4. What are the solid contents of a block of stone 3 ft. 2′ long, 2 ft. 3′ 6′′ wide, and 4 ft. 2' high?

Ans. 30 cu. ft. 2′ 10′′ 2′′.

5. What is the surface measure of the foregoing block? Ans. 60 sq ft.

3. DIVISION OF DUODECIMALS.

1. Divide 22 ft. 3' 9" by 5 ft. 3'.

SOLUTION.

Explanation.-5 ft. are contained in 22

5 ft. 3' 22 ft. 3' 9' 4 ft. 3' ft. 4 times. Multiplying the entire

21 ft. 0'

1 ft. 3' 9/

1 ft. 3' 9/

divisor by 4, and subtracting, we have 1 ft. 3'9'' remaining. Reducing 1 ft. to primes, we have 1 ft. 3′ 9′′ 15′9′′; the divisor is contained in this number 3' times; multiplying as before and subtracting, we have no remainder. Hence, 4 ft. 3' is the quotient.

2. If a floor contain 216 sq. ft. 5′ 10′′ 6"", and is 10 ft. 6' wide, how long is it? Ans. 20 ft. 7′ 5′′.

3. A block of marble contains 112 cu. ft. 5' 4", and is 5 ft. 4' wide and 3 ft. 10' thick: how long is it? Ans. 5 ft. 6'.

4. A board contains 26 sq. ft. 6", and is 12 ft. 6′ long: how wide is it? Ans. 2 ft. 1'.

5. A plank is 5' thick, 20 ft. 2' long, and contains 14 cu. ft. 8"": how wide is it? Ans. 1 ft. 8'.

CHAPTER XVII.

MEASUREMENTS.

329. The process of finding the measure of lines, surfaces and volumes is called Mensuration. The rules of Mensuration depend on the principles of Geometry. Many of them are of much importance, and are therefore given here without an attempt to explain them by means of geometrical demonstration.

DEFINITIONS.-1. LINES.

330. A Line is that which has length, but no breadth or thickness.

331. A Straight Line is one whose direction does not change.

332. A Curved Line is one whose direction changes at every point.

333. Parallel Lines are those which extend in the same direction, and are equally distant from each other at all points.

334. A Horizontal Line is a line parallel with the horizon or water-level.

2. ANGLES.

335. An Angle is the difference in the direction of two lines meeting at a common point, called the vertex.

336. A Right Angle is one in which the two lines are perpendicular to each other.

337. An Obtuse Angle is one greater than

a right angle.

338. An Acute Angle is one less than a right angle.

339. A Polygon is a figure bounded by straight lines. The distance around a polygon is called its perimeter.

340. A polygon of three sides is called a triangle; of four sides, a quadrilateral; of five, a pentagon, etc.

341. The Area of a polygon is the surface included within the lines which bound it.

3. TRIANGLES.

342. A Triangle is a polygon of three sides and three angles; as, A B C. The side A B in the figure is called the base; B C, the altitude; and AC the hypothenuse.

343. A Right-angled Triangle is one which has one right angle.

B

344. An Equilateral Triangle is one whose sides are equal.

345. An Isosceles Triangle has two sides equal.

346. A Scalene Triangle has all its sides unequal.

The following are the rules for the measurements of triangles:

It was proved in Denominate Numbers (Art. 107) that the area of a parallelogram is equal to the product of its base and altitude, and since a triangle is half a parallelogram, we derive the following

RULE.

1. To find the area of a triangle when the base and the altitude are given, take half the product of the base and altitude.

Ex. 1. What is the area of a triangle whose base is 16 in. and altitude 25 in.?

2. What is the surface of the gable end of a house 30 ft. wide and 12 ft. 6 in. high? Ans. 187 sq. ft. 3. How many acres in a triangular field whose base is 920 ft. and perpendicular 816 ft. Ans. 8 A. 98 P. 1991 sq. ft. 4. What will it cost to paint the two gables of a house 25 ft. wide and 8 ft. 3 in. high, at 6 cts. a square foot. Ans. $12.371⁄2.

2. When the three sides of a triangle are given, to find the area, from half the sum of the sides subtract each side separately; multiply the three remainders together, and this product by the half sum of the sides; the square root of the product is the area.

1. What is the area of a triangle whose sides are 20, 40 and 36 ft. respectively? Ans. 359.2-sq. ft.

2. What is the area of an equilateral triangle whose sides are each 60 ft.? Ans. 1559-sq. ft.

3. A man owns a triangular field at the junction of two roads; its sides are 40 rd., 60 rd. and 50 rd. respectively: how many acres does it contain? Ans. 6 A. 32.16 P.

The following principles have been established by Geometry: 1. The square described on the hypothenuse of a rightangled triangle is equal to the sum of the squares described on the two other sides.

2. The square of the base, or of the perpendicular, of a rightangled triangle is equal to the square of the hypothenuse, less the square of the other side.

Hence, the following rule to find the hypothenuse:

RULE.

To the square of the base add the square of the perpendicular, and extract the square root of this sum.

Ex. 1. What is the hypothenuse of a right-angled triangle whose base is 20 and the perpendicular 25? Ans. 32.015+.

SOLUTION.

202+252=400+ 625 = 1025

1025=32.015+

2. If the two sides of a rectangular field are 500 ft. and 700 ft. respectively, how far from one corner to the opposite one? Ans. 860+ ft.

3. If the sides of a house are 40 ft. and 30 ft. respectively, what must be the length from one corner to the opposite that the corners may be perfect right angles? Ans. 50 ft.

4. The height of a gable is 8 ft., and the width of the house 30 ft. how long must the rafters be that they may project over the edge 2 ft. ? Ans. 19 ft.

5. Two vessels start at the same port, one sailing south at the rate of 12 mi. an hour, the other east at the rate of 10 mi. an hour: how far apart will they be at the end of 72 hr.? Ans. 1124.67 + mi.

From the second principle the following rule, to find the base or the perpendicular when the hypothenuse and one of the other sides are given, is derived:

RULE.

From the square of the hypothenuse subtract the square of the given side; the square root of the remainder is the other side. Ex. 1. A ladder 40 ft. long, whose foot is 10 ft. from a building, reaches to a window: how high is the window from the ground? Ans. 38.73 ft.

402-102-1600-100-1500

1500-38.73-ft.

2. A man standing 200 ft. from the foot of a tree shoots a bird from the top; the bullet travels 250 ft.: how high is the tree if the man's eye was 5 ft. higher than the foot of the tree? Ans. 155 ft.

3. A boat in crossing a stream 620 ft. wide goes 800 ft.: how far does it drift down stream? Ans. 505.57 ft. 4. The rafters on a house are 20 ft. long, the width of the gable is 30 ft., the rafters project 2 ft. the gable?

what is the height of Ans. 9.95 ft.

5. A room is 18 ft. long, 15 ft. wide and 10 ft. high: what is the distance from an upper corner to the opposite lower

one?

Ans. 25.475 ft.