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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 127v, page id 00068199:0254
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=254
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83
iudo
 3
2 6:
s
i e
 
et e
s
C et c
1 
t 5 1 eo
 ut = D 
EX+
3   F  
= E 5.F.
= id 10
32 2.2t
EE
8∞ 
= 5
5 112d
570 
E Ese
5- 1
+ 350
F5 
ux =. B)
∞L. C
2 a2
E 2
E u
+  cL=
1 = = E 
XbE.
b0: 60a = 10. 3u_ S gota - a D0. b1 a = l10. 31 F 6)
+
5+ Y
+
e e 
2.=2
 c
12 3
E En X
c 8
5 1
o + las
8
53  F d
= 1
5 83
a i 3
∞ das
8+8
15)C
85.3
8 d) 5p = X ∞ F. = L
s
05=
 8 29a 
E +t 2 3e
Se
8 83F e. = 280
 Ne Ex his ratiu . g0: 10r30 = G a. 2 = ir: 31. quid si etiam foret - 0
t - 8
00
E
(0 d1: 8
Lc 
x his ratio 00. 40 :, 1030 = 00. 4+,:, 11. 21 quod si et sed et  Ct 00. 41G:, 10. 21 + u. 20. itaque daturtatio
etrum inter
uiicic onstans intest   h ba a
G8, adeoque X fiet constans contra intentum
et a O3(Z a at 5 = a (24. Est autem (G - 20. 2OXX + 2. 20. 2 + 2t Uaa
O eatr  + lu 2G- 40nX +a lax a q ad et 56 - 6ona + biax + 6r aa. Undere 
interpretando fut 20. 20 XX +2. 20. 118 +2 a   60apaat 00. 6 ax + 00 62 a p  ibe a 2oaa
alteram tantemus rationem, fintus +el 
sed ex aquternntundo, ftat n
+  e
C2= 1 = ∞ =  E +d
3F5 :
sit  00 60 = 10. rop. + 00. qul,: a Eprins) 20 2o aa -00. 4olp,: ap
ssepctia ab habintdinischen amt
t. 2 c
et e
2
10 1XX 110 raper usi an 0o. soatr. t ae biaax + 00. bra
ao tinsre ommune illo
 aaitu
5 ute
et 
 et
8 det
 5. dd ∞ X
d 
5
et 5
i
5:
3
ia  Diet pro 20 20po, 2.20. X
roctilxi
nC 40 4 a2 p trrcngum e a B sto  l. 2ut. 142  = h.  A   Sia  26 p 1 a 1. n 10 :
absi bbra M2F Et 00- 52a = 11 31 aP +004 21
Et similem caceum indituendo pax m aqu. quae drato, simibs habe latur valares iprorum 40, 4l. 43. F Intum aua l, p, ponendo M, 4, et
siet 40 = H31 a4 20a4 - 10. 30197 : 29100 et 4 = 214a - 11. 3199, : 29 m 00 et 4_ = 2. 20. 21 aa = (10. 31 + H. 30) 17, : 2 moo Et binos valores
ipsorum 40, 42, 41, aequambinter se fiet 2020 aa lp = 10. 3049 p = 20. 20 a a 9 - 1030 ppogn  et tote 10.30: 20. 20 =  jaa t p. - qni : B 4 pg (pn - 91) = 11. 31 : 21. 21
ex 19. 31 + H. 30: 2. 20. 21 sed ita productum ex D et D habet constantem ationem ad 30. Ergo et inter se, et , ut O ad 9, et D ad Q, et 50 ad 20; etc. adeoque
et 218 ad GE, et ila fieret  constans 
contentum
gamus diversos modos et Hecusa calerer probum essee, vel ipsa prosut, etsi deside taca contrntia reductia ne lastat
Quodsi conjungamus diversos modos, _ et quoad l, aequemus Beq uni producenti, ut DQ alteri; sed quond m aequemus 94 uni pro ducenti, et OD alteri
ert 40 = 20 (10 a a - )pp): 200 pl, et rursus 40 = 20 aa - 10. 3099,: 2gn 2. 00 9a. Et fiet 10. 2oaagm - 30. 20 pp qui  20 20ca pl- 10. 3099p ita
videtur rumpi incantamentum  reductionis ad constantem, nisi restituatur ex necessatate trium aequationum auxiliat ideamus Eritpa jure
et  42 = XC1a a - 3Ipp): 200pl = 21. 2104 - 11. 3199, :2. 00g. Unde 11. 21 aaqm -31. 21 pp gm = 21. 21 a a fl - 11. 3199pl. sandem d pari 
rursus jare, ob  4 - (10. 21 + 11. 20)aa - (30. 21 +31. 20)pp, : 2. 00 pl = 2. 20. 21 aa _ (0. 31 F 11. 3. p, : 2. 00 qm. unde n0. Et cagm - 30. 21 ppqu  2. 20. 21 a apl. 103 f l
Ex aegu Cexnovissime prima est 30= 20(20 aa pl - roaagud: ( 1049 pl - (20. pp qu) Et ex secunda, 31  2HL 1 49 21 (21 aap l - 11 a a 9a) : (149p l - 2ipp auo)
qui valores substituendi sunt in aequ. 3. _d qua Hto et 1 H. 20) aaqu 2 con ccapt vocemus 0000r, et ex aeq. 3 fieit  5opu (el t n t H p (X 2o2
a= Unde fit 20p9  (20opl ionnqn nL CH(1gl - Z1pn )  21 (21 aapl naa qu) qu (10gl rapn) = 00 aY (nau pl -  aagn, 109l - 2 0pn)
ivaq
aapus
Si aequ. 3 dividatur per 20. 21. reducetur ad 10720, 11: 21, 30-20, 31: 2 ide ope aeg 5 et 6 tollantum 30: 20,   spe ipsanm 10: 20 est 31. 21 ope ipsius 11: 2
et prodisit  aequatio ultima  in qua solae erunt binae quantitates 10: 20 et 11: 21. Hau Harum loco seribamas pro 10: 20 seribagnis 59: 00, et pro H1: 21, scrilamus 51: 00. +taqu dividam
et perpa. ita CX aeq. 3 fit 5  (5B +51) aagu (0: a0 +91 2) pp an = 2 aapt (50 e) 50 +5) (aa  - (20 + 31) pm-  2aa l - ( 31 + 51. 3.) 9  Hic lex est
unimmnrinturis iata
avoto
_ H   permutentur cum 3i B et pra,  itemque1 et aequi forme ex 1. et 3. permutentur cum Leq et 4b. r Ex aeq. fit 3n  a a L Z aam:, 5 gl - im perpe d
16 aat st aa :, s1 gt - pnc. compondi causa sit  F  q Ausu - g e pn = ah. Et in aequ. 6 subsituendo valores ipsorum 0 et 31
ex aeqp. 7 t 8 fiet 50 +419 -
2eaf - at
+ ne- 
Sias- a
J_) ah = 2aC- (. ae - 50 a9. 5 + ac l taqe 50) af Hie 50 e3 servunt inter se legem Homngeneoru
5o af =A
EbF ah
nceus ens partes id quibus plures sunt ductus sinctia Dubit (+ 60. 51 Cf 50. 51 f9 P00. 00R
itaque si divellamus aequationem ( 1 in d
litora lextiolateris pende
-60C-509, 51f -0olisu ( 10, 2 (50f - 0oh, 51f. - 0oh) 00  = (00. 51- 50. 51, 51 f- ot.
-(00C- 519, 50f -0uh)
Requatio 10, compendio 52ga +h = D  ) 00. 50 ef _50. 51fg 40
20052lf 42. 50. 51f
+00 sigh 12. 00. 00 e
indu
ab
00. 50. 51 gh(= fu sv. 51. 5259 +00. 5+ s1 Pf F 2, 00. 50. 19h = 00. 01 529h = 5051. 32f9 +00. 52. 52 ef + 2. 00. su. s1 (t 9 = v0. 00. s2 gh = 7
sed Terit Sacefg +00. si t et
- 50.5051 fg +00.5050ef-00,00 50 eh
Et fit tandem 50.51
00. 51 91
00. 51. 50gh.
f-99n +2C, : g, 9h - e
itaque tam Z quam 7, non alia habent ingredientia quam SD, S1, et 50+51 = 52
+800. 5051 ef = 2, 0050. 51 ef
Th = f, ex priore sint (50+51)g00e = 529, ex posteriore  2.00  ph( - 201 50. 51f
(20029) 2.00000C
200e  uie
Erog Euoegus
vabemus ergo absoluti tam valorem ipsius 50 + 51, equam ipsius Si. Unde problema est planem. Si sit  ab et 50+51 = C, fit S1 = ab: 50 et 50+ 31 = C dat 5050 + ab = G0C
t 50 = =C _C+ L H(CC - 4ab) unde si Et 5t = EC + L d (cc - 4 ab). st itiaqe si s0 et si desideremus rationalles, opus erit ut sit cc + 4 ab est autem sie
C a erg e ec= (D eB et (quod dedet aequari ipsi 4ab) - D, eth ggh trees,:(qgh es,  Et fit egh _ees et n os ggh erees, seu
3ef +est egh + gyA jam e= al q f = gl ia g anep h - pm: a. Ergo ef Il et gh - na et fiet H(3e+ h) = mn (etq) seu C: = gl9M tll, :, 3l- num
est autem p=ata, ergo erit 4 atChata m: (gmm hll, - +mppll. a L) (ea m pel ap). Ex seu q= aa p, 3H- mm,:, aaM- pllm
sed quia ponenda ce 4ab -0, fieret  10. 20 =  Non licet autem destrutre irrationalitatem faciend cc = gab, nam fieret 50 =C-5
= 11: 21 quod nolim, ob caetem quae inde consequentur. Itaque sec videndum an (CC= 4 ab possit fien quantitas rationalis ope. ipsarum p et 4 quae ad huc sunt arbitrariae.
eut aeerque- 4e. efh -gyh +ree Fd = AH quadpto, et 1 - eth- g9t+ 2 ees = H.H. Est autem e- at f- a g-am h - jm Ent eft ef = U gh = mm
oh- ef
Et fiet emm et = hll + gtum +(2 eH: emm= ell = KK. aa seu aa emm- aahll = emmkk - ellKK, et fit
oh- cet
e ga, gnim -hll, :, mon (KK = aa) =H (KK + a a) et explicatis e, g, h fint  d: q = (aamm-PBl  ap) tum (Fr   -UCKKtaa) Unde tandem fit
g= a, mntetea nm (KK - aa)= U(Fir. +ad PCaamm ppIC). quae est quantitas rationalis ubi p et R sumi possunt pro arbitrio. Sed quia
homonim ieterins non ntin silire alato, it poinde, do ver norentum na ua reoiannus lait, ot numeris tandem comprotudus
vertatur
s 
∞C
1 = +E
s
E5.
2 +11
PE0
l.
e
didis
es=