I recently acquired this battery jar (on left) from Bob Berry, who is selling items from the Jim Fielding estate. Markings and size as follows:

(top section) LD / PATENTED / JULY 4th 1871 (bottom section) F.C. OTTO & SONS / JERSEY CITY 5 3/8" tall and 4" across at the base

This jar is old, heavy and crude with some terrific orange peel surfaces. I showed it to a couple of seasoned collectors who had never seen anything like it. Online research revealed a couple that had sold on ICON and eBay, but nothing about their original application. I finally stumbled on this old Etsy listing that definitely identifies this jar as part of a quack medical device:

https://www.etsy.com/listing/153184068/antique-quack-medical-device-the-phenix?show_sold_out_detail=1&ref=nla_listing_details

Interestingly, the device has a Whitall Tatum marking! Here is part of the Etsy description:

"The box is stamped on the inside front lip - Pat. Aug 18, 1883 - Whitall. Tatum Co. - 5071. The battery glass jar and the black lid are both embossed with F.G. Otto & Sons New York."

I wonder if Whitall Tatum manufactured this entire device including the jar? It's an interesting mystery. I'm just happy to have found the definitive purpose of this VERY cool jar!