# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068007:0008
# Permalink to this reading: https://leibnizlegible.com/page/00068007:0008?run=73
# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 10
# Folio 4v, page id 00068007:0008
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 3r); this page is its left half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068007&page=8
# Source image: https://digitale-sammlungen.gwlb.de/content/00068007/jpgs/default/00000008.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 94 lines, status machine
# Lines: 94 recognised, mean confidence 0.81
# 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.

potest etiam tolli litera, duas aequationes considecando semper ut pares dum inferior
multiplicatur, et ita depumendo
_ 10X + 11 = 0
20x + 21 =0 fit 11 20 10r t ∞0 10. 21 - 11. 22 = 0
+1130 o CtortuFio 22 - 0   M 2CX  2I = 0.
 haculm ex ot eo de  oa (e nun emuera apgiari esectinu
IOX uXFIXII=o W tuXruxtIS =O F1B2I4 1B. 2X +1023 =0et vslent tn OX 412 2X +11. 24 = 0
Eugo res deneducta est ad (10) X + ( + (12) = 0 et (20X ( 21) + (2) =0 fietque: (0) -71021 611 = 0 ut (19 - F.10 23
et (20= +10 23 (2) = f 11 23 (22) - + 12. 24. Ut ante habetur Ergo lex explicandi Et nolandum  semper primum
valorem in secunda explicatione, et altimum in prima coincidere. Hic 10, 11, 12 explicantur
prisi explicatione 20, 21,22. videamus jam an in reductione si aqu. dualus parbus
set et 20X etC. cyntiocatur larpar 10X et. et 20X et
Nempe  exdnabus hox ec etroxe
applicate sipia resultans aeque vertio siat X, erit: + 10 21 2(+ 11.22 - 378. 282  1  = 0. Hac explicale
∞10. 12. 21.21 + 10, 12. 20. 2
 + 11. 12. 20. 2
it2. 202= 12. 1. 2020  serest ne in altioribus veurrat depressio 
itaque opus depressione: Maeterea ascendetur
   etC. eodem modo so habent 10 eri,
us
c q. sirum
non necessarium plititem
Et proinde explicula  a umme timcum
10 et 21 item Eodem
iter pretemur ad  a 1ox eu = et 20X e
adeoque et valoris, et sublationis literarum
Considerandum etiam inquisitionem eois divisoris, posset absolvi sola
binl ur Termini 32.
+ 2t C -10Y n ne Et
inquisitione solis liter aequationibus, simplicibus inter se conjunctis hoc modo:
sint duae aequ. 10X + uX + 1X +13 =0 et 20X + 21X + 22X +23 20 hamm qua
quaeritur valor ipsius X simplex. id est quaeritur communis divisor qui sit 30X + 31 =0
quaeruntur ergo aequs duae, in duct duisr  3, det 297. et 2 Nempe
sit 40X + 41X+ 42 = d et  5oX +51X + 52 = 0 ita ut (3 in 4 det 1 et 3ins det 2
fit 30. 40 X3 + 30. 41X + 30. 42X
+ 21. 40 431. 41 +31. 47
10.60X + 11.60X +1260 X + 13.6020
30, 50X + 30. 51 X + 30. SIX
+ 31. 50 + 31. 51. + 31. 52
20. 70(+ 21. 70. X + 22.7cX + 25.70 =0
negligamus 10, 20, 30, 40, 50, 60, quasi esse ut 1. reperietur esse 31 - 11. 3P2 + 12. 31 - 13 = 0
et 313213F + 2. 31 ~23 = 0 sed ita nihil actum; nam hinc tollere 21 seu
ipsius 31 valorem simplicem inverire aeque difficile ac valorem simplicem ipsius X
Aatur aequatia facta
ex() X et   ia r explicationes n fet aequatio similis 
sint due aequationes ()X es
Sed rursus videndum quomodo habita aeq. c5
ailicet facta est ex  Ex g9 (6) Xnat et (7), X mar per explicationes binas
ducenda ex aqy. 1 et 2.
fit aeq. similis (B), inde per explicationes binas fit aeq. 5. Et ipsa aequatio 
habita ex aequus Xm, et (10) Xm 3 unde sit aeq. similis (11), et hunc per binas explication
sit aeg. 8. Et ita semper redmendo ad minores, donec veniatur ad x. Atque ita
Incipiama ab 10X + 1 = 0 conbinata cum alius quotcum isi aliis incipiendo  _ a 20X + 21 =.
res Analytice exhibetur. At synthetice hoc modo
Tollatur X fiet aq (3) inde interprelandoque 20 per 10. 21 - 11. 20, et 21 per 1. 22 - 12. 20, et ita porro continue, progedica
et ae habebitur aeq. liberata ab X, pro aq. X 20X etc. 20X ell
20X et. etc cum
jam procedendum ad aequ. conbinandas cum x: et quidem  hbr 2.
 proponitur 10 X + UX +12 = 0 et 20X + 21 = 0 facta est ex (Dx +  et 20X + 21
10 () ponendo 20 H- H 20 102 20. 11 - 21. 10 seu - 10. 21 + 11. 20; et pro 11 porendo 20. 12- 22. 10, seu -10. 22 +11. 20
aeq. 1. cobinatis.
Hin (x 10X + 11X+B, ect 20x+2, fent  fet aq libera ab X unde sidaretur 19. X cum X,
x eau aequala libera prodiret ex aeq. 6. per explicationes n + 1 directas, seu ipsius 20. porro
Hhic datis 10X3 + 1X L M X + 13Eet 20x + 2X + 22 0 fit 29. (9)
Rem actu exhobeamus.
 a ho per tomuro, etc. 1
 ex aeq. 5 fiet aeq. (7), explicationibus tinis indiretis, 10 pet 20. 1 - 21. 10 seu ) 1. 20- 10. 21
Sit 20X +2X+ 22X+23X +24 =0. Unde projunctis ceq1. R 6 et 8 fiet 4eq. (9) explication bus sinis directis, 20 per 10. 2 - 120, eu
10X+n = 0 20X12120,
sit 1003+ux+XtBounde
1,(=10,   XI. 20
Lionret  
(10 11 4r1. 20  - (-1B 5 11 . 2B, o herm , noitua a- Iutr. 20
110 2
-12 20
410. -u Z, F1D. 2 - M. 297 HoM-11. 20XE
Ao LLOSLIO  1Batur  -102+2 2BX -102 trro, 1o
 Fu 19t411210
si remse Es, . ( - - . - .  X t1 =0 2 Var H =0
1CXFI =O et 20XL + 2IXY 2=0
unde opicatione ua Aatvnosta Lsit equalio 10. 10. 22 - 10. 12. 20 - 10. 11. 21 + 11. 11. 20 =0 Expis qua tuilit X ex 10K + H = 0
ox nxrune mmamrduit ii ag 9 isieret exeq.  sxplica   r
et ex TeK +O. Et axpaeq. ex aeq. V explicationibus binis ipserum 10,11, fit aq(3) quo Ellil x ex
quae lohit X ex loXZuMX =oet 20X + 2X n = V
et exsqe. 3. ex plicationibus sinis ipsorum 24, 1, 22. fit. cq () quo la lit x  - - - . - ex 10 rx FrX F o et 20 4 3  1 1 , 1 BI-
ex 10X uXKE
1 per = 10. 11+ 12120 12 per - 10. 23 + 17. 20, et
(1120 2 pe 41 12. 12 20 reph Ho. 23 - 15 20 et
equares teducantur ad
 is oa e ci tu ere
10 X +140. 20x  =o. 1   0X  + 10.
10X + 11 20 2Xmx + =
10XXux 2X tuXM =o
Hg Berite