THE PLAIN SCALE.

One of these instruments is represented in the annexed figure, being such a one as is usually supplied with a pocket. case of instruments. It is made of ivory, six inches long, and one inch and three quarters broad. On the face of the instrument represented in the engraving, a protractor is formed round three of its edges for readily setting off angles. In

using this protractor, the fourth edge, which is quite plain, with the exception of a single stroke in the middle, is to be made to coincide with the line from which the angle is to be set off, and the stroke in the middle with the point in this line, at which the angle is to be set off; a mark is then to be made with the pricking point, at the point of the paper which coincides with the stroke on the protractor, marked with the number of degrees in the angle required to be drawn; and, the protractor being now removed, a straight line is to be drawn through the given point in the given line and the point thus pricked off. The instrument has on the same face the two diagonal scales already described (p. 10), and on the opposite face scales of equal parts, and several of the protracting scales already described (pp. 14-16), according to the purposes to which the scale is to be applied: thus, for laying down an ordinary survey, we merely require scales of equal parts, and a line of chords, and these consequently are all the lines placed on many of the instruments in the pocket cases; but for projecting maps, lines of sines, tangents and semitangents are required; for dialling, the dialling lines; and for the purposes of the navigator, the lines of rhumbs, and longitudes, the whole of Gunter's lines already described, and two lines of meridional, and equal parts to be used together in laying down distances, &c., upon Mercator's charts. The plain scale is sometimes fitted with rollers, as represented in our engraving, making it at the same time a convenient small parallel rule.

THE SECTOR.

scale. By its aid all questions in proportion may be solved; lines may be divided either equally or unequally into any number of parts that may be desired; the angular functions, viz., chords, sines, tangents, &c., may be set off or measured to any radius whatever; plans and drawings may be reduced or enlarged in any required proportion; and, in short, every operation in geometrical drawing may be performed by the aid of this instrument and the compasses only.

The name sector is derived from the tenth definition of the third Book of Euclid, in which this name is given to the figure contained by two radii of a circle, and the circumference between them. The instrument consists of two equal rulers, called legs, which represent the two radii, moveable about the center of a joint, which center represents the center of the circle. The legs can consequently be opened so as to contain any angle whatever, or completely opened out until their edges come into the same straight line.

Sectors are made of different sizes, and their length is usually denominated from that of the legs when shut together. Thus, a sector of six inches, such as is supplied in the common pocket cases of instruments, forms a rule of twelve inches, when opened; and this circumstance is taken advantage of, by filling up the spaces not occupied by the sectoral lines with such lines as it is most important to lay down upon a greater length than the six-inch plain scale will admit. Among these the most usual are (1) the lines of logarithmic numbers, sines, and tangents already described (pp. 2528); (2) a scale of 12 inches, in which each inch is divided into ten equal parts; and (3) a foot divided into ten equal primary divisions, each of which is subdivided into ten equal parts, so that the whole is divided into 100 equal parts. The last-mentioned is called the decimal scale, and is placed on the edge of the instrument.

The sectoral lines proceed in pairs from the center, one line of each pair on either leg, and are, upon one face of the instrument, a pair of scales of equal parts, called the line of lines, and marked L; a pair of lines of chords, marked c; a pair of lines of secants, marked s; a pair of lines of polygons, marked POL.

Upon the other face, the sectoral lines are—a pair of lines of sines, marked s; a pair of lines of tangents up to 45°, marked T; and a second line of tangents to a lesser radius, extending from 45° to 75°

Each pair of sectoral lines, except the line of polygons, should be so adjusted as to make equal angles at the center, so that the distances from the center to the corresponding divisions of any pair of lines, and the transverse distance between these divisions, may always form similar triangles. On many instruments, however, the pairs of lines of secants, and of tangents from 45° to 75°, make angles at the center equal to one another, but unequal to the angle made by all the other pairs of lines.

The solution of questions on the sector is said to be simple, when the work is begun and ended upon the same pair of lines; compound, when the operation is begun upon one pair of lines and finished upon another.

In a compound solution the two pairs of lines used must make equal angles at the center, and, consequently, in the exceptional case mentioned above, the lines of secants and of tangents above 45° cannot be used in connection with the other sectoral lines *

When a measure is taken on any of the sectoral lines beginning at the center, it is called a lateral distance; but, when a measure is taken from any point on 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 contained within three parallel lines, the innermost being the line on which the points of the compasses are to be placed, because this is the only line of the three which goes to the center, and is therefore the sectoral line.

On the Principle of the Use of the Sectoral Lines.-1st. In the case of a Simple Solution.-Let the lines a B, A c represent a pair of sectoral lines,

* Since, however, secant : rad. :: rad. : cosine;

and tangent rad. :: rad. : cotangent;

D

E

the line of sines may be used with the other sectoral lines in place of the line of secants, and the line of tangents less than 45° in place of the line of tangents greater than 45°, the complements of the angles being taken upon these tines, in either case, instead of the angles themselves. See page 41.

and B C, D E, any two transverse distances taken on this pair of lines; then, from the construction of the instrument, we have A B equal to a C, and A D equal to A E, so that A B AC:: A D A E, and the triangles A B C and A D E have the angle a A common, and the sides about this common angle propor tional (Euc. vi. prop. 6); they are, therefore, similar and

AB BC::AD: DE.

In the case of a compound solution, the angles at a are equal. but not common, and the reasoning is, in all other respects, exactly the same.

USES OF THE LINE OF LINES.

To find a Fourth Proportional to three given Lines.-Set off from the center a lateral distance equal to the first term, and open the sector till the transverse distance at the division thus found, expressing the first term, is equal to the second term; again, extend to a point whose lateral distance from the center is equal to the third term, and the transverse distance at this point will be the fourth term required

If the legs of the sector will not open far enough to make the lateral distance of the second term a transverse distance at the division expressing the first term, take any aliquot part of the second term, which can conveniently be made such transverse distance, and the transverse distance at the third term will be the same aliquot part of the fourth proportional required.

A third proportional to two given lines is found by taking a third line equal to the second, and finding the fourth proportional to the three lines.

Example.-To find a fourth proportional to the numbers 2, 5, and 10. Open the sector till the lateral distance of the second term 5 becomes the transverse distance at 2, the first term; then the transverse distance at 10 will extend, as a lateral distance, from the center to 25, the fourth proportional required.

To bisect a given Straight Line.-Take the extent of the line in the compasses, and open the sector till this extent is а ransverse between 10 and 10 on the line of lines: then the transverse distance from 5 to 5, on the same pair of sectoral lines, gives the half of the line, and this extent set off from either end will bisect it.

To divide a Straight Line into any Number of equal Parts.— 1. When the number of parts are a power of 2, the operations

are best performed by continual bisection. Thus, let it be required to divide the line A B into sixteen equal parts. 1. Make A B a transverse distance between 10 and 10 on the

line of lines; then take off the transverse distance of 5 and 5, and set it off from A or B to 8, and A B will be divided into two equal parts at the division 8. 2. Make A 8 a transverse distance at 10, and then the transverse distance at 5, set off from a or* 8 at 4, and from в or* 8 at 12, will divide the line into four equal parts at the divisions 4, 8, and 12. 3. Make the extent A 4 a transverse distance at 10, and the transverse distance at 5 will again bisect each of the parts last set off, and divide the whole line into eight equal parts at the divisions 2, 4, 6, 8, 10, 12, and 14. Each of these may be again bisected by taking the transverse distance at 21 or 2.5, that is, at the middle division between the 2 and the 3 upon the line of lines, and the line will be divided as required.

When the divisions become smaller than can be conveniently bisected by the method just explained, the operation may still be continued to any required extent by taking the extent of an odd number of the divisions already obtained as the transverse distance of 10 and 10, and setting off the half of this extent, or the transverse distance at 5, from the several divisions already obtained. Thus, in the preceding example, by making the extent of three of the divisions, or five, or seven, a transverse distance at 10, the transverse distance at 5, set off from the several divisions already obtained, will divide A B into 32 equal parts.

2. When the number of parts is not a power of 2, the opera tions cannot all be performed by bisections; but still, by a judicious selection of the parts into which the line is first divided, many of the after operations may be performed by bisections. Example.-Let it be required to divide the line

A B into seven equal parts. 1. Make the whole extent, A B, & transverse distance between 7 and 7 on the line of lines; then

* Greater accuracy is obtained by setting off the distance from both ends of the extent to be bisected, and then, in case the two points so found do not accurately coincide, taking the middle point between them, as near as the eye can judge, for the true point of bisection.