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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 140r, page id 00068199:0279
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voctam,  quaeritur fdx sCaxtam, Extan) quadratura rationali. Fiat XXtam = OC)D artan = D98
C erit ((X Xtam, XX+an) = OD8
o=2ct + 2t + 2 18
X = 11 Vh2 + u2 1t5 + 13 (46 +0 8 = 41tq + 42 N2 + 4318
90 = 21. 2lat +, 2t. 22a lt4 + 2. 21. 231 46 + 2. 21. 23 (46 10D-
1.2
23. 23a8
00D = 2121. 30 at6 L 2. 21. 23 30a (S + 2. 21. 2231 a 2 16 4 2. 21 22. 312t 126
2. 22
23. 23
2. 22. 23. 31346.  +2. 22. 23. 312a4 n
2. 21. 22. 312 ah2 + 2. 21. 22. 300ab.. + 2.4. 23 300a6 1..
2. 21. 23. 313at8
2. 22 23. 313a276
am) 113. 11346
123. 12326
100. 100 84
am
+2, 4. 23. 31 at
+2. 22. 23. 300a8...
+ 21. 21. 312at.. +4. 4. 313at ... + 21.4. 323at..
22. 2
2723 
2.2
3.23 S
aequum coincidentiandum cum
22. ve
23. 23 X
+ 2. 112. 100. 8 Th   2. 13. 100G 1h6 + 2. 123. 1009128
= 2. U3. 123.816. - + 2. 112. 1232..
+ 2. 112. 113 t
3a
mMederi tamen fortasse malo liceret faciendo, quidem 312 = 313= 323, (quod etiam jam huilicapius catculi aequatio coincidit prion
jam tumlicet et 212932 et 712 = 113 = 173, et 21. 22 = at 23 = 22. 23
et 291222 12tn27 g.2+ 2ne
ab = ac = bc fit  1B  113- N= 122 et
21 21.22 = 21. 23 = 22. 23 t 31L = 313 -323 ita. in atraque aequatione primaria una irra-
112
13
13
a1. 42 = 91. 43 - 42. 43
112
113
123
iIX +12y + 1Z + 10 =0
=20. 13 (2b. 13 X _ 22. 139,-23,132
+10. 291f23. 32 2
= 20. 135 22.33)
-(123. 30) fl2. 23
2 20. 335- 22. 13)
+ 11. 237p+23. 3
=21. 13542237
=az. 3115 F 1217
l. 21.335
per X, 9, Z, in elligemus Z, H, 6
 et +.
113
123
tionalium coincidentiatio efficeretur
2F. 23
21x + 22y + 23Z+20 =0
3lid
= 22. 13
112
qq
1. 42 Em
H3 41. 43
+2330 + 23,31X 7 + 232932y
du - 10. 225153
+13. 20.43
- 12. 23f13) 30
3
quoda
3AX + 32y +53 + 30 =0
20. 33 -2183  -27.33 y
+10. 25 (e oqu
=13, 20() 32
+ 13. 2252330
+ 12, 20 23 7
+ 11. 23 (27. 32
= 11. 22. (23,. 33
 (2. 21.43332
F19. 21. 22.3
= 12. 27. 23 3
13. 20. 22,33)
00D = xxtam = X +lam, X am, et _D4Z = X + - ar, x - 1. qu
X 13, 22. 23, 3
F. 12. 21. 63737
13. 21. 22. 33)
ergo D aequale vel proportionale alterutri, v. g. D = BX + βf an ergo i 112, 113, 123 sunt ipsis 312, 319,323
proportionales, et seu B. 12 = 72, B2  3: β123 = 323., ut  β. 100 + β1 - am = 3008
Prsus 00 = X - l + a sed  par jue foret D = B)X F 9) suam, unde ficret
V112 = 312, et ita porro ergo ) = β. ego foret (-am = - s an vel sinu tale quod sed hoc non licet
itaque talis resolutio non procedit.
an ergo
X F V.am is rursi resolvere in binas, et X
Ar ergo licebit. X+ daam, et caetera rersu resolvere in bina, quorum quodlibet sit quale O vel E, siet
X. F Y-am = f9. X = 1-am = hl, X + 1. an = mn
X - l- an = p 9
siet et similiter D resolvemus in 5t, et fiet OC st = fghl, quorum quodlibet constabit
possum  satere fghl. si jum extribus irrationalibus simplicibus.
foret
et D seu 5
quol eremus 3D = fg, seu X + Nnam extrahibilis, fieret etiam X- nam extratibulis ergo
Rursus ste8 = mupus. ubi portebit face
non licebit facere st = ma, vel p9, quia feet Xalam - x+ Lan vel tale alius vid. Ergo
eu quadratus, et foret D X- 1-am.
faliam
sus ergo ma p aut tale quid seu
et X 1 an habebunt divisorem communem sed nq vel onsi - eg. ita
seu) in ed
d
s
x lam
videamus ergo an haec duo simul quad praestari possint, ut X+ Jam sit quadratus
X + f-an,
an, et X- s an habeant divisorem communem. intio ad hibermu sotu duas
et ut tidnall
u
empe ves
 s b9 +cZ bv + cZ in AV. + m Z
ectpabit CmZZ + bm + cl) vZ sit et sit  Liap + blovtemZZ + (bm +Cl) v
 sit de pro ductum ex
= quadrali ab v +7, quod est 0 + 2Z + 2VZ, et fiat bimt cl = 2β
blvo + cmZZ  + Yam = v B0D + B2Zs quod licet ponendo v et Z esse irrationales
simplices
X+ Ynam =  sit + X = b sB + 16 in f  + 91S, = s +c96 1(bgt . NGs
hoc aequandum quadato a hv + KZ
nam
V.am
Si saleremq 21 = 22 = 23
et 312 = 313 = 323
et (12 = 113 = 123
coxxtam
omnes irrationales binis
binis aequationibus effici possent, una
foret 2521. 312 = 112. 112-
altera foret
2. 21. 21. 3005 + 21. 21. 312 (t + 4 +6)
aequ. 2. 12. 1008
Et similiter ob xxtan = DEG
fierent bina aequatione
41. 41. 312.  = 112. 42. (Erg . X- 4
sed possunt esset. AeC 4 = - 21
et 2. 41. ar sed ita semoa a posteria
sed ita fieret 41. 41 foret = 21. 2
et proinde coinciderent dd ut 94
adeoque etiam xx +am, et xxtam
43 licet
+XX] = 112. 112G7
contra hyspothesin