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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 321r, page id 00068196:0637
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 V vib, vrC = ito
v = i Hm L 123 Imn
_V = +EUlum + BL 14. 1123. 123Im  An q Hac quantas
iss unumta 4. ter ns rem.
+a vo = 1e af lm
+ 2. 112. 13af M B SIn
 + rapus An
siet sit aq m fiet p = Et 3. 12. 13lmm +2. 112. 125 mM. + 112. 123 afm
+ 1122 125 aesfmm + 
aequ. 12 gulmm + 6. 12. 125 4glmnn + 12 afg glim + ae994
e u  ore
112L mm + 6. 12 15 nm +ag
opsitet jum ut 9 possit haberi rationaliter
etamsi non integre
compendio sit e q9 + a5. =0
fiet q- argu r aarr-darrs
seu qq = ar + aarr -5),:2
Ergo Ex his patet si l, m, n sint integri simplices, etiam q fore talem, nam quia ipt valoris 
1a mar at
iis gu in integris, inventi non alius esse potest dirisor quam , qui non est ratiopelis. In, qui non est
otitit
dum per in potest dividi numerator
Hinc  i a pro aequ.  et  effi-
cienda tantum oporeet, assumer l, m, n, g per X ratioraliter integre simple
=3X + 3a q= 40X + g1 a ubi sunt et habemus hincliteras
adeoque oporeet q fieri rationalem integrum
sine lileris artiligrias 8. sed aequato  hocrndo explicata ascendet ad gradum X adeoque opus est
si imalimus p quoque  esse integram, cumque assumere pro duplice, manente a simplice, potnissemus
aequationibus destrnctitiis Z, ut aequatio fiat identica
in  ob irrationales ascendentur ad
dX et opus esset aequationibus
rationalbius comparatitiis 5 in summa novem sed litera praeter 10 supra dictas accederea
identitasfacere duas aequationes, quibus termin omparantur et
tres ex valore ipsius p pens nempe soxx +st ax +S2 aa = ap itaque videtur salis affore
X, opusque esset  comparatitiis aegqu. 4 ob ratiorales ascenderetur
literarum arbitratiarum.
pah tak sin extr sit extrahibilis
B = ak : 2Y et aXK+ 47 =4 In = 4 aehV
btih - aXL seu 2YY =aussaa hih . A Kla) + a t
Aliter videamus quibus casibus formula 
ssit 2, a (3 + V sin L taa ββ + 2 a β) An fit
In
+e diu
strisamus conpentio u apret ag W2t H 50. 240 X = 40 s0 X + 40. si a XX + 40.5
compericus sintums
noo redeamus ad cum modum qui rem red
_n ln = WUX 10. 20XX + 10.  a
aax + 41 s2 aa
Lnrjaa la mn = 20 30 XX + 20. 31 ax + E1. 31 aa
rntionales integer
sed valar debet es
po. 125 Um = 11E qul1 min l. 1  112. 123 mm = 99. seu 9- + 2. 112. 123M
 -4. vi. 135 m  pt a 4. 12 123 a fmn 4 115. 123 auffrum 4 123 ggmmnn + 123 af. g9mn + agqu
t destruatiu ubi U, et habebitur
Senquin q - 7m. fini
4.112 25 mEp + 4. 1127, 123 af m + gt - 6. 112. 125 49 mnn + 112 afgum
=4. 12. 1236mmn - 4. 112 12 a fmmn -112. 125 aa f + 125. 7 mmun + 123 7agma. +. 7a.g
4. 112 125 mnn F 4. 11E. 123 uf  F mim - 6. 112. 125. 7 mun ( 112. Zaf m
mmn mnun mun mn nd
4. 115 923 afmC 112. 11X 193 afm connescit
et 2. 12 123 mun - 6. 12. 125. 4. 112. 123 my  = - 20. 2. n a mmn
et ( 4. 12. 125 tuml - 512. 125 mmul= 123 mm n + 125 as mmon-n antuzasmn+ ag. t. aass
sit ergo jam m = 10n + Ua et l = 20n + 2la
cd hoc praestari non sb in rationalibus interus simplis, quia mmnt ninus assurgit
Ergo redeundo ad aeq.  explicemus dia per m
a = 20m + 21 a et fiil + 4. 12 125. 112+ I M + 4 12. 15
huut sepem aequationes ut destrnatur
Me, M mt n m n, m
liferae 10711; 20, 21; 30,31;
ubis  ex ipsis, 6, an atolli poles
Itaq rspes ad huc est successe.
la. = 6. 112. 123 In qq
-6. 12. 123 Igen mm
=112t uumno
e extrahibee
aili durit gaeis dclet al
datur m, quaeruntur b et c. fit at = bbb + cem et g= 2bC. Vanur a, so
vilitur ast semlit prdit ab
 vel potius in integris
n es aar qua ubi cem ducto facunt am
af abb+cem, diodo per n, socisque donec residuum quotiens sit quad alus et
quid si hoc loco, ubi omnia debent esse rationalia procedatur, more Vietao ad imutatiorem numerosae ptotestatum
solution
residuus etiam quadratus quorum latera invicem
at +029)  B4. 1. n Pn ex ita non denominato
ex frahendo nocet
 m  umtum au )u ga biltinibi und u viaetu.  quod ede
dacere suetimman rere dili tntonatum e tau cotiet
vider jam, si qdebet esse rationalis
atiam aque deberese
b q ab+ csum = acbb m +2abc Jam = as taagtam
lietaeo extrahamus. Ha
luu pomudsa et bistars. sid quae pisto altus asingit calido
nn  intigre simple