PERMUTATIONS.

381. PERMUTATION is the arranging of a given number of things in every possible order of succession.

382. PROBLEM. To find the number of permutations of a given number of things.

The single letter, a, can have but 1 position, i. e. it cannot stand either before or after itself; the 2 letters, a and b, furnish the 2 permutations,

Sabr

bas

the number of which is expressed by the product of

1 X 2 = 2; and if a 3d letter, c, be introduced, we have

cab, C b a

a cb, b c a a b C, bac

; i. e. the new letter, c, may stand 1st, 2d, or 3d

in each of the 2 permutations of a and b; hence the number of permutations of 3 things is expressed by the product, 1 × 2 × 3 =6. If a 4th letter, d, be taken, it may stand as 1st, 2d, 3d, or 4th, in each of the 6 permutations of a, b, and c, and, of course, furnish 4 times 6= = 1 × 2 × 3 × 424 permutations.

By the above, it is evident that the number of permutations

RULE. Form the series of numbers, 1, 2, 3, 4, etc., up to the number of things to be permuted, and their continued product will be the number of permutations.

Ex. 1. How many different integral numbers may be "expressed by writing the 9 significant digits in succession, each figure to be taken once, and but once, in each number? Ans. 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 2. In how many different orders may a family of 10 persons seat themselves around the tea table?

362880.

381. What is Permutation? 382. Object of the Problem? Rule?

MENSURATION.

383. MENSURATION is the art of measuring lines, surfaces, and solids.

The principles are all Geometrical, and are very numerous. A few only of the more simple are here presented. 384. Two parallel lines are everywhere equally distant from each other.

When two lines meet so as to form equal angles, the lines are perpendicular to each other and the angles are right angles. A right angle contains 90°.

An acute angle is an angle of less than

90°.

An obtuse angle is an angle of more than 90°.

Two lines are oblique to each other when

they meet so as to form acute or obtuse angles, and the angles are oblique angles.

385. A TRIANGLE is a plane figure which

is bounded by three lines.

The base of a triangle (or any other figure)

is the side on which it is supposed to stand.

The altitude of a triangle is the perpendicular

distance from the angle opposite the base to the base, or to the base extended.

386. PROBLEM 1. To find the area of a triangle: RULE. Multiply the base by half the altitude.

Ex. 1. The base of a triangle is 7 inches and the altitude 8 inches; what is its area?

Ans. 28sq. in.

2. The base is 8ft. and the hight 11ft.; what is the area?

383. What is Mensuration? 384. What of two parallel lines? What is a right angle? An acute angle? Obtuse angle? What are oblique lines? Oblique angles? 385. What is a Triangle? Its base? Its altitude? 386. Rule for finding its area?

387. A QUADRILATERAL or QUADRANGLE is a plane figure, having four sides and four angles.

There are three kinds of quadrilaterals, viz.:

1st. Trapeziums, none of whose sides are parallel;

2d. Trapezoids, as A B CD, only one pair of whose sides are parallel; and,

The diagonal of a figure is a line which joins two opposite angles, as A C in the above trapezoid, and B D in the parallelogram. The altitude of a trapezoid or parallelogram is the perpendicular between two parallel sides.

388.

PROBLEM 2. To find the area of a trapezium:

RULE. Draw a diagonal dividing the trapezium into two triangles, and find the area of each triangle by Problem 1. The sum of these triangles will be the area of the trapezium.

Ex. What is the area of a trapezium, one of whose diagonals is 20 inches, and the length of the perpendiculars let fall upon it, from the other angles of the trapezium, 6 and 8 inches? Ans. 140sq. in.

389. PROBLEM 3. To find the area of a trapezoid: RULE. Multiply the half sum of the parallel sides by the altitude, and the product will be the area.

387. What is a Quadrilateral? How many kinds? What is a trapezium? Trapezoid? Parallelogram? What is the diagonal of a figure? Altitude of a trapezoid? Of a parallelogram? 388. Rule for finding the area of a trapezium? 389. Rule for finding the area of a trapezoid?

Ex. 1. The parallel sides of a trapezoid are 10 and 12 feet, and its altitude is 6 feet; what is its area? Ans. 66sq. ft.

2. What is the area of a board, whose length is 10ft., the wider end being 2ft. and the narrower 18 inches in width? 390. PROBLEM 4. To find the area of a parallelogram:

RULE. Multiply the base by the altitude, and the product is the area.

Ex. 1. What is the area of a rectangular field, whose length is 40 rods, and altitude or width 8 rods? Ans. 2 acres.

2. The base of a parallelogram is 6 feet, and the altitude 4 feet; what is its area?

391. A POLYGON is a plain figure bounded by straight lines.

NOTE 1. Three straight lines, at least, are required to bound a polygon. The lines which bound a polygon, taken together, are called the perimeter of the polygon.

A polygon of 5 sides is called a pentagon; of 6, a hexagon ; 7, a heptagon; 8, an octagon; 9, a nonagon; 10, a decagon; 11, an undecagon; 12, a dodecagon; etc.

NOTE 2. A polygon may be divided into triangles by drawing diagonals, and then its area may be found by Problem 1.

392. PROBLEM 5. To find the area of a circle when the radius and circumference are given (Art. 109 and 361):

RULE 1. Multiply the circumference by half the radius; or, RULE 2. Multiply the square of the radius by 3.141592, and the product is the area.

Ex. 1. What is the area of a circle, whose radius is 6 and circumference 37.699104? Ans. 113.097312.

2. What is the area of a circle whose radius is 10?

390. Rule for finding the area of a parallelogram? 391. What is a Polygon? Note 1? Perimeter of a polygon? Name the different polygons? 392. Rule for finding the area of a circle? Second Rule?

393. A PRISM is a solid that has two similar, equal, parallel faces, called bases, and all its other faces parallelograms.

NOTE. A prism is triangular, quadrangular, pentagonal, etc., according as its bases are triangles, quadrangles, pentagons, etc.

A CYLINDER is a round body whose diameter is the same throughout its entire length, and whose ends or bases are equal, parallel circles.

394. PROBLEM 6. To find the surface of a prism or cylinder:

RULE. Multiply the perimeter or circumference of the base by the length of the solid, and to the product add the area of the two ends.

Ex. 1. What is the surface of a prism, whose length is 10 inches and base 4 inches square? Ans. 192sq. in. 2. What is the surface of a cylinder, whose length is 20 feet and diameter 4 feet?

395. PROBLEM 7. To find the solid contents of a prism or cylinder:

RULE. Multiply the area of the base by the altitude.

Ex. 1. What are the contents of a cylinder, whose length is 20 inches and whose diameter is 10 inches?

Ans. 1570.796c. in. 2. What are the contents of a quadrangular prism, whose length is 25 feet and whose base is 3 feet square?

396. A PYRAMID is a solid, having a polygonal face, called the base, and all its other faces are triangles which meet at a common point, called the vertex of the pyramid. The slant hight is the distance from the vertex to the middle of one side of the base.

393. What is a Prism? A Cylinder? 394. Rule for finding the surface of a prism or cylinder? 395. Rule for finding the contents of a prism or cylinder? 396. What is a Pyramid? Its vertex? Slant hight?