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The nature of this scale will be better undera stood by considering its construction. For this purpose :
First. Draw eleven parallel lines at equal distances ; divide the upper of these lines into such a number of equal parts, as the scale to be expressed is intended to contain; from each of these divisions draw perpendicular lines through the eleven parallels.
Secondly. Subdivide the first of these divisions into ten equal parts, both in the upper and lower lines.
Thirdly. Subdivide again each of these subdivisions, hy drawing diagonal lines from the 10th below to the 9th above; from the 8th below to the 7th above; and so on, till from the first below to the O above; by these lines each of the small divisions is divided into ten parts, and, consequently, the whole first space into 100 equal parts; for, as each of the subdivisions is one-tenth part of the whole first space or division, so each parallel above it is one-tenth of such subdivision, and, consequently, one-hundreth part of the whole first space: and if there be ten of the larger divisions, one-thousandth part of the whole space.
If, therefore, the larger divisions be accounted as units, the first subdivisions will be tenth parts of an unit, and the second, marked by the diagonal upon the parallels, bundreth parts of the unit. But, if we suppose the larger divisions to be tens, the first subdivisions will be units, and the second tenths. If the larger are hundreds, then will the first he tens, and the second units.
The numbers therefore, 576, 57,6, 5,76, are all expressible by the same extent of the compasses : thus setting one foot in the number five of the larger divisions, extend the other along the sixth parallel to the seventh diagonal. For, if the five
larger divisions be taken for 500, seven of the first subdivisions will be 70, which upon the sixth parallel, taking in six of the second subdivisions for units, makes the whole number 576. Or, if the five larger divisions be taken for five tens, or 50, geven of the first subdivisions will be seven units, and the six second subdivisions upon the sixth parallel, will be six tenths of an unit. Lastly, if the five larger divisions be only esteemed as five units, then will the seven first subdivisions be seven tenths, and the six second subdivisions be the six hundredth parts of an unit.
Of the line of chords. This line is used to set off an angle from a given point in any right line, or to measure the quantity of an angle already laid down.
Thus to draw a line that shall make with another line an angle, containing a given number of degrees, suppose 40 degrees.
Open your compasses to the extent of 60 degrees upon the line of chords, (which is always equal to the radius of the circle of projection,) and setting one foot in the angular point, with that extent describe an arch; then taking the extent of 40 degrees from the said chord line, set it off from the given line on the arch described; a right line drawn from the given point, through the point marked upon the arch, will form the required angle.
The degrees contained in an angle already laid down, are found nearly in the same manner; for instance, to measure an angle. From the centre describe an arch with the chord of 60 degrees, and the length of the arch, contained between the lines measured on the line of chords, will give the number of degrees contained in the angle.
If the number of degrees are more than 90, they must be measured upon the chords at twice: thus, if 120 degrees were to be practised,60 may be taken from the chords, and those degrees be laid of
twice upon the arch. Degrees taken from the chords are always to be counted from the beginning of the scale.
Of the rhumb line. This is, in fact, a line of chords constructed to a quadrant divided into eight parts or points of the compass, in order to facilitate the work of the navigator in laying down a ship's course.
Of the line of longitudes. The line of longitudes is a line divided into sixty unequal parts, and so applied to the line of chords, as to shew, by inspection, the number of equatorial miles contained in a degree on any parallel of latitude. The graduated line of chords is necessary, in order to shew the latitudes; the line of longitude shews the quantity of a degree on each parallel in sixtieth parts of an equatorial degree, that is, miles.
The lines of tangents, semitangents, and secants, serve to find the centres and poles of projected circles in the stereographical projection of the sphere.
The line of sines is principally used for the orthographic projection of the sphere.
The lines of latitudes and hours are used conjointly, and serve very readily to mark the hour lines in the construction of dials ; they are generally on the most complete sorts of scales and sectors; for the uses of which see treatises on dialling,
OF THE PROTRACTOR.
This is an instrument used to protract, or lay down an angle containing any number of degrees, or to find how many degrees are contained in any given angle. There are two kinds put into cases of mathematical drawing instruments ; one in the form of a semicircle, the other in the form of a parallelogram. The circle is undoubtedly the only natural measure of angles ; when a straight line is therefore used, the divisions thereon are derived
from a circle, or its properties, and the straight line is made use of for some relative convenience : it is thus the parallelogram is often used as a protractor, instead of the semicircle, because it is in some cases more convenient, and that other scales, &c. may be placed upon it.
The semicircular protractor, is divided into 180 equal parts or degrees, which are numbered at every tenth degree each way, for the conveniency of reckoning either from the right towards the left, or from the left towards the right; or the more easily to lay down an angle from either end of the line, beginning at each end with 10, 20, &c. and proceeding to 180 degrees. The edge is the diameter of the semicircle, and the mark in the middle points out the centre, in a protractor in the form of a parallelogram : the divisions are as in the semicircular one, numbered both ways; the blank side represents the diameter of a circle. The side of the protractor to be applied to the paper is made flat, and that whereon the degrees are marked, is chamfered or sloped away to the edge, that an angle may be more easily measured, and the divisions set off with greater exactness.
Application of the protractor to use. 1. A number of degrees being given, to protract, or lay down an angle, whose measure shall be equal thereto.
Thus, to lay down an angle of 60 degrees from the point of a line, apply the diameter of the protractor to the line, so that the centre thereof may coincide exactly with the extremity; then with a protracting pin make a fine dot against 60 upon the limb of the protractor ; now remove the protractor, and draw a line from the extremity through that point, and the angle contains the given number of degrees.
2. To find the number of degrees contained in a given angle.
Place the centre of the protractor upon the angular point, and the fiducial edge, or diameter, exactly upon the line ; then the degree upon the limb that is cut by the line will be the measure of the · given angle, which, in the present instance, is found to be 60 degrees.
3. From a given point in a line, to erect a perpendicular to that line.
Apply the protractor to the line, so that the centre may coincide with the given point, and the division marked 90 may be cut by the line; then a line drawn against the diameter of the protractor will be the perpendicular required.
OF PARALLEL RULES,
Parallel lines occur so continually in every species of mathematical drawing, that it is no.wonder so many instruments have been contrived to delineate them with more expedition than could be effected by the general geometrical methods. For this purpose, rules of various constructions have been made ; and particularly recommended by their inventors; their use however is so apparent as to need no explanation.
The scale generally used is a ruler of two feet in length, having drawn upon it equal parts, chords, sines, tangents, secants, &c. These are contained on one side of the scale, and the other side contains the logarithms of these numbers. Mr. Edmund Gunter was the first who applied the logarithms of numbers, and of sines and tangents to straight lines drawn on a scale or ruler; with which, proportions in common numbers, and trigonometry, may
be solved by the application of a pair of compasses