on p. 402, of an annuity of $1 for that number of years? What is the present value of an annuity of $400?

7. A person left $5000 for the poor of his native town. How great was the perpetuity realized from it, at 6%?

8. How much will an annuity of $100 amount to, in 8 years, at 8% simple interest? How much, at 6% compound interest ? Last ans. $989.75. 9. A soldier 57 years old, having a pension of $80 a year, agreed to sell it for cash at 10% less than its present value, compound interest being allowed at 7%. How much should he receive? Ans. $680.16.

10. How much a year must be invested for a boy 11 years old, that the sums thus invested, with compound interest at 5%, may make a total of $10000 by the time he becomes of age ? Ans. $795.05.

11. An annuity of $350 was left to A, and one of $550 to B, by the same person; both were to run 12 years. Allowing compound interest, at 6%, by how much would the amount of A's exceed that of B's in the given time?

Ans. $3373.99.

CHAPTER XXXIV.

MENSURATION.

682. Mensuration is that branch which enables us from certain data to calculate the length of lines, the areas of surfaces, and the solid contents of bodies. Its rules, as presented in Arithmetic, are derived from Geometry; and to the latter recourse must be had for the reasoning on which they are based.

Several rules of Mensuration have been already given; those, for instance, relating to the sides of right-angled triangles (Art. 648), to the areas of rectangles (Art. 333), and to the solid contents of cubes (Art. 339). Some of the others that are most important will now be presented.

683. Parallelograms.-A Parallelogram is a four-sided figure that has its opposite sides equal and parallel; as, Figs. 1, 2, 3.

(1)

(2)

(8)

A

A

A

B B

B

A rectangle (Figs. 1, 2) is a parallelogram whose angles are right angles. A square (Fig. 1) is a parallelogram whose angles are right angles and whose sides are equal.

The Base of a parallelogram is the side on which it stands. Its Altitude is the perpendicular distance from its base to the opposite side; as, AB in the figures above.

684. The Diagonal of a parallelogram A, B is a line connecting two of its opposite corners; as, A D.

C

The diagonal of a rectangle, as will be seen in the figure, divides it into two equal right-angled triangles, of either of which the diagonal is the hypothenuse. The diagonal, therefore, can be found according to Art. 648, by adding the square of the base to the square of the altitude and extracting the square root of the sum.

In like manner, the diagonal of a square equals the square root of twice the square of its side.

685. RULES.-I. To find the area of a parallelogram, multiply the base by the altitude.

II. To find the area of a square when the diagonal is given, divide the square of the diagonal by 2.

1. The base of a parallelogram being 2 chains and its altitude 2 rods, what is its area?

2. How many tiles 6 inches square will be required for paving a rectangular court 15 feet by 30?

3. What distance will be saved by taking a diagonal path across a square field containing 10 acres, in stead of following the two sides? Ans. 23.43rd. 4. What is the area of a square court, the line connecting its two opposite corners being 50 feet long?

686. Triangles.-The Altitude of

a Triangle is a perpendicular drawn from one of its angles to the base, or the base produced; as, CD.

D

D

RULES.-I. To find the area of a triangle, multiply its base by half its altitude.

Or, when the three sides are given, from half their sum subtract each side separately, multiply together the three remainders and the half sum, and extract the square root of their product.

II. To find the area of a right-angled triangle, when the hypothenuse is given and the other two sides are equal, divide the square of the hypothenuse by 4.

The area of a rectangle is the product of its length and breadth. But a rectangle, as shown in Art. 684, is divided by its diagonal into two cqual triangles; the area of each of these, therefore, is half the product of the length and breadth of the rectangle,—or, what is the same thing, half the product of the base and altitude of either triangle, according to the first rule just given.

5. The base of a triangle being 5 ft., and its altitude 9 inches, what is its area?

6. How much land is there in a triangular field whose sides are 4, 5, and 6 rods long? Ans. 9.92+ sq. rd.

7. What is the area of a right-angled triangle, if its hypothenuse is 40 feet, and the other two sides are equal?

687. Circles.-The Circumference, Diameter, and Radius of a Circle, are defined in Art. 348.

RULES.-I. To find the circumference of a circle, multiply the diameter by 3.14159.

II. To find the diameter, multiply the circumference by .3183.

III. To find the area, multiply 4 of the circumference by the diameter: or, the square of the circumference by .07958: or, the square of the diameter by .7854.

IV. To find the side of a square which shall have an area equal to that of a given circle, multiply the circumference of the circle by .282096, or its diameter by .8862.

V. To find the area of a ring included between two circles having the same centre (see Figure in Art. 349), subtract the square of the diameter of the smaller circle from the square of the diameter of the greater, and multiply the difference by .7854.

The ratio of the circumference to the diameter can not be exactly expressed in figures, but is approximately expressed as above.-Multiplying the circumference by .3183, to find the diameter, is equivalent to dividing it by what?-Whence do the decimal multipliers in Rule IV. arise?—In the case covered by Rule V., if the circumferences are given in stead of the diameters, what is the best way of proceeding?

8. The diameter of the sun being 852584 miles, what is its circumference ?

9. What is the diameter of a circle whose circumference is 20 feet? What is its area? Last ans. 31.832 sq. ft.

10. What space will a cow have to feed on, if she is fastened to a stake with a rope 25 feet long?

11. How far is it around a circle containing a square mile? How far is it around a square having that area? First ans. 3.54 + mi.

12. A circular plot 100 rods in diameter contains how many square rods?

13. Within the plot just mentioned, a circular pond is laid off, with its circumference everywhere 5 rods distant from the circumference of the plot. What is the area of the pond? How many square rods rounding the pond?

are in the ring surLast ans. 1492.26 sq. rd.

14. How many times will a wheel 5 ft. across revolve in going a mile? Ans. 336.13+ times.

15. How much space is contained in a quadrant, the sides of which are each 12 in. ? Ans. 113.0976 sq. in. 16. If the equatorial diameter of the earth is 7925.83 miles in length, how long is a degree of longitude at the equator? Ans. 69.1658563+ mi.

17. A man wishes to lay off a square flower-bed having the same area as a circular bed 100 ft. in circumference; how long must each side of the square be?

18. A walk 2 ft. wide is laid out round a circular plot which has a radius of 10 feet. What is the area of the walk? Ans. 138.2304 sq. ft.

body whose ends or parallel plane figures,

688. Prisms.-A Prism is a bases are any similar, equal, and and whose sides are parallelograms.

The Altitude of a prism is the perpendicular distance between its bases.

A prism whose bases are triangles (see Figure) is called a Triangular Prism.

If the bases and sides are equal squares, the prism becomes a cube.

The Perimeter of a triangle or other plane figure is the sum of its sides.

RULES.-I. To find the convex surface of a prism,

multiply the perimeter of the base by the altitude.