Needle lies between, (the North, South, Eaft and Weft being called the four Cardinal Points, and are graved on the Bottom of the Box,) putting down those Points together by their initial Letters, and thereto annexing the Number of Degrees, counting from the North or South, as before, thus: If the Point of your Needle lies between the North and Eaft, North and Weft, South and Eaft, or South and Weft Points in the Bottom of the Box, then put down NE, NW, SE, or SW, annexing thereto the number of Degrees cut by the Needle on the Side of the Ring, counting from the North or South, as before.

But if the Needle Points exactly to the North, South, Eaft, or Weft, you are then to write down N, S, E, or W, without annexing any Degree. This is the Manner of taking Field Notes, whereby the Content of Ground may be universally determined by Calculation; and they are faid to be taken by the Quarter'd Compass, or by the four Nineties.

To find the Number of Degrees, contained in any given Angle.

Set up your Inftrument at the Angular Point, and thence direct the Sights along each Leg of the Angle, and note down thier refpective Bearings as before; the Difference of thefe Bearings, if lefs than 180, will be the Quantity of Degrees contained in the given Angle; but if more, take it from 360, and the Remainder will be the Degrees contained in the given Angle.

OF

OF THE

THEODOLITE.

T

HIS Inftrument is a Circle, commonly of

Brafs, of ten or twelve Inches in Diameter, whose Limb is divided into 360 Degrees, and those again are fubdivided into smaller Parts as the Magnitude of it will admit; fometimes by equal Divifions, and fometimes by Diagonals, drawn from one concentric Circle of the Limb to another.

In the Middle is fixed a Circumferentor, with a Needle; but this is of little or no Ufe, except in finding a Meridian Line, or the proper Situation of the Land.

Over the Brafs Circle is a Pair of Sights, fixed to a moveable Index, which turns on the Center of the Inftrument, and upon which the Circumferentor Box is placed.

This Inftrument will either give the Angles of the Field, or the Bearing of every stationary Diftance Line, from the Meridian; as the Circumfe rentor and Quaered Compafs do.

First then, To take the Angles of the Field.

Plate VI. Fig. 6.

Lay the Ends of your Index to 360, and 180;

X 2

turn

turn the whole about with the 360 from you, direct the Sights from A to G, and fcrew the Inftrument faft, direct them from A, to cut the Object at B; the Degree then cut by that End of the Index which is oppofite to you, will be the Quantity of the Angle GAB, to place in your Field-Book; to which annex the Measure of the Line AB, in Chains and Links: Set up your Inftrument at B, unscrew it, and lay the Ends of your Index to 360, and 180; turn the whole about with the 360 from you, or 180 next you, till you cut the Object at A; fcrew the Inftrument faft, and direct your Sights to the Object at C, and the Degree then cut by that End of the Index which is oppofite to you, will be the Quantity of the Angle ABC. Thus proceed from Station to Station, ftill laying the Index to 360, turning it from you, and obferving the Object at the foregoing Station, fcrewing the Inftrument faft, and obferving the Object at the following Station, and counting the Degrees to the oppofite End of the Index, will give you the Quantity of each refpective Angle.

LEMM A.

All the Angles of any Polygon, are equal to twice as many Right-Angles as there are Sides lefs by four. Thus, all the Angles A, B, C, D, E, F, G, are equal to twice as many Right-Angles, as there are Sides in the Figure, lefs by four,

Plate VI. Fig. 6.

Let the Polygon be difpofed into Triangles, by Lines drawn from any affigned Point H within it, as by the Lines HA, HB, HC, &c. It is evident

then

then (by Theo. 5. Sect. 1.) that the three Angles of each Triangle are equal to two right; and confequently, that the Angles in all the Triangles, are twice as many right ones as there are Sides: But all the Angles about the Point H, are equal to four right (by Cor. 2. Theo. 1. Sect. 1.) therefore the remaining Angles are equal to twice as many right ones as there are Sides in the Figure, abating four. Q. E. D.

SCHOLIUM.

;

Hence we may know if the Angles of a Survey be truly taken: For if their Sum be equal to twice as many Right-Angles, as there are Stations, abating four Right-Angles, you may conclude that the Angles were truly taken, otherwife not.

If you take the Bearing of any Line with the Circumferentor, that Bearing will be the Number of Degrees the Line is from the North; confequently the North must be a like Number of Degrees from the Line, and thus the North, and of course the South, as well as the Eaft and Weft, or the Situation of the Land, is obtained.

Secondly, To take the Bearing of each refpective Line from the Meridian; or to perform the Office of the Circumferentor, or Quarter'd Compass, by the Theodolite.

Set your Inftrument at the firft Station, and lay the Index to 360 and 180, with the Flower-de-Luce of the Box next the 360; unscrew the Inftrument, and turn the whole about, till the North and South Points of the Needle cut the North and South Points in the Box; then fcrew it faft, fo is the Inftrument

North,

North, and South, abftracted of the Variation.

The Circumferentor Box may be then taken off.

Direct the Sights to the Object at the fecond Station, and the Degree cut by the oppofite End of the Index will be the Bearing of that Line from the North, and the fame that the Circumferentor would give.

After having measured the ftationary Distance, fet up your Inftrument at the second Station, unfcrew it, and fet either End of the Index to the Degree of the last Line, and turning the whole about with that Degree towards you, direct your Sights to an Object at the foregoing Station, and fcrew the Inftrument faft; it will then be parallel to its former Situation, and confequently North and South Direct then your Sights to an Object at the following Station, and the Degree cut by the oppofite End of the Index, will be the Bearing of that Line.

In like Manner you may proceed through the whole.

If the Brafs Circle be divided into four Nineties, from 360 and 180, and the Letters N, S, E, W, be applied to them; the Bearings may be obtained by putting down the Letters the far or oppofite End of the Index lies between, and annexing thereto the Degrees from the N or S; and this is the fame as the Quarter'd Compass.

If you keep the Compafs Box on, to fee the mutual Agreement of the two Inftruments; after having fixed the Theodolite North and South, as

before ;