PROBLEM X.

To find the Area of a Triangle.

RULE.

Multiply the base of the given triangle into half its perpendicular height; or half the base into the whole perpendicular, and the product will be the answer.

Ex. 1. Required the area of a triangle whose base, or longest side is 36 inches, and the perpendicular height 16 inches.

Ans. 36 X 8 288 inches. 2. Required the area of a triangular garden, whose base, or Yongest side is 15.6 rods, and the perpendicular opposite the base is 9 rods. Ans. 70.2 rods.

PROBLEM XI.

To find the convex surface of a Cylinder.*

*Diffinition. A Cylinder is a round body whose bases are circles, like a round column or stick of timber of equal bigness. from end to end.

RULE.

Multiply the length into the circumference of the base.

Ex. 1. How many square feet in the superficial contents of a cylinder which is 42 feet long, and 15 inches in diameter.

Ans. 42 X 1.25 X 3.14159 = 164.933 square feet. 2. Required the convex surface of a cylindrical stick of timber, whose axis is 5 feet, and the diameter 7 inches.

PROBLEM XII.

To find the solidity of a Cylinder.

RULE.

Ans. 1520 inches.

Find the area of the base (by Prob. VII.) which multiply into the length, and the product will be the solid contents.

1. What is the solid contents of a round stick of timber whose diameter is 18 inches, and length 20 feet?

144)5089-3920(35-343 solid feet, Ans.

2. What is the solidity of cylinder, whose length is 121, and diameter 45.2? Ans. 45.2 X 7854X121 194156.6

PROBLEM XIII.

To find the solidity of a CONE.

Deffinition. A CONE is a solid whose base is a circle, from which it decreases gradually to a point in the top, called the VERTEX.

A line drawn from the vertex, perpendicular to the base, is called the height of the cone.

RULE.

Multiply the area of the base by the height, and of the product will be the content.

Ex. 1. What is the solidity of a cone, whose height is 12 feet 6 inches, and the diameter of the base 2 feet 6 inches? 2.52 X 7854 X 12.53 20.453125 feet, Ans. 2. Required the solidity of a conical monument, that is 9 feet high, and the diameter of its base 24 feet.

PROBLEM XIV.

Ans. 14-726250 feet.

To find the solidity of a Frustrum of a cone.

Deffinition. A FRUSTRUM of a cone is what remains after any portion of the upper end is cut off, by a plane paralell to the base.

RULE.

Add together the areas of the two ends, and the square root of the product of these areas; and multiply the sum by of the perpendicular height, and the result will be the solid con

tent.

OR

2. Divide the difference of the cubes of the diameters of the two ends, by the difference of the diameters, and this quotient, being multiplied by 7854 and again by of the height, will give the solidity.

EXAMPLES.

1. Required the solidity of a frustrum of a cone, whose altitude, or height is 18 feet, the greatest diameter 8 feet, and the least 4 feet. BY THE 1st. RULE.

8.3—43 = 448 (8—4) = 112 X 7854 X 6 =

527-7888 in. Ans.

The latter method, in many cases, will be found preferable to the former in point of expidition.

2. What is the content of the frustrum of a conical block, whose height is 20 inches, and the diameter of its two ends 28 and 20 inches ? Ans. 9131.5840

The number of gallons or bushels which a vessel will contain may be found, by calculating the capacity in inches, and then dividing by the number of inches in 1 gallon or bushel; as by the following

TABLE OF SOLID MEASURE.

1 cubic foot of pure water weighs 1000 ounces, Avojɛ

dupois, or 621 pounds.

EXAMPLES.

1. What is the capacity of a conical cistern, which is 9 feet deep, 4 feet in diameter at the bottom, and 3 feet at the top? Ans. 87.18 cubic feet X 7.4805* 652.15 wine gallons.

2. How many gallons of ale can be put into a vat in the form of a conic frustrum, if the larger diameter be 7 feet, the smaller diameter 6 feet and the depth 8 feet? Ans. 1886.5458 gallons.

3. There is a cistern in a distillery whose altitude is 10 feet, the greater diameter 14 feet, and the smaller diameter 12 feet; required its capacity in hogsheads.

1 4 3 — 12 3 —— 14—12 X 7854 X 10 X 7·480563:

Ans. 157.918193 hhd.

PROBLEM XV.

To find the surface of a Sphere.

Definition. A SPHERE, or globe is a round solid body, in the center of which is a point, from which all lines drawn to the surface are equal.

RULE.

Multiply the diameter by the circumference.

Note. In like manner, the convex surface of any zone or segment is found by multiplying its height by the whole circumference of the sphere.

*Note. When the capacity is in feet, multiply by 7.4805, because 17.4805 the number of wine gallons in 1 cubic foot: When the ale gallon is required, multiply the feet by 6-1276, because 112 1729= 6.1276; but if the capacity be calculated in inches divide by the number of cubic inches, in the gallon.

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EXAMPLES.

1. What is the convex surface of a sphere, whose diameter is 7 inches, and circumference 22 inches?

Ans. 7 X 22 154 in.

2 Required the surface of a globe, whose diameter or axis, is 24 inches. 24 X 3.14159 X 24 1809 5616 inches, Ans. 3. Considering the earth as a sphere, whose circumference is 25000 miles; how many square miles are there on its surface? Ans. 198943750 sq. miles. 4. The axis of a sphere being 42 inches, what is the convex superficies of the segment, whose height is 9 inches?

Ans. 42 X 3·14156 X 9 = 1187 5248 inches. 5. If the circumference of the sun be 2800000 miles, what is the surface? Ans. 2495547600000 sq. miles.

PROBLEM XVI.

To find the solidity of a SPHERE.

RULE.

1. Multiply the cube of the diameter by 5236.

OR

2. Multiply the square of the diameter by of the circumference.

OR

3. Multiply the surface by of the the diameter.

EXAMPLES.

1. What is the solidity of a sphere, whose diameter is 1 foot? 123 X 5236

904-7808 inches, Ans.

2. What is the solid content of a sphere 4 feet 6 inches in. iameter ? Ans. 47.7130500 feet. 3. Required the number of solid miles contained in the earth, supposing its circumference to be 25000 miles. Ans. 263858149120 miles.

4. How many wine gallons will fill a hollow sphere 2 fect & inches in diameter ?

The capacity is 9.9288 feet X 7.4805 = 1 .hhd. 11.27 gallons, 5. How many gallons of water may be put into a hollow. sphere that is 4 feet in diameter, and what will be the weight of the water?

Note. The numbers 3.14159, 7854, 5236, should be made perfectly familliar. The first expresses the ratio of the cir cumference of a circle to the diameter; the second, the ratio of the area of a circle to the square of the diameter; and the third, the ratio of the solidity of a sphere to the cube of the di

ameter.

The second is, and the third is of the first.

Ans. 205.33832704 gallons, and the weight is 12833.64544 lb. 6. If the diameter of the moon be 2180 miles, what is her solidity? Ans. 5424600000 miles. When the solidity of a sphere is given, the diameter may be found by dividing the solidity by .5236, and extracting the cube root of the quotient.

52 36

=

7. What is the diameter of a sphere, whose solidity is 65.45 cubic feet? 39455 feet Ans. 8. What must be the diameter of a sphere, to contain 1052 gallons of wine? Ans. 3 feet.

9. Required the diameter of a globe, to contain 16755 pounds of water. Ans. 8 feet. 10. How many globes that are 3 inches each in diameter, are equal to another globe whose diameter is 12 inches?

Ans. 64.

Note. The solid contents of similar figures are in proportion to each other, as the cades of their homologous sides, or diameters. Euc. El.

12. If a cannon ball 6 inches in diameter, weigh 32lb. what will another ball weigh, whose diameter is 3 inches?

63 216 and 33 = 27, then as 216: 32 :: 27: 4 lb. Ans. 13. If a metalic globe 8 inches in diameter, weigh 72 lb. what will be the weight of a globe of the same metal, whose diameter shall be 4 inches? Ans. 9 lb. 14. If a globe of silver 3 inches in diameter, be worth $150; how many such globes will be equal in value to $9600?

Ans. 64.

ANNUITIES, OR PENSIONS.

AN ANNUITY, is a sum of money payable every year, for a certain number of years, or forever.

When the debtor keeps the annuity in his own hands beyond the time of payment, it is said to be in ARREARS.

The sum of all the annuities for the time they have been forborn together with the interest due upon each, is called the AMOUNT,

If an annuity be bought off, or paid all at once, at the begin ning of the first year, the price, which ought to be given for i is called the PRESENT WORTH.

To find the amount of an ANNUITY at SIMPLE INTEREST.