A. By the rule of three; say, as the decrease of the angle, at a given distance from the base is to that distance :: so is the length of the base: to the whole perpendicular of the angle, or to the point, where the two sides will meet.

Q. Having the area of a triangle, and the length of the base given, how can you find the perpendicular?

A. Divide the area by half the base, and the quotient will be the perpendicular: or, if you divide the area by half the perpendicular, the quotient will be the base.

EXAMPLES.

1. In the annexed angle, the base A B, is 40 rods; the diameter C D, is 32 rods and the perpendicular distance from A B, to C D, is 16 rods; what is the length of the whole perpendicular, or to the point o, where the two sides meet?

Operation.

40-32-8 the decrease at C D.

Then as 8: 16: 40: 80 the whole perpendicular, or answer, 80 rods.

2. If the base of an angle be 140, and at the perpendicular distance of 35, the width be 124, what is the length of the perpendicular?

3. A surveyor, in laying out a new road, came to a river and marshy ground that he could not pass, the distance across which he must necessarily ascertain. He therefore takes a station at A, and observing a large tree on the opposite side, measures in a direct line towards it, to the margin of the marsh, 20 rods, to D; he then measures off from A to B, 16 rods; from B, he ranges towards the tree, and sticks a stake at O; on measuring from O to D, he finds the distance to be 10 rods; it is required to find the distance across the river and marsh, from D to the tree?

Ans. 33 rods.

Ans. 306.

10

20

4. It is required to lay out a garden in the form of a right angled triangle, the area of which shall be 864 yards, and the base 48; what must be the length of the perpendicular, and hypothenuse? Ans. Perpendicular, 36 yds. Hypothenuse, 60. 5. A surveyor, in running and measuring a line between two towns, is interrupted by a pond of water; he therefore measures directly back from the edge of the water, 36 rods; and ranges to, and notes an object on the opposite side of the pond; he then measures off, perpendicularly to this line, 36 rods, and ranging to the same object on the opposite side of the pond, he measures directly towards it, to the water's edge, when he finds the distance from this station, to the first one, to be 28 Ans. 126 rods. rods; what is the distance across the pond?

ARTICLE V. TO MEASURE REGULAR POLYGONS.

Q. What are Regular Polygons?

A. They are figures, whose sides and angles are all equal, and denominated from their number of sides, which are from 5 to 12, and sometimes more.

Q. How can you form a regular polygon of any given number of sides?

A. Draw a circle of the diameter of the required polygon, and divide the circumference into as many equal parts as the polygon has sides, and the quotient will show the length of each side, which may be set off and marked accordingly.

Q. What is the RULE for finding the area of a polygon? A. Multiply the length of one of the sides, by the number of sides, then multiply this product by the length of a perpendicu lar, drawn from the centre of the figure, to the middle of one of the sides, and half the product will be the area of the polygon.

a

ARTICLE VI. TO MEASURE A CIRCLE AND ITS PARTS.

Q. What particular parts are necessary to be noticed in a circle?

A. The Circumference, the Diameter, and the Radius or Semidiameter.

Q. What is the circumference of a circle?

A. It is the curve line that forms it, or the distance round it.
Q. What is the diameter of a circle?

A. It is a straight line, passing through its centre from one side to the other, and is the longest straight line that can be drawn in it.

Q. What is the radius, or semidiameter of a circle?

A. It is a line drawn from the centre to the circumference, or half the diameter.

Q. What proportion does the diameter of a circle bear to its circumference ?

A. The diameter bears the same proportion to its circumference, that 7 bears to 22; or (which is a little more exact) that 113 bears to 355.

Q. If the diameter of a circle be given, how do you find the circumference?

A. By the rule of three; say, as 7: 22 or as 113: 355 :: so is the diameter to the circumference.

Q. If the circumference be given, how do you find the diameter ?

A. Change the order of the proportion, and say, as 22 : 7 or as 355: 113 :: so is the circumference to the diameter. Q. Having the circumference and diameter of a circle given, by what RULE do you find its area?

A. Multiply half the circumference by half the diameter, and the product will be the area.

Q. If the diameter alone be given, how can you find the area, without finding the circumference ?

A. Square the diameter, and multiply this square, by the decimal,7854, and the product will be the area.

Q. If the circumference alone be given, how can you find the area, without finding the diameter ?

A. Multiply the square of the circumference by the decimal ,07958, and the product will be the area.

The following short rules will be found sufficiently correct for all common purposes, and convenient in many instances, for both instructor and scholar, as well as for others; viz.:

1. If the diameter of a circle be multiplied by the decimal ,886, the product will be the side of a square of equal area.

2. If the circumference of a circle be multiplied by,318, the product will be the diameter.

3. If the circumference of a circle be multiplied by,282, the product will be the side of an equal square.

4. If the side of any square be multiplied by 1,128, the product will be the diameter of a circle of equal area.

5. If the area of any figure be multiplied by 1,273, the square root of the product will be the diameter of an equal circle; or extract the square root of the area, and multiply this root by 1,128, and the product will be the diameter.

EXAMPLES.

1. If the diameter of a circle be 14 inches, what is the cir. cumference, and what the area?

As 7 22 14 : 44 inches, the circumference.
Then half the circumference is 22
Half the diameter is

7

Ans. 154 inches area.

2. If the circumference of a circle be 154, what is its diam. eter, and superficial area? Ans. Diameter 49. Area 18861. 3. There is a circular field, whose diameter is 44 rods; how many rods of fence will enclose it, and if planted with corn, and were to yield 35 bushels to the acre, how many bushels would it produce? Ans. It will require 138 rods of fence. It would yield 332 bushels, 3 pecks.

4. The arms of a wind-mill are 24 feet, from the centre of motion, to their extremities; how large a circle do they describe in their rotation, and how many yards of canvas, of a yard wide, would cover the whole surface? Ans. They will describe a circle 154 ft. circuit, and it would require 279 yds. 1 qr. 319 nails.

5. There is a wheel, 4 feet in diameter; what must be the diameter of another wheel, whose circumference is just twice as much; and what proportion do their areas bear to each other? Ans. The second must be 8 feet diameter, and the area of it, just 4 times as much.

6. The diameter of the annexed outer circle is 32 inches; it is required to draw a circle within it, which shall enclose just one half of its area; what must be its diameter ?

Ans. 22,62 inches.

7. There is a square whose side measures 64 inches; what is the diameter of a circle, that will contain the same area? Ans. 72 inches, 2".

1

8. A gentleman has a garden containing 144 square rods, and wishes to lay out a circle in the centre, that shall contain just of the whole area; what must be the diameter of the circle? Ans. 6,769+ rods.

9. There is a square meadow, containing 8 acres; if a man commence mowing in the centre, and mows round in a circle, how many rods in diameter must he mow, to cut down one half the meadow ? Ans. 28,5+ rods.

10. There is a cubic cistern, containing 2500 gallons, and a gallon measure is computed to contain 231 cubic inches; what must be the diameter of a round cistern, of equal depth, to contain the same quantity? Ans. 7 ft. 10 inches.

11. There is a circle, 60 inches in diame ter; it is required to draw three circles with. in it, that shall divide it into four equal parts; what will be the diameter of each inner circle? Ans. Diameter of the smaller circle, 30 inches. Diameter of the second circle, 42,42. Diameter of the third circle, 51,96.

The area of circles are to each other, as the squares of their diameters. When the diameter is 1, the area found to be,7854 decimal. Therefore it is, that the square of the di ameter, multiplied by the decimal,7854, gives the area of any circle.

ARTICLE VII. TO MEASURE AN ELLIPSIS OR OVAL.

Q. What is an ellipsis or oval?

A. It is a long circular figure, having

two unequal diameters, as seen in the an-
nexed figure.

Q. How are the two diameters of an
ellipsis distinguished?

A. The longest diameter is called the transverse, and the shortest is called the conjugate diameter.

Q. How do you find the area of an ellipsis?

A. Multiply the two diameters together, then multiply this product by the decimal,7854, and the last product will be the

area.

EXAMPLES.

1. If the two diameters of an ellipsis are 24 and 36 fect, what is its area? Ans. 678 feet, 7 inches.