Cours d'analyse de l'École Royale Polytechnique; Première Partie. Analyse algébrique¶
Cauchy, A. -. (1821). Cours d'analyse de l'École Royale Polytechnique; Première Partie. Analyse algébrique: Analyse algébrique.
Cited by¶
4 citations across 4 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Completeness
- **Completeness is the no-gaps-in-the-structure principle that names the condition under which a system's internal processes (convergence, deduction, coverage, the construction of canonical extensions) have their natural terminations inside the system itself rather than escaping to a larger ambient structure.**
This sourceFoundational early formulation of the limit-and-continuity framework for real-valued functions; states the Cauchy convergence criterion (the originating articulation of the no-gaps idea in analysis) that anticipates the later epsilon-delta condition, though the criterion was made fully rigorous only with the Cantor/Dedekind completion of 1872.
Supported in partVerified against the publisher's abstract
Cauchy's Cours d'analyse is the foundational articulation of the limit and convergence framework, supplying only the analytic convergence limb of the no-gaps principle the claim states.
“Cauchy was a pioneer of mathematical analysis, a branch of mathematics concerned with the idea of a limit, whether of a sequence or of a function.”
- **Completeness is the no-gaps-in-the-structure principle that names the condition under which a system's internal processes (convergence, deduction, coverage, the construction of canonical extensions) have their natural terminations inside the system itself rather than escaping to a larger ambient structure.**
- Continuity
- … is the no-sudden-jumps principle: a mapping or process for which arbitrarily small changes in input produce arbitrarily small changes in output, formally captured by the epsilon-delta condition for real-valued functions (`∀ε > 0, ∃δ > 0 : |x - x₀| < δ ⟹ |f(x) - f(x₀)| < ε`) due to Cauchy (1821) and Weierstrass (1872)
This sourceFoundational early formulation of the limit-and-continuity framework for real-valued functions; gives the first rigorous limit-based definition of continuity and convergence, anticipating the later Weierstrassian epsilon-delta condition.
Supported in partVerified against the publisher's abstract
The abstract confirms Cauchy's Cours d'Analyse founded the rigorous limit-based treatment of real functions, but says nothing of the epsilon-delta condition the claim assigns to him.
“Cauchy was a pioneer of mathematical analysis, a branch of mathematics concerned with the idea of a limit, whether of a sequence or of a function.”
- … is the no-sudden-jumps principle: a mapping or process for which arbitrarily small changes in input produce arbitrarily small changes in output, formally captured by the epsilon-delta condition for real-valued functions (`∀ε > 0, ∃δ > 0 : |x - x₀| < δ ⟹ |f(x) - f(x₀)| < ε`) due to Cauchy (1821) and Weierstrass (1872)
- Convergence
- **Convergence is the limit-approach principle: a sequence or process is said to converge when its elements eventually enter and remain within every neighborhood of a target limit, formally captured by the epsilon-N condition for sequences (`∀ε > 0, ∃N : n ≥ N ⟹ d(xₙ, x) < ε`)
This source(Originating treatment of the modern sequence-convergence framework, including the Cauchy criterion — a sequence converges in `ℝ` iff it is Cauchy — and the basic theory of convergence of series. Same source publication as the continuity citation; the convergence treatment is distinct enough to warrant a separate inline marker, but the bibliographic entry consolidates in B3 verification.)
- **Convergence is the limit-approach principle: a sequence or process is said to converge when its elements eventually enter and remain within every neighborhood of a target limit, formally captured by the epsilon-N condition for sequences (`∀ε > 0, ∃N : n ≥ N ⟹ d(xₙ, x) < ε`)
- Infinity
- The rigorous formalization of limits in analysis, pioneered by Cauchy (1821)
This source(Foundational early formulation of the limit-and-continuity framework for real-valued functions; introduces the limit-based definition of continuity that anticipates the later Weierstrassian epsilon-delta condition.)
Supported in partVerified against the publisher's abstract
The publisher's abstract for the 1821 Cours d'analyse credits Cauchy with pioneering mathematical analysis around the idea of a limit, which supports the attribution but not the phrase 'rigorous formalization of limits'.
“Cauchy was a pioneer of mathematical analysis, a branch of mathematics concerned with the idea of a limit, whether of a sequence or of a function.”
- The rigorous formalization of limits in analysis, pioneered by Cauchy (1821)
Verification¶
Does it exist? Not checked yet. This entry carries no identifier to resolve. It was extracted from the citation as written in the article, normalized, and deduplicated against the rest of the registry.
Does it back the claim? Read against the text for 3 of 4 citations: 3 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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Links previously used in the corpus¶
Before the registry existed this work was also linked 1 other way.
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