We have "demonstrations over chilblains" and such things.

It seems to be a curtailed way of saying "demonstrations of the power of Christian-Science Truth over the fiction which masquerades under the name of Chilblains." The children, as well as the adults, share in the blessings of the Science.

"Through the study of the 'little book' they are learning how to be healthful, peaceful, and wise." Sometimes they are cured of their little claims by the professional healer, and sometimes more advanced children say over the formula and cure themselves.
A little Far-Western girl of nine, equipped with an adult vocabulary, states her age and says, "I thought I would write a demonstration to you." She had a claim, derived from getting flung over a pony's head and landed on a rockpile.

She saved herself from disaster by remembering to say "God is All" while she was in the air.

I couldn't have done it.
I shouldn't even have thought of it.