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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 253r, page id 00068196:0501
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X + m + m X + n = pX + qX +r
X F 12m X + 2 n X + mmx + 2MnX +nn 1 =
- p X = - 2pgX = 2prX
= 9q X = 2gr X = YY
X x bx4 + cX + dX + ex + f
fit AC m= 2b - L pp n = LC + p9
Yr(E nn-5) = q CC + 209 + 4 pp 49 - f
94 - L bb ab pp + 1 p 2p s ce + repus a +ppgu - f - d
rursus rr = _animun- amne + ee,: qq = mmun-ane teC, :nim -2p
seu er =. bb - 2bpp + pt _2, cc + EC P9 + 4 ppgqu = b. pp C + apqu e + ee,: qq =  cet repqi ospus
hX + lX + mX + n = pX + qX + rX + s
hh X + 2hlX + 2hmX + 2h nX
+ U X + 2lmX +2lnX
= ppX + 2pX + 2prX + ap SX
+ mnXX + LIMX X + XN
+X t vX + WX +y X + ZX + CX +9
pp = 1 + hh p = hl ergo 1 + hh = ht HI
2hlr + gq = b + 2hm + H
2hl5 + 2gr = c + 2hn + 2Im
295 + rr = d + 2ln +mm
225 = C + 2mn,
Ss = f + nn
=
+ 9q X+ + 29r X + 295X
+ rrX +2r5X + 5
1 X + vx + bx + CX + d X + C X +5 _
t
+M + y. +Z. + β +G
h = 49 - b - 1, : Em - 2 1r = c + 2lm - 2gr, : 2l5 = 2n
G 49 - b - 11 2l5 = In a =  2m - 21r. C+ 2lm - 297
seu n = q9 - b - l 2l5 = 2m - 2Er C + 2m - 2gr: 299. b- d
rurus n = 295 +rr - mm - 3 : 3l
et n =, 2Y5 - e: 2m
aequando duosposteriores ipsuus R valores fit
Am 295 + rr - mm - d = l Ers - C 1
et aequando priorem posteriori, fit qq - b - H 2lsm = m Em - EIr C+ elm - 29r = 49 - b  EYs- e
harum duarum aequationum ae fit 5 = D. m + dm = e l = mrr,:  29m -2Y
et rursus s = n. Em - 2r E + ulm - 2gr = e qg - b - H :  9 - b - H. 2l m - 2Y
qui duo valores ipsius saequandi sunt inter se tet p  =  e amm + 4r
rursus hh = 1 : H - 1 = q9 - b - u: 2m - 2lr, quae est ceq. f
Rursus nn = 5s -f = 4rrss- 4rse f ee: 4mm, habemus ergo novum valorem
ipsius, S quem aequando uni ex prioribus nova prodibit aequatio e
Nns aequationes D, 8, et resolvendo eva prodibit equatio calculus opinor satis ulte
(pergamus in exemplo proxipur aq. 5. erit ay L ax +XX. Ejus differentialis esst- adu  = adx+ 2x ax et fit.
assurget
 dy: ax a+EX, : - a aequando aeq. 4 sit aa + 3aX+2XX- da, verum hi valor non consentit priori
jam invento, nec hos ducit ad inveniendam e; jam supra  ex 5 et 2 fiet  ( 2X: a + X: aa= e-X
quos duos valores aeq porio XX +Z ax  Zaa= _S aa seu X =   15 : 2 = 3:2 quo valore
substituto in aeq. 4, habetitur et e e = XS X: aa + 2X: a + E s lgcunde aretum ad
pariter 8 ceptrum circuli, qui Logarithanicam osculatur, simulus punctum osculationis
videamus. Mothodi bonitatem. probemus in casu jam noto, nempe cum parabolam osculatur circulus
sit x sit  ra  yu et qyyl  yy- caxt XX  ft 2aX FXX = va
et sit H) centum circulit et HB ∞ (et BC et ABLX et AH = b, et AB = b - X
et latus rectur paribila 24) Hit at caX = gq  bbL ∞X
fit 2ab- raxFyy et py + XX=  exprore eg fit adeL yay
utamur solis 2 et 3 ex sit 29dy x ax seu ay du x X. y ergo
ex 3 et  fit x9  d. 
en xa
rabu 2ax Bt rabo ExI XX = yy
fit in parbola a
X
et pacg  Ex Isii quex bad eyaus
seu atx -q. r:: a. y seu ay+- 9y
 es at ae  a
et ex 2 fil dy: txL= x 9. et
r. BP, a D: ga PZ: ZR:. 4: Y
et, adque +. Xdy +y dx - aay = 0 et ly: x. 4: 9:: y: g- a -
n Re AC. AQ, q
ZR
A a= a g = 3F su P. = 4u a, 3 fit
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