# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068197:0073
# Permalink to this reading: https://leibnizlegible.com/page/00068197:0073?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 37r, page id 00068197:0073
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=73
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000073.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 87 lines, status machine
# Lines: 87 recognised, mean confidence 0.79
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# After the first blank line: the recognised lines in reading order, one per line.

(dx (xx - bb, XN- CC) quadratuna rationali sit X = f Av +, ltv + 
= (compendio EV: 2 fiet B sN = HSin taacoco + 2afcoAtv et 77 = tv + p p + 22b stv
Quaeritur
et XF = aa ses= lb 12: 13 = fa F vyscot 33 .  uiaaguo trgscoste. : 
et RX -00=  aasus- 2 12 b tvb - 2bb
et Xx- bb, XX- ec = aN +  bbcc J - Dabb 51513): 72
XX = cC =
 45u+ - +5+ HW+ a ct + 6aafftvcoco
XX bb, N.CCG
+bbcc  = +bbcettwv bba  +6 bbcc trip Y
cav . Eccr HtD
216
+ 4bbec 2to Av + 4bbc VHtr
em +299 -
b 3s3I- ysto bbsset staso to - 2bb 25 ssen qu At - ubb 1s tn rascrtte quam e exia sua
IccHE aacocos
UaeSt pId. Cm 12g9 -6 aa - 9.d_ 29.1+-yt 
compendio sit gge bb + ec = gg et f = a et  a = bbcc; bb + Cc - 2Aa = qu (b- c) =
acozancocoto P
Ecc
fieca t cLCH acolt
=aahhetvo r.
nhubna-
4aagu ovcct
C-raahh (a tu A
+4u (co3q AV
-2aaggc0 2 E 31 t V. ,. 37
_ Reducamus co et  ad c0 + 2 ut ad co2, quia ed et V servent interse loges justitiae, Et C+  vocetur  
gcoco
fint XX - bb, XX - CC, =
+at0
= Cot ++ ScocoBIB + 4corpx = coo + 2coV. Hinc coco t it x=  2c02
4
et cot t W= W E(ScocoYV) 4 coc +(Brocorxi
at fint t
et co + = B oco -corp = - 3c 
94
a B+- qu eopicoco 
raahh E sto +a s+vZ. 5
aagraapplooto
Hujus tractionis  denominator, est extrahibilis quadratice, talis ergo efficiendus est
1X- bb, XX - CC =
H.
e
74
aa ti. HoVam tBB2ac 
Numerator. is identificetur quadrata a latere aa rZ + _5 AZ,
quod est: et vZ + e 3 EtZ + 2e4gEZ Av
Aequetur ergo irrationalis pars irrationali, et rationalis rationali. Et ad aequandas irrationales
vocetur sit ( cum necesse sit aliquam ex his g, 8, Z esse in cratione ad B Videtur praeferenda
ex irrationalibus, fiet
aequaliois rationalium
litera Z, cum prae caeteris se distinguat, et fieri poterit 
482. El Z = 4 a β(3 = aatiht v- na pcor. sed ita habetimus
β3(β, β, quod non
placet, nalimque habeiri solum B et BB. Itaque praestat  alterntiam G vel Z  redici
ad β3, et sit E = β: C, et fiet ex rationalis aequatio Z = 247 β3. aatit. lB. mop caes
Et ex rationalibus fiet aβ- 4 co v)BB =aahhsHto - 2 aupp tB corp = eBB eeHtZBB + e~Z
sit l0 = 6P 10s + nas +naa et _ colp = 20)5 + nast 2raa et 4Z = 3055+ 31 a5+3aa, fiet
cocorp
+aentv b
cahtr 110)+naS + iraa = 00. 2aβ3β. jam ut formula aliqua velut 190s+ 91 as+ graa sit ideatifica-
bilis quadrato 80. 80 S + 2.90. 81 a5+ 81. 8laa. fiet 20 l90.92 = 01. 91
seu 4. 01.01, 90.92 = 01. 01. 91. 91 seu 4. ot 9 c 4. 90. 92 = 91. 91. seu fiet
aapp 1 205+21 a(+ 22 aa
B(. 30(+31 a(+32aa
adeoque firt qu (H aath + 21 aapp +31 e) = Loaahh. + 202app + 30e3C, auh 1aatis + 22a app + 32e l
Hutus aequationis ope ex datis habetur l. ut ita jam possit haberi l pro inventa ex ipsis bet C
80 = 101.90. 81 = (01.92)
Ergo B = s  (10aahhi + aapp XB1) + a Y (12 aahh + 22aapp + 32 C3)), quae quantitates
et eut a4βt ta β ( (200) = 805 + sia
constantes et inverta  haberi possunt pro datis Inde jam veniendum esset ad aeq. 2. et extractio
primum pro BB habendo, deniue pro habendo  institui possit: imo su
sed prostit fortasse pro aequatione utraque procedere astrmenda explicanto corp per raff 20BC+ capo nraa
sed caeteras uteras.  t, V, Z. _, explicando singulas per B et adribendo numeros fictos, quaelibet
ex litens habebit certum valorem cumpo pro eo  assumamus namen, eum prolitens substituendo in
aeqq. ) et 2 odibil valor laius aequationis qui ponendus loco ipsius 0, etsi revera de inde
haberi debeat pro. ita aequatio utraque  vera esit etiam in numeris veris, dende
quia dnstituendo sunt, sed hoc fortusso rectius differetur prr aequationeb identificatorias, que
do
fiant vera actualiter si jam constat hoc modo expli aequationes errore calculi calere,
quod etiam objectione novenaria et undenaria comprobari poterit; supererit, ut quaevis
aequatio divellatur in suas id entificatorias, eaeque singulae simil iter ad nundios veros reduantur
loeo 0, ponendo valorem in numeris veris. Hoc facto licebit venire ad calculos, qui semper compro-
Explicotur aqu l per β. erut ouni bodletur terminus absolutus et divisis om
valor _β per 5 repertus ex aeq. 2. substituatur in aeq. 2. debet ipsa addi identica
bationem suam secum ferent.