113 
M E N S U R A T I O N. 
Ing the re&angle AC for its upper fide, and OS (which is 
perpendicular to B Q) for the depth or thicknefs ; B Q 
'being parallel to O C. 
Suppofe the planes or Tides A O C G and RBCG, and 
alfo the end ARBO, are trapezoids, and the latter any¬ 
how inclined to the two former; and let the plane B P D 
be parallel to the fide RAG. Then the whole wedge B G 
•will be divided into a triangular prifm ARB P D G, and 
the pyramid PB O C D, the latter having B for its vertex, 
the trapezoid D P O C for its bafe, and O S the perpendi¬ 
cular height of B above the bafe. Then, if AI is the 
perpendicular diftance of A O and C G, the area of the 
parallelogram AD will be APX AI. And APX AIx*OS 
is the content of the prifm ARBPDG, OS being the depth 
of the parallelopiped. And P ° - — -—X AI is the area of 
, PO+CD 
the trapezoid PC, the bafe of the pyramid. And--- 
XAIXfOS, the content of the pyramid. 
But APxAIXlOS the content of the prifm, is the 
fame as 3AP multiplied by the reilangle or produft 
POxDC 
AIXiOS- And ---XAIX^OS the content of the 
pyramid, the fame as PO + DC multiplied by the reftan- 
o le AI X^S. Therefore the fum of both Or 3 AP-J-PO-l-DC 
multiplied by the re&angle AIx£OSis the content of the 
wedge BG. But AP+PO is equal to AO ; and AP+DC 
equal to G C; alfo A P is equal to R B. Therefore 
AOri-GC-l-RB is equal to 3AP + PO + DC. And confe- 
quently the fum AO+GC+RB multiplied by the pro¬ 
duft AIX^OS is the content of the wedge; OS being the 
perpendicular diftance of RB from the face AC. 
Ex. Let AO—4, GC=3, RB=:^, the perpendicular 
diftance (AI) of AO and GC=i2, and OS the perpendi¬ 
cular height of the face AC above RB=3J feet. Then 
4q-3+2|=9|, and 9|Xi*Xy=9£x 7=66| cubic feet, 
■the content. 
To find the Solidity of a Cone. —Multiply the area of the 
bafe by the perpendicular height, ana | of the produft 
will be the folidity. 
Ex. Required the folidity of the cone ABC, fig. iS. the 
diameter AB of the bafe being 12 feet, and the perpendi¬ 
cular altitude DC 18 feet 6 inches.—Here 7854Xi2 s = 
•7854 X144= 113-0976, the area of the bale; and 
r 1 3 - 0976 Xil^ 9 l 3 £ 56 = feetj the f 0 l idity 
.3 3 
required. 
To find the Solidity of a Sphere, or Globe. —Multiply the 
cube of the diameter by '5236, or the cube of the circum¬ 
ference by -016887, and the produft will be the folidity. 
Ex. What is the folidity of a globe whofe diameter is 
25 inches ?—Here 25 3 X'5236= 2 5X25X25X'5 2 3 6 = 62 5 
X 2S+’5236 = is625 X‘5 2 36 = 8i8i-2s cubic inches, the 
-folidity required. 
To find the Solidity or Superficies of any Regular Body. 
—1. Multiply the tabular folidity, in the following Table, 
by the cube of the linear edge of the body, and the produft 
will be the folidity. 2. Multiply the tabular area, by the 
fquare of the linear edge, and.the produft will be theyit- 
perficies. 
Solidities and Surfaces of Regular Bodies, when the linear 
edge of each is 1. 
No. of 
Sides. 
Names. 
Solidities. 
Surfaces. 
4 
Tetraedrcn 
0-1178511 
i’7;20 ;o8 
6 
Hexaedron 
rooooooo 
6'ooooooo 
8 
OClaedron 
0-4714045 
3-4641016 
12 
Dodecaedron 
7'66;ii8q 
20-6457288 
20 
Icofaedron 
2-1816949 
8-6602540 
Voi,. XV. No. 1028. 
Ex. What is the folidity and luperficies of the tctrac- 
dron A B C D, fig. 19. whole linear edge is eight inches ?—■ 
Here - i 178511 x 8 3 ='i 178511 X 512 = 60-3397632 inches, 
the folidity ; and i - 73205o8 X 8 2 = 17320508 X 64 = 
110 8512512 inches, the fuperficies. 
Of MEASURING TIMBER. 
Various methods are nfed in order to obtain the dimen- 
fions of Handing trees; but their girts and altitudes may 
be found moft correftly by means of a ladder, and a long 
lfaff divided into feet and inches, and numbered from the 
top to the bottom on one fide, and from the bottom to 
the top on the oppofite fide, for the convenience of taking 
dimenlions. The ladder will enable you to take the girt 
at or near the middle of the tree; and with the Half you 
may meafure the boughs and the upper part of the trunk. 
The lower part of the trunk may generally be meafured 
by a tape. 
Divide the arc of the quadrant of a circle of any conve¬ 
nient radius, into 90 equal parts, or degrees ; and figure 
them at every 10 degrees thus; 10, 20, 30, See. to 90. Upon 
that radius which is contiguous to 90 degrees, place two 
fmall brals fights ; and from the centre or angular point 
fufpend a plummet. Fix the quadrant to a fquare ftaft’, 
of a convenient length for ufe, by means of a nail palling 
through both ; and upon the end of this nail ferew a fmall 
nut, lb that the quadrant may be made fait to the ftaft' at 
pleafure. The neck of the nail lhould be round, fo that 
the quadrant may be readily turned upon it; but that part 
which is contained within the Half lhould be fquare, to 
prevent the nail from turning round with the quadrant. 
Then, in order to find the height of a tree, ferew the qua¬ 
drant fait to the ftaft', fo that the plummet may hang ex¬ 
actly at 45 degrees, when the ftaft' is perpendicular to the 
horizon. Move the ftaft' backward or forward, always 
keeping it perpendicular, until you can fee the top of the 
tree through both the lights; meafure the diftance between 
the bottom of the ftaft and the bottom of the tree, to 
which add the height of your eye; and the fum will be 
the height of the tree, fuppoling the ground to be hori¬ 
zontal. 
If, by reafon of impediments, you cannot retire fo far 
from the tree as directed above, make the quadrant fall, 
fo that the plummet may hang at 63! degrees when you 
view the top of the tree through both the fights; then 
twice your diftance from the tree added to the height of 
your eye, will give the height of the tree. If this alfo be 
impracticable, take any angle of altitude, and meafure the 
diftance to the bottom of the tree; then, by a fcale of equal 
parts, draw a line equal to the meafure-diftance ; and at 
one end of this line ereCt a perpendicular, and at the other 
end make an angle equal to the angle of altitude. Mea¬ 
fure the perpendicular of the right-angled triangle thus 
formed, by the lame fcale from which the bafe was taken, 
to which add the height of your eye; and you will obtain 
the height of the tree. 
Divide a fquare ftaft, AB, fig. 20. of about feven or 
eight feet in length, into feet and inches, for the conve¬ 
nience of meafuring the diftance between the place of ob- 
fervation and the tree, or taking any other dimenlions. 
Upon one fide of this ftaft', at a commodious diftance from 
the bottom, fix a rectangular board, CDEF, whofe length 
DE is exaCtly equal to twice its breadth CD, which breadth 
maybe about four or five inches. At C and D fix fights, 
or fmall iron pins ; and alfo at G and E ; making D G 
and GE each equal to CD. Then, when the top of a tree 
is feen through the fights at C and G, the tree's height is 
equal to your diftance from its bottom added to the height 
of your eye ; but, if feen through the fight at C and E, its 
height is equal to twice your diftance from its bottom, ad¬ 
ding the fame height as before. 
In making an obfervation with this inftrument, it ought 
to be fixed perpendicularly to the horizon, which may be 
done by means of a plummet fufpended from n. In taking 
the altitude of a tree growing upon an inclined plantq 
G g you 
