How this instrument works
A catenary is the curve a perfectly flexible chain or cable takes when it hangs freely between two supports, held up by nothing but its own weight. The defining equation, y = a·cosh(x ⁄ a), falls out of a straightforward physics argument: at every point along the hanging chain, the horizontal component of tension is the same constant, while the vertical component grows with the length of chain hanging below it. Balancing those two forces produces a differential equation whose solution is the hyperbolic cosine — not an approximation, but the exact shape gravity settles the chain into.
It is tempting to call this shape a parabola, and Galileo did exactly that in 1638 — a guess disproved a few decades later, with the correct hyperbolic-cosine solution published in 1691 by Leibniz, Huygens, and Johann Bernoulli, each independently answering a challenge posed by Jacob Bernoulli. The mix-up persists today for a good reason: a suspension bridge's main cables, which carry a deck loaded uniformly per horizontal foot rather than per foot of cable, really are parabolic, while a freely hanging chain, a power line, or a necklace draped between two fingers is a true catenary.
Near its lowest point the two shapes are nearly impossible to tell apart, which is exactly why the mix-up survives: expand cosh(x ⁄ a) as a series and the first correction term is x² ⁄ (2a²), a parabola in disguise. Move further from center, though, and the catenary pulls away — hyperbolic cosine grows exponentially while a parabola only grows quadratically, so a long enough span always exposes the difference. The parameter a is not decorative: it equals the horizontal tension divided by the chain's weight per unit length, and it is also exactly the height of the curve's lowest point above whatever baseline y = 0 marks.
- Type the chain's tension-to-weight ratio into the a (catenary parameter) field — larger values give a slacker, more open curve.
- Enter the horizontal distance from the curve's lowest point into the x field; negative values work too, since the curve mirrors itself on both sides.
- Read y for the height of the curve at that x, measured on the same baseline the lowest point sits on.
- Recompute y for a spread of x values, positive and negative, to trace the full sag between two supports.
- Compare two different a values side by side to see a tighter cable (smaller a) curve more sharply than a slacker one (larger a).
Worked example — a cable with parameter a = 2
Take a hanging cable whose parameter works out to a = 2, in whatever length unit the supports are measured in, and look one unit to the side of its lowest point, at x = 1. The formula gives y = 2 × cosh(1 ⁄ 2) = 2 × cosh(0.5) = 2 × 1.1276259652063807 = 2.2552519304127614 — the height of the cable at that point, measured on the same baseline as the lowest point itself.
The lowest point of this same cable sits at height y = a = 2 exactly, since cosh(0) = 1. One unit further along, the cable has climbed to about 2.2553, a rise of roughly a quarter unit — a sag profile no parabola sharing the same curvature at the very bottom would reproduce a few units further out, where the true catenary keeps climbing exponentially rather than quadratically.
Questions
Is a catenary curve the same as a parabola?
No, though the two look almost identical near the bottom of the sag. A catenary follows y = a·cosh(x⁄a); a parabola only matches its curvature right at the vertex. A freely hanging chain is a true catenary, but a suspension bridge's main cable, loaded uniformly per horizontal foot by the deck rather than per foot of cable, is actually parabolic.
Where does the formula y = a·cosh(x ⁄ a) come from?
It comes from balancing forces along the hanging chain: the horizontal pull of tension stays constant everywhere, while the chain's own weight bends the curve according to how much chain hangs below each point. That balance is a differential equation whose exact solution is the hyperbolic cosine, not an approximation to it.
What does the parameter a actually represent?
Physically, a is the horizontal tension in the chain divided by its weight per unit length, so a stronger pull relative to weight gives a larger a and a flatter curve. Mathematically, a also equals y at x = 0, the height of the curve's lowest point above whichever baseline y = 0 marks.
Did Galileo really get the catenary wrong?
Yes — in 1638 he proposed that a hanging chain traces a parabola, a reasonable-looking guess that turned out to be false. The correct hyperbolic-cosine equation was worked out independently in 1691 by Leibniz, Huygens, and Johann Bernoulli, answering a challenge Jacob Bernoulli had posed the year before.
Why is the Gateway Arch called a catenary arch?
Because, inverted, a catenary puts every point of an arch into pure compression with no bending stress — turn a hanging chain upside down and it becomes a self-supporting arch. The St. Louis Gateway Arch closely approximates this inverted-catenary shape, weighted slightly to account for its own tapering cross-section.