Ex. 2. If the polar circle extends 23° 27′ 36′′ from the pole find the convex surface of either frigid zone.

Ans., 8,128,252 square miles.

Ex. 3. On the same suppositions, find the surface of each of the temperate zones.

Ans., 51,056,587 square miles.

PROBLEM X.

(128.) To find the solidity of a spherical segment with one base.

RULE.

Multiply half the height of the segment by the area of the base, and the cube of the height by .5236, and add the two products.

See Geometry, Prop. 9, B. X.

Ex. 1. What is the solidity of either frigid zone, supposing the earth to be 7912 miles in diameter, the polar circles extending 23° 27′ 36′′ from the poles?

Ans., 1,292,390,176 cubic miles. (129.) The solidity of a spherical segment of two bases is the difference between two spherical segments, each having a single base.

Ex. 2. On the same supposition as in Ex. 1, find the solidity of either temperate zone.

Ans., 55,032,766,543 cubic miles

Ex. 3. Find the solidity of the torrid zone.

Ans., 146,682,491,911 cubic miles

PROBLEM XI.

(130.) To find the area of a spherical triangle.

RULE.

Compute the surface of the quadrantal triangle, or one eighth of the surface of the sphere. From the sum of the three angles subtract two right angles; divide the remainder by 90, and multiply the quotient by the quadrantal triangle. See Geometry, Prop. 20, B. IX.

Ex. 1. What is the area of a triangle on a sphere whose diameter is 10 feet, if the angles are 55°, 60°, and 85° ? Ans., 8.7266 square feet.

Ex. 2. If the angles of a spherical triangle measured on the surface of the earth are 78° 4′ 10′′, 59° 50′ 54′′, and 42° 5′ 37′′, what is the area of the triangle, supposing the earth a sphere, of which the diameter is 7912 miles?

Ans., 3110.794 square miles.

If the excess of the angles above two right angles is expressed in seconds, we must divide it by 90 degrees also expressed in seconds; that is, by 324,000.

PROBLEM XII.

(131.) To find the area of a spherical polygon.

RULE.

Compute the surface of the quadrantal triangle. From the sum of all the angles subtract the product of two right angles by the number of sides less two; divide the remainder by 90, and multiply the quotient by the quadrantal triangle. See Geometry, Prop. 21, B. IX.

Ex. 1. What is the area of a spherical polygon of 5 sides on a sphere whose diameter is 10 feet, supposing the sum of the angles to be 640 degrees?

BOOK IV.

SURVEYING.

(132.) THE term Surveying includes the measurement of heights and distances, the determination of the area of portions of the earth's surface, and their delineation upon paper.

Since the earth is spherical, its surface is not a plane surface, and if large portions of the earth are to be measured, the curvature must be taken into account; but in ordinary surveying, the portions of the earth are supposed to be so small that the curvature may be neglected. The parts surveyed are therefore regarded as plane figures.

(133.) If a plummet be freely suspended by a line, and allowed to come to a state of, rest, this line is called a vertical iine.

Every plane passing through a vertical line is a vertical plane.

A line perpendicular to a vertical line is a horizontal line. A plane perpendicular to a vertical line is a horizontal plane.

A vertical angle is one the plane of whose sides is vertical. A horizontal angle is one the plane of whose sides is horizontal.

An angle of elevation is a vertical angle having one side horizontal and the other an ascending

line, as the angle BAD.

An angle of depression is a vertical angle having one side horizontal and the other a descending line, as the angle CDA.

(134.) When distances are to be found A

C

D

B

by trigonometrical computation, it is necessary to measure at least one line upon the ground, and also as many angles as may be necessary to render three parts of every triangle known.

In the measurement of lines, the unit commonly employed by surveyors is a chain four rods or sixty-six feet in length, called Gunter's Chain, from the name of the inventor. This chain is divided into 100 links. Sometimes a half chain is used, containing 50 links.

(135.) To measure a horizontal line.

To mark the termination of the chain in measuring, ten iron pins should be provided, about a foot in length.

Let the person who is to go foremost in carrying the chain, and who is called the leader, take one end of the chain and the ten pins; and let another person take the other end of the chain, and hold it at the beginning of the line to be measured. When the leader has advanced until the chain is stretched tight, he must set down one pin at the end of the chain, the other person taking care that the chain is in the direction of the line to be measured. Then measure a second chain in the same manner, and so on until all the marking pins are exhausted. A record should then be made that ten chains have been measured, after which the marking pins should be returned to the leader, and the measurement continued as before until the whole line has been passed over.

It is generally agreed to refer all surfaces to a horizontal plane. Hence, when an inclined surface, like the side of a hill, is to be measured, the chain should be maintained in a horizontal position. For this purpose, in ascending a hill, the hind end of the chain should be raised from the ground until it is on a level with the fore end, and should be held vertically over the termination of the preceding chain. In descending a hill, the fore end of the chain should be raised in the same

manner.

INSTRUMENTS FOR MEASURING ANGLES.

In measuring angles, some instrument is used which contains a portion of a graduated circle divided into degrees and minutes. These instruments may be adapted to measuring

either horizontal or vertical angles. The instrument most frequently employed for ineasuring horizontal angles is called

THE SURVEYOR'S COMPASS.

(136.) The principal parts of this instrument are a compassbox, a magnetic needle, two sights, and a stand for its support. The compass-box, ABC, is circular, generally about six inches in diameter, and at its center is a small pin on which the magnetic needle is balanced. The circumference of the box is divided into degrees, and sometimes to half degrees; and the degrees are numbered from the extremities of a diameter both ways to 90°. The sights, DE, FG, are placed at right angles

to the plane of the graduated circle, and in each of these there is a large and small aperture for convenience of observation The instrument, when used, is mounted on a tripod, or a single staff pointed with iron at the bottom, so that it may be firmly placed in the ground.

Sometimes two spirit levels, H and K, are attached, to indicate when the plane of the graduated circle is brought into a horizontal position.

(137.) When the magnetic needle is supported so as to turn freely, and is allowed to come to a state of rest, the direction it assumes is called the magnetic meridian, one end of the needle indicating the north point and the other the south.

A horizontal line perpendicular to a meridian is an east and west line