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# Piece: https://leibnizlegible.com/api/records/59856/text
# Catalogue record: 59856 — [Aufz. zur Rektifikation der Kegelschnitte]
# Incipit: Si q sit latus transversum et r latus rectum sectionis conicae
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59856&katnr=57567
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 401–402 (from the shelfmark LH 35, 13, 1 Bl. 401-402): 4 pages, canvases 797–800, page ids 00068196:0797 to 00068196:0800
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=797
# Lines: 143 recognised on 3 of 4 pages, mean confidence 0.83
# 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.
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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 401r — 00068196:0797
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 402v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000797.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 57 lines, status machine
sectionis lonicae
Si g sit tatus transversum et Y latus rectum C et Z sit
abscissa ex axe inde a centro, et y ordinata ad axem, et hn = ra
=qu. 4 + 4. r yy vel yy- + h5F ih 77
492
et firt dx + du = dy sut atrgy: Siz +2y)  + 414,  trg,. squati
Ute
conicae
H
E.
Ut igitur Elementum curvae revocemus ad
quantitatem rationorem, tentemus
generalitur aequare quodrato  alicquamlibet ex his duabus
formulis: b + cZZ et b + ez7 aequare quadrato  per
calculum indefinitum, ita ut indefinita non ingrediatur vinclum.
Non possunt b et C aut b et e simul esse negativae, nam tota
formula esset negativa unde radix foret impossibilis. Ergo si
b est negativn necessario c et e sunt affirmativa; Etsi c est
negativa, b est affirmativai et si  e est negativa, b est affirmativa
Si utamur TCS y erit b semper affirmativa. Si T6Z, erit b
affirmativa in Ellipsi, negativa in Hyperbola. Itaque in Ellipsi
b semper est affirmatisa, sive Ted y sivo Tu Zutamur. Videamus
aftirmativam,
sC
inc ina L  utamu sive pro Z sive proy.
1b
 2  yV +ZZVV seu pro , ponendo n et divedendo
B
siet CZ V Et sit B + cZZ= b + 2 Sb ZV +ZZVY, seu pro 2 Vb ponendo n et
et sitb+ CZ
dividendo per Z, fiet CZ = nv + ZvV. seu Z= nv: c- vV atque isa formula
b+ c2Z aequata est quadrato  quod sufficeret si sola staret, sed
formmua b+ ezZ etiam aequanda est quadrato, et eodem modo procedendo
fiet Z = ncd: e= coco. Et aequationes 3 et 4 aequandae sunt inter se
(67
quod ut fiat facilius, ponatus co XV fiet 1. C vv xic xxrv. seu
C C XKov LEXC, XDD seu VD= XC- e; X- XX
C11Z
10)
vel ponamus XC -e = l et X = p , et X- 1 = (t fiet
cra
5. 6
vv = 1. pp son  v = 1. (+) p8 jam per 10 et, et. fit =0
p251 (t et per 9 et 10 fit pSC C t fint rursus vD= M XX
(14)1
et aequando B et 9 fiet: XC - C: X  = mmiX sen CXX- ex = nimX-um
quae aequatio simplicior est prioribus, ascendit enim incogni indetermina ter
qantilas mnex ad solidum, quae prius ad quadrato quadratum. Erqo
Mm  CXX - CX. X~1 sed video nos nihil lumatos. Redeamus ad
aequatines 3 et 4 fiet enV - nVCoCd = cn Cd - nCOVV. explicentur
Y et Cd per  m et p. it destruatur  p3 et mo et p
sit V=  m+daK edF 2M +2P+4 et c = 5m+ 6p +7
avcoco
am 23. b74 2. 6 +3. 3.6R np +64p el
i
3. 6. bise 401
lego rns at als sete ao  a iandti eis a itum aseo vita i ba

## Folio 401v — 00068196:0798
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 402r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000798.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 50 lines, status machine
sed preotus. redeamus ad aegih. et 4 et ponauu  con k. D
a = V: X fiet 1: c on = BI: X: e -vv: xX. seu
I: c = BB = K : CXX - V: seu CxX - VD = CX L XVV
folli termini p3, et Mp, et P. ergo sid
ad heflendum pfiar
(10)
 ad tollendum mp L3.3. B = O ergo
vel 3 vel 1320 iu
a
e
fiet 23 3.3. 13 + bis 2.3. 13
sequadem
2)
0
utramque assequitur 11 et 12, ponendo 320. tandem ad tollendum
op, fiet e = 0 sed hoc absindum sequeretur enim tam
quam 13 esse e seu abesse p. ititur ad efflicie sic potius
tenta adum ad efficiendas aequationes I1 et 12 ponatur 13 =0
(t 2=0 et fiet  ut alud ad tollendum p, hoc ad tollendum
mpp. jam ut tollatur p, fiet - 3.3 = - 448 14. 3.3
explicerum ergo e VCCd = C - pCVV
povendo V = 2m + 3p + 4 sC = 22. 3 M + 3p + 7 per
exam +es. 3p + er. 1. A. 5. M  3 p bs Z( 3 n p - bis 223 mp
4 22. 8 M - 4. 3 p
is
177
bis 27. 3 m = bic33. 7p
+
Abi 5(Z. 2. 3. 4. 7 m . bis 3.+7p
-2. penfran -
3
8.p
C2. 2.3M, C3p +5t -  n  . 23 nt p 4 2. 2. 7 M
bis 2p  bis 2.13 m p. bi 2 3  2. 2.2 n ( 3. p - 3. 3. 7. P
dsi
= C.2m + X3P + e4 - 7: 37 2M - 2. 5, 3 Mp - 2. 37, 4 m 5. 2 mp 3.3P= 34p
=bis 22. X n p = bis L. 23mp - bis 22. 4 1 p - bis 2.2.7: 3, 24) -6y227. 37 mp
= b2s 2. 2. 7: 3, 4 M = bis 3. 7. 2mp = bi s 3 7. 3 p - big 3. 7. 4. p 72m
Solvamus in numeris ut deinde facilior sit calculus. et
equationem e v + vcoco =-CCd + CdVV dividamus in duas
ut sit c0 = se et v = sc. seu sit e = 4 et C = 9 fiet c0 = 2
et V = 3. et ita stabitur: - 4. 3 + 3. 2. 2 = 9. 2 + 2. 3.3
sit jam 3 = _ 11. 5 + 12.7 = 136
et 2 = 01. 5+027 - 17
2= 01. 5 + o2. 5 + 17 4 bis 0t or. 5.7 = by 91. 17. 5 - bij v2. 17. 7
bis 24(:3, M - bis 2. 4. 3 Ip - bi ) 2. 4. 7 M - bi 5. f. 4. 2. 2 Mp bis 3. 4p , 2.4. 2
=4. 2.2: 3, M - 4 3 p 7427
72. 3P = 7. 4.
15. 52
 e g.

## Folio 402r — 00068196:0799
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 401v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000799.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 36 lines, status machine
fier. 4ue 5 +4. 127 4. 136 fiet
=4. 11. 5 = 4. 12.7+ 4. 13682 
4
a2
+ 17. 11. 5 + 17. 12. 7 - 172 136 = bis D1. 02. 5. 7. 115 + by 01. 02. 5. I +2. obis 1021
-bi5 01. 17.5, 115 - bi5 01. 17. 5. 12. 7 + bis 01. 17. 5. 136 - bis O217. J. 11. G-bis 02. 17. 1. 12. I1b, 52 177
l- 9. 01. 5. =9. 02. 1 + 9. 17
+1I. 5.015 + 11.5, 02.7 - 15. 17 + 1237. 915 + 12. 7. o2.7 - 12. 7. qY
+136:015 + 136.827 -136, 17) + b.5 11. 146. 5.7 01. 5F 11. 11. 5. 702. 7. L 11. 16. 5717
-bis. 11. 1365015 - bis H. 136. 5. 02.Y + bes 11. 1365. 17 - bis 12, 136. J op5 - bis. 12. 136. 702.7. +biy 12. 1367. 17
2 p 9l73 fiet
fret 02, 12 = 12. 02 seu 12 =02
rursus ob 5. 7 firt 02. 11 + bis 01. 02. 02 = 02. 01 + b, 5 11. 0202
seu n + bi5 01 = O1 + b15 11. Ergo 01 = 11
Tandem ut toltabur 72
fiel: +02. 136 4602. 17. 02 = 02 17 + bis 02. 136. 02 seu 136+ b,517 = 17+b.51
seu 136 = 17, sed ita sequeretur V = co. quod est contra
hinc
intentum daretur enim relatio inter C et e, quae tamen jum est data
ut igitur rem consequamur consentancum erit fractionem
assumi pro valore ipsius V et cd, et quidem nominatore
ipsius posito communi, non ascendemus altius, et tamen
plures habebimus arbitrarias. Resunta aequatione
Yco =CV = cOVV- cCd ponendo 10= v.x fit:
CxX WD = CX XeV, et faciendo V = 3 et X = 2
nempe scribamus 3 - 11. 5 + 12.7 = 136,: 06.5+08.) =85
et 2= 01. 5+ 02. 7- 17,: 06. 5+087 - 85
34
+46
85
et manebit aequatio ut ante; nisi quod pro  substituetur D ducta in qua
diatum nominatoris; et pro S substituetur 6 duct. in quedadem md
CE est jam quadratum nume ratoris: e
162.5408. 7 + 854 bis 06. 08. 5. 7 - bis 06. 85. 5 - bis 08. 85. 7. Egopro D scubetur
-06. 5. 4.91. 5 -063 5. 4. 12. 7 4062.5. 41360082 7. 411 5- 08. 7. 4. 12. 7 4 08. 7. +. 3G85 411 85 m1t
+952. 4. 136 4

## Folio 402v — 00068196:0800
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 401r); this page is its left half.
# No recognised text on this page (recognized).