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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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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 ...
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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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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 ...
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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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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 ...
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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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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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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 ...
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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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. ...
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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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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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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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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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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 ...
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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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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,...
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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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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-...
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When do we see for the first time the use of the Cartesian plane?

I want to see an exact image of the first use of the Cartesian plane. I guess it came the first time with Descartes.
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Why is calculus missing from Newton's Principia?

I'm not suggesting that Newton did not discover calculus - the question is written this way to express my surprise that the Principia does not use the methods of calculus (or 'fluxions'). He instead ...
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Why did the ancient Greeks originally become interested in conic sections?

How much is known, or can be conjectured, about why the Greeks originally became interested in the somewhat arbitrary construction of intersecting a plane with a cone? The folklore that I've heard is ...
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Five perfect solids

Has anyone ever considered the connection between the five perfect solids and the three most important music intervals of 2, 1.5, 1.25 and their two counterparts, 1.33333 and 1.6? The tetrahedron ...
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Who discovered the duality between platonic solids?

As it is well known, every platonic solid has a dual (obtained by interchanging vertices and faces), which also happens to be a platonic solid. I would like to know who was the first person to ...
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Discovery and Development of Coordinate Systems

I'm very interested to know how coordinate systems were discovered and why mathematicians discovered them? Actually I want to know what things motivated mathematicians to discover and develop ...
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Who was the first to calculate $\pi$?

I am very interested in the history of $\pi$. I am first trying to find out who calculated it. Many sources have different answers, from the ancient Egyptians, to Archimedes, to the Babylonians. I ...
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Who calculated for the first time the volume (and surface area) of the sphere exactly?

As we know, even Archimedes did soon some experimental calculations. My question were, who calculated first time the exact formulas ($V=\frac{4\pi}{3}r^3$, $A=4\pi r^2$)? As I know, these formulas ...
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Who came up with the “proof” that all triangles are isosceles?

"All triangles are isosceles" is a famous geometric fallacy (see below). Unlike many other fallacies its flaw is subtle and hard to spot, so it is often used as a cautionary example against the "...
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About Archimedes methods in the discovered palimpsest

I think Archimedes had some great non-infinitesimal methods for discovering the area and volume of shapes. Some very visual methods involving his method of exhaustion for the volume of a sphere for ...
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Why were geometers dissatisfied with the parallel postulate?

Euclid himself already treats it with gloves, it has an unusually precise formulation, and is not used in the first 28 propositions of the Elements. Why? Did he doubt it? It's not like Euclid was a ...
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Why is the Pythagorean Theorem so ubiquitous?

We all know the Pythagorean Theorem is one of the most fundamental formulas in mathematics, but it is very interesting to me that this ratio shows up as often as it does. It seems to have been ...