contain 100° instead of 90; and the circumference will then contain 400° instead of 360. 100' instead of 60 =1° 100′′ —1'. The convenience of a decimal division we have experienced in this country in our system of Federal money. The French are likely, despite the despotism of custom, to enjoy the same advantage in all denominate numbers.

7. Another method of expressing the magnitude of angles is as follows.

A distance at pleasure is laid off from the vertex of the angle upon one of the sides, and a perpendicular there drawn to this side till it meets the other side of the angle. The ratio of this perpendicular to the distance from its foot to the vertex, serves to indicate the size of the angle.

For example, if the line BCDE be perpendicular to the line AB, and BC be one fourth AB, the angle BAC is said to be an angle of 1. If BD be one half AB, the angle BAD is said to be an angle of. If BE be equal to AB, BAE is said to be an angle of 1; and so on for other magnitudes. An angle of 1 is plainly half a right angle, or 45°.

A

E

B

This kind of measurement is much used by engineers, to express the degree of slope in excavations and embankments.

8. The protractor which we are now prepared to describe, is an instrument for drawing upon paper an angle of any given number of degrees.

This instrument is made in a variety of forms; sometimes with a full circle divided into degrees, sometimes comprising only a semicircle, sometimes upon a rectangular rule having not the circumference but the radii drawn, as they would be through the divisions of the circumference if it were actually described. The first kind is made usually of brass, or of silver, which is less liable to corrosion, and communicates no unpleasant odor to the hands. It has a metallic radius move

able about the centre of the circle and extending beyond the circumference. This prolonged radius serves to point out the number of degrees, and is armed with a sharp pin under the outer extremity for the purpose of pricking the paper, so that when the instrument is removed a line may be drawn with pencil through this point, and that upon which the centre was placed; which line shall be a radius corresponding to the number of degrees at which the instrument was set.

which is the one most commonly seen, is a semi-circle of brass, (or other metal,) having the greater part of the interior cut out to render the instrument less heavy.

The semi-circumference is divided into degrees by marks made in the metal, and these are numbered from 0° to 180 (the number in a semi-circumference) both ways, in order that the counting may commence with convenience at either end.

The degrees are also sometimes divided into half degrees, and lines of different length are employed to mark more distinctly every five and every ten degrees.*

The centre is marked by a notch in the straight side of the instrument, which side is a diameter of the semi-circle.t

* Such a division of instruments is termed graduation.

+ This instrument may be made out of paper, and a large one so made is very

accurate.

9. In order to explain the use of the instrument here described, suppose it be required to draw at the point a in the line AB a line making with AB an angle of 22°.

paper at the point c against the 22d division of the protractor, and a line joining c and A will form with AB the angle required.

10. We are now prepared to construct triangles when three parts are given, the angles in degrees and the sides in feet, yards, or other linear units.

In order to show the practical utility of trigonometry at the same time that we explain the solution of a triangle, let us take the following problem in the calculation of distances to inaccessible objects.

Suppose a fort situated upon an island, and a light-house upon the main shore, and let the distance from the light-house to the nearest salient of the fort be required.

Measure a line along the shore

of any length at pleasure, say 500 yards, beginning at the light-house. Then if two lines be imagined to be drawn from the extremities of the line just measured, to the salient of the fort, a large triangle will be formed having its two longest sides resting upon the sea. If now the angles which these two sides form with the first side, which we will call the base, could be de

791yds

105

500 yds

termined by observation upon the shore, there would be known in this triangle a side and the two adjacent angles, which would be sufficient data to construct the triangle on a small scale, and to obtain the length of the required side ex tending from the light-house to the salient of the fort.

A somewhat rude instrument for the purpose of observing such angles as those alluded to above, might be easily made.

closed at one end except a very small orifice, and having two threads crossing at right angles in the centre of the other end, so that in looking through the tube with the eye at the small orifice, the line of sight may coincide with the axis. Let this apparatus be mounted upon a three legged stand called a tripod, so that the plane of the circle shall be horizontal then, by placing the instrument thus formed at the light-house, in the example above, and sighting with the tube, first to a staff at the other extremity of the base, and then to the salient of the fort, keeping the circle stationary, the number of degrees passed over upon its circumference by the tin tube will indicate the angle of the triangle at the light-house. This angle we shall suppose to be 1051. The angle at the other extremity of the base might be found in the same manner, and suppose it 47°.*

To construct the triangle with these data, draw on paper a line AB, and make it equal in length to five hundred divisions of some scale of equal parts. Then draw an indefinite line AC, making with AB an angle of 10310. Also lay off in a similar manner at the point в an

B

* The instrument here described is of course very rude. It was deemed not advisable to encumber the work with a detailed description of more accurate instruments, which belongs properly to a treatise on surveying.

+ This may be done conveniently by taking 50 divisions, and considering each division as equal to ten.

angle of 47°, and the two lines AC and BC will meet at c. Take the line AC in the dividers and apply them to the scale. The number of equal parts upon the scale between the feet of the dividers, will show the number of yards from the lighthouse to the fort. This number is 791.

If the angle at c were required, it might be measured by applying to it the protractor; or it is equal to 180°—(A+B.) The side B c if among the sought parts might also be measured from the scale.

11. The instrument described above may be rendered suitable for application to the determination of heights. If a round bar be made to project horizontally from the top of the tripod, so that the graduated circular frame can be suspended by the socket at its centre in a vertical position, it will then serve to measure angles in a vertical plane.*

To show the use of the instrument thus prepared, take the following problem.

Required the height of a tower which stands upon horizontal ground, and the base of which is accessible.

tance the base line; at the extremity of the base line place the instrument arranged for taking vertical angles; suspend a plumb line from the centre of the circle, and the point 90° distant from that in which the plumb line cuts the circumference will be the point through which a horizontal radius would pass. Then sight with the tube to the top of the tower; the number of degrees between the tube and the horizontal radius just mentioned, will be the measure of the angle included between a line drawn to the top of the tower and the base line; let this number be 30°. Constructing a right

* A vertical plane is one perpendicular to the surface of the earth.