A Question Pertaining to Benjamin Peirce and Odd Perfect Numbers

Question

It is known, but perhaps not too well known, that Benjamin Peirce demonstrated in 1832 that an odd perfect number must have at least four distinct prime divisors. In fact, he did so before a mathematician named Nocco proved the case of 3 in the early 1860s. Moreover, more than 50 years passed before J. J. Sylvester and Clement Servais independently produced the same result in 1888 that Peirce did in 1832.

Sylvester has been credited with ushering in the modern era of research on odd perfect numbers. Peirce's contribution seems to have disappeared in obscurity, though I know that it was published in something called the $New \, York \, Mathematical \, Diary$.

Does anybody know how Peirce proved his result?

Many thanks.

Answer 1

Following MJD's lead in his comment, I found Peirce's paper in the Internet Archive, via the following hyperlinks:

Start page (267)

End page (277)

To quote part of the second footnote from this JSTOR hyperlink:

In a paper on perfect numbers by Benjamin Peirce, "Mathematical Instructor in Harvard University", The New York Mathematical Diary, no. 13, vol. 2, pp. 267-277, 1832, it is shown that there can be no odd perfect number "included in the forms $a^r$, $a^r b^s$, $a^r b^s c^t$, where $a$, $b$, and $c$ are prime numbers and greater than unity."