To be more specific, the book would be H. M. Belden's Ballads and Songs Collected by the Missouri Folk-Lore Society (1955). The song is number 408 in the Roud Folk Song index, and Laws number Q30. It's of English origin, of unknown authorship, and very common; I sometimes think that the few collectors who don't report finding it just thought "Oh, no, not that one again", and didn't bother mentioning it! It has also circulated quite widely in songbooks since the 1850s (and on broadsides for a bit longer), which would in part account for the fact that it varies so little.

The entry at The Traditional Ballad Index takes it back as far as 1845, when it appeared in the log of the ship Elizabeth out of New Bedford (Gale Huntington, Songs the Whaleman Sang, 1964). Farmer's Boy, The [Laws Q30]

You can see a text from Robert Bell's Ancient Poems, Ballads, and Songs of the Peasantry of England (a revision of James Dixon's 1846 book of the same name) at:

The Farmer's Boy

The editors considered it likely to be a product of the earlier part of the 18th century, but opinion nowadays would tend to place it later; perhaps in the early 19th. Broadside catalogues take the date back to around 1832, beyond which point there don't seem to be any references to it; it may not be much older than that.There are a number of broadside examples (probably all 19th century) at Bodleian Library Broadside Ballads; these are to be found under the titles of The Farmer's Boy and (perhaps its earlier name) The Lucky Farmer's Boy. There are also some later Irish editions, titled The Farmer's Lucky Boy; and a slight re-write, The Lucky Factory Boy.