I mean why was a separate term "continued proportion" created simply if $$a:b::b:d$$ Why could it just not have been called proportion? Is there any bigger use of continued proportion at a bigger level hence a separate term "continued proportion" was given to it?

  • $\begingroup$ Such proportions arise naturally all the time, at all levels of mathematics and its applications. Because of this, in the past somebody studied them and gave them a name because it made it easier to discuss them. Enough people found it useful that the name survived. $\endgroup$ – Paul Sinclair Nov 24 '18 at 18:12
  • $\begingroup$ What does the notation $a:b::b:d$ mean? – $a, b, c, d, \ldots$ are “in continued proportion” if $a:b = b:c = c:d = \ldots$, is that what your question is about? What exactly would you call “proportion” instead of “continued proportion”? – In its present form, your question is unclear to me. $\endgroup$ – Martin R Nov 25 '18 at 11:48

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