A Tangled Tale · Chapter
§2: Balbus’ Essay explained
Balbus poses a puzzle about a solid immersed in water, where the water rises through an infinite series of decreasing distances (2 inches, 1 inch, 1/2 inch, etc.), and he concludes the water will rise without limit.
What happens
Balbus poses a puzzle about a solid immersed in water, where the water rises through an infinite series of decreasing distances (2 inches, 1 inch, 1/2 inch, etc.), and he concludes the water will rise without limit. The solution explains that the series never reaches 4 inches, as each term leaves a remainder equal to the last term. Three answers are received, with Tympanum, Old King Cole, and Vindex offering correct identifications of the fallacy.
Themes in this chapter
Mathematical Reasoning
The chapter examines an infinite geometric series and the fallacy of assuming an infinite sum implies an unbounded total.
Logic and Paradox
The problem is directly linked to Zeno's paradox of Achilles and the Tortoise, highlighting logical pitfalls in infinite processes.
Characters to notice
- Balbus
Author of the essay posing the water-immersion problem.
- Tympanum
Respondent who suggests the stick problem is a blind and jokingly proposes a practical test.
- Old King Cole
Correctly identifies the series as a decreasing Geometrical Progression.
- Vindex
Correctly identifies the fallacy as that of 'Achilles and the Tortoise.'
Key passages
“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.”
Balbus claims that immersing a solid in water causes the water level to increase by an endless sequence of decreasing amounts (2, 1, 1/2, ...), and he infers that the water will keep rising indefinitely.
Sets up the central puzzle of the chapter.
“This series can never reach 4 inches, since, however many terms we take, we are always short of 4 inches by an amount equal to the last term taken.”
The infinite sum of the series never exceeds 4 inches, because after each term the remaining gap to 4 inches is exactly the size of the last term added.
Explains the resolution of the paradox.
“I trust Tympanum will not test this in his own person, by taking the place of the man in Balbus’ Essay! He would infallibly be drowned.”
The narrator hopes Tympanum does not actually try the experiment himself, as he would surely drown.
A humorous warning directed at Tympanum's suggestion.