# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068201:0417
# Permalink to this reading: https://leibnizlegible.com/page/00068201:0417?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 209r, page id 00068201:0417
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 210v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068201&page=417
# Source image: https://digitale-sammlungen.gwlb.de/content/00068201/jpgs/default/00000417.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 86 lines, status machine
# Lines: 86 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.

 satis agnosco aequationes pares reduci posse ad impares proxime inferiores. Sed nondum rationem id
praestandi in altioribus ostendit quisquam, sit aequatio quadrato (ubica.  + PX + 9X + 1X + 5X +  quam
multiplicando p ejus ra dices per   13 b, fiet: B X 
10 1
s
lX abse bl X 1qb +OX  bl X +st (t LX, H
10V W pso b1 V  10 b  + X, sObP. .  st VI H +H Obl V + NOI 1 sive
b. X + pb eb1. _ r 9b, b1 Y7 , Yi H+ s11  bL, H = Hls∞ bV. _ + ∞t 10. ego e Yn.
Tentemus jam efficere ut ex hac aequatione extrahi possit radi quadratica; interventu aequationis surde solidae.
sit scilicet OH b _ + c_ + d Y + e. et D ⊓ fH3 +gY + h_ + l qua = Y  o erit
et  D2 O - Y cujus equationis num latus, nempe dinitrum eAt quadratum, superest, ut dextrum quoque
quadratum reddatur. Nempe quantitas O = aequanda est quadratio, ex ciletur qntitus  O + D, quod erit 8 + 207 
et sit (6) - = ( (9) + 2DO + D  et explicando, trminosque respondentes sibi subscribendo, ita stabit:
D2( f~6 + 2F9_5 +25 _ yt V = gl3 + 2hlN + 1
nI 
92.
291 + h
2007 2bf1 2c+. 2d f. ref. 2eg. 2ch el
2b9.
2c9. 2d9. 28h.
2bh. 24h. 2cl.
de
 = V p - pb. v. 1X at pr. Y -lLy -p spbr s 11 g ry - 1
primum termines primis, f+bn0 ex terminis ultinus l+ er o.
Ex terminis secundis, - 2ct - pbs b1 Ex terminis penultimis +2d t s b n vurte quantitatum destructa
Ex terminis tertiis. 9 2dB + 2c9H - 4b 13b
Ex terminis antegenultimis: n + 2d h + 2 el  - sl +O bl
Sunt hic tria aequationum paria et respondent =c +b - g
dupt
d +l
+h
Ex  duabus postertonvus primi nedinis, fiet:
CA P1b1 et dii 3 + 2p 9 1661 + 4b 18)b12
Ex duabus posterioribus secundi ordinis, fiet:
de - + s0BV  -c n st 15t + h 4 rth s0b 1 + sl. 10b1
Acquemus duos valores tum is siu uc, tum ipsius d, ex pi fiet:
=pl (0bl  h - 21h 13B1 + 5l10 b.1
- +b 13b1 B g + 2 p9 1Bb1  9b + b1
 Ex senis fiet:
Superest t aequatio novissima ex terminis mediis, ubi nutandum, qua f+bH0. et C + fB, quantitatem
et
hanc 2fl + 20f + 2b1 trilaciter explicari pesse. Nam si adhibeamus  51 brjo. fiet 22es H substitaamus
sclicet vutas, frei. (2bl ∞ 21b) + 2bI B 2bl  = Al +2C+ + 2bl. Et aequatio ita stabit:
29h - +9 1(3bV + ph (0Bl + 2b1 p r b1 Tres ergo habetmus aequationes concinna satis ratiime concoruentes
et pro iis ae indeterminatas: h. g. b. l. Et quanquam non dubitem elegantem quandam in natura rerum
latere constructionem, Conicarum fortasse superficierum ese, quoniam tamen doctrina de locis ad superficiem nondum satis excussa
est, cogemur via commini incedere, ac primum ex novissima aequatione, gF h BF
ex prima trium novisimarum
TI
-p1 13) b,  29+ p 15) b1
rbl + +9 1(5)b7  s0
s
29 + p s0b2t
4+ 2t s0 bl pui - p1 13621, 2 2g rp 19b21, + 2+ sB bl a rH +267 b1
+r bI + tg 136
Quos duos absolutos ipsius h valores aequande.  ac per crucem multiplicando habebimus equationem:
r BB + 2rbtg 10b1 + F5 b10b = 5Ob 15 49 + agp1 b1+ pLVCBV, t s agrbl + 2tp h rbl tatgul tgl ps
AY
.
Habemus ergo aequationem ubi h non amplus extis et 9 non ascendit ultra quadratum. Porro pro 9 quadrato substituamus ejus
valorem ex secunda trium aequationum novissimarum, habebimus valurem purum ipsius g. Nempre fiet:
+4.
G1 X bY - +6 (5C) bL - + bg bl 1(9)b L, pV_4. =+bb p b Hl 3- 2t pbl rbl 44tb1, - +b 16612
=ir bl + sbl, + Lb ap b1, - p La, -rp. b. B + apo  + 2 +. Dbl 3 r b v - 4 F bl 2p 1(2) bl + 2F l p. B) bl
Unde sine calulo ulteriore statim ipsius h valor es efformari potest, servatis tantum substitutionibus upra notatis. Porro si bl sumanus pro certa
hpaL +B
Quare compendii carsa scribamus
B7B3+ L. et
. 4 = 9p AB+ . . jam vero invenimus hp er H et etgea ubi pro, 9 substituendo ejus valorem
VetS e+ BB+ 5
VB2B3+ 0L + s
sed viteo id non procedere, praeterquam enim quod B ascenderet fiet hps +2bi rxpreo   L +B
t quod scitiret
 
2cAB re n EpsOo B, EB rov LH yL + SP+BB +sI
Eui malo remedium haberi forte posse, est et anud nutum needestruendo d vel D. ope arbitririae BL) praeter hoc inqua
quantitatibus affectae; adeoque ope assumtae BL. non potest iuveniri pure B+L, quod tamen necessarium
Bud matum quod scilicet _ B. et L. non sunt iisiem
erut. Rectios ergo faciemus L esse multiplam ipsius B, et omnia
diridi pobunt pe Binempe: GFum 12 1E.  3. sic 8H _ x et BB. at h _ B  BC + 0 B ubi coanescit B, quare sic quoque nihil agitur, 
_ et eommet
16
Unde jam terminos terminis conserendo,
habebimus aequationes sequentes:
unde plurimum deminecetur cacul
na in sequent. b