# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068197:0065
# Permalink to this reading: https://leibnizlegible.com/page/00068197:0065?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folio 33r, page id 00068197:0065
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 34v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=65
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000065.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 90 lines, status machine
# Lines: 90 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.

Quaeritur p dx LCxx - bbD, Xx- cc pro dradratura conica Rationali
sit x = f stv + aco = Ast compendio fiet XX, obb =  a a N) - bbme, et Xx -cc = aa A,2- ee mei
Be
Ver+ V
 rabb, in ( -ec  a A  aa MA ep + Hbccpp, e
aaCC
ee
Efficiendum est ut Numorator silere hujus valoris efi quem poductum habet sit extrasibilis
Et primum fie videamus in possimus reddere ex rationalem deinde an extrahibilem
(4frvaco +40fc3G (++V)
+abbcc tv + 2p3) V
Numerator explicatus ent B B a co +b aa sft vedco)
+2fc Atv: n compendio ast + 3tv: eu
bba ttob6bbcc  
+fFF∞ Faacoco
Et explicando XX - bb = - lib t v - bb V
= 2bb1 s+V
+2afio stn,: n
concendio a a + Hv: r
quadvatice
XX - cc = +ff + v +aacoco
Efficiendum est ut productum ex ipsis ex - bb et AX- cc, habeat Numeratorem  eus extra-
- 2cc stv
-cc tv = cc 24
hibilem, uti denominator est extrahibilis. Et quo facilius obtineamus extrahibilitatem, tentemus
an obtineri possit si faciamus hunc umeratorem rationalem, seu ut in producto evanescat
irrationalitas. Qued ptaestabimus, si  toefficientes partiun irrationaliu sit sibi
ippositer ser silite. Pasu  x  atIp seu ase C (
Exclerquea teductum ade b productum autem est, aa XB ac 5o3 tZ)tp + a  7 A1V. itaque
Unde prodeunt
+a 7
uit tollatur irrationalis fidt fiat aa 3 + a  Av  ~ ag_ gH
duae aegrationes aa a 3 = gs et 2a  = -297, et produtum enit , aa- 4 3 ) t V 
ob a 3 = gse fiet (5f tv + aagi co) (8- a) = g (9b8 - ac) + v + 1p y)
Eob a 1 = -92 fiit - afgeo + bbgs p = aa fco - ac P. per hanc aequatiorem fit 2p =  afio (at): ste c
Undo rx priore aequ. fiet fet ((9bs ace) + (a -g 5 tB = (g. aua
(acC - gbb + (9 - a) ff) t V = ((a-9) au + aa f qusat 4) coco
qu (9bb+ acc 4
Sed non potest tv esse in ratione constante ad coco, quia n posito esse simplices,
ut hic supponitur, forent etiam in ratione constante inter se, et Atv mutaretur
in rationalem, quoad Variantes, quod non licet non succedere aliunde constat. imox fi constans
Ergo si velimus tollere ration irrationale consi in producto altius assurgendum est,
et  Et cum productum sit ex a s +2d Atx in a3 +2 AV sit aa 5 H 477 = hG
aa N1 + 4 2Y +v + Za N 7 A Fn, oportebit esse  7Z = - 5 Y
et auiso ++
compendio f f tV + aacoco vocetur aeg et L v - ip  = aY, fiet (  tago - aaque lbY
a S =  aaqu bbr et a B = aaq = cer et fiet (aaq - cer) (afco - bb1) = -(aaq bbr) (afco - ce 
+ 2a43
seu fiet  afBay = aabb. 1P9 - acc fcY + bbcc r Y - - a 7 co9 + aac clsq + abbfcor - bt cc 21
fit anfbon uad t at r:
n
seu 2 a fcog f 2bbcer ip = aa (bb + C) 19 + a C bb + Cc) F cor et explicando
22ftva +2afco tbbC et vip - 2aabb cep = a bbtCC, afftvi + a cocoast- pu
seu _ bb + cc  (bb + CC) H  2bbCd)  D WN + 2bbCC 13 + aa cocor =0
(1bb + cc) af - 2 a 5)  Co +B - 2 a25 c03 = af(lbt ce) c 1 1p
aco - se si esset bbcc ad f ut f ad _ et (bb + CC) a5 -  2 bbCC ad (bb + CC) a f. - 2a, ut fad
posset dividi haec squatio per fV - aco. Et foret. Ebbec = aaf seu +bc = af, sed rursus
asbe
ebb + ec a 2bC
Daa (bbt ec) bc = 2635
+ (bb +cC) abc - 2bbcC a = (bb +CC) Abbcc: a = 2 VC: a
et porendo b + esse et fieret a = bb ce er posito + ~ + sit a qu( ba C) = a bc, a a (bb+CC) - 2bb cc
Cui aequationi satisfit porendo aa = be, fieret enim aa qu (b -c) = BC qu (b = c) seu an = bc- Unde
ta. Et divide Ent erguus 2bbcc = 2af = df = ga seu 9 - 2bb,  2a35, : f- a = 2a -2a, a (1.)
2 aa far s) ( F C0) F (bb +c - 2bc) ap ) txer dividerdo totum per co, fut
= 2a Et Ablu CC - 2bcb a a =  h(a - si) fit h = 2 qu b -c,: a- 5
vioeo re non bene sucredere, sive enm ponas f= a, sive = - a non potest in omnibus tribus
2 (bb. + cc - 2bc)
termlnis quaesitum obtineri, ut sinunt sit divisio ubique fieri possit per +co, aliculi eum fet
sit f- - a fete
(bl + cc) Wo
_ a purci) a teset
quoo si non succedit haec depressio, dificls s vix est ut spetari
per e+ lo, aliti per _- co
possitus rem  praestari posse. Habemus prostandam unem aequationem 3ti gradus
et irrationalem tollendam, et unam oti ob iationales habenq antem
junde sunt aequationes destractititiae +g = C, h sunt anteque assumtae
emnt E, isco, B. E
C v L, B, e, c0, et ob ti tundi grades, rurtus, es hn 10. bue 1 
explican per B, p lossuntem
uo
aa 52+ 4 7 Av
Aatai 7C
a 2B
a 2b +a
aa utac rad + 2ab
F + bd + cd
19 - co 1 _coro
Aat C)Co
cop +