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CHAPTER XV.

MENSURATION.

DEFINITIONS.

308. If a block of wood or stone be cut in the shape represented in Fig. 1, it will have six flat faces.

FIG. 1.

Each face of the block is called a surface; and if these faces be made smooth by polishing, so that, when a straight edge is applied to any one of them, the straight edge in every part will touch the surface, the faces are called plane surfaces or planes.

309. The sharp edge in which any two of these surfaces meet is called a line.

310. The place at which any three of these lines meet is called a point.

311. The distance between the right and left faces is a dimension, called the length; the distance between the front face and the back face is another dimension, called the breadth (or width); the distance between the top and bottom face is a third dimension, called the thickness (height or depth).

A solid, therefore, has three dimensions, length, breadth,

[graphic]

and thickness.

312. The surface of a solid is no part of the solid. It is simply the boundary or limit of the solid. So that if any number of flat surfaces be put together, they will have no thickness, but will coincide and form only one surface.

A surface, therefore, has only two dimensions, length and breadth.

313. A line is no part of a surface. It is simply a boundary or limit of the surface. So that, if any number of straight lines be put together, they will have no thickness and no width, but will coincide and form only one line.

A line, therefore, has only one dimension, length.

314. A point is no part of a line. It is simply the limit of the line. So that, if any number of points be put together, they will have no length, breadth, or thickness, but will coincide and form a single point.

A point, therefore, has no dimension, but denotes position simply.

C

D

315. A point is represented to the eye by a fine dot, and named by a letter, as A (Fig. 2); a line is named by two letters, placed one at each end, as BF; a surface is represented and named by the lines which bound it, as BCDF; a solid is represented by the faces which bound it.

B

FIG. 2.

316. A line which has the same direction throughout is called a straight line, as A

[blocks in formation]

B

D

E

318. A line composed of

G

several straight lines lying

FIG. 3.

F

H

in different directions is called a broken line, as EF.

319. A line composed of straight and curved lines is called a mixed line, as GH.

320. A plane surface or a plane is a surface in which, if any two points be taken, the straight line joining these two points will lie wholly within the surface.

321. A curved surface is a surface no part of which is plane.

322. Since all straight lines which pass through the same point in the same direction coincide, the position of a straight line is known if its direction and one of the points are known.

Since all straight lines connecting two points coincide, the position of a straight line is known if two points of the line are known.

323. Of all lines between two points, the shortest is the straight line; and the straight line is called the distance between the two points.

324. When two straight lines cross each other, they are said to intersect, and the point of crossing is called their point of intersection.

325. To draw a straight line on paper between two points, we place the straight edge of a ruler so that it touches the two points; then draw the line with a pencil, keeping the pencil in contact with the ruler.

FIG. 4.

326. To draw a straight line on wood, we chalk a cord, and stretch it across two points, through which the line is to pass. While the cord is held firmly at two points, it is pulled away from the wood, and then let go. It strikes the wood, and leaves a white trace, which is a straight line.

327. To measure the distance between two points on paper, we employ an instrument called a pair of compasses (Fig. 4), or a divided rule.

ANGLES.

328. The figure formed by two straight lines, limited at their point of meeting, is called an angle,

as BAC (Fig. 5).

The point of meeting A is the vertex of the angle, the lines AB and AC are the sides of the angle.

C

[blocks in formation]

An angle is designated by placing a letter at its vertex and one at each of its sides. In naming the angle, we name the letter at the vertex, or the three letters, putting the letter at the vertex between the other two, as BAC. An angle is also designated by putting a small letter within the angle, and close to the vertex.

329. The magnitude of an angle depends wholly upon the extent of opening of its sides, and not upon their length. Thus, if the sides of the angle BAC, namely AB and AC, be prolonged, their extent of opening will not be altered, and the size of the angle will not be altered; but, if the side AB remains fixed, and the side AC is made to turn about the vertex A, the angle increases or decreases according as the moving side is turned from or towards AB. 330. Two angles are equal when they can be applied the one to the other so as to coincide. Thus, the two angles BAC and EDF are equal, if, when the vertex A is A placed upon D, and the line AB upon DE, the line AC falls falls below DF, the angle BAC is less than the angle EDF. If AC falls above DF, the angle BAC is greater than the angle EDF

C

F

Ᏼ Ꭰ
FIG. 6.

E

upon the line DF. If AC

331. When two angles have a common vertex and a common side, they are called adjacent angles. Thus, the angles FCB and FCE (Fig. 7) are adjacent angles.

332. A line is perpendicular to another when it makes with this other two equal adjacent angles, called right angles.

Α

Suppose the line CF(Fig. 7) can turn freely about the point C. Suppose also that CF lies upon CB. FIf we make CF turn towards the left about

E

D

FIG. 7.

B

C, the angle BCF at first will be less than the angle ACF. But the angle BCF will increase, and the angle ACF diminish as CF moves to the left about the point C. There will be one position CE in which the two angles BCE and ACE will be equal. In this position EC is said to be perpendicular to AB, and the angles ECB and ECA are called right angles.

333. The sum of all the angles about a point in a straight line, situated on the same side of the straight line, is equal to two right angles (180°), and therefore the sum of all the angles about a point in a plane (on both sides of a straight line drawn through the point) is equal to four right angles (360°).

334. An acute angle is an angle less than a right angle, as BCF (Fig. 7).

335. An obtuse angle is an angle greater than a right angle, as ACF (Fig. 7).

336. If the sum of two angles is equal to a right angle (90°), each angle is called the complement of the other. Thus the angles BCF and FCE (Fig. 7) are complementary angles.

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