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# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 7
# Folio 1r, page id 00068025:0001
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Tangentium applicatio
ad nemprorum series
ut in quadraturis
19+
mo B A
Cdt B
5
= (3)
BD
b
. ci
i (C)
19
it et
β4 (30)
ergo
jam
At
to
Nb 1671.
rogressiones Numerorum Geometrice tractandae
nel roc
+E)   Quaenitur Sigum ex seu solygonum regulatum,
 progressionis harmnicae. Intervalla terminorum sint
 B C equalia ABF B(B) ete  β rpsa Unitus sit AD
(CD o mod
5((B ACBado L
(
P2)
reliqui termini BC, (B)(C) B) (CD ernut t vatr et
BncheX CD erigatur in econtralias ipsis B, patles recta DE,
in qua sumantur DE
aequales jungantur
AD in punctis F
Z.
DE) D(E). (ED ipsis 1B, n(B) A(B) ex
BC, 2B(5) B(ED, quae secubunt opses
F.  er hun AF. ACL A()) et
erc adeoque ipsae BC, (B) (C) et. tantum
ipsis sumendae sunt aequale adra
Ex cujuslibet ordinate termino, ut CC, inioidinatam proxire majorem BC
demittatur perpendicularus CL F (B) B or est B odstanti. erunt ipso
LC hel) CC) erae differentiae pamrun
poredL et.
sit Bp V. erit BC p a  embo nentrat
va
2CB.
x
YC N 
Ducantur Tangentes e figurae, seu producantuer platera potusgom huhusmum, utrique
Axiiscil. AB, et ei contunito, sanrant in() exo se om
Erit ( (B
BCF, Z H 
ex
(B(C)
Emo CD 
BY
(es ) se sur (BCF
+C
3 + βy
yrovne
ilva
1 d. Ergoeut (8)(Y) B
 et BCH BF BF B β. Eodem modo: (B)(1)
(BC) LC C A  (B (1)13   ( er B) (1)  2B
 et generaliter (BT p L (1) BB
6723
virBy   E Ergo (B7)  VHGB, AB)
hy B semper ergo Eodem modo ABD(TDD A(B). et ita in caeleris
Quaeritur jam Ab. intervallum scilicet inter tangentes et punctum A, in age conningato
Erct A sed statim patet fore AB 2 BC. quia ATB2BY.
Jungantur AC. A((CD. Erit summa omnium A(B in  B. dupla 
spatii A C (C) etq(CD A, at eadem dupla spi B omnibus BC. Ergo spatum
spatium. - - - . . .. spatio BC... (C) (B, aequale quod aliunde spatea, quia
et a A(BD_L N B ABC. perpetuo Ideo in figura hannonica nihil hinc duci
potest. At videamus an non in aliis hac methodo quae in Geometricis
nihi utilis fuit, aliquod detegi queat:
2(C)
68
((B)
Aer
sit ACr1. GB1 GA AB BFD) (∞B) (YF1 B)
BC
GBB y.
SF BCB
SE B SC_BB) (F1B) (VF 1C) (yC)
LCp
Habetur autem
2
n
X+2y+ BL 1
SLC. Ergo ad habendam
2yB+ B2
24O
3= 2y. ∞y+7
si omnes iiinfiuitum quaerantur SC eut summa corum fil  1. sifutaumentim d. +2G P 7
0+DB1-7
s
O∞
x B 1D.
Say (4B tantum opu haberi
SBtA
- s. (37 +s(G.B Y S B0. Ergo
H 1 - y = s0. Ergo f(  D+
3 A (-
s
SB01 ( - 8 - sD
rgo SB B B - B = 1 + 2 + s0 qui calcutus si rectus habebitur (O. adeoque et s3.
y
∞ L
quae quaerebatur. Eademque methodus ad caeteras potentias poterit applicaria et ni fallor
alibi probavi ex datae s. dari summan finitur
Altera trethodus, quae Geomehia est, et lalitudinem sive
nempe BY
BCM2, .
et fei AB 2) +B
2γ +B
75+0 N ( + B1
Drxyry A By
Eitque A8
A, 3+Y. lni si addatur ABB V - B
B ec 1) eF B
2Y + 5 37+  svendo D op
it AC.
yy1B
varsBtyor Quorum fumma si habeptur, potenit aterum
signum
haberi area spatii. Et nohbile quod ad eandem editur lic
BB)
vC
A(EV
1((E)
(40140 4c)
6d5C)
e