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Questions tagged [euclidean-geometry]

For questions about the mathematical study of shapes and space based on the works of Euclid.

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1answer
144 views

I can't comprehend the sentence in Euclid Elements [closed]

I am Korean, and I thought I can understand majority of english sentences, but this is really hard to translate literally for me. Even though I asked it to my English teacher, he did not know either. ...
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85 views

Did Ostrogradsky dismiss Lobachevsky's book on non-Euclidean geometry “because the world is obviously Euclidean”?

I read in a book of popularization of Mathematics that in 1830 Mikhail Ostrogradsky wrote, about non-Euclidean Geometry, that he did not see why anyone would care about that, since the world is ...
3
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1answer
131 views

Were people aware of the “mistakes” in Euclid's Elements before the start of the formalization of Mathematics?

For example, in proposition 1, Euclid assumes that the instersection of the two circles exist, when he shouldn't have. This, among many other things, was corrected quite recently (by Hilbert and ...
6
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1answer
108 views

How did the use of the word “origin” become commonplace in geometry?

My understanding is that in Cartesian geometry, all coordinate axes of an n-dimensional space may intersect at one point. I would like to know how that point--whether (0, 0), (0,0,0), ... -- came to ...
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2answers
110 views

Which is the earliest written record of hexagonal tesselation of the plane?

I am wondering which is the earliest record of the fact that the plane can be tiled by regular hexagons (in addition to triangles and squares, which may be slightly more obvious). Had a look in the ...
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1answer
1k views

When Indian mathematicians learn of Euclid's Elements?

Transfer of mathematical knowledge from India to Europe (such as a positional number system with zero) allowed Europeans to develop arithmetic. But was there also a reverse direction (probably via ...
2
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2answers
174 views

What made Euclid/Heron define line as a length without breadth and point as that with no part?

A point is that of which there is no part. And a line is a length without breadth.$^1$ If above definition on point, expresses on point as to be indivisible length, as seems to be expressed in ...
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1answer
139 views

What theorem of Sophus Lie on the number of geometries is H. Poincaré referring to?

In this quotation from Henri Poincaré's essay "Non-Euclidean Geometry" published in Nature in 1892 (No. 1165, Vol 45, p. 406), he refers to a theorem of Sophus Lie. Does anyone know a source for this ...
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1answer
77 views

What's the relationship between Aristotle's theory of elements and motion and geometry?

I'm having a hard time gathering my thoughts about this. I'm trying to find a connection or some sort of relation between the first 3 axioms (postulates) of Euclidean geometry (though around Aristotle'...
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2answers
149 views

Euclid’s Proposition I.3 overused?

[ Question copied from https://math.stackexchange.com/questions/2541170/euclid-s-proposition-i-3-overused ] Although the references to postulates, axioms, and previous propositions are not part of ...
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1answer
79 views

What geometric results were first proven by assuming all real numbers are rational?

Pythagoras and his followers believed that all magnitudes are commensurable; that is, the ratio of two magnitudes of the same kind, like two lengths or two areas, is equal to the ratio of natural ...
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2answers
147 views

What are historical applications of geometry to measuring distances beyond human reach?

I am searching for books and articles about applications of Geometry, in particular to the problem of computing distances and lengths which are apparently beyond human reach. As an example, consider ...
5
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1answer
62 views

What was the old system of using right circular cones to solve problems about circles in the plane?

[I asked this originally at the Math Stack Exchange, and they suggested I also ask about it here.] I heard about this from a college professor but haven't ever been able to find any other mention of ...
3
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1answer
176 views

Does any extant Greek text prove that the area of an inscribed regular polygon increases with the number of sides?

Does any extant Greek text prove that the area of a regular polygon inscribed in a fixed circle increases with the number of sides in the polygon? I can't find such a proposition in Euclid, but the ...
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0answers
89 views

Were the ancient Greeks aware of the “topology” of (Euclidean) space?

Related to a more mathematically inclined question, I'd like to ask the following question: The ancient Greeks made use of infinite arguments and processes (limits), e.g. in the method of exhaustion ...
4
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2answers
122 views

What was the relation between Euclid's points and Democritus' atoms?

Geometry as described in Euclid's Elements originated roughly at the same same time as Democritus described his atomic theory. I wonder how close these two points of view were related at those times: ...
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1answer
89 views

Lengths as equivalence classes

From Wikipedia on cardinal numbers: The oldest definition of the cardinality of a set $X$ (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the class $[X]$ of all sets ...
2
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1answer
103 views

Why didn't Euclid try to assign numbers to lengths?

Preliminary note: With "Euclid" I don't mean a person but the mathematicians of the Euclidean period of which Euclid (if he had been one person) was a representative. I imagine that Euclid could have ...
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2answers
77 views

Did Euclid consider circle segments as another magnitude?

[I adapted the question to reflect what I've learned from Alexandre's answer: that Euclid never talks of lengths and areas but only of line segments and figures (like squares). The question itself ...
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2answers
184 views

How did the integer degrees angles counting being first adopted in geometry and mathematics? [duplicate]

The purpose of this question is trying to know originally how did counting in integer degrees angles from (one degree to $360$ degrees) being adopted basically in geometry, despite the impossibility ...
4
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1answer
169 views

Who originated the concept of making the point dimensionless?

Over the years I read different versions of how the point in geometry (and subsequently in maths) came to be defined as an abstract, dimensionless entity. I read that it was Architas who influenced ...
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1answer
142 views

mathematicians attempts at proving Euclid postulate

Is there a list of all the people who attempted to prove the parallel postulate (also known as the fifth postulate or the Euclid axiom) in Euclidean geometry? Wikipedia has a page on the subject but ...
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1answer
123 views

Straightedge and compass

According to most discussions of Euclid's Elements, this work - and indeed, much of Ancient Greek geometry - should be seen as engaged in the game of figuring out what can be done with straightedge ...
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2answers
72 views

Whether Euclid considered squares to be rectangles

When I look up 'that which is right-angled but not equilateral' there are translations that show the word before the above phrase to 'oblong', some that show 'rectangle' and some that show both ...
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3answers
1k views

Compass and straightedge: why?

Why is it that, in ancient Greece, mathematicians tried to solve geometrical problems using compass and straightedge only and, apparently, only if that failed, they tried to use other tools? Note that ...
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1answer
281 views

What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?

Euclid's Elements could tout to have the longest and most famed publishing history of any book ever written. First written in 300 B.C., Euclid's Elements became the standard text from which ...
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1answer
240 views

Why didn't Euclid's Elements treat conic sections?

There's a well known treatise by Apollonius on conic sections, but these objects are absent in Euclid's Elements. Why? If I were to guess, I'd say that conic sections cannot be constructed using a ...
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1answer
105 views

Did Euclid formulate his definitions/postulates/common before or after writing all his theorems?

Did Euclid formulate his definitions/postulates/common notions at the beginning of Book I of the Elements before or after writing the 465 theorems of the Elements? cf.: Michael J. Crowe, “Ten ...
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3answers
2k views

How was geometry with 3 dimensions discovered/invented?

I wondered if back in the time of ancient Greeks mathematicians, 3D geometry was discovered as result of plane geometry? (Was there anything in the axioms of plane geometry that indicated existence of ...
3
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3answers
812 views

How did Aristotle influence Euclid?

In other words, how is Aristotle's logic represented in Euclid's Elements? I have read many articles where Euclid's Elements is linked to Aristotle's logic, but I do not understand, and I can't find ...
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1answer
73 views

Motivation of Infinite Series

What is the historical motivation of infinite series? According to Wikipedia, they are arose separately by Newton, Leibniz and Somayaji.
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4answers
550 views

Why did Euclid Avoid Using the 5th Postulate?

In Euclid's elements, some of the theorems (e.g. SAA congruence) can be proven using the parallel postulate, much easier than without it. But it seems that Euclid has intentionally avoided using it, ...
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0answers
62 views

What is the earliest attested mention of the fact that a parallelepiped in Euclidean 3-space can be decomposed into six tetrahedra?

The question is in the title. A pictorial representation of what this is about is the following: (created with GeoGebra and GIMP) The orthoschemes named after Ludwig Schläfli are very relevant but ...
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1answer
514 views

When did people start accepting $\mathbf{R}^{2}$ as “the plane”?

The standard presentation of "coordinatizing the plane" in 19th century British textbooks on geometry (Salmon, Smith, Besant, and many more) take the plane as being rigorously (at the time) ...
4
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1answer
251 views

Who classified plane isometries first?

There are only four types of plane isometries: translations, rotations, reflections, and glide reflections. My question is: who was the first person who proved this? I asked this question personally ...
4
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2answers
319 views

What did the ratio of two magnitudes mean to ancient Greek mathematicians?

My understanding is that magnitudes to ancient Greeks meant the actual line segments and plane regions (not the size of the line segment or the area of the plane region), the concept of ratio was then ...
4
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0answers
189 views

When was the British Flag Theorem discovered or proven?

The British Flag Theorem is a fancy name for a law relating distances from the corners of a rectangle to an arbitrary point. The wikipedia article is small and has no history section. Could not find ...
3
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1answer
530 views

What is the history of angle quintisection (division into five equal parts)?

I was reading lately that the quintisection of an angle is possible with paper folding (origami). Now, in contrast to the trisection of an angle, a problem which was discussed historically, and was ...
3
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1answer
168 views

Why did mathematicians have a hard time accepting Euclid's 5th postulate as a postulate? [duplicate]

Many had tried in vain to prove Euclid's parallel postulate using the existing axioms and theorems. But my question is that what is it about the parallel postulate that made it seem so much like a ...
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1answer
107 views

Straight line is the shortest of curves, who proved?

I am curious, when and by whom it was proved that straight line is the shortest of measurable curves connecting two given points.
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3answers
106 views

History of the quadrature of curvelinear figures prior to the middle ages

Hippocrates was able to construct the quadrature of three different lunes. Euler found two more squarable lunes. Tschebatorew and Durodnow showed that these five are the only squarable lunes. ...
3
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2answers
262 views

What were the applications of ellipses before elliptical orbits were discovered?

I'm interested in the history of ellipses. When were they discovered, what uses (if any) did they have before the true shape of orbits were found (by Kepler I think)? There are some interesting ways ...
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1answer
592 views

When and who was the first mathematicians to prove rigorously that $\sqrt[3]{2}$ was impossible number? [closed]

The purpose of the question is to understand why the number $\sqrt[3]{2}$, that was proven rigorously by ancient Greek is an impossible number (even at infinity), by their three famous impossibility ...
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2answers
723 views

Who discovered integer triangles with one angle trisecting another?

When & who was the first mathematician to discover the following simple triangle with a unique property that it has one angle is equal to one third of another angle in the same triangle? The ...
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2answers
3k views

On Einstein's proof of the so-called Pythagorean theorem

Part I In E. Maor's book [2, p. 117] we read that, somewhere in his Autobiographical Notes, Einstein wrote this: An uncle told me about the Pythagorean theorem before the holy geometry booklet had ...
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2answers
536 views

History of impact of non-Euclidean geometry on math, philosophy, and the public

I'm interested in the impact of the discovery of non-Euclidean geometry on math, philosophy, and the attitudes of the general public. I don't know anything about how things changed right after the ...
3
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1answer
368 views

How did Saint Vincent prove the logarithmic property of areas under hyperbolas?

How did Saint Vincent prove that if $\frac{a}{b} = \frac{c}{d}$, then the area of a hyperbola $y = \frac{1}{x}$ from $a$ to $b$ equals the area from $c$ to $d$? What references (pdfs, links, books) ...
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1answer
1k views

The origin of quadratic equation in actual practice

I read that in ancient times the quadratic equation of this kind $$x^2+10x=39$$ had been solved long ago. I read that this kind of equation originated in the geometric question of "Given an area of 39,...
3
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1answer
224 views

Classical source for theorem on three parallel lines cut by two transversals

I am trying to find a classical source for the following theorem about parallel lines and transversals: If three parallel lines are cut by two transversals, then the segments between the ...
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2answers
718 views

Did Dieudonné say “Euclid must go!” or “Down with Euclid! Death to triangles!”?

In his famous address at the Royaumont Seminar in 1959, Jean Dieudonné famously called for the elimination of Euclidean geometry from the secondary school curriculum. In the published (English-...