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

HYDRODYNAMICÆ tera ab, bc, cd, de, & ef, ſingulæ ſingulis æquales; jungantur lineis rectis
puncta B, C, D, E & F cum punctis a, b, c, d, & e: his ita factis, ſi ſuperfi-
cies plana rurſus in cylindricam convolvatur, junctis lineis A F & a f, coinci-
dentibuſque punctis A & a; B & b & c. fiet ut lineæ a B, b C, c D & c. in ſuper-
ficie cylindrica lineam continuam forment, quæ ipſa erit ſpiralis deſiderata. Ad
faciliorem intellectum in utraque figura puncta homologa communibus litteris
diſtinxi.

172.1.

Fig. 52.
(1.)
Fig. 52.
(2.)

(II.) Propoſitus jam fuerit cylindrus M a f N (Fig. 52. (1)), habens ad
ductum ſpiralis modo deſcriptæ circumflexum canalem, cujus diametrum ve-
luti infinite parvum cenſebimus ratione diametri ad cylindrum pertinentis: at-
que ſic habebitur cochlea Archimedis, quâ ſi uti velimus ad elevandas aquas ex
M in N, cylindrus erit horizontem verſus inclinandus, & ita quidem ut an-
gulus a M H (interceptus inter diametrum baſeos M a, quæ eſt in plano verti-
cali, & horizontalem M H) ſit major quam angulus s a o, quem faciunt tan-
gentes circuli & ſpiralis in communi puncto a. Deinde converſo cylindro cir-
ca axem ſuum in directione a g h M s aquæ influent per inferius canalis circum-
ducti orificium effluentque per ſuperius.

(III) Ut naturam hujus elevationis recte intelligamus, tria ſe nobis of-
ferunt puncta in qualibet ſpiralis helice examinanda, nempe puncta o, p & q,
quorum primum o maxime diſtat ab horizonte, alterum p eidem proximum eſt,
& q in eadem altitudine poſitum eſt cum puncto o in helice proxime inferio-
ri ſumto: per ſingula puncta o ducta eſt recta g n; per puncta p recta h m & per
puncta q recta s t. Situs vero harum linearum determinabuntur in ſequentibus.

(IV) Sit radius, qui pertinet ad baſin cylindri, = 1 ſumatur-
que pro ſinu toto; ſinus anguli sao = m, ejuſdemque coſinus = M, ſinus an-
guli a M H = n, ejuſdemque coſinus = N; arcus a g = X; coſinus illius arcus
= x, erit perpendiculum ex o in horizontem demiſſum, nempe o r = {mNX/M}
+ n (1 + x). Quia vero or maxima eſt, fit {mNdX/M} + ndx = o, & cum ex
natura circuli ſit dX = {-dx/√1 - xx}, erit {- mNdx/M√(1 - xx)} + ndx = o, ergo
√1 - xx = {mN/Mn}. Eſt igitur ſinus arcus quæſiti a g = {mN/Mn} aut coſinus

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