SURVEYING. 
- Propomon VII. “To ascertain the area 
of tlie irregular six-sided figure, or hexagon, 
(fig. 7) A B C D EF.” In this, some of the 
angles point inward. First draw the line C E, 
which will divide the figure into two trape- 
zia ; viz. C EB A and C E F D, next divide 
each of these trapezia by the diagonals B E 
and' CD, into two triangles respectively : the 
areas of the four triangles, B A E, B C E, 
FCD, and CED, will, when added toge- 
ther, exhibit the contents of the whole 
figure. 
Proposilion VIII. “ To measure the irre- 
gular field A B CD E, fig. 8.” The figure 
here given has two curved sides, one of 
which projects, the other which inflects: 
the ordinary parts, which can be divided 
into triangles, are worked in the manner 
already shown ; but the curved parts must 
be measured in the following mode ; Draw 
the line ED, and from it make three, or 
more, off-sets to the curved part ; take from 
E to 1, as a base, and half the depth of the 
ofi'-set 1, as an altitude ; multiply them to- 
gether : then take from 1 to 2, as a base, 
and the mean of the depths of the off sets 
2 and 3, for the altitude ; multiply these also 
together: do the same for the space between 
2 and 3, and calculate the end, between 3 
and D, as was done from E to 1 ; the sum 
of their several products, added together, 
will show the area of the curve. As the 
other curve bends inward, draw the line 
AE, and treat it the same as was done 
regarding E D : then, considering the entire 
triangle, A E D, as a part of the field, com- 
pute its contents, and deduct from it the 
measures taken by means of the off-sets 
4, 5, 6 : the residue, added to the contents 
of the curve from E to D, and of the trian- 
gles ABC, and ACD, wall show the area 
of the whole figure A B C D E. It is ob- 
vious that, in this manner, the extent of 
water may be deducted from the area of 
any field. 
The next figure. No. 10, shows the me- 
thod of surveying with a plain-table, which 
usually stands upon three legs, and has a 
compass attached to one side. There is a 
box- wood frame that fits on the board of 
the plain-table, and is graduated with 360 
degrees. This serves to show the direction 
of any line from the centre of the board, 
where there is a brass stud, or plate, let in ; 
and it also compresses the paper so as to 
prevent its shifting. To this instrument 
there is a brass rule of two feet long, with 
ends turned up at right angles, in which are 
slits, or Sights, to direct the surveyor’s eye. 
He places the rule so as to touch the brass 
centre, arid directing it to any particular 
point, observes the angle it makes, accord- 
ing to the index on the box-wood frame, 
while an assistant measures the distance, 
from the centre under the plain-table, to 
the point observed. The surveyor draws a 
tine on the paper, in the direction of the 
brass rule, from the exact centre of the 
plain-table, and notes down at the side of 
that line, what its length may be in chains, 
links, &c. according to Gunter’s scale ; or 
else in yards and feet, as in familiar measure- 
ment. Thus, in fig. 10, A represents the 
plain table, placed in the centre of the field- 
BCDE: fg is the rule with sights: the 
figures written on the sides of the lines pro- 
ceeding to the four corners, express their 
several distances from the centrg of the 
plain-table. This mode of surveying is pecu- 
liarly suited to small surveys ; especially to 
the interiors of enclosed places ; and has the 
advantage of forming the plan on the paper, 
as the survey proceeds : for the number of 
yards, &c. being set off, from the scale on 
the, brass rnle, on the several directing lines, 
as from the centre A to B, C, D, and E, 
respectively ; and their lengths being deter- 
mined by their several due measures, the 
lines BC, CD, D E, EB, will give a true 
fac-simile of the shape of the field. The 
contents must be ascertained by dividing 
the field, as . before explained, into triangles, 
whose conjoint measurements will amount 
to its contents. 
Although the plain-table is not a sufficient 
instrument for general purposes, it is in the 
foregoing instance extremely convenient: 
its use may be extended, under due pre- 
caution, to ascertaining the distances of re- 
mote, or of inaccessible, objects, situated 
on tlie same level with itself. But tor such 
purposes a theodolite, standard-triangle, cir- 
cumferentor, or some instrument capable 
of taking heights, as well as levels, is ordi- 
narily employed. The following proposition 
will illustrate the above point. 
Proposition XI. To ascertain the dis- 
tance of an object at C, from the point B, 
fig. 11.” Draw the base line, B A, in any 
convenient direction, and from each station 
(commonly marked by surveyors © ) ob- 
serve what angle is madej viz. at B, ascer- 
tain the value (or extent) of the angle CBA, 
and at A, the value of the angle CAB: 
take care to be very correct as to the 
measurement of your base line ; which we 
will take at 120 yards. Now, tlie points 
A and B being established at a certain dis- 
