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, seven 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 great

er 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.

GUNTER'S SCALE.

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

only. The method is founded on this property, That the logarithms of the terms of equal ratios are equidifferent. This was called Gunter's Proportion, and Gunter's Line; hence the scale is generally called the Gunter.

Of the Logarithmical Lines, or Gunter's Scale. The logarithmical lines, on Gunter's scale, are the eight following:

SRhumb, or fine rhumbs, is a line containing the logarithms of the natural sines of every point and quarter point of the compass, numbered from a brass pin on the right hand towards the left with 8,7, 6, 5, 4, 3, 2, 1.

TRhumb, or tangent rhumbs, also corresponds to the logarithm of the tangent of every point and quarter point of the compass. This line is numbered from near the middle of the scale with 1. 2. 3. 4 towards the right hand, and back again with the numbers 5, 6, 7 from the right hand towards the left. To take off any number of points below four, we must begin at 1 and count towards the right hand; but to take off any number of points above four, we must begin at four and count towards the left hand.

Numbers, or the line of numbers, is numbered from the left hand of the scale towards the right with 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 which stands exactly in the middle of the scale; the numbers then go on 2, 3, 4, 5, 6, 7, 8, 9, 10 which stands at the right hand end of the scale. These two equal parts of the scale are divided equally, the distance between the first or left hand 1, and the first 2, 3, 4, &c. is exactly equal to the distance between the middle I and the numbers 2, 3, 4, &c. which follow it. subdivisions of these scales are likewise similar, viz. they are each one-tenth of the primary divisions, and are distinguished by lines of about half the length of the primary divisions.

The