Take the given line between the compasses; apply the two feet transversely to the scale concerned, and slide the feet along till they both rest on like divisions on both legs; then will those divisions show the degrees and parts corresponding to the given line.

To find the length of a versed sine to a given number of degrees, and a given radius.

Make the transverse distance of 90 and 90 on the sines equal to the given radius.

Take the transverse distance of the sine complement of the given degrees.

If the given degrees are less than 90, the difference between the sine complement and the radius gives the versed sine.

If the given degrees are more than 90, the sum of the sine complement and the radius gives the versed sine.

To open the legs of the sector so that the corresponding double scales of lines chords, sines, and tangents may make each a right angle.

On the lines, make the lateral distance 10 a distance between 8 on one leg and 6 on the other leg.

On the sines, make the lateral distance 90 a transverse distance from 45 to 45; or from 40 to 50; or from 30 to 60; or from the sine of any degrees to their complement.

Or on the sines, make the lateral distance of 45 a transverse distance between 30 and 30.

1

OF THE PLAIN SCALE.

The divisions laid down on the plain scale are of two kinds, the one having more immediate relation to the circle and its properties, the other being merely concerned with dividing straight lines.

Though arches of a circle are the most natural measure of an angle, yet in many cases right lines are substituted, as being more convenient; for the comparison of one right line with another is more natural and easy than the comparison of a right line with a curve: hence it is usual to measure the quantities of angles, not by the arch itself, which is described on the angular point, but by certain lines described about that arch.

The lines laid down on the plain scales for the measuring of angles, or the protracting scales, are, 1, A line of chords marked CHO. 2. A line of sines marked SIN., of tangents marked TAN., of semitangents marked ST., and of secants marked SEC.; this last is often upon the same line as the sines, because its gra

There are two other scales, namely, the rhumbs marked RU and longitudes marked LON. Scales of latitude and hours are sometimes put upon the plain scale; but as dialling is now but seldom studied, they are only made to order.

The divisions used for measuring straight lines are called scales of equal parts, and are of various lengths for the conve nience of delineating any figure of a larger or smaller size, according to the fancy or purposes of the draughtsman. They are, indeed, nothing more than a measure in miniature for laying down upon paper, &c. any known measure, 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 simply, 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 show 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 ten equal parts, which small parts are those which give a name to the scale: thus it is called a scale of 20, when 20 of these divisions are equal to one inch. If, therefore, these less divisions be taken as units, and each repreɛents 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 35, 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 sixth subdivision, reckoning backwards or towards the left hand. Then will this extent represent 8, 36, or 360 miles or leagues, &c. and bear the same proportion in the plan

To adapt these scales to feet and inches, the first primary division is often duodecimally divided by the upper line; therefore, to lay down any number of feet and inches, as, for instance, 8 feet 8 inches, extend the compasses from 8 of the larger to 8 of the upper small ones, and that distance laid down on the plan will represent 8 feet 8 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 there is usually half an inch divided into 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-hundredth 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 a unit, and the second, marked by the diagonal upon the parallels, hundredth 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 5 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 a 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 a 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 lay

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 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