§ XLII. PERMUTATION.

ART. 323. PERMUTATION is the process of finding the different orders in which may be arranged a given number of things.

ART. 324. To find the number of different arrangements that can be made of any given number of things.

Ex. 1. How many different numbers may be formed from the figures of the following number, 432, making use of three figures in each number? Ans. 6.

FIRST OPERATION.

In the 1st operation,

4 32, 423, 342, 324, 243, 234. we have made all the

SECOND OPERATION.

1 × 2 × 3 = 6.

different arrangements that can be made of the given figures, and find the number to be 6. In

the second operation, the same result is obtained by simply multiplying together the first three of the digits, a number equal to the number of figures to be arranged. Hence the following

RULE. Multiply together all the terms of the natural series of numbers, from 1 up to the given number, and the last product will be the answer required.

EXAMPLES FOR PRACTICE.

2. My family consists of nine persons, and each person has his particular seat around my table. Now, if their situations were to be changed once each day, for how many days could they be seated in a different position?

Ans. 362880 days, or 994 years 70 days.

3. On a certain shelf in my library there are 12 books. If a person should remove them without noticing their order, what would be the probability of his replacing them in the same position they were at first? Ans. 1 to 479001600. 4. How many words can be made from the letters in the word "Embargo," provided that any arrangement of them may be used, and that all the letters shall be taken each time?

Ans. 5040 words.

QUESTIONS. Art. 323. What is permutation? Art. 324. What is the rule for finding the number of arrangements that can be made of any given number of things?

§ XLIII. MENSURATION OF SURFACES.

ART. 325. A SURFACE is a magnitude, which has length and breadth without thickness.

The surface or superficial contents of a figure are called its

area.

ART. 326. An ANGLE is the inclination or opening of two lines, which meet in a point.

A right angle is an angle formed by one line falling perpendicularly on another, and it contains 90 degrees; as A B C.

An acute angle is an angle less than a right angle, or less than 90 degrees; as E B C.

An obtuse angle is an angle greater than a right angle, or more than 90 degrees; as FB C.

THE TRIANGLE.

ART. 327. A TRIANGLE is a figure having three sides and three angles. It receives the particular names of an equilateral triangle, isosceles triangle, and scalene triangle.

It is also called a right-angled triangle when it has one right angle; an acute-angled triangle, when it has all its angles acute; and an obtuse-angled triangle, when it has one obtuse angle.

The base of a triangle, or other plane figure, is the lowest side, or that which is parallel to the horizon; as C D.

The altitude of a triangle is a line drawn from one of its angles perpendicular to its opposite side or base; as A B.

An equilateral triangle is a figure which has its three sides equal.

QUESTIONS. Art. 325. What is a surface? What are the superficial contents of a figure called? Art. 326. What is an angle? What is a right angle? An acute angle? An obtuse angle? Art. 327. What is a triangle? What particular names does it receive? When is it called a right-angled triangle? When an acute-angled triangle? When an obtuse-angled triangle? What is the base of a triangle? What the altitude? What is an equilateral triangle ?

An isosceles triangle is a figure which has two of its sides equal.

A scalene triangle is a figure which has its three sides unequal.

A right-angled triangle is a figure having three sides and three angles, one of which is a right angle.

ART. 328. To find the area of a triangle.

RULE 1. Multiply the base by half the altitude, and the product will be the area.

Or,

RULE 2. - Add the three sides together, take half that sum, and from this subtract each side separately; then multiply the half of the sum and these remainders together, and the square root of this product will be the area.

1. What are the contents of a triangle, whose base is 24 feet, and whose perpendicular height is 18 feet? Ans. 216 feet. 2. What are the contents of a triangular piece of land, whose sides are 50 rods, 60 rods, and 70 rods?

Ans. 1469.69+ rods.

THE QUADRILATERAL.

ART. 329. A QUADRILATERAL is a figure having four sides, and consequently four angles. It comprehends the rectangle, square, rhombus, rhomboid, trapezium, and trapezoid.

ART. 330. A PARALLELOGRAM is any quadrilateral whose opposite sides are parallel. It takes the particular names of rectangle, square, rhombus, and rhomboid.

The altitude of a parallelogram is a perpendicular line drawn between its opposite sides; as C D in the rhomboid.

QUESTIONS.-What is an isosceles triangle? A scalene triangle? A rightangled triangle? Art. 328. What is the first rule for finding the area of a triangle? What the second? Art. 329. What is a quadrilateral? What figures does it comprehend? — Art. 330. What is a parallelogram? What particular names does it take? What is the altitude of a parallelogram ?

A rectangle is a right-angled parallelogram, whose opposite sides are equal.

A square is a parallelogram, having four equal sides and four right angles.

A rhomboid is an oblique-angled parallelogram, whose opposite sides are equal.

A rhombus is an oblique-angled parallelogram, having all its sides equal.

ART. 331. To find the area of a parallelogram.

RULE.

Multiply the base by the altitude, and the product will be the

area.

1. What are the contents of a board 25 feet long and 3 feet wide? Ans. 75 square feet. 2. What is the difference between the contents of two floors; one is 37 feet long and 27 feet wide, and the other is 40 feet long and 20 feet wide? Ans. 199 square feet.

3. The base of a rhombus is 15 feet, and its perpendicular height is 12 feet; what are its contents?

Ans. 180 square feet.

ART. 332, A TRAPEZOID is a quadrilateral, which has only one pair of its opposite sides parallel.

ART. 333. To find the area of a trapezoid.

RULE.-Multiply half of the sum of the parallel sides by the altitude, and the product is the area.

1. What is the area of a trapezoid, the longer parallel side

QUESTIONS. What is a rectangle? A square? A rhomboid? A rhombus? -Art. 331. What is the rule for finding the area of a parallelogram? Art. 332. What is a trapezoid? — Art. 333. What is the rule for finding the area of a trapezoid ?

being 482 feet, the shorter 324 feet, and the altitude 216 feet? Ans. 87048 square feet.

2. What is the area of a plank, whose length is 22 feet, the width of the wider end being 28 inches, and of the narrower 2ɩ inches? Ans. 44 square feet.

ART. 334. A TRAPEZIUM is a quadrilateral, which has neither two of its opposite sides parallel.

A diagonal is a line joining any two opposite angles of a quadrilateral; as E F.

ART. 335. To find the area of a trapezium.

RULE.

E

F

Divide the trapezium into two triangles by a diagonal, and hen find the areas of these triangles; their sum will be the area of the trapezium.

1. What is the area of a trapezium, whose diagonal is 65 feet, and the length of the perpendiculars let fall upon it are 14 and 18 feet? Ans. 1040 square feet.

2. What is the area of a trapezium, whose diagonal is 125 rods, and the length of the perpendiculars let fall upon it are 70 and 85 rods? Ans. 9687.5 square rods.

THE POLYGON.

ART. 336. A POLYGON is any rectilineal figure having more than four sides and four angles. It takes the particular names of pentagon, which is a polygon of five sides; hexagon, one of six sides; heptagon, one of seven sides; octagon, one of eight sides; nonagon, one of nine sides; decagon, one of ten sides; undecagon, one of eleven sides; and dodecagon, one of twelve sides.

ART. 337, A REGULAR POLYGON is a plane rectilineal figure, which has all its sides and all its angles equal.

The perimeter of a polygon is the sum of all its sides.

ART. 338. To find the area of a regular polygon.

QUESTIONS. Art. 334. What is a trapezium? What is a diagonal? Art. 335. What is the rule for finding the area of a trapezium? Art. 336. What is a polygon? What particular names does it take? - Art. 337. What is a regular polygon?