Many authors use mathematical metaphors with great effect. A recent book, “Once Upon A Prime” by Sarah Hart, describes the wondrous connections between mathematics and literature. As a particular example, she discusses the relevance of the cycloid curve in the work of Herman Melville. The book Moby-Dick, first published in 1851, opens with the words “Call me Ishmael”, and ends with Ishmael as the sole survivor of the voyage.
Moby-Dick
In Moby-Dick, Melville employed many mathematical images, and he used the properties of a mathematical curve called the cycloid — the curve traced by a point on the rim of a rolling wheel — to explain Ishmael’s observation of the motions of a soapstone in a large cauldron.
Melville had limited education, but he was fortunate to attend the Albany Academy, where he was a pupil of Joseph Henry, a renowned scientist, later the first Secretary of the Smithsonian Institution (the SI unit of inductance is named for Henry). Henry’s teaching had a profound and lasting impact on Melville, who retained a fascination for mathematics (Bellini, 2022).
Bellini (2022) describes an incident in Chapter XCVI of Moby-Dick, in which Ishmael is cleaning a “try-pot”, a large cauldron in which blubber is boiled to make whale-oil. These pots are kept clean by scrubbing them with a heavy soapstone. As Ishmael stood within the pot, scrubbing away, he was struck by
“ … the remarkable fact, that in geometry all bodies gliding along the cycloid, my
soapstone for example, will descend from any point in precisely the same time.”
Ishmael is referring here to an interesting property of the cycloid, the curve generated by a point on the rim of a rolling circular wheel. Ishmael notices the “equal-time” property of the inner surface and, even with his limited knowledge, seems to conclude that the try-pot must have a cross-section in the form of a cycloid.
The cycloid played a pivotal role in the development of calculus and was studied intensively by several leading mathematicians in the seventeenth century. Christiaan Huygens found that, if a point mass is placed anywhere on a cycloid curve and moves without friction, it reaches the lowest point in “precisely the same time”, as observed by Ishmael. For this reason, Huygens called the curve the tautochrone (meaning same time), and he used it in the design of an accurate pendulum clock.
The Brachistochrone
The cycloid has another remarkable property, which gives it another name. It was Johann Bernoulli who discovered that a particle moving under gravity between any two points A and B in a vertical plane, takes the least time if the trajectory is a cycloid. This property led Bernoulli to name the cycloid a brachistochrone (from the Greek Brachistos, shortest, and Chronos, time).
Having solved this problem, Bernoulli issued it as a challenge to other European mathematicians. One of the most noteworthy solutions, allegedly found in a single evening, was that of Isaac Newton. Here is how Bernoulli announced his challenge:
Mechanical-Geometrical Problem on the Curve of Quickest Descent.
‘To determine the curve jointing two given points, at different distances from the horizontal and
not on the same vertical line along which a mobile particle, acted upon by its own weight and
starting its motion from the upper point, descends most rapidly to the lower point.’
Bernoulli was delighted to discover that his brachistochrone problem involved the same curve, the cycloid, as Huygens’ tautochrone problem. As quoted by Kimball (1955):
“You will be petrified with astonishment when I say that precisely
this cycloid, the tautochrone of Huygens, is our brachistochrone …
Furthermore, I think it is noteworthy that this identity is found
only under the hypothesis of Galileo, so that even from this
we may conjecture that nature wanted it to be thus.”
For a full and interesting account of the brachistochrone problem, see Kimball (1955).
Sources
- Federico Bellini, 2022: “Melville’s Curves: Mathematics and the Melvillean Imagination. Measuring a Cycloid in Moby-Dick”, Miranda [Online], 26 | 2022. URL. DOI.
- Kimball, William S., 1955: A History of the Brachistochrone. Pi Mu Epsilon Journal, 2, No. 2, pp. 57-77. JSTOR.
- Sarah Hart, 2003: Once Upon A Prime. Mudlark, x+290pp. ISBN: 978-0-0086-0108-9.
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