#histmath — Public Fediverse posts

Friedrich Schiller's (1759–1805) poem ‘Archimedes and the Student’ (see 1st attached image for typeset text):

To Archimedes came an inquisitive youth
“Initiate me,” he said to him, “into the divine science,
That bore such splendid fruit for the nation
And shielded the walls of the city from the sambuca!”
“Divine you call the science? It is,” replied the sage,
“But it was so, my son, even before it served the state.
If you want only fruit from her, even mortals can provide it;
Who courts the goddess, seeks not in her the woman.”

(The sambuca was a ship-mounted siege engine; see 2nd attached image. During the Roman siege of Syracuse, it failed in the face of the war-machines designed by Archimedes.)

In 1808, Carl Friedrich Gauss (1777–1855) became director of the observatory at Göttingen and in his inaugural lecture declared that mathematics in general and astronomy in particular had a value — at least in part aesthetic — that was prior to and independent of any utility:

‘The happy great minds who created and expanded astronomy as well as the other beautiful parts of mathematics were certainly not inspired by the prospect of future use: they searched the truth for its own sake and found in the very success of their efforts their reward and their happiness. I cannot avoid at this point reminding you of ARCHIMEDES […]. You must all know the beautiful poem by SCHILLER.’

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[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

#Archimedes #Schiller #Gauss #poetry #HistMath

Evangelista Torricelli’s (1608–47) solid is defined by rotating the hyperbola $y = 1/x$ about the $x$ axis and truncating it at $x=1$ (see attached image).

It has infinite length and infinite surface area but finite volume.

This counter-intuitive discovery caused philosophical disturbance, for it seemed to violate the distinction between finite and infinite.

Torricelli, foreseeing the scrutiny to which his work would be subjected, took the precaution of preempting some criticisms by supplying two different proofs, one by ‘indivisibles’, one by exhaustion.

But René Descartes (1596–1650) seems not to have been provoked to any philosophical objections and thought that Torricelli's discovery was beautiful.

Henry Needler (fl. 1690–1718), a perhaps slightly obscure figure who foreshadowed 18th-century discussions of the sublime, seemed to be impressed by the solid's ‘Grandeur and Magnificence’ and thought that it would ‘afford the greatest Delight and Satisfaction to curious Minds’.

(Today, Torricelli's solid is also called ‘Gabriel's horn’ or ‘Torricelli's trumpet’.)

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#infinite #Descartes #HistMath #HistPhil #Torricelli #MathematicalBeauty #sublime #aesthetics

Max Dehn (1878–1952) said that Archimedes’ (c.287–212 BCE) discovery that the surface area of a sphere was four times its great circle was the one of the most beautiful results of Greek mathematics.

Archimedes himself had a high opinion of this result and two others in his two books ‘On the Sphere and the Cylinder’: that the volume and surface area of a sphere and a cylinder exactly circumscribing it are in the ratio $2 : 3$. One can add a cone fitting inside the cylinder to have ratios $1 : 2 : 3$ (see 1st attached image).

It has been suggested that Archimedes’ conjectures for these ratios may have been guided by a conscious or unconscious search for beautiful integer ratios between geometric configurations. There is no direct evidence for this motivation, but Archimedes’ work seems to exhibit a preference for small integer ratios.

According to Plutarch, Archimedes desired that his tomb should be marked by a cylinder enclosing a sphere and an inscription of the ratio of the one to the other; Cicero related how he had sought out Archimedes’ tomb and found a column just so inscribed (see 2nd attached image).

[Each day of February, I intend to post an interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

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#MathematicalBeauty #HistMath #Archimedes #Plutarch #Cicero #geometry #aesthetics