SURVEYING. 
jre a ; or, half the circumference multiplied by 
the radius = area ; viz. AaB X A b — area. 
Circular Ring. Between two concentric circles 
multiply the sum of the diameters by their dif- 
ference, and that product by .7854, and the half 
product — area : (fig. 11.) 
j AC -j- DB X AC — DB X -7854 — area. 
Segment of a Circle, or Other cuwillneal figure. 
Divide the line OP (fig. 12.) into any even num- 
ber of equal parts, as O a, ab, be, &c.; and let 
perpendiculars be raised from these points. Put 
B for the sum of a c J., c4, and other even ordi- 
nates, and C for the sum of the others ; then 
four times B X twice C, X the common distance 
between the ordinates, = three times the area : 
4 b " I - 2 c 
t h a t is, — A x D (the common distance) 
= area. 
A mean breadth may readily be found, by di- 
viding the whole measure of the ordinates by 
the number of them, accounting the ends parts 
of such number; which mean breadth multiplied 
by the length, will be = area. 
Ellipse. Multiply continually together the two 
uses and the decimal .7854, and the product = 
area ; viz. AC X BD X .7854 = area. (fig. 13.) 
All pieces of land are found to be of some 
one of the forms before described, or composed 
*f two or more of them ; and the general rule 
for finding the content of any such compounded 
figure is, to divide it into as many of the fore- 
going simple figures as the case requires ; to 
measure such lines and angles in the field as may 
be necessary to determine the content of each 
single figure ; and the sum of the whole will be 
= area. 
The Chain. The most general instrument 
which a land-surveyor employs, is the chain. — 
Chains of sundry. lengths and dimensions were 
invented in former days ; but that which was 
most approved of, and is now in general use, 
was invented by the Rev. Edmund Gunter , about 
180 years since, and is composed of 100 links 
of strong iron wire, each link 7.92 inches; con- 
sequently the whole chain is 22 yards, or 4 poles 
in length. Hence it appears to be peculiarly 
well adapted to the measuring of land ; as 10 
square chains (that is, 10 chains in length and 
1 in breadth, or 5 in length and 2 in breadth, 
or of any other dimensions in such proportion), 
are exactly an acre. 
The accompaniments to the measuring-chain 
are a 6taff or rod, of the tenth part of a chain, 
called an ofl'-set staff, divided into ten parts, 
answering to ten links of the chain, by which 
short distances are measured ; to which staff a 
rectangular cross may be affixed, to set off the 
direction of lines perpendicular to a general 
line. Picket staves to set up in the angles of 
fields are necessary ; and ten arrows of strong 
wire, which are employed by the measurer’s 
assistant at each chain’s length. 
The dimensions of all lines on the land are 
taken in chains, or, rather, the links of a chain ; 
and the contents are found in square acres, 
roods, and perches. The acre, we have before 
observed, contains 4 square roods ; a rood con- 
tains 40 square perch.es. In one square acre are 
100.000 square links ; in a square rood are 
25.000 square links ; and in a square perch are 
625 square links. 
By an ordinance of the 35th of Edw. I., as 
well as by a statute of the 34th of Hen. VIII., it 
is ordered, that the perch should be 16^ feet ; 
but custom, however, permits perches of dif- 
ferent lengths to prevail, in sundry parts of the 
kingdom : for instance, in Lancashire the cus- 
tomary perch is 21 feet in length; in Cheshire 
and Staffordshire, 24 feet; in Dorsetshire, 15^ 
feet ; in Somerset and Devon, 15 feet ; and in 
Cornwall the customary perch is 13 feet. 
To reduce the statute measure to either of the 
customary measures, the following rules will 
apply : — first, if the customary is smaller than 
the statute, as the Devonshire for instance, sgy, 
as the square of 15 is to an acre, or number of 
statute acres, so is the square of 16.5 to the 
number of customary acres : — secondly, if the 
customary is the larger measure, as the Che- 
shire for instance, say, as the square of 24 is to 
an acre, or number of acres, so is the square of 
16-J to the number of acres customary. 
Before a measurer begins his work in the 
field, he should consider what lines are necessary 
to be measured for obtaining the content ; 
taking such as require the least walking forward 
and backward. 
Having carefully measured such lines as will 
reduce the field to some of the simple figures 
before-mentioned, with such of their measuring 
lines as may be necessary, he will be enabled to 
find the content of each part, by the rules laid 
down in the former part of this article. 
We would observe, that a measurer may di- 
vide the same field different ways, and obtain 
the content thereof by each. For instance, the 
field ABCDF. (fig. 14), may be divided into a 
trapezium ABCD, and a triangle ADE. 
Or, it may be divided into four triangles, (as in 
fig. 15. 
It may also be divided into four triangles 
AE<7, ED5, C de, and Bed, and two trapezoids 
D bde, and AB.rc; as in fig. 16. 
Or, into three triangles AE<7, EBC, BC«, and 
one trapezoid AaBc ; as in fig. 17. 
Land-measurers are much in the practice of 
taking such lines only in the field as will enable 
them to draw a geometrical plot thereof by 
some scale of equal parts ; and by taking such 
measure-lines on the plot, by the same scale, 
they calculate the content with less trouble than 
by taking all such measure-lines in the field, as 
may be necessary to reduce the same to trian- 
gles, trapezia, or other simple figures. 
The calculations for the quantity of land in 
the same field, by the four respective methods 
of taking the dimensions, will stand as follow : 
Big. 16. 
Triangle AEa = A a x ” 
260 X 180 = 46300 
Triangle ED5 = DJ X Ei = 
450 X 330 = 148500 
Triangle CDr ~ de X Cc = 
470 x 50 — 23500 
Tiiangle Bed — Be x cd — 
320 X 60 = 19200 
Trapezoid D bde — D/> -j- de X bd 
— 450 4- 470 X 560 
Trapezoid AaB*. = An -j- Ik x ac 
— 260 4- 3¥cT x 622 
515208 
- 360700- 
)m3900 
Fie*- 14. 
Trapezium ABCD 
Jib — |— De X AC 
2 
Triangle ADE = 
460 -J- 440 X 
AD X El * 
1020 
=s 459000 
780 X 251 
97890 
5.56890 
5 acr. 2 rds. 1 1 per. for the answer. 
It is unnecessary to divide the square links of 
each small part by 2 ; as the double content may 
be carried on, and the aggregate, from thence 
arising, be divided by 2, once for all. 
Iig. 15. 
Trapezium AF.DC = De -f- X EC 
~ ‘29d -j- 330 X 1020 = 634440 
Triangle ABC = AC x B b = 
1020 X 470 = 479400 
5 acr. 2 rds. 1 1 perches, the answer, 
as before. 
5 C 3 
2.2768 
H.072 - 
5 acr. 2 rds. 11 perches, as before. 11.120 
Fig. 17. 
Triangle AE a — An x E a — 
330 x 10= 23100 
Triangle EDC = EC xW = 
1020 X 292 = 297840 
Triangle BCc = B; x Ce = 
624 X 390 = 243460' 
Trapezoid ABca = A a -j- Be X ae 
— 634 -j- 330 X 570 = 549480 
)1 1 13§8C» 
~5.56 940 
2.27760 
5 acr. 2 rds. 1 1 perches, as before. 11 ,1040~ 
We have hitherto confined our consideration 
to such figures only, whose few sides are straight 
lines of considerable length ; but, as the general 
boundaries of many pieces of land consist of 
short indentations, it is necessary to avoid the 
tediousneSs of computing the contents of a mul- 
tude of small triangles and trapezoids ; to find 
such equalizing lines as shall constitute a tri- 
angle, or other figure, of equal area with the 
sum of all such triangles and trapezoids com- 
bined. 
Suppose, then, that an irregular boundary of 
a field is of the form of fig. 18, composed of 
two triangles and four trapezoids. 
Draw the line AB, and at A erect a perpen- 
dicular AC. — Lay a parallel ruler from A to e, 
the third point. Move the upper part of the 
rule to b, and note where it cuts the perpendi- 
cular, as at 1. — From this point 1, lay the ruler 
to d\ bring its lower part down to c, and note 
where it cuts the perpendicular, at 2. From 2 
lay the rule to e, and move it upwards to d, and 
mark the perpendicular at 3. — From thence lay 
the rule to fi, and bring it down to e, and mark 
the perpendicular at 4. — From this point lay the 
rule to B, and raise it to/, and mark the per- 
pendicular at 5. — From 5 draw the line 5B ; 
then will the triangle, AB5, be equal in area to 
the aggregate of the two triangles and four 
trapezoids. 
Example. Suppose that, on some well gradu- 
ated scale, the base of the triangle Agb, was 
found to be 185, and perpendicular, 1 10; the 
base, gh, of the adjoining trapezoid 250, and 
sum of its perpendiculars 160; the base, hi, of 
the next trapezoid is 120, and its perpendicu- 
lars 180; the base ik 325, and the perpendicu- 
lars of that trapezoid 190; the base kl of the 
next trapezoid 300, and the perpendicular A 
thereof 349 ; the base of the latter triangle, /B, 
630, and its perpendicular, Ifi, 289 ; and that the 
content of the whole is required. 
Suppose also, the content of the triangle ABC; 
whose base AB, by the same scale, is 1810, anti 
perpendicular AC, is 238,^ is required. 
