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# Piece: https://leibnizlegible.com/api/records/67886/text
# Catalogue record: 67886 — Theorema memorabile
# Incipit: [Fig. & Formeln]
# Date: 1674
# Akademie-Ausgabe: AA VII,7 N. 018
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=67886&katnr=Cc2+849%2F2
# AA VII,7: the volume in the Repositorium des Leibniz-Archivs (CC BY-NC 4.0): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-bd-vii-7
# In the Akademie-Ausgabe: AA VII,7, p. 247 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/LAA-BdVII7.pdf#page=247
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 167–168 (from the shelfmark LH 35, 12, 2 Bl. 167-168): 4 pages, canvases 333–336, page ids 00068195:0333 to 00068195:0336
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=333
# Lines: 307 recognised on 4 of 4 pages, mean confidence 0.73
# 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 167r — 00068195:0333
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 168v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000333.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 78 lines, status machine
CX N X, YX s. LA  AB, CZ
Notabilis haec solutio, ob
CB5 ABY
por
DY FIAB+ rCE
X R et
V Xa.
item 2X 27
VE
X
FX ASyY
es. 25 2
2sA
ex
ubi minime didebatur
sum im Methodi de
maximis et Minimis, necessarum
VNDXa, Y SXE = XE
ap
A
sve XSyY
XC. 1Z
X+ XC
X x.
X5ESZ
22
D
e 50
7
seu
sZr  ZY + + XEr
 Ser B 2+ X+ rtC
s unde XE 2 +  + XC F Z+X+ X
e
er
unde tanden XC p 47X
 - ∞6 Y yX + XY + XEF
Attera aequalio est: XC + AE  Xa, inde XE 
lide XE FL 
 Ergo 1ZX H X a 
sive 47 + Y, a. sive Z ( a - Y
In parabola distantiam
2
q
a vertice in axe sumtam,
Thcorcma est memorabile, abscissum Riametri aequari secundariae primaritu aequari, quarta
parti excessus  latu rectius praris superes latus rectum sisecundarm Unde
consequitur seuper latus rectum primarium secundario quolibet mus
esse
lentandus est calculus generalis, pro omnibus sectionibus conicis. Ac primum consta generaliter
in omni sectione Conica, quis sit valor ipsius LB, vel etiam ipsius BH si ponatur CH perperi
dicularis ad curvam in C. Nam ostensum saepius si CB5 2 24 (∞Y)S 44
esse BH A s si CB(d 2bZ + b ∞ esse BH p, b + b 7.
r1 
sive
Eodem modo in omni sectione conica  CX ⊓ LBH
v
posita C.  X erit FX, x +b ZX
unds fiet: posites X EAC 1 2b2 qt
6r e + 2b17C+
F
B)
VE A bXC +b Z XE
u
forro YE + EB, 1 e AB(), C+ (2)+X
Ad
b4
5n02
 A+gt +bg0 + 2be 4. 2l. ge p 2b2+ 2bX + 2be 4 12 f2bysa tit  BX,
32+  ac demius im Chiplicat omnibus per 267 + 6Ze, et destunctis utrinque:  . . . . (F 2bxC +be
jam YE + XC fet lod Etx a 2X + 22=+ X, 2Z + Z (quod in Parabuladaret e B +X
ponatur jam
in et ZX  Prax FaBx ponatur. Ergo.  b + 2bt2 + C
1242
 i  f a 2  .
a
Ders
acqu

## Folio 167v — 00068195:0334
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 168r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000334.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 64 lines, status machine
sx
raX a as
DaX au
sepria Liby - v - Ely - l +H 1 Sray + ay + g. eti
lur A B applicata circuli quaesitis A, appurcata unt sectionis Conicae
datae. g. excessus ipsius A, super B.
Notandam autem si B est applicata circuli, b+ h, aequari ejus radio.
quod ita demonstro: ponatur V +l Z et fiet B,, 2bZ - y0 s
poncndo B r X, habebimus ab7 - Za + h F XY vel 2b3-3 b ∞ X -h  b2
Et ponendo Ct + b +Z et I Y, fier v B x 1 be vel
vel 6 Z= 2tZ F b + h  X  Vel +b +Z5
Ergo b+ h B a.
jam resumta aequatione Be A + g, quadrataque habebimus BF) A + g + 2Ag
vel B A= g r 2A9, et ordinando expereandoque
-. 
2b V + 2bl
Fit
q S ra p)a a  Y
82 +9 man ca)
1
+9, eran 2a ca, liX et
2a
1 N 2A9 vel O5 Dy
+
en
- gu
Et quadrando
ory 2077 -38 7
D
+49aga 
 san esto data aquatio cubica 
in
ap ar  a, termines
omres habens completos:, abtollatur ad quadraticam, multiplicando
per incognitam, fietque 
qua ut patet terminus ultimus deest. eritus Q5 o. et omnia per Y multiplicata.
deoque per _ 4 tum terminum habebitur g p =aY valor purus ipsius g
b erit si  P + l + a per terminum secundum
Sed inde vemueiuu
d a Tocder, quia ut termino tertio valor pujus ipsius d substitiatu
habebitur valor ipsius g bis, quod esse non debet.
Y ul pro
substituamus 9, 4+ l caeteris omnibus retentis.
Sed nec res procedet si ponatur e vatere aliquid quua habet et D nihl b habebitur
exia bis valor absolutus ipsius
Eix frd 1 9r d bBi) =   (2y) b e) ragnt rah fas trahys +a sv
y aty. 2a y-
+rat.. 1a 4 0y4 10D4 204y 207y+
r2C.
318 C
sabugetmr idu lim D.
+62
(496
+49 -+9 G, + 4g
er ninitur er l, ope termnri seaudi
se temmunus aedimus desit er semndus si pro Termino secundo sub sti tuatur
8 ejus valer n. fit ap0- n - ag
aer 0+gt
4 ap O - n. 8 - 49 in loun L sit pusito cat c4 - 38. Ergo Laca C   n B g2- a9. X 5) ar O andaliola
si abtet baninus altimus valebit Y et 29l e SI. s apD n. B = X AG et nCa B  - a 16 - g
aeim
destincto ahobus. g habebitur aeglb + B9I 4Y 3b + 8+o
no
4b

## Folio 168r — 00068195:0335
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 167v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000335.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 91 lines, status machine
-y  Blela pi Sray +a + raty + ti
0192 100y3. 6 29∞y 7 P
ta
Ita habetur prunum valor ipsius ge absolute.
Da..
4ga 2ga - qu al
rota
jam conflrendo dios vutores ipsius , fiet: ex Cermino tertio ed quinto;
fiet  29a 2ap O ih p aba 4g. al + a 5D (he sb2 a
4
abax iba p ratia  - 2a + ba - ra iba + a
2aq
exaxes
In sectione conica sit cujus vertex A. abscissa AX
sit ordinata YX, angulum ad axem faciens datum
XYB, vel ita scilitet ut ratio rectarum. XB, et
BY Ritidata E. Datur recta AX F. datur
saso Ergo VBr 4 Ergo YBe X bg, ut cadeligo
Frax (+) 2ab + ax + (D) a abx + a b p bq
YBF a AX p aX B +
D Data ZB, quae est 2aZ+Z
XB, quae ent
+19
 X B b. Ergo ABe (. Jam
XB
c
posito AB1Z datur et
15 2aZ 147 et YX eut paz + 2q + aia2+ ca
Hoc jai ind solationem problematum transferamus: ad Erculum separatim appliemus
+ 1
Lray cuns applicata esto 2by -y + l ⊓ YB
ergo datur et XB s, E Seby -y 1a et VX cud g
y
2by
NX erit S 2bys -y + Ca + 2ab3     . e 
Jam faciemus: sabus
B
quadrando Rordinando fret: ut yV 2b V + la
Pone ca pu c
L43
qu
=5
VNoZ A
aib + cala Eis L be a. si D50.
2a
-qu
2ac8
sed incommodum vetus redit quod
scilicet raeter quod
duo valores puri ejusdem incignike
habentur,
Ci malo ut quaeratur renedum omipo  trnsferatur in platus B, C in B pro y, ponatur y ati
9. 1S.25 1ag+ qga, fietque pradiamdeo + ordiu ando
tea y.
1, .. 911 - 24
q2 
G12 + hS
=gr
7 2gB Et quadiando:
∞yt 1ODY 102y ed Y .
Hnc sequitur ante ombia bp- h. (sublato ido temmino aequalionis sensi 0 0)
sed sed hinc etiam eo non sublato alia absunditas, nimorum valor ipsius
opobis explicatus per h. in termino 2do, e in termino ato
3)a -Eig
692 (ybt
ejus ndcu seu  ()  A quaeritur valor rectae YX ex recta AN AL
ponatur AV, rI V. Ergo p  e, et LB  et A6 Sy- .
XE,. SiNEa + a AE et A ED B EXE, A AD. AE. Ergo
AD- AE ⊓
Ari ar
f. 47 (Ex AD ddtur X Du sa
12) Cntet subtrncta YD ExD C
XLH DY +q
t 
XY ( VF XE-FE
A
ec 18 a  cr a  q. V ysi 
igo d y+
Eror est qua e ve uon ul AB, sid AE, invula sipl. X..
a 6y  = 7
ag +
2a V
diba
XB
esta
2n
2414
AAX7 +2
fietqe, 2by + 10n - y = h + 2tius
13
1203

## Folio 168v — 00068195:0336
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 167r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000336.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 74 lines, status machine
r
1 b  ( ragua 2ah + ag + 2 h. + ca hy, Et quadundo
Fa y na V +hah p 24.B. Et rursus quadrando.
+rah +a h
L
..
Unde
cer Cy 30 Dy 107y +2d8y +4
o
- 1
Heq u ilnde qu e1 La r8
cecca
y
Conflrendo teminum tertum et ultrum ap C - 4g
a p  41 6 b69 - 8 ap1 0 20
+C
tasia
Aeq. 2. (
Aeg. 3
c
1Z1 
Erisque b as d- 4lg  p 
i.
po 17 a5 0 - 4 l92
u
16g 8apg
ep  0  + 16b+9 8 a pb g = a5 O  cnius lono ponatu
dem per tenn. 3tum exit. b4
dinisique mitus pot-
es
ati 22C
b rgr
F-- 701
CL
=2Olq, vel multiplicitore ipsis
A
ap divilis omnibus per
a
14 quippe cognito
posito a
veis valor ipsiu (2 209, habebitur
Rtat adhuc dlia aequalio pro vulore ipsius h et ber et ea summenda ex termino ullimo.
ponatur jam b  Ed + V a li   habulinus notorm ips. (- b C ap - bqH 1- C
4C)
Erisque b otc 1 c. Atque ta vator ejsshabebitur pure
Uende et valor iptius h. pure habebitur p pi term 2d. Unde ha tennium seandum
atque ordinando.
b y0
qui
.
o Lo
3
2.
32 10
ato
data sectionis conicae datae
later param etro, invenite dia-
netrum illi parametro respon
dentem
Error pulo aliquis adimissus cuca
humerum supialum , deberet
ibi esse 4, loco 2, quod
ex applicatione ad paracoram
est AB. b is latus rect param eter principalis, Judico Applicetur e ad Erculum
in
O, est parameter daso
e est labus plar versum
prncipate Laliu ransversi secundari nulta habetur ratio
ac denique fiet. +b9
n
+
2b
1 +.
Z ao Aequatio Problenatis: