# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/438/text
# Catalogue record: 438 — [Aufz. zur Ellipse]
# Incipit: 2ax - [a/q]xx Sit Ellipsis APΠ cuius axis AΠ; Centrum in eius medio
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=438&katnr=58014
# Work: https://leibnizlegible.com/work/00068199
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folios: Bl. 172 (from the shelfmark LH 35, 13, 2c Bl. 172): 2 pages, canvases 343–344, page ids 00068199:0343 to 00068199:0344
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=343
# Lines: 111 recognised on 2 of 2 pages, mean confidence 0.75
# 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.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# 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 172r — 00068199:0343
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000343.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 77 lines, status machine
ydy = adx - 4 X dx
dx = ydy
a x
XX - 29X +94=
 yy +49, et
q qux = 
qy Lgy
seu
X= q f99
yy
y= R2aX = a XX
sit Ellipsis A BF cujus axis AI. Centrum
in ejus medio C. duo Foci S, F
FL semilabus rectum a AC, q
SL + LF = A 24 Ergo SL = ge29- a
FF Et SL2 = LF = SF
V
= 499 4gataa aa
ergo SF = 2199 - a9 vel si SF sit f
fiet 49q - s,: 49 = a seu a = 4 - (: 49)
seu differentia 242a = f: 29, seu differentia inter latus
traus versum est recte seu distantia focorum est media
proportionalis inter differentiam lateris transpersi et
o recti ipsumqus latus transversum vel latus rectum
truusverium
es differentia inter latus rectum, et ipsi nte
latus transversum pilateri transverso, distantiaque focorum tertiam propor
et
et ax= a-as ationacem.
Honale
strea AI
et . ax = a 1 21) Area in Ellipsi est sit AB, X et PB, y, fiet
Vaa = qy
sx
AdB
yy - 2aX - = Xk eritque area ABP = (rax-xx dx
SA
sut s qu- 2x Etu A5
sa est
ACA SC = 1
qu yf+sy dx, et quia ydus= ax quax quaereus  per y
XX . 29X +99 = 4 94 - 2 HH fit X = 9 - Y9q - 9 H)
et x = a - 3gg - 9 xX = a - Yaa = = XX)
ergo ydque = (adx = adX H raa -  Hy Ax seu de- Idy: saa-  yy
Angulus exprimatur ASB exprinatur per rationem SB ad. BP, seu
4+
S
Area vero
2 (9+ =
qtas seu SA vocetur 9 fiat 9 - X: y= Z: a et E(g - X) y + sydx =coa
quaeritur relatio inter Z et co. Tollatur y fiet y - X: Trax- A xx = Z: a
et L(9. X) rap -  Xx + (Scax. a XX c) = coa prioce mod. Ex priore
aequatione habetur
2. fel enim g9- 29X + XX :, 2a X - a xx = ZZ: aa
ZaX -  XX = g. X, a: Z p
et
fint Z(99- 29X +XX) a. Z) + seXa: Z dxida
sed (g- X)) aa = 2 aX ZZ = a XX Z
+ggaa seu XX - (daa +2
la  2Z
aa F a
(l Cait a 7417
e
X
sen aaXX+ a ZZ XX = 2aa9X - 2427
+
paq
sen XX - 2 a c 
aa ta
aagu aat qu
aae (4 au + 4 2
aa +.44
s
aat
aata aatra
(l lang + a35) daages an
1

## Folio 172v — 00068199:0344
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000344.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 34 lines, status machine
et sit LK
AF, CA = q
C F-
ALF = 2p. AF - X A.
 AF q _b
2Y4 = 2Yb AB X B)
2YX = ZXX = yy
F P = Y(Gy + AF- AB
e
J. b)X
Sp2 = G - x + b + 2YX - FXY
seu  Sp = 2YX - X XX + q9 + 29b + bb + X+ - 2(9+ b) x
et
SP + PF = 24. 
XSp = 2YX - ZXx + 4q + 29b + bb + xx C E(GtbX 24X -2bc
+ PFP = 2YX - X XX + 9q = 2 9b + bb + XX
1442
seu Fp = gy + g - b -X seu 2rX - Y XX + 49 + 29b + bbt Xx E qa
u
=29X + 26x
225p. PF =  (XF (1 - Y) + (2Y - 29) x + g9+b) - (2) 2bx + 29b
ex Lut
2Y ( (ex + 29X + 49 +b, 4) - (4bb (X
L(9- X)y + sydx = coa. posito yg = 2aX - a xx
it 92 + Zf et f = 2920, 29
vE rax - 2 xx = XZ XxZZ: aa fiet 2a - a X = X
et 29 - X = XAZ qXZ
et dx et y = XZ: a =. 2009Z:, 9e2 + a et dx =
et X 2aaq. aa t2
et ydx = -B  9 Z2 dZ: (2972 + a
AT, F AA, 29 CV. Vh
atu
209: 972ta
aq. 292 . qua