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# Piece: https://leibnizlegible.com/api/records/37420/text
# Catalogue record: 37420 — Quadratura Circuli et Hyperbolae arithmetica Logarithmi
# Incipit: paucis logarithmis v.g. omnium numerorum usque ad 100, datis, fieri po
# Date: 1676-07
# Akademie-Ausgabe: AA VII,6 N. 032
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=37420&katnr=Cc2+1485
# AA VII,6: 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-6
# In the Akademie-Ausgabe: AA VII,6, p. 406 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/LAA-BdVII6.pdf#page=406
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 431–432 (from the shelfmark LH 35, 13, 1 Bl. 431-432): 4 pages, canvases 857–860, page ids 00068196:0857 to 00068196:0860
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=857
# Lines: 266 recognised on 3 of 4 pages, mean confidence 0.71
# 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 431r — 00068196:0857
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 432v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000857.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 83 lines, status machine
jilii 676
Ccm panci loganchuris, vius ib mmnium immnerorumm usque ad 100, datis feri possent
us additiones subtrnctiones inetiplicationes et divisiones etiam numerorum majorum, ut pso
5i
pritimet. sit:
329728 mullipl. per 15629. fiet 32 100 + 9 728. per 5 1000, +634
cogantm seu fer ab + C per de+ f. nlliplicando: abde  Ede + fab 5C.  ut invemiatuo abde
addantur eorum logunthm. sed hic jam deberet ex dato logarictuo investig facile posse numeru. quod fiet tas
modo, dividatur loganthmus ex a bde per mmerum aliquem donec fiat tam paroque ut reperiatur in Tabula
nostra exigna logaritimorum usque ad 100, quod facillumo negolio fieri potest, et tuna quaeratur
logarithmi numerus, qui in se tolies ductus quot in divisore sunt unutales dabit ammeru abde. eode
modo habebitur, numeru cde, et fab. et fc. sed haec inciositatis canssrii pruxi enim ines
numero et contra inveniumu numeri exdato cogarichuro tunc quaeramus floganthum numeri
cum saepe adhibendus est ducendus in seipsum, extrahenda ex eo radix alicae
 
difficiles, e ut dato numero H. erit l,E na  Lnt + n n erc. et contra ni fallor
ex lato logaribhmoent np a 2 + B-B eu ita facile omnia ad magnam satis exacti
las
u
V D. is numerus addatur vel dividatur et mingatur vel anglatur, vel etiam multiplicetur
aut dividatur;  t habebitur logarichum numdira 4, vel Ctnue puogadi opue etiti polent quodam
numerus. Et quoniam exactitudinis causa forte ad
modo praecedens, quasi cogurithmus invenire ad invienteu inveniendos sequenles; quod morum
qraehitur eumi logarichuus ipsius n. per regulam D. isponturesse is est
no +n3 et
aZ
s mueciplicatus
per 3. dabit Tubum ergo 3n = 3n eri logarithinus cuciter ab 1. id est eo nomnihit major
hunc videntur novae quodam orri posse. considerationes. ut si incirias es  ab  est
1n labn p nn.
L ab n sn. IL ab lab n 1 2 l∞3 on 1. n 94n
P n 
1 n -X+ n
P n = n, 1 1
labu e n - n + 17  n4  ad -n
lab np n- n + ne- nd 1 n1o- n
nora qne u
oa 5) i in  ( on 32  30 ab. aret 4
1 7- = + nd
n+ + ne - n
H 2n -2 + 2Y= 2nt + 2nd  210
labur) n. n6 + n d - n’ +n 5 = n 8 p, 20n   +313- nt  31 - qud
enor
Ex hit o aequationibus tam miris prira possunt duci  adhuc multa ut quonia
ex addito logarithumo n et n. fit loganitemus n3, hinc, et ita porro
n = n +  = n  +  J etq surum a logumchrioram ab ominitus in potestatibus, sed ista
+n
1 72- nt +n
sine fine varia rectius eui considerare ellifferentia
 intem logaridimum  er n, dare logarichmum n
 diffentium inter loganthimum IB et e, dare e elit loqimthum
l ab n p) n L n + n3 -n - n + n + n - nd. sed jam video non ita despondre
 nt
4 n43
+
no
licere pro artitrio
sed hinc jam liquid patitrius quoni am cogarichini 4 ab 1. ab rest. ab r3 ab n Y etc.
sunt progressionis Arthmeticae, etm omninim summadae qualis in quadrato primi dimidiato.
lab n + (a n+ la  etc  n  n + p3 rnt
M e
tu. Sunamus poteuntus
cavendum ne eremus: constat si sint AE. AE) pregressionis Geometricae fora E() et EC spatia equalia
el
et generaliter sd sint AE uut numeri fore spatia D. EF C ut logantimos
ex + B +L  en. et tebC est autem e , n-,
ergo si  ab 14 e H.  =  3 ida  5 a quod e t 2 n p 117 2gu
possumus aliu etraro viai sinuus
ex dato arcum invenires. si soinventis sins et
arcus ex ose invicem valoribus
ex eodem scil. tangente seminiu, aequationem conermm
ex eoveus scil. tangenteiaer refellor. aru seu
dere rationalem, ut sujus est ex tangente semialius est
245 t 4. _ 13 + + +
Spt =2 t3 +ats
e
+ +9 tu et arus autem et 2t - 21 f 2 f - 2t verC  a 4BF
Et tentandu in au
den
reddipossit pure. Item Eadem prorsus mirantu
in aren, quae in logarictinis nam dapliandus est tantum areus, et triplicandus ot temen aliae runsus nulti-
plices orientur expressiones um arcus tum laeterdum.Hinc etiam mova sux pro aequationibus
datis omnibus ope tutarum spamum a loq ans hrmotium neselvendes  comgonendo sut let egnud

## Folio 431v — 00068196:0858
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 432r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000858.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 115 lines, status machine
etc. Negligantur rliqua
 alieque enuntiandur
ex his aliosque dinta se compositis et innesiplicatis. frmari poteri aequatio similis
cognita tgant inventione ipsius e, ex dato logarictmo, hinc ope ejus methodi
aequatio proposita statimraddi pottis puta mon tamen sic habebitur pura infinita, sed
tantum quantum libuerit appropinquate licebit ve eoram ex labielis ope
e
nodo b
Juentio Trigonometria sine labulis aventid logaritimorum sine Tabulis, et
contra; inventio logarichmocum ex numeris antiratibus sinuum langentium B secantur
serem infuitam. Methodus quamlibet curvam Analyticam reducendi ad
ute  b
aqualiones iderticas, prius per vatas quantitate mltiplicetas
1e
nyl
A u
eds ame ab 1+ C pdE te neue ligeantur reliqua
uv
 logaritentique invenri potent numerus + e eodum V
vei est
dta solum Canonem sinuum.
BI
1 2
ll
pc, 5uY
aA, cujus resiluttorum detur ex  logarichuis, quoniam vi, pareim etiam
sis hiet sum numeris simmum tangentium e secantiam non inventis, methodo diversa
bea quae adhibetur Methodus omnem aequationem propositam reddendi puram
luncuriam analiticam aequivitentem rationalam, et per consequens
vadietura onnidum figurarum per series infinitas Methodus resolvendi
oaniem aequationem propositam per solas Tabulas logaritmorum, aute etram
t
bganeI
(2 1, 2, 3, 4,  1 2, 3, 4, 5, 6,2
2,3S
a
sin ex arci 
rest. 4 , 57b. 
1eu, 9
e
I v. B Ab t resus
sin  a. arcui quod est absurdum
 a
ver
GW
posito arcu a,  sit sunus.  sinet 
lese
sin e 
sumsin HI  = 
opent seque retur
sit ergo abstracte: ac sin p
AEHy ABp2y
et BC13
dna
a vi
sin 
NIm F
na
a
a
23b 45C GZ8 39,E
e
 234b 456C 670 89, 10, e 1
tam Sa s4 Is. _ sinum i semper aduno edripigua rttionet differre alibus ostensu
ova forter comparatio jum x v
1+y
idg
diffeculia simum facilis Nam
fieri
i
+1P
=C)
H
34,5
nas)
1,23 1
ad
EAB
Xy
ersi
BB
c
17, 3
2B-
n  2y 2y+B
11y d
ls
ey
+y+2yβ
s
sib 1 2y
seu
11db
seu posita 1 pi 8.
fii as dsmtroda x ee
23
ed
t
1iry
g est quantitus cujus summa haberi potest. Nam S piegtII irgo
(. quae est summa arcuum in off sino, est q
112
Ergos CyC2 qu e oo sin 1
 rye
la
+y
ercuspi 294 1y +195- 197. Seuu arcpise
(Si nanevoa in dift. taps. semarus p  areri t
 at 17 = hy 3+ γ 29 et. quaeritur sius ex dato arcu. 60p  193 +  y - 2y et
rejere a pi 9= y. br eg _ p Y 27. Ex priore 2y3 p 8-6. sed quoninita fiet
ie1rg
puy nesio
t 291 277
s
ermt X p 2g a. sedila sabstitatio for e palieus quoaiimm rd alte drctra imnatus Ergo pro 8V

## Folio 432r — 00068196:0859
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 431v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000859.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 68 lines, status machine
Enm sumanus differentiam senus et radii, nempe ea est 1 -0 vocemus eam d  et pet
1 1 - 2y + 293 - 29+ 19. et rad arcus 1 2y, - 2 1 + 293 - 2y7  2y3 - 27 est
5+1- 2y hini nihil
S mau eut s fi a et ye -8 ergo rs p a seu d 1a
a p 17 5H 1-19 ay3. g p a 7 g (y. N g 27- 24 7 7 esp a. et s poeriy
si tangens 19, auy a
ex duplus arius erit
cirgu Ea6Y ergo a  a 12X et 2y expriore yt 1y-a.
a H 1y-
exposteriore ist  e a. Ego 2y -60 pi 
(2)d
) v7.
0vB 27.- 193
q
r
x
arc P 2.
si  arc prus = 27
1x . Y - a ex 3.
eut ap 7- 3 + 3 3 en
jam si langens sit y. tangens arcus duli dupli H G. est SH
est autem 2a p 8. 8 + 8- 8 erc.
1- y
1+
1-7
+ nyt etc. 1 2y . 25 + 95 8. 2y
5 6)1-y
Et ita porro in infinitum, quae Hyperbolicis consonant.
2a7 1
3
2a 1 27 - 292+ 2y5 - 2y) ex.
sei ra p 27 8y3
15]
4a  
1 - (7)
(C3)
8ac =
a 7(VD a e (Cyn
1- ( 1))
ar(
ar y Ex aC a r (3.
posterioi appropinquatione semper exactior priore. Verum notandum quod hoc modo ec non assurgi
potest in infraitum nec infinituplicari potest arcus ejusdem circuli, nisi sit ipse circulus infimtai magnitudinis
(H 8H, Z. ergo G 8y F, 2y. et
itaque contra potius divisione utendum:
Dy + 29 1 9 et V +Zy H et y + 1 y +  H
et y +  H Sp+. a Y⊓ 19+1 -1 3
8
arc  8
arc 7 + (8) ure ri 4(9)
7 a 1941 -.
1is
an 9 8( er
angulum in data ratione secare: J sit  a A 7 - 3 + y - y et.
sit quaeritur sit alius arcu  X B  - B + 85 - D7 etC sit  aB ab. erit
.
t
bN8. 87 +95- 87 et a r 18- b0. b8 - b0 p y= 3 +3,
quaentur 8.
a e ari 29.
jam
ap
a J2v. a Rey esy
are
api 19. 2
auc 4
V p 2.
cop

## Folio 432v — 00068196:0860
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 431r); this page is its left half.
# No recognised text on this page (recognized).