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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 96r, page id 00068198:0191
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C
(3
Xdy = plX = 0. Ergo et ydy - ydx, :  XX = 0 Ergo y: X = a
Eadem conclusio
sic obtinetur. d Ex aeq. 1. fiet - dy: y = dx: x =0 Ergo -log. X + log. y = loq. a
Ergo leg. X= log a + log. y. Ergo log. H = log aX Ergo y = ax.
a xdx = b ydy + cxt + eydx =0 Ergo 0 = + a XX + L byy + s c xdu + eydx superest
ergo tantum ut conerum, ita explicare X et y ut summari possit quantilas si. sub virculo
quem in finem fiat yZ et in X + nV et cy= max + ndv et exdy + eyd
 Ex mCxdx nCX.  + OmX me xdx + nevdx ubi deberet fieri _nc= ne sed sence
non est in potestate, cum c et e sint data. itaque fiat  y= mZ + nB, et dy maZ+ nan
et X = fZ+ gV et dx in f 4Z 8 dv et exdys + eydx fiet ( por, 3, 4, 5, 6)  5mzdq + faz dv + gmvaque
+ +r Bdo inque fiat X = 11 Z + 12V dx = 11 dZ + 12 dv y = 21Z + 22V et dy = 29 dz+ 22 dv
fiet Cx dys 4 ey dx (per 3. 4. 5. 6.) = c, 11. 21Z _ 4 122. rdv + 11 22 Z dv + 12. 21 vdZ  + C, 11. 21 ZdZ + 11 22 BA Z + 2, 21 ZdB. + 12. reb ap
ubi _ in qua aequalione deb et esse c. 11. 22 + C. 12. 21 = c. 12.21 + C. 11. 22 seu ( 1. 22 - 12. 21 C = 11. 22-12. 21 C
quod ut succedat, debet fieri ae 11. 2c. = 12. 21. Unde majoris facilitatis causa
suffecerit ponere 1  22.   V - 12 = 21 - 22  1 et ex oj. 7. foi Cx dy + eydx
CZdqu + c x Av. + 5 Zdv +  vize
do + e ∞dZ
abstitito inaq 2. siet 0- a Xx a + bgy tpec E () Z Ubi si ponerem
eequ.d
eZiq + C vio
1 =- 1L = 21 =22 fieret X = y, quod est absardum. Itaque ponamus -1 et 21 =1
et 11 = a: b et 22 = b: a fict x=  v et y  Z + Cdpobia et ex ∞y 7 fies
exdy +egdx +c Zd2 + fiet X = Za: b +. v et y = Z + vb: a. et ex ag. 7. fie L
Cxdy + ey Ax= Ca. b, Z4 + C bia, var + CZdv + C vdque. Et sendy teydu = Fere (a o Z2 + b. a vv + ZV
eoque valore substitato in aeq. 2. fiet: d =  a KX = Lb yy L CFC a. 26  = b. a WV + Z  unde tillendta
literas Z et v, ope aequationum 12 et B, ut redeant sola literae X et y
EE Juvaniamus valores Z et V. nempe ex 12 et 13 in inaeg. 12 substituendo pro Z ejus
sed hini sit
valrrem n aeg. 13. fiet n = IX - 9a: b + v sed hrc est absurdum, sit enim X = ya: b
contra hypothes in sed et similiter si quaeramus Z fiet .  y - b. aX + Z vurtus ut cg aeg. 18
quam aquationem substitueulo in Ex oy. 18 fit  Hy eD d dx dyab. quam substitu endo
ui  6quationes  18 et 20 in ag. 2. fiet aaydy: bb. + by dy + cydy a: b + e ydys a: b0
quae aequatio non procedit, ergo non licet assumere squ., 11. B ed cauus ergo ad aegue. 10.
et fius ope quo quia 22= 12. 21:91 quo substatuamus hunc valorem in 59. 5. et siet
y = 219 +  12. 212I V. mancule aeq. 3 ut ante. Ergoope 5y. 38 et 12 quaerendo v et Z
e
s ap
d quanta valeres 9 at 4 fit v = x: H = y: 2) ,: 12. 1 1E. 21 : 1 22 et pari jure
siet  qu Xs y: H = X. 21 jam ope ag. 3 et 5. quarendo v et Z, per
Unde aquando valcres
19)
= X: 11 = p. 12: 1 et 2∞ 5y. 5. fiet Z = y: 21 = v. 22.21.
epoje 
ex 0y. 13. et , siet X: 11 = y: 21 , : 12: 11 - 22: 2 = v seu v = 2IX - 11y, : 12. 21  - 1. 22
et similiter et hunc datorem substituendo in Z Fil  arg. 13. fit Z = X: 14  12. 21: HX + 12G, : 12. e1 - 11. 22
11. 13. 21
seu 2 (X. 12. 21.1 = X. 22(- 12. 21: 11D + 12, 9: 12. 21 - 11. 22 quod consentib.
ubi ex aeg. 11 substitutudo 12. 21: 11 pro 22 fier 7 - X. 12. 21 + 9. 12. 11: 11. 12. 21 - 1 a2
quod est absardum, nam Z fieret infinita adeoque ipsa aequalio, 10. non est permissa
Quad nocumento imposterm
ac quunde nihil lacramur per aequationes 3 et5
esse devel ue explicatinibus fidamus, nisi  ipsas literas substitutas, pursus possums
invenire per inito  hab nisi post cactam ciparationem literas substitutas rursus
possmus invenire per initio positas. Hic autem succedere rem non posse vel certe substitutione
u
sumus modi puc juvane vel initio maevideri porerat, quias inventa aequatione qualis essati b
parsus interpretendo ipsas Z et v saer X et g. prodiret aequatio per solam a et y
quae si quam tamen suppuosuipus quae situm non praestare.
quid ergo si facesenus XE aBZ +I2 V+ 10∞ + ierV + 112DZ et y = 211Z+ 20I V+ 223 =  L 2127
ubi sed ita u hos valores substiedo cam differentialibu substituendo inaeq. 1 vel 2
quanitate exdus + eydx Assurgeretur ad gradum  ZZa in quo gradum
sunt formae Z2dy, vedque AD dy, _ 2Vde et adest praeerea Zav, et vdque nempe omna gradi
numem termino qu haus eq hic accom accom modandoum auatur numr summt numeri terminoru 
 est 3+2 bis, id est 10 sunt ergo termini 1e accommodandi
arbtrariae autem sunt etiam 10, et praeterea in valoribus X et Z. praeterea assumemus
gradu, et ge, Z3 et et gradus 2
Z  X o21ZZV + 012ZWp + ∞dB74 2S 020  012V. et hujus quaeramu dasserer
formula dita ex cxdus abeydx substitu
me tantum 10
to  r  eseum 3  piber