# The Originator of Cobweb Diagrams

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.

According to Alessandro Rosa, An episodic history of the staircased iteration diagram", ANTIQUITATES MATHEMATICAE Vol. 15(1) 2021, p. 3–90, doi: 10.14708/am.v15i1.7056, a version of the diagram can be traced back to Adrien-Marie Legendre, in 1822.