whose vertices the degrees of the graduated circle are numbered, is in the same vertical plane with the hairs of the sights.
The compass box is turned about its centre, without moving the plate AB, by the milled screw L; it is fastened to the plate AB, by the screw P.
Now, the 0 of the vernier coinciding with the 0 of the arc HI, if the needle does not stand at one of the lines of division of the graduated circle, let the whole degrees be read; then turn the compass box, by means of the screw L, until the needle points exactly to the line which marked the whole degrees, the space passed over by the 0 of the vernier indicates the minutes that are to be added.
Compasses are often constructed, not only for the purpose of ascertaining the angles which lines form with the magnetic meridian, but also, for the accurate measurement of horizonmade shorter, a vernier is attached to it, and it is mounted on a graduated circle, similar to the horizontal limb of the theodolite. This limb must always be made horizontal before measuring horizontal angles.
OF INSTRUMENTS FOR PLOTTING. 126. A plot or plan of a piece of ground is the accurate delineation, on paper, of its bounding or other principal lines
or such a representation of it, that all the lines and angles drawn on the paper have the same mutual relation as their corresponding lines and angles on the ground. In making such representations certain drawing instruments are used, some of which are now to be deseribed.
The use of the common compasses, or dividers, as well as that of the plain rule, is too obvious to need particular explanation.
OF THE SEMICIRCULAR PROTRACTOR. 127. This instrument is used to lay down, or protract angles, and also to measure the angles included, by lines given in position, upon paper.
It is a brass semicircle ACB, (Pl. 5, Fig. 1, divided to half degrees; the degrees are numbered to 180, both ways, from A and B. There is a small notch at the middle of the diameter AB, to show the centre of the protractor.
128. To lay off, with the protractor, any angle, at a given point of a right line.--Place the diameter AB on the line, so that the centre shall fall on the given 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 given point: this line makes with the given line the required angle.
OF THE CIRCULAR PROTRACTOR. 129. This instrument consists of a brass circular limb, (Pl. 5, Fig. 2) of about six inches diameter, with a moveable
screw at the other extremity B, with a concealed cog-wheel that works with the cogs of the limb, and thus moves the index AB, about the centre of the protractor. At the centre of the protractor, is a small circular glass plate, on which two lines are cut; the point of their intersection, is the exact centre of the instrument. The limb is generally divided to half degrees; the degrees are numbered from 0 to 360.
At the 0 point, and at the opposite extremity of the diameter passing through that point, are small lines on the inner edge of the limb; the two extremities of the diameter, perpendicular to this latter, are also designated in the same way.
Two angular pieces of brass, each having a small and sharp steel pin at its extremity, are fastened to the index, and revolve freely around the lines ab and cd. The small screws a, b, c, and d, move them in the directions of the lines ab, cd, for the purpose of bringing the steel pins exactly into the line which passes through the 0 of the index, and the centre of the protractor.
The vernier is generally divided into 30 parts, by means of which accurate readings can be made to l'.
130. To lay off an angle with this protractor.—Let its centre be placed over the angular point, and the diameter passing through 0 and 180° on the given line : turn the screw that works the index until the 0 of the vernier coincides with the division corresponding to the given angle; then let the anguJar brass pieces be turned down; the points dotted by the steel pins will show the direction of the line making the required angle with the given line.
If this line does not pass through the angular point, the pins are out of place, as they ought to be in the line passing through the centre and the 0 of the vernier, in which case they must be adjusted by the screws a, b, c, and d.
OF THE SCALE OF CHORDS. 131. If a circle be described with any radius, and a diameter be drawn, and from either of its extremities arcs of 1, 2, 3, 4, 5, &c. to 90° be laid off, and the corresponding chords drawn; then, if the lengths of these chords be laid
down accurately on a scale, such scale is called a Scale of Chords.
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 sectors which belong to cases of mathematical instruments, and marked CHO.
132. To lay off an angle with the scale of chords.- Take a radius equal to the chord marked 60, and with the angular point for a centre describe an arc of a circle ; then, take from the scale, the chord of the given arc, and apply it from the point where the arc before described intersects the given line ; the line drawn through the extremity of the chord and the angular point, makes, with the given line, the required angle.
OF SCALES OF EQUAL PARTS. 133. A scale of equal parts is one which shows any number of equal distances, or spaces. If a line of a given length, one inch for example, be divided into any number of equal parts, say thirty, such a scale is called a scale of thirty parts to an inch, and the same, whatever be the length of the line divided.
If a line, on the ground, of a given length, estimated in feet, yards, or rods, is to be laid down on paper, let a number of parts, equal to the number of feet, yards, or rods, in the given line, be taken from the scale, with a pair of dividers, and extended upon the paper.
If a number of lines be laid down after the same manner, their lengths on the paper will obviously have the same proportion as the lines on the ground, and if they be plotted with the further condition of making with each other the same angles as they formed on the ground, the plan on the paper will be, in every respect, similar to the figure which it represents.
If a line upon the paper, of one inch in length, represents any the plan or plot is said to be made upon a scale of twenty feet to the inch: and similarly, whatever be the relation of the numbers that represent corresponding parts.
Of The DIAGONAL SCALE. This scale is thus constructed. Take a line, AD, (Pl. 5, Fig. 3,) equal in length to , }, }, or 1 inch, or of any other convenient dimension, and describe on it the square ADCB. Divide the sides AD, AB, each into ten equal parts, join A and a, and draw the other nine parallels as in the figure. Produce DA, and lay off the distance AD any convenient number of times, from A to F, from F to G, from G to E, &c., and number the points of division 1, 2, 3, &c. Then divide the line DC into ten equal parts, and through the points of division draw parallels to ED, as in the figure. Now, the small divi. sions on the line AD are each one tenih (.1) of that line, and therefore .1 of AF, or FG. It follows from the proportion of similar triangles, that the part of the line .01, intercepted between the lines AB and Aa, is equal to .l of Ba; the like intercepted part of the line .02, equal to .2 of Ba, the like intercepted part of .03, equal to .3 of Ba, and similarly for the other parallels.
These latter spaces, being tenths of the divisions Al, 12, &c. which are themselves tenths of AD, are hundredths of the line AB or AD.
Naming the line AD the unit of the scale, if it were required to take in the dividers the value of the unit and any number of tenths, let the dividers be placed at F, and extend to the figure between A and D, which designates the tenths. If two or more units were required, the dividers must be placed at that point on the line AE, which is designated by a number equal to the number of units. If units, tenths, and hundredths, are wanted, place the dividers at the intersection of the proper ţine of units with the line designating the hundredths, and then extend them to the intersection of the line designating the tenths with this line of hundredths: thus, for the number 2.44 the dividers are placed at b, and extended to