Answers to Knot IX explained

What happens

Themes in this chapter

• Mathematical Reasoning

  The chapter focuses on solving mathematical problems involving geometry, series, and displacement, with detailed critique of correct and incorrect reasoning.

• Humor and Absurdity

  The author uses playful and sarcastic language to mock incorrect solutions, such as calling Tympanum's reasoning 'very terrible' and joking about drowning.

Characters to notice

• Balbus

  Balbus is the author of the Essay Problem about a solid immersed in water causing an infinite series of rises.

• Tympanum

  Tympanum submits incorrect solutions to both the Buckets Problem and the Garden Problem, and is criticized for his reasoning.

• Old King Cole

  Old King Cole correctly identifies the series in Balbus' Essay as a decreasing Geometrical Progression and is listed in the class list.

• Vindex

  Vindex correctly identifies the fallacy in Balbus' Essay as that of 'Achilles and the Tortoise' and provides a neat proof for the Garden Problem.

• Hecla

  Hecla submits a flawed solution to the Buckets Problem and is noted for making two mistakes that cancel each other in the Garden Problem.

Key passages

• “Lardner states that a solid, immersed in a fluid, displaces an amount equal to itself in bulk. How can this be true of a small bucket floating in a larger one?”

  Lardner claims that a submerged solid pushes aside a volume of fluid equal to its own volume. How does this apply when a small bucket floats inside a larger bucket?

  This is the core question of the Buckets Problem, challenging a common physics principle.

• “Balbus states that if a certain solid be immersed in a certain vessel of water, the water will rise through a series of distances, two inches, one inch, half an inch, etc., which series has no end. He concludes that the water will rise without limit. Is this true?”

  Balbus says that immersing a solid in water causes the water level to rise by an infinite series of decreasing distances: 2 inches, 1 inch, 0.5 inch, and so on. He argues the water will rise indefinitely. Is that correct?

  This presents the paradox of an infinite series with a finite sum, similar to Zeno's paradox.

• “An oblong garden, half a yard longer than wide, consists entirely of a gravel-walk, spirally arranged, a yard wide and 3,630 yards long. Find the dimensions of the garden.”

  A rectangular garden is half a yard longer than it is wide. It is completely covered by a spiral gravel path that is 1 yard wide and 3,630 yards long. What are the garden's dimensions?

  This is the Garden Problem, solved by setting up a quadratic equation based on area.