Chiasmus is a literary technique in which either words, ideas, or grammatical structures are repeated in an inverted way. In 1787, Anglican bishop Robert Lowth discovered that chiasmus is used in the Bible. For example, in Matthew 19:30 the words "first" and "last" get repeated in reverse order:
But many that are first
shall be last;
and the last
shall be first.
Another example is found in Psalms 124:7 where the words "escaped" and "snare" create an inverted parallelism:
Our soul is escaped as a bird
out of the snare of the fowlers:
the snare is broken,
and we are escaped.
Chiasmus can be unintentional at times, but there are other instances where it's obviously done on purpose, such as in Isaiah 6:10:
Make the heart of this people fat,
and make their ears heavy,
and shut their eyes;
lest they see with their eyes,
and hear with their ears,
and understand with their heart, and convert, and be healed.
Here, the pattern is heart, ears, eyes, eyes, ears, heart.
In the late 1960's, John W. Welch, the founder of the Mormon apologetic group Foundation for Ancient Research and Mormon Studies (FARMS), discovered that there is chiasmus in the Book of Mormon. For example, Mosiah 5:10-12:
And now . . . whosoever shall not take upon him the name of Christ
must be called by some other name;
therefore, he findeth himself on the left hand of God.
I would that ye should remember also, that this is the name . . .
that never should be blotted out,
except it be through transgression;
therefore, take heed that ye do not transgress,
that the name be not blotted out of your hearts . . .
I would that ye should remember to retain the name . . .
that ye are not found on the left hand of God,
but that ye hear and know the voice by which ye shall be called,
and also, the name by which he shall call you.
Welch claims that it would have been very unlikely for Joseph Smith, who produced the Book of Mormon, to have known about chiasmus, therefor, the existence of chiasmus in the Book of Mormon proves that Joseph Smith didn't write it.
Does this mean that the Book of Mormon really was written by ancient Jews living on the American continent?
In his book, Early Mormonism and the Magic World View, D. Michael Quinn obliterates Welch's claims in a footnote. Quinn points out that Robert Lowth's study of chiasmus in the Bible from 1787 was available in an American edition by 1815, well before the 1830 publication of the Book of Mormon. Welch himself admits this in his 1970 master's thesis, although he hasn't acknowledged it since.
Chiasmus received much more recognition with the first American edition of Thomas Hartwell Horne's Introduction To the Critical Study and Knowledge of the Holy Scriptures in 1825. Not only was Horne's book advertised in Palmyra's newspaper (which Joseph Smith was known to have read), but the 1818 London edition of the book was for sale by 1820 in the Canandaigua Bookstore (which also had a lending library) only nine miles from Smith's home.
Far from being nearly impossible, Joseph Smith being aware of chiasmus is actually rather likely. He could have found out about it by going to a nearby library or bookstore, by noticing the advertisement in the newspaper he always read, or by having a conversation with someone who'd heard about it. Sorry John, but chiasmus does not prove the Book of Mormon is true.