This a very terse proof. It is stated "The same method can also be used to prove that e cannot be a root of a second degree equation with rational coefficients", but if I recall correctly, the generalization for the irrationality of $e^2$ isn't entirely straightforward, mainly due to the presence of powers of 2 in the terms of the expansion.
Off topic, but can you tell me whether you're using a browser extension to render $e^2$ (and if so which one!), or if you're simply using the $ character as a delimiter for humans who read and write a lot of LaTeX code?
I always found it weird that despite being developed at CERN and most widely used in the early years by universities and research labs, HTML never developed a decent standard for transmitting/rendering math equations (I know that there's a math tag now, but it doesn't seem to get much use). Especially given that Latex would seem to provide a pretty widely known de-facto standard for the syntax.
1) HTML is derived directly from SGML which was used to handle similar markup issues in printed texts. One of the reasons why HTML was successful was because there were already similar tools in place from SGML that could be quickly applied to handle processing and correctness. Adding math would require building a number of additional tools.
2) TeX by itself doesn't resolve very well into the DOM tree which is what parsers use to organize the markup data internally. You need a markup language that is similar to HTML if you want to make it easy for browsers, parsers, and libraries to adopt the new standard.
This may be a little ridiculous, but I have a bookmarklet that I click on sometimes to render most LaTeX in the current window.
I use this because I'm a mathematician and I chat with some other mathematicians through Slack frequently, and this doesn't have good LaTeX support. But by clicking this, it loads MathJax from the MathJax CDN to render. We also use this on the chatrooms associated to Math.StackExchange and MathOverflow, which is why it's called ChatJax.
As pointed out in the annotation, isn't the case $e^2$ just exactly the same proof but for the special case $c=0$? And in any case, if $e^2$ were rational, then $e$ could only be rational or a quadratic irrationality.
The way math papers and proofs are written is fascinating, because it reveals so much about how people read mathematical notation. Much of the older notation was more explicitly used as shorthand for words or phrases, rather than being a language in its own right (as it is today). (And old papers were written out in natural language, which seems so hard to read by today's standards.)
I especially like how certain symbols we regularly use nowadays weren't always known that way. For example, the set of integers is now written as \mathbb{Z} , but used to be written as a fraktur "Z" with a bar above.
"was" considered?... In French, traditionally, the inequality signs do not include the verb to be; that is how I learned it some 40 years ago. Thus "<" is "plus petit que" although some people nowadays would take it to include the verb to be as in "est plus petit que", mimicking the English usage.
I still see this, particularly when it would be awkward to create a new symbol: "when the discriminant is ≤ 1". I feel that Halmos [1] wouldn't have liked it, but it does lessen the distance between what is written and the equation the reader will want to have in his mind.
Funny how my geometry teacher back in high school didn't offer that up as a means of completing any of the proofs we had to do. It would have made finishing my homework so much faster.
