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In the sines. The transverse distance of 90 and 90 will be the radius sought.
In the lower tangents. The transverse distance of 45 and 45, near the end of the sector, will be the radius sought.
In the upper tangents. The transverse distance of 45 and 45, taken towards the centre of the sector on the line of upper tangents, will be the centre sought.
In the socant. The transverse distance of O and 0, or the beginning of the secants, near the centre of the sector, will be the radius sought.
Given the radius and any line representing a sine, tangent, or secant; to find the degrees corresponding to that line.
Solution. Set the sector to the given radius, according as a sine, or tangent, or secant is concerned.
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 shew the degrees and parts corresponding to the
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 corres
ponding 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 eight on one leg, and six 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.
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 measures 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 semitantangents marked st. and of secants marked sec. this last is often upon the same line as the sines, because its gradations do not begin till the sines end.
There are two other scales, namely, the rhumbs, marked Ru. and longitudes, marked con. 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 convenience of delineating any figure of a large or smaller size, according to the fancy or purposes of the draughts-man. 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 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 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 desser 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 44, 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, ti 36, 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 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 cach of these subdivisions, by drawing diagonal lines from the 10th below to the 9th above; from the 8:1 below to the 7th above; and so on, till from the first below to the 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 isone-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