
This small corner contains a minor unpublished paper of mine, New Proofs of the Irrationality of e^{2} and e^{4}, which I submitted to the MAA Monthly back in 2002. A referee (see added note  of more than twenty years later  below) pointed out that the main idea was to be found in Liouville, a point evidently lost on no less a person than C. L. Siegel (see his classic book on transcendental numbers, a book that was one of my paper's references). (Originally I had written: "I also prepared an accompanying Maple worksheet", but I have decided to remove it for two reasons: only readers who have the software Maple could open it, and  while I could convert that worksheet to pdf format (so that anyone could read it), the file extension ends up being Ziegler's Proofs from the BOOK. I sent my proof to Günter Ziegler (before I heard from the Monthly Editor), and he liked it, to the extent that he was going to include it in the 3rd edition of their book. After I heard from the Monthly, I informed Günter Ziegler, who included Liouville's argument in the 3rd edition, thanking me for bringing it to their attention. While I was disappointed to have been anticipated by Liouville, I was more pleased to receive an autographed copy of the third edition from Günter. Some comments of late Nov. 2023. Doing some website editing, I have looked at this page for the first time since Günter kindly sent me his autographed book. I was tempted to just let the page stand as it was (here is how the page appeared in 2019, for which I have to thank the remarkable WaybackMachine), but  simply to get it off my chest, for I do not expect any readers at this stage  I want to make some comments. One. The proofs offered in my paper were ones that I made at a time when I worked in the Mathematics Department of Manchester University, when I included them  as no more than a small aside  in an advanced course that I gave to faculty on transcendental numbers. And, while I certainly thought my proofs to be original, my personal standards were such that I did not consider they would be worthy of publication, and thus I didn't write them up for possible acceptance anywhere. Two. Nevertheless, when shortly afterwards I attended the 1974 British Mathematical Colloquium in Sussex I decided to present them in a 15minute talk at the Number Theory Splinter Group. My entirely undemanding talk was wellreceived (perhaps because everyone would have followed everything!), and I even received thankyous afterwards from no less personages than David Masser (then a rising star) and the supernova JeanPierre Serre (he who could have tossed off such proofs when he was only five years old). What did their behaviour represent? Simply this: good manners. Three. It was only in 2002  on reading the remark by Aigner and Ziegler (that no elementary proof  along the lines of the universally known proof of the irrationality of e  that no such proof, according to them, had been given for the known irrationality of e^{2}) in the second edition of their book, that I decided to write up what I'd had since 197274, and submit it to the obvious place: the Monthly. In the meantime  while awaiting a response from the Monthly I sent my proofs to A+Z. (Z. warmly responded). Four. In the course of time an offensive/supercilious referee's report was sent to me by the then Editor of the Monthly  the Editor wasn't offensive, it was the (as per practice) anonymous referee. I didn't keep it I should add, but I very much remember its content/tone, and it went something like this: the author is clearly ignorant of the history. All he had to do was go into his library and there find Liouville's proof (dating from the mid 1800s)... . Five. What could a wellmannered, sensitive referee have written as a report? I would suggest something along these lines:

Contact details. jbcosgrave at gmail.com
