Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
Text
Text Image
Image
XML
Thumbnail overview
Document information
None
Concordance
Notes
Handwritten
Figures
Content
Thumbnails
Table of Notes
<
1 - 3
[out of range]
>
<
1 - 3
[out of range]
>
page
|<
<
(187)
of 361
>
>|
<
echo
version
="
1.0RC
">
<
text
xml:lang
="
la
"
type
="
free
">
<
div
xml:id
="
echoid-div222
"
type
="
section
"
level
="
1
"
n
="
175
">
<
pb
o
="
187
"
file
="
0201
"
n
="
201
"
rhead
="
SECTIO NONA.
"/>
</
div
>
<
div
xml:id
="
echoid-div223
"
type
="
section
"
level
="
1
"
n
="
176
">
<
head
xml:id
="
echoid-head224
"
xml:space
="
preserve
">Scholium 2.</
head
>
<
p
>
<
s
xml:id
="
echoid-s5384
"
xml:space
="
preserve
">(VIII) Apparet quidem poſt levem rei contemplationem eò majorem
<
lb
/>
eſſe rationem inter arcum helicis o p q & </
s
>
<
s
xml:id
="
echoid-s5385
"
xml:space
="
preserve
">integram helicem a 1 b, id eſt, inter
<
lb
/>
g & </
s
>
<
s
xml:id
="
echoid-s5386
"
xml:space
="
preserve
">h, atque proinde eo majorem ceteris paribus aquæ quantitatem ſingulis
<
lb
/>
revolutionibus ejici, quo minor eſt angulus s a o & </
s
>
<
s
xml:id
="
echoid-s5387
"
xml:space
="
preserve
">quo major angulus a M H,
<
lb
/>
ſeu quo minor eſt diſtantia inter duas proximas helices & </
s
>
<
s
xml:id
="
echoid-s5388
"
xml:space
="
preserve
">quo magis cochlea
<
lb
/>
verſus horizontem inclinat: </
s
>
<
s
xml:id
="
echoid-s5389
"
xml:space
="
preserve
">Veram autem illam rationem algebraice expri-
<
lb
/>
mere non licet: </
s
>
<
s
xml:id
="
echoid-s5390
"
xml:space
="
preserve
">In omni tamen caſu particulari id facili appropinquatione
<
lb
/>
obtinetur.</
s
>
<
s
xml:id
="
echoid-s5391
"
xml:space
="
preserve
"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5392
"
xml:space
="
preserve
">Exemplum præcedentis regulæ deſumam à cochlea, qualem Vitruvius ad-
<
lb
/>
hibere & </
s
>
<
s
xml:id
="
echoid-s5393
"
xml:space
="
preserve
">conſtruere docet. </
s
>
<
s
xml:id
="
echoid-s5394
"
xml:space
="
preserve
">Facit autem angulum s a o ſemirectum & </
s
>
<
s
xml:id
="
echoid-s5395
"
xml:space
="
preserve
">ſic
<
lb
/>
m = M = √{1/2} = o, 70710: </
s
>
<
s
xml:id
="
echoid-s5396
"
xml:space
="
preserve
">deinde inter N G & </
s
>
<
s
xml:id
="
echoid-s5397
"
xml:space
="
preserve
">M G rationem ſtatuit,
<
lb
/>
quæ eſt ut 3 ad 4; </
s
>
<
s
xml:id
="
echoid-s5398
"
xml:space
="
preserve
">inde deducitur angulus G N M vel a M H = 53
<
emph
style
="
super
">0</
emph
>
, 8
<
emph
style
="
super
">1</
emph
>
, ejus-
<
lb
/>
que ſinus n = o, 80000 atque conſinus N = o, 60000: </
s
>
<
s
xml:id
="
echoid-s5399
"
xml:space
="
preserve
">ergo (per art. </
s
>
<
s
xml:id
="
echoid-s5400
"
xml:space
="
preserve
">III.)
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s5401
"
xml:space
="
preserve
">eſt ſinus arcus a g altiſſimum punctum o definientis = {m N/M n} = {3/4}, ipſeque
<
lb
/>
arcus a g = 48
<
emph
style
="
super
">0</
emph
>
, 35
<
emph
style
="
super
">1</
emph
>
. </
s
>
<
s
xml:id
="
echoid-s5402
"
xml:space
="
preserve
">Debet adeoque vi regulæ art. </
s
>
<
s
xml:id
="
echoid-s5403
"
xml:space
="
preserve
">VII. </
s
>
<
s
xml:id
="
echoid-s5404
"
xml:space
="
preserve
">arcus extra aquam
<
lb
/>
eminens in fundo eſſe 97
<
emph
style
="
super
">0</
emph
>
, 10
<
emph
style
="
super
">1</
emph
>
; </
s
>
<
s
xml:id
="
echoid-s5405
"
xml:space
="
preserve
">immergeturque arcus 262
<
emph
style
="
super
">0</
emph
>
, 50
<
emph
style
="
super
">1</
emph
>
.</
s
>
<
s
xml:id
="
echoid-s5406
"
xml:space
="
preserve
"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5407
"
xml:space
="
preserve
">Ut jam præterea definiamus rationem inter arcum helicis o p q & </
s
>
<
s
xml:id
="
echoid-s5408
"
xml:space
="
preserve
">helicem
<
lb
/>
integram a 1 b, notandum eſt, eandem eſſe illam rationem, quæ intercedit in-
<
lb
/>
ter arcum circularem g h M s & </
s
>
<
s
xml:id
="
echoid-s5409
"
xml:space
="
preserve
">circumferentiam circuli, quod ex figura ſocia
<
lb
/>
manifeſtum eſt. </
s
>
<
s
xml:id
="
echoid-s5410
"
xml:space
="
preserve
">Determinatur autem arcus g h M s hunc in modum. </
s
>
<
s
xml:id
="
echoid-s5411
"
xml:space
="
preserve
">Eſt nem-
<
lb
/>
pe arc. </
s
>
<
s
xml:id
="
echoid-s5412
"
xml:space
="
preserve
">g h M s = arc. </
s
>
<
s
xml:id
="
echoid-s5413
"
xml:space
="
preserve
">a g h M s - arc. </
s
>
<
s
xml:id
="
echoid-s5414
"
xml:space
="
preserve
">a g. </
s
>
<
s
xml:id
="
echoid-s5415
"
xml:space
="
preserve
">Sed vidimus in articulo tertio, ſi ex
<
lb
/>
quocunque puncto ſpiralis, veluti o & </
s
>
<
s
xml:id
="
echoid-s5416
"
xml:space
="
preserve
">q perpendicula ad horizontem punctum
<
lb
/>
M radentem demittantur, qualia ſunt o r & </
s
>
<
s
xml:id
="
echoid-s5417
"
xml:space
="
preserve
">q x, fore iſtud perpendiculum
<
lb
/>
= {mNX/M} + n (1 + x) ſeu in noſtro caſu = o, 60000 X + o, 80000(1 + x),
<
lb
/>
denotante X arcum circularem, puncto in ſpirali aſſumto reſponden-
<
lb
/>
tem, nempe arcum a g aut arc. </
s
>
<
s
xml:id
="
echoid-s5418
"
xml:space
="
preserve
">a g h M s & </
s
>
<
s
xml:id
="
echoid-s5419
"
xml:space
="
preserve
">x ſignificante ejusdem arcus co-
<
lb
/>
ſinum. </
s
>
<
s
xml:id
="
echoid-s5420
"
xml:space
="
preserve
">Eſt vero arc. </
s
>
<
s
xml:id
="
echoid-s5421
"
xml:space
="
preserve
">a g = 48
<
emph
style
="
super
">0</
emph
>
, 35
<
emph
style
="
super
">1</
emph
>
= (quia radius exprimitur unitate)
<
lb
/>
o, 84797, ejuſque coſinus = o, 66153: </
s
>
<
s
xml:id
="
echoid-s5422
"
xml:space
="
preserve
">Igitur in noſtro caſu fit or =
<
lb
/>
o, 50878 + 1, 32922 = 1, 83800. </
s
>
<
s
xml:id
="
echoid-s5423
"
xml:space
="
preserve
">Quia porro puncta o & </
s
>
<
s
xml:id
="
echoid-s5424
"
xml:space
="
preserve
">q ſunt in eadem
<
lb
/>
altitudine poſita, atque lineæ o r & </
s
>
<
s
xml:id
="
echoid-s5425
"
xml:space
="
preserve
">q x inter ſe æquales, apparet quæſtionem
<
lb
/>
nunc eo eſſe reductam, ut alius arcus a g h M s inveniatur puncto q </
s
>
</
p
>
</
div
>
</
text
>
</
echo
>