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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 4
# Folio 88r, page id 00068049:0177
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sbX + CC
sext s
Ergo
ex+A  bxta
A2begh.
a
O
yy= bxtcegu + exiss ih + 2
exts 
bx +c
bbxx + ybCCX + C + CCXX + 2efX + 4 + 2ghb CX + 2gh cc ex + 2gh sCff
+ bb9
sen  icekt. ) XX
=
Abccqeue
+2et
reccan
e
120t
+ e
 c2 si
 AbX
C5=0 et RCg9+ Fhh L Cgh+ fgh =0-
ergo s
99EE= -c. Et fit cgg. cht
199-
(jametin quil seniens ad dimensionem
Unea onice esst ejusdemformae
+h ceffghi sequt per qy. hit
bexX + bff2X + cc ff (sed ita
XX +, lEccgG: XX - cc
7 2c89
terninus al ter
eccs.
Ham  comparemus, cum formanqnerati ejusdem jormae, nempet
comalam
 eXX + mX + n,: px + qxr 
Et quia video g et A_ esse arbitrarias iuntiles
Sed
x sed video admistis g et h, ipsas o et e esse arbitrarias imitiles, itaque
reu sic de integro ordiamur y = seCX+C + hS
seu re  XX +2CGGX + cC gq
1)+29h1.
seu yy =
+ 2fth.
+ 2C gh
23
+ffHh
+2cf gh
XX + PX + cf
∞+ F
X+C
X+f
vy = ) +Cg) lilx+ suh , 29t
(4) Px am x tan
XX + pX +a9
X + f
(4)
A+C
(Vx + am Xtan
ex + px + aq
 Comparando jam ipsam formulam 3 cum formula generali 4 habebimus:
9+h = l et cg+ fh = an
et C+ f = p et 1f = ag. reservatan
ultimm oquatione comparatoria ipsius aem cum respondente, fit A= l-g f = p -c
et et ex aeg. 6. fit cg + lp - el = gp + c9  an et ex 8. fit Cp - cc = ag. et
et Lspp. 9 - aq + lp: 2 + an,:, 2 spp: qu aisex -
et sS 1 spp. 4 =aq - lp. 2 - an, : 2 spp. 4 - aq et f = - App. 4- ag + p. 2
Restat, 2C99 + 2fhs + 2Cgh + 2fgh = aM jam g9+ gh = 9l per 5
(20
et fil 2c99 +2Cgh = 2cgl et ex aeq. 11 et 2cg = an -lp + c1. exgo ex 7 et 20
17
fit 2c99+2Cgh =,  aln = IIp + Cll. Et eodem modo hht gh = hl. Ergo
(24)
fit 2 fhh + 2fgh = 2fhl, jam ex 6, et 20 est 2th = danf an + lp + cl. ergo
ex 27 et 24 fit 2fhh + 2fgh = a ln + IIp + cll. Ergo ex 17 per 21 et 25. fit
6)
2aln = aM seu 2In = am. Quod est perpulchrum et simplex
nisi qubtres
incam ensionem x
am seritum in
Ellipseos et Hyperbolae, ac superficiem Com scaleni, si quidem hacmethode licet
7
pro curva  Ellipseos aut Hyperbulae, summandae  ordinatae y, quarum aequatibes
y b: 4 +Bv0
posito b esse axem majorem seu latus primarum transversum, et e
(29
fiat v = X+Y fiet vV XX + eA 21X + rr, ,t 1 37. per 29 fiet y= + exx +r ee rX + eerr
(20, bt: 4 =  vr
vel y = (+ eem XX + Neera X + cerr
(30)
B: 4
= XX - 2 IBYX =, . 4
a quam aequationem comparando cum sg. 14)
(32 AXX + 2YX + rr
(22) = bb: 4
et p
fit H- + ec: bb aa faciamus  =e
(34)
(37
fiet I1 = + p
X ubi pateb ob 49+ hh + 19h = l et ob ccgg + ffhh + 2Cgh =hn. t necessario
imadinarius evitare volumus
adebere esse += =, adeoque l= saa et, et n = Yaarr = b: 4 adeoque oportet
aar esse majorem quam bb. Igitur hic quidem calculus restringitur hactenus
ad cllipsia; etsi ad Hyperbolam quoque i transferri possit interventu imaginariarum
quae nostros calculos non impediunt. Sed sufficiet nunc remuin Hincobola tentari
exaea 26 fieri debet.
E ( (4) Placet tamen calculum generaliter
prosequi. Ex aeq. 26. fieri debet ha atrr f Ba=  L2ra et sit (4aa r + b  (grraa)
erto fit b =0 quod est absurdum, itaque comparatio non procedit Et opponit
sese successui natura rerum. Utamur ergs quadam praetransformatione et
idaatnrs ad . 27. danA c9. 27. cum equencibus