#histmath — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #histmath, aggregated by home.social.
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#ConseilPodcast Avoir raison avec... Sophie Germain | France Culture
Une série en cinq épisodes, très intéressante pour les amateurs et amatrices d'histoire des mathématiques. Les premières invitées sont les excellentes Jenny Boucard et Isabelle Lémonon (Centre François Viète, Nantes).
https://www.radiofrance.fr/franceculture/podcasts/serie-avoir-raison-avec-sophie-germain
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#ConseilPodcast Avoir raison avec... Sophie Germain | France Culture
Une série en cinq épisodes, très intéressante pour les amateurs et amatrices d'histoire des mathématiques. Les premières invitées sont les excellentes Jenny Boucard et Isabelle Lémonon (Centre François Viète, Nantes).
https://www.radiofrance.fr/franceculture/podcasts/serie-avoir-raison-avec-sophie-germain
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Some nice 18th-century mathematical typography: pages from Edmund Halley’s 1710 Latin translation of Apollonius’ ‘Conics’, showing Propositions X and XI of Book I. Prop. XI defines the term ‘parabola’.
The Greek and Latin texts are in parallel columns, with the Greek in the inner columns of each two-page spread. Small diagrams, like the ones on these pages, are placed in the middle, with the text of both columns wrapping around them. Some of the narrow lines of Latin next to the diagrams are widely spaced, but generally the typographer has done an excellent job of fitting the text around the diagrams.
Books I–IV are translated from the Greek, books V–VII (which are not extant in Greek) from the Arabic. Halley also tried to reconstruct the lost book VIII.
The book is available on Gallica: https://gallica.bnf.fr/ark:/12148/bpt6k9760213g
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Some nice 18th-century mathematical typography: pages from Edmund Halley’s 1710 Latin translation of Apollonius’ ‘Conics’, showing Propositions X and XI of Book I. Prop. XI defines the term ‘parabola’.
The Greek and Latin texts are in parallel columns, with the Greek in the inner columns of each two-page spread. Small diagrams, like the ones on these pages, are placed in the middle, with the text of both columns wrapping around them. Some of the narrow lines of Latin next to the diagrams are widely spaced, but generally the typographer has done an excellent job of fitting the text around the diagrams.
Books I–IV are translated from the Greek, books V–VII (which are not extant in Greek) from the Arabic. Halley also tried to reconstruct the lost book VIII.
The book is available on Gallica: https://gallica.bnf.fr/ark:/12148/bpt6k9760213g
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Big update to ‘Form & Number: A History of Mathematical Beauty’: the PDFs are now *tagged* (plus various minor improvements).
A #TaggedPDF contains hidden extra semantic information to assist screen-reading software etc., including alt text for diagrams+images (in the case of ‘F&N’, 552 pieces of alt text were required, some quite lengthy).
The new version is available (#OpenAccess as always) at: https://archive.org/details/cain_formandnumber_ebook_large
Feedback from users of screen readers would be very welcome!
Creating this version has been possible because of the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/] (thank you to the LaTeX team and other contributors!). I have been adapting my own #LuaLaTeX styles [https://codeberg.org/ajcain/minos] to use the new interfaces.
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#accessibility #a11y #MathArt #HistMath #HistSci #MathematicalBeauty #aesthetics #typography #TeXLaTeX
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Big update to ‘Form & Number: A History of Mathematical Beauty’: the PDFs are now *tagged* (plus various minor improvements).
A #TaggedPDF contains hidden extra semantic information to assist screen-reading software etc., including alt text for diagrams+images (in the case of ‘F&N’, 552 pieces of alt text were required, some quite lengthy).
The new version is available (#OpenAccess as always) at: https://archive.org/details/cain_formandnumber_ebook_large
Feedback from users of screen readers would be very welcome!
Creating this version has been possible because of the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/] (thank you to the LaTeX team and other contributors!). I have been adapting my own #LuaLaTeX styles [https://codeberg.org/ajcain/minos] to use the new interfaces.
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#accessibility #a11y #MathArt #HistMath #HistSci #MathematicalBeauty #aesthetics #typography #TeXLaTeX
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RE: https://hcommons.social/@bho/116770184058726900
One thing I would add to this article is that the Biodiversity Heritage Library has scans of many older general scientific journals (not just those related to biology/life sciences), which are invaluable to anyone working in the history of science. I have used it frequently in my work on the history of mathematics.
Many of the same journals are available elsewhere, e.g. archive.org. But finding particular journal volumes on archive.org can be hit-or-miss, depending on how complete+correct the metadata is. The BHL lists the volumes for each journal.
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RE: https://hcommons.social/@bho/116770184058726900
One thing I would add to this article is that the Biodiversity Heritage Library has scans of many older general scientific journals (not just those related to biology/life sciences), which are invaluable to anyone working in the history of science. I have used it frequently in my work on the history of mathematics.
Many of the same journals are available elsewhere, e.g. archive.org. But finding particular journal volumes on archive.org can be hit-or-miss, depending on how complete+correct the metadata is. The BHL lists the volumes for each journal.
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A short #BookReview of ‘Form & Number: A History of Mathematical Beauty’ in ‘Mathematics Magazine’ has been pointed out to me: https://doi.org/10.1080/0025570X.2025.2587000 (not open-access)
I am quite pleased by the reviewer's evaluation :-) ‘a very careful and absolutely thorough survey of the “rich heritage of scholarship” about both beauty in mathematics and the study of that topic’.
There was a more detailed review by Viktor Blåsjö in ‘TUGboat’ last year: https://doi.org/10.47397/tb/46-2/tb143reviews-cain (open-access)
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A short #BookReview of ‘Form & Number: A History of Mathematical Beauty’ in ‘Mathematics Magazine’ has been pointed out to me: https://doi.org/10.1080/0025570X.2025.2587000 (not open-access)
I am quite pleased by the reviewer's evaluation :-) ‘a very careful and absolutely thorough survey of the “rich heritage of scholarship” about both beauty in mathematics and the study of that topic’.
There was a more detailed review by Viktor Blåsjö in ‘TUGboat’ last year: https://doi.org/10.47397/tb/46-2/tb143reviews-cain (open-access)
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Each day of February I posted a fact/image/anecdote about the aesthetics of mathematics, which seemed to provoke a certain amount of interest.
The posts are all collected on my personal website, with minor fixes and improvements (including vector versions of diagrams).
The index is here: https://ajcain.codeberg.page/posts/2026-03-01-aesthetics-of-mathematics.html
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Each day of February I posted a fact/image/anecdote about the aesthetics of mathematics, which seemed to provoke a certain amount of interest.
The posts are all collected on my personal website, with minor fixes and improvements (including vector versions of diagrams).
The index is here: https://ajcain.codeberg.page/posts/2026-03-01-aesthetics-of-mathematics.html
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[Auto-promo] Dans ma boîte aux lettres ce matin, le dernier numéro de la revue Tangente – l'aventure mathématique, dans lequel je signe un dossier d'une douzaine de pages sur Gabriel Cramer, savant genevois de la première moitié du XVIIIe siècle..
Si ce numéro tombe entre les mains d'amateurs ou amatrices d'histoire et de culture mathématique au sens large (en bibliothèque, au CDI...) qui passeraient par ici : n'hésitez pas à me faire signe et me livrer vos impressions !
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[Auto-promo] Dans ma boîte aux lettres ce matin, le dernier numéro de la revue Tangente – l'aventure mathématique, dans lequel je signe un dossier d'une douzaine de pages sur Gabriel Cramer, savant genevois de la première moitié du XVIIIe siècle..
Si ce numéro tombe entre les mains d'amateurs ou amatrices d'histoire et de culture mathématique au sens large (en bibliothèque, au CDI...) qui passeraient par ici : n'hésitez pas à me faire signe et me livrer vos impressions !
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The philosopher, biologist, and political theorist Herbert Spencer (1820–1903) has a minor but curious role in the history of mathematical beauty, because of comments he made about Monge’s theorem, which states:
For any three circles in a plane, none contained within another, the intersections of the outside tangents of the three pairs of circles are collinear. (See attached image.)
Spencer said that when he thought of it he was
‘struck by its beauty at the same time that it excites feelings of wonder and of awe: the fact that apparently unrelated circles should in every case be held together by this plexus of relations, seeming so utterly incomprehensible.’
However, Spencer’s reaction of wonder and of awe may ultimately have been born of his limited mathematical ability.
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The philosopher, biologist, and political theorist Herbert Spencer (1820–1903) has a minor but curious role in the history of mathematical beauty, because of comments he made about Monge’s theorem, which states:
For any three circles in a plane, none contained within another, the intersections of the outside tangents of the three pairs of circles are collinear. (See attached image.)
Spencer said that when he thought of it he was
‘struck by its beauty at the same time that it excites feelings of wonder and of awe: the fact that apparently unrelated circles should in every case be held together by this plexus of relations, seeming so utterly incomprehensible.’
However, Spencer’s reaction of wonder and of awe may ultimately have been born of his limited mathematical ability.
1/3
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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.]
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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.]
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The idea that the ‘golden’ ratio — $1.61803\ldots:1$ — has applications in visual art and architecture does not go back any further than the 2nd edition (1799–1802) of Jean-Étienne Montucla's (1725–99) (generally superb) ‘Histoire des Mathématiques’, in which he made the **incorrect** statement that Luca Pacioli's (c.1447–1517) book ‘Divina Proportione’ included illustrations of the ratio's application to architecture and font design.
This was shortly after the earliest known appearance of the term ‘golden section’ in Johann Samuel Traugott Gehler’s (1751–95) general scientific dictionary ‘Physikalisches Wörterbuch’.
The golden ratio was then taken up by Adolph Zeising (1810–76) as the basis for a system of aesthetic proportion in his book ‘New Theory of the Proportions of the Human Body’ (1854), where he argued — apparently to his own satisfaction — that his system agreed with the proportions of many masterpieces of art.
The psychologist Gustav Fechner (1801–87) made a much-misreported experiment in which people were asked to choose the most aesthetically pleasing of various rectangles (shown in the attached image). The most popular choice was the 34 ∶ 21 rectangle, whose proportions approximate the golden ratio. Fechner's conclusion was only that **a range of rectangles**, including the golden ratio rectangle, were considered most pleasing.
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#GoldenRatio #GoldenSection #DivineProportion #HistMath #aesthetics #Zeising #Fechner
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The idea that the ‘golden’ ratio — $1.61803\ldots:1$ — has applications in visual art and architecture does not go back any further than the 2nd edition (1799–1802) of Jean-Étienne Montucla's (1725–99) (generally superb) ‘Histoire des Mathématiques’, in which he made the **incorrect** statement that Luca Pacioli's (c.1447–1517) book ‘Divina Proportione’ included illustrations of the ratio's application to architecture and font design.
This was shortly after the earliest known appearance of the term ‘golden section’ in Johann Samuel Traugott Gehler’s (1751–95) general scientific dictionary ‘Physikalisches Wörterbuch’.
The golden ratio was then taken up by Adolph Zeising (1810–76) as the basis for a system of aesthetic proportion in his book ‘New Theory of the Proportions of the Human Body’ (1854), where he argued — apparently to his own satisfaction — that his system agreed with the proportions of many masterpieces of art.
The psychologist Gustav Fechner (1801–87) made a much-misreported experiment in which people were asked to choose the most aesthetically pleasing of various rectangles (shown in the attached image). The most popular choice was the 34 ∶ 21 rectangle, whose proportions approximate the golden ratio. Fechner's conclusion was only that **a range of rectangles**, including the golden ratio rectangle, were considered most pleasing.
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#GoldenRatio #GoldenSection #DivineProportion #HistMath #aesthetics #Zeising #Fechner
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Gottfried Wilhelm Leibniz's (1646–1716) first great mathematical achievement was the ‘arithmetic quadrature’ of the circle through his alternating series: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
He communicated the result to his mathematical mentor Christiaan Huygens (1629–95), who thought it ‘very beautiful and very pleasing’. Isaac Newton (1642–1726) welcomed Leibniz’s work as ‘very elegant’. Leibniz himself wrote that there was no simpler or more beautiful way of expressing the area of a circle using rational numbers.
In a short note concerning the beauty of theorems, Leibniz wrote:
‘Theorems are not intelligible except by their signs or characters. […] The beauty of theorems consists in the beautiful arrangement of their characters.’
To illustrate ‘beautiful arrangement of characters’, Leibniz gave the example of a theorem concerning Berthet’s curve (shown in red in 1st attached image). The detail of its definition is not important here, but it is defined with reference to an arc AC centred at B.
Leibniz's result was a way of finding the tangent to the curve at E: take the tangent to the arc at its intersection with BE (i.e., at D), and find the point F such that FD ∶ DE ∶∶ EB ∶ BD. Then EF is the desired tangent (see 1st attached image).
Why is there a ‘beautiful arrangement of characters’? Because the proportion FD ∶ DE ∶∶ EB ∶ BD is easily remembered via a mnemonic: one can draw the path FD ⋅ DE ⋅ EB ⋅ BD without raising one's pen (2nd attached image).
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#geometry #Leibniz #Huygens #Newton #MathematicalBeauty #aesthetics #HistMath
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Gottfried Wilhelm Leibniz's (1646–1716) first great mathematical achievement was the ‘arithmetic quadrature’ of the circle through his alternating series: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
He communicated the result to his mathematical mentor Christiaan Huygens (1629–95), who thought it ‘very beautiful and very pleasing’. Isaac Newton (1642–1726) welcomed Leibniz’s work as ‘very elegant’. Leibniz himself wrote that there was no simpler or more beautiful way of expressing the area of a circle using rational numbers.
In a short note concerning the beauty of theorems, Leibniz wrote:
‘Theorems are not intelligible except by their signs or characters. […] The beauty of theorems consists in the beautiful arrangement of their characters.’
To illustrate ‘beautiful arrangement of characters’, Leibniz gave the example of a theorem concerning Berthet’s curve (shown in red in 1st attached image). The detail of its definition is not important here, but it is defined with reference to an arc AC centred at B.
Leibniz's result was a way of finding the tangent to the curve at E: take the tangent to the arc at its intersection with BE (i.e., at D), and find the point F such that FD ∶ DE ∶∶ EB ∶ BD. Then EF is the desired tangent (see 1st attached image).
Why is there a ‘beautiful arrangement of characters’? Because the proportion FD ∶ DE ∶∶ EB ∶ BD is easily remembered via a mnemonic: one can draw the path FD ⋅ DE ⋅ EB ⋅ BD without raising one's pen (2nd attached image).
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#geometry #Leibniz #Huygens #Newton #MathematicalBeauty #aesthetics #HistMath
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An enduring locus of mathematical beauty in the seventeenth century concerned curves like the cycloid and the catenary.
A cycloid is the path followed by a point on the circumference of a circle rolling along a straight line (see attached image).
Christopher Wren (1632–1723) proved that the arc length of the cycloid is four times the diameter of its generating circle.
Christiaan Huygens (1629–95) thought Wren's work ‘really beautiful’. Blaise Pascal (1623–62) also called it ‘beautiful’ (even though he also seemed to repudiate any true notion of mathematical beauty in his ‘Pensées’.
Huygens proved that an inverted cycloid was the ‘tautochrone’: the curve along which a body starting from rest and freely accelerated by uniform gravity reaches the lowest point in the same time, independently of its starting point.
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#cycloid #tautochrone #Huygens #HistMath #MathematicalBeauty #Pascal
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An enduring locus of mathematical beauty in the seventeenth century concerned curves like the cycloid and the catenary.
A cycloid is the path followed by a point on the circumference of a circle rolling along a straight line (see attached image).
Christopher Wren (1632–1723) proved that the arc length of the cycloid is four times the diameter of its generating circle.
Christiaan Huygens (1629–95) thought Wren's work ‘really beautiful’. Blaise Pascal (1623–62) also called it ‘beautiful’ (even though he also seemed to repudiate any true notion of mathematical beauty in his ‘Pensées’.
Huygens proved that an inverted cycloid was the ‘tautochrone’: the curve along which a body starting from rest and freely accelerated by uniform gravity reaches the lowest point in the same time, independently of its starting point.
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#cycloid #tautochrone #Huygens #HistMath #MathematicalBeauty #Pascal
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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
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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’.)
1/2
#infinite #Descartes #HistMath #HistPhil #Torricelli #MathematicalBeauty #sublime #aesthetics
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Number theory was the one area of mathematics on which Pierre de Fermat (1607–65) worked throughout his life, and he found it ‘very beautiful and very subtle’.
Among other results, he said that the Polygonal Number Theorem (which asserts that every natural number is the sum of at most $n$ $n$-gonal numbers) was ‘a most beautiful and wholly general proposition […] this marvellous proposition’.
(He offered no proof of this result, but claimed to have one in a marginal note to Diophantus' Arithmetica; this was the same book in which he noted what became known as Fermat's Last Theorem.)
Fermat also seems to have counted magic squares and analogous configurations as part of number theory, and wrote that: ‘I know hardly anything more beautiful in arithmetic than these numbers that some call planetary and others magic’. (The term ‘planetary’ is derived from certain treatises linking the magic squares to planets used in talismans.)
He said he had found a rule to find magic cubes (one of his examples is in the attached image) and also determined how many different ways each such cube can be arranged, which he called ‘one of the most beautiful things in arithmetic’.
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[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
#Fermat #NumberTheory #HistMath #MathematicalBeauty #MagicSquare
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Number theory was the one area of mathematics on which Pierre de Fermat (1607–65) worked throughout his life, and he found it ‘very beautiful and very subtle’.
Among other results, he said that the Polygonal Number Theorem (which asserts that every natural number is the sum of at most $n$ $n$-gonal numbers) was ‘a most beautiful and wholly general proposition […] this marvellous proposition’.
(He offered no proof of this result, but claimed to have one in a marginal note to Diophantus' Arithmetica; this was the same book in which he noted what became known as Fermat's Last Theorem.)
Fermat also seems to have counted magic squares and analogous configurations as part of number theory, and wrote that: ‘I know hardly anything more beautiful in arithmetic than these numbers that some call planetary and others magic’. (The term ‘planetary’ is derived from certain treatises linking the magic squares to planets used in talismans.)
He said he had found a rule to find magic cubes (one of his examples is in the attached image) and also determined how many different ways each such cube can be arranged, which he called ‘one of the most beautiful things in arithmetic’.
1/2
[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
#Fermat #NumberTheory #HistMath #MathematicalBeauty #MagicSquare
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François Viète (= Franciscus Vieta; 1540–1603) Viète described as ‘elegant and very beautiful [elegans & perpulchræ]’ the result shown in the original Latin and using his original notation in the 1st attached image.
Translated into English and into modern algebraic notation, Viète's result says: If, for an unknown $a$ and known $b$, $d$, $g$, $h$, and $k$, the equality shown in the 2nd attached image holds, then $a$ is either $b$, $d$, $g$, $h$, or $k$.
A mathematician today, working with this modern notation, would presumably notice that if one sets $a = b$ and multiplies out the brackets on the left hand side, all terms except $bdghk$ cancel; by symmetry, the same reasoning applies for $d$, $g$, $h$, $k$.
Thus, to modern eyes, the result collapses into near-triviality and seems to deserve little if any aesthetic praise. Perhaps the notation Viète had to work with made the result seem more mysterious and thus more aesthetically pleasing.
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François Viète (= Franciscus Vieta; 1540–1603) Viète described as ‘elegant and very beautiful [elegans & perpulchræ]’ the result shown in the original Latin and using his original notation in the 1st attached image.
Translated into English and into modern algebraic notation, Viète's result says: If, for an unknown $a$ and known $b$, $d$, $g$, $h$, and $k$, the equality shown in the 2nd attached image holds, then $a$ is either $b$, $d$, $g$, $h$, or $k$.
A mathematician today, working with this modern notation, would presumably notice that if one sets $a = b$ and multiplies out the brackets on the left hand side, all terms except $bdghk$ cancel; by symmetry, the same reasoning applies for $d$, $g$, $h$, $k$.
Thus, to modern eyes, the result collapses into near-triviality and seems to deserve little if any aesthetic praise. Perhaps the notation Viète had to work with made the result seem more mysterious and thus more aesthetically pleasing.
1/3
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The thirteen archimedean solids are the polyhedra (other than the five regular solids) all the faces of which are regular polygons and where for each pair of vertices some symmetry transformation carries one vertex to the other (see 1st attached image).
According to Pappus (fl. c.300–c.350 CE), who wrote a half-millennium later, Archimedes discovered them. The context of Pappus' report suggests that Archimedes was seeking polyhedra inscribable in spheres.
Archimedes excluded the infinite classes of prisms and anti-prisms, in which two n-gons are joined by squares or equilateral triangles (2nd attached image). Although they satisfy the definition, and are technically inscribable in spheres, they are somehow not ‘sphere-like’.
This suggests that Archimedes may have been influenced by the aesthetic preference for circles and spheres that descended from Pythagoras.
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#ArchimedeanSolids #RegularSolids #polyhedra #Archimedes #Pappus #HistMath
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The thirteen archimedean solids are the polyhedra (other than the five regular solids) all the faces of which are regular polygons and where for each pair of vertices some symmetry transformation carries one vertex to the other (see 1st attached image).
According to Pappus (fl. c.300–c.350 CE), who wrote a half-millennium later, Archimedes discovered them. The context of Pappus' report suggests that Archimedes was seeking polyhedra inscribable in spheres.
Archimedes excluded the infinite classes of prisms and anti-prisms, in which two n-gons are joined by squares or equilateral triangles (2nd attached image). Although they satisfy the definition, and are technically inscribable in spheres, they are somehow not ‘sphere-like’.
This suggests that Archimedes may have been influenced by the aesthetic preference for circles and spheres that descended from Pythagoras.
1/3
#ArchimedeanSolids #RegularSolids #polyhedra #Archimedes #Pappus #HistMath
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Abū Sahl al-Kūhī (or al-Qūhī; fl. c.970–c.1000), who was regarded by contemporaries as the ‘Master of his age in the art of geometry’, once wrote of his motivation for considering a problem:
‘Having completed the construction of a regular heptagon in a circle, we set out to investigate another proposition, one more beautiful [ʾaḥsan أحسن], deeper, more opaque and more difficult to find out than the construction of the heptagon […]. This is the construction of an equilateral pentagon in a known square.’
Al-Kūhī's construction is shown in the attached diagrams. The red and blue curves are hyperbolae, both with latus rectum equal to 2AG, and with major axes NS and PI. The segments AE and DK (found using the the intersection of the hyperbolae) are equal to the required sides of the pentagon and a simple translation moves them into position.
(Note that the pentagon is only *equilateral*, not *equiangular*, and so is not regular.)
#MathematicalBeauty #geometry #HistMath #conics
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Abū Sahl al-Kūhī (or al-Qūhī; fl. c.970–c.1000), who was regarded by contemporaries as the ‘Master of his age in the art of geometry’, once wrote of his motivation for considering a problem:
‘Having completed the construction of a regular heptagon in a circle, we set out to investigate another proposition, one more beautiful [ʾaḥsan أحسن], deeper, more opaque and more difficult to find out than the construction of the heptagon […]. This is the construction of an equilateral pentagon in a known square.’
Al-Kūhī's construction is shown in the attached diagrams. The red and blue curves are hyperbolae, both with latus rectum equal to 2AG, and with major axes NS and PI. The segments AE and DK (found using the the intersection of the hyperbolae) are equal to the required sides of the pentagon and a simple translation moves them into position.
(Note that the pentagon is only *equilateral*, not *equiangular*, and so is not regular.)
#MathematicalBeauty #geometry #HistMath #conics
1/4
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Abū’l-Wafāʾ al-Būzjānī (940–77/8 CE) wrote one of the earliest extant treatises dedicated to magic squares, focused on constructions. He repeatedly referred to the aesthetic value of the methods of he described.
For instance, he wrote about a method of constructing a magic square of order 4:
‘It is possible to arrive at the magic arrangement in this square by means of methods without displacement showing regularity and elegance [niẓām wa-tartīb ḥasan نظام وترتيب حسن]’ (trans. Sesiano)
Such a method with ‘regularity and elegance’ was: (1) place the number 1 in a corner, 2 and 3 adjacent to the opposite corner, and 4 diagonally adjacent to 1; (2) place 5 to 8 in reverse order in positions horizontally symmetrically opposite to 1 to 4; (3) place $17 − n$ diagonally two places away from $n$ for $n = 1,\ldots,8$ (see attached image).
1/4
[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
#MagicSquare #HistMath #MathematicalElegance #elegance #aesthetics
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Abū’l-Wafāʾ al-Būzjānī (940–77/8 CE) wrote one of the earliest extant treatises dedicated to magic squares, focused on constructions. He repeatedly referred to the aesthetic value of the methods of he described.
For instance, he wrote about a method of constructing a magic square of order 4:
‘It is possible to arrive at the magic arrangement in this square by means of methods without displacement showing regularity and elegance [niẓām wa-tartīb ḥasan نظام وترتيب حسن]’ (trans. Sesiano)
Such a method with ‘regularity and elegance’ was: (1) place the number 1 in a corner, 2 and 3 adjacent to the opposite corner, and 4 diagonally adjacent to 1; (2) place 5 to 8 in reverse order in positions horizontally symmetrically opposite to 1 to 4; (3) place $17 − n$ diagonally two places away from $n$ for $n = 1,\ldots,8$ (see attached image).
1/4
[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
#MagicSquare #HistMath #MathematicalElegance #elegance #aesthetics
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Leonardo Pisano (c.1170–after 1240), dubbed ‘Fibonacci’, thought that Archimedes' proof that π was between $3\frac{10}{71}$ and $3\frac{1}{7}$ was beautiful [pulcra].
Archimedes' proof proceeds by calculating approximate ratios of the perimeters of 96-gons circumscribed about and inscribed in a circle to the diameter of that circle, implicitly starting with dodecagons and repeatedly bisecting edges to obtain 24-, 48-, and then 96-gons (see attached image).
Fibonacci’s judgement seems to be the earliest extant description of a *proof* as beautiful in the European tradition. [Al-Nasawī (fl. 1029–44) had earlier described a proof as beautiful.]
But there is a twist in the story...
1/3
#MathematicalBeauty #BeautifulProof #HistMath #Fibonacci #Archimedes #Pi
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Leonardo Pisano (c.1170–after 1240), dubbed ‘Fibonacci’, thought that Archimedes' proof that π was between $3\frac{10}{71}$ and $3\frac{1}{7}$ was beautiful [pulcra].
Archimedes' proof proceeds by calculating approximate ratios of the perimeters of 96-gons circumscribed about and inscribed in a circle to the diameter of that circle, implicitly starting with dodecagons and repeatedly bisecting edges to obtain 24-, 48-, and then 96-gons (see attached image).
Fibonacci’s judgement seems to be the earliest extant description of a *proof* as beautiful in the European tradition. [Al-Nasawī (fl. 1029–44) had earlier described a proof as beautiful.]
But there is a twist in the story...
1/3
#MathematicalBeauty #BeautifulProof #HistMath #Fibonacci #Archimedes #Pi
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‘Form & Number: A History of Mathematical Beauty’ has been updated: https://archive.org/details/cain_formandnumber_ebook_large
Quite a few typos have been fixed, and there are *many* small but important typographical improvements, mostly to kerning and spacing.
Karl Berry, the treasurer of the TeX Users Group and one of the editors of its journal ‘TUGboat’, has been doing a frankly heroic job of proof-reading the book with a typographer's eye — thank you Karl!
(The new PDF is available for download immediately; archive.org will re-generate images for the in-browser reader during the next few hours.)
Attached is a preview of a two-page spread from the print variant of ‘Form & Number’, showing the start of a section on the Ikhwān al-Ṣafāʾ and a diagram of the numerical structure of their cosmology.
#MathematicalBeauty #HistMath #MathArt #aesthetics #InternetArchive
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‘Form & Number: A History of Mathematical Beauty’ has been updated: https://archive.org/details/cain_formandnumber_ebook_large
Quite a few typos have been fixed, and there are *many* small but important typographical improvements, mostly to kerning and spacing.
Karl Berry, the treasurer of the TeX Users Group and one of the editors of its journal ‘TUGboat’, has been doing a frankly heroic job of proof-reading the book with a typographer's eye — thank you Karl!
(The new PDF is available for download immediately; archive.org will re-generate images for the in-browser reader during the next few hours.)
Attached is a preview of a two-page spread from the print variant of ‘Form & Number’, showing the start of a section on the Ikhwān al-Ṣafāʾ and a diagram of the numerical structure of their cosmology.
#MathematicalBeauty #HistMath #MathArt #aesthetics #InternetArchive
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According to the biography by Diogenes Laertius, Pythagoras (c.570–c.490 BCE) ‘held that the most beautiful figure is the sphere among solids, and the circle among plane figures’.
This aesthetic preference for the circle and sphere can be traced through thinkers like Plato (who, according to later writers, set the problem of describing the movements of the heavens using uniform circular motions), Cicero (106–43 BCE), and Proclus (410/12–485 CE), and into the middle ages.
Thomas Bradwardine (1290/1300–1349), one of the mediaeval ‘Oxford calculators’, was obviously influenced by this tradition when he wrote that the circle ‘is the first and most perfect of figures, the simplest and most regular, the most capacious and the most beautiful of figures’.
But Bradwardine then presented evidence that he saw as attesting to the beauty and perfection of the circle: (1) the construction to find the centre of a circle by bisecting a diameter found as the perpendicular bisector of a chord; (2) that the intersections of six equally-spaced radii with the circumference define a regular hexagon; (3) that exactly six circles of equal size can touch a given circle (see attached image).
For Bradwardine, the perfection of the circle was thus linked to the perfection of the number 6 = 1+2+3: the construction involves six intersections with the circle; the hexagon is made up of six lines; the third result involves six outer circles.
1/2
#MathematicalBeauty #HistMath #Pythagoras #Bradwardine #geometry #aesthetics #PerfectNumber
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According to the biography by Diogenes Laertius, Pythagoras (c.570–c.490 BCE) ‘held that the most beautiful figure is the sphere among solids, and the circle among plane figures’.
This aesthetic preference for the circle and sphere can be traced through thinkers like Plato (who, according to later writers, set the problem of describing the movements of the heavens using uniform circular motions), Cicero (106–43 BCE), and Proclus (410/12–485 CE), and into the middle ages.
Thomas Bradwardine (1290/1300–1349), one of the mediaeval ‘Oxford calculators’, was obviously influenced by this tradition when he wrote that the circle ‘is the first and most perfect of figures, the simplest and most regular, the most capacious and the most beautiful of figures’.
But Bradwardine then presented evidence that he saw as attesting to the beauty and perfection of the circle: (1) the construction to find the centre of a circle by bisecting a diameter found as the perpendicular bisector of a chord; (2) that the intersections of six equally-spaced radii with the circumference define a regular hexagon; (3) that exactly six circles of equal size can touch a given circle (see attached image).
For Bradwardine, the perfection of the circle was thus linked to the perfection of the number 6 = 1+2+3: the construction involves six intersections with the circle; the hexagon is made up of six lines; the third result involves six outer circles.
1/2
#MathematicalBeauty #HistMath #Pythagoras #Bradwardine #geometry #aesthetics #PerfectNumber
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As noted in a previous post, Archimedes thought highly of the result that the ratio of either the volumes or surface areas of a cone, a sphere, and a cylinder exactly circumscribing them is $1:2:3$.
So did others: three centuries later, the architect Nicon (d.149/50 CE), father of the philosopher and physician Galen (129–c.210/217 CE), thought it fitting to point out the ratio of the configuration in a public inscription in his city, Pergamon:
‘the cone, the sphere, the cylinder.
If a cylinder encloses the other two shapes,
[...]
Competition the principle and in solids
the progression $1 ∶ 2 ∶ 3$,
a noble, divine equalization,
but also mutual interdependence
of the solids, always in the ratio $1 ∶ 2 ∶ 3$.
They should be beautiful and wonderful,
the three solid shapes’Nicon doubtless admired these ratios as an architect: a sphere inside a cylinder brings to mind the Pantheon at Rome, of which the Temple of Zeus Asclepius Soter in Pergamon was a half-scale copy. These buildings were designed so that a basically cylindrical rotunda was crowned with a hemispherical dome under which a sphere would fit (see attached image).
[Each day of February, I am posting a story/image/fact/anecdote related to the aesthetics of mathematics.]
1/2
#MathematicalBeauty #HistMath #Archimedes #geometry #architecture #aesthetics
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As noted in a previous post, Archimedes thought highly of the result that the ratio of either the volumes or surface areas of a cone, a sphere, and a cylinder exactly circumscribing them is $1:2:3$.
So did others: three centuries later, the architect Nicon (d.149/50 CE), father of the philosopher and physician Galen (129–c.210/217 CE), thought it fitting to point out the ratio of the configuration in a public inscription in his city, Pergamon:
‘the cone, the sphere, the cylinder.
If a cylinder encloses the other two shapes,
[...]
Competition the principle and in solids
the progression $1 ∶ 2 ∶ 3$,
a noble, divine equalization,
but also mutual interdependence
of the solids, always in the ratio $1 ∶ 2 ∶ 3$.
They should be beautiful and wonderful,
the three solid shapes’Nicon doubtless admired these ratios as an architect: a sphere inside a cylinder brings to mind the Pantheon at Rome, of which the Temple of Zeus Asclepius Soter in Pergamon was a half-scale copy. These buildings were designed so that a basically cylindrical rotunda was crowned with a hemispherical dome under which a sphere would fit (see attached image).
[Each day of February, I am posting a story/image/fact/anecdote related to the aesthetics of mathematics.]
1/2
#MathematicalBeauty #HistMath #Archimedes #geometry #architecture #aesthetics
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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.]
1/2
#MathematicalBeauty #HistMath #Archimedes #Plutarch #Cicero #geometry #aesthetics
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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.]
1/2
#MathematicalBeauty #HistMath #Archimedes #Plutarch #Cicero #geometry #aesthetics
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The ‘Conics’ of Apollonius of Perga (c.260–c.190 BCE) became the standard text for ‘conic sections’ — the curves formed by the intersection of a plane and a cone, namely an ellipse, parabola, or hyperbola, depending on the angle of the plane relative to the slope of the cone (see attached image).
In the preface to the ‘Conics’, Apollonius wrote:
‘The third book contains many incredible theorems of use for the construction of solid loci and for limits of possibility of which the greatest part and the most beautiful [kallista κάλλιστα] are new.’
This quotation is triply important in the historiography of mathematical beauty: (1) it is the earliest extant description of a mathematical theorem as ‘beautiful’; (2) it is the earliest extant application of the term ‘beautiful’ to mathematics by a mathematician; and (3) it is the unique extant use of the term ‘beautiful’ to describe theorems by an ancient Greek mathematician.
(There is much discussion of the beauty of mathematics in ancient Greek thought, but it normally applies to the objects or concepts of mathematics.)
[Each day of February, I intend to post an interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
1/3
#MathematicalBeauty #HistMath #Conic #ConicSection #geometry #aesthetics
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The ‘Conics’ of Apollonius of Perga (c.260–c.190 BCE) became the standard text for ‘conic sections’ — the curves formed by the intersection of a plane and a cone, namely an ellipse, parabola, or hyperbola, depending on the angle of the plane relative to the slope of the cone (see attached image).
In the preface to the ‘Conics’, Apollonius wrote:
‘The third book contains many incredible theorems of use for the construction of solid loci and for limits of possibility of which the greatest part and the most beautiful [kallista κάλλιστα] are new.’
This quotation is triply important in the historiography of mathematical beauty: (1) it is the earliest extant description of a mathematical theorem as ‘beautiful’; (2) it is the earliest extant application of the term ‘beautiful’ to mathematics by a mathematician; and (3) it is the unique extant use of the term ‘beautiful’ to describe theorems by an ancient Greek mathematician.
(There is much discussion of the beauty of mathematics in ancient Greek thought, but it normally applies to the objects or concepts of mathematics.)
[Each day of February, I intend to post an interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
1/3
#MathematicalBeauty #HistMath #Conic #ConicSection #geometry #aesthetics
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Tout juste paru chez College Publications (Londres) après une loooongue période de maturation :
Circulations des mathématiques dans et par les journaux — Histoire, territoires, publics
Cet ouvrage résulte des travaux du collectif international réuni autour du projet ANR « Cirmath » qui a réuni plus de 30 contributrices et contributeurs de près d’une dizaine de pays.
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Tout juste paru chez College Publications (Londres) après une loooongue période de maturation :
Circulations des mathématiques dans et par les journaux — Histoire, territoires, publics
Cet ouvrage résulte des travaux du collectif international réuni autour du projet ANR « Cirmath » qui a réuni plus de 30 contributrices et contributeurs de près d’une dizaine de pays.
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Mathématiques en Chine : à la racine du calcul | Sciences Chrono France Culture
Karine Chemla au micro d'Antoine Beauchamp !
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Mathématiques en Chine : à la racine du calcul | Sciences Chrono France Culture
Karine Chemla au micro d'Antoine Beauchamp !