MEN
173
MEN
X.—To find the solidity of the segment of a
Cl,E —s' = p(rtf — ; where p is equal to 4 X
’^eorem. 3,1416: and *
Sc;
l E 2-a _
and r, as in the preceding2 a 2
= .7854 (^+^11’T'l . \ 2 3 /
b(>i n 18 ? n 'y differs from Rule 2. Theorem IX. in the basediin? ® lrc ular. It gives the contents independent of the* Tn ter °f the great circle.
doin e EOrem XI—To find the solidity of a truncated square8e ct i ’ ln< kp en dent ^e radius, or diameter, of the vertical
uon.
8 = a 4 a 5
, " g— ; where a is the altitude, and s the side, of
® square base.
T
a hemi^^ M —To find the solidity of the frustum of8
th,
•^854 (a s 1 — ; being the same as in the
last
^ ore m, except the multiplier .7854.
c ateif »° Rem XIII—To find the solidity of a hollow trun-8 e ^quare dome.
hstvw 0 ^ D * — < ^ 2 ) > where 'd is the side of the square base,8qa are R opposite external surfaces, d the side of theth e a n-. b j 8e between the opposite internal surfaces, and a
1 URe • supposing the heel to be equally thick.
8 ph er -°, RKM XIY.—To find the solidity of a hollow hemi-•^854 rU8tUm ‘
e * e et>tiL a J Da ~ ; being the same as Theorem XIII.,
•p "® the multiplier .7854.
8 -n. h?** XY.—To find the solidity of a paraboloid.
2 > where A is the height, and b the area of the base.
4 —To find the solidity of the frustum of
8 5;
^2 ^ > where a and b are the areas of the two ends.
s a XVII.—To find the solidity of an hyperboloid.
"" —— 4 m
t w g A ; where a and b are the areas of the
*i ^ filing «
^ill ^ area sect i° n ' This
co n ' 1 a ' S ° serve to measure a sphere, spheroid, para-1 these boj’i P^ rarn *d, pyramoid, or any segment, or frustum
■Vote* on tfe preceding Theorems.
HtjLj. g Rectification of Curves.
8in r ’ Theorem VII. by Huygens , is very neatlyE es8 «d by ® ^ — c
fon ni j * jj— 5 but the half chord must either be
^ ^J^Uy, by bisecting the chord by a perpendicular,0p ei>at ; ° l " e half chord, or by a very operose arithmetical
bsti“It.
%
‘fitioti = 2 A — c, then will c + d + — be the
. °f th e 45
'^tior, . , ai- e : this affords a very easy geometrical con-
c/ ^y?ecl„ VlZ ’ if to the chord of the whole arc be added.Srf,? 1 " of twice the l — —-' ' J -
„ :,' tce of twice the chord of the half arc and the
* lle whole arc , and one-third of the said differenced will b e nenrh , fh „ -c *■- * 1
nearly the length of the arc, for c + d
+ (2A —
c) +
2 A — ,3
2 T
8 A
Rules 3, 4, 5, 6. Theorems VII. VIII. IX. and X. wereinvented by Mr. Nicholson. The circumstance which gaverise to them was a stone weir, or dam, which he had designedand superintended at Denton Holm Head, over the riverCaldew, near Carlisle. The form of the weir was that ofthe segment of a circle, of which the chord was 200 feet, andthe versed sine 22 feet; when the work was completed, thecontractor for the mason’s work was very desirous to have itmeasured, but at that time, being early in the spring of 1810,the river was flooded so high, that the water ran two feetabove the top of the weir ; now having the true dimensions,as above, and an exact section of the work, it was onlynecessary to find the length of the arc, and then the solidcontents were easily computed : but, in order to obtain thisend, he found that the calculations by the rule of Huygenswould require too much trouble for common business, andtherefore, as well for present convenience as for anythingthat might happen in future of a like nature, the following
6 d — v
formulas were invented : l = c
or t = c +
6 d — 5 »’
4 vc _3 c 2 + 10 v*
6 d — 5 v ~ °3 c* + 2 v ,t
8 v 7 c 6 s 2 + 5 v*
OT L = ° + 3o> + 2/ = C 6 s 2 + t, 2 ’
4 v 2 c
or L = C + — -j.
6 tr + tr
The second formula, or Rule 4, is derived from the first,by substituting the value of d in terms of c and v; andRule 5 is also derived from Rule 3, by substituting the valueof d in terms of s and v ; so that, being derived from eachother, they will all give the very same result; and to showhow near this is to the truth, the investigation is as follows:
_ 6 d — v 6 d — v
For c ;- : — =2 (dv — » ) 2 X —;-the
6 d — 5 v
6 d — 5 v
quantity 2 (d v — f 2 ) 2 , being the value of c, in terms ofd and v:
but2 0* = 2d“t^ x (i — Yd~ §V’ &C -)
,6 d — v 2 v 5 v*
and - = 1 +-U -, &c.
6 d — 5 v T 5d r 9<f’
Hence 2 (dv — t> 2 )^ X
6 d — v6 d — 5 v
= 2 d*v^ X
i '- h - Zp *) *
= 2<&vi X (l + gL +
but 2dhvi x ( 1 + ~-z + V , &c.^ is known to ex-\ o d 40 cr J
press the arc of a circle, whose diameter is d, and its versedsine v ; now this last series only differs from the former in
the third term, in being less ; the excess being
45 <P
Now, to show how far these rules may be depended uponin practice, Mr. Nicholson calculated the following table,