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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 41v, page id 00067993:0082
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supplendum adhuc foret quomodo, sinus et tagentes
anificiales possint, habem ex arcubus et contra
sit  radius  arcu a
Est autem tangens A sitsisimus versus X. arcus d
aadus hariusm (fit sdx: Y2X -XX = a sivus complementi 1 = X = C et bujus relil
munda et logarithuu sinus compl. est sdx: 1- X = _PZ fit d = ex. inX
et da = dx: S2x -XX fie 1Z. da = 1-X. T2X-XX son - c. Sin ee
sine Colligamus plurinia in unum. AC Radius 1, arcus a sio
vexsus sirus versus X n omple  sinus complinentur ec fouenti ts
vel AM
secansc
rectus Y. et secans e ATlangens F. F tangens complimenti B
E langen arcus dimidii d sinus arcus dimidii duplus seu Morda areus dati 
(egius duplum AH)
sinus artificialis   sinus complementi arificialis e, langens
roctu
 cmo E = 1. Y
C = 1 -X Y=  Six -X =  - cc. C C: 1: : 1: C. go C = 1: c
artificialis
_ t: Y: : 1: C. seu t = r: C D Ergo O = C: Y et pronde +D = 1
X: Y:: d: 1 (triangula enim CAF et LBA similia sunt)   a: seu d = x: 1
et  a = N: Y, et CN dmi sinus coplementi arcus dimidius est
est dimirus ipsius EL, chordae areus quae dato AL ad semicirculum deest
CYda = X
quia deXIr = aX. FEXXX
fit  D2 = R: 2 X. seu  2d - Xd = x seu X = 2 d: 14 9
arus a = s, dx: (1x - x
3 = (1X, IX V xc dx: drc: r: c. Ergo ex- d de der:c
D- dG: AiE
est anter Y = iur ergo dy = = tr et dr- . Y = dx. s ex XX
sa: d P = dx idr:: t: 1 seu da = + dJ
jam a = (_t: 1FF. ergo da = at 1IF F erjo d P = dt: + + +5
e P = s. vtrir
51 =  P - Z X log. da =  log. X - _ 7
da = + dP. + = da: dP. Ergo dt = _ d P ida = da ddB, : d pr = a. 1+ aa ide
E d
Sd  A dr = x
et d seu dP ada = da adp-= a dp + da et ponendu di p =0
sit dP + Ita sed par jure sit d2. sed pari modo habetur
larus relatio inter sim consinum artificialem, et arcum
complementalem Q- a = (a) et quia d() = = da et et desa) = ada
ergo fiet d B dea + da du  da dd B = a  d B Z d 3 - a d X + da
d 1 t da + la F = dt fit dda + dda t + 2da t dt = ddt
unde fit   = dt: a - 1 = ddt - dda = 2dat dt: dda
adeoque fib dt dda = a dda = a ddt = dda a) - 2 + a da dt
si jam dda =0, fit ddt: 2 + dt =  da. Ergo 25 +da = log. dt
Yda = X dad t rdda + da dr = dx et dx = drt.
Erg, r  EtB  de = d. V 2x. XX
Errdi potius da 1d P = +
dB: la = C: Y = B. seu dP: da = +. Ergo targens
ad Ergo summa tangentium ad arcum dat sinum artificialem
ita habemus pelationem inter sinum artificialem et arcum
seu mod inveniendi sinum artificialem rati arcus per seriem
(2:  - Xx = - 11 1 4X x 1: i X + 1. 14 X. Ergo
s+C: = +C = log _ 1=Z Dlog. 1+ C. seu log 1= C 0 1+XC
seu loganshmus ralinis in qua ordinata (ecat diametrum
sid - ∞0 1. 1 X + seu log. 1. XX et sex:, xe = log. si 
sulgnes
est, simus artifficialis posito acesse sinumiversum
Sitaque sdx:  CC, hibsis ec at invemre arcum dP de 1i x.
Vivennus an hinc ex sim 
jam ex da texux. Ergo
dP. da V2x XC : 1 XX = da
(dr:  D. ergo dP = dr: - Y = dr. C ergo
dd T = P. Ergo ddq f + dadt =
aqate t pa  etide in cot  B= sax. 1t
da = dx: Y.
Ergo poiteret da diidere per t, at haberetar 1F
8
sode: C
sin dat = 0. fit
u -2tda = log dt: 1+ +2
Ergo tda = log 11.
et quin de cgnslans
(da=
loganhinus)
posit
anthmericae fi
=2t da = loy. da
et 
qu’est
cantis ergi
pronessionis
da
o secans arrificalis