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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 69r, page id 00068200:0139
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uli
1
2 plicis - 2 X +yy
Splicis
sit AB diameter circuli et AC. CD. DE ete arcus circuli quoteunque
semilinumse
semicireto minores, inter se aequales, et ab opposito diametri termino
E. dacantur rect
complementorum, arclum AC, Ad. A
jam Radius circuli vocetur I. seu unitas, AC sinus complementi
anguli simper seu arcus AC vocetur, V. et BD sinus complemen
arcus duplisderit yy - 2, et BE s complementi arcus tripli, AE,
chorda
erit e - 3y, et BF sinus complementi arcus quadmpli, erit
y- 4yy +2. et ita porro Progressio erit talis
Thordae complementi sunt angulorum dimidiorum sinus complement
+2. ipsa scilicet diameter
son simplicis X + 2
elili
c ve
X -3y X + y
Aplicis +2 X =ayy X + y
itaqe progroessio ita stabit
subteesa crescentes deroscne heoremat. Grometricum ex
areuum arithmetice a
6tuplicis =2 X +ggy X -by X y
 - yy X +12) + -7y x + y
etc
mi
circulo deciescentiumus subtense
A DA. BC bB. 8E
Splicis X +5y X -sy x + y
etc
chorda
etc
quo hic calculus ductus: BA dia
meter ad Diametro proximam BC.
ut quaecunque BE, duplicata ad
Summur duarum quas in ter
ot Bd. BF Hoest alo
haec sublensam ps hat ab rithmeticaq qui
Arithimetica locum habet ratio aequa
litatis dupli ad summam, hic ratio
quae ib ot alia constans, plaeter ratio
nem aequalitatis
demonstratio si arcubus Ae Ad, AEH sumantur in alter
ernicirendo aequaliAG, AH Ai HK. et jungantur CG, CH. DH, DI
eI, EF EK patet Ex esse meonam subtensam inter dua
quas in terjacet, Ei et E
Et EI. EK FX. secent diamet
in. N. M. L. nempe N et L _ angulis rectis, in medio ips
FK.
antum anguli EKF. et 1EK aequales interse
et angulo ABC qui
arcu. Ergo  BA. BC:: KM: KL:. ME, ENH Erqu i: KE: KLt EH
2HE. KE+
Manifeste satis apparet Numerorum progressio qu’infinitum, et
generalis ratio, ex dato. Sinu complementi alicujus arcus, invenendi
sinum complementi arcus tet dimidii, e trientis, quartae partis,
quintae, sextae, etc
ex eodem
Theorema memorabile, ut AC ad AD, ita Ad ad AC + AE, Itaqiacs
sint progessi ex ponatur curva in qua abscissae LM, LN, LP, sint ut arcus
seu anguli AC, AD, AB, etc et ordinatae MZ NR, PS, sint
ut subtensae AC AD, AC, ec erit haec proprietas curvae,
et Gse ad. NB +TK
inter opera
vieta
ut Mq sit  ad NR, ut NR ad MZ + pS. su ip se
seu caequ. bb-aa, : a
MQ. NR. aPQ Tysiet bb aequ aa + ac. Est sergo continuatio
analogiae quae est in recta extrema et media ratione sectae.  erit
ags.  equ. bb + bd seu d aequ, cc- bb: B, seu b bbea +a (bbaa: ab
a
12a
e
1
nasus et dd aequ cet ce seu eaqu d cat seu b rba qua
seu ad 4ba 4ba +ba + a ala 2b a Va: a
+
rursus a: b :: c: b + d :: bb = aa: ab + ad suit  ab aqu ab
seu daeque b 2ab,: a. Rursus a: b:: d: C+ C:: b- a b: ab- a+ 
fint: C aequ. b23a b + a et a: b:: Cf d+f:: b- 3 ab + a: ab- 2a b + a f
a et fiet f aequ. b3- 4 a b + 3a b: a
ita pono a: b 2: f: et g e fer:: b- 4ab+ 3a b: ab- 3ab + a + ag
fiet Gaequ. B- 3 a b + 6ab - a
Et habetur progressio
subtensapareus simpli sit 2
dupli-
Etit tripli
Quad rupli
quintupli
sextupli
a
NB componi hic possent aliae
aequationes Ax. gr. data
lA + mB F n C aequ. Y invenire b
poterit solvi aequalio talis ex Tabulis
sed non sine labore opinor, quem tamen
limites reddent facitiorem, et haec
opinor set suit Methodus pellis, aut
certe aliqua similis, so lvendi
equatio nes ex Tabulis
chorda
fipns compla
septupli
b+ = 3b2 + 1
Octupli-
Noncaph
b7 - 6b + 10
10 - 3b +56t- 106+