Precursors of continued fractions in a geometric guise go back to ancient Greece, geometric version of producing successive terms of a continued fraction expansion was known as anthyphairesis. Fowler is perhaps the most thorough scholar of ratios and anthyphairesis. Here is from his Approximation Technique, and its Use by Wallis and Taylor:
"This procedure can be used to approximate a wide class of numbers whose values are not known in advance, like roots and logarithms (as in Taylor's calculation, below), or it can be used to find more convenient approximations to a number for which some accurate but inconvenient approximation is given (as in Wallis' calculation).
This mediant has a long history going back to classical Greece: a special case
of it is to be found at Plato's Parmenides 154b-d, and, while the inequality
does not occur in Euclid's Elements, the case of equality was treated as V 12
and VII 12. It was then stated and proved in Pappus, Collection VIII 8. Chuquet
rediscovered it and called it 'la rigle des nombres moyens" in his Triparty en
la Science des Nombres (1484). It was, we shall see, adduced by both Wallis
and, in a variant form, by Taylor. It was the first theorem enunciated in Cauchy's Cours d'Analyse (1821). It is also, inter alia, the basic generating property of Farey series. Thus we have an enduring motif of mathematics."
This second paragraph inspired Guthery to write a monograph A Motif of Mathematics dedicated specifically to the mediant, its significance and history, from ancient roots to Farey series and the Riemann hypothesis.