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Triangle, and multiplying the Half-Sum of the Sides of each into the Differences betwixt the Half-Sum and each Side continually; then extract the Square Root of the Product, F according to the Rule of Art. 57. Thus having obtained the Areas of all the Triangles, add them together, and this will be the Area of the given Figure. In the Triangle AFB the Bafe FA is 100, and the Perpendicular Ba 49; in the Triangle F BE the Bafe BE is 92, and the Perpendicular F d 52; in the Triangle EBD the Bafe BE is the fame as before, and the Perpendicular De 44; and in the Triangle DCB the Bafe DC is 80, and the Perpendicular Bb 38; by which the Area of each may be found, by Art. 56, as follows.

50 Half A F.

49 Perpendicular Ba.

450

200

2450 Area of AF B.

46 Half BE.
44 Perpendicular D c.

E

D

46 Half BE.
52 Perpendicular F d.

92 230

2392 Area of FBE,

38 Pependicular Bb.
40 Half DC.

1520 Area of DCB,

184

184

2024 Arca of EBD,

2450.

2392

2024

1520

9186 Area of the Figure ABCDEF.

In dividing any irregular Figure into Triangles, the Triangles will be lefs by two than the Number of the Sides of the Figure,

and

and the Diagonals lefs by three than the Number of Sides. Thus, in the prefent Cafe, the Figure has fix Sides, and is divided into four Triangles, FAB, FBE, EBD, and DBC, by three Diagonals, BF, BE, and BD.

Article 60. To meafure any regular Polygon.

Definition. A regular Polygon is a Figure whofe Sides and Angles are all equal; they are ufually denominated from the Number of their Sides: Thus, a Figure of five equal Sides is called a regular Pentagon; of fix Sides, an Hexagon, &c.

Rule. Multiply the Length of one of the Sides by the Number of Sides; then multiply this Product by the Half of a Perpendicular let fall from the Center of the Figure to the Middle of one of the Sides, and the Product is the Area of the Polygon.

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But, for the more ready measuring regular Polygons, the following Table, containing Multipliers for all regular Figures from the Triangle to the Dodecagon, or Figure of twelve Sides, will be of Ufe to the Learner. The Rule for making the Table will be fhewn at the End of the Trigonometry.

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Now, if a Side of any of these Figures be given, square the Side, and multiply it by the Multiplier in the Table, and the Product is the Area of the Figure.

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Exam. 1. The Square of the Side is 9025, which being multiplied by 1.72, the Multiplier in the Table, omitting the other Figures, as the third Place of Decimals is a Cypher, the Product is 15523, for the Area of the Pentagon, as before.

The other Example is done in the fame Manner, only there I have taken three Places of Decimals in the Multiplier, the third Place being 8, otherwife two Places are fufficient in moft Cafes..

I

To

To Measure a Circle and its Parts.

What a Circle is, and how defcribed, is fo well known that it can't be neceffary to define it, any farther than to acquaint the Reader with the Names of the feveral Parts to be measured or referred to in measuring.

In the annexed Circle A B C D, the Arch-Line ABCD is called the Periphery, the Length of which is called the Circumference: Any Line, as DB or AC, paffing thro' the Center E, cuts the Circle into D two equal Parts, called Semicircles or Half-Circles; and fuch Lines are called Diameters of the Circle: If two Diameters are drawn thro' a Circle, at right Angles to each

A

E

B

C

other, then is the Circle divided into four equal Parts, called Quadrants: Half the Diameter, as E B, is called the Radius or Semidiameter.

Article 61. The Diameter of a Circle being given, to find the Circumference,

Rule. This may be done by either of the following Proportions; in whole Numbers, as 7 is to 22, or in Decimals, as I is to 3.14159, fo is the Diameter of a Circle to the Circumference.

Exam. A Circle whofe Diameter is 17, to find the Circumference?

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The Answers are nearly the fame in both; but, for greater Expedition, the Multiplier in the Decimals may be reduced one Figure, and 3.1416 may be used without any fenfible Difference; by which the Diameter of a Circle being multiplied, the Product is the Circumference: The firft Term in the Proportion being 1, the Work of Divifion is rendered ufelefs.

Article 62. The Circumference of a Circle being given, to find the Diameter.

Rule. As 22 is to 7, or as I is .31831, fo is the Circumference of a Circle to the Diameter.

Exam. The Circumference of a Circle being 473, to find the Diameter ?

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Article 63. To find the Area of a Circle.

Rule. Multiply Half the Diameter by Half the Circumference, and the Product is the Area.

If the Diameter is given, find the Circumference by Art. 61 ; if the Circumference is given, find the Diameter by Art. 62. Exam. A Circle whofe Diameter is 17, and Circumference 53.4, to find the Area?

26.7 Half the Circumference.

8.5 Half the Diameter.

1335

2136

226.95 Area of the given Circle.

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