A cobweb diagram is a visualization tool that allows one to qualitatively study the iterates of a self-map of the real line based on the graph of the function; here is an example:
(Here the map is the Boole's transformation $x\mapsto x-1/x$. The link to the interactive Desmos graph is https://www.desmos.com/calculator/20yftwjmys.)
Other names for this construct include: zig-zag, staircase, graphical analysis, Verhulst diagram,..., the latter being from https://en.wikipedia.org/wiki/Cobweb_plot (the other terms are from e.g. Rudin, Devaney, Hasselblatt-Katok, Strogatz).
Question: Who first used cobweb diagrams in a publication? In particular, I would be interested in the verification or falsification of the attribution to Verhulst.
Added: According to the paper Prof. Friedland referenced, Verhulst never used cobweb diagrams, at least in his work on the logistic map; which seems reasonably attributable to him. Since the logistic map is a popular choice to introduce cobweb diagram, it seems there has been a transference of attribution.
Added: To continue, here is a relevant page from Legendre's 1816 monograph Essai sur la théorie des nombres (2e) (https://gallica.bnf.fr/ark:/12148/bpt6k62826k/f609.item.zoom#):
The context is approximation of roots; and two arbitrary curves are allowed (see also https://math.stackexchange.com/q/4435177/169085). Rosa, in the paper referenced by Prof. Friedland also mentions Bidone's work which precedes that of Legendre, but also that it's a possibility that already this was folklore. Rosa claims (p.35) that Fourier and Galois, together with Legendre, are regarded as the pioneers of the method, due to their interest and contributions.