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# Piece: https://leibnizlegible.com/api/records/43048/text
# Catalogue record: 43048 — [Math. Aufz.]
# Incipit: x aeqv. a. a+b a+b+c a+b+c+d
# Date: 1680
# Other printings: vgl. LAA III,3 N.40-56
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=43048&katnr=39685
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 173–174 (from the shelfmark LH 35, 12, 2 Bl. 173-174 (174v°-173r°(39684;39686)): 4 pages, canvases 345–348, page ids 00068195:0345 to 00068195:0348
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=345
# Lines: 289 recognised on 4 of 4 pages, mean confidence 0.76
# 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 173r — 00068195:0345
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000345.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 82 lines, status machine
e
1d
1 +
i
6
6. 16 od. L 615
1
+
AAE
a baent Yhio
utn Y
+ lutum
Aucuorui cumr um ve
9. atn 99 utd 2+2 + (∞p -) 1 +
(9 plv. 9. ab n 2p - 1 s + 2_ + 1 s
1)
aln (2)+ W _E + ( -) 1-pE -)V
 ii ip v
1pv + bv +
s
2-7
1
t
9up
+ 1
es
est
+dd d alu
. v + + Vb + YY d +
a9+3∞+ qu
v+CC- LXC + kexb al + X
19 3F equ + squ ut v C
i9 + su + qu. k
13+9+x
b0
. ati g9V
si itio qu
0+ 3+9+ v. X
vo +h2. ntv XX
.v. atv X
3d + pC + 337 99 + bV
3+9 +V
1 +V
v et v x
Saa - XX
ECaequ. X. aequX CxXX
rax - X aequ. Z7
Abax -  aequ 0
X aeqo ( A7
EC Dumanimum
Dercribatur hemicirculus cujut centrum in axe porabole
et trauciens ses vertinem 83, ita ut maximium in eo
quad rectanguen eCD aequatus
lus sumptomm Cunca suncto quolibet uut D, darcributum
sica
Auiribatua temicivinens cujus centrum in anc et trancieus
sir paralioea berticem, ita uut marcmu in eo rectanquu
Cum. ECD aequatur AB quadratu. secabit vel tangit ille
Curvam in D,
nam AB9∞ ECD ergo AB sin BC ∞ ECD in BC ad art DC Cubo
porro eadent omnia guncta temicirculi intra curbam, nam ei ex–
Ara cadant, recetur Clova in alio suatirea funito, tit GH
at erit ea natura frarabolae ABqau BT. ∞ T. H iubo
Eed. EF H minut et CCD quia manianum, ergo etiam aniae
ipei equa quadrato etB, et loco conce humpta conanum aetiue
quiam
Audine Bp, erit EF A in Bs ad et Y B cribuo prrimor AB qu ad
in BF, hoc ert quam cubu ex GF quave omnia in remicirculo soun
intra Curbam cadant, et illa inten diverris locio sio Circulims
eccani porsit. Quod dumonitrandum crat
N - 41
a alud uy. de A
4. cio e 2
V- deat
Mo
V = dC
ult
i x + 1E+ b2 + 91 B alia E
V - (4 + (4 - 4 alit B - 9c + 9 alo ∞7 1173
t 1 - 3 Y -4 M  at alc,. atro F V. ab  tra , t9. ab  0Z
3 ali a 0. 8. ato 4a - 3E. Y . abx adio  _E Y
ab0 a0.

## Folio 173v — 00068195:0346
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 174r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000346.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 56 lines, status machine
(48) per aequi 181
CV - Ceo aqu. 1s - ycd + xZ + X 1
(50) f) per aqu. 29
ev+ XV aequ. Cy + Zy
v. Z.
(9) per aequ. 23.
deo - XWo aequi dy-1y
X. y co. 1 fghsug. L. X. y
Laequ. 2f + 39 + 4h F 5y. + 6X + 7y1 42- + c0-90 1 2n - cv +v7 2, 3g
= (51
c aequ. 12f + 139 + 14h + 15  + 16X + 19 y - 11
(2)
1 aequ. 22f + 239 + 24h + 25 + 26x + 27 y+2
534)
V aequ 3af + 339 +3gh + 35  + 36X + 37y+ 31
si jam in locuni Z. co. p. in aegqu. 48. 49. 50. substituantur valores ex
3b
4
aequationibus 51. 52. 53(53) Habebimus tres novas aequationes, nempe 54. 55. 56,
Habemus praterea duas adhuc aequationes ad cirulum empe
s Habemus praeterea duas adhuc aequationes pro pari co. V. aequationem 20.
et pro pari Z. V. aequationem 22. ubi etiam valores substituendo,
habebimus aequationes, 57. 58. Quas multiplicabimus singulas per literam
quandam arbitrariam, nempe 57 per literam 8, et 5 ut fiat aequatio 59.
et 58 per literam 9. ut fiat aequatio 6.
jam aequationes 54. 55. 56. 59. 60. addamus ut inde fiat una
aequatio 61.
fg. fh. fl. fx. fy.
quae aequatio 61 habebit terminos sequentes.
gh. Gl. 9X. gy
El. hX. h.d
X
ff. g9. hih. l l. xx. yy. fg _ fh. fl. g h. gl. hl. fx. fy. gx gyj. h. v. h. S. lxoly. Xy
33912 314 f. 9. h. l. X. Y. Yn
19
XX + cy + d aequ. 0
CXX + Cd X +C∞ ∞qu. 0
+
+
a. XX + byy + CXY + dX + e Y + C aequ 0.
f + Yy
g + 2f X + cy
5. f19 ft35x +gy +cy
24
3+2 aequ. 0
coefficiens
(9)Fy + 2 ff 1399 + 4 hilit SI sequ. 0
19) fy + 12 8 + 1399 + 1+h h. + 15  aequ0 4
29fy + 23 f + 1399 + 2 4tik 125 l Xqu. 0
39fy + 32f +339913 etili +35 (+ qu 18
2 + 12 + 22 +32 ae qu. 0. + 3+3e2 3+ 33 aequ. 0.
4+ 19 + 24 +34 4qu. 0. 5
9+ 19 +29 +39 aqu. 0
ys
 Karyy
C2. XX. Fyy (Xy) . X. y.

## Folio 174r — 00068195:0347
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 173v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000347.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 64 lines, status machine
B. inaequ. 24. et 2b. explicetur Z. v. per X. y. G. h et per arbitianas
e
ilemque componendo 24 et co explicatas. habebitur aequatu i qua
non amplius assum
incognilas, uttra simplie
habit
reu
xel q  3) not iem sitis oe sutineiendo 18 conanoo solum ut et
deo = Xco aequ 3y = 1s seu d vrco aequ nco - 
in trbus aequationibus, 18. 23. 29. sunt termini v. c0. y. . yc. XZ. x. yV
 77 rc- reg + qu. 2a - xX
ero 72 94 Xv
22u (c) +Euu (aap t p
vo + Cc - 21v aequ. datte - Z7. - vo (1). (taa) - 1  20
Sumantur tres aequationes ad conitum: 3Xy   41y + 5 aequ. 0
et 410 XyF n y + nB + 3c + 14 y0 + 1X 164Z yZ1 ay i + aguo
+ 20 xy + e y + 22V + 23C0 + 24yc0 + 25Xco + 16XZ +27y E H8 xprey r  25 yv + 2AxV + 2W aequo
Malhplientur aequalio 50 per l, s1 per m. 52 per n. et inde fiat una quae compretur
Cum aequatione 18.
lExg + ty
Morgemus im avanco aga L AEmaerAmZmmrbqu
n2Ox, n 2. nn ng a24 ne n2b
n 17 nrs nius n2
n2f
n2 1
Comperando cum equ. 19. fit. 12 +m 1 F n 20 aqu e. lamune aqu. 0. marn aqu c. mi3n qtC
mi tnra aqu. - 1. m en 2s aqu. 0. ms anbuqu( miraniq ao m 18t M 2B eo () negi neg ipu o
mis+nquomitentam irtaW + l aqu.0
(nt  mis
 aequa 2 aqu per sy
n
  ago  20 Equ. - 12 - m 10
5)
per 5q n
(79
(68)
27aqu. = m17
per 6, n
19a- mius 2s ugo mis ieuju im 1t
21 aequ = 14 - m 1
per 55. n
per6q
per bg
per ls
mago- c. mu 23ago tcum1
per 56 n
51)
persu
 2qugo - ( - mit 26 4s A1 = mib 28 aequ C1) = m 18. E aqu (  e - 15 - m.d
per 58
per 60.
peror
n
pere 
Loco aequationis 53. scribatur alia  ponendo At pro l. (n pro M H pron, et lao 1s porendum
eaque  aequatio sc comparetur cum aequ. 23.. et podibnt aequationes
(8 vico, y
Cv- cco aqu YY- y id frZ ens X
5, 0)
quinque aequationes:
ev + XV aequ Cy+ Zy
+ Y X
cocot ce -2o aqu. aa  re 
TExG+ l4y

## Folio 174v — 00068195:0348
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000348.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 87 lines, status machine
dZco  ett v aequ. A. Z - v aequ. B. C0v aequ C
Zoaqu. BH1 vep aequ. A + e p B. Ergo V aeqo B - AC C. - 1 
EXZ et ZCo aequ B + d B - A aqu. B es- bd + Bd - A
Vago C
co
-d
Zaqu. _1h + 2g + 3l + am
e -d
 aequ d B- A
et-dip. Ergo
quaegod B - A
Coaequ. BC eB - A
Eequ. vC- a
dB- A
nom cunouvasmamne uB. 1. nas ravros
Ernmnos) arct  eane an in g 1. rungos nonger vogut
mn enommers in ummo anub AE  egem nnt po 30g 55 
 
in ut et s mermo meno Hd 1r pr 15 cr 743 pa +Es 
ag nimone uemng ran e Ess egacennt ugnntg wres
e Cemont
sumu mwg mmunranio v.b3 ero muno H L 3. rs
agu HL a it Et n  ngaga vanguro in 110 +o
tt egnunch wonech ergo in g) angoret eneon va
 i moe mogan nesur ymennarges nnch vne enon aanch
e
07dE prr  3Eb  (33∞38  E p caro (38 a bE)Y muie
∞ un unian)
vi rebung guo teguros gacen bes est angute (33 unu
autuma 0 d nance vio . uri perogognans ans
cromnes po erao in ugu enn engnnmoh ausogenns
gise
anpugarot per juggauo aumra mamns tas
aus, 6 Cdliutrapo
nout
er in crouuramuo 17 t1 108 mongar os rrmatso
ogngmemesoorer in unnepea intim engunemmr ausugurres
 mnuao
0 X9 DB Min X
0 Atu X - Xv9
L2. abiv X = v2
Xps abos. X. alid.3 3
XX - vv
codequ. C
a.
a + b.
atb+c. a+b +c+8
aa + bb +cc + dd + ec.
aa
y. ab + ac + bc
Laqu. abc + abd + Bc8 + bed
X equ. 4X + 3Xy - 3 + a.
X + pXX + gX + Y aqu. 0.
X aqu. P + FP +
X aequ. a.
A+ aequ. 2y + aa
X a + b + C +0
ab aequ y
aAbb aequ.
1b aequ.
a67 aequ.
ab flc +dea
a+ - 1
a V-
+a4 4 a t-1
SI equ. 2 
a (= 2 - 1 (=)
+
2aa ((2) aqu bl
1 + s+2 + 31 - (02 aequ. b. 16.
((2) + 2F2 aequ bb aequ. 6.
tngns.
Data tangente arcum invenire
de aqu.
arcus a radius r.
- + + t - + + l’e
3YY
5r 7r6 9r9
quadr 1 - 1 + 1 = 1 + L = 1 et
+ L et
a + - b  Va = s- b
e
et =
a 12- i. L
e
a
a et =1