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equal parts, viz. the Triangle e az to the Triangle abz (byTheorem 10;) at the point z make the Angle nzb to the Angle F, draw n in Parallel to zb, and of equal Length to z b, draw m b Parallel to nz and of equal Length to n z, then is the Parallelogram in nzb equal to the Triangle e a b. Now because it is required that a Paral lelogram fhould have its fide equal to the Line A B, continue the Line n m, and make m x eqal to AB,continue the Line z b,and make br equal to mx, and join, the Line x r; from the point x and b, draw the prickt Line at pleafure, continue the Line nz to interfect at the point o, draw o y Parallel to z r, and continue the Line mb to interfect at p, making py equal and Parallel to b r, and join the Line ry; then is the Parallelogram bry p, equal to the Parallelogram nm bz, by Theorem 11. In like manner make the Parallelogram py qs equal to the Triangle EBD; likewife make the parallelogram sqfh equal to the Triangle DBC,then will the Parallelogram br f h be equal to the Right-line Figure ABCDE QEF and QED.
If it were required to lay out a Rightangled Parallelogram equal to the fame Right-lined Figure, and the Side equal to the Line AB, let the Right-lined Figure ABCDE contain 12 Acres, adding 5 Cy phers to it, the fame Field will contain 1200000 Square Links: Let the Line AB re
prefent the Length of a Hedge given containing 60 Chains or 6000 Links; Divide 1200000 by 6000 and the Quotient will be 2000 Links or 20 Chains for the Breadth. of the Right-Angled Parallelogram; for 2000 Links multiplyed by 6000 Links, produces 1200000 Square Links, or Square Acres.
To reduce Statute-measure to Cuftomary-measure, and the Contrary.
Although an Acre of Land by Statute is to contain 160 Square Perches, of 16 Feet and in the Perch; yet in fome places of the Nation, through long Custom, there is at this day other Perches ufed, as 18, 20, 24, and 28 Feet to the Perch; it is therefore neceffary to fhew how to reduce Statute-meafure to Cuftoinary, &c.
Suppofe therefore you would reduce. Statute measure to Wood-land-measure of 18 Feet to the Perch, then fay
As the Square of the greater Perch of 18 Feet, is to the Square of the leffer Perch of 16 Feet and, fo is the Content in Acres according to the leffer Perch, to the Con tent in Acres, according to the greater Perch..
Let it therefore be required to reduce 36 Acres, 2 Roods, 10 Perches, at 16 Feet and to the Perch, into Wood-land-measure of 18 Feet to the Perch.
Firft, You muft obferve that the Square of 16 Feet andis Decimally 272, 25, and the Square of 18 Feet is 324; then I reduce the 36 Acres 2 Roods, 10 Perches, into Perches, which makes 5850, then i multiply the fame by the Square of the leffer Perch, 272. 25, and the Product is 1592662. 50, being divided by the Square of the greater Perch 324, the Quotient is 4915,625, which is 30 Acres, 2 Roods, 25: Perches for the Answer.
But fuppofe you would reduce Woodland-measure into S atute-measure, then say,
As the Square of the leffer Perch of 16 Feet and is to the Square of the greater Perch 18 Feet; fo is the Content in Acres according to the greater Perch to the Content in Acres, according to the leffer. Perch..
How to caft up the Content of any Plot, in Acres, Roods, and Perches.
Fig. 112. Admit the following Figure noted with the Letters A, B, C, D, E, F, G, H, I, be the plot of a Field, whofe
Content in Acres, Roods and Perches is required.
Fift, Then (in all fuch Cafes ) divide your Plot into Trapezia's and Triangles; accordingly this Figure is divided into one Trapezium, as K, D, P, I, and four Triangles; for finding the Area of all which, begin with any one firft, and multiply the whole of the Bafe by one half of the Perpendicular, or (which is alone) the whole of the Perpendicular, by the of the Bafe the product either of the Content of that Triangle and then Sum up all the Area's of the feveral Triangles to her, gives you. the Content of the whole Plot.
Letan. Area or
A. R. P.
Trap. KDPI 7-2-25
Trian. DEF 1--3---00
The Area of the whole Field.
But the moft exact way of all, is to multiply the Length of the Bafe of each Triangle, by the Length of the Perpendicular; the Sum Total of all the Triangles being halved, gives the true Area of the whole Field in Square Links, (or Chains and Links)
Links,) which may be reduced at laft (by the former Dire&tions) into Acres, Roods and Percher.
Of laying ont New Lands.
A certain Quantity of Acres being given, how to lay out the fame in a Square Figure.
Annex to the number of Acres given five Cyphers, which will turn the Acres into Links; then from the Number thus increased, extract the Square Root, which fhall be the fide of the proposed Square.
Example. Suppose the Number given to be joo Acres, which I am to lay in a Square Figure ; I join to the 100, five Cyphers, and then it is 10000000 Square Links, the Root of which is 3162 nearest, or 31 Chains 62 Links the Length of one side of the Square.
Again, If I were to cut out of a CornField one Square Acre, Ladd to one five Cyphers, and then it is 100000, the Root of which is 3. Chains 16 Links, and fomething more, for the side of the Acre.