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# Piece: https://leibnizlegible.com/api/records/9862/text
# Catalogue record: 9862 — für Otto Mencke zu Johann Christoph Sturm: G.G.L. De commensurabilitate et incommensurabilitate ex seriebus infinitis colligenda, occasione
# Incipit: Ignota mihi fuit objectio Sturmiana extans p. 181 Matheseos enucleatae
# Date: 1693
# Sender: Leibniz
# Akademie-Ausgabe: AA II,2 N. 252
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=9862&katnr=40765
# AA II,2: its series at Leibniz-Forschungsstelle Münster (online after accepting its terms): https://www.uni-muenster.de/Leibniz/seite5.html
# Work: https://leibnizlegible.com/work/00068101
# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 6
# Folios: Bl. 1 (from the shelfmark LH 35, 8, 6 Bl. 1): 2 pages, canvases 1–2, page ids 00068101:0001 to 00068101:0002
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068101&page=1
# Lines: 119 recognised on 2 of 2 pages, mean confidence 0.88
# 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 1r — 00068101:0001
# Source image: https://digitale-sammlungen.gwlb.de/content/00068101/jpgs/default/00000001.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 99 lines, status machine
nisa abganus
G. G. L De commensurabilitate et incommensurabilitate, ex seriebus
infinitis inf celligenda, occasione respansio objectionis V. Cl. Joh. chr. sturiii
Ignota mihi fuit objectio sturmiana extans p. 181. Matheseos enucleata
donec a Celeberrimo Guglielmino per communem amicum doctissimum et
fficii plenissimum, Autonium Magliabecchium indicaretur, Dixeram ego
Quad rato circumscripto existente 1.
Circulum fore 1 -  + 1 = 1 + 1. - 1 etc.
hic celeterrimus sturmius, subdubitans tamen), objicit: posse videri
hanc summam esse uno numero effabilem, quandoquidem partes omnes sint
ractiones ad communem denominatorem revocabiles. Et tamen notas
me ipsum fateri circulum non esse commensurabilem quadrato. Idque ipse
etiam ex eo probari putat, quod circuli area possit concipi, ut summa infini–
tarum ordinatarum irrationalium. . Ad haec respondeo duo: unum
principaliter, alteram incidenter, et si a duo assumsisse virum
egregium quae non sunt admittenda; primo omne aggregatum termi
multitudia
secundo
e norum rationalium numero infinitorum esse rationale
omne aggregatum terminerum numero infinitorum irrationalium
multituumero infinitorum esse irrationale. Utrumque instantias reftdmi
it possum dane seriem infinidam numerorum
ratio. Nam ver paimum, quia dari possunt series infinite rationales
contrarias patitur; nam
cuivis numero etamfinito irrationali dato aequales exempli causa si
quadrati latus sit 1. poterit diagoaii valor V3, exprimi tali serie:
13
1.2.
seu 12 = 3 - 1 + B
8
empe
3.5
ubi si series continuetur aliquo usque, erroram i
1. 3.5.7
l. 2. 34.5;32
siest
minor proximo termino, seu primo omissorum. alternatum
justo major vel justo minor; et ita continuata serie
-
pinor dato quovis ut experienti patebi
Adeoque toba
muvum fructiones
flicissim dam possunt infinita ordinatae irra
series rationalium aequatur sihi quantitati irrationali ineque
errorerit
e
13. 5. 7. 8 E
ia
1. 2. 3. 4. 5. 6; 64
tionales componentes aveam rationalem. Exemplo sit
parabola cujus ordin parametro existente rationali, et partibus
axis assintis qundlucnlibet parbo intervullo differ infinite arvo intervalla
semper aequali differentibus, tunc ordinatae regulariter (hoc est certis
eix abscissis
tantum exceptis) erunt irrationales et tamen exiis compenetur area
rationalis, quippe quae constat duabus tertiis rectanguli citcumscipti
ordinata, sit suae abscissae commen
locjum habet
itaque mallem suir doctissimus et humanissimus
si ultima
surabilis idemie infinitis iatias parabolociditu aliisque siguris
posito, quod erit rationale
mihi dubitationem suam ante editionem communi casset
actias quam rectoribus,  ista non satis in haec mustepia
theorempte
admissis sequibus a sequius aliquid de meo demonstratione
quamquam illadatot maguis
suspicandi occasionem probuisset
Gedmetris probatum, extra aleam positinsit et praeterea a dimio
olannam veritus sintemae Hovomalis calculo lon atque indubitata
monstratione nitatur qde qua duarum Regiarum societatum Mathematicis
insignibus satis constat. praetereat a dontinis 0 sannam calculo longissime
experimento
satis lonse producto per experimentum nem
comprobarit
e
b
2
2
l. 2.3.
16
rec adlicet termit
continue addat.
revocando ffactiones
ad communem dec
nominatorem,
totam seniem
unquam absolves
an nec proinde
colligis hac ra-
tiones potest
totam seriem
esse rationalem
etsi a rationit
quantitate
ad eam quan-
tumvis accedi
possit
ere

## Folio 1v — 00068101:0002
# Source image: https://digitale-sammlungen.gwlb.de/content/00068101/jpgs/default/00000002.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 20 lines, status machine
Libenter tamen excuso virum doctissimum, in re praesertim
tam perplexa cum Analysis infinitiau non dum satis
doctrina infiniti nondum satis explicata publice extet. Et simile quid
contigisse video ei qui relationem de Geometria practica domini
Osannam insernit Diario Gallico insertam composuit. Is enim
cum retulisset quae de theoremate meo _d. Osannam dissimulato
licet nomine met) cujus tamem et fhesremet e
ille s+ inseruerat. subs
nanemst inserverat suo inic libro, subirt hinc consta
t
o
posse etiam pertinacissimo cuivis circulum quadrato esse
diametri, esse commensurabilem. Quod tamen minime sequis ex
dictis manifestum esta
e
X
ea
et
ui
jo