III. For the Triangle F E G. half E F, 9.50

0.977723 Logarithm of GN, 5.82

0.764923 the Area of FEG, 55.29

1.742646 SABCF

2

98.86
The Arca of FCDE is

299.5 1
FEG S
F

55.29
The Area of ABCDEFG

453.66 V. Of a Circle A B C D, whose Diameter B C let be 14.00. Fig. XV. The Proportion of the Diameter of any Circle, to the Circum

ference thereof, is (in the least Terms) as 7 to 22 : But in greater
Numbers, as 113 to 355 ; and of these I shall make use.

I. By the Diameter, to find the Circumference.
As 113 : 355 : : 14 Diam. : 43.98 Cir.
Logar. of 113

2.053078 Logar. of 355

2.550228 Logar. of 14.00 the Diameter

1.146128

3.696356 The Logar. of 43.98 the Circumference

1.643278
II. By the Circumference, to find the Diameter.
As 355 : 113 :: 43.98 Circum. : 14.00 Diam.
Logar. of 355

2.550228 Logar. of 113

2.053078 Logar. of 43.98 Circ.

1.643278

3.696356 Logar, of 14.00, the Diam.

1.146128

III. By

Is to 355.

III. By the Diameter, to find the Area.

Fig.XVI. As 113 in 4 (viz. 452.)

2.655138

2.550228 So is the Diam. 14 in 14 (viz. 196)

2.292 256

4.842484
To the Logar. of 159.94, the Area

2.187346
IV. By the Circumference, to find the Area.
As 355 + 4 (viz. 113)

3.152288
Is to 113

2.053078 So is 43.98 + 43.98, Area Squa.

1.643255 The same again

1.643255

5.339588
To the Logar. of the Area 153.49

2.187300
V. By the Diameter, to find the Area.
As 10 : 8.862 : : 14 Di. : 12.41 : : 12.41 : 154.00
Log. 8.862

0.947532

1.146128 Logar. 12.41

1.093660 The double of it

2.187320 Log. of 154 the Area.

Log. 14

CHA P. IV.

I

OF GEODÆ CIA: Or, Land Measuring.
1. How (by any Graduated Instrument) to take the true Plot of

any large Piece of Ground, as Common Field, Park, Wood, &c.
N going round about a Field to Survey it, there are two ways;

for in going round about it, you must either go on the inside or
on the outside, and sometimes you may be constrained to go fome-
times within, and sometimes without.
· Let A B ĆDEF, be such a Field to be surveyed and plotted. Fig.

1. Begin at any angle thereof, as at A, and there setting up XVII. your Instrument, lay the Index and Sights upon the Diameter thereof; and turning it about, direct the sights to B, and there

Сс

screw

Fig. screw it fast; and turn the Index about, till through the Sights, XVII. you see the Angle at F; and there note what Degrees of the In

sirument were cut by the Index, which we will suppose to be 300 Deg. for the Quantity of the Exierior Angle F A B, without the Field : Or, 60 Deg. for the Quantity of the Interior dn. gle. within the Field: Then measure the Side A F, which fuppose to be 41.00 Pole or Perches, and the Side A B 30.00 Perches. These Diftances, with the Quantity of the Angle before observed 300, or 60 Deg. set down in a Book or Paper, as you see in the following Table.

2. Remove your Instrument to B, and laying the Index upon the Diameter, turn the Instrument about, till through the Sights you see the place where the Instrument last stood at A, and there

screw it fast ; and turn the Index about, 'till, through the Sighis, you fee the Angle C, and note the Degrees which the Index cutteth; which are here 145 Deg. for the Exterior, or 215 Deg. for the Interior Angle A BC; and the Distance B C measured is 28.40 Perches: Both which note down as before.

3. Remove your Instrument to C, and placing the Index on the Diameter, look back to B, and there screw it fift; and turn the Index about, till

you see the Angle at D; where the Index cutteth 270.25 Deg, for the Exterior, or 89.75 Deg. for the Interior Angic BCD, and the measured Diftance CD is 23:40 D:g.

4. Remove the Instrument to D, the Index on the Diameter, and look back to C, and then screw it falt; then turn the Index about till by the Sights you see the Angle at E; the Index cutting 263 Deg. for the Exterior, or 97 Deg. for the Interior Angle CDE, the Measured Distance D E being 28.00 Perches: Which note down as the other.

5. Place your Instrument at E, the Index on the Diameter, took back to D, and then screw it fast, and turn the Index about till through the Sights you see the Angle at F, the Index there. cutting 220 Deg, for the Exterior, or 140 D:g. for the Interior Angle D E F; the Measured Distance E F being 42.80 Perchesa.

6. Place the Instrument at F, and the Index lying on the Diameter, look back to E, and then screw it; and turn the Index a. bout, till through the sights you see the Angle at A, (or the Place where you firlt placed your Inftrument) where the Index cuts 210 Deg. for the Exterior, or'150 Deg. for the Interior Angle EF A, the Measured Distance F Á being 41.00 Perches, as before

. All which being noted down according to these Obfervations, and Measured, will stand as in this Table.

The

:

Fig.
Exterior.

Perches.
Inferior. 1

XVII.
A
300.00
60.00

A to B-30.CO
B
145.00 215.00

The Di

B to C-28.40
C 270.25
89.75

C to D-23.40 he Angle at D. 263.00 97.00

stance
from

D to E-28.00
E 220.00 140.00

E to F -42.80
F 210.00 150.00 l

-F to A-41.00 These Observations of Sides and Angles, as they were taken in le Field, being noted down in a Book or Paper, as in this Table; Plot of the Field may be drawn upon Paper or Parchment, by ae following Directions.

To Protract the former Work. 1. Upon a Sheet of Paper or Parchment draw a Line A B, to ontain 30.00 Perches, of any Scale : And upon the end thereof 1, place the Centre of a Protracter, laying the Diameter of it upin the Line A B: -Then, the Exterior _Angle at A, being 00.00 Deg. make a Mark against 300.00 Deg. of your Protra. Por ; and through that Point, and the Point A, draw a Right Line, ownwards, as A F, to contain 41.00 Perches, of the fame Scale.

2. Apply the Centre of the Protractor to the Point B, and the Diameter upon the Line A B; then against 145.00 Deg: (the Anle at B) make the Mark, and through that Mark, and B, draw a Right Line B C, to contain 28.40 Perches.

3. Lay the Centre of the Protraltor upon C, and the Diameter ipon B°C, and against 270.25 Deg, make a Mark ; through which, ind the Point C, draw the Line CD, to contain 23.40 Perches.

4. Apply the Centre of the Protractor to the point D, and its Diameter to the Line CD; making a Mark against 263.00 Dig. hrough which, and the Point D, draw the Line D E, to contain 28.00 Perches.

5. Lay the Center of the Protra&tor upon the Point E, and its Diameter upon DE, and against 220.00 Deg. make á Mark ; through which, and the Point E, draw a Right Line; which (if jou have committed no former Error) will cut the Line, first drawn, in the Point F: And, by this Means, you will have upon your Paper, the exa&t Figure of your Piece of Ground: Which you may caft up, and find the Quantity thereof in Perches, (by the Directions of the Third Chapter hereof) which being divided by 160, will give the Content in Acres and odd Perches. Сс 2:

In

Fig. In this Example, we have wrought by the Exterior, (or Outward) XVII. Angles : But if you would Protract the same by the Interior (or

Inward) Angles; it is done, by subftracting the Exterior Angle from 360 Deg. and the Remainder will be the Interior Angle: So, the Angle at A, being 300 Deg. that substracted from 360 Deg. leaves 60 Deg. for the Interior Angle at A: And fo of all the rest, as they are set down in the Table.

CHA P. V.

Of STEREOMETRIA:

OR,

Logar.27.04

Mensuration of Solids.
Fig.
XVIII.

I. Of the Cube ABCDE F, whose Side is 5.20 Foot.
As 1 : 5.20 : : 5.20 27.04 :: 27.04 : 140.61 Foot.
Lógar. 5.20

0.716003
Logar. 5.20

0.716003

1.432006 Logar. 5.20

0.7 16003 Log. of 140.61

2.148009 Equal to the Solidity of the Cube in Feet.

This might have been done more easily, for the Logar. of the XIX.

Side 5.20, which is 0.716003, multiplied by 3, would have pro-
duced 2.148009, the Logar. of 140.61, the Solidity.
1. Of a Long Cube or Parallelipipedon, GHIKL, whose Side

of the Square at the End, K I, is 1.27 Foot, and of its Length
GH, 5:32 Foot.

As 1 : 1.27 : : 127 : 1,61 :: 5-32 : 8.58 Foot.
Log. 1.27

0.103804
Log. 1.27

0.103804 Log. 5.32

0.725911 Log. 8.58

0.933519 The Solid Consent,

Fig.

III. Of