# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/175906937X:0059
# Permalink to this reading: https://leibnizlegible.com/page/175906937X:0059?run=74
# Title: Ps. 2 (shelfmarks Leibn. Marg. 178, Stück 2; Leibn. Marg. 178)
# Folio 35, page id 175906937X:0059
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=175906937X&page=59
# Source image: https://digitale-sammlungen.gwlb.de/content/175906937X/jpgs/default/00000059.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 69 lines, status machine
# Lines: 69 recognised, mean confidence 0.80
# 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.

1 + 2 a AbC
ex dirio.
led: 1, d. b,
viac +b
. ab + bb,
c + abbc.
C3 46=
e
cet
tiris
IC, Quam
e
cum iptis
cyjulque
red I
eb
e
: ipluus
-
lorum ope
C ad inre
et
DuS
bInOI
de adhiberi
eriandum
acor dirio
 1 dl19
adhiber
CliBNOIS
licali
NIA THESE OS IIHLVERSALTS.
dicali. Uc quia incer divilores quancicaris a3 b. + a ab b reperitur
quadratum a a, pocerir V a b. + a a bb, diridendo per a as reduci
ad a V ab L bb.
Sic EE cum a b - a a bb L 2 a abc = abcc = a b = bbCC
2 b3C-F b+ pro dirisore habeat quoque quadratum a a L 2 a6 +
CC =- 2 a b = 2bc + bb, POterIEpEO
V a3 b - a abb + 2 a ab c F abcc - a b3 - = bbC6 -= 2 b3c + b0
Ecribi a + c = b V ab F bb. Similiter cum numerus 75 incer di-
rilores quoque habeat quadracum numerum 25, reduci poterit
V75 aa ad5 a V 3. Ica _, quia x200 diridi potest per numeros
Quadratos +, 16, 25, 100, et 400; Poterit proV 1200a a bb
Ccribi 2 ab V 300 vel 4 abV 75, rel5 abV 45, vel r0abV 12.
vel denique 20 ab V 3.
Ouod si inter dirisores praeter unitarem quadratum nullum
aut cubus ec. reperiatur, non poterit data quantitas praecedenti
modo reduci, nisi relis eam in formam fractionis delignare. LIF
quia 20 praeterunitarem quadracum nullum inter dirilores ad
miccit, poterit V10aa, diridendo 10 per aliquod quadratum.
uElubet, ut 4, 25, 200, dc. denotari hoc pacto: 2 a V ?, vel
5 aV =, Vel 10aV 163_C.
Sciendum denique, quod, licet hae quancitates omnes per le
consideratae surdae existane, tamen inter se collarae duorum sint
generum: aliae enim dicuntur Commensurabiles seu COImuNi-
cantes; alia vero Incommensurabiles seu non Communicantes.
Communicantes lune, qua attinitarem habences cum ouanti-
catibusrationalibus, aut etiam numeris, inTeF SC SUINEUE quantICaS
racionalis ad quanticatem rarionalem, Peu licut numerus ad nu-
MerUIm.
Non Communicantes vero sune, quarum unius ad alteram re
latio non est ut quantitatis rationalis ad quanticacem rationaleim,
QHE nUIeri ad NUIerUI.
Racio aurem dignoscendi communicantes a non communican-
cibus est, si, postquam ad simplicissimos terminos lune reducta,
reveriantur inter le ese ut ouantitas rationalis ad OuanLIEAEEM Fa
cionalem, aUE numerUS ad HUICFUM. VEV 75 a a X V 27 a a tunt
Communicantes, quia dirisione per V 3, maximum earum com-
MII-