As with many things today there is great dispute about what works what doesn't, etc. The 19th century also had such differing opinions. Below you will find two quotes one that's not to excited about Herschel's table and another that agrees.
This comes from the Meteorological tables and climatology of Vermont ©1877 by Hiram Adolphus Cutting
There is another class of signs which some believe in, that are merely superstitions, having no foundation in fact. The Hindoos have their rain gods, the South Sea Islanders their wind conjurers, and the negroes of Africa their rain doctors ; and previously we had our weather Almanacs and our Herschel's weather tables, one just as good as the other. Not many }'ears ago that celebrated Herschel's weather table, which Herschel never saw, was considered almost infallible, and Thomas' Almanac quite so ; but all enlightened people, unless some whose age has outgrown science, discard them. For the last thirty years our storms have taken place without regard to moon's quarters. We have had 2,668 storms, divided as follows: at new moon, 660; first quarter, 664; full moon, 668; last quarter, 676. This shows very plainly that the moon has nothing to do with storms. If the generally received idea was true, what little difference there is goes directly against it. The truth is that the moon has so little, if any, influence upon the weather, that men have never found out which way it is ; and I can say, without fear of contradiction, that rain and wind doctors, and Hindoo gods, have just as much to do with the weather as this weather table, and no more.
English Mechanic and World of Science ©1883
THE REPUDIATED WEATHER TABLE.
—Let Us now examine Herschel's weather table, as improved by Dr. Adam Clarke. Sunday, January 30th, 1881, n
I find that Saturday and Sunday moons coincide with the worst parts of the weather according to the weather table, and the 20-day period agrees with both. The eastern edge of the storm path I»"es over the British Islands, so that storms occasionally miss us altogether. Full moon, 7th IMober, 1881, came when the moon was ascendant jour days past perigee. Full moon, 25th November, 1882, will come four days past perigee. New moon, October 23rd, 1881, came seven days past apogee. New moon, December 10th. 1882, will come seven days past apogee. Full moon, December 24th, 1832, comes seven days past perigee. We shall see how the conditions agree this time. W. M. Gardner.