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# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 17
# Folio 1v, page id 00068035:0002
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Curva cujus natura
D
sit eq
ae v pX
De Demonstrandum est e FC
C tangens
Calculus haec docet: Zty Xa et tp X
Ergo tn3
Ly
a
W0
Demonstrare ita quaeramus: Ostendendum est si sit ATBAB. totam TC(C)
cadere extra curvam, AC(C) quam supepnimus ad easdem partes cavum
Cadat si ficri potest intra curvams eamque seced in duobus punctis C. (C) exqribus demittantur
rectae CB, (C) (B) in axem. ostendendum est er sit B(8) rC0. Ex hat 
AB 3 erit AB) p y +C et ex natura currae B(C) H y + Z
H DDC Unde auserendo BC ,, (8) D. restabit
N c0 + 2.4
23
a 42
C erc DC. et eri DC
C0 A
id est ex hypothesi V. Quod est absurdum,
v 2,2
eet
ai-
vo 2
ce
Potentiae a Binomid aut residuo conque altissimus  num quocunque  proximque a sumo gradus
summa dignitas alterutrius nominis mullipsa est sub numer potentiae exponente.
velminu
A+ B. Sit C aequalis AA B. erit quadratum a C aequale quadrato ab A, plus duploAinu
plus quadrato a B. el
et Cubus a C aequalis cubo ab A, plus triplo quadrato A in
B plus triplo A in B quadratum plus B cubus H
Et quadrato quadratum a C, aequab
quairato quadrato ab A, plus quadrapto cubo ab YA in B, etc. Paet ergo
A sit C aequalis A + B. erit C  idem quod A + 2AB + B.
Ubi patet in A et 2 AB reperiri 2.
in C et 3 A B reperiri 3,
et ita in celeris.
A3 + 3 AB + AB + B3
A+ + 4A B etc.
Si a binami potentia alterius nominis alteratrius sum Membra omnia potentiae a binomio
exceptis duabus summis partium potentiis si potentia
detur
sit curra sejus naturae, ut
in rem ad angulos rectos demissae
ordinatae sint. interex
exponente
Directric inde a vertice abscissarum
partium as multiplicata secundum numerum quemcunque, et inde a vertice versus
ordinatam in axe  ei sumatur portio abscisse quam minerus exponens indicat,
jungaturque nictum hujus paationis recta jungens hujus portionis extremum versus
ordinatam cadit, et punctum curvae ex quo ordinata ducta est, curvam tanget.
vel 
Sit curva proposita ACC. cujus vertex A. Punctum aliquod C. exquo in
directricem Axem ABB) demittatur ordinata, perpendicularis BC,  ( Sint aliud punctum (C) ex quo
ent
sintque ordinatae. BC et (8) (C) inter se in multiplicata exempli causa duplicata, tripli-
rectrice
cata aliave ratione ipsarum AB, A(B) ib exe inde a vertice per ordinatas
abscissarum. Sumatur portio AT inde a vertice ina directrice versus ordinatam.
BC, ita, ut sit A T portio dimidia tertiave, aut alia (secundum numerum scilicet exp
nentemr qui rationis multiplicationem exponit) abscissae AB, Dico quod juncta
TC.  curvam tanget.
secet si fieri potest in duobus punctis, C. (C). ducanturque duae ordinatae,
CB (C) (8) quarum differentia
Erit Tex
s
D(C) et abscissarum differentia B(B) vel CD
tum
 etem st
DCCd
cditum et enissas ut differentia
et dcrentia earigu
ond HB, pus ut Ae
i B1
eri Bqi erit ABC  ACub. A B plus triplum Quad. AB in, B(B) pl sriptum AB in triplum quadr. B(B), plus cubs 16)
ad cubus a Bab AB et differentia ipsarum (B)(C) er BC nempe D(C) Cd ipsas
a pradictarum
e en B(B) plus quad AB
Adeoque D(C) erit ad B(B) vel ad DC, ut B Quad. AB in B(B), + AB in 3 grad. B(B) tenb BB)
ad B(B), id est a ab lata communi altitudine B(B) ut   AB plus B(B) (quad AB, AB in 3B(B) + Euad B(B
ad umtatem seu parametrum E. Adeoque erit D() ad ratio rC (C)D ad DC, sive CB adBF
in differentia um potentiarum, nempe et (Z quad ABiM BB) + AB in 39 uad B(B) + cub B B))
ad ipsus potentias. Porro differentia potentiarum abscissis est ad ipsam abscissarum
differentiam BB Et quoniam potentiae omnes per olineas designari possunt
ergo nota obiter absundum sive
impossibile esse,  c
+y 9-, 2
111. 3
-
2
con
ere siit jo
AB,y. BCHX
Qugdratum
en
et
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