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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 40r, page id 00068198:0079
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4 seu 22 = 
grbb
sit conicae centricae centrum C vertex A, curda AV. Focus vortici
proximus F, ordinata ad ab F ad quem FGer  Erit FG semilatus
rectum et AC semilatus transversum. sit AX, X; et CX, Z;
et XY y et FG a et AC b. fiet 2ax + (a : b) XX - u
unde ubi tantum observatur, quod b in Hyper
bola est quantitas effirmativa sed in Ellipsi quantitas negativa
 A X = Z - b, et fiet 2aZ - 2ab + a 2Z - 2 a Z + ab = yy
seu fit (a: b) 22 -  ab = gy . Unde (a: b) ZdZ = ydy seu fiet
dy: dz = (7. y) (a: b) Et dydys. AZ (22, 9y) (aa : bb) Et dy dy + d2dzue, : dz dz = Z2 (ac. bb) + By,: yy
Curve
Eodem
modo dz: dy = (y: Z) (b: a). Et dedz . dydys = (yy: 27) ab: 4) et dz dze + dydys,: dydus =
gy (bb: 4d) +22,: 22.  1  r dy du) = d ((b   gy +bb seu fiet
(dydus + d2dZ) = dy (((6b. (+ eb. 1) yy + bb, :, 1b: a) gy +bb)
secundum priorem fibdq (ZZ BC) - b2C, atb), :, b,  ) b., at b)22 - Bato
Seu de (22 = bb bi arb), ZZ bb): (L2- bb, veb:, at b))
s
Et rementum
Tautum Ergo opus est ut dz 1 (Z2- bbib: arb), 27- bb) seu dq sele
seu d
a
nales simpliciores. pro a, et b substituamus Y et 9, quia a et b aliter
bb
mox usurpamus et fiet summandum
Eudem modo mocetanus cum
d
aequatione valore 6.
4 (3t  - 24  992Z + 9(9. p 4)
IC riq
22- 96, 1(9:, r+4)
Nempe sit ad (yy +abbi a+b),: (a:, atbdyy tab (, a +b))
seu du (yy + ab (a:, atb), yy + ab): yy + ab, s(a: at b)
e et denique mutata a in Y, et b in q, fiet
G. (3 + .V + pp G (p: Y.1))
sendemque de
(dydyt d2 1)=
ides
pi 
P yy +, Ver:. r+4)
in Hyperbola aequilatera se primaria, compendiotor
posito Y esse latus rectur
9 trans versum, et
D Z y, esse
coordinatas ad duos
axes primarios transe
untes per cenbrum
Sin Hyperbola autem q esse affirma
in fillor prodest prodit valbr ad Asymplitum
xy= a ff 3 = ff ax et dy- de xx tivam, in Ellipsi nogativam
a suae molis
et
et dydy = f+ dxdx: X et 
dydy + dxdx = 5 4 (5+ + X  olXdx, : X
sed sit p ubi oblique CY. E, Y = CY. s Eis Y agatur normalis YX ad Cy
Sidydus + dedx) = dx (f+, X) eXX
et sit XV: XE = f. phet fiet X et ZY_. y =  f: (V+pp) seu : CC - yfs.
sed C∞ = _CX + XE = X + yff:f. Et fiat ff:, Xot yp. f = yp. l.. seu dem
fiet EAp5F+= PX + yt, yp et fit X =  fB. _ B 1H rpp - fyy): py
B
seu fiet  f (x 1 yp, f   _fYCG) - fX+py Jy se
ff (f+ pP) = FXy + pyY, ubi posita p = 0 reditur ad As  Xy. sed
hoc loco fit X = ff YB +pp) - pyy,: Fy, et fit hinc
ab dx= fxdu +fydx = = 2pyd X seu dX - frdys : (5t 2p) y compindii lausa
f+ Ep vocetur m, et fpp) rocetur n, erit X = fn- py. fy et erit
seu dx = - fn +H pyy, dy: 4m7 y
et
a
dx = - afx dy : ny.
et dxdx =g fmm 2 (pim Gs + (sp. na) y: t
unct2 2(payY + (pptmM, ina) y,: y
app.
et dx Ax tdydys  a anpttyy + porin
:m yy. Vestam)
ubi pro irrationali compendiose scribendo (y + ab yu + aC), et
ppenm
seu xAx + dy dy) 289 (V
in summando evitabuntur ircationales imaginariae, si sit qac- bb affirma
tiba 4dc = 4 fh: ppX et aabb = a nn pps:, p + mupp +m, et
+ac bb erit at pet asann 4nnpest qu Errinu). Ergo debet esse
petyn maje auc bb erit (esa napp) + 4f4,  ( 4 mpp f:, qu Cpp tinn)
Ss1 PP + S3 tqu
seu (tac - bb)  efsum: ppron   + 2ff fstpp.:, f5 + 2 pp
ateunun (t 4f nupp - 4 nupp s,: qu (pp + MM) adeoque
VAaC aabb) = +2H ran:, mn + pp serB: (s Astrp) 1sreod.
2Hs + Spp F 4Ps
et 4 ac= aabb erit
et