It bears mentioning that liquid soap is likely older than you think: William Sheppard of New York City, given credit on Wikipedia and in a number of pop history books for inventing liquid soap in 1865, was in fact only patenting an improvement on the already-existing concept of liquid soap, making it thicker and probably closer to the consistency we expect today. And while the narrative usually makes it sound as though nobody really did anything with this until the twentieth century, this is not the case. The Era Formulary: 5000 Formulas for Druggists (1893), for instance, provided the men who ran drug stores - who had to create a number of their own products on request, rather than simply dispensing them all premade - with a selection of recipes for different solid and liquid soaps. (They start here.) Were they meant for hands, faces, or dishes? Maybe it was mostly intended for laundry.

So, if not using the liquid soaps, what did people use to clean dishes in the nineteenth century? Well ... for the most part, they used a block of soap. Instructions for washing dishes rarely went into the practice of creating suds, assuming that any woman looking for advice in cleaning could do that much on her own; from what they do say, it seems that one would generally stab a piece of solid soap with a fork and use that to swish it around in the water. (By the 1890s, women could put the soap in a soap shaker instead of using a fork.) Rubbing with sand was recommended before washing wooden items like rolling pins, and pots and pans that really needed to be scoured also were scrubbed with it. Baking soda was sometimes recommended for more stubborn grease/stains on dishes, for cleaning drains, or as a cheaper alternative for regular soap, but on the whole, block soap seems to have been the default.

Most of the advice is fairly boring and straightforward. Catharine Esther Beecher's Treatise on Domestic Economy (1843) explained that you should scrape off the food, then wash the nicest pieces in the sudsy washtub with the cleanest washcloth, then rinse them in your second tub of fresh water, then lay them out to dry; then refresh your hot, soapy water and wash your greasy dishes with a dedicated washcloth (rinse and dry); then refresh the water again and wash the kitchen tools. While later sets of instructions weren't carbon copies of this, in essentials they were the same: make the water soapy, try not to pollute it too much and get new soapy water when it's too dirty to clean with (a serious issue to keep an eye on before running water), rinse them before the suds dry, and let them drip dry afterward or rub them with a clean cloth.