Subject: Re: [HM] Bolzano, Cauchy, Epsilon, Delta (3/5)
From: Joao Filipe Queiro (jfqueiro@mat.uc.pt)
Date: Fri Feb 04 2000 - 06:08:23 EST
Dear list members,
I refer to the following passage of the recent article "Bolzano, Cauchy,
Epsilon, Delta", by Walter Felscher;
"... yet neither cares to mention the "there exists delta" - presumably as
their authors considered it as obvious that an approximation, once
achieved, would finally progress to the better. The difference between the
two authors is the use they make of their definitions: Cauchy wanted to
exhibit more than d'Alembert, wanted to prove consequences of that
definition, and to this end he discovered the delta and its role."
I would argue that a perfect epsilon-delta argument appears in the treatise
'Principios Mathematicos' (Lisbon 1790), by Jose Anastacio da Cunha. This
book appeared in French translation in 1811 and again in 1816.
In Chapter XV (on the differential calculus), Cunha defines an
infinitesimal: "a variable which can always admit values than any proposed
quantity". Proposition I of this chapter states that a polynomial in x with
zero constant-term is an infinitesimal if x is.
The proof runs as follows: Let Ax + Bx^2 + Cx^3 + Dx^4 + ... be the
polynomial. Denote by n the number of coefficients, and let P be any
quantity larger than each one of them. Let Q be any proposed quantity, and
take x < Q/nP and < 1. Then Q/n > Px^k for all k up to n, whence Q > Ax +
Bx^2 + Cx^3 + Dx^4 + ... .
[Clearly all numbers involved here are assumed to be positive. Here and in
some other passages in the book, no distinction is made between a number
and its absolute value, but conditions like 'Let a be positive' also appear
when they are essential.]
Cunha's 'Principios Mathematicos' are interesting in that the author wants
to present much of the mathematics known at the time in a systematic way,
as a body governed by carefully chosen axioms and definitions. This led to
remarkable "foundational" work on series [the Bolzano-Cauchy criterion is
presented as the definition for series convergence, and used in proofs], on
exponentials [a^b being *defined* by exp(b.log a), with exp and log
presented as series, previously studied for convergence], on differentials,
and so on. The style is very concise throughout.
A reference is:
J. F. Queiro, Jose Anastacio da Cunha: A forgotten forerunner, The
Mathematical Intelligencer 10 (1988), 38-43.
An expanded version of this appeared in: Matematica Universitaria
(Brazilian Math. Society), 14 (1992), 5-27.
Other references on Cunha, taken from the University of St Andrews archive:
1.Biography in Dictionary of Scientific Biography (New York 1970-1990).
Articles:
2.A N Bogolyubov, The views of J A da Cunha in the domain of mechanics
(Russian), in Studies in the history of mathematics 19 'Nauka' (Moscow,
1974), 177-187.
3.A L Duarte and J C e Silva, Some comments on the 'Mathematical
principles' of Jose Anastacio da Cunha (Portuguese), Proceedings of the
XIIth Portuguese-Spanish Conference on Mathematics II (Braga, 1987), 274-289.
4.A J Franco de Oliveira, Anastacio da Cunha and the Concept of Convergent
Series, Archive for History of Exact Science 39 (1988), 1-12.
6.A P Youschkevitch, J A da Cunha et les fondements de l'analyse
infinitesimale, Revue d'histoire des sciences et leur applications XXVI
(1973), 3-22.
7.A P Youschkevitch, C F Gauss et J A da Cunha (French), Rev. Histoire Sci.
Appl. 31 (4) (1978), 327-332.
8.A P Youschkevitch, C F Gauss and J A da Cunha (Russian), Istor.-Mat.
Issled. 24 (1979), 186-190, 387-388.
9.A P Youschkevitch, J A da Cunha and problems on the foundations of
mathematical analysis (Russian), in Studies in the history of mathematics
18 'Nauka' (Moscow, 1973), 157-175, 337.
10.A P Youschkevitch, J A da Cunha et les fondements de l'analyse
infinitesimale, Rev. Histoire Sci. Appl. 26 (1) (1973), 3-22.
Both the 1790 Portuguese edition and the French translation of the
'Principios' were published in facsimile by the Coimbra Math. Department in
1987 (200 years after Cunha's death; the 1790 book was published
posthumously). Copies are still available. The French translation is in
general reliable, but faulty at some crucial passages, like the convergence
definition for series. This is analysed in the Queiro and Franco de
Oliveira references above.
With best wishes,
Joao Filipe Queiro
University of Coimbra
Portugal
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