MENSURATION. 
planes j of solids, or substantial objects ; 
and tlie lengths, breadths, &c. of various 
figures ; either collectively or abstractedly. 
The mensuration of a plane superficies, or 
surface, lying level between its several 
boundaries, is easy ; when the figure is re- 
gular, such as a square, or a parallelogram, 
the height, multiplied by the breadth, will 
give the stiperficial contents. Thus, if a 
table be 5 feet 2 inches in length, by 4 feet 
1 inch in breadth, multiply 62, (the number 
of inches in 5 feet 2 inches) by 49, (the 
number of inches in 4 feet 1 inch), the re- 
sult will shew tlie number of square inehes; 
which, being divided by 144, (the number 
of square inches in a square foot), will exhi- 
bit the number of square feet on the sur- 
face of the table. Whatever balance may 
remain, may either be left as fractional, or 
hundred and forty-fourth parts ; or, being 
divided by 36, may be made to show the 
numbers of quarters of square feet, beyond 
tlie integers produced by the first division. 
For instance, multiply 62 inches 
by 49 inches 
558 
' 248 / 
Divide by 144) 3038 (21 : 
288 
lo8 
144 
14 
In regard to triangles, their bases multi- 
plied by half their heights, or their heights 
by half their bases, will give the superficial 
measure. But it is necessary to caution 
our readers not to measure by the oblique 
line of a triangle, considering it as the alti- 
tude : a reference to the article Geome- 
try will show, that the height of a triangle 
is taken by means of a perpendicular to the 
base, limited by a parallel to the latter, 
which exactly includes the apex, or sum- 
mit. 
Any rectilinear figure may have its sur- 
face estimated, however numerous tlie sides 
may be; simply dividing it into triangles, 
by drawing lines from one angle to another, 
but taking care that no cross lines be made : 
thus, if a triangle should be equally subdi- 
vided, it may be done by one line, which 
must, however, be drawn from any one 
point to the centre of the opposite face. A 
four-sided figure will be divided into two 
triangles, by one oblique line connecting 
the two opposite angles : a five-sided figure 
(or pentagon) by two lines, cutting as it 
were one triangle out of the middle, and 
making one on each side ; a six-sided figure 
(or hexagon) will require three diagonals, 
which will make four triangles ; and so on 
to any extent, and however long, or short, 
the several sides may be respectively. 
With respect to the form and properties 
of various figures, we refer our readers to 
the head of Geometry, where all that re- 
lates thereto is pointed out, and the com- 
mutations they undergo, when their con- 
tents or areas are measured, will be dis- 
tinctly seen. 
The most essential figure is the circle, 
of which matlieraaticians conceive it impos- 
sible to ascertain tlie area with perfect pre- 
cision, except by the aid of logarithmic and 
algebraic demonstration. It may be suffi- 
cient in this place to state, that 8 of the 
diameter, will give the side of a square, 
whose area will be correspondent with that 
of a circle having 10 for its diameter. 
Therefore as the diameter may be easily di- 
vided, either arithmetically*, or mechani- 
cally, into ten equal parts, and one of tliose 
parts into seventeen ; by taking 8 integers, 
and 10 of the 17 th portions, the side of such 
a square may be easily demonstrated. 
Where a circle is small, its scale may be 
extended by an oblique line, which may be 
made to any extent, as shewn in the fig. 7, 
Plate X. Miscel. where A B is the diameter 
of a circle, and A C the oblique line, lying 
between the perpendiculars that would fall 
on A B. If A C be divided into any num- 
ber of parts, perpendiculars drawn from 
AB to the points of division, as ab,cd, 
will divide the diameter exactly, in the 
same proportions as A C is divided. The 
radius, or semidiameter, of a circle also gives 
us the means of forming a square corres- 
ponding with its area. Having drawn the 
w'hole diameter, A B, fig. 8, take the ra- 
dius, C B, and set it off from B to D ; from 
which measure another radius, at right an- 
gles with C B, to wherever it may fall, i. e. 
at E, on the diameter : the hypothennse, 
B E, will give the side of the square sought. 
We have been particular in describing 
this process, because so many cireular or 
cylindrical figures come under the mea- 
surer’s consideration ; whether they be mir- 
rors, arched passages, columns, &c. The 
contents of a pillar are easily ascertained, 
even though its diameter may be perpetually 
varying ; for if we take the diameter in dif- 
ferent parts, and strike a mean between 
every two adjoining measurements, and 
multiply that mean area by the depth, or 
