Clairaut's "most important and profound discovery"
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From: James Dow Allen on 11 Feb 2010 13:33 On Feb 11, 11:30 pm, Chip Eastham <hardm...(a)gmail.com> wrote: > On Feb 11, 11:10 am, James Dow Allen <jdallen2...(a)yahoo.com> wrote: > > On Feb 11, 7:36 pm, David Bernier <david...(a)videotron.ca> wrote: > > > Google Books has a scan of the second edition of Clairaut's > > > "Theorie de la lune" (1765): > > > <http://books.google.com/books?id=U1EVAAAAQAAJ> > > > I wonder if you thought you were being helpful. :-) > > I thought he was! It clarified the controversy to the > point where I felt grateful sci.math is not inundated > by cranks complaining about Newton's theory of gravity. > > Clairaut's book is in French, but it cost me nothing > to download. OK, thanks! I get confused with all the different formats of books on the Internet, many of which are difficult for me pay-or-not. (I will be getting a faster Internet connection soon, but as of now, slowness makes a lot of simple things impossible.) But given how rusty my French is, the length of the book, and the difficulty of the math, I'm not sure I could answer my own question if I had the whole book in front of me now! Euler's comment: > "the most important and profound discovery that has > ever been made in mathematics." seems pretty extreme! Surely there's a straightforward synopsis-for-the-layman of Clairaut's discovery more accessible than a long French book! James
From: David Bernier on 12 Feb 2010 04:13 James Dow Allen wrote: > On Feb 11, 7:36 pm, David Bernier <david...(a)videotron.ca> wrote: >> James Dow Allen wrote: >>> One reads in >>> http://www-history.mcs.st-andrews.ac.uk/Biographies/Clairaut.html >>> about Alexis Clairaut's 1752 paper "Theorie de la Lune" on the >>> three-body problem and that Euler called this >>> "the most important and profound discovery that has >>> ever been made in mathematics." >>> Even allowing for a bit of hyperbole, this sounds like strong praise >>> from Euler! Can anyone summarize Clairaut's "profound" discovery >>> for the layman? >> From the MacTutor biography of Clairaut: >> [snip] >> Google Books has a scan of the second edition of Clairaut's >> "Theorie de la lune" (1765): >> <http://books.google.com/books?id=U1EVAAAAQAAJ> > > You quote a slightly longer excerpt from the webpage I'd already > cited. Then post a link to a 161-page pay-per-view book. > > I wonder if you thought you were being helpful. :-) I was doing my best. I know for example that there are different lunar months of varying lengths. One is the anomalistic month, the average time between one apogee of the moon and the next. In Newton's Principia, it's written: "The apse of the moon is about twice as swift" or: "Apsis lunae est duplo velocior circiter" [ Newton's Principae, Ed. 2 and 3. ] I think this meant: "The apse of the moon is [in fact] about twice as swift [as my calculations on perturbations show]." Clairaut may have contributed to better perturbation calculations, including the motion of the apses. David
From: David Bernier on 12 Feb 2010 04:55 David Bernier wrote: > James Dow Allen wrote: >> On Feb 11, 7:36 pm, David Bernier <david...(a)videotron.ca> wrote: >>> James Dow Allen wrote: >>>> One reads in >>>> http://www-history.mcs.st-andrews.ac.uk/Biographies/Clairaut.html >>>> about Alexis Clairaut's 1752 paper "Theorie de la Lune" on the >>>> three-body problem and that Euler called this >>>> "the most important and profound discovery that has >>>> ever been made in mathematics." >>>> Even allowing for a bit of hyperbole, this sounds like strong praise >>>> from Euler! Can anyone summarize Clairaut's "profound" discovery >>>> for the layman? >>> From the MacTutor biography of Clairaut: >>> [snip] >>> Google Books has a scan of the second edition of Clairaut's >>> "Theorie de la lune" (1765): >>> <http://books.google.com/books?id=U1EVAAAAQAAJ> >> >> You quote a slightly longer excerpt from the webpage I'd already >> cited. Then post a link to a 161-page pay-per-view book. >> >> I wonder if you thought you were being helpful. :-) > > I was doing my best. I know for example that there are different lunar > months of varying lengths. One is the anomalistic month, the average > time between one apogee of the moon and the next. > > In Newton's Principia, it's written: > "The apse of the moon is about twice as swift" or: > "Apsis lunae est duplo velocior circiter" > [ Newton's Principae, Ed. 2 and 3. ] > > I think this meant: > "The apse of the moon is [in fact] about twice as swift [as my > calculations on perturbations show]." > > Clairaut may have contributed to better perturbation > calculations, including the motion of the apses. [...] Based on an MAA guest column by Dominic Klyve, it seems that is the case. Ref.: < http://www.maa.org/editorial/euler/Dec2009.pdf >
From: Marko Amnell on 12 Feb 2010 06:48 "Marko Amnell" <marko.amnell(a)kolumbus.fi> wrote in message 7tkqhoFvm0U1(a)mid.individual.net... > > "David Bernier" <david250(a)videotron.ca> wrote in message > hl363d0r2(a)news3.newsguy.com... > >> Clairaut may have contributed to better perturbation >> calculations, including the motion of the apses. > > Yes, Clairaut was able to explain the motion of the > apse (a semicircular or polygonal projection of a plane) Err, rather the apse or apsis, the point of maximum or minimum distance in an elliptical orbit of an object from the centre of attraction. Sorry for the brain short circuit...
From: James Dow Allen on 12 Feb 2010 14:39
On Feb 12, 4:55 pm, David Bernier <david...(a)videotron.ca> wrote: > David Bernier wrote: > > Clairaut may have contributed to better perturbation > > calculations, including the motion of the apses. > [...] > Based on an MAA guest column by Dominic Klyve, it seems > that is the case. > > Ref.: > <http://www.maa.org/editorial/euler/Dec2009.pdf> Thank you very much for this link! *Very* interesting ... and even *I* could understand it! It answers my question, and tells a fun story to boot. (One piece that caught my eye was where Euler experimented (anonymously!) with deriving an inverse-square-distance force (d^-2) as the difference between positive and negative inverse-distance forces (d^-1). This reminded me of something I'd only recently become aware of: the way a magnet gives an inverse-cube-distance attraction (d^-3) because its net force is the difference of the inverse-square forces from the two poles.) One particularly amazing part of the story is where Euler and d'Alembert independently conjectured a huge protrusion from the hidden side of the moon as one way to explain the anomaly! Apparently Clairaut's "most important and profound discovery" was simply that the approximation d(x^2) = 2 dx + (dx)^2 = 2 dx caused trouble; one could not afford to discard the (dx)^2 term. But I'm still surprised this warranted such hyperbole. Thanks again for the link (and sorry for acting irritable earlier). James Dow Allen |