Squaring the circle

by John Q on July 4, 2005

Reading John and Belle’s blog, not the place I would usually look for unfamiliar maths results, I discovered that the circle can be squared in Gauss-Bolyai-Lobachevsky space . Checking a bit further, I found this was not a new result but was shown by Bolyai back in C19.

I haven’t found a link that shows how the construction was done, though. Can someone point me in the right direction, please?



Kevin Donoghue 07.04.05 at 6:57 pm

Looks like a library might be the best bet (the original was in Latin though):

“Bolyai gives also … a geometrical construction, in hyperbolic space, for the quadrature of the circle….”



John Quiggin 07.04.05 at 7:34 pm

Hmm, a library, now there’s an idea. But I’d still need the name of a book.


ogged 07.04.05 at 7:46 pm

We have to do all the work, don’t we? The link in the comments at John & Belle’s goes to this page, where the claim is cited to a book by Jeremy Gray.


James Wimberley 07.05.05 at 2:20 am

There must be an academic ethnic joke about Hungarians in this, on the lines of “a Hungarian is someone who goes into a revolving door behind you and comes out ahead of you.”


dsquared 07.05.05 at 4:20 am

I can’t believe that those pikers Gauss and Bolyai got their name on “Gauss-Bolyai-Lobachevsky space”. One man deserves the credit …


John Quiggin 07.05.05 at 5:26 am

And Nikolai Ivanovich Lobachevsky is his name!


Kieran Healy 07.05.05 at 6:12 am

Can someone point me in the right direction, please?

I have a friend in Minsk, who has a friend in Pinsk…


Creative Commonist 07.05.05 at 6:24 am

Bolyai János
The theory of space / János Bolyai ; With introduction, comments, and addenda edited by Ferenc Kárteszi ; Supplement by Barna Szénássy ;
Budapest, Akadémiai Kiadó, 1987
This is an English translation of the Latin original.
Bonola, Roberto
Non-Euclidean geometry, a critical and historical study of its developments. / Bonola, Roberto Carslaw, H. S, Halsted, G. B, Bolyai, John, Lobachevski, Nicholas
Translation with additional appendices by H. S. Carslaw. Supplement containing the G. B. Halsted translations of “The science of absolute space” by John Bolyai and “The theory of parallels” by Nicholas Lobachevski.
Dover Publications, Inc., New York, 1955
You may contact the Mathematical Institute of the Hungarian Academy of Sciences: math@renyi.hu


Jacques Distler 07.05.05 at 10:04 am

Since you don’t accept trackbacks, look here.


Robert 07.05.05 at 10:29 am

am never forget the day my first book is published.
Every chapter I stole from somewhere else.
Index I copy from old Vladivostok telephone directory.
This book was sensational!
Pravda – well, Pravda – Pravda said: (Russian double-talk)
It stinks.
But Izvestia! Izvestia said: (Russian double-talk)
It stinks.
Metro-Goldwyn-Moskva buys movie rights for six million rubles,
Changing title to ‘The Eternal Triangle’,
With Brigitte Bardot playing part of hypotenuse.


James Wimberley 07.06.05 at 4:40 pm

The Halsted translation of Bolyai’s “Theory of absolute space” is on the Web at http://www.hti.umich.edu/cgi/b/broker20/broker20?verb=Display&protocol=CGM&ver=1.0&identifier=oai:lib.umich.edu:ACA7845.0001.001


Walt Pohl 07.06.05 at 6:46 pm

Jeremy Gray has another book that’s really good (though probably more technical than the book on non-Euclidean geometry), called
Linear Differential Equations and Group Theory from Riemann to Poincare
. It shows how geometric functions of a complex variable naturally are. By telling the story historically, it reads like the mathematical analogue of a mystery novel. At first you only see glimpses of what’s going on, and then bigger and bigger hints, until finally everything is revealed.

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