# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00067993:0042
# Permalink to this reading: https://leibnizlegible.com/page/00067993:0042?run=39
# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 21v, page id 00067993:0042
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 22r); this page is its left half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00067993&page=42
# Source image: https://digitale-sammlungen.gwlb.de/content/00067993/jpgs/default/00000042.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 39, 2026-08-27T09:55:26Z — 82 lines, status machine
# Lines: 82 recognised, mean confidence 0.90
# 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.

N. au lieu de
L s faut
la suite.
Unde intelligi poterit, non aliter  obtinere, quam quatenus in hanc resolvi
potest. Possem differentiam multis illustribusque exemplis ostendere d si
locus pateretur. Sed malim id e lectores suo potius experimento discere
quam meis verbis, sentient autem quantus inveniendie campus pateat, ubi
hoc unum recte perceperint, figuram curvilineam omnem nihil aliud quam
polygonum laterum numero infinitorum, magnitudine infinite parvorum esse
quodsi Cavalerius, imo ipse Cartesius satis considerassent majora dedissent
aut  sperassent, qn
Hactenus figuras Analyticas simplices generaliter tracta vimus, ipsa
demonstrationis nostrae generalitate invitati: ad quadraturam vero Circuli
Arithmeticam non nisi Paraboloidibus opus habemus, imo ex toto paraboloidum
ordine, non nisi illis quae rationales sunt. Quarum naturam generalem utile
erit hac propositione complecti
PROPNITIO XXIV.
 in Trilineo  Paraboloidis rationalis ABCA, fig. 6
abscissa posita ADB, ordinata eBCC, Parametro AP
valor ordinata aod Parabolam
Bd
BoC aequalit
Quadraticam
(2) AB
1) A P
Cubicam
BI AB
(12) AP
quadratoa qua
1 AB. etc.
B) AP
it enim Nam exosatin a blolidm eratiahalia
Namsi 1B, C aequal. (t AB vel   1. B
et 3B2C aequal
erit 1BDC ut
ad 2B.C  ad
Cd AP
B AP
( A2B vel 13) A2B
B AP
I AP
1 A.B  vel ut _ A. B
B A. B ad () A. B
erit
Ergo Ordinatae ut
Qvadrata vel cubi abscissarum, etc. adeoque ex definitione
erunt ad Parabolam quadraticam, cubicam, etc. Sive ad Paraboloidem
rationalem in universum.
P20 50SI 7 i0 X ∞Y2
uisdem positis portioni Trilineae ABC A sig Earea valebit.
in Parabola
ps AB ∞ B AB, B AB etc
L) AP) a +B AP sH AP
Nimirum in Parabolae Quadraticae, area A2B2C, CA, seu, summa rectarum B. C
Trilineo
B. C, vel rectarum _A, B. D A= B gvel summa omnium quadratorum ab A. B. A2 B
trienti
 AP
B) AP
divisa per rectam constantem sive parametrum AD aequabitur Tubi ab ultima, abscissa
A. B, fer Param etrum divise,   A. B, et
i Paraboloidibus
ita de saeteris. Nam ex prop. 2B. spatium ALB, CFA est ad rectangulum
A2B2C, a ut co ad cote. sunt autem hoc loco ordinata in ratio ne duplicata,
triplicata, aut quadmeplicata, etc. abscissarum, ergo ordinatarum exponens eo hoc
toto, est 1. ad abscissarum,, 2 vel 3 vel 4 etc. Ergo nea spatium est ad rectangut
ut YL el 1 ad 1+ 2 vel 143 vel 1+4, etc. seu spatium est rectanguli
pars tertia, quarta, quinta, etc. Acat Cumque ordinata 2B2C sit
1) A1B. res. B AB  etc (prop. 24) eit rectangulum icta B2C, ex ordinala
F AP a () AP
31 AP 4 F) AP 5 ()AP
Ex iis quae hadenus de figurarum Analyticarum  simplicium quadraturi
diximus, poteramus hoc solo theoremate contenti esse adeotu rl quaata
praesertim cum id ap ex aliisquoque scriptoribus possit hauriri: si praeter quadraturam
circuli Arithmeticam nihil dare constituissemus; sed quoniam tot alia ex eodem fonte
rectanguli triplum vel quadruplum vel quituplum etc. erit TAMB (B A. B. L A. B. ex
Quod asserebatur. .
Scholium
A 1B2C in abs cissam A ArB eritR W A 1B  A 2B etc. et spatium A2b. be)
1) AY
et ainsi tousjours dans
() AP