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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 51r, page id 00068201:0101
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y +b + taz in y+ C+ taZ, = gy + by + bc. H (2y+ b) taZ videndum un possit
identificari aaequari (quadrato a v +taZ =) v0 + ft Z + 2 vt taZ; fiet v = (yt6) a -27 + abt c)
et gy + by +bc = (00 + AZ =) aggyy + ad B + a ab   HZ, fist aa -ss
a
fiet Z = seu affgu + affby + abcff + aaffZ = agy t a by + a bb + f Z
inde Z = a (f5-aa) yy + a (ff- pa) b y + abCH,: fs (ff - aa)
sit C = - b seu b +c = 0 fiet Z = a yy D - abB
et
1, b6
EEbd
28c
Nempe 9 + b + saq in y - b + sac = yg - bb +aZ + 2y saz = vv + 1Z + 2t utaz fit v - a y
ft- aa
Et vy - bb taz = ac gy + ffZ, vel affyy - affbb + aa HZ = a gy t f Z, fit atfradius - ap 1
seu q = 4 9y - a bb. Comprobemus actu ipso
ft (. aa)
yt b + at (e s  ma) in gub + at (3 . 3t aa) dat yy. bb + navus - aabb + 2ya . . . . . )
st
H-aa
quod debet esse quadratum a 2, 2 + f1 (23 - b) = au gu + Hyy - Et 1b + 2y a 1(. . . . )
quod consentit priori. Haec ergo res succedit nam yy - bb L aagus - aubb = yy + aagus -ff bb
f
st-aa
+t aa
F Hua
Esto t _ C0 = 104 + 1DQ fit coco = 09 + D8 + 2 1OD. Debet coco + b in coco- ab,
aqua  iden reddi extrahibile, fit D et idenlificari quadrato a v + f 1OD, fint ay = OG + D8
et l = 4(09 + D9) et tandem 00D = E4 qu (O+ D) abb
ff- aa
quod ut praestetur identificatorie, e considerandum est C) non ascendere nisi ad secundum potet gravum
sed valorem ejus nisi coerceatur ascendere ad quartum. evercitio suplex, est, una ut e fiat constans verb. g. =f
baa seu C = h(C + D)+ L
a Yes. aa) ut  = 4 (O+D) -
a
unde  () +D = (h. + 4) (() + 5) Ergo h + a = 1 et hh + aa = ah seu bt. +
1fmaa
et fiet CD = qu (0+)
Cd = 1+0 + 1FD sit coco = 50+ fD +  lo t ab in pt c0+ = ff C fqu (O+B) + afs∞D + aff (O+V) YOD
seu hih =ah + L aa = L aa - a. seu h = =  +   Y - 3
et coco + ab in 1oco - ab = cot - nabb = ff qu (D1) +afOD- aabb
+4f (0+ 5) 107
formula identificanda quadrato a aB. + gYB D, quod est aavv + g90D. + 2agv 11OD, fit V = 2H (O +3)
() - 4
) qu (O+ D)  aabl
99 - 4 f
 qu (3+D) +990D, adeoque tandem
Eo
= OD. seu fF Qu (O+ D) - aabb
= OD
gg- Hss.
 ilH. da)
ab -)
sit _ h L (e13) + L cab  =  et a ((0+ 5) = n Hegg. 2H)
a gg, a)
tiet hF + sed ita addendo duos vulores, ut misus prodeat O+ D, deb. L esse hf + af = 1
na
ta
et h - = = 0. Ergo hh = aa et f et hhf + aaf = aan, seu fR- aah
Y(g9- 275) compendo A Zaa
utque adeo f = F D isi h = a et f=  g et ab
Fiet. L. O. + D 2O+ ZD + m = O seu O+D + m = 20 seu C = D + 2M
et = C + = D - m = D sen 0+ D - 2m = 42) et D = C -2m quae
consentiunt. Seo inritur abjurdum, quia I(gg - 4f) -0.
res ergo hac via non sucedit.
Alter Redeundum ad OD = sf 44 qu (0 +8) - aabe
H-aa
si retineamus Q inconsta variantem, sed faciamus ut  qu (0+ O +) sit constans = n
ea _ et sit O = h sM C + h ab et B - a
a aa t
a 1 (. aa)
= n = -2a
sit cd = 10Q + (DG et cocd = 02+ 3Q  28 son et ponendo C+ D = n fint coco = n8 + 29(0D.
5t0)
a a 
tr rtf. aa)
9G(4, +) 1. (O+ 3) + (98 4 - 4)03 - bb
= 2 1
(: aa)
ubi in (4 + aa) qu (0 D) + = OD destrui debent altiora, ut fint has haec quantitas aequalis
ac au
fit v = 42Q, O+ D,: af et
ab
h(. an)
Ergo ut O + D sit = constanti M, fiet p, fiet 2 + 4 = 0. seu hh = =aa seu h = a f- 1, et
ab
et _c in gg + 940B + et Cot - nabb = 4Q,  8+8 + 4816D ~ xv aa + 2av . 1OD
Constantiu quod fiat explicando C) et D per 8. restabit EG = A C) = bb 2 0 explicando 0 et 3 per 84
quod; a praetentari pt
D = 20X + 21 a firt habe
A-4aa = ag
e
e
acy
seu (5+- 4 aa) qu (O+ D) + 4 OD sit 9 = X et O = 10X + 11a, et
erit (C+) HaC ,. 10Xit 2. 10
4Op =
20. 20X + 2. 20. 2 aX + 21. 2 ca
et erit (11. 11 + 21 21 + 2. 11. 21) aq  = am
(ap.)  F.A 0. 20, ta. 20. 2) ax + 2. 11. 2100
quod ut
oustru
erut
4 10.20
aa
aa Y 10. 20XX +  10. 2 ax L H. 2aa
seu 1010+ 2020+2 120+4H
+3( +2H) 10. 2020 ad
et ob ex m - H ( (io. conpar 1 , A p. 1 4 20. 21 + o. 21 + 11 2) (1+ 25) =0
20 ax
Batur 1121. datur et B pr ha
fiet  . ut ua Funa tas 4. 10 20  01
et 1021 + 11. 20 =0 et -11. 21 act. b)
eu 1o - - 24 seu 1e =- 1
ergo datur et H per fia
ex 2 fit B. 11+20 21 20 eu =-
Ergo 10 = - 2
Ergo . 1i daFf Ag. 5 dut 11 et 21
aeq. 3 dat 10 vel 20
Ergo fit 11 = 21. Ergo 11= bY