1
MEN
198
MEN
Thus calling the whole length l , and the part to be cut off l,we have:
2d s n/ (d* + 4 ds) — 2s + d :: l : l =
J CL
X ( f (d 3 + 4 d s) — 2 s + d).
Example .—Using still the same numbers as in the prece-ding examples, we have, s = 16, d — 12, and l = 32.
I = — d X (^(*+ 4di)
2 s + d) =
32
2 X 12
X
(v/(12* + 4 X 12 X 16) — 2
X 16 + 12) = - X
(V (144 + 768) - 20) - X (v' (912) — 20) = | X
(30.1993377length to be cut off.
20) = - x 10.199337 = 13.5991198 the
O
Therefore the length of the remaining part is 32 —13.5991168 = 18.4008832.
And if s be taken from the root, viz. f (d 2 + 4 ds) halfthe remainder will be the girt of the section. Therefore,according to the numbers given, the girt of the section
f (<f* + 4rfs) — s 30.1993377 — 16
= 7.0996688.
2 ~ 2Hence the girt in the middle of the greater part is14 4. 7 099660
-^- = 10.549834, whose fourth part is 2.637458;
and consequently the content of the same part is 2.637458*X 18.40083 = 128, the very same as the whole tree mea-sures to, notwithstanding above one-third part is cut off thetrue length.
“ Note .—The principles of these last three Problems arealso applicable to the new, or second rule, in page 361, andindeed to any other approximate rule, or such as is notfounded on the rule for the frustum of a cone.
“ Demonstration. —Using still the same notation, weshall have L = (l — 2) X (g + x) 3 ; hence, instead of x,
, . . . , d z l
substituting its value — + g, we obtain z — —- xL 2 (i
(f (4 s + d). d) — 2 s + d).
“ And hence x — f (4 s + d). d) — J s. Q. e. d.”Rules have already been given for measuring the areas ofcircular segments, which may at least be depended upon tothree places of figures, and which we have thought suffi-ciently correct for most practical purposes; but where greateraccuracy is required, the following Table will carry theapproximation to five or six figures.
The construction which comes after the Table dependsupon the following series for the segment of a circle.
The area of the segment is a +
5 v
Ibi^ 3 c v
7 v
9 v
5 t>v 7 ev „ . 4 v -
+ ——, &c. where a = —- f v v, b, the second, c the
11 v
13 v
third, &c. See also Rules 3 and 4, Theorem XI., page 171,which are the same in principle as this.
A Table
et*
Of the Areas of the Segments of a Circle, whose cilu "‘~ lt glis Unity, and supposed to be divided into 1000 e< iParts.
.001.002.003.004.005.006.007.008.009.010, .011.012.013.014.015.016.017.018.019.020.021.022.023.024.025.026.027.028.029.030.031.032.033.034.035.036.037
• 038. 039.040
• 041.042.043.044.045.046.047
• 048.049.050.051.052
• 053
• 054
• 055
• 056
• 057
• 058
• 059
• 060•061• 062•063
• 064
• 065.066.067.068.069.070.071.072.073.074
it Area Seg.
Heigh
t Area Seg.
Heigl
t Area Seg.
. 000042
.075
. 026761
. 149
.073161
.000119
.076
. 027289
. 150
.073874
.000219
.077
. 027821
• 151
.074587
.000337
.078
. 028356
. 152
. 075306
.000470
.079
. 028894
. 153
.076026
.000618
.080
. 029435
. 154
.070740
.000779
•081
029979
. 155
.077469
.000951
.082
. 030526
. 156
.078194
.001135'’
.083
. 031076
. 157
.078921
.001329"'
.084
. 031629
. 158
.079649
.001533
.085
.031186
. 159
.080380
.001746
.086
. 032745
. 160
081112
.001968
.087
.033307
. 161
.081846
.092199
.088
. 033872
. 162
.082582
. 002438
.089
.034441
. 163
.083320
. 002685
.090
.035011
. 104
.084059
.002940
.091
. 035585
. 165
084801
.003202
.092
. 036162
. 166
085544
.003-171
.093
. 036741
. 167
086289
.003748
.094
. 037323
. 168
087036
.004031
.095
. 037909
. 169
.087785
.004322
.096
. 038497
. 170
088535
.004618
.097
. 039087
. 171
.089287
.004921
.098
. 039680
. 172
090041
.005230
.099
.040276
. 173
090797
. 005546
.100
. 040875
.174
.091554
.005867
.101
. 041476
. 175
.092313
.006194
. 102
. 042080
. 176
.093074
.006527
.103
. 042687
. 177
.093836
. 006865
.104
. 043296
. 178
094601
. 007209
.105
. 043908
. 179
095666
.007558
. 106
. 044522
. 180
.096134
.007913
.107
045139
. 181
096903 .
. 008273
.108
045789
. 182
096674 .
. 008638
.109
046381
. 183
. 098447 .
.009008
.110
047005
. 184
.099221 .
. 009383
. Ill
047632
. 185
.099997 .
.009763
.112
048262
. 186
.100774 .
010148
. 113
048894
. 187
.101553 .
.010537
.114
049528
. 188
.102334 .
010931
. 115
050165
. 189
.103116 -
011330
.116
050804
.190
.103000 .
.011734
.117
051046
. 191
. 104685 .
012142
. 118
052090
.192
.105472 .
012554
. 119
052736
.193
.106261 •
012971
. 120
053380
. 194
.107051 .
013392
. 121
054036
. 195
. 107842 •
013818
. 122
054689
.190
. 10S636 •
014247
. 123
055345
.197
. 109430 .
014681
. 124
.056003
. 198
.110226 .
.015119
. 125
056663
. 199
.111024 •
.015561
. 126
057326
. 200
.111823 ■
.016007
. 127
057991
.201
.112624 .
016457
. 128
058658
.202
113426 .'
016911
. 129
.059327
.203
114230 .i
.017369
. 130
.059999
.204
115065
017831
. 131
.060672
.205
115842
018296
. 132
.061348
.206
116650 ‘
018766
.133
062026
.207
117460 i
.019239
. 134
062707
.208
118271 A
.019716
• 135
003389
.209
119083 .
.020196
. 136
064074
.210
119897 .1
.020680
. 137
064700
.211
120712
021168
. 138
065449
.212
121529
.021659
. 139
066149
.213
122347 .-
022154
. 140
066833
.214
123167
022652
. 141
067528
215
123988 .*
.023154
. 142
068225
.216
124810 -'2
.023659
.143
068924
.217
125634 .2
.024168
.144
069625
218
126459 .-
.024680
.145
070328
.219
127285 - 2
.025195
146
071033
.220
128113 .2
.025714
.147
071741
221
128942 .2
.026236
.148
072450
.222
129773 .2
Height
.224. 225. 226.227.228.229.230.231232.233234.235.236.237.238.239.240.241.242.243.244.245
250
253
254
255
256
257
258
Area'
228 :1S»
:SSt
.133943. J 34784
’ l35b ti
• 1364 ??•137307
• 138130. I 3899 f
• 140688•141587. 1423« 7.143238.144091. 144944• 14579 ?
• DOOfiJ
■ I 4 ? 5 ';
• l48fi
14923 ?1300911509^151816152680
■ 13*$*^-.«0
• 1 56 Ijq
157019
- 157890
■ 15876^
• 159686
• itiOalO
l6l*g
16230 3
165780166663167546
168433
• 1 8031.2
• 17108 9• 1719/6• 1728 ?'
• 173738•174640
&
: S?
: S
1827181836'
1^11834 ')I8b3-j187 *#
188(4918904 ;,' 1809 f
■i$l
•'jgi
268