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# Piece: https://leibnizlegible.com/api/records/14987/text
# Catalogue record: 14987 — [Math. Unters.]
# Incipit: [Anf. fehlt] Sit aeqvatio proposita 20xx+02yy+11xy+10ax+01ay+00aa=0
# Date: 1695
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=14987&katnr=39692
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 183–186 (from the shelfmark LH 35, 12, 2 Bl. 183-186): 8 pages, canvases 365–372, page ids 00068195:0365 to 00068195:0372
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=365
# Lines: 442 recognised on 8 of 8 pages, mean confidence 0.83
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 183r — 00068195:0365
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val -gy +d
l
3e1 0115
det
i 
p 

## Folio 183v — 00068195:0366
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Sit Aequatio 20XX + O2yy + uXy + 10aXtoray +00 aa = 0. Ita praestanda in
proposita
rationalibus ut prodeat identica si fiat X: a = 10OV = 10Ia,: 200 P + 20. a et y: a = 300B+ hoa,: 200V + 201 a
Et secundum hos valores explicando Aequationem propositam, debet prodire atia cujus termini evanescant
20XX = 20, 100100V + 2. 100. 101 a V + 101. 101aa
Oryy = 02, 300. 3001∞ +2. 300. 301 = + 301.301 
I1Xy = 11, 100. 300.. +1.
101.300.-
10ax, = 10, 100. 200.. +1. 100201.. + 101.201..
101. 200.
O1 ay = 01, 200. 300. +1.
. 201. 301
+ 201. 301 oi
00aa = 00, 200. 200. +2. 200. 201.. + 201. 201..
compendio 20. 100402. 300 =I1. 100. 300 0. 402 et 10, 100 +01. 300- 50: 401
sit 20. 101+ 02. 301 + 11. 101. 301=00.412 10. 101401. 301=00. 4N
2. 20100.101 +2. 02. 300. 301 F 11. 100. 209 + 11. 101. 300 -00. 4012
fiet (1mo) 208 + 401. 200 + 402 =0 Ex termine summo
(20) 201 + 41. 201 + 412 =0
(3tio) 2.200. 201 + 401. 200 +4012 =0
medio
o = 33 4  - 2.10. 6) = 2.313
Adde compendia ad aeqqu. 12, 15, 16
Sed quoniam arbitrariae videntur sufficere faciamus 4020. Et ex aego. 1. fiet 200 = - 401. unde ex 0g. 3.
seu 201= 4012- 402 :9 2401 - 4 11  (ep ag. 2) - 411 + Y(4. (- 9. 412), : 2
faciamus calculi magis deprimendi causa 4112 - 4. 412 = (8)  = o fiet  ex nq. 6 et y 2t = tnoz 201 = (9) = = +11: 2
habebimus 2. 401.201 - 4012 + 411. 201 +4012
1 4012 - 401 = o 401. 411 = = 4112 ob _ qu 4 fiet 308: 100 = 11 m0 = - 11 = Yp5 - 4. 20. 02) , 02 = 12mo per compend - 102: 000
at ob aqu. 8 fiet 10. 10P + 0r. 30 P + 2. 10. 01. 101 301 = 4. 00. 20. 105 4, 00. 02301 4, 00. 11. 101.301. unde per compendium.
adeoque 301 : 104 = 1910 = - 313 + L (313. 313 - 31. 33),: 33 = (15t) - 2 00
 explicatis 4012, 401, 481 et pro 300: 100, Idem 301. 10l substitutis ex agq. 12 et valonibus compendiosis fiet
fiet 31. 1014 33. 301 + 2. 313, 101. 301 = 13tio=
4002 
G cell.
sa
2, v1 1. 000. 102
∞2104 000. 0
42. 1081.
+ O1. 112. 112...
=2. O1. 102.102.
-4. 1001. 000. 102... +2. 1081. 000112...
ne sit 301: 100 = 101: 100. Et priora duo videns c jege eeter Neque huie malo
videntur hypotheses nostrae restringentes esse obnoxiae. (301: 300 = C112. 108: 000. 100 per 
multiplicando scilicet tum 301 : 101, tum ejus valorem est aeq. 15 per 101: 100. Ergo si 301: 300 esset
idem quod 101: 100, foret 112:000 = 1: 1 seu hoc saem (nisi in certis casibus specialibus
quos bransformatione evitares licet) non patitur aeq. 15.. superest ut ostendamus negi
gereratim saltem) esse 301: 300 aut 101 : 100 = 201: 200. Nam ex ag9.5 et 9
habetur 202201: 200 = V 4 (119n0 = 411 : 2401 = 10 101 +01. 3013: 
2(10. 100 + 01. 300)= 10. 000. 101 + 01. 112. 101, :2 (10. 000. 100 + 01. 102100 quod 
(10,000 + 01. 112, : 2 (10. 000. + 01. 102)) 101: 100. Ergo non aequat regulititer) psi 101 : 100
sed nec aequatur (regulariter) ipsi 301: 300, seu per 18 ipsi 112. 101: 300. 100
alioqui foret 10. 000 +01 112, : 2 (10.000 +01. 102) = 112: 000, quod utique regu-
Veniamus ad exemplum quo iudigemus sico =a (C +aXX, :, aC + ag XZ)
lariter locum non habet
quaeritur (CD)dX. siat   ZZ =  thX et fiet ( 2+Z = hdX L ch
ot du = h dx: Y2RX et Cix AZ =hat (aE + 2FhX, :, (a C + aghX) 2hX) debet
ergo a C FrafghiX, ac + raGhX, 2hX aequari quadrate. Quoisi ergo res praestari
otest faciendo X = 100V rioIa,:, 20ov+ roia, su effici potest, ut  aa C+ rfhX, aact 2h X
Liet 02 = -A1., et 20 = mn: a et n 20 et 01 =0 et 10 = ac con + t et  men,ic et 00 = aa
quadrato quod sittjus fietque
per 5 erit 200 = C=(M+n: C) 100. Et per 9 erit 201 = - (mt 2 (m +n: 2C) 101
33= = 4. 44 : CC sed 313 =0. Ergo per aqu. 15 fiet 301: 101 seu 112:000
10. 10 seu 101 -C 2mod + nirueumqus + ammus - m M L- a nin (bmmnid 41e n +(3 mmuB)
seu  majus quam um + ns + Bm + B n. si MLn hoc non proc
+12 mmnon resse majus quam 44 (in + n) quodsi alterutra m, vel n sit quantitas negativa
saltem nmmi
(A (aa : CC)q (m -n: C)):  (= 4aa : cc) seu A (m - n,: C) a(a: C): (- 4da :cc) =  f m. n ). 2a = 1 : a
Denique ex aequ. 16 fiet + 4. ma 1100. 181 - 2. si + _) 100. 108 tq.  F) 101. 101 =0
rp. velbreniter A. 20 - 2. 10. 10) 106. 101 +  101. 101 - 2 = 0
Et 101  = - 2. 20+10. 10 + ((4. 20 4+102 - 4. 20. 10. 19 6 - 2 10t)
ubi videndum an fieri Bossit ab irrationalis sit quantitas reali
non iideo quid obstet, sed si ambae sint affirmativae debet esse
d obi mo
nam soli
sed quia Quia 11 =0 fiet mn. 100 = a a. 3 d = cc. 402. Ergo si 402 =0 fi et 300: 100 = Imn: a = 102:000
sed 412 = Mn. 1012 - 4a. 3012, : cC 401 = mn (m+n,: C) 100 41 = (m +n, : C) 101 Ergo
Ergo 201: 200 = _. 101 1 00 101 : 2. 100. porro 31 = 4 mn- mm - hm - 2mn, : Cc se
seu 31 = qu (m - n, : C)
=4 mn (mmenn)
jus
aequetur quadrato, facile effici potest, ut deinde et iha aequetur quadrato. Par fh scribemusn
prorfg scuibamus am, et pro 29h scribamus an, et ac + nX, ac + nX debet aequari
mEX ju + mnXX +acmX + aa d  =0. itaque
unde 101 : 100 = (B, 2) 17mo = - 59, + Y  50. 51): 51 oe
ubi tantum cavendum ne sit 101: 100, neque 301: 300, aequ. 201: 200
 qu et denique ne sit 301: 300 = 101: 100.
Ubi tantum cavendum ne sit 101 : 100 = 201. 200, neque 301. 300 = 201 : 200, ac demque
42 001. 1o2 12 ...
40. 100 + g1. 102 +,GD1. 100. 101 - = d
100. 100

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Sed cum successus videatur mirus, et pene, incredibilis; operae pretium, est exemplum
ipsum Fractare speciali seu proprio calculi, ut facilius appareat si quis adsit lapsus
 ergo aequatio - aayy + mn XX +ac (mn) X + aa cc = 0. Hanc ut praestemus prationalter
ad lustitutum nostrum, sumamus _ X: a = 100V + 10Ia,: 200V+ 201a. et y. a = 300Y + 7019) :, 200V + 201 a. Fiat
+ 201. 201..
1. 101. 200..
aequ. 0.
Ex termino si
200cC + C (mtn) 100. 200 + _ 100 mn = 300.  a=v
ubisi faciamus 100 mn -300 aa =v. fiet
300: 100 = Vmn : a et 200 = - 100, m+n,
Ex Termino summo destructo fiet
201CC - + 101 (mtn) 201 C + 10emn - 301aa = 0
fiet 201C = C - = 101 (m +n) + L Soi qu m An)- toinm + 4. 3ota)
aequ. 0 ubi ut destruamus quod est in vinculo radicali fiet ,onm - 2toi mitaedoia
2. 200.
quae aequatio est imaginaria, sed hoc nos non moramur, et fiet
. 301 : 101 = m - n, Y-1, : 2 et fiet 201: 10, = t1): 2C
Rursus ex Termino medius destruendo est 201 = (t300. 30l aa - 2. 190. 101 mn - 101. 200 (n + n) C, 1 2 20
seu  100. 101 gu (m+n)  = 100 (mn)  (m = n) D 101 = 2. 100. 101 mn + 101. 100 (mAn: C) om + r) C
opparet divisis omnibus per  100. 105 evanescere
sitasrrationalem, aut ullam novam ad hibes
restrictionem;  sed maneat 201 c = = 101 (mtr) + L 1 (101101 qu ( m - n) + 4. 301.301 a4)
Eno non licet in valere ipsius 201. Destinete
sed ex Termino medio fil 201 = 2. 300. 301 aa - 2100. 101mn - 101. 200(M + n)C ; 2. 200 ∞C + 100 (n + n) C
seu 201C =  2. 100 mn) 301 - 2. 100. 101 mn + 101. 19qu (m 12), :,  2 man) Y + 1e (u i)
seu 201C = 2. 100. 301 armn + q (mm. + nn) 100. 101, : - f. 100 (m + n) 
= ∞= 101 (mFn) + L (1oi qu (Mn) + 4. 301 aa)
(1430
seu 2. 100. 301 a tmn = 2. 100 101 mn (4 100. 101 qusm tn) = (10B. 101 qu ( +n) = 100 (m + 1) L(io qu mg
4. 100. 30i aa mn) + 4 G00dioi mn _ 100 iqu emn. n) +(30) vitum + nn)  4 100i 30i m aa
_. tot. et amn Amn 4. 30 mntnn) aa + 8. 101. 30amnumn + 101 qu (mm. nn) = 0
seu 
= 8. (1003101. 30lamnima
ioimmnn
et fiet 2. 301 a =- 2. N. masma +)  m id - (qu (mm. na) - 4 mmnn) nm + rn)). n
F 200. 200 Cc + 200C. W n) 1009 + 105. 100 a f - 300. 300 aa = 0
00 : 100. = sf: a) - taf: a
201. 201CC + 201C 1019 8 a(, 101. 101 af = 0. Ergo 201C - - 5 1619 + 5 110199.  as
Ergo 200C + 100g = 0 seu 200: 100 = = g: c
301. 301 aa
2 200, 201cc f201 c. 1009 + 101. 200c9 + 2. 100. 101 af - 2. 300. 301 aa =
Hs ex deniue ex permino medio est
X = - = 11 + Y (i- 4. 12)) q pro 300 et pro 700 ponendo valores per 100, et 300 a = 100 Yaf 200C - 
fis -(2. 1009201C+201C. 1009) 101. 10099 + 2. 100. 1014f - 2. 100 1a f 301a = 0
et divisis omnibus per 100 fit 201C9 = - 1099 + 2. 101af - 2. 301 a 1a f
Ergo ex priori valore ipsius 201C fit (- 101 99 + (  (10i gg - 4. 10r a5 + 4. 301 a a)-
sit 9g aaf = ah, et fiet
9 (101 ah + 4. 301 aa)   101 ah - 4. 3oratas, et quadrando
=(2, 10199 + 4 101af - 4. 301 a ra f
utrobique siet roj atigu + 4. 50i aague - (10i aatt) + 16 20i a3f + 8 101. 30+ aa h lat
10X = - 4 11 4 3 (1. 4.10. 13)
aequ.0
100. 101 (m +n) C (- 1m 4 1). 20)
10. 10XX + G. 10X. 11 +10.12
200. 200.. +2. 200201.
AX + IX + Z 1 = L, 11 - 12)
19XX FIIX + 12 =0
rqu 411 = 10. 12
seu 62
XX + IPX + 1220
10XX + 10X. 11 + L. 11
Aayy = 24, 300. 300VP +2. 300301 aV + 301. 301 aa1
+mnXX = +mn, 100. 100.. + 2. 100. 1011. +101. 101..
raesnm) X = csmn), 100. 200 .- + 1. 100. 201 .. 4 101. 201..
taacC = cc,
estructo
Te
44 2elaa 9g
ccuce
seu (1019 = 4. 10i afgg + 4. 301 aagge = (10 q -(8) 101 a599 + 8 agg Saf. 10130) + rt s 3eur. B 
4. Ze aavt
1o1 aatihreroias
B F16. 10C a t = 32. 101. 301a f (af + 16301 af bi 
Sit jam Aeg ag - 4 af = ah fiet 4. 301  ah 8, 4 4. 10 afah  8. 101 30r a atratat Z0
301 aR - 101 sa f =0 videamus au res actu succedat
Is vulo Ergo 201C = L 101G +  10, Cah + a(af vat)) sei 2ot  toc (:2C) g+ Hatita aalsi
seu
200 =  100g: C  300 300 = 100 raf ia 301 =10Naf: a
Em XB: a = 100B + 101a, :, = 100(9: C)D + 1O1 C2C) 9+(ah + 4 a a f : h)1)
seu 201 = 101 (: 2c),94 Hah + 4 aa (f: h))
sit formula af xx + BC9X + CC, ubi pro X: a substituatur vator 100∞ + 11a, : -100G: d)v+ 201
afxX = aB p0o po V + 2. 100af101 aV = 101a s1o1 aa
G 109.10)
F10099..
1. 10059201 _.. + 101. 201 aa C9
1. 10099101().
= (2.) 10029. 201 . . + 201 201 aa cc 
100. hocas
n 100. 101 39
ubi 20+ postea etam expli
Sed hinc ortur absurdim
nam si 301: 101 = taf: a
fiet 301: 101 = 300: 100
quod non licet. Itaque
res non procedit. 
ta y esset in ratione data
ad X, contra hypothesin
C9X=
cc =

## Folio 184v — 00068195:0368
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n
p
n
GOTTFRIED WILHELM LEIBNIZ BIBLIOT
eeeeeeee
EDERSACHSISCHE LANDESBIBLIOTHEK

## Folio 185r — 00068195:0369
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rationalis integra
cum formula secundi gradus aequanda est quadrato, res succedit
literam capitatem explicando per formulam rationalem sed non integram
cujus numeraior est secundi, sed Denominator simplicis gradus integer. Verum
ita haec ipsa quantitas non rursus aequari potest  quadrato, quod tamen
desideraretur si quadrato aequanda esset formula rationalis integra
tertii greadus quae scapitalilitera posita X) reduci potest reducta intelliq
ad  +. . XX +. X. Nam fit XX +. X + . X, ducta in x. Hac autem
memodo ubi post quam  Xx+. P + X quadrato aequata est, superesset
X ruit aquanda  quadrato.
sane si fiat XX + pX+ aq - gy
posset fieri y = X + V, et fieret Xx + px + aq = XX + 2XV+ vV, unda
fieret PD X = aq. V V ,:  2B. p quod 4 aequare quadrato non licet
sit ergo aequatio  20XX +02VY+ uxy+ 1oXa + Diya + ooaa=
C, et y = l4+ vV,: 20- p
prostanda in rationalibus, tentendemus, faciendo
20XX = 20, 100, 100 vV + 2. 100. 101 aV. + 101. 101aa
Orgy = 02, 30. 300.. 2. 300. 301
I1Xy = 11 7 100. 300.. + 1_100. 301.
104X = 10, 100. 200..
o1ay = 01, 200. 300..
1. 100. 201.
100 V + 101a - 300P + 301a
a 200V + 201a a 200V + 2o1a
Videamus an tolli possuul tres termini
ope e arbitrariarum, sexant
saltem quinque. Suppond nu semper
adesse, atque adeo adesse et ho
ac 30) hiuc ergo inchoeruus
sed videtur res non debere sucederes
sublato V, dabitur
20100 + 1p. 100. 300 ,: 600. 200 10100 = 220. 100. 101
00aa = 00, 200.300. 2. 200. 201..
1E200
i
et
10. 100. 201 7 : - 2. 201 - 101
10101  10. 100, 20,7 (10. 100
o13od, 1. p. 100. 3. 
 inttem
quos duos valores aequando
o1. 300, 201 f201. 700
Or. 308 513o1 A emale n   4 - v0 200 5 4 00 ∞9 200. 201 45101 f + 401 301
20. 100. 2. 201 + 20. 108. 7(1601 + 2 11. 109 201 30
00, 2. 20. 100.101
0. 101. 2. 200 + 20. 10115100
02. 20r52300
00, 2. 20. 100. 101
200bd 4 000be + 000 se su 200-
o ed = +20. 10822B
Eodem modo ex med Termino
medio et ultimo habebitur, valor
+2. H. 101. 200. 301
et tantum permutando
500 - 10. 101. 200)
D 201b + 000C 201 = 200D 4000C
p 200=vobe rrote
doo vetioote. Et pam pro 291 = veo D facte b0
(20.108 2210
10 02. 308
prot  000 (1. 10430t3l
630) 270
Ex termino caedio est 
+ 00 fro tooni
00 (2. 25. 10010
100. 201 + 200 (10. 101) +281 (9. 300)
o 300hrs
v1. 3005C
20. 100101 + II. 1od. 30. M.
eodem modo
o. 105 + 02. 5. 3013 10. 10p.
4012 =5 411 = 10. 101401.
mno adio fiet 09oe poartot ) r 19
2. 20. 100. 101
1. 11. 102. 300
(2. 10r )) 101
Ees, rem reducamus ad casum utiliorem, possumus facere 1 = C 10) LH.
seu cd+ et-
2.ocde + bC HCC) + 1
o1. 3015 scst bd.
5120. d. Fl. 02. 30r - 10.
ha, (ut 200 = 10dC: f = 000 C: d
 e a mne e

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=aagy sGXX + ac gX + aa cC = 0 sit praestanda rationcilibus sampacibus
X: a = 100V + 101a, :, 200V + 201a y: a = 3000+ 301a, : 200V 4201a
seu X: a = 0.  y: a = d. E fiet - na aa 8P + asaa b0 + acgat + aa cC = 0
84
seufiet = aa DD +af  + cg0Q+ c 2Z- 8
=aa DD = aa, 300. 300VV + 2. 300. 301 aV + 301. 301 aa
+af 6b = +af, 100. 100
+C90Q-
rcc94
+ 2. 100. 101
+C9, 100.200
+CCg 200.200
+ 1 109, 201
1. 101, 200
+ 101. 101
+ 2, 200. 201
+ 101. 201
+201. 201
Ex termino summo 20d ec + 200C 1009 + 100 af - 300aa = 0 haec aequatio divellatur
in duas 200 cc + 200C. 1009 = 0 et 100 af - 300 aa =0. Ex priore fiet
200 = = 100g: C Ex posteriore fiet 300 = 100 laf: a
et fiet 200cc + 200c. 1009 + 100 a f- 300 aa = 0
aequ. 100 9) c  - 100(9: 109 + 100 af - 100 af: aa) aa =
Ex Termino ultimo fit: 201. 201 cC +201C. 1019 +101 af - 301 au = 0
fiet 201C =  101g + 3 V(101g9 = 4. 101 af + 4. 301 aa)
2 201CC = L101g9 + = 101 gg -  101 af + 301 aa -  101g Y (0igg - 2. 10ias
+ L191g1C..
201C.1019 = - = 10199
+iorat=
-30aa
4. 3otaa
+ 10i af - 30raa
Rursus Ex Termino medio fit o = 2. 200c201 CH + 101g. 20C + 1009. 201C + 2.10101a f. - 2. too goraa
quod succedit
seu explicando 200, + 300 fit 0 = - (2.100g. 201C = 100. 10199 (4 1009. 201G + 2. 10010145.- 2. 100. 301  saf
Ergo 201C = = 10199 + 2. 101a5- 2. 301 a Vaf, : 9 = - L 109g +  (. ..)
seu (2)101g9 + 4. 101 af - 4. 301 a 1a f = (- 10199 + 9Y(.. .). Et utrinqe quadrando
+1018 - (8) 101 af gg + 8. 101301 aggat = (101 9 4. 101 a599, + 4301 aag9
+ 16 10l aaff +1 6301 a f = 32. 101. 301 aa fraf
sit gg- 4af = ah et ex aequatione hac fiet:
+810i afat +  301 aa aki = 02 101. 301 ahr a taf 0
0
seu omnibus divisis per ato, fit 105 af + 301’aa = 2.101. 3011 4f =
seu 101 4af - 301 a =0 seu 301 = 101 faf: a sed hoc non
licet, fieret enim 301: 101 = 300: 100

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du
15005 tartu

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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 110 lines, status machine
fiat 20. 100 +02 300 + 11. 100.300 = 402.  ubi 2 in fine ob duos 1 otales
10. 100 + O1. 30=8
Et 2. 20. 100. 101 +2
10. 101+01. 301 = 00, 411. (4 30101+02. 30 4 11. 101. 301 = 00, 912
scilicet permutatio interm, et.. O inhrinetalib
et aequando valores 2402. 201 + 411. 402 = 2 4012. 200 + 401. 201. 200  + 401. 4012 + 40,2 201
Ergo 200 = 12. 4  401) 201 + 411. 402 - 401. 4012,: 9012 +401. 201 qui tersins valor
si aequetur secundo, habebitur aequatio in qua 201 assurget ad gradum secundum. Sed videarum
an r deprimere liceat sed deprimere dem licet melius hoc modo: Sint duae aequationes
00. 200 + 00. 40]. 200 +004920
205 + 401. 201 + 413 20
2.00200. 201 + (929 00. 401. 200 +90. 4012 =0
ba 00. 9I1. 201
ubi aeqq. 12 oriuntur ex destructionibus terminorum
auserendo aequ. 2, a summa aequationum ieth fiet
(
sae
200 +201  2200. 201 = 4012- 402 seu 200  - 201
(5) CB -  401 +  1(401. 401 - 4. 402) sd si conjungamus aegqu. 4, 5, 6
 habemus solum duas
2
20) et 26) - 1. 411 + 1 Y(4 11. 411 - 4. 4 12)
200 + 201 vocando et 2001, fiet
se d sint interventus irratio seid a
Addendo aequ 1 et 2 et 3
20 et 4uo 4 200. C + 401. 200 4, 4: 201 + (2. 401. 411. 2011 201 = 4. 200. 201 + 4012 + 2. 2 e 
eu -401 (401. 200 +402) 4411 (411. 201 + 412) (- 401. 411 (401. 200 +411. 201 4 4012), aequ.
(D (401. 4u (200. 2101) + (20, an. 200 + 402. 412) + 4012 4. 4017 (401. 200 + 2 1. 201 + 4012)
ita res omnis reducta ad 200 ut 201. non nis
simpricem gradum facientes. Nam ordinando
401. 200 +401 402 qu. 2. 401. 411. 200 + 2. 401. 211. pu12
2. 401. 4112. 201
40B. 200 2. 401. 41. 200 + 4. 401. 912. 200 = 4. 4012. 401. 200 + 40 (. 402 = 4. 401. 4u. ton + 4 402 112 ( 4013 - ( pon
4113. 201 4912. 412
413. 201 - 2. 401. 412.201 4. 411. 903. 201 - 4. 4012. 411. 201 4 411441
compendio inde fiat 830. 200 +840. 201 +801 =0 et 690 200 1640 201 + 601 = 0
et loco aequationis 0- 200 - 201 - 1 (4012 - (402+ 412)) scibatur 200 - 201 = 152,  = 0 vel
generalius 630. 200 + 640. 201 + (60) = 0. unde fit - 200 =+540. 201 + 501, : 530 = 640. 201 +601, 63
ergo 540. 69D. 201 + 501. 630- = 530. 640. 201+ 530. 601. Et 20 = 501. 630- 530. 601, :, 530. 640 - 540. 630
Et pari jare 200 permulando. 4. et a3. fiet 200 = -501. 640 +540. 601, :. - 540. 680 +530.640
ubi jam explicationefacta et pro 630, 640, 601, ponendo +1, - 1, - X2, fiet iunde ut
200 = 501- 540152, :, - 530 -540 pt 201 = 501453015E, :. - 530-540 quos duos
C + qu (530+540)4012
6= 2. 501. 501 + 501. 5301522530. 540. 52   A (570+ 540) 501 (401 4 91) - (BB 0n 20) (210530 - 401. 340) 152 7
valores substituendo in aequatione rda, fiet
ubi magnum habevitur compendium, si destnatur 152, quid fiet
si faciamus 2.501.530 - 2. 501. 540 = (530+ 840) (411. 530 - 401. 540)
2501. 540 152
atque adeo 2. 501. 501 - 2. 530,540 (9012 - 402 - 412) = (530+540) 501 (4014 411) + qu (530+540)4017 =0
sed haec omnia forte brevius consequemur, utendo Canone ilum a me posito pro tollendis literis ex duabus
aequationibus secundi gradus  Redeamus ergo ad aequationes 1,2 et 3. Et per aequ. 2. erit
261 = - 401 . 200 - 4012,: 2. 200 +411. quo valore substituto in 0g. 3. fiet
+ 401. 401. 200. 200 + 2. 401. 4012. 200 +40121. 4012
= 2. 411. 4012
+ 4. 411. 412
Unde ex canone alibia me reperto fit.
=0 seu compendii causa 00 + 8t. 200 + 82 = 0
cui conjungende aequationem primam, quam umfor
mitatis causa sic scribemus 70. 20047. 200 +7
80. 10. 82. 82 + 70. 7. 31. 8 -10H. _ =0 ubi res plurima
+72, 12. 30. 80
71. 71. 80. 32 =71. 17. 80. 
= 20. h. 81. 82 =0 sed fortasse
2. 70. R. 80.  tantum +70. 72818
H. n. Ve. B fiet s
coptraheretur, si fieret 1B. et - 7280 =0. restaret
et optimum quidem videtur
aq ita foret 412 =  + L 401 41 - = 40i, ob 20 = 0
fatis ert ad contrahendum facere 82 vel 80 =0
82i GI = + 11 + 11.71 - 4. 70.72),: 2.10 seu 8 g = 401 + Y(401. 401 = 4. 402),: 2
401 + 3 (40i - 2409)
70. 82. 32+ 72. 81. 81 = b1. 81. 82 = 0
4012. 4012 - 4112. 401284112 (2401. 411 - 4401
2. 401. 4012 - 2. 411. 4112-401. 4112+ 4 (.3, 401. 411 -  405)
401+ S(40- 4. 402)
seu 4012 = 4112. 4012 - L 411.401 + L 411.401
Hinc aequatione plana habebitur 4012 N llideamus j 411. quae cum 4012 sola ex ha cinculum radicale est
Aia = 2. 401.9012 - 2. 411. 4012 (= 401. 413+ (2) 401. 4112 - 41. 401
bendit ad cubum, quodest sisto altius
 sed videndum an ex ipsis 402, 401, 411, , 4012 possitit inveniri quantitates 100,300, 10130
Nam 20 = (00 . 81) 461 - (10: B1)100 301 = (00. 1) 411 - (D: () 101 sid musus
et quidem plane. Et apparet rem succidere
Ex qua et unde 2. 20. 100. 101 + 2. 02.  qu (oN: 31) 401. 411 + (00. 10: 012) (101. 401 + 100. 411) + H1. 100 00: 01 ) 41 br1. 100 (10. 8,) 101
per et
nteg adhibitas ubi pro 100, 101, substiturndo valores fit relatio inter ipsas
assimulas 401, 411, 402, 4012 nam sufficientes adsunt siliceret facere
Si liceret facere 902 =0, diretur 300: 100 ex datis et fiet 200 = - 401
esset aequationi I. Hinc ex aeq. 2 fieret - 2. 401. 20) - 40, + 411. 201 + 4012. Unde fiet
201 = 4012 - 40i,:- 41 + 2401 = - 411 + Y (411 - 4. 412),: 2
ex 402 et 412 habetur
hons satis factum 300 + per 100, ita
quod si faceremus
et ficet tandem
habentur 20, et 30
101. sed praeteren
4V- 4. 412 =0 duretur haberetur 301: 101 ex datis
4012 -, 401 = 4112 + 2. 401. 41 sex A6 q0c qui q gr unde elici
poterit ratio 14 ad 101 ad 100. sed intis Haec succedent modo ne fiat 300: 100, ut 301 ad 101
vel 301 ad 100, ut 200 ad 100. Et primum si sit 402 = 0 f et
di
2; 100 = - 11 + H(1 2- 4. o 20.02), : 2. 02. porro ob 415 - 412 = 0 fiet 10. 10124 01. 30 i + 2.10. 01. 101 301 2
4. 10. 1012 +402. 305+ 4. 11. 101.301 sit 4. 20- 10 = 3 Et 4. 02 -0 = 39 Et 4 . 1= 2. 10. 01 = 2. 31. atque it
neque fiat 101ad 100
g in 31. 1012 4 33, 301 +, 313. 101. 301. unde 201. 161A = _ = 373 + YC3132 - 31. 33)
nisi egitis cesibus
100 sit 301 = 112 101:
semper in fallor niabilibus coincidit cum
bre substituto, . 000. 1  100
+2. 401. 411. fro1310. 10d. 100. 101 +2. 62. 112. 102. 100. 10141
e ea a ui is