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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 28v, page id 00068197:0056
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A X, X. XV, y
AE, 9
BD, h
vy: Zlh = AVE: ACE = AYE: L99. Ergo gu = hoh, gX - XX.
Hinc si fiat rg hh fiet yy = rX - =XX. porro cum h, sit minor
quam 9. ser h: q ratiominoris erit et r. 9 ratio ad huc
itaqueY potest esse una ex upplicatis lipseos. sit FR,. fiet r = YX - XXX,
hine quaeratur X fiet   qx 74 -gr = XX - gX. seu  39. X-
C494. ) 1d = (t  H) = CF  (L49 i hh) = CF = ( FH
Cum ergo quadratum ab FC sit differentia quadraturum FB etBC, vel
Quadratorum ab FB, et ab , erit FB, L9. Ergo F est focus
 CY _ est =9 - X Vocetur Z, et fiet Lx erit X = Lg -7 et, et ex
gy = Lrq= et
minoris quir r: q = hh. 94
Et latus rectum est ordinatim applicatu ad axem in foco
2(+ E 42) - YZq = = r9 - X7 =  tt - hiti
qu
si est esset B, ei Ex masto cpseos F agatur nolmatis in aer cd jngatum Bd normalis  et occurens in V
et x CV vocetur v erit D - 2h - y, et feret  a Lhh - hv + bD = gy = Llit - hhZZ. et fiet Z2 = 4a v- 4qvD
plu 22= h9v - 2 W. sed prostat uti literii y et Z, ubi coordinatae i agibus ve
sumuntur, et a absunt simplices lariata. Poro Aris compug atus empue mineestidiveto e eutu an
quaeritur fa tagay + dz d2) seu Elementum curvae Ellipseos, fiet: est autem ydy = -y dZ
et duys dy =  he yque deder ,: 12hb - sitqque, adeoque dpdus + traz - 1 2dq, bili 23 1 - 12 2
itaque summore oportt AZ H 99- ∞e22,: 99. 94
ht yn =gaco et  et 7d7 = 2aded t d=
= 9122
 dydus +, drah = + g- eez 2g
Aadeo:, (3 laco s dz
d2d
F agged, aco Adesque de = a dco, ap ace + q 9972
A Et fiet dn = a du C1G4 4 raeeto
Y(aco, 99 aecco, +q aqqco
ta
az au
 itaque extrchibilis aedtenda est quadatice formula aco acoco-=  aqqeo-  vel ducendo invicem
quqqo.
4qued +qBac, quod totum dividamus per ee qq, tint C-44 (eergqu) nacoco +
Qarqus coco
Uhius sermulae vallies sunt acal, acd- 4 99, aed -9. ee 60, h et cD,  ibeC
Ia Hypert, A5, 9, CA, G latus rectum Y. sit tih = ce AX g. X, y
19 = ry -Ixa LYyy
yu = rX + YXX. sit Z - L9+X se fet - 24
+F
seu yy = XZ2 - L Yaseu gy = tih 2 + gtk
jungatur AB, ejque paractia dualu aXG occurrer ipsi  axi conpigato in e exsi
erit X Y, seu y, medi proportionalis inter DG et GB, seu inter summam
disferentiam ipsatum
itaq generaliter yy=
et
CB, et CG. idque tam in Ayibola quam in Ellessi reum es
22Lth,. ubi in Hy perbola
ydy = + tih 7dZ et dZdZ + dijdus = d2d2, Z2 +th F  B F, in Cllipsi minas est-.
seu d ez t ayaus - drc d N = p A  a q + ee 2,:, + 4922 + B L 90 7 4 223:,  nt
Ergo generalis haec est construssio pro Ellipsi et Hexperbo lice curveatemento, de
Z + fil
aut sot
s mbiguitate in valore ipsius
n
e in Ellipsi es distantia focorum quod sic ostendo; FB qu =- BCqu = FC qu, ident est quod 294 - L hh = FC qu.
 Ergo oC = 49 - hh= AFC qu = FO. qu.
ad quaeramus distuntiam focorum generaliter, tam pro Ellipsi quam pro Hyperbola
in aequatione si sit generli gy + ln q2 +  lh, sit y= LY, siet  vr + hh d Z ( hh. sec. 1Y = h: 49,
et fiit Zhh = + 22 + 419, seu 47Z = 49 F hh. Poro  27 = bis CF _  = FC
Ero FOqu = 49 + hn = ee, seu C = FC. itaque in . Hyperbola non minus quam in Ellipsi
e = YCga =+ lin) est distantia focorum
ape directo, seu latere transverso
Haec in Ellipsi est minor, in Hyperbola major
aa : 7 = 9 fi de = - a ly:    d8 d 5  de 1e = des s (a + 84):  B 3.t 2Z = A at, fiet X- f ak
et dus idh L  et ft s(d 9des + d3 de) = d dX sC at + aaxK, - .a  d(at tate)
eK
quod aequandum   Rk tatu = KK WV fit ( siet a + akk = k10c0
1
RK. = K. WW +  . W0
Zl -aa. = qua) W- 4 Ergo rest redit eo
ut. 15 - 4 a aequetur quadinto quo sit si aZ + ak  qu Caa + K K) suit aK AA = a M L XK  aa
44  4. sit 19: e = f et sit  4 = 3 a f - 4 fint 99+ f = 25
H= Apura +.
sin H(2
+ ff.) =. 5
Sit +hhf9e - ee
11