Note. One acute angle of a right-angled triangle being the complement of the other, or the sum of the two acute angles being equal to 90°, when one of the acute angles is given, the other is also given.

Solution.-Extend the compass

from 90°, or radius, on the line of A sines to the number of degrees in

either of the acute angles, and that extent will reach back wards, on the line of numbers, from the hypothenuse to the side opposite this angle. Example.-Given the hypothenuse AB=250, and the angle a= = 35° 30'.

Extend from 90° to 35° 30' on the line of sines, and this extent will reach from 250 to 145 on the line of numbers.

Extend from 90° to 54° 30′ on the line of sines, and this extent will reach from 250 to 203.5 on the line of numbers

90° 0'

BAC 35 30

ABC 54 30

.. BC 145

.. AC=203.5

Case 2. The angles, and one side being given, to find the hypothenuse, and the other side.

Solution.-Extend from the angle opposite the given side to 90°, or radius, on the line of sines, and this extent will reach forwards from the given side to the hypothenuse on the line of numbers. Again, extend from the angle opposite the given side to the angle opposite the required side, and this extent will reach in the same direction on the line of numbers, from the given side to the required side. Or, extend from radius, or 45°, on the line of tangents, to the angle opposite the required side, and the extent will reach, in the same direction on the line of numbers, from the given side to the required side; recollecting that, when the angle is greater than 45°, the extent is to be taken on the scale backwards from rad. or 45° to the complement of the angle, but is to be reckoned a forward distance, the logarithmic tangents of angles greater than 45° exceeding the logarithmic tangents of 45°, or radius, by as much as the logarithmic tangents of their complements fall short of it. Example.-Given the angle a = 35° 30′ and side Ac=203·5.

Extend from 54° 30' to 90°, or rad., upon the line of sines, and this extent will reach forwards from 203.5 to 250 on the line of numbers

Again, extend from 54° 30′ backwards to 35° 30′, on the line of sines, and this extent will reach backwards from 203.5 to 145 on the line of numbers.

90° 0'

BA0=35 30

ABC-54 30 ... AB=250

.. BC=145

Or extend backwards from 45°, rad., to 35° 30′ on the line of tangents,

and this extent will reach backwards from 203.5 to 145 on the line of numbers, as before*.

Case 3. The hypothenuse and one side being given, to find the angles and the other side.

Solution.-Extend from the hypothenuse to the given side. on the line of numbers, and this extent will reach from 90 or rad. to the angle opposite the given side upon the line of sines. The other angle is the complement of this. Extend upon the line of sines from the rad. to the angle last found, which is opposite the required side, and this extent will reach from the hypothenuse to the required side. Example.--Given the hypothenuse AB=250, and the side AC=203·5. Extend backwards from 250 to 203.5 on the line of numbers, and this extent will reach from 90° to 54° 30′ on the line of sines

90° 0'

... ABC 54 30

Extend from 90 to 35° 30′ on the line of sines, and this extent will reach backwards from 250 to 145 on the line of numbers

BAC 35 30

... BC=145

Case 4. The two sides being given, to find the angles and the hypothenuse.

Solution.-Extend from one side to the other upon the line of numbers, and this extent will reach backwards upon the line of tangents from rad. to the least angle, and to the same point, considered as a forward distance, representing the greatest angle, which is the complement of the least. Again, extend on the line of sines from one of the angles just found to rad., and this extent will reach from the side opposite the angle taken to the hypothenuse. Example.-Given Ac=203·5 and BC145.

Extend backwards upon the line of numbers from 203.5 to 145, and this extent will reach backwards from 45° to 35° 30′ on the line of tangents, which is the angle opposite the side 145

If we measure forwards from 145 w 203.5, then from rad. to 35° 30' is to be considered a forward distance, and the angle to be taken as the complement of 35° 30′, that is, 54° 30', which is the angle opposite the side 203.5.

Again, extend from 33° 38′ to 90° on the line of sines, and this extent will reach from 145 to 250 upon the line of numbers

90° 0'

.. BAC=35 30

.. ABC 54 30

=

.. A B = 250

* The property that tan. : rad. :: sine cosine, may be made a test of the accuracy of the scale, since the distance from 45 to any angle upon the `ine of tangents ought to be the same as the distance from the angle to its complement upon the line of sines.

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 o; 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;

E

and tangent: rad. :: rad. : cotangent;

E

C

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 lines, in either case, instead of the angles themselves. See page 41.