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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 29r, page id 00068197:0057
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OD in OE in 3( = 00DDG Esto ax = O D, aX = bb = 09 et aX - cc = DG
erit productum ex his tribus ax, ax- bb, ax- cc aequale quadrato.
Quia ax = OD erit OD - bb = O∞, seu _ = 1B - bb,: O  Ergo O0D + bbD = ODD+c
sit O =
et 0D- cC = DG seu D =OD -CC3: D vel + 6OD - CDDa +bbD
207 + 12a
et 2
= 300 +40Z
a
602+56a
602+sba
Aoet +(3O D = 20. 20ZZ + 2. 20, 12 aZ + 12. 12aa, 30v + 40Z, +aa
a
Fiet o= -(9DD = 40. 4072 + 2. 40. 30 Zv  40. tovB, 12a + 20Z,=aa
+ bb D = 60. 6027 +2. 60.56aZ +5656aa, 30v + 40Z, + bl
-cc0 = 60.6022 +2, 60.56 a7, + 56.56aa, 124 +20Z7 = cc
in hac equatione destruamus terminos Z3 VZZ et aZZ, ut solum restet Z
fundamentalis
Habebimus tres aequationes destructitias
O= 20. 20. 40 aa = 20. 40.40 aa + 60. 60.40 bb - 60. 60. 20Cc
(1
0= 20. 20. 30aa - 320. 40. 30 aa + 60,60. 30 bb
2
02) 220. 40. 12 aa - 40. 40. 12a a + 2. 40. 56. 60 bb - 12. 60 60cC. 2, 20. 5bboe, aZZ
Ex aequ. 1. fiet 60.60 (40bb -20CC) = 20. 40 aa (40 -20)
60. 60bb = 20 aa (2. 40 -20)
2.56.60 (40bb -20cC) = 12, 60 60cC +40. 40 4a - 2. 20.40aa
2
Aequando binos valores ipsins 60. 60 ex agq. 1 et 2 firt 40 bb (40- 29 = (2. 40-20)b0bb- 20cc)
3
seu (40. 40bb) (20.40bb) = (2340. 40bb +20. 20cC- 2. 40. 901c (20. 40b6)
unde siet 40. 40 - 2. 40 20 eC + 20. 20Ct = 20. 20c- 20. 20 bbcc
62
tandemque 40:20 = cc  + C Y (cC - bb),: bb. compendi,
causa  HCC- bb) vocetur f, et fiet 40, 20 = c (C+): bb
Et licebit facere 40 = c (c+Y) et 20 = bb. Et hos valores
substituendo in aequ. 2, fiet 60. 60 = aa (2cc +  2Cf = bb) vel
60. 60 = aa, cc _ + ff + 2cf adeoque tandem 60 = + a (c + f)
Superest aequatio tertia, in qua substituedi sunt valores numerorus
Inventorum 20, 40, 60, supersuntque quaerendi 12 et 56. itaque ex eq. 3.
fiee (onsidero autem quia 60 et 40, habent communem divisorem C+ f, posse
inter explicandum aequationem 3, simul fieri hanc depressionem, et fiet
12. 56a bbc ( + 2. 56abCf ((2. 56 abb C0) = 12, aa C t aacf t ec aac +ccaaf. raalbc
vel + scab 56 bb f = 12 a   (a Cc - bb) =) aufs + a ef, et
divisione facta per f, fiet tandem +56 bb = 12a (f+C). Unde fieri
Quod si faceremus
potest 12 = =bb et 56 = a (f + c) = 60.
F esse + fieret ut 56 =60, ita 12 = 20, quod non licet nam fieret
d: a =  itaque debet esse = idem quod - et fiet 12 = - bb. Sed
Hlabemas ergo valres intentos, prore nata setise simplices
20 dbc 122 - bb e se = a Cets) = 44 =C e
sed ita fiet 60 = = a (c+ f) adeoque 60 = - 50
Et ex
sed ita fit e=-
si omissa divulsione  consem u ratirem 40: 25 seu fiassemus 40= 20, ce tcf b9
a fint bb Z - bba
ZSolba f)
videamus an itam eadem difficultas occurrat, et ex atr et ex 2 fiet 60. 60b ea
-569 = 5ba
2.X 20g onoa - 28. 20 A2. 20. 20 (CC + fod:bb) seu 60. 60bb: 20 20ap =
bb - bb + 2cc + 2c. f,  seu cA H + 2tf et +60. b, 20 a = c + f et 60 = Za, C+F: bb
CC = 12, aa cccest aacc fstraacces tec ccaace + ccaqet rceaaet 
Hos valaes o et 60, substituam in cer. 3. eoanescet in ex 20, et fiet
Se sbt = 12, 2 au C  raa aace (a- bb7 =) 2aacff+ Laiff + (2.a a CC= bb) est
25oabbC
sbacbbi Hets
urau
pato
etragus cs
 aees doseets us adt nast in P ti t ena
 l constanti
b
Aquod nescio
au liceat.
Quorsi