6

8. What is the present worth of an annuity of $80 to continue forever, at per cent.?

616. To find the present worth of an annuity in reversion.

Find the present worth of the annuity from the present time till its termination; also find its present worth for the time before it commences; the difference between these two results will be the present worth required.

9. What is the present worth of $79.625 at 5 per cent., to commence in 4 years and continue 6 years? Ang. $332.50.

PERMUTATIONS AND COMBINATIONS.

617. By Permutations is meant the changes which may be made in the arrangement of any given number of things.

The term combinations, denotes the taking of a less number of things out of a greater, without regard to their order or position.

618. To find how many permutations or changes may be made in the arrangement of any given number of things.

Multiply together all the terms of the natural series of numbers from 1 up to the given number, and the product will be the answer. 1. How many changes may be rung on 5 bells?

Ans. 120.

2. How many different ways may a class of 8 pupils be arranged? 3. How many different ways may a family of 9 children be seated? 4. How many ways may the letters in the word arithmetic, be arranged? 5. A club of 12 persons agreed to dine with a landlord as long as he could seat them differently at the table: how long did their engagement last?

619. To find how many combinations may be made out of any given number of different things by taking a given number of them at a time.

Take the series of numbers, beginning at the number of things given, and decreasing by 1 till the number of terms is equal to the number of things taken at a time; the product of all the terms will be the answer required.

6. How many different words can be formed of 9 letters, taking 3 at a time? Solution.-9X8X7-504. Ans. 504 words.

7. How many numbers can be expressed by the 9 digits, taking 5 at a time? 2. How many words of 6 letters each can be formed out of the 26 letters of the alphabet, on the supposition that consonants will form a word?

SECTION XIX.

APPLICATION OF ARITHMETIC TO GEOMETRY.

620. In the preceding sections abstract numbers have been applied to concrete substances, or to objects in general, considered arithmetically. On the same principle, geometrical magnitudes may be compared or measured by means of the numbers representing their dimensions. (Arts. 7, 516. Obs. 3.)

OBS. The measurement of magnitudes is commonly called mensuration.

MENSURATION OF SURFACES.

621. In the measurement of surfaces, it is customary to assume a square as the measuring unit, whose side is a linear unit of the same name. (Leg. IV. 4. Sch. Art. 257. Obs. 2.)

Note. For the demonstration of the following principles, see references. 622. To find the area of a parallelogram, also of a square. Multiply the length by the breadth. (Art. 285, Leg. IV.

5.)

OBS. When the area and one side of a rectangle are given, the other side is found by dividing the area by the given side. (Art. 156.)

1. How many acres in a field 240 rods long, and 180 rods wide?

2. How many acres in a square field the length of whose side is 340 rods?

3. If the diagonal of a square is 100 rods, what is its area?

4. A rectangular farm of 320 acres, is a mile wide: what is its length?

623. To find the area of a rhombus. (Leg. I. Def. 18. IV. 5.) Multiply the length by the altitude or perpendicular height. 5. Find the area of a rhombus whose length is 20 ft., and its altitude 18 ft. 624. To find the area of a trapezium. (Leg. IV. 7.)

Multiply half the sum of the parallel sides by the altitude.

6. Find the area of a trapezium the lengths of whose parallel sides are 27 ft. and 31 ft., and whose altitude is 15 ft.

625. To find the area of a triangle. (Leg. IV. 6.)

Multiply the base by half the altitude or perpendicular height. 7. Find the area of a triangle whose base is 50 ft., and its altitude 44 ft.

1

626. To find the area of a triangle, the three sides being given. From half the sum of the three sides subtract each side respectively; then multiply together half the sum and the three remainders, and extract the square root of the product.

9. What is the area of a triangle whose sides are 20, 30, and 40 ft.? 10. How many acres in a triangle whose sides are each 40 rods?

A 627. To find the circumference of a circle from its diameter. Multiply the diameter by 3.14159. (Leg. V. 11. Sch.)

Note. The circumference of a circle is a curve line, all the points of which are equally distant from a point within, called the centre. The diameter of a circle is a straight line which passes through the centre, and is terminated on both sides by the circumference. The radius or semi-diameter is a straight line drawn from the centre to the circumference.

11. What is the circumference of a circle, whose diameter is 20 ft.? 12. What is the circumference of a circle, whose diameter is 45 rods?

628. To find the diameter of a circle from its circumference. Divide the circumference by 3.14159.

OBS. The diameter of a circle may also be found by dividing the area by 7854, and extracting the square root of the quotient.

13. What is the diameter of a circle, whose circumference is 314.159 ft.?

629. To find the area of a circle. (Leg. V. 11.)

Multiply half the circumference by half the diameter; or, multiply the square of the diameter by the decimal .7854.

15. What is the area of a circle, whose diameter is 50 rods?

16. Find the area of a circle 200 ft. in diameter, and 628.318 ft. in circum.

630. To find the side of the greatest square that can be inscribed in a circle of a given diameter.

Divide the square of the given diameter by 2, and extract the square root of the quotient. (Art. 581. Obs. 1.)

17. The diameter of a round table is 4 ft.; what is the side of the greatest square table which can be made from it?

631. To find the side of the greatest equilateral triangle that can be inscribed in a circle of a given diameter.

Multiply the given diameter by 1.73205. (Leg. V. 4. Sch.)

18. Required the side of an equilateral triangle inscribed in a circle of 20 ft. diameter.

MEASUREMENT OF SOLIDS.

632. In the measurement of solids it is customary to assume a cube as the measuring unit, whose sides are squares of the same name. (Art. 258. Obs. 2.)

633. To find the solidity of bodies whose sides are perpendicular to each other.

Multiply the length, breadth, and thickness together. (Art. 286.)

OBS. When the contents of a solid body and two of its sides are given, the other side is found by dividing the contents by the product of the two given sides. (Art. 159.)

1. What are the contents of a stick of timber 4 ft. square, and 85 ft. long? 2. What is the capacity of a cubical vessel, 14 ft. 8 in. deep?

634. To find the solidity of a prism.

Multiply the area of the base by the height. (Leg. VII. 12.) OBS. This rule is applicable to all prisms, triangular, quadrangular, pentagɔnal, &c., also to all parallelopipedons, whether rectangular or oblique.

3. Find the solidity of a prism 461 ft. high, whose base is 7 ft. square?

635. To find the lateral surface of a right prism.

Multiply the length by the perimeter of its base. (Leg. VII. 5.) OBS. If we add the areas of both ends to the lateral surface, the sum will be the whole surface of the prism.

4. What is the surface of a triangular prism, whose sides are each 3 ft., and its length 12 ft.?

636. To find the solidity of a pyramid and cone.

Multiply the area of the base by of the height. (Leg. VII. 18.) 5. What is the solidity of a pyramid 100 ft. high, whose base is 40 ft. square? 6. What is the solidity of a cone 150 ft. high, whose base is 15 ft. in diameter ?

637. To find the lateral or convex surface of a regular pyramid, or cone. (Leg. VII. 16, VIII. 3.)

Multiply the perimeter of the base by the slant-height.

7. What is the lateral surface of a regular pyramid, whose slant-height is, 15 ft., and base is 30 ft. square?

8. What is the convex surface of a right cone, whose slant-height is 94 ft. and the perimeter of its base 37 ft. ?

638. To find the solidity of a frustum of a pyramid and cone. To the sum of the arcas of the two ends, add the square root of the product of these arcas; then multiply this sum by of the perpendicular height. (Leg. VII. 19. Sch., VIII. 6.)

9. If the two ends of the frustum of a pyramid are 3 ft and 2 ft. square, and the height is 12 ft., what is its solidity?

639. The convex surface of a frustum of a pyramid and cone is found by multiplying half the sum of the circumferences of the two ends by the slant-height. (Leg. VII. 17, VIII. 5.)

10. If the circumferences of the two ends of the frustum of a cone are 18 ft. and 14 ft., and its slant-height 11 ft., what is its convex surface?

640. To find the solidity of a cylinder.

Multiply the area of the base by the height. (Leg. VIII. 2.)

11. Find the solidity of a cylinder 10 ft. in diameter, and 35 ft. high. 12. Find the solidity of a cylinder 100 ft. in circumference, and 150 ft. high641. To find the convex surface of a cylinder.

Multiply the circumference of the base by the height. (Leg. VIII. 1.) 13. Find the convex surface of a cylinder 5 yds. in diameter, and 5 yds. long. 642. To find the convex surface of a sphere or globe. Multiply the circumference by the diameter. (Leg. VIII. 9.) 14. What is the surface of a globe 18 inches in diameter?

15. If the diameter of the moon is 2162 miles, what is its surface?

643. To find the solidity of a sphere or globe.

Multiply the surface by of the diameter. (Leg. VIII. 11.) 16. Find the solidity of a globe 15 inches in diameter.

17. The diameter of the moon is 2162 miles: what is its solidity?

MEASUREMENT OF LUMBER.

644. The area of a board is found by multiplying the length into the mean readth. (Arts. 622, 623.)

The solid contents of hewn or square timber are found by multiplying the sengih into the mean breadth and depth.

The solid contents of round timber are found by multiplying the length by the mean girt or circumference.

Obs. 1. The mean breadth of a tapering board is found by measuring it in the middle, or by taking the sum of the breadths of the two ends.