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Questions tagged [geometry]

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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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48 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 ...
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Earliest drawings of the plots of trigonometric functions

[Even though this question may seem as a duplicate of this question about the History of sine function, I'd like to ask it again - with a more specific title and a more specific focus (on specific ...
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1answer
44 views

Centroid in Babylonian Mathematics

Are there any problems in Babylonian mathematics that deal with finding the centroid of some plane figure?
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70 views

When and why was inversive geometry created/studied?

I have been revisiting math from my highschool through undergrad. I picked Courant’s excellent What is Mathematics? The flow is well so far. However, in one of the chapters he introduces inversion - ...
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What is the purpose of Gauss in his letter to János Bolyai when he mentioned “howling Boeotians”?

I'm wondering what is the purpose of Gauss in his letter to János Bolyai when he mentioned "howling Boeotians"? Is it related to some scientists or philosophers that opposed non-Euclidean geometry? I ...
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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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2answers
121 views

What mathematical techniques Gauss used in order to tessellate the unit disk?

This question is a continuation of my previously posted question: Was Gauss aware of the non-euclidean implications of his work on moduler forms?, and is based on the information given in John ...
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1answer
198 views

What are the modern connections of the Pentagramma Mirificum studied by Gauss?

In the last years, i read a lot about a mathematical object that was discovered by John Napier in 1620 and explored much more deeply by Gauss, who called this "Pentagramma Mirificum" (latin for "the ...
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1answer
146 views

Sphericity of Earth from lunar eclipses - is Aristotle's argument valid?

Aristotle is often credited with proving the sphericity of Earth from the fact that the shadow of the Earth on the moon during lunar eclipses is always an arc of a round circle (as opposed to arcs of ...
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167 views

How old is the Pythagorean Theorem? [closed]

More specifically, what is the oldest evidence of human awareness of what we now call the Pythagorean Theorem? The phrase, "evidence of human awareness" was used to exclude a different question of ...
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Priority on lemniscate of Gerono?

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). Historically who has priority ...
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Indiana Pi Bill: Other attempts to establish mathematical truth by legislative fiat?

Wiki: The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, one of the most notorious attempts to establish mathematical truth by legislative ...
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108 views

Was a regular heptagon ever constructed by ancient Greeks?

Today it is well known that a regular heptagon cannot be constructed with straightedge and compass, since it would require to solve an equation of third degree which is not possible with the standard ...
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1answer
99 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
90 views

Who was first to recognize the link between (synthetic) elliptic geometry and geometry on the sphere?

Riemann was the first to talk about elliptic geometry: Bolyai and Lobačevskij (even Gauss too) studied only hyperbolic geometry. But of course some theorems of (planar) elliptic geometry were known ...
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2answers
66 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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134 views

Definitions of continuity pre-Dedekind

In his article on "Kant's Theory of Geometry", Michael Friedman claims that: (...) before Dedekind mathematicians would commonly give what we call the definition of denseness when explaining what ...
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When was this projective property of an ellipse's directrix known?

I stumbled on this image from here: It's mentioned w.r.t. to the elliptical orbits of planets and how the focus-directrix property comes into play. It's an interesting POV but when was such a thing ...
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367 views

How was the focus/directrix property of conic sections discovered?

I've always thought that defining conic sections by a locus of points w.r.t the ratio of the distance to the focus and directrix was always "too artificial" - how does one actually discover this ...
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2answers
282 views

What astronomical reason led to the creation of the trigonometric sine and cosine?

Sines and cosines are commonly introduced as ratios of sides of a triangle with its hypotenuse and attributed to ancient Indian scholars. However, I've never actually thought of the reason for ...
3
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1answer
159 views

What is the origin of French/Burmester's curves?

French curves are a set of curvilinear rulers used in industrial design, before the advent of CAD, when everything still had to be drawn by hands. The most popular set of such rulers is made up of 3 ...
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99 views

Comparison of Bolyai's and Gauss's volume integrals for the orthoscheme tetrahedron in hyperbolic space

After a recent advance i made, I've succeeded in deriving Bolyai's result from Gauss's formula (for one special case, as we will see) up to one troubling coefficient. This advancement was made after ...
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1answer
96 views

Why is there some doubt whether or not Gauss saw the pseudosphere as the embodiment of hyperbolic geometry?

I read a lot of historical articles that doubt the possibility that Gauss saw in the pseudosphere the realization of hyperbolic geometry; that geodetic triangles on the pseudosphere obey the same ...
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3answers
646 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 ...
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Who popularized using $x,y$ to label axes and $z$ as complex variable?

I believe although it was Rene Descartes who popularize using $a,b,c$ as constants and $x,y,z$ as variables, he wasn't the one who associated $x$ and $y$ with axes labeling. Also, how did using $z$ ...
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Has Euclid stated Cauchy's theorem?

Cauchy's Rigidity theorem says that if the corresponding faces of two convex polytopes are isometric (congruent) then the polytopes are related by a (proper or improper) motion. Cauchy's biography (...
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3answers
472 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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3answers
209 views

What did Lobachevsky do?

It is often said that he discovered non-Euclidean geometry. But in which sense? I am reading the book 'geometry' by Brannan et al. They use the disk model as an example of hyperbolic geometry. Did ...
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2answers
236 views

What topological ideas did Gauss introduce to his student Möbius?

Recently I found a website with good historical information about the contributions of Gauss to Analysis situs (the old term for topology). The site is in German so I made a Google translate to ...
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2answers
231 views

What did Farcas Bolyai write to his son?

There are famous quotes about what Farcas Bolyai wrote to his son Janos to persuade him not to study the "theory of parallels " or what is now known as hyperbolic geometry But not all translation of ...
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1answer
362 views

How did Eratosthenes knew the exact time of the day?

Eratosthenes measured the radius of the Earth with an incredibly accuracy. To do it, you need to measure the length of the shadows from 2 different cities at the same time of the day. Then knowing ...
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Is there any evidence supporting this claim about Cassini and his ovals?

The Wikipedia article for Cassini ovals claims in the introduction that "Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval." This ...
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Earliest presentation of a 3-D permutahedron?

Below is a picture of a 3-D permutahedron sundial by Stefano Buonsignori (16th century) in the Medici collection presented by Museo Galileo. The permutahedra / permutohedra and the closely related ...
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153 views

What exactly did Poincaré mean by 'simply connected'?

I've been reading John Stillwell's translation of the famous Analysis Situs and have become confused about the exact meaning of 'simply connected' in Poincaré's language. On page 7 (in the ...
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3answers
152 views

Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?

This idea provides the foundations of algebraic geometry now; and they have certainly gone down the rabbit hole with it. As a student studying this subject, I have always found it such a great leap to ...
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1answer
170 views

Books or articles criticizing Benoit Mandelbrot? (fractals)

I'm researching about fractals history and one of its main contributor and promoter Benoit Mandelbrot. As far as I'm concerned, when he published his first book about this subject in 1975, he was ...
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1answer
88 views

Origin of the latitude 36 of Eratosthenes

How did Eratosthenes come up with the latitude 36 line, also called 36th parallel north, in the Mediterranean world? Rhodes was one of the navel points in his calculations and even today N36.00 goes ...
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1answer
244 views

A knot cannot be tied in 4-dimensions, but when was this conjectured and proven?

Today it has been shown that a 1-dimensional object in 4-dimensional space cannot be tied into a knot. But I would like to know who first conjectured this and when? And when was it proven? (P.S., is ...
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1answer
144 views

How did the Arabs determine the longitudes of cities?

I am reading the book by Berggren, 'Episodes in the mathematics of medieval islam'. An important problem is determing the direction of Mecca with respect to a local city. The book introduced a method ...
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2answers
167 views

When did mathematician start to draw figures from equation?

I know that when solving geometric problem, Descartes used variables $x,y$ and derived equation such as $y^2=cy-\frac{cxy}{b}+ay-ac$. Conversely, in algebraic geometry, an arbitrary polynomial $F(X_1,....
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1answer
479 views

What was the best approximation of π known to ancient Babylonians?

Wikipedia's Babylonian mathematics says that the ancient Babylonians usually used a round value for $\pi$ (3). But they knew a more precise value: Babylonian texts usually approximated π≈3, ...
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317 views

Where was the word “pencil” first used in (projective) geometry and what is the reason behind this curious name?

The title is pretty self-explanatory: A pencil in projective (or algebraic) geometry is the family of all lines through a point. The above-linked website tells me that Cremona, on page x of Elements ...
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2k views

What is the etymology behind sine, cosine, tangent, etc.?

I heard somewhere that it was actually a mistake in translation. What's the correct story?
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763 views

How was Newton's classification of cubic curves completed?

According to what I have read, using Newton’s methods there are 78 different families of cubic curves. Newton discovered 72 of them while “subsequent research identified another 6”. This paper ...
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4answers
596 views

What are the origins of the study of symmetry as a subject in itself?

Symmetry has become a central concept in mathematics. The Euclidean concept of similarity is an example of symmetry, but similarity was not a subject of study in itself. Q: How did symmetry come to ...
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102 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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797 views

What is Ptolemy holding in his picture on Wikipedia?

I would like to know the name of the device Ptolemy is holding in his picture
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244 views

What is meant by higher order infinitesimals in the works of Galileo and Cavalieri

According to Boyer, Salviati introduces the idea of a higher order infinitesimal on the “third day” in Galileo’s Two Chief Systems of 1632. They are introduced in order to counter Simplicio’s ...
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1answer
74 views

Early discoveries combining groups and geometry?

More specifically: When were the symmetries of polygons/solids first presented as groups in Cayley tables? Textbooks often use the symmetries of polygons/solids to introduce group theory, however, I ...