Questions tagged [euclidean-geometry]

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

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30
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3answers
3k views

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 ...
22
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1answer
2k views

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 ...
22
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1answer
795 views

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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4answers
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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 ...
16
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2answers
8k views

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 ...
12
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1answer
502 views

Concerning the measurement of the Earth's circumference by Eratosthenes

I. In an episode of his Cosmos series, Carl Sagan discussed the brilliant argument whereby Eratosthenes purportedly estimated the circumference of our planet. At one point of the episode, Carl Sagan ...
12
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1answer
535 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) ...
11
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1answer
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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,...
9
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1answer
183 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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4answers
706 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, ...
8
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1answer
757 views

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 ...
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635 views

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 ...
8
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2answers
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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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394 views

When do we see for the first time the use of the Cartesian coordinates?

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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422 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 ...
7
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2answers
852 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-...
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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 ...
6
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4answers
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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 ...
6
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1answer
329 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 ...
6
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3answers
374 views

How the Cross-Ratio appears in the Work of Pappus?

The cross-ratio of four collinear points $A,B,C,D$ in the Euclidean Plane is defined by $$(A,B,C,D) = \frac{AC}{AD}\frac{BD}{BC}$$ And the wikipedia article states it already appeared in the works ...
6
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1answer
118 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 ...
6
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2answers
770 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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4answers
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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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1answer
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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 ...
5
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1answer
306 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 ...
5
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1answer
64 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 ...
5
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1answer
105 views

Were epicycloids from astronomy acceptable curves in Greek geometry?

My simplified historical understanding is as follows. Euclidean geometry accepted a limited number of geometrical objects (straight-edge and compass constructions, conics). Descartes' Géométrie ...
5
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2answers
88 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 ...
5
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1answer
215 views

Do we have any in depth material on Baudhayana?

I know he wasn't strictly a mathematician, but as I understand, Baudhayana recorded information on (what we know as) the Pythagorean Theorem and other geometrical properties like $\sqrt{2}$, in the ...
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322 views

What were the applications of conic sections before Kepler?

When recently asked for a practical application of parabolas, I responded by talking about objects in free-fall. Afterwards as I was re-thinking this conversation it occurred to me that an object in ...
4
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2answers
96 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 ...
4
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2answers
117 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 ...
4
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2answers
254 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: ...
4
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2answers
162 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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3answers
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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 ...
4
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2answers
323 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 ...
4
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1answer
220 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 ...
4
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2answers
464 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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1answer
465 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) ...
4
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1answer
118 views

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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0answers
65 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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0answers
211 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
757 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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2answers
571 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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2answers
107 views

Inscribing equilateral triangle in square — mistake in historical work by Abu'l-Wafa Al-Buzjani?

(I asked this question in the general Mathematics forum, but I have been advised to post it here instead -- or as well.) In David Wells's "Curious and Interesting Puzzles", Penguin, 1992, ...
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1answer
88 views

Who first gave a definition of congruent triangles?

Who was the first to define congruent triangles? I couldn't find the definition in Euclid's Elements.
3
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1answer
156 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 ...
3
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1answer
163 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 ...
3
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1answer
97 views

Who discovered the thin lens equation $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$?

According to Weisstein's webpage it was Halley in 1693 (quoting Steinhaus); but I've also seen it attributed to Cotes, Huygens, even Gauss (eg Britannica). Wikipedia's History of Optics does not give ...
3
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1answer
163 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 ...