I understand that $\cosh(x)$ traces the catenary curve and points $(\cosh(t), \sinh(t))$ give points on a unit hyperbola. And the rectangular hyperbola $xy = 1$ is the basis of the curve for the natural log.

What I am struggling with is the proving their equivalence with one another, from a historical point of view, without using modern calculus. I want to derive these proofs myself for the sake of clarity, but haven't found any good reference/source to aid in the matter.

  1. I know $e^{i\theta} = \cos \theta + i\cdot \sin\theta$. Algebraic manipulation allows me to come up with definitions of $\cosh\theta$ and $\sinh\theta$ as $\frac{(e^x \pm e^{-x})}{2(i)}$. How do I prove that these points lie on a hyperbola? This manipulation provides me a definition but not a proof.
  2. Riccati's derivation of hyperbolic functions predates that of the definition above. Although this page tries to explain it, the figures are so bad I have no idea what's really happening and how were they hyperbolic functions first discovered. How do I go from here to their modern definition in #1 above?
  3. Is there a relation between the above and the natural log? Sure looks like it. The area under the unit rectangular hyperbola gives the natural log like shown here. Any way to tie this to the above or is it an independent result?
  4. What's the proof that the graph of $\cosh\theta$ produces a catenary curve? How did we accept this proof initially? Saying that the equation graphs a catenary is one thing, but how was it proved to be a catenary? How do I tie #1-3 together into this?

Is it possible to develop an intuition of how all these concepts relate to each other, without resorting to calculus? I believe these discoveries predate Newtonian/Leibniz' Calculus - I'd like to trace my path through history just to get a more intuitive grasp of the evolution of these ideas and how they all fit together.

PS: It's alright if calculus is part of the solution - I'd prefer to know if it was possible to prove all this without calculus and if so how. Calculus, makes life easier but clouds the historical development/evolution/undertanding IMHO.

  • $\begingroup$ See Catenary and Catenary: History. The "discovery" does not predate calculus: without it, Galileo conjectured that a hanging cord is an approximate parabola. $\endgroup$ Commented Dec 20, 2017 at 14:50
  • $\begingroup$ Here's an answer explaining the connection between the natural logarithm and hyperbolic sine and cosine. (BTW, this answer gives geometric interpretations of all the trig functions.) $\endgroup$
    – Blue
    Commented Aug 21, 2018 at 22:19

1 Answer 1


I don't think it's possible to define the hyperbolic functions without calculus. I've seen the derivation of the definitions of sinhx and coshx. It's around 5 pages worth and starts off with the area under a hyperbola. If you wish to see this derivation let me know.

  • $\begingroup$ I'd make this answer a comment $\endgroup$
    – PhD
    Commented Dec 19, 2017 at 23:41
  • 1
    $\begingroup$ @PhD people with too few reputation points are not allowed to comment. $\endgroup$ Commented Dec 20, 2017 at 9:27

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