# Leibniz Legible — https://leibnizlegible.com
# Page: https://leibnizlegible.com/page/00068196:0761
# Permalink to this reading: https://leibnizlegible.com/page/00068196:0761?run=74
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 383r, page id 00068196:0761
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 382v); this page is its right half.
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=761
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000761.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 98 lines, status machine
# Lines: 98 recognised, mean confidence 0.79
# 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: the recognised lines in reading order, one per line.

de smendis fit C. Zl  1 2 Ni2 = leg C 1) ( 2. 27 ( 3) + log (. .1 F A 2a 7 et
1. Z. n. 2 = log ( L. 1)  X  2a ) ( Ha1) 4 log (- 1. 1) ( A. 2a 7 (a- 1) = 4 -na (51 4,  (p: 2) 4 - ua,:, 4. + c
ita habetur a, arcus quaesitus ex radio, pel peripheria, intervenientibus cogari thmis imposibitabus
et ita habetitur arcu
imaginariis similiter possumus tollere. ~. per
sinci
seti constructu
tam exporeatit
ircus irculi
t ponentiatem
tud, ulmuifi fk E, a, et s stud stundamicut esto rli st an tse Bbna
idem quadando ductam ex quasi Legaribimis, quae si hanc corpegetietur, nec identica esset, posset tolli a ex
i
Habeo et achuc aliam expressionem excention
sub ata Z, posset postiemo toihi et etiam ex aequatione et haberetur vtor ipsius IC. seu
Alia ad huc ratione eadem consiqui loferet ex hoc ipso carculo
 et quodem maltis diversis modis, dum non facimus qu o sed sumpmus alim arcum cognitae ad
adus cmsi sa   nt   s pue e     a    d
trausferento in
easann
inta ab imogi
perpieriae
peripheria mors rationis.
Secans minimam ex caeteris secantibus ad suum crcum rationem habens spiralem quadrapsis dividit in
lib. 6. prop. 3
uncto ubi maxima
tangens complementi ejus arcus ad quem ejusdem secans minimam habet rationem, est arcui aequa eidem aequalis
e ipius n D vel BC, sumti pro 1 log. sit 0.
ipsius
si ipsins AB 1. seu 1. log. pit v. ipsins Ad loq. erit CBdEC  i
vel si AB, Ad, AF sint puogressionis Geometricae erunt EB, 9d prog
eruat EB, GD, progressionis arithmeticae, adeoque Et erunt BF et EH aequales
1ta 25 X
itaque si AE sint numeri FB sunt logarithmi
et AE = 1 X, fiet BF = (dx : 1+X
1  (it - - X + XX - X + X+ Erx fx:
sit BE, Z
Ergo S = +X - LXx + 5, 43- : x + + X = L. X   et C
 sed BH 2 similiter BLE E si BL, sit X erit LK=  1 :91 - X.
ergo BK = = sax = - ( + x + L xx + L x t x +  X = log. 1 - x
(X  it do = eg eodem modo.
sit  g = XX, fint = (2Xdx = log. _ (1 ~ XX)
Cup  X + X+  + X  X + et et f
nampro X substituendo -x
= -1, LXX = L X + L X + L X + eic _ = log  , - XX
loqu eF (1- XX) = log (1 + X) + lg (1 - X) = s du -5 dx
CB  Cs  H - L Xx + 1. x -  X 2 + 2, X -  H = -,  nx + L X 4  n et
- 1X - L XX = L Xe - = x - + X - 1 xC
sed  fdx + sax = log (IH X) - log _(1 -x) seu = log (IX, :, 1 - x)
= (IX, 1 = X + 1LX = 25 dx = log (1+ X, :, 1 - X) = 2, + X + L X +  X +  Xet
I=-XX
LX _ - ZXX + L x - = X + ese + 2X + L XX +  X + L p et C
1+X (- 1
 +22 = - xx fit X - 7 f. 1 et dy ~ dx Y 1, et fiet Ef 2 Yt S dZ = log 1 - x Y- 1
AB. X. AEZ. AX
de) vel 25 de C t, log  sit - XX - Z2 fiet (~ X X = Z: f-1
et dx =d3:  et liet 
i2 (dx
dZ = log 1+ F=, X
-p 1-1 ItXX
1-1 + 8) = arcui, posito Z esse tangentem seniarcus et 1 esse adio
1-1 -
 = 1-1, loc
seu e sit arcu X, et lpugcas adeoque si tangens Z arcus X, fiet
(d X = 1 Y-t, l09 (- 1 + Z, : 1- 1 - Z) seu 2a: sut = - 2∞ f-1 =
seu 28: f- 1 = - f- 1. 28 = log (1 - 1 + 2:, (- -Z), adeoque fret
 f- 1+ Z e = (= 1-  2a) = 
2∞: Y - 1
Ib liet Ih + Z = 2p - Z  et ( 1( - h): t  + h = Z = Y., b2a : 1 - - 1
n si fiat S(It 22). quaenitur Z tinus com versus est 2 
2L2:, 1+ 22
12a: -1 + 1
posito Q esse t angentem arcus dimidii X: are = quo casu. jam tque est tangent
_ :  =1 : b= +1 posito e esse arcum duolum ejus
L 2ZZ:, 1+ Z2 est ratio sinco versi ad areum ubi quaritur 
 Y
diseren ali cabula de: d3
(e+ Z2 a32aZ A ZZ (1 3) du - , a (2  d2) = 0
Pda 27da d 23 = 9 seo da: a = 1d Z: , 21 2
apatis sigu
2ZX d Z ( 273a12) -
ita hoc casu erit arcus aequalis sangenti semiatimenti, at finus aequalis distantiae centri
sid de =  2dZ :, 1+ 23
 E (1YZ) ∞ 2Zd2
Eda (elit
us
et fiet ∞ =
avitati
X: Li  B
uto esse ererea in culculo no
s hE +  2) dZ, : (. 1423 - 231. 1-
qu (sn1 +Z) G 11 -312
est ejus latitudo, seu ubi parallela diamstro per punctum sectionis ducta cspitalem langi
secans est
sinuiversus 
etiam pro ax sum rartur
iac
2. 1 -1 d