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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 84v, page id 00068198:0168
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ab +  + ed multiplicetur per g et fieto agb +oy+ gedus comparandacum
 agb + admy + aumdus Ergo; m = ge: a et am = gc; a = gde + edque l. a.  seu 9c = 9det ea
et dg: g = c: e - de: e seu l0g ge = S(c: C) vel logg = fC. C) - log e vel  splie-ge
Heui 9= R. d: e et summabile est agb + gcy+ gedy, ejusque summa est
paramne javeer cuius
a(gb + amy = a sgb + gey  = a (gb +  et
Quod si ergo esset 0 = ab + cy + edy, fiet o- a s9b + C.   + tiseut y. Edhass
ei
si aequatio sit aab tacy taedy + fyy + K ydu F0 comparanda erit haec aequatio
Cum ista o- a ab + acy + ae dus + a a31gy + 2a 31ydy. posito Q qualem invenimus. ita enim
multiplicando aequationem per g. cum qualem inbenimus per cete, summabilis erit tota
iet pro comparatione ergo
residua fiet 31 =, dK: 2a, et d3. = gdK + X aq.  = 9 f(: a et fit do: g = dK: K Ff: K
r
aequatio et fiet 0 = a (gb + RS  + _ 31 ajy
acedqe sed supra jam habeatur dg:g. Ergo prodit  deferminatio aequationis datae, qualis
seu qualificatio coeffici se quomodo nimirum se coefficientes invicem habere debeant,
idem est pro alia quaecunque  ejusdem
aequatione cui eadem applicari possent, ut: 4 b + cy + edue + f y tX yy dy = 0.
seu fit c: C -2f: R = de: e - dR: B
tantum pro 2 ponendo 3 2 f: K ponendo 3f. R. Et in universum si aequatio constet
ex duabus partibus, quarum net utraque fiat summabilis per multiplicationem
Ut si esset b + cyy + eidy + fy + R pydy =0 fieret 2c: C - 3f: R - de: C - d H: K
fieri etiam
fieri etiam potest, ut tres partes fiant summabiles per multiplicationem
potest. Ep. gr. si sit b + cy + edy + f yy + K ydy + 1y + lysydy =0, tunc majore
opus est  determinatione, et fit c: e - de: e = 2f: R = dK: E = 31: l = dl: l = d9: 9
potest eliam aliter obtineri  multiplicatio summativa. Ut si sit brcytetus + syudus
eriique g multiplicator
ab rcu + edus edsyy + Efydus o= aabm t aemus taemdu d L ad fyystrasys e
+cnGy + en ydu
debet Ven yy + enydy ~ adoc gy + au ydy seu 11 = en: a et du - edn + nde; a = Cn: a
+abny
ac my + aemdus F aa d21yF aa 21 dy. et fiet
Hod debes
am+n
21 = em: a et d21 = edm + mde, a = cm + bn,: a, seu eam + mde = bn
c
ab n u
fiet dn:n = c: e = de: e superest
et habetur in ex inventa n. Et b, c, e, f manent arbitiariae
sit aequatio data 12. f 1 + 21 dy + 30gu +40 ydy- conferenda aequationi 2t
e
50 = sen - 2af: 40 = cn = adf, : 30 seu 30en - 2a f_ = 40 cn - 40 a df et n = 2. 30a f - 4 adf. 30C. X
50 = 2C. 30af- (de ad f (= 2. 30eaD + 2 (40  af, ;, B0 e - 4bc 40) seu 50 = ) a, 2cf - edf, :, 30e - 40c
jam 9B piorro 50 = Ron + Abn, : 11
pursus 50 = Rem : 21. Ergo fit m = a 214, 2cf - edf,:; (30c-40c) e
= _em: 21 et fit 2Cm + 21 Nbn = ugem seu m =  21bn:, 11C-2iC adeoque n
m - 24b, 2. 30f - 40df, : (1e - 21C, 308- 40C) = 214, 2 Cf = ed f, 2, (30 - 40c) e quod duos valores aequando fit
superest  bin (= 12.50) = 12 em: 21 seu b = 12e: 21
cui ae quationi accedit unde prob ponendo valorem in praecedente fie
b, 2. 30f - 40df = 2C f - edf, 11e - 21C
DeC  (12: 21), 2. 30f-40d5 = 2c f - edf, 11C = 21C
ipsint b et ipsios m, et ita res redibit ad tres aequationes pro f, e, C quas
m= 214, 2Cf - edf, :, (30c - 40c, e)
et edm +m de =bn
Teulamentum generale hoc est, quod pro omni aequatione diffelansent ali
divellere formulam duorum terminorum quidiffe qui differentia unius intelligi
possunt, in duas partes summabiles v. g.  my +. ndu divelli debet in has duas
praestare difficiluus fuerit.
Generatior
meua inservire
 _ ny dsn: y)-  du: y -nde
partes dds y + fdy et dgy+ gdus fietus df + du = m
divellatur in agygdus + ba pt quu
ipsius cyt edue
tenlaturque. hoc esset nimium e Republica petere
b+ y+ edus summatali, seu multiplicatione
my+na
Cydn-ndy, gy sit ergo aequation byy +acy + aedis
seu y-ndys: gus
per 9: 7y, fiet b9 + (a9G tagedys) (.9y) =0 et erit agct agedus,: gu c
Omnia quae hactenus deduxim
sed d(n: y
summabiles
ut fiat
o quae multiplicetu
uidentisi dadnus Mantys. yy
et dn = gdec dg,: a = gc: a unde -dg: g = +c: C + de: e
y5b9 + aan = 0 seu ysbgH- age =0
quod utique non est
sit aequatio m mogtaan + ab do fiat v - py: a. 79 unde hinc
fit ge= an et fit ni-ge: a
prodibit in ppyy aa taan + py et fiet m P p yy mepp: aa) yy + 2nsp: a)9y + nq
seu
a ap y +abd
+ aar
1+a4 red procedit. sit ergo v-
et filt (b9 taan: y
Contemnendum
ed hic nihil agimius nam opus est aequatione quae dat
+abdy
∞taq,: yt Y et fiat
mpp yy + Empagut mangqu +  - aa bgdy imppys F impag + maaque (+ pydy + rp dy = 0
ad nec 
t tollatur terminus simplex. Iliter ergo sit
gepurrt obrdq abg er fuciendo.d aan yy traanry t aanri
yy dp
+aan yy fia
ab  P
efariusus
Er. qq) +  b d(: q
rediuns pareta
positae difficultat
pydy - agdy
+ adg
rapu
ueq potelo pu3
aeqr poni 9 constans, quoties n: m es
n
+ abdo
+ ra dq
fieret q lanme uc ergo putussesset
ut
duo
alsi ebo na voli thata  abere
n = 2, 30a f- 4 ad), : 30c - 40c, dn: n = c: C - de: e
ande aequandi duo qualores
et f+g = n, sed hoc frustia