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# Title: Phoronomia, Sive De Viribus Et Motibus Corporum Solidorum Et Fluidorum Libri Duo (shelfmark Leibn. Marg. 182)
# Folio 269, page id 1066518432:0297
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DE VIRTBOSETMOTTDVS CORPORDM. LIB. II. 269
S CH OL ZO N.
467. Flaec propositio plurimum conducie reductioni elemento.
rum sectorum tiyperbolicorum ad simpliciora elementa l0g. mica. Nam
si AO =a; OD = m; OR = X; EB = Y; AF = Z, _E AF =11. AF.
que his positis, exit
T. y = V (∞∞ = aa), AF = aV (X∞ =aa: XX); adeoque Ff = a’ dX:
vaer (xX = aa), e FFI. TTB = a dX: XaV (∞∞ = aa). Inde quia trianq.
BOB: Friang. FOf =OFE: OA, reperietur duplum trianguli BOB =
aadx: V (XX = aa). TIOC ideM BFCVIUS inVentum FIISTCE ODC RUM. IV.
S. 466. uti ostendimus 2. BOb =OT. b0; nam b0 = ad∞: V (XX = aa)
ec OT = aa: X, ergO OT. b0 = aadX: XV (∞X = aa) = aadX: V (X X -
aa). Ta KHI = (aX = aV (XX = a4)): X, ec CFI vel OT = aa: X;
atque triangula RHICBE KOD similia TUnE, erGOm =aa: X - V (∞X -
aa) ec x = (aa L mm): 2m, qui ralor substitutus in praecedenti ex-
prectione elementi BOB, producet 2BOb = aadm: m.
TI. Ouia x = a + W _ dx = du, crit 2BOb = aadx: V (∞X = a1) =
aadu: V (2aW = Wn), X W + a = (aa = mm): 2m, vel W = (M = a 2: 2M,
qui ralor in Formula aadu: V (2au = un) suffectus, dabit iterum
2BOb = aadm: m.
TTT. Propter triangula similia OAF _ OFB invenietur ay = XZ,
hyperbola vero praebet y = Y (XX = aa); unde ∞ =V (aa - 2Z), CX
criangulis vero similibus XTTCI _E KOD elicitur am -mZ = aV (aa
=22) X Z =amm -- a: mm + aa, qui ralor ipsius 2 ejusque cle-
menti ad2: aa = 22 = 2BOb substitutus, dabit etiamnum 2BOb =
aadm: M.
Similes regulas reducendorum sectorum hyperbolicorum ad lo-
sarichmos ex lua methodo generali reducendi quadraturas curra-
rum, quarum ordinatae Fractionibus rationalibus per abscillam rarie
affectam ex datas, exhibitis exprimuntur, ad logarithmos, jam pri-
dem elicuit Vir Celeb. Joh. Bernoulli in Comm. Acad. Res. Scient.
Paril. 1702. d. T3 Decemb. e in Act. Lipl. 2703. P25. 30. _c3I. Cdetc-
rum, et si praecedens theorema nostrum de lirperbolatantum requila-
cera agit, paucis tamen mutatis ad quascunque hrperbolas Potest
extendi.