EXAMPLES FOR PRACTICE.

In how many ways can the following be arranged?

6. In how many different ways can 10 cars be arranged in making up a train?

CASE II.

Art. 463. To find the possible number of arrangements of a given number of things, taken a given less number at a time.

=

Each of the four letters b, c, d, f, prefixed to each of the other three, would make three arrangements by twos; in all, 4 X 3 12 arrangements by twos. Again, if to each of the 12 arrangements by twos, each of the remaining two letters be prefixed, 4 X 3 X 2 = 24 arrangements by threes will result.

Rule.-Multiply together the numbers of the decreasing natural series, whose greatest term is the number of objects, and whose number of terms is the number of objects taken at a time.

EXAMPLES FOR PRACTICE.

1. How many whole numbers can be expressed by the nine digits, taken three at a time?

2. How many, taken six at a time? 3. How many, taken five at a time? 4. In how many ways can written, taken two at a time?

Ans. 504. Ans. 90720.

the letters of the alphabet be Three at a time?

COMBINATIONS.

Art. 464. Combinations are groups so formed from the members of a given number of things that all the members of one group are not the same as those of another.

Art. 465. To find the number of possible combinations of a given number of things, taken a given number at a time.

Rule.-Divide the number of possible permutations of the whole number, taken the given number at a time, by the number of possible permutations of as many things as are taken at a time.

EXAMPLES FOR PRACTICE.

1. How many combinations can be made up of the nine digits, three in a set?

2. How many, six in a set?

Ans. 84.

Ans. 126.

Ans. 252.

4. Of seven musical notes, three in a set?

3. Of ten letters, five in a set?

5. Of eight persons, four in a set?

6. Of the letters in Pittsburgh, six in a set? 7. Of twelve letters, seven in a set?

Ans. 35.

Art. 466.

magnitudes.

CHAPTER XX.

MENSURATION.

Mensuration is the art of measuring

In reference to the kind of magnitude, mensuration is of lines, surfaces, and solids.

The measurement of lines is linear measure. (See Arts. 205-211.)

MENSURATION OF SURFACES.

Art. 467. In mensuration of surface, (see Arts. 212216,) the definitions are given, and the processes are demonstrated by geometry.

A figure is a surface limited by a line or lines.

In reference to their limiting lines, figures are rectilinear, or curvilinear.

A rectilinear figure is a figure limited by straight lines. A curvilinear figure is a figure limited by a curved line, or curved lines.

Regular surfaces are either plane or curved.

A plane surface is a surface such that a straight line between any two of its points lies wholly in the surface.

A curved surface is a surface which is neither plane, nor composed of plane surfaces.

The usual unit of measure for both plane and curved surfaces is the area of a plane surface, called a square.

The base of a figure is the side on which it is supposed to rest.

Art. 468. In rectilinear figures

A polygon is any rectilinear figure.

A regular polygon is a polygon of equal sides and equal angles.

An irregular polygon is a polygon of unequal sides and unequal angles.

A triangle is a plane figure of three sides.

An equilateral triangle is a triangle having three equal sides.

An isosceles triangle is a triangle having two equal sides. A scalene triangle is a triangle having no equal sides. The altitude of a triangle is the shortest distance from the vertex of one of its angles to the opposite side taken as a base.

A quadrilateral is a plane figure of four sides.

A parallelogram is a quadrilateral having its opposite sides parallel and equal.

A rectangle is a quadrilateral whose angles are right angles.

A square is a rectangle of equal sides.

A rhombus, or rhomb, is a parallelogram of equal sides and oblique angles.

A rhomboid is a parallelogram whose opposite sides only are equal, and whose angles are oblique.

A trapezium is a quadrilateral having no parallel sides. A trapezoid is a quadrilateral having only two parallel sides.

A pentagon is a polygon of five sides; a hexagon, of six; a heptagon, of seven; an octagon, of eight; a nonagon, of nine; a decagon, of ten; an undecagon, of eleven; a duodecagon, of twelve.

A diagonal is a line which joins two angles of a figure which are not adjacent.

The perimeter of a figure is the sum of its sides.

A line is inscribed in a circle when its ends are in the circumference. A polygon is inscribed in a circle when its sides are inscribed. A circle is inscribed in a polygon when the

circumference touches each side. In the last two cases the outer figure circumscribes the inner.

The radius of a circle is a straight line whose ends are at the centre and circumference.

The diameter of a circle is a straight line passing through the centre and having its ends in the circumference.

The centre of a regular polygon is the centre of either the inscribed or circumscribed circles.

The radius of a regular polygon is the radius of the circumscribed circle.

The apothem of a regular polygon is the radius of the inscribed circle.

The altitude of a parallelogram or trapezoid is the perpendicular distance between two parallel sides when one of them is taken as a base.

Art. 469. In mensuration of surfaces and solids the factors must express linear units of the same kind. The product expresses, in surface measure, square units of the same name as the linear, and, in solid measure, cubic units of the same name as the linear.

Art. 470. To find the area of a parallelogram.

Rule.-Multiply the base by the altitude.

1. How many sq. yd. in a sidewalk 18 yd. long and 9 ft. wide?

Ans. 54.

2. What is the area of a parallelogram whose base is 21 ft. and altitude 15 ft.?

3. How many more acres in a piece of land 120 rods square, than in a rectangular piece 32 rods long and 30 rods wide?

Art. 471. To find the area of a triangle.

Ans. 84 A.

NOTE.-A triangle is half of a parallelogram of equal base and altitude.