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323. Permutation is the process of finding the different orders in which a given number of things may be arranged.
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?
In the 1st operation, we 432, 42 3,3 4 2, 3 2 4, 2 4 3, 2 3 4. have made all the different
arrangements that can be
made of the given figures, 1 X 2 X 3 = 6.
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
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.
323. What is permutation ?-324. What is the rule for finding the number of arrangements that can be made of any given number of things ?
MENSURATION OF SURFACES.
325. A Surface is that which has length and breadth without thickness.
The Area of a figure is its surface or superficial contents.
A Plane is that in which, if any two points be taken, the straight line that joins them will lie wholly in it.
326. An Angle is the inclination or opening of two lines, which meet in a point.
The Vertex of an angle is the point of meeting of the lines forming the angle.
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 BC
An Obtuse Angle is an angle greater than a right angle, or more than 90 degrees; as F B C.
327. A Triangle is a plane figure having three sides and three angles.
It receives the particular names of an equilateral triangle, isosceles triangle, and scalene triangle ; also, of a right-angled triangle, ocute-angled triangle, and obtuse-angled triangle.
The Base of a triangle, or other plane figure, is one of its sides, on which it may be supposed to stand ; 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 one which has its three sides equal.
325. What is a surface? What are the superficial contents of a figure called ? — 326. What is an angle?. A right angle? An acute angle? An obtuse angle ? — 327. What is a triangle? What particular names does it receive ? When is it called a right-angled triangle ? An acute-angled triangle ? An obtuse-angled triangle? What is the base of a triangle? The altitude ? What is an equilateral triangle ?
An Isosceles Triangle is one which has two of its eides equal.
A Scalene Triangle is one which has its three sides unequal.
A Right-Angled Triangle is one which has a right Angle.
NOTE.- An acute-angled triangle is one which has an acute angle, and an obtuse-angled triangle is one having an obtuse angle.
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 hight 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.
QUADRILATERALS. 329.. A Quadrilateral is a plane figure having four sides, and consequently four angles.
It comprehends the rectangle, square, rhombus, rhomboid, trapezium, and trapezoid.
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 any two of its opposite sides ; as Č D in the rhomboid.
327. What is an isosceles triangle? A scalene triangle? A right-angled triangle? -328. The first rule for finding the area of a triangle ? The second ? - 329. What is a quadrilateral ? What figures does it comprehend? - 330 What is a parallelogram? What particular names does it take? The altitude of a parallelogram?
A Rectangle is any right-angled parallelogram.
A Square is a parallelogram, having equal sides and right angles.
A Rhomboid is an oblique-angled parallelogram.
A Rhombus is an oblique-angled parallelogram, having all its sides equal.
NOTE. — An oblique angle is one either acute or obtuse. 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; the one being 37 feet long and 27 feet wide, and the other 40 feet long and 20 feet wide ?
feet. 3. The base of a rhombus is 15 feet, and its perpendicular hight is 12 feet; what are its contents ?
Ans. 180 square feet.
332. A Trapezoid is a quadrilateral which has only two of its sides parallel.
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 being 482 feet, the shorter 324 feet, and the altitude 216 feet?
330. What is a rectangle? A square ? A rhomboid ? A rhombus ? 331. The rule for finding the area of a parallelogram? 332. What is a trapezoid ? -333. What is the rule for finding the area of a trapezoid ?
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 20 inches ?
feet. 334. A Trapezium is a quadrilateral, which has no two of its sides parallel.
A Diagonal is a straight line which joins the vertices of any two opposite angles of a quadrilateral ; as E F.
335. To find the AREA of a trapezium.
RULE. - Divide the trapezium into two triangles by a diagonal, and then find the areas of these triangles ; their sum will be the area of the trepezium.
1. What is the area of a trapezium, whose diagonal is 65 feet, and the lengths 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 lengths of the perpendiculars let fall upon it are 70 and 85 rods ?
Ans. 9687.5 square rods.
336. A Polygon is any figure bounded by straight lines.
It takes the particular names of pentagon, when it 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.
337. A Regular Polygon is one which has all its sides and all its angles equal.
The Perimeter of a polygon is the broken line which bounds it.
338. To find the AREA of a regular polygon.
RULE. — Multiply the perimeter by half the perpendicular let fall from the center on one of its sides, and the product will be the area.
334. What is a trapezium? What is a diagonal ? – 335. The rule for finding the area of a trapezium ? — 336. What is a polygon? What particular names does it take? — 337. What is a regular polygon?