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# Sinkiewicz G.I. On History of Epsilontics. Antiquitates Mathematicae. 2016. Vol 10. P. 183-204. Digital Object Identifier (DOI): 10.14708/am.v10i0.805.

Authors:

## Abstract

This is a review of the genesis of ε–δ language in works of mathematicians of the 19th century. It shows that although the symbols ε and δ were initially introduced in 1823 by Cauchy, no functional relationship for δ as a function of ε was ever specified by Cauchy. It was only in 1861 that the epsilon-delta method manifested itself to the full in Weierstrass’ definition of a limit. The article gives various interpretations of these issues later provided by mathematicians. This article presents the text of the same author which is slightly redone and translated into English
ANTIQUITATES MATHEMATICAE
Vol. 10(1) 2016, p. xx–zz
doi: 10.14708/am.v10i0.805
δ
 δ
δ 
Galina Ivanovna Sinkeviq
x
lim ∆P
∆x
(x+ 1)µ=a
µ
nth
lim
x=c
lim
xc
fx x
x+i f(x+i)
i
f(x+i) = fx +iP
P=f(x+i)fx
ii
i
f(x+i)f(x)
i
x i x
x+i
i= 0 0
0
i1
i x
x=a x =k f(x)
A K x =a
x=k b, c, d, e,
B, C, D, E
KE
ke
EA
ea
EA
ea<KA
ka<KE
ke
x, f 0(x)f(x+i)f(x)
i
x x +i
x+i=zf(z)f(x)
zx=p f(z) =
f(x) + p(zx)
f(z) = f(x) + f0(x)(zx) + p0(zx)2
f(z) = f(x) + f0(x)(zx) + f00(x)
2(zx)2+p00(x)
2(zx)3
f(z) = f(x)+f0(x)(zx)+ f00(x)
2(zx)2+f000(x)
23(zx)3+p000(x)
23(zx)4
f(x)f0(x)
δ
f(x)x
x
x
x a
f(x+a)f(x)
a x
f(x)x
x
f(x+a)f(x)
a f(x)
x
f(x)
x x
x
x f(x+1)f(x)
kf(x)
x
k 
x
f(x+ 1) f(x)k
h x h
k k +
n
f(h+1)f(h), f (h2)f(h+1), . . . , f (h+n)f(h+
n1) f(h+n)f(h)
n
k k+f(h+n)f(h)
n=k+a
a, +
h+n=x
f(x)f(h)
xh=k+a
f(x) = f(h)+(xh)(k+a)
f(x)
x=f(h)
x+1h
x(k+a)
x
n
h x
f(h)
x,h
x
k+a a
+f(x)
x
k k +
klim f(x)
x=k= lim[f(x+ 1) f(x)]
x±∞
δ 
δ
f(x)
x
f(x+i)f(x)x
f(x)
x
y x
y=f(x)
y x
y=f(x)∆y y
∆x x
y+∆y =f(x+∆x)
∆y =f(x+∆x)f(x)
h i
a=i
hh
∆x ∆y
f(x)
∆x ∆x =i=
ah ∆y f(x+i)f(x)f(x+ah)f(x)
f(x)
x f(x+i)f(x)
f(x)
f(x)
x
f(x)x=
x0, x =X A B
f(X)f(x0)
Xx0
A B
δ, 
i
δ x x0X
f(x+i)f(x)
if0(x)
f0(x) +
x1x2. . . xn1
x0X X x0
x1x0, x2x1, . . . , X xn1δ
f(x1)f(x0)
x1x0
f(x2)f(x1)
x2x1
f(X)f(xn1)
Xxn1
f0(x0) f0(x0) +  f0(x1) f0(x1) +
A B +
f(X)f(x0)
Xx0A B +
f(X)f(x0)
Xx0
A B
A < f0(x) < f(x+i)f(x)
i< f0(x) +  < B i < δ
 δ
δ
δ δ
xn
a f(xn)
f(a)
δ
δ x
f(x+i)f(x) = pi +qi2+ri3+. . .
f x x +i
θ < 1f(x+i)f(x) = if 0(x+θi)
inf
x[x0,X]f0(x)f(X)f(x0)
Xx0sup
x[x0,X]
f0(x) ()
x0X
f(x)
x x =x0x=X b
f(x0f(X)f(x) = b
x0X
[x0, X]
f
x0X 
δ i
δ x x0X
f0(x)f(x+i)f(x)
if0(x) +
x x0X
[x0, X]
f(x)
x
δ
 δ
δ < 
δ < 
lim
x=
pn=
δ
 δ
δ
f(x)x x
x x+h f(x+h)
f(x+h)f(x)
x x+h
δ h
h δ f(x+h)f(x)
 δ
 δ
δ
δ
f(x)x0
 η()|h| ≤ η()
|f(x0+h)f(x0)| ≤  f(x)
(a, b) η()x0
x0(a, b)
 δ
 δ
 δ
δ 
 δ
N. Aleksandrova. Matematiqeskie terminy. Sprav-
oqnik Moskva: Vysxa
xkola
I. Baxmakova. O roli interpretacii v istorii
matematiki
Istoriko-matematiqeskie issledovani. Moskva: Nauka
B . Belhost. Ogsten Koxi
Moskva: Nauka
O. Koxi. Algebriqeski\$i analiz
F. val~d, V. Grigor~ev, A. Il~in
O. Koxi. Kratkoe izloenie urokov o differ-
encial~nom i integral~nom isqislenii
Bunkovski\$i
S. Demidov. “Zakon nepreryvnosti”Le\$ibnica i pontie
nepreryvnosti funkcii u \$ilera
Istoriko-matematiqeskie
issledovani
A. Dorofeeva A. Formirovanie ponti nepreryvno\$i
funkcii Is-
tori i metodologi estestvennyh nauk. Moskva
P. Dgak Pontie predela i irracional~nogo qisla, koncep-
cii Xarl Mere i Karla Ve\$ierxtrassa
Istoriko-matematiqeskie issledovani
L. \$iler. Vvedenie v analiz beskoneqno malyh
Moskva: Fizmatgiz
L. \$iler. Nastavlenie po differenciaǉnomu isqisleni.
L. \$iler. Differenciaǉ-
noe isqislenie Moskva-
A. Lebeg. Integrirovanie i otyskanie primitivnyh
funkci\$i.
G.I. Sinkeviq. Uliss Dini i pontie nepreryvnosti
Istori nauki i tehh-
niki. Moskva
G.I. Sinkeviq. Genrih duard Ge\$ine. Teori funkci\$i.
Matematiqeskoe mod-
elirovanie, qislennye metody i kompleksy programm. St.-Peterburg
G.I. Sinkeviq. Razvitie ponti nepreryvnosti u X.
Mere. Trudy Me-
dunarodnyh Kolmogorovskih qteni\$i
roslavl~
G.I. Sinkeviq. K istorii psilontiki
Matematika v vysxem obrazovanii
A.xkeviq. Istori matematiki
A.xkeviq. Moskva: Nauka
A.xkeviq. L. Karno i konkurs Berlinsko\$i Akademii
Nauk 1786 na temu o matematiqesko\$i teorii beskoneqnogo
Istoriko-matematiqeskie issledovani
A. xkeviq. Hrestomati po istorii matematiki.
Matematiqeski\$i analiz
A.xkeviq. Moskva: Prosvewenie
A. xkeviq Razvitie ponti predela do K.
Ve\$ierxtrassa
Istoriko-matematiqeskie issledovania. Moskva: Nauka
 δ
 δ
δ 
Vtoraja Krasnoarmejskaja ul.
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Article
We explore Leibniz's understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus of Newton and Leibniz have been a target of numerous criticisms. Some of the critics believed to have found logical fallacies in its foundations. We present a detailed textual analysis of Leibniz's seminal text Cum Prodiisset, and argue that Leibniz's system for differential calculus was free of contradictions.