All Questions
Tagged with ho.history-overview euclidean-geometry
22 questions
7
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1
answer
1k
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An unpublished calculation of Gauss and the icosahedral group
According to p. 68 of Paul Stackel's essay "Gauss as geometer" (which deals with "complex quantities with more than two units") , Gauss calculated the coordinates of the vertices ...
- 2,099
1
vote
1
answer
253
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"On models of elementary elliptic geometry"
While perusing p. 237 of the 3rd ed. of Marvin Greenberg's book on Euclidean and non-Euclidean geometries, I learned that it can actually be proven that "all possible models of hyperbolic ...
- 8,842
11
votes
3
answers
557
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Was the small Desargues Theorem known to ancient Greeks?
My question concerns the classical Desargues Theorem and its simplest version
The small Desargues Theorem: Let $A$, $B$, $C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
- 41.8k
2
votes
0
answers
59
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Historical question about tangent lines to disjoint circles
It is pretty well known that two disjoint circles have 4 different lines that are simultaneously tangent to both circles. There are constructions with ruler and compass available in many books, but I ...
5
votes
0
answers
267
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Are there any neusis-hard/neusis-complete problems?
I have lately been enjoying Richeson's Tales of Impossibility (see MAA review), an accessible book on the famous problems of Euclidean geometry including angle trisection/cube doubling/heptagon ...
- 2,185
18
votes
2
answers
1k
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Emergence of the orthogonal group
Do we know what mathematician first considered, and perhaps named, what we call the group $\mathrm O(n)$, or $\mathrm{SO}(n)$, for some $n>3$?
I mean it specifically as group (not Lie algebra) ...
- 31.5k
4
votes
1
answer
216
views
Is this elementary formula for the parabolic segment new?
Recently (May 2020) a formula for the area of the parabolic segment (i.e. the region enclosed by a parabola and a line), in terms of the coefficients of the Cartesian equations, has been published by ...
34
votes
5
answers
3k
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Open problems from antiquity solved with analytic geometry
E. T. Bell wrote in Men of Mathematics:
Though the idea behind it all is childishly simple, yet the method of analytic geometry is so powerful that very ordinary boys of seventeen can use it to prove
...
5
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2
answers
355
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Frégier and Frégier's Theorem
A curious and interesting gem is Frégier's theorem, quoted here from David Wells:
Choose any point $P$ on a conic, and make it the vertex of a right
angle which rotates about $P$. Then the ...
- 3,443
3
votes
1
answer
610
views
Problem Understanding Euclid Book 10 Proposition 1 [closed]
this is embarrassing, but I am having trouble reading through Proposition 1 of Book 10 of Euclid's elements. I'm struggling with Euclid's terminology and don't have a clear picture of what divisions ...
- 595
11
votes
1
answer
480
views
Yau's problem: Construct a triangle given a side, an angle, and an angle bisector
In Shing-Tung Yau's autobiography The Shape of a Life, he mentions a problem that he came up with as a teenager.
Suppose you know the length of one side of a triangle, one angle, and the length of ...
- 82.6k
3
votes
1
answer
807
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History of the Taxonomy of Quadrilaterals
Question:
how did the classification of quadrilaterals come into being? Was there a single major contributor who coined terms like "rectangle", "square", "trapez/ium/oid"...
- 13.2k
29
votes
2
answers
2k
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Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?
In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before:
quoted from Leo Corry, Modern algebra, German original:
Why did Dedekind doubt that $(\...
- 9,720
4
votes
1
answer
829
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Constructibility of the regular 17-gon [closed]
There is a standard construction of a regular heptadecagon by H.W. Richmond (1893) (https://en.wikipedia.org/wiki/Heptadecagon ) (As anyone knows, it was Gauss who found out that it is possible to do ...
- 61
4
votes
1
answer
420
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Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?
I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
- 57
2
votes
1
answer
565
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Pasch axiom and Pythagorean field condition?
I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true.
Since posting the question, ...
- 16.6k
18
votes
1
answer
875
views
What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?
As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
- 15.3k
6
votes
1
answer
208
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Did Lucas discover Lucas circles?
MathWord's article on Lucas circles traces the name to a little-known 1973 publication. These interesting circles have found their way into several 21st century publications, including the online ...
- 3,443
2
votes
0
answers
189
views
Examples of Geometric Constructions in Higher Dimensions
The classical problem of geometric construction seems to be restricted to planar Euclidean Geometry with straight edge and compass as the only admissible "construction-tools".
I would like to know,...
- 13.2k
7
votes
2
answers
1k
views
Sine and Archimedes' derivation of the area of the circle
The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...
- 313
18
votes
4
answers
16k
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The Ramanujan Problems
I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here.
In the Wikipedia page on Ramanujan (current ...
- 829
5
votes
2
answers
447
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Historical question re: ellipses obtained by certain geometrical constructions
I am a faculty member in the Forensic Science Program at PennState (UP). I am trying to obtain information of a historical nature concerning two closely related topics. I seek historical references ...
