FRUSTUM. 
er of fimi lar folids, when the inclination of the uo to 
the axis, and the altitude of the fruftum, are give 
In the parabolic conoid this difference vaitee: ie fruf- 
tum being always equal to a cylinder of the fame height, 
upon'the fection of the conoid that bifects the altitude of 
the fruftum, and is parallel to its bafes. 
Ina {phere, the fruftum is always lefs than the cylinder’ 
by one-fourth part of a pag 
height with the fruftum, or b 
cone, 0 i e jiame 
e half of-a 
a 
diameter equal to that height ; le “this difference 2 is s alwaye 
the as n all {pheres, when the altitude of the fruftum is 
give 
in the cone, the fruftum aie exceeds the cylinder 
by one-fourth part of the content of a fimilar cone that has 
the fame height with the fruftum 
In the hyperbolic conoid this ee is the fame as in the 
cone generated bya triangle oc e, (fig. 93 
axis oc, the afymptote oc, and a efpendicular ce, the al- 
titude of the frultum and the inclination of the axis to their 
bafes being the fame in both. 
In the {pheroid (fig. 94. ) the neers exceeds the fruf- 
the fa 
ee, and ay spake between e asin the 
e CDrd, the plane Drd, o a being fuppofed 
ee alle 1 to thofe which terminate tie fruftum. In different 
in nelination of thofe ‘planes, when the aleaids of the fruf- 
tum is given, that difference is rec ciprocally as the cube of 
the ce B4&, .which is thé conjugate o 
But if the altitude of the fru 
alfo varied fo as to be reciprocally proportional to the 
then the difference between the fruftum and 
cylinder will - always of the fame maemndes in the fame 
fpheroid or conoid. 
When 2 aeiaeon of the axis of the folid to the 
planes. that . ie the fruftum is give 
etween the fruftum and a indes in th 
odies, is as the cube of their common altitude. 
laurin’s Plicn, Introd. p. 24, 25. 
The rules above given, and others of et rai kind, are 
of ufe in the menfuration of timber 
Mac- 
of the one being 15 inches, and each 
fide of the other oO ae the. length along the fide meafuring 
24 = ? (fz. 9 
15 x = = 225, the greater bale and 6 x 6= 3 
ei lef, ae 15 x 6= 90, thei ne Their fum is 351, 
d 3d of it is 117. But AB — DE = 7h —3= 48 
=AF,and ~AD -~AF = V24x 127 — “4b? 
= 2847,9649 inches = D F, the perpendicular. Confequently, 
417 x 287, 9649 = 33691,8933 inches = 19,49762 feet 
= folidity. 
If a cafk, confifting of two equal conic fruftum 
jad together at the bafes, have its ae diameter 28 inches, 
its head-diameter 20 inches, and length 40 inches; how 
many. gallons of wine will it hold? 
ere 20° 
54 = 314,16 
= the area at the 
ev And 28° x 
bung circle, 
nd 20 x 28 x 47854 = 439,824 
a 
- 39854 == 615,7536 = the area of the 
== their mean pro- 
fum is 136957 16 
3 
nd its se thied part is 456,5792 
which multiplied by 40, the length ‘ef both fruftums | tos 
gether produces 18263168 folid sii AG Oth divided 
by 231, the inches in a wine ‘gallon, th 
13 
231=3xX 7X 11 
‘ 1 
69,6746 
ves = — 3 wine galtons, 
ty 20 7462 40 92618 = 17446 
X 10,472 = 18263,1 168, The folidity 2 oe 
3- What is the folidity of each of the frigid zones of the 
earth: the axis being 7957% miles, and half the breadth, 
or arc D A, (fig. 92.) of the zone being 233 degrees? 
tabular radius : 39783 = radius of 
= tab. verfed fine of 234 degrees 
330,0074946, the verfed fine or height of the fegment, 
Then 15236h* x 6 x S50:0074940% 
X 23213,2350108 = 1323679710, the content 
By ruler. As 1: 39783 2: 43987491 = ‘abular fine 
of 2335 degrees : 1586, aes 26, the ae of the bafe. 
ae 152364 x 3r' — AF 15236 330 0074946 
x 1660544086 = 192 s680z00165, the foliaits , 
What is the folidity of the fr uftum ofa Falietes the 
re of whofe great end is tour-feet, the diameter of the 
lefs three feet, and the neigh 25 eee > Here R* = 7? ae 4 
xX 197086 = 2°41 
x 39927 
required. 
. What is the folidity of each temperate zone of t 
x pee . 664 
tude, and the diameter of the i 
The radius of the top appears mg the third ome to be- 
1586,572 peed ia as 1: 39783 :: ,9170601 = tabular. 
fine of 663 degre cages: 86750538, the radins of the bafe. 
ae isi ey is 20622955. 
49136 x 2062,2955 
a L eae 
— 3 — 
» 5° 1,570 
32; oie i . folidity of the pares 
ddd + ihb x 145768 b = oo oe x 22602955 
x 31,5708 = 55877778668, the content. as befor 
a here i orm ©: hood ds Fruftam m, or 
inches ; 3 
= = 
Therefore 2D? rs +a x Cy an. 
= 2124 X z2= 
Then, Gace the ale 
282 cubie thee and the wine 
31, we have pen 2528 > 231 = 00) 288 wine- 
axes of an oblong f{pheroid be 50 and 30, 
quired the Sona of a fruftum whofe ends are perpendicular 
to the fixed axis, and one of them pafling through the'centre 
the height of the fruftum being zoinches. Here sha 3h a 
3 v 
3 x 25° — 207 ; ‘ 
er X 3st4159 X 20 x 15% = 
12 
1475 xX 3914159 x > = 11121,23799 = 
required. For a variety of other a Sou: w 
Hutton’s Menforstion, part iii. S$ 5 
TIMBER, 
FRUTEX. ex Suace, 
xX phrr= 
the content 
refer to 
os UGING —_ 
FRU. 
