SE C T. V.

Containing Four new concife Methods of determining the Areas of Right Lined Figures univerfally, or by Galculation With fome necellary previous Definitions and Theorems.

CALCULATION.

1.

DEFINITIONS.

Plate VIII. Fig. 7.

ERIDIANS are North and South Lines,

M which are fuppofed to pass through every

Station, running parallel to each other.

II. The Difference of Latitude, or the Northing or Southing of any ftationary Line, is the Distance that one End of the Line is North or South from the other End; or it is the Distance which is intercepted on the Meridian, between the Beginning of the stationary Line, and a Perpendicular drawn from the other End to that Meridian. Thus, if NS be a Meridian Line paffing through the Point A of the Line AB, then is Ab the Difference of Latitude, or Southing of that Line.

Ff2

III. The

III. The Departure of any stationary Line, is the nearest Distance from one End of the Line, to a Meridian paffing through the other End. Thus, Bb is the Departure, or Eafting of the Line AB.

But if CB be a Meridian, and the Measure of the ftationary Distance be taken from B to A; then is BC the Difference of Latitude, or Northing, and AC the Departure, or Wefting of the Line BA.

IV. That Meridian which paffes through the first Station, is fometimes called the first Meridian; and fometimes it is a Meridian paffing on the Eaft or Weft Side of the Map, at the Distance of the Breadth thereof, from Eaft to Weft, set off from the firft Station.

V. The Meridian Distance of any Sation is the Distance thereof from the first Meridian, whether it be fuppofed to pass through the first Sation, or on the Eaft or Weft Side of the Map.

THE O. I.

In every Survey which is truly taken, the Sum of the Northings will be equal to that of the Southings; and the Sum of the Eaftings equal to that of the Weftings.

Plate IX. Fig. 1.

Let a beefg reprefent a Plot, or Parcel of Land. Let a be the firft Station, b the second, c the third, &c. Let NS be a Meridian Line, then will all

Lines

Lines parallel thereto, which pass through the feveral Stations, be Meridians alfo; as ao, bs, cd, &c. and the Lines bo, cs, de, &c. perpendicular to thofe, will be Eaft or Weft Lines, or Departures.

The Northings ei+go+hq=ao+bs+cd+fr the Southings. Thus:

Let the Figure be compleated; then it is plain that go+hq+rk=ao+bs+cd, and ei-rk=fr. If to the former Part of this firft Equation ei-rk be added, and fr to the latter, then go+hq+ei=ao+ bs+cd+fr; that is, the Sum of the Northings is equal to that of the Southings.

The Eaftings cs+qa=ob+de+if+rg+oh, the Weftings. Thus :

For aq+yo (az) =de+if+rg+oh, and, bo=cs-yo. If to the former Part of this firft Equation cs-yo be added, and bo to the latter, then cs+aq=ob+ de+if+rg+oh; that is, the Sum of the Eaftings is equal to that of the Weftings. QE. D.

SCHOLIUM.

This Theorem is of Use to prove, whether the Field-Work be truly taken, or not: For if the Sum of the Northings be equal to that of the Southings, and the Sum of the Eaftings be equal to that of the Weftings, the Field-Work is truly taken, otherwife not.

Since the Proof and Certainty of a Field-Work depend on this Truth, it will be neceffary to fhew how the Difference of Latitude, and Departure,

for