SECTION V.

GEOMETRICAL CONSTRUCTION OF TRIANGLES, BY THE PLANE SCALE.

ART. 156. To facilitate the construction of geometrical figures, a number of graduated lines are put upon the common two feet scale; one side of which is called the Plane Scale, and the other side, Gunter's Scale. The most important of these are the scales of equal parts, and the line of chords. In forming a given triangle, or any other right lined figure, the parts which must be made to agree with the conditions proposed, are the lines, and the angles. For the former, a scale of equal parts is used; for the latter, a line of chords.

157. The line on the upper side of the plane scale, is divided into inches and tenths of an inch. Beneath this, on the left hand, are two diagonal scales of equal parts,* divided into inches and half inches, by perpendicular lines. On the larger scale, one of the inches is divided into tenths, by lines which pass obliquely across, so as to intersect the parallel lines which run from right to left. The use of the oblique lines is to measure hundredths of an inch, by inclining more and more to the right, as they cross each of the parallels.

To take off, for instance, an extent of 3 inches, 4 tenths, and 6 hundredths;

Place one foot of the compasses at the intersection of the perpendicular line marked 3 with the parallel line marked 6, and the other foot at the intersection of the latter with the oblique line marked 4.

The other diagonal scale is of the same nature. The divisions are smaller, and are numbered from left to right.

158. In geometrical constructions, what is often required, is to make a figure, not equal to a given one, but only similar. Now figures are similar which have equal angles, and the

*These lines are not represented in the plate, as the learner is supposed to have the scale before him.

sides about the equal angles proportional. (Euc. Def. 1. 6.) Thus a land surveyor, in plotting a field, makes the several lines in his plan to have the same proportion to each other, as the sides of the field. For this purpose, a scale of equal parts may be used, of any dimensions whatever. If the sides of the field are 2, 5, 7, and 10 rods, and the lines in the plan are 2, 5, 7, and 10 inches, and if the angles are the same in each, the figures are similar. One is a copy of the other, upon a smaller scale.

So any two right lined figures are similar, if the angles are the same in both, and if the number of smaller parts in each side of one, is equal to the number of larger parts in the corresponding sides of the other. The several divisions on the scale of equal parts may, therefore, be considered as representing any measures of length, as feet, rods, miles, &c. All that is necessary is, that the scale be not changed, in the construction of the same figure; and that the several divisions and subdivisions be properly proportioned to each other. If the larger divisions, on the diagonal scale, are units, the smaller ones are tenths and hundredths. If the larger are tens, the smaller are units and tenths.

159. In laying down an angle, of a given number of degrees, it is necessary to measure it. Now the proper measure of an angle is an arc of a circle. (Art. 74.) And the measure of an arc, where the radius is given, is its chord. For the chord is the distance, in a straight line, from one end of the arc to the other. Thus the chord AB (Fig. 33.) is a measure of the arc ADB, and of the angle ACB.

To form the line of chords, a circle is described, and the lengths of its chords determined for every degree of the quadrant. These measures are put on the plane scale, on the line marked CHO.

160. The chord of 60° is equal to radius. (Art. 95.) In laying down or measuring an angle, therefore, an arc must be drawn, with a radius which is equal to the extent from 0 to 60 on the line of chords. There are generally, on the scale, two lines of chords. Either of these may be used; but the angle must be measured by the same line from which the radius is taken.

161. To make an angle, then, of a given number of degrees; From one end of a straight line as a centre, and with a radius equal to the chord of 60° on the line of chords, describe an arc of a circle cutting the straight line. From the point of intersection, extend the chord of the given number

and

of degrees, applying the other extremity to the arc; through the place of meeting, draw the other line from the the angular point.

If the given angle is obtuse, take from the scale the chord of half the number of degrees, and apply it twice to the arc. Or make use of the chords of any two arcs whose sum is equal to the given number of degrees.

A right angle may be constructed, by drawing a perpendicular without using the line of chords.

Ex. 1. To make an angle of 32 degrees. (Fig. 33.) With the point C, in the line CH, for a centre, and with the chord of 60° for radius, describe the arc ADF. Extend the chord of 32° from A to B; and through B, draw the line BC. Then is ACB an angle of 32 degrees.

2. To make an angle of 140 degrees. (Fig. 34.) On the line CH, with the chord of 60°, describe the arc ADF; and extend the chord of 70o from A to D, and from D to B. The arc ADB 70° × 2=140°.

On the other hand,

162. To measure an angle; On the angular point as a centre, and with the chord of 60° for radius, describe an arc to cut the two lines which include the angle. The distance between the points of intersection, applied to the line of chords, will give the measure of the angle in degrees. the angle be obtuse, divide the arc into two parts.

If

Des

The dis

Ex. 1. To measure the angle ACB. (Fig. 33.) cribe the arc ADF cutting the lines CH and CB. tance AB will extend 32° on the line of chords. 2. To measure the angle ACB. (Fig. 34.) arc ADB into two parts, either equal or unequal, and measure each part, by applying its chord to the scale. The sum of the two will be 140o.

Divide the

163. Besides the lines of chords, and 'of equal parts, on the plane scale; there are also lines of natural sines, tangents, and secants, marked Sin. Tan. & Sec. of semitangents marked S. T. of longitude marked Lon. or M. L. of Rhumbs mark ed Rhu. or Rum. &c. These are not necessary in trigonometrical constructions. Some of them are used in Navigation; and some of them, in the projections of the Sphere..

164. In Navigation, the quadrant, instead of being graduated in the usual manner, is divided into eight portions, call ed Rhumbs. The Rhumb line,on the scale, is a line of chords, divided into rhumbs and quarter-rhumbs, instead of degrees. 165. The line of Longitude is intended to show the num

ber of geographical miles in a degree of longitude, at different distances from the equator. It is placed over the line of chords, with the numbers in an inverted order: so that the figure above shows the length of a degree of longitude, in any latitude denoted by the figure below*. Thus at the equa. tor, where the latitude is 0, a degree of longitude is 60 geographical miles. In latitude 40, it is 46 miles; in latitude 60, 30 miles, &c.

166. The graduation on the line of secants begins where the line of sines ends. For the greatest sine is only equal to radius; but the secant of the least arc is greater than radius.

167. The semitangents are the tangents of half the given arcs. Thus the semitangent of 20° is the tangent of 10°. The line of semitangents is used in one of the projections of the sphere.

168. In the construction of triangles, the sides and angles which are given are laid down according to the directions in arts. 158, 161. The parts required are then measured, according to arts. 158, 162. The following problems correspond with the four cases of oblique angled triangles; (Art. 148.) but are equally adapted to right angled triangles.

169. PROB. I. The angles and one side of a triangle being given; to find, by construction, the other two sides. Draw the given side.

of the given angles.

From the ends of it, lay off two Extend the other sides till they intersect; and then measure their lengths, on a scale of equal parts.

Ex. 1. Given the side b 32 rods, (Fig. 27.) the angle A 56° 20′, and the angle C 49° 10'; to construct the triangle, and find the lengths of the sides a and c.

From a scale of equal parts, make b=32. With the line of chords, make, at one end of b, an angle of 56° 20', and at the other end, an angle of 49° 10'. From A and C, draw the lines a and c, till they meet in B. Their lengths, on the same scale from which b was taken, will be 25 and 27.

2. In a right angled triangle, (Fig. 17.) given the hypoth* Sometimes the line of longitude is placed under the line of chords.

L

enuse 90, and the angle A 32° 20', to find the base and per pendicular.

Draw the hypothenuse, and make the angle A equal to 32° 20'. Extend AB indefinitely, and make the angle C equal to the complement of A; or let fall a perpendicular from C upon AB. (Euc. 12. 1.) The length of AB will be 76, of BC 48.

3. Given the side AC 68, the angle A 124°, and the angle C 379; to construct the triangle.

170. PROB. II. Two sides and an opposite angle being given, to find the remaining side, and the other two angles.

Draw one of the given sides; from one end of it, lay off the given angle; and extend a line indefinitely, for the required side. From the other end of the first side, with the remaining given side for radius, describe an are cutting the indefinite line. The point of intersection will be the end of the required side.

If the side opposite the given angle be less than the other given side, the case will be ambiguous. (Art. 152.)

Ex. 1. Given the angle A 63° 35' (Fig. 29.) the sideb32, and the side a 36.

Draw the side b 32, make the angle A 63° 35', and extend AB indefinitely. From C, with 36 for radius, describe an arc cutting the indefinite line in B, and draw the side a from B to C. The side AB will be 36 nearly, the angle B 52° 45' and C 63° 391.

21. Given the angle A (Fig. 28.) 35° 20'; the opposite side a 25, and the side b 35.

Draw the side b 35, make the angle A 35° 20′ and extend AH indefinitely. From C, with radius 25, describe an arc cutting AH in B and B'. Draw CB and CB', and two triangles will be formed, ABC and AB'C, each corresponding with the conditions of the problem.

3. Given the angle A 116°, the opposite side a 38, and the side b 26; to construct the triangle.

171. PROB. III. Two sides and the included angle being given; to find the other side and angles.

Draw one of the given sides. From one end of it, lay off the given angle, and draw the other given side. Then connect the extremities of this and the first line.

Ex. 1. Given the angle A (Fig. 30.) 26° 14', the side & 78, and the side c 106; to find B, C, and a.