Extracted Text
HEDI-2010-02.pdf
2
It can be seen that he much preferred the
education of his students than the small satisfaction
derived from astonishment; he never believed that he had
truly done enough for Science if he did not feel that that
he had added new truths to enrich it and the exposure of
the simplicity of the idea which lead him there.
…
Of the sixteen professors attached to the Saint
Petersburg Academy eight were trained under him and all
are known through their works and have been awarded
various academic distinctions and are proud to add the
title of Euler's disciples.
Condorcet goes on to mention some of the people who
studied under Euler, his two sons, Lexell, and Fuss in particular,
and that Fuss married one of Euler's granddaughters.
Except for disclaiming "the small satisfaction derived
from astonishment," this does little to tell us about how Euler taught or why he was effective. The
stories that Condorcet relates about:
• Euler’s students taking dictation,
• Euler reading things written large on a tablet or chalkboard,
• Euler wearing a shiny track in the table as he used it to guide himself while he paced around it
and talked to his students,
tell us very little about how Euler actually taught.
Euler last attended a meeting of the St. Petersburg Academy on January 16, 1777, after which he
sent his papers in to the Academy with his assistants. One of the portraits of Euler, shown above, has a
sub-portrait, a smaller rectangle beneath the oval of the main portrait. The sub-portrait shows two men,
one with pen and paper, sitting at a table. Apparently it pictures Euler dictating to one of his assistants,
probably his son, Johann Albrecht, because Euler himself could no longer read or write.
Internal evidence
I want to cite two kinds of evidence about Euler's teaching in St. Petersburg: 1
° data from the
Adversariis mathematicis, or Mathematical daybook [E806]; and 2° subjective observations from
reading several of Euler's late papers.
The Adversariis mathematicis.
The
Adversariis mathematicis was a series of three notebooks kept in the foyer of the St.
Petersburg Academy. Members used the
Adversariis as a kind of chat room or virtual seminar to show
their colleagues what they were working on and to announce preliminary results. Eventually, the
3
Adversariis filled three notebooks totaling 776 pages. Less than 30% of their contents, amounting to
111 entries, appeared in the
Opera posthuma in 1862. [E805] They are sprinkled about several volumes
of the
Opera omnia according to the subjects of the notes.
Most of the notes are dull and technical. Many are wrong. Some are dead ends, sometimes
intriguing, but ultimately doomed. For example, Note 67 is a 17-page joint effort by J. A. Euler, Lexell,
Fuss and Krafft to solve Fermat's Last Theorem. They get stuck on the same technical points of unique
factorization that befuddled 19th century mathematicians.
Note 24 is a contribution by Krafft, noting that both
x
2
+x+17 and x
2
+x+41 give nothing
but prime numbers for small values of
x.
Note 104 is by J. A. Euler, and essentially he rewrites Ptolmey's theorem about the sides and
diagonals of a cyclic quadrilateral in terms of sines, and then extends it to "infinitely large" circles, i. e.
straight lines. It is not clear if he noticed that what he gets is a theorem in geometric algebra found in
Euclid's
Elements, book II.
Note 96 is a proof by Nicolas Fuss of the elementary properties of the Euler
ϕ function. Notes
82 and 83 are about magic squares and Greco-Latin squares. Dozens are about Diophantine equations,
especially those related to quadratic reciprocity.
The full text of the
Adversariis mathematicis has not been published, and I've not seen any
description of what fills the more than 70% of the notebooks that were not published in 1862. We can
only speculate.
By the time these notes were published in 1862, 79 years after Euler's death, the results were all
quite stale, but only a couple would have been interesting even if they had been disseminated earlier.
I tabulated who contributed to the
Adversariis mathematiciis:
number pages
Fuss 26
68
Lexell 23 70
JA Euler 17 51
Krafft 11 42
Golovin 3 6
Euler 1 2
unsigned 56
Note that the number of signatures doesn't add up to 55, because several notes, like Note 67,
were joint efforts with more than one signature. Half the notes were unsigned.
What does this say about Euler's teaching? It looks to me like Euler used the
Adversariis as a
proxy for a graduate seminar. It is as if his five students would read papers from earlier in Euler's
career, work through the difficulties, either alone or together, and demonstrate their mastery of the
material (or ineptitude) by proving old results in new ways, working through examples, filling in details
and extending and generalizing the results. The style closely resembles the way Johann Bernoulli had
taught Euler some sixty years earlier.
4
Late papers
When Euler died in 1783, he left a legacy of over 200 unpublished papers, virtually all of which
he wrote after returning to St. Petersburg in 1766. Only a few of them were important. Let's look at a
couple of examples.
In
Investigatio quarundam serierum quae ad rationem peripheriae circuli ad diametrum vero
proxime definiendam maxime sunt accommodatae
, "Investigation of certain series which are designed to
approximate the true ratio of the circumference
of a circle to its diameter very closely," [E705,
Sandifer Feb 2009] Euler repeats work from
De variis modis circuli quadraturam numeris
proxime exprimendi
," On several means of
expressing the quadrature of area of a circle
very accurately." [E74] The first paper
showed how to use the Machin equations,
arctan1=arctan
1
a
+arctan
1
b
to give fast converging approximations to π.
The second paper does the same thing, but
engineers the series so they have easy
denominators that are powers of 2, 5 and 10.
Sur l'effet de la réfraction dans les observations terrestres, "On the effect of refraction on
terrestrial observations," [E502] shows how to correct for the way the varying density of the atmosphere
bends light and affects surveying the heights of mountains. It repeats much of the material from De la
réfraction de la lumière en passant par l'atmosphère selon les divers degrés tant de la chaleur que de
l'élasticité de l'air
, "On the refraction of light passing through the atmosphere due to the different
degrees of heat as well as the elasticity of the air," [E219] where he studied the same refraction for
astronomical observations.
Many of his other late papers are like this, especially his late number theory, revisiting earlier
problems with a new twist or complication, a different proof (not necessarily better) or a longer
example. The papers are often a bit choppy and the examples less interesting than in his earlier papers.
For a long time, I thought that these were signs of Euler declining as he aged. That may be true, but
now I think that these are also the fingerprints of Euler's students. It seems likely that Euler would
revisit an old paper and ask his students to see what they could do with it. They could have asked him
questions, harvested ideas for extensions, and then, with his guidance, write papers under Euler's name.
This would not be much different from a master artist who had his students fill in the backgrounds of his
paintings.
Punch line
Euler learned at the feet of Johann Bernoulli, who had Euler "read the masters." Euler read
difficult mathematics and Bernoulli helped him when he got stuck.
5
We now know some details of Euler's classroom teaching during his first St. Petersburg years,
but we have no evidence or testimonies about what kind of teacher he was.
We only know a few details about Euler's teaching in Berlin. The setting wasn't a formal, but it
foreshadowed his later teaching style.
In his late years, I propose that Euler was able to teach in the style under which he himself had
learned. He had learned guided by the principle
Read the masters.
He taught under the style
Read me, read me. I am Euler and I am your teacher in all things.
References:
[Condorcet 1786] Condorcet, Marie Jean Antoine Nicolas de Caritat, Marquis de, Eloge de M. Euler, Hist. Acad. Royale des
Sciences 1783, p. 37-68, Paris 1786. Translation by John S. D. Glaus, http://www-history.mcs.st-
and.ac.uk/Extras/Euler_elogium.html.
[Dunham 1999] Dunham, William, Euler: The Master of Us All, Mathematical Association of America, 1999.
[E74] Euler, Leonhard, De variis modis circuli quadraturam numeris proxime exprimendi, Commentarii academiae
scientiarum Petropolitanae 9 (1744), pp. 222-236. Reprinted in Opera omnia I.14, pp. 245-259. Available online at
EulerArchive.org.
[E219] Euler, Leonhard, De la refraction de la lumiere en passant par l'atmosphere selon les divers degres tant de la chaleur
que de l'elasticite de l'air, Memoires de l'academie des sciences de Berlin 10 (1756), pp. 131-172. Reprinted in
Opera omnia III.5, pp. 185–217. Available online at EulerArchive.org.
[E502] Euler, Leonhard, Sur l'effet de la refraction dans les observations terrestres, Acta Academiae Scientarum Imperialis
Petropolitinae (1780), pp. 129-158. Reprinted in Opera omnia III.5, pp. 370–395. Available online at
EulerArchive.org.
[E705] Euler, Leonhard, Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime
definiendam maxime sunt accommodatae, Nova Acta Academiae Scientarum Imperialis Petropolitinae 11 (1798),
pp. 133-149. Reprinted in Opera omnia I.16, pp. 1–20. Available online at EulerArchive.org.
[E805] Euler, Leonhard, Opera Postuma mathematica et physica, 2 vols., St. Petersburg Academy of Science, 1862.
Reprinted in Opera omnia, various volumes. Individual articles available online at EulerArchive.org.
[E806] Euler, Leonhard, Fragmenta arithmetica ex Adversariis mathematicis depromta, in [E805], pp. 157-266. Reprinted in
Opera omnia, various volumes. Available online at EulerArchive.org.
[Rice 2003] Rice, Adrian, Brought to book: the curious story of Guglielmo Libri (1803-69), Newsletter of the European
Mathematical Society, 48, June 2003, p. 12–14.
[Sandifer Feb 2009] Sandifer, C. Edward, How Euler Did It, No. 64, maa.org.
[Sandifer Jan 2010] Sandifer, C. Edward, How Euler Did It, No. 74, maa.org.
[WikiQuote] http://en.wikiquote.org/wiki/Pierre-Simon_Laplace
Ed Sandifer (
SandiferE@wcsu.edu) is Professor of Mathematics at Western Connecticut State
University in Danbury, CT. He is an avid marathon runner, with 37 Boston Marathons on his shoes, and
he is Secretary of The Euler Society (www.EulerSociety.org). His first book, The Early Mathematics of
Leonhard Euler, was published by the MAA in December 2006, as part of the celebrations of Euler’s
tercentennial in 2007. The MAA published a collection of forty How Euler Did It columns in June
2007.
How Euler Did It is updated each month.
Copyright ©2010 Ed Sandifer
It can be seen that he much preferred the
education of his students than the small satisfaction
derived from astonishment; he never believed that he had
truly done enough for Science if he did not feel that that
he had added new truths to enrich it and the exposure of
the simplicity of the idea which lead him there.
…
Of the sixteen professors attached to the Saint
Petersburg Academy eight were trained under him and all
are known through their works and have been awarded
various academic distinctions and are proud to add the
title of Euler's disciples.
Condorcet goes on to mention some of the people who
studied under Euler, his two sons, Lexell, and Fuss in particular,
and that Fuss married one of Euler's granddaughters.
Except for disclaiming "the small satisfaction derived
from astonishment," this does little to tell us about how Euler taught or why he was effective. The
stories that Condorcet relates about:
• Euler’s students taking dictation,
• Euler reading things written large on a tablet or chalkboard,
• Euler wearing a shiny track in the table as he used it to guide himself while he paced around it
and talked to his students,
tell us very little about how Euler actually taught.
Euler last attended a meeting of the St. Petersburg Academy on January 16, 1777, after which he
sent his papers in to the Academy with his assistants. One of the portraits of Euler, shown above, has a
sub-portrait, a smaller rectangle beneath the oval of the main portrait. The sub-portrait shows two men,
one with pen and paper, sitting at a table. Apparently it pictures Euler dictating to one of his assistants,
probably his son, Johann Albrecht, because Euler himself could no longer read or write.
Internal evidence
I want to cite two kinds of evidence about Euler's teaching in St. Petersburg: 1
° data from the
Adversariis mathematicis, or Mathematical daybook [E806]; and 2° subjective observations from
reading several of Euler's late papers.
The Adversariis mathematicis.
The
Adversariis mathematicis was a series of three notebooks kept in the foyer of the St.
Petersburg Academy. Members used the
Adversariis as a kind of chat room or virtual seminar to show
their colleagues what they were working on and to announce preliminary results. Eventually, the
3
Adversariis filled three notebooks totaling 776 pages. Less than 30% of their contents, amounting to
111 entries, appeared in the
Opera posthuma in 1862. [E805] They are sprinkled about several volumes
of the
Opera omnia according to the subjects of the notes.
Most of the notes are dull and technical. Many are wrong. Some are dead ends, sometimes
intriguing, but ultimately doomed. For example, Note 67 is a 17-page joint effort by J. A. Euler, Lexell,
Fuss and Krafft to solve Fermat's Last Theorem. They get stuck on the same technical points of unique
factorization that befuddled 19th century mathematicians.
Note 24 is a contribution by Krafft, noting that both
x
2
+x+17 and x
2
+x+41 give nothing
but prime numbers for small values of
x.
Note 104 is by J. A. Euler, and essentially he rewrites Ptolmey's theorem about the sides and
diagonals of a cyclic quadrilateral in terms of sines, and then extends it to "infinitely large" circles, i. e.
straight lines. It is not clear if he noticed that what he gets is a theorem in geometric algebra found in
Euclid's
Elements, book II.
Note 96 is a proof by Nicolas Fuss of the elementary properties of the Euler
ϕ function. Notes
82 and 83 are about magic squares and Greco-Latin squares. Dozens are about Diophantine equations,
especially those related to quadratic reciprocity.
The full text of the
Adversariis mathematicis has not been published, and I've not seen any
description of what fills the more than 70% of the notebooks that were not published in 1862. We can
only speculate.
By the time these notes were published in 1862, 79 years after Euler's death, the results were all
quite stale, but only a couple would have been interesting even if they had been disseminated earlier.
I tabulated who contributed to the
Adversariis mathematiciis:
number pages
Fuss 26
68
Lexell 23 70
JA Euler 17 51
Krafft 11 42
Golovin 3 6
Euler 1 2
unsigned 56
Note that the number of signatures doesn't add up to 55, because several notes, like Note 67,
were joint efforts with more than one signature. Half the notes were unsigned.
What does this say about Euler's teaching? It looks to me like Euler used the
Adversariis as a
proxy for a graduate seminar. It is as if his five students would read papers from earlier in Euler's
career, work through the difficulties, either alone or together, and demonstrate their mastery of the
material (or ineptitude) by proving old results in new ways, working through examples, filling in details
and extending and generalizing the results. The style closely resembles the way Johann Bernoulli had
taught Euler some sixty years earlier.
4
Late papers
When Euler died in 1783, he left a legacy of over 200 unpublished papers, virtually all of which
he wrote after returning to St. Petersburg in 1766. Only a few of them were important. Let's look at a
couple of examples.
In
Investigatio quarundam serierum quae ad rationem peripheriae circuli ad diametrum vero
proxime definiendam maxime sunt accommodatae
, "Investigation of certain series which are designed to
approximate the true ratio of the circumference
of a circle to its diameter very closely," [E705,
Sandifer Feb 2009] Euler repeats work from
De variis modis circuli quadraturam numeris
proxime exprimendi
," On several means of
expressing the quadrature of area of a circle
very accurately." [E74] The first paper
showed how to use the Machin equations,
arctan1=arctan
1
a
+arctan
1
b
to give fast converging approximations to π.
The second paper does the same thing, but
engineers the series so they have easy
denominators that are powers of 2, 5 and 10.
Sur l'effet de la réfraction dans les observations terrestres, "On the effect of refraction on
terrestrial observations," [E502] shows how to correct for the way the varying density of the atmosphere
bends light and affects surveying the heights of mountains. It repeats much of the material from De la
réfraction de la lumière en passant par l'atmosphère selon les divers degrés tant de la chaleur que de
l'élasticité de l'air
, "On the refraction of light passing through the atmosphere due to the different
degrees of heat as well as the elasticity of the air," [E219] where he studied the same refraction for
astronomical observations.
Many of his other late papers are like this, especially his late number theory, revisiting earlier
problems with a new twist or complication, a different proof (not necessarily better) or a longer
example. The papers are often a bit choppy and the examples less interesting than in his earlier papers.
For a long time, I thought that these were signs of Euler declining as he aged. That may be true, but
now I think that these are also the fingerprints of Euler's students. It seems likely that Euler would
revisit an old paper and ask his students to see what they could do with it. They could have asked him
questions, harvested ideas for extensions, and then, with his guidance, write papers under Euler's name.
This would not be much different from a master artist who had his students fill in the backgrounds of his
paintings.
Punch line
Euler learned at the feet of Johann Bernoulli, who had Euler "read the masters." Euler read
difficult mathematics and Bernoulli helped him when he got stuck.
5
We now know some details of Euler's classroom teaching during his first St. Petersburg years,
but we have no evidence or testimonies about what kind of teacher he was.
We only know a few details about Euler's teaching in Berlin. The setting wasn't a formal, but it
foreshadowed his later teaching style.
In his late years, I propose that Euler was able to teach in the style under which he himself had
learned. He had learned guided by the principle
Read the masters.
He taught under the style
Read me, read me. I am Euler and I am your teacher in all things.
References:
[Condorcet 1786] Condorcet, Marie Jean Antoine Nicolas de Caritat, Marquis de, Eloge de M. Euler, Hist. Acad. Royale des
Sciences 1783, p. 37-68, Paris 1786. Translation by John S. D. Glaus, http://www-history.mcs.st-
and.ac.uk/Extras/Euler_elogium.html.
[Dunham 1999] Dunham, William, Euler: The Master of Us All, Mathematical Association of America, 1999.
[E74] Euler, Leonhard, De variis modis circuli quadraturam numeris proxime exprimendi, Commentarii academiae
scientiarum Petropolitanae 9 (1744), pp. 222-236. Reprinted in Opera omnia I.14, pp. 245-259. Available online at
EulerArchive.org.
[E219] Euler, Leonhard, De la refraction de la lumiere en passant par l'atmosphere selon les divers degres tant de la chaleur
que de l'elasticite de l'air, Memoires de l'academie des sciences de Berlin 10 (1756), pp. 131-172. Reprinted in
Opera omnia III.5, pp. 185–217. Available online at EulerArchive.org.
[E502] Euler, Leonhard, Sur l'effet de la refraction dans les observations terrestres, Acta Academiae Scientarum Imperialis
Petropolitinae (1780), pp. 129-158. Reprinted in Opera omnia III.5, pp. 370–395. Available online at
EulerArchive.org.
[E705] Euler, Leonhard, Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime
definiendam maxime sunt accommodatae, Nova Acta Academiae Scientarum Imperialis Petropolitinae 11 (1798),
pp. 133-149. Reprinted in Opera omnia I.16, pp. 1–20. Available online at EulerArchive.org.
[E805] Euler, Leonhard, Opera Postuma mathematica et physica, 2 vols., St. Petersburg Academy of Science, 1862.
Reprinted in Opera omnia, various volumes. Individual articles available online at EulerArchive.org.
[E806] Euler, Leonhard, Fragmenta arithmetica ex Adversariis mathematicis depromta, in [E805], pp. 157-266. Reprinted in
Opera omnia, various volumes. Available online at EulerArchive.org.
[Rice 2003] Rice, Adrian, Brought to book: the curious story of Guglielmo Libri (1803-69), Newsletter of the European
Mathematical Society, 48, June 2003, p. 12–14.
[Sandifer Feb 2009] Sandifer, C. Edward, How Euler Did It, No. 64, maa.org.
[Sandifer Jan 2010] Sandifer, C. Edward, How Euler Did It, No. 74, maa.org.
[WikiQuote] http://en.wikiquote.org/wiki/Pierre-Simon_Laplace
Ed Sandifer (
SandiferE@wcsu.edu) is Professor of Mathematics at Western Connecticut State
University in Danbury, CT. He is an avid marathon runner, with 37 Boston Marathons on his shoes, and
he is Secretary of The Euler Society (www.EulerSociety.org). His first book, The Early Mathematics of
Leonhard Euler, was published by the MAA in December 2006, as part of the celebrations of Euler’s
tercentennial in 2007. The MAA published a collection of forty How Euler Did It columns in June
2007.
How Euler Did It is updated each month.
Copyright ©2010 Ed Sandifer