In the introduction to his A Third Book of French Country Dances or Cotillons, Gherardi complained:

‘The reason of the little Improvement (generally speaking) hitherto made in the Cotillons, has been and is, doubtless, owing to the obscure and unintelligible method of explaining the Figure; for, to this Day, Masters have generally adopted Terms made use of for the English Country Dances; which, inadequate as they must appear to be in pointing out the Figure, leave the Dancer totally in the dark with respect to what he ought to do himself, or cause his Partner to perform.’

Gherardi’s answer to this problem was to repeat what he had done in his second book, ‘I think it not improper to explain them, both by Representation and Words’. So, he again used diagrams to make the figures as clear as possible. He chose to both explain and illustrate nine figures, including: simple chassé across; chassé dessus et dessous; chassé double (for which he gave two diagrams). He also showed some more complex figures, for which I will give only his diagrams:

Did he need to use all this ingenuity to keep his cotillons interesting and, above all, novel and thus fashionable?

After his explanations and illustrations, Gherardi was careful to add:

‘I recommend to the Lovers of the French Country Dances, or Cotillons, a careful and frequent consideration of these Figures, & also of those in my last Book, … in order to fix them strongly in their Memory.’

He ended his introduction by reminding his readers that ‘Mr. Gherardi’s Academy is begun for the Winter’. Gherardi’s books were not so much self-help manuals as advertisements.