Some web sites (Tolkienbooks.net and Tolkien Library) include The Oxford Book of Greek Verse in Translation in the JRRT bibliographies.

Tolkienbooks.net lists the work under "Other Credits", saying that "Tolkien helped with 'Introduction Part II - Greek Poetry in Translation'.", and gives 1944 as the publication year.

Tolkien Library gives 1938 as the publication year.

Amazon.co.uk also has 1938 as the publication year, saying it was the 2nd edition.

The Oxford Book of Greek Verse in Translation is not included in the section "Published Writings by JRRT" in the JRRT Companion and Guide (I'm eagerly awaiting the promised new edition of the JRRT: A Descriptive Bibliography!). Can anyone point me to a published source which includes a description of this book in relation Tolkien's alleged contribution?

Was there an edition of this book published before 1938 (the 1st edition?), and did Tolkien not contribute to this edition?

Thank you!

In my 1938 edition from The Claredon Press is written:
"[...] Mr. E. A. Barber and Mr. D. L. Page have given valuable help in the Notes; Professor J. R. R. Tolkien and Mr. W. H. Shewring in the second part of the Introduction; [...]"
I don't know of an earlier edition, but I know there exists a blue version from the same year that prints 2nd ed., the 1944 edition was already the 5th printing (if I'm correct).

I have a 1942 edition (which has the same acknowledgement to Tolkien quoted by Beren) and states

First Published January 1938
Reprinted February 1938, April 1938
1942

That suggests (as Beren says) that 1944 would be at least a 5th printing.

Confirm
I have a 1944 5th impression

My mistake at TolkienBooks.net. Thanks for pointing out the error.

The first printing of 1938 gives no indication of a previous edition in the bibiliography, so I would presume that some sellers list the second printing as a second edition.

Amazon is not entirely reliable as a source for publication dates of old books. There is a copy on there published in 1000!