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Anyone know the history of using hyperbolic functions to parameterize the parabola? Wolfram hasn't really helped. Anyone know where to look?

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    $\begingroup$ For those interested, this was previously asked in Mathematics Stack Exchanged (with a bit more context), which I answered there before realizing (maybe 30 minutes after I posted my answer there) that it had subsequently asked here. $\endgroup$ Jun 26, 2021 at 14:05

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For the old version of the question, who first used the terms:

I got this at Earliest Known Uses of Some of the Words of Mathematics

HYPERBOLIC SINE and HYPERBOLIC COSINE. Vincenzo Riccati (1707-1775) introduced hyperbolic functions in volume I of his Opuscula ad Res Physicas et Mathematicas pertinentia of 1757. Presumably he used these terms, since he used the notation Sh x and Ch x.

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  • $\begingroup$ Google immediately gives: Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. So Lambert was also involved. $\endgroup$
    – markvs
    Jun 25, 2021 at 21:36
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    $\begingroup$ maa.org/sites/default/files/321922717729.pdf.bannered.pdf $\endgroup$
    – markvs
    Jun 25, 2021 at 21:40
  • $\begingroup$ I recall an old copy of Janke & Emde. They used $\operatorname{Sin}$ and $\operatorname{Cos}$ for the trig functions, with $\frak{Sin}$ and $\frak{Cos}$ for the hyperbolic functions. $\endgroup$ Nov 16, 2023 at 22:30

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