Each pair of these lines (except the line of polygons) is so adjusted as to make equal angles at the centre, and consequently at whatever distance the sector be opened, the angles will be always respectively equal. That is, the distance between 10 and 10 on the line of lines, will be equal to 60 and 60 on the line of chords, 90 and 90 on the line of sines, and 45 and 45 on the line of tangents.

Besides the sectoral scales, there are others on each face, placed parallel to the outward edges, and used as those of the common plain scale. There are on the one face, 1. A line of inches. 2. A line of latitudes. 3. A line of hours. 4. A line of inclination of meridians. 5. A line of chords. On the other face, three logarithmic scales, namely, one of numbers, one of sines, and one of tangents; these are used when the sector is fully opened, the legs forming one line.

To read and estimate the divisions on the sectoral lines. The value of the divisions on most of the lines are determined by the figures adjacent to them; these proceed by tens, which constitute the divisions of the first order, and are numbered accordingly; but the value of the divisions on the line of lines, that are distinguished by figures, is entirely arbitrary, and may represent any value that is given to them; hence the figures 1, 2, 3, 4, &c. may denote either 10, 20, 30, 40; or 100, 200, 300, 400, and so on.

The line of lines is divided into ten equal parts, numbered 1, 2, 3, to 10; these may be called divisions of the first order; each of these are again subdivided into 10 other equal parts, which may be called divisions of the second order; and each of these is divided into two equal parts, forming divisions of the third order.

'The divisions on all the scales are contained between four parallel lines; those of the first order

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extend to the most distant; those of the third, to the least; those of the second to the intermediate parallel.

When the whole line of lines represents 100, the divisions of the first order, or those to which the figures are annexed, represent tens; those of the second order, units; those of the third order, the halves of these units. If the whole line represents ten, then the divisions of the first order are units; those of thesecond, tenths, and the thirds, twentieths.

In the line of tangents, the divisions to which the numbers are affixed, are the degrees expressed by those numbers. Every fifth degree is denoted by a line somewhat longer than the rest; between every number and each fifth degree, there are four divisions, longer than the intermediate adjacent ones, these are whole degrees; the shorter ones, or those of the third order, are 30 minutes.

From the centre, to 60 degrees, the line of sines is divided like the line of tangents; from 60 to 70, it is divided only to every degree; from 70 to 80, to every two degrees; from 80 to 90, the division must be estimated by the eye.

The divisions on the line of chords are to be estimated in the same manner as the tangents.

The lesser line of tangents is graduated every two degrees from 45 to 50; but from 50 to 60, to every degree; from 60 to the end, to half degrees.

The line of secants from 0 to 10, is to be estimated by the eye; from 20 to 50 it is divided to every two degrees; from 50 to 60, to every degree; and from 60 to the end, to every half degree.

The solution of questions on the sector is said to be simple, when the work is begun and ended on the same line; compound, when the operation begins on one line, and is finished on the other.

The operation varies also by the manner in which the compasses are applied to the sector. If a mea

sure be taken on any of the sectoral lines, beginning at the centre, it is called a lateral distance. But if the measure be taken from any point in one line, to its corresponding point on the line of the same denomination, on the other leg, it is called a transverse or parallel distance.

The divisions of each sectoral line are bounded by three parallel lines; the inermost of these is that on which the points of the compasses are to be placed, because this alone is the line which goes to the centre, and is alone, therefore, the sectoral line.

We shall now proceed to give a few general instances of the manner of operating with the sector. Multiplication by the line of lines. Make the lateral distance of one of the factors the parallel distance of 10; then the parallel distance of the other factor is the product.

Example. Multiply 5 by 6, extend the compasses from the centre of the sector to 5 on the primary divisions, and open the sector till this distance become the parallel distance from 10 to 10 on the same divisions; then the parallel distance from 6 to 6, extended from the centre of the sector, shall reach to 3, which is now to be reckoned 30. At the same opening of the sector, the parallel distance of 7 shall reach from the centre to 35, that of 8 shall reach from the centre to 40, &c.

Division by the line of lines. Make the lateral distance of the dividend the parallel distance of the divisor, the parallel distance of 10 is the quotient. Thus, to divide 30 by 5, make the lateral distance of 30, viz. 3 on the primary divisions, the parallel distance of 5 of the same divisions; then the parallel distance of 10, extended from the centre, shall reach to 6.

Proportion by the line of lines. distance of the second term the

Make the lateral parallel distance

of the first term; the parallel distance of the third term is the fourth proportional.

Example. To find a fourth proportional to 8, 4, and 6, take the lateral distance of 4, and make it the parallel distance of 8; then the parallel distance of 6, extended from the centre, shall reach to the fourth proportional 3.

In the same manner a third proportional is found to two numbers. Thus, to find a third proportional to 8 and 4, the sector remaining as in the former example, the parallel distance of 4, extended from the centre, shall reach to the third proportional 2. In all these cases, if the number to be made a parallel distance be too great for the sector, some aliquot part of it is to be taken, and the answer multiplied by the number by which the first number was divided. Thus, if it were required to find a fourth proportional to 4, 8, and 6; because the lateral distance of the second term 8 cannot be made the parallel distance of the first term 4, take the lateral distance of 4, viz. the half of 8, and make it the parallel distance of the first term 4; then the parallel distance of the third term 6, shall reach from the centre to 6, viz. the half of 12. Any other aliquot part of a number may be used in the same way. In like manner, if the number proposed be too small to be made the parallel distance, it may be multiplied by some number, and the answer is to be divided by the same number.

To protract angles by the line of Chords. Case 1. When the given degrees are under 60. 1. With any radius on a centre, describe the arch. 2. Make the same radius a transverse distance between 60 and 60 on the line of chords. 3. Take out the transverse distance of the given degrees, and lay this on the arch, which will mark out the angular distance required.

Case 2. When the given degrees are more than

60. 1. Open the sector, and describe the arch as before. 2. Take or of the given degrees, and take the transverse distance of this or, and lay it off twice, if the degrees were halved, three times if the third was used as a transverse distance.

Case 3. When the required angle is less than 6 degrees; suppose 3. 1. Open the sector to the given radius, and describe the arch as before. 2. Set off the radius. 3. Set off the chord of 57 degrees backwards, which will give the arc of three degrees.

Given the radius of a circle, suppose equal to two inches, required the sine and tangent of 28° 30' to that radius.

Solution. Open the sector so that the transverse distance of 90 and 90 on the sines, or of 45 and 45 on the tangents, may be equal to the given radius, viz. two inches; then will the transverse distance of 38° 30′, taken from the sines, be the length of that sine to the given radius; or if taken from the tangents; will be the length of that tangent to the given radius.

But if the secant of 28° 30′ was required?

Make the given radius, two inches, a transverse distance to 0 and 0, at the beginning of the line of secants; and then take the transverse distance of the degrees wanted, viz. 28° 30'.

A tangent greater than 45° (suppose 60o ) is found thus.

Make the given radius, suppose two inches, a transverse distance to 45 and 45 at the beginning of the scale of upper tangents; and then the required number 60° 00' may be taken from this scale.

Given the length of the sine, tangent or secant of any degrees; to find the length of the radius to that sine, tangent, or secant.

Make the given length a transverse distance to its given degrees on its respective scale: then,