Reading Cauchy's lecture on the derivative

In summary, the conversation discusses Cauchy's lecture on the derivative and the techniques used to prove the limit. While Cauchy's result is true, it is not particularly rigorous and later mathematicians, such as Karl Weierstrass, developed more rigorous techniques for proving limits. Cauchy's intuition and willingness to polish his arguments are still highly regarded and serve as a foundation for modern mathematics.
  • #1
henpen
50
0
Reading Cauchy's lecture on the derivative, I see he goes from this limit

[tex]\large \lim _{h \rightarrow 0} \frac{1}{ \log_A((A^h)^{\frac{1}{A^h-1}})}[/tex]
To this one [itex]A^h= 1+\beta[/itex]
[tex]\large \lim _{\beta \rightarrow 0} \frac{1}{ \log_A((1+ \beta)^{\frac{1}{\beta}})}=\frac{1}{\log_A(e)}[/tex]

I understand the intuition behind the technique, and the result. However, is this variable-change rigorous? How can we be sure that [itex] \beta =A^h-1 [/itex] tends to 0 in the same way as [itex] h[/itex] does, or do we just need to know that when [itex] h=0 \Rightarrow \beta=0[/itex], so the limit will be the same?
 
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  • #2
No, it isn't particularly rigorous.
Cauchy made great headway in his time to elucidate how proofs ought to be made, but it was a really tough, and interesting history how mathematicians eventually managed to make their arguments as fully rigorous as the demands of our day has become.
When it comes to the tricky business of limits, the first mathematician who systematically developed a rigorous technique for this was Karl Weierstrass, working some decades later than Cauchy.

In this case, for example, I seriously doubt that at Cauchy's time, a rigorous definition of the exponential for all real numered exponents had been made, along with proofs based on continuous variable substition and how continuity of limits with respect to kernels of functions should be proven.

That being said, Cauchy's result is true. :smile:
 
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  • #3
arildno said:
No, it isn't particularly rigorous.
Cauchy made great headway in his time to elucidate how proofs ought to be made, but it was a really tough, and interesting history how mathematicians eventually managed to make their arguments as fully rigorous as the demands of our day has become.
When it comes to the tricky business of limits, the first mathematician who systematically developed a rigorous technique for this was Karl Weierstrass, working some decades later than Cauchy.

In this case, for example, I seriously doubt that at Cauchy's time, a rigorous definition of the exponential for all real numered exponents had been made, along with proofs based on continuous variable substition and how continuity of limits with respect to kernels of functions should be proven.

That being said, Cauchy's result is true. :smile:

Thank you. It seems my inner pedant is becoming more mathematical.
 
  • #4
Proofs are, really, a laborious polishing act. Geniuses like Cauchy had great intuition (that's why so much of his results are still regarded as true!), and a willingness to polish their arguments to the utmost of their ability. It is upon their shoulders we all stand, removing a bit of floss and dross that remained within their arguments..:smile:
 

1. What is Cauchy's lecture on the derivative?

Cauchy's lecture on the derivative is a famous mathematical work written by French mathematician Augustin-Louis Cauchy in 1823. It presents a rigorous and comprehensive definition of the derivative, a fundamental concept in calculus.

2. Why is Cauchy's lecture on the derivative important?

Cauchy's lecture on the derivative is important because it lays the foundation for modern calculus and analysis. It provides a clear and precise definition of the derivative, which is essential for understanding more advanced mathematical concepts and applications.

3. How did Cauchy's lecture on the derivative contribute to mathematics?

Cauchy's lecture on the derivative made significant contributions to mathematics by providing a rigorous and formal definition of the derivative. It also helped to resolve some of the existing discrepancies and controversies surrounding the concept of the derivative at the time.

4. Who is Augustin-Louis Cauchy?

Augustin-Louis Cauchy (1789-1857) was a French mathematician who made significant contributions to various fields of mathematics, including calculus, analysis, and number theory. He is best known for his work on the foundations of calculus and his rigorous approach to mathematical analysis.

5. Is Cauchy's lecture on the derivative still relevant today?

Yes, Cauchy's lecture on the derivative is still relevant today. The concepts and definitions presented in the lecture are still used in modern calculus and analysis, and it is considered a classic and influential work in the field of mathematics.

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