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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 54v, page id 00068197:0108
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Ex eodem fundamento tentemus modum panto simpliciorem. Sit
paritur ut XX + ab in Xx + ab producat quadratum. sit x = f10 +9+D
at 550 +99D + ac  F. fiCB Xx + ac
erit Xx+ab S s1C + 90D + ab 8 + 1s9 1C)
22.5.
Ex  palu dum nit 5B02. + abf508 +
et C  acfF. 
Frff9dDD
4455996)
abst DE
acgus n
aab
ffb
09D 14f9, OD
ab
ac q
24
Sit he quadra Hujus Fractionis Numerator sit quadratum a latere n +YO
aab aabc 24
 (10Z + i1a) 1C + (20X + 21 a) D ergo cointui coincidet cum ile cum 1ozC
Ergo coincidnt Numerater iile cum (10. 10227 1. 11 a Z + B. H aa) ( + (2. 10. 20Z2F ,. 1. 2
(20. 28(Z+ 2. 20. 21 47 + 21 21aa))
sed hoc hoc productum deprimi debet, cum sit ubique uno grad altius supradicto Nu-
1n
meratore. et quidem in parte rationali hoc fieri posset, dum in 1 (18. 10 +2002)C
nam in 10. 10R + 20. 20) tolletur pars summa faciendo coefficientem ipsius _X in C esse
oppositur loefficientis ipsius X in D sed in ponte parte irrationali, terminus 2. 10. 2072
qui est summus tolli non patest. Ergo aliter proaedendum est Nempe numerator
fractionis ex productae ex valoribus ipsor Xxt ab et XX. + ac sit fiat potius
Quadratum la Latere am sO + an La posito m et n esse constantes
Ergo idem numerator coincidentiari debet cum amme +OD Cm VOD
Et quia f et g possunt poni a, nec prosunt, fiet aenn Y
 aC + aD Hb +Ce = _= mn
et a ()B + a DD + 6a C D + (b + c) ∞ 8 + (b + C) D∞ + bq  E = mmO + un D
Hinc hab ex 5g. 1. habetur valor ipsius Q, qui substituendus est in aequatione secundus
Deinde in eadem pot secunda ponatur 1= aZ et D = 10Z tc pZ + ag.
re
Harum quatuor literarum m, n, p, 9, ope Toltaturte destruantur omnes termini aequationis 2ds
sed quia una adhuc superest atera arbitraria, videamus an ejus ope tolli possit 10
val ipse terminus fBC, vrt terminus summus in loc ipso +eDE
Nempe ex positis fieret  x = asCQ +, a +DQ,: E. jam d+Og: g d2 10q. 
iam dy= M. y = e Y 2 = dy. Ergo d yl: 2 = = y = 1.2 dy = dy : 2 dy Erpo d 09 = 0de + 940, .2108
e fict x vel d V60 = (6 19 +9d0 10g:, 209 Eeodem modo habetur d1G, et ita sin
ix = a (0d8+ 916) sC  + a (D++ 2d5) 138
2DE8
2024
-
104+ 198, a d
5AF
Hac quantitas, differentialis ducendae est in radicem(xx tab, Xxtac) i id est
in am tad + an faD,: Q, et dabit saepius t0DG et deberet esse
20D dG +D8 d0 +0ZdC 20 quod daret aequa tione 2di gladus identicam, sed ad
eam praestandam non adsunt sufficientia data
Ergo fiat radix quaesitu am sot + an L SDF, cuii quadratum addumoqu + raamatus, et
aana DE
_:  ma, quo datore ipsius 5. Sed quia sit haberetur quarta,
ut  sit  in ratione constante ex Z. sed ita e haberetur per C et D. 5 valor substituae
liet t-a + aD t (b + C CC
in Atera aequalit  a D +aDD, +6a OD + (b+ C)CD + +) C8 4 (b+C) 1 9+ b.98- minus s. C
pru est
ixpitionala 
4G
et habebitur relatio aequatio in qua C et D servant leges homogeneorum inter se, sed talis aequatio non
inservit qui it  et D haberen relationem constantem imer se, contrantentum
ca
Itaque redeundum video ad modum quem prososui sub (m
paginae praecedentis, ecubi merae adhibentur D)
(  1DG et ita dx postea nihil turbare potest, et puto posse poni t =0
quod etian contrahit calculum et conseructionem