MEN
177
MEN
.5236
13000
15708000
5236
The
6806.8000 = .5236 (4 ad + 2d 2 )
24 ] eg e a ” swer ought to be 6830.4507 ; the error being aboutGruber an trut ^i which is very trifling in so great a
diirie n ai ^' tecture > we ver y seldom have domes of such largecan n l0nS 5 an< ^ va ^ ue a f° ot of plaster, or painting,fo re j J(j V<!r 6e of any great consequence. This rule will there-The p'!® c i.ent for every practical purpose.face s 0 ,.° ow * n g table contains the areas of the curved sur-forn^j 0 ®pheroids of various proportions, according to thePariso a X (4 a d + d 2 ) and to the real series; the com-
th e „ n ^'0 results will enable us to judge of the truth ofswer as found by the formula.
_
A
D
Area.
Difference.
Series
formula ...
40
40
40
40
5026.56 j5026.56!
none, being equal.
Series
formula
50
50
40
40
5882.63 \5864.32 J
18.31 defective.
Series
formula ...
60
60
40
40
6867.30 16702.08 j
65.22 defective.
^bovg j. S now try if any addition can be made to amend thetr Uth . ™J nu ' a > winch always gives the result less than thee ^Ual t Q e know, that the quantity to be added must be*be Sl)m Nothing when a is equal to d; we also know that^eir p ® the squares of two quantities is greater than twice6u rn of t i u ? t » an d when these quantities are equal, that thej. r efor e le » r Sf l uares a nd twice their product will be equal:( * 1 8 > er e , 1 ® ^ is greater than 2 ad; and the greater the
a 9 a nd et - Ween a an, i d, the more the sum of the squaress 8 ’ Whe ex ceed twice the product, 2 a d, of a and d:I* d* of ° ne l * le c l uant i ,; i e s> as d, is nothing in the sum,'V t j’ p ° i-i'e squares, the quantity a 2 only remains, and iny :| lue 0 » Product 2 ad, the whole vanishes. Therefore theet w een _
ofV t 0 “r d d ' S S reater -
' ^ e se q , °w what this difference is, by finding the valueTet t],| lantIt ies in the same terms :
'•'on ft £ reat er, d = a + x;
+ + 2 ax + x>;
A ^'n,2a f rJ? + 2 aX + X '-
et ' v «en a ^ d 1 — 2 ad will be greater, as the difference
and
a b ,
! a (a + x) = 2 a 2 -f* 2 a x :
now 2 a 2 + 2 a x + & is greater than 2 a 2 + 2 a x by oe 2 ;that is, the sum of the squares, a? + d 2 , of a and d, is greaterthan 2 ad by the square of the difference of a and d; there-fore the value of a 3 + d 2 — 2 ad will be greater, as a isgreater than d. But to show that the square of the differ-ence will not be very great in common cases, let a = 4,and d = 3, then the square of the difference is only 1; thatis, when the difference is 1, the square of the difference willalso be 1.
Let us now return to the subject, by adding the quantityproposed to .5236 (4 a d + 2 d 2 ) ; it will be found by addinga 2 + d 2 — 2 a d to .5236 (4 a d -f 2 d 2 ) that the sum will be
too great;divided by
addition of
therefore let thesome number:i 2 + dp — 2 ad
quantity a 2 + d 2 — 2 ad beon trial it appears that the
to .5236 (4 a d + 2 d 2 ) will give
Rules
A
D
Areas.
Difference.
Series .
40
40
5026.56 1
nothing.
Formula ...
40
40
5026.56 J
Series .
Formula ...
50
50
40
40
5882.635880.8 !
1.83 in defect.
Series .
60
40
6767 30 7
1.44 in excess.
Formula ...
60
40
6768.74 J
c ~ (a
a b c — ■
•tier.
«m ent
‘of
:nt . of the solid, or the solidity of a shell, the thick-- , ch is x : consequently, if the increment, 2 b e x +
°b t ai n 9 + . 2 « b x, of the solid be divided by x, we shall then8 oli<j . ? b c + 2ac + 2 ab + 2 a* for the surface of ther 8 l >o’tJ 1 ;' U * he quantity 2 ar 1 is indefinitely small, and appearsf ° r e g f, e ln comparison of26c + 2a« + 2aJ; there-
^ly .+ 2ae + 2 a b, or 2 a 6 + 2 a c + 2 6 c is
• 52 S6 r9 qua i to the surface of the solid ; and consequently,Hbq' a b + 2 ac + 26c), or 1.0472 (ab + ac + i c)'’gl ,. earl y equal to the surface of the ellipsoid.
2 z
VII. Rule 1; by adding-
a result very near to the truth, at the same time that itfurnishes a very easy formula for architectural purposes, inthe mensuration of domes, as will be seen in the followingtable.
This formula will therefore now stand .5236 (4 ad + 2d 2 )
a* + d 2 — 2 ad a 2 4- d 2 —2 a d
-)-----= 1.0472 (4 a d + 2 d 2 ) + —-
which may again be reduced to this form .8805 (2 ad + d 2 )a* + 2 d 2
f? *
This table shows the formula sufficiently exact for anypractical purpose, at least in all useful propositions of archi-tecture ; and therefore confirms Theorem VIII. Rule 2. Ifa dome upon an elliptic plan, rising half the minor axis, berequired to be measured, then the rule for such a dome
a 2 4- 2 d
will be .4402 (2 a d + d 2 ) + -—-.
The approximating formula 2 X .5236 (a b -f- a c + b c),Theorem IX. Rule 1, may be thus confirmed : let the threedimensions of the ellipsoid be a, b, c, and let x be a verysmall increment to be added to a, b, and c, respectively ; alsoa very small decrement to be taken from a, b, and c, respect-ively ; now the solidity of an ellipsoid is .5236 ab c; then
+ *) x (5 + x) X (c + x) = abc + bcx + acx + cx 2 + abx + bx 2 + ax 2 + x 2 , nearly ;
= (as — x) x (5 — x) x ( c — x) = a b c — box — acx + cx 2 —abx+bx 2 + ax 2 — a: 3 , nearly ;
therefore 2 bcx + 2acx +2 ab x + 2 x 3 is the
Corollary 1. —Hence 2 ab + 2ac + 2 be will be thesurface of a rectangular prism, exactly.
Corollary 2. —Let a, b, and c, be each equal to d; then1.0472 (ab + ac+bc) becomes 1.0472 X 3 d 2 = 3.1416d 2 = 4 x .7854 d\ as it ought to be : but the greater thedifference between a, b, and c, the more will the error be.This rule may be corrected in the same manner as Theorem-f J 2 -j-c 2 — ab—ac — be
■; which