188. A man wishes to build a sidewalk 6' wide and 150' long. The planks are 1''X8" S. 1 S. 1 E. He puts in 3 sleepers 4"x4" under the planks for nailing. How many board feet must he order of each lot, allowing 10% waste?

189. How many board feet of 1"X3"' tongue-and-groove flooring will be required to lay a floor 14'X 16', allowing for face measure and 5% for waste?

190. Fig. 44 shows the first-floor plan of a small cottage. How many feet B.M. of 1" X4" flooring will be required, including the porches, allowing 5% for waste?

191. Fig. 45 shows the plan of an ordinary ridge roof. How many feet B.M. will be required to sheath the roof using 1"X6" sheathing spaced 2" apart?

192. Fig. 46 shows the forms for a concrete job in which 11"X8'' shiplap is used. How many feet B.M. of shiplap will be required?

193. How much material will be required to make the cupboard shown in Fig. 47?

194. Figure the lumber required to make the kitchen cabinet shown in Fig. 48. Give the number of pieces and the B.M.

195. How much material will be required to niake the workbench shown in Fig. 49. Give the number of pieces and the B.M.

196. How many thousand shingles will be required for the roof shown in Fig. 50, if they are laid 41" to the weather?

197. How many bundles of shingles will be required for the garage shown in Fig. 51, if they are laid 41" to the weather?

198. Make out the complete lumber bill in pieces for the shed shown in Fig. 52. Give the number of pieces of each different length.

199. Make out the complete lumber bill for the barn shown in Fig. 53.

CHAPTER XIII

BUILDERS' GEOMETRY. GEOMETRIC CONSTRUC

TIONS. ANGLES AND ANGULAR MEASURE.
ELLIPSE. POLYGONS.

136. Use of Geometry. The carpenter and woodworker has to deal largely with points, straight and curved lines, plane and curved surfaces and various kinds of solids. Geometry is a study of the properties, construction and measurement of lines, surfaces and solids. A thorough knowledge of geometry is very useful.

137. Geometric Points and Lines. A point has no dimensions; it merely has position. In marking points with a pencil or scriber, we must give them some size in order to see them, but theoretically they are only imaginary.

A line has only one dimension, that of length. A line drawn with a pencil or chalk has some width, but in theory it should have no width. Lines may be straight or curved. A straight line is one that does not change its direction. A curved line is one which changes its direction at every point. A broken line consists of a series of straight lines variously directed and joined together.

138. Geometric Surfaces and Solids. A surface has two dimensions—length and breadth. A plane surface is one which will wholly contain a straight line no matter in which direction the line is laid in the plane. A curved surface is one which changes its direction in accordance

with a given law. The surface of a cylinder or sphere is a curved surface.

The definition of a plane surface gives a practical method of testing for a true plane. If a surface has a warp or wind, it may easily be detected by laying a straightedge on the surface in different positions and sighting for unevenness. If the plane is true, the straightedge will lie wholly in the surface in whatever position it may be placed. This method of testing is illustrated in Fig. 54,

Fig. 54.—Testing a Surface.

A solid is space completely surrounded by surfaces. A solid has length, breadth, and thickness.

139. Geometric Angles. An angle is formed by two straight lines which meet at a point. The point is called the vertex of the angle. A right angle is one in which the two intersecting lines are perpendicular to each other. The angle between the two edges of a steel square is a right angle. An acute angle is one in which the lines make less than a right angle. An obtuse angle is one in which the lines make more than a right angle.

An angle is designated by letters placed at the point or vertex and on the sides or legs. The letter at the vertex is always written between the letters representing the sides.

а

In Fig. 55 the angle DAB is a right angle, as is also the angle BAC, for the reason that the line AB is perpendicular to the line DC at the point A. The angles EOF and GOH are both acute angles because each is less than a right angle. The angles EOG and FOH are both obtuse angles because each is greater than a right angle.

140. Circular or Angular Measure. In Chapter X the circle was studied in part and some of its properties were investigated. In this chapter additional definitions and properties will be considered. The arc of a circle is a

part of its circumference. The part of the circumference between the points A and C in Fig. 56 is called the arc ABC. A chord is a straight line connecting two points on the circumference of a circle. The straight line joining the points A and C in Fig. 56 is spoken of as the chord ADC.

Circular arcs are measured in degrees. In a complete circle there are 360 degrees. In a fourth of a circumference or quadrant, there are 90 degrees of arc. An angle at the center is measured by the number of degrees of arc intercepted between the sides of the angle; the number of degrees in the arc gives the number of degrees in the angle. For precise measurements degrees are divided into minutes and minutes are again divided into seconds.

There are 60 seconds in one minute and 60 minutes in one degree. Degrees, minutes, and seconds are designated by the symbols ", respectively. Thus 34 degrees, 42 minutes and 30 seconds are written 34° 42' 30''.

141. Sectors and Segments. The area included between two radii and the arc is called a sector.

In Fig. 57 the area AOBC is a sector of the circle. A segment is that part of a circle which is included between an arc

and its chord. Thus in Fig. 57 the shaded portion ACBD is a segment.

142. To Bisect a Line and to Erect a Perpendicular. To bisect a line means to divide it into two equal parts. Ordinarily it would only be necessary to measure the total length of the line, divide this measure by two and lay off the half distance on the line. By the methods of geometry we not only can bisect a line without actually measuring it, but we can also erect a perpendicular line, or a line making a right angle with the first, at the middle point.