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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 14r, page id 00068197:0027
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Saltem quoad variabiles)
vel Hgy + bG +aC)
Possumus rationaliter exhibere 1(gy + ab). Csed difficultatem reperio
ranonalitatem ipsius CLZram) CNQ + ap)
naliter  lC vy t ab
nta Z+ aapt)    + (ZZ + fZ+ ag
d
yy + ac)
vel quod eodem redit ad iapi) + Hea
hoc modo
cl Z+ am) + Y (n A + ap)
et dge compendio y = fact + 1 a D. fiet gy - a0 + 2 a tOV, et gyt ab ducend
in gy + ac, fiet d + 4 aa 83 1 (3.
aa D
+ 2aa 3D
aa 8fC
+ aa bC
+ 2aa b7)
videamus an hoc possit coincidentiar
quadrato a Ba Z +OD, qu ubi e est variam
quod est taa 8 + 2a9810)
+9937
coincidentiando irrationale membrum irrationali fiet raa (O+ D) + 2aa lbtC) = 2a94 se
fiet 8 Ca ()  = 2a (O+ D) + a Cb+C). Ergo hunc valorem substituendo in coinciden
membrorum rationalium, fieti
O + DD + 20∞ + (b+C) (O + D) + bc = 4 aa qu (0+ D) + 4ca (b+C) (C) +) + aa qu (b +C) + 99C
effitiamus ut g D + at qu Cb+c) bc, dividi possit per O+ D, seu sit (3+ D) 6. ita tota aequatio
SY dividi poterit per (3 + D sit O = 10D + ua et S = 20D tsra, et per aq. +. fiet
2) 10 D) + 44 1a D + aa (b+ C3 - bc ~ 10. 20 DD + 10 2 4D + p. 2 aa + 20DD + 21 aD, hinc
C) inveniatur rationaliter
+ H. 20 a D
prodeunt tres quae aequatio ita praestanda est, ut
integre per D, quod ut habeatur ponatur 10C) + 2D + ita, in 300 +20d + 44a
quae aequatio ita prestabitur ad scopum, si fiet () = 10s + na, et ex aq 7 fiet
(+ 18 +3) + 2. 18. H a D + H. Ht aa
(++a) +00. 00 37 +7.
00
la
+2. 0. 10 DDY, + 1. 00. dD + v0 1br)a
 + 1. 00. 10 (b1C) 
+ 92 00 10 DD + 4, 00. 11 a
aa
Ex termino medio seu Y e 0D, fit
+ 10 =
1+44
Eex 03 fit 10. 10 + 2 00. 
equatio
400 00 bc
f
id est singuli ter
+ (6V a 44:
ODD + 0D + 0D
id est singuli termini aequandi
10 =  1, (99. a0) 00. a + (4aa - 1) (00. 07b+ cH +
+ ae qus H.)
+44 00. 1a + Aga  (+2. 00. 11a)
aa
nihito
  400. 001b +C))
.
2. 11a + 00(b + C)
200
110. + 00. 00 = 0 sex 10- - 00 + 99. ca , f. u(t 19 2
2. 8
-8aa + 2  - 00.
1+ 4 aa
10. = b. 10 + ac
10.10 + b. 10 1 bb = ac F = b1
10 = - = b + +(ac + = bb
habemu duos valores ipsius 10 quos aequando habebitur  1
sed mlasus habetur i ex termino ultiaro, seu oD
vedetue compendiosias procedemus hoc modo:  loco aequationis + per compendium cribamus
12 9B + C 1230D + 1245 + nona  CD  =0 fiat 0 = 10D + ua,
fiet 122 10. 10D8 + 2. 122. 10. 11 a D + 122. 11. 11aa (
122. 00. 00 V  B BgCC_1
123. 00. 10DD + 1. 123, 09. 11 a)
~01 +01+0d
Ex termino priuo DD est 10too 
te
+ 1. 121 00 10aD + 121: 00. 11Ma
ns et H1 = 123, 00 + 2. 12210 (u 3. 1Z =
seu 10 = + =3 00 Bo (9 m2. 122
diesC..
Aursus xquos duos valores ut aequemus fiet 12+ 00. 0 ~190.  
redio
00.00 L 121, 00. 1
C . ob e4 123. 00. 11 4 12100.003:, 2. 122 H 1 1121 00 Ex primo 10 10 +. 23 00 10 t B 00. 00
+ 1, 121. 00. 00 a D + 120.00500 aa
- 121. 00.00 1. 121. 00. 00 123 (B121. 50. 00 1) (25. 4 11) 173. 00  122. 00
2. 122
400
SV
( L∞. 16.   +00. Lu io 
4. R
3. 1
qui duo vulores ipsius et sunt aequandi, ubi utrobius evanestit 00, quod adhisumius solum ut homogener
(123.00 Es22) n300) (2. 122109 Y(3 - q 2)
tas melins appareret jam restat una adhuc in cognita 4 vel 49, ubi compendio pro q4
trbamus tar. Latet autem que in ipsis 1, 120, 12, 122, 123
120 = aa (bb + 2bC + CC) - bc et 121 - (4aa - 1) 1b + C) et 122 = (4aa - 1)
et 123 = 94 + (84a - 2)
a
ad
nam est
et 123 - (d ag- 2)
ubi b et c, sunt datae q est quaesita, et quia ubique
comparando duas valores ipsius 1. fiet
-2121. 122 + 121. 123 (121. 121 - 4. 120. 121) ubi ilud comm
est q, ideo pro 44e possumus ponere ar
si bene calculavimus fiet
((15 - 4. 125)
ubi lud commode evenit, ut unaquadratione
llatur irrationalitas; et fit
ubi pro 120, 121, 122, 12
t blg dev et fit, 7. 120. 1233 = 120. 132 123 4 121. 121. 122. 123 = 2. 121. 121. 122. 122 =0
substituendo valores, in quibus latet qqe vel ar, habebitur aequatio pro que vel r, quae vvidetur pro r assupere
ad sextum gradum, credibile tamen est, si res obtineri potest calculum esse depressionis capacem; si non in
universum, certe in casibus notabilibus; et licet q non sit quantitas rationalis, nihil tamen refert,
a seopam nostrum, modo sublata sit irrationalitas ex variantibus