Full text: Bernoulli, Daniel: Hydrodynamica s. de viribus et motibus fluidorum commentarii

176. Scholium 2.

(VIII) Apparet quidem poſt levem rei contemplationem eò majorem
eſſe rationem inter arcum helicis o p q & integram helicem a 1 b, id eſt, inter
g & h, atque proinde eo majorem ceteris paribus aquæ quantitatem ſingulis
revolutionibus ejici, quo minor eſt angulus s a o & quo major angulus a M H,
ſeu quo minor eſt diſtantia inter duas proximas helices & quo magis cochlea
verſus horizontem inclinat: Veram autem illam rationem algebraice expri-
mere non licet: In omni tamen caſu particulari id facili appropinquatione
obtinetur.

Exemplum præcedentis regulæ deſumam à cochlea, qualem Vitruvius ad-
hibere & conſtruere docet. Facit autem angulum s a o ſemirectum & ſic
m = M = √{1/2} = o, 70710: deinde inter N G & M G rationem ſtatuit,
quæ eſt ut 3 ad 4; inde deducitur angulus G N M vel a M H = 53 0 , 8 1 , ejus-
que ſinus n = o, 80000 atque conſinus N = o, 60000: ergo (per art. III.) eſt ſinus arcus a g altiſſimum punctum o definientis = {m N/M n} = {3/4}, ipſeque
arcus a g = 48 0 , 35 1 . Debet adeoque vi regulæ art. VII. arcus extra aquam
eminens in fundo eſſe 97 0 , 10 1 ; immergeturque arcus 262 0 , 50 1 .

Ut jam præterea definiamus rationem inter arcum helicis o p q & helicem
integram a 1 b, notandum eſt, eandem eſſe illam rationem, quæ intercedit in-
ter arcum circularem g h M s & circumferentiam circuli, quod ex figura ſocia
manifeſtum eſt. Determinatur autem arcus g h M s hunc in modum. Eſt nem-
pe arc. g h M s = arc. a g h M s - arc. a g. Sed vidimus in articulo tertio, ſi ex
quocunque puncto ſpiralis, veluti o & q perpendicula ad horizontem punctum
M radentem demittantur, qualia ſunt o r & q x, fore iſtud perpendiculum
= {mNX/M} + n (1 + x) ſeu in noſtro caſu = o, 60000 X + o, 80000(1 + x),
denotante X arcum circularem, puncto in ſpirali aſſumto reſponden-
tem, nempe arcum a g aut arc. a g h M s & x ſignificante ejusdem arcus co-
ſinum. Eſt vero arc. a g = 48 0 , 35 1 = (quia radius exprimitur unitate)
o, 84797, ejuſque coſinus = o, 66153: Igitur in noſtro caſu fit or =
o, 50878 + 1, 32922 = 1, 83800. Quia porro puncta o & q ſunt in eadem
altitudine poſita, atque lineæ o r & q x inter ſe æquales, apparet quæſtionem
nunc eo eſſe reductam, ut alius arcus a g h M s inveniatur puncto q reſpondens,

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