Kepler’s barrels

Sam Hartburn orders wine by the barrel, but wonders if she’s getting the most wine


Kepler went out shopping for a barrel full of wine,
He had a problem with the price the merchant would assign,
The merchant used a measure stick to calculate the price,
Into the barrel it did drop,
  Through a hole at centre top,
  And down and left until it stopped,
He thought it imprecise.

He said there’d be a certain price for a barrel short and fat,
And another that was tall and thin would cost the same as that,
But while the short fat barrel would hold lots and lots of wine,
The tall thin one would hold much less,
  This observation caused distress,
  Kepler started to obsess,
He didn’t think it fine.

Though Kepler paid the merchant then, he never could forget,
That inconsistent pricing, well, it really made him fret,
Day after day he pondered as he went about his work,
He just had to investigate,
  The fairness of the merchant’s rate,
  He could no more procrastinate,
Or he would go berserk.

For any given pricing, how much wine could one consume?
What measurement of barrel gave the maximum volume?
He noted that a cylinder was almost the same shape,
As the barrel he’d been using,
  For his celebratory boozing,
  And so he began his musing,
He could approximate.

For cylinders of different heights he worked out the diameter,
Assuming that the merchant’s measure was a fixed parameter,
Trying different values gave a groundbreaking conclusion,
Diameter, times by root 2,
  Would give the perfect height value,
  A maxed-out volume would ensue,
It was no illusion.

Then Kepler found out something new that really made him cheer,
All of Austria’s barrels had this ratio, or near,
So the barrel that he’d purchased had a fair price after all,
A few might slightly deviate,
  But Kepler could appreciate,
  The volume change that this creates,
Is imperceptible.

But the story doesn’t end here, because Kepler’s observation,
That the points around the max have but a tiny deviation,
Inspired Fermat’s theorem about stationary points,
It might not seem so obvious,
  But this led on to calculus,
  A field meritorious,
That never disappoints.

Sam is a freelance proofreader and copy-editor and a hobbyist maths geek. She likes to make maths rhyme.

More from Chalkdust