sure, as chains, yards, feet, &c. each part on the scale answering to one foot, one yard, &c, and the plan will be larger or smaller, as the scale contains a smaller or a greater number of parts in an inch. Hence a variety of scales is useful to lay down lines of any required length, and of a convenient proportion with respect to the size of the drawing. If none of the scales happen to suit the purpose, recourse should be had to the line of lines on the sector; for, by the different openings of that instrument, a line of any length may be divided into as many equal parts as any person chooses.

Scales of equal parts are divided into two kinds, the one simple, the other diagonally divided.

Six of the simply divided scales are generally placed one above another upon the same rule; they are divided into as many equal parts as the length of the rule will admit of; the numbers placed on the right hand, shew how many parts in an inch each scale is divided into. The upper scale is sometimes shortened for the sake of introducing another called the line of chords.

The first of the larger, or primary divisions, on every scale is subdivided into 10 equal parts, which small parts are those which give a naine to the scale: thus it is called a scale of 20, when 20 of these divisions are equal to one inch. If, therefore, these lesser divisions be taken as units, and each represents one league, one mile, one chain, or one yard, &c. then will the larger divisions be so many tens; but if the subdivisions are supposed to be tens, the larger divisions will be hundreds.

To illustrate this, suppose it were required to set off from either of the scales of equal parts, 36, or 360 parts, either miles or leagues. Set one foot of your compasses on 3, among the larger or primary divisions, and open the other point till it falls on the 6th subdivision, reckoning backwards

or towards the left hand. Then will this extent represent, 336, or 360 miles or leagues, &c. and bear the same proportion in the plan as the line measured does to the thing represented.

To adapt these scales to feet and inches, the first primary division is often duodecimally divided by an upper line; therefore, to lay down any number of feet and inches, as for instance, eight feet eight inches, extend the compasses from eight of the larger to eight of the upper small ones, and that distance laid down on the plan will represent eight feet eight inches.

Of the scale of equal parts diagonally divided. The use of this scale is the same as those already described. But by it a plane may be more accurately divided than by the former; for any one of the larger divisions may by this be subdivided into 100 equal parts; and, therefore, if the scale contains 10 of the larger divisions, any number under 1000 may be laid down with accuracy.

The diagonal scale is seldom placed on the same side of the rule with the other plotting scale. The first division of the diagonal scale, if it be a foot long, is generally an inch divided into 100 equal parts, and at the opposite end there is usually half an inch divided into an 100 equal parts. If the scale be six inches long, one end has commonly half an inch, the other a quarter of an inch subdivided into 100 equal parts.

The nature of this scale will be better understood 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, by 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 0 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, hundreth 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 be 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, 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 off 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 show, 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 show the latitudes; the line of longitude shows 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 pre