Containing Five various Methods by which the Areas of Right-lined Figures may be determined, two of which were never yet publiked,

T

DEFINITION.

HE Area or Content of any Plane Surface in Perches, is the Number of Square Perches, that Surface contains.

Plate VII. Fig. 1.

Let ABCD reprefent a Rectangular Parallelogram, or Oblong: Let the Side AB, or DC, contain 8 equal Parts; and the Side AD, or BC, three of fuch Parts: Let the Line AB be moved in the Direction of AD, till it has come to EF; where AE, or BF, (the Distance of it from its first Situation) may be equal to one of the equal Parts. Here 'tis evident, that the generated Oblong ABEF, will contain as many Squares as the Side AB contains equal Parts, which are 8; each Square having for its Side one of the equal Parts, into which AB, or AD, is divided. Again, let AB move on till it comes to GH, fo as GE, or HF, may be equal to AE, or BF; then it is plain that the Oblong AGHB, will contain twice as many

Squares

Squares as the Side AB contains Equal Parts. After the fame Manner it will appear, that the Oblong ADCB will contain three times as many Squares as the Side AB contains equal Parts; and in general that every Rectangular Parallelogram, whether Square or Oblong, contains as many Squares as the Product of the Number of equal Parts in the Base, multiplied into the Number of the fame equal Parts in the Height, contains Units, each Square having for its Side one of the equal Parts.

Hence arifes the Solution of the following Problems.

PROB. I.

To find the Content of a Square Piece of Ground,

1. Multiply the Bafe in Perches, into the Perpendicular in Perches, (or fquare the Bafe) the Product will be the Content in Perches and because 160 Perches make an Acre, it must thence follow, that

Any Area, or Content in Perches, being divided by 160, will give the Content in Acres: the remaining Perches, if more than 40, being divided by 40, will give the Roods, and the laft Remain der, if any, will be Perches.

Or thus:

2. Square the Side in Four-Pole Chains and Links, and the Product will be fquare Four-Pole Chains and Links; divide this by 10, or cut off one more than the Decimals, which are five in all, from the Right towards the Left: The Figures

refting

refting to the Left are Acres, because 10 square Four-Pole Chains make an Acre, and the remaining Figures are Decimal Parts of an Acre. Multiply the five Figures to the Right by 4, cutting 5 Figures from the Product, and if any Figure be to the Left of them, it is a Rood, or Roods; multiply the laft cut off Figures by 40, cutting off five, or (which is the fame thing) by 4, cutting off four; and the remaining Figures to the Left, if any, are Perches.

1. The firft Part is plain, from confidering that a Piece of Ground in a fquare Form, whofe Side is a Perch, muft contain a Perch of Ground; and that 40 fuch Perches make a Rood, or Stang, and four Roods an Acre; or which is the fame Thing, that 160 fquare Perches make an Acre, as before.

2. A fquare Four-Pole Chain (that is a Piece of Ground four Poles or Perches every Way) muft contain 16 fquare Perches: and fince 160 Perches make an Acre, therefore 10 times 16 Perches, or 10 fquare Four-Pole Chains make an Acre.

Note, that the Chains given, or required, in any of the following Problems, are fuppofed Two-Pole Chains, that Chain being moft commonly used in this Kingdom.

EXAMPLES.

C. L.

Let ABCD be a fquare Field, whose Side is 14. 29;

I demand the Content in Acres.

By

C. L.

By Problem 4. Section 3. 14. 29, are equal to

29.16 Perches.

29.16

17496

2916

26244

5832

C. L.

A. R. P.

160)850.3056(5. 1. 10. Content.

40)50( 1 Rood

10 Perches.

Or thus:

C. L.

14.29 is equal to 7.29 of Four-Pole Chains, by

7.29

Prob. 1. Sect. 3.

It is required to lay down a Map of this Piece of Ground, by a Scale of twenty Perches to an Inch.

Take 29.16 the Perches of the given Side, from the small Diagonal on the common Surveying Scale, where 20 fmall, or two of the large Divisions are an Inch; make a Square whofe Side is that Length, (by Prob. 9. Sect. 1.) and it is done.

PRO B. II.

To find the Side of a Square, whofe Content is given.

Extract the Square Root of the given Content in Perches, and you have the Side in Perches, and confequently in Chains.

It is required to lay out a fquare Piece of Ground which fhall contain 12A. 3R. 16P. Required the Number of Chains in each Side of the Square and to lay down a Map of it, by a Scale of 40 Perches to an Inch.

A. R. P. 12. 3. 16

4

;

51 40

C. L.

2056(45.34 Perch, which is 22. 33, by Prob.

85)456

903)3100

9064)39100

[6. Sect. 3.

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