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. Make the lateral distance of the second term the 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 pro

portional 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 450 (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 600 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, 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 sec

tor on the line of upper tangents, will be the centre sought.

In the secant. The transverse distance of 0 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 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 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 Cно. 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 gradations do not begin till the sines end.

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 convenience of delineating any figure of a large 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 mea