In volume 8 of the Michaux Herbarium lies this lovely specimen, which Michaux called Polygonum hydropiper. That species name is no longer recognized by ITIS and has been replaced by the name Persicaria hydropiper though the USDA sticks with Polygonum hydropiper. I'm not making any declarations as to whether that is the exact determination of this specimen; we're nowhere near volume 8 yet.
What interests me is this note up at the top right corner of the page:
Apparently the plant is very peppery, hence the specific name "water pepper." It was found in Pennsylvania, the Carolinas, and Illinois - that's easy. But that last line?
"Arse weed des Américains" is one of the more evocative phrases I've encountered in these specimens. What on earth can it mean?
Googling the term first results in a bunch of hits from cannabis growers. There's this rather bizarre blog account, which I hesitate to credit. This plant name book provides a list of common names for P. hydropiper, which include Arse-smart and Arsenick. Merriam Webster confirms the Arsesmart usage for this species. It's apparently a synonym for Smartweed, which is another popular common name for P. hydropiper.
Which brings us to the reason for the name, courtesy of WebMD: Smartweed, or arsesmart, scientifically known as Polygonum hydropiper, is a plant widely used as a medicine for bleeding hemorrhoids. It's also used to treat uterine bleeding and diarrhea. The plant contains large amounts of vitamin K, which help blood to clot. The active ingredient is known as polygonic acid. It was of interest to pharmacists in the 19th century. Researchers in Bangladesh have recently proven that the plant does have antibacterial properties, validating its use in traditional medicine.