Infinitely small quantities defined by Cauchy

Question

I am reading Cauchy's book about analysis [0] and there he defines an infinitesimal quantity of first order as

$k\alpha$ or at least $k\alpha(1±\epsilon)$,

an infinitesimal quantity of second order as

$k\alpha^2$ or at least $k\alpha^2(1±\epsilon)$,

...

where $k$ denotes a finite quantity different from zero and $\epsilon$ denotes a variable number that decreases indefinitely with the numerical value of $\alpha$. What is this "or at least $k\alpha^n(1±\epsilon)$"? And what is the motivation behind it?

[0] Cauchy’s Cours d’analyse: An Annotated Translation - Bradley and Sandifer

Thanks

Answer 1

For pity's sake OP, don't try to learn analysis from the Cours d'Analyse. There's at least one serious and subtle error -- see here), and it is not by modern standards rigorous. That said, it's a fascinating book, especially if you are interested in Cauchy's ideas about infinitesimals.

In the première partie chapitre II, Cauchy introduces an 'infinitely small quantity' $\alpha$ which he says means 'a variable whose numerical values decrease indefinitely.' When such an $\alpha$ is fixed we can talk about infinitely small quantities of first order, second order, and so on, which are related to powers of $\alpha$.

Cauchy first says that a variable quantity is called infinitely small of the first order if its ratio with $\alpha$ converges to a finite nonzero limit (as $\alpha$ decreases to zero). You should regard that as Cauchy's definition. One example of an infinitely small quantity of first order is $k\alpha$ where $k$ is finite and nonzero.

More generally we can restate the definition by writing down the most general quantity which satisfies it. A quantity whose ratio with $\alpha$ tends to $k$ takes the form $$ \alpha k (1+\epsilon)$$ where $\epsilon$ is some quantity that tends to zero as $\alpha$ does. So a quantity is infinitely small of the first order iff it has that form for some finite nonzero $k$. The reason for the $\pm$ is probably that Cauchy tends to deal with infinitesimal quantities which are always positive (all the examples at the start of that chapter are).

Answer 2

Cauchy is interested in generalizing $k\alpha^2$ to $k\alpha^2(1±\epsilon)$ because he has in mind polynomial expressions in the infinitesimal $\alpha$. Any such polynomial would take the form $k\alpha^n(1±\epsilon)$ rather than merely $\alpha^n$ (where $\epsilon$ is another infinitesimal). Cauchy has a whole series of results on such infinitesimals in his Cours d'Analyse, as analyzed in this article.