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# Piece: https://leibnizlegible.com/api/records/412/text
# Catalogue record: 412 — Ratio 1 ad 1 - 1/3 + 1/5 - 177 + etc. ad modum quaerendae communes mensurae investigata
# Incipit: + 1 - (1 - 1/3 etc) = 1/3 - 1/5 etc
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=412&katnr=57988
# Work: https://leibnizlegible.com/work/00068199
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folios: Bl. 140v (from the shelfmark LH 35, 13, 2c Bl. 140v°): 1 page, canvas 280, page ids 00068199:0280 to 00068199:0280
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=280
# Lines: 112 recognised on 1 of 1 pages, mean confidence 0.79
# 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.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
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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 140v — 00068199:0280
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000280.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 112 lines, status machine
X + M. X- M
XFU, X 2,  prodnt
seu reddendam extrahibilem, liat  X sit quo X=
q
ex _b s 4 a6, in 5 1t +9Y6
debet esse X+m, = _0D X - m = x st Xtn = sp X n= bp X+ n = 5Q. X n = +t e
st = 5)
ita XX - um in XX -nn, = De B in 7OOg COV in D44; quod libet autem exhis s,t,  C, Q es
six compositum ex duabus irrationalibus simplicibas.
inolli caeteum intelligo aequalia vel quasi, id test, aequatia,
Effici potest primum, ut sit Xt m = B00
 effici etium potest, ut Xam sit productum ex robus, quomunum dividat xon, alterum n
vel non nisi
dratione constanti differentia
Z_, possit intelligi producta
ubi A et p ente, HC et 6o quaesitie fit et B++ sh+ V 1,
quae deo habeant communem oivisorem
videamus an generaliter quaevis formuta ut
fit 1Gt + o  L (12 Y + 5V ) eti f ures oro procedit
ex uit duab. nt + 
x = bdt + cYB, sut +916= b5t + c96 + bg + 15) st6 0 - hd
(+ V
t+56
3 (X + n = 00 = Eih + Hti S + iila 1hS. fiet Bbg+ ( et = 2hta et BbsG + Bc96+ (m = HEHt5
anisut X-m = 5 sit bft + C90 - n + ba + C F)    t 5 = pYL + 456 + (p5 + qY) 746 fiet
g + cf = p5 + qV et bf. h + c98 - m = prt t. q5
X +n =5q seu
b5h+ C95 +n + (b9 +cfI YhC=
PBh+ do . + (pro +qu) 146
psh+ 9lo, Y1q+ S+ 1d
tit Vst + cg6 n = p pBh + quo s et botes  1
X - n = 2Q sin rut + 51C, v. 1E + 10. - rrt+ s1os + (ro + 51). 1E6.
unde sit bFt + c9t = n = rnG + sio 6 et. Xbg +s _ = rW + 5V
bgf.) hK q = p5. + qu  pro + qu  ru + sD
 2d: D = 5-9,:, p. h pV (5 - 91: p. r) + qB - p5 + qY
pariin ps-rq, : ptrq
vip. -r = pCps + rq; rCP)+Y4). seu
prs qudo pro duebus valoribus ut  2hkaB, inter quae postitur
H his ergo aequationibus non videtur elb difficurtas, sed vieniendum ad illas ubi explicare oportet v et 6
fit p5. V( pao +pqu ran = pps + jgr.
habemus ergo v et W absolute. Restaut duae ag9.
8 1b5 - pr) ZD = m, :, 45- cg, quae aequatio
sit H = ut 7Zt A 8 = AZ +  a + (5bf7 + Bbs da + (3
divellenia est in duas, ne tu habeatur per constantes, et fer Bbf  hh, 45- cg = bf -pY, KK = (3C9
etico rics
3b5 BCA 8 = (Bbf - hh) t + BM 3 =, XK-
Rursusque
C (b+ - pB) t +n; : qu - c9 = 15. rv) Z. n,  5u-
4-9u +cg
KK = bt BCG = prKK + pr BCg ex Bq. sit 2q B95 + KK = 2 βC
3, 95 - cg d = 1, KK βC) Eodem modo ex n, hatebitur
Exaquor sit in fit bf sid C lfc9 - p CYW + B C9B(2 b5 quo _C b fCg)  rqov + rego et ex ( 1t sit (5-quo2C
Sed ut c valores ducti ex m, et ducti et n connectantur inter se, opus est valorem ipsius. S ex aeq. 8 si placet, aequa valori
Ex sq. 9 fit 3bf95 ( β0fC9) = hhqu = tihcG.
ea
ipsius X ex aeq.  ubi et divelli aequationem in duas et sint  bt -pr, qu- c9 = t - pv, qu -cu et -M, qu- cg = +n, - c9
ex proretiot bf quo (= bs C9) - pr quo + prcg = bf q5 b feg - pa5V + pcgD et m: n  q -c9, :, cg - qu super sunt ergo
17,18, 19, 20. Ex aeq. 18 fiet per 6
2 squ (5 - 4), pstru ex 20 et (8. fit
15, 16, 17, 18, 19, 20, pons oamen D et 10 ape 5 et 6 dollantur ex
W = 3C9:, suq = nCCg. 95) + egm am in lvaim 19 pro ad   ponendio valo rest, ex 18. fiet.
194. 1= 2c
abtEgu - 2prcgq = bf95 = prcd 5 - pcgv, 5- 9. seu v = 2bfC99+ 3prC99 7 b5 95(3, 9 =p 5C95,  91 
Restan o et 20 runtenda ex 17 tollantur, ita hhabebunus gaa
literas B, b, c, f, 9, h, K, p, 9, r, 5. Vidramus an in 17 liceat facere p5 = rg.
praestandis
ita omnes calculi valde contraherentur t
sed tamen et valde mutarentur,  ita _ fieret quadratus et hunc fieret vel pa r = 0 et 5- 9 -0 ex aeq q. 3et b vel quod mallem, pro utraque
(23L= (EX5) pari P5 trg, :, pe5 - r9
iceret
una aequatio p5 +rq = 0, unde revera nihil supposstisemus. quia aepono suppositio non diminuisset numerum aequationum. Verum enim vero non didentur simul
stare pose hae aeqque p5 - rq = 0 et pstrq  nam fiet p5 = rq = - rq, ergo frt Y et quaansus fiet r920 et r vel 9 = 0 et p5 = 0 Ergo p orl q = 0, qui
tert p et 9 sint ipsis r et 5 aequel
pi ee qu pro gn. se
Ergo p5 = r9 =8, necu sune multum
Eaperior aeg. 7. manet sine mutatione sed in aeq. 8 pro pr fit pp pro 4) fit 4. id fot etiam in aeq. 9, 10, 15, 16 sed ita in aeq.  8 fit vnt
omnia non sunt permittenda. Ergo fiereto facere oporlet p = r, et 5
d 
itq utiue tutemus  et s substituem
= 0 vel 9 = 0, faciamus 9 =0 et videamus quampda horc jam Formetur calculus. fit +er. ()RK = p 2q = proxqu= po
an respective proportionales
seu KK = - 344.p 4. ex 17
to perq. a nn
sit imentica eua succedit sponte sua. aequ 1B etam jamhabetur. Ex aeq. 19 fiit Gguo- peoco = b fqupaaB et divo
bflo - p bsq - PI D. Etaqu. 20 dabit m n = 49. - qu seu m. H q: W Num ergo pautatim omnes exhris valores definiamus
A =0 r - p s=g 10 = -gn: m = Cparig reformatam) q(bf- pV): (bs- pD) fietq Cop - bf)n (b p∞) M. et fiet hinc
e
ca =0, id est v
Ex 7 fiet
1x aq. 15fiet Abf -ntra - b5β + Pp (3. Ergo
Clit = lit Y BPF, seu ( = tih + PP, : 2βf. Ex 3 fit per valorem ipsius W0 fit 264 = - 4(n: m) D + qu, seu v = 2 (2 + (2: m)) p
v. bF Cm  n) =  + PD, in ex + q 1b suit KX = (344. EX 1q. c6 funt KK. = = β.
sed sup paulo ant. v = (th + βpp _, : 22) (m +n) 3: jm +pp. fint  P + ppm 3 2( (Em +n) = hit + Bpp, m + h. m
unde habetur h per B et β. superest ex
et ex2 sit hq  t = pF BP9 = qu-  seu h = (P. V- B. unde
14.. - Cpp + Ppp, a+ a, m. Ergo p = - m ted 9 e 5 manent artihrariae
ex aequ. mmpera fit pd + ppm, 2 (3 (en +n) =
561 + Sc
ex h uvat tum 5 -  +
1 - L - E ut B 
Do neu vabo 1ra munt etam 1, - E - 5, 15 1 6, monb larm 1 2d  - E +  - E
E CE = E - 10 = - 3 = (90 4+ E- E) 2 E+ E- E
3 = - E = 30E E+ E - + = (405 - E) - 210 4 - 2 + 2 - 2
312 - E+ E -  = (313 5: ∞5) - 13 E- + +
0B3- E, invab 1 sulum 20 , +  E -  mon Geeror non (30 E, - +) = 23E-  +
2105- E, _ = (2 E, - 1) -
esolirsprant oersare Hanmumes eg vavet unenm ep. 313 + E, - 5, + E, a1 pr 1 arjore
+
u m
iu
3
56